THE UNIVE. S ITY OF l'i[IC^HIGIN I rDUSTRY PRO" 0J RAIIf OF THE CO.LLELE OF EENG IN:E ERI, RELATIONSHIP.3ETWEENT UNSTEADY STATE WELL PER.FORi`J[LAiTCE ATND INS ITU RESERVOIR- CH ARACTERISTICS Maurice C. Miller A dissertation submitted in partial fulfillment of the requirements for the deg:ree of Doctor of Phlilosophy inl the University of Mlichigan 1963 July, 1963 IP-627

AC KNTOW LEDGEIE NTS The author wishes to express his sincere appreciation to the following individuals and organizations for their contributions to this research: Professor Donald L. Katz, co-chairman of the doctoral committee, for his advice and interest in the formulation of the problems investigated in the course of this research and for his aid in securing field wrell test data. Associate Professor M. Rasin Tek, co-chairm.ian of the doctoral committee, for his generous donation of time in discussing the mathematical problems wrhich arose in the course of this research. Professors G. Brymer Williams, Robert L. Kadlec and Bernard A. Galler, committee members, for their interest and valuable discussions. Doctors J. E. Briggs, S. C. Jones, R. Lo Nielsen, and J,. R. Street and Messrs. H, B. Kristinsson, D. A. Saville, and J. L. Skinner for their interest and helpful suggestions. t ions. The Michigan Gas Associa.tion and American Gas Association fellowships for their financial support. The Northern Illinois Gas Company for their donation of aquifer pump test data. Messrs. R. E. Carroll, D. L. Danford. and the personnel of the Industry Program of the College of Engineering for their aid and production of the final form of this dissertation, ii

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS....... ii LIST OF TABLES........v.... i LIST OF FIGURES........... Vii LIST OF APPEND ICES....... *.,. x NOMENC LATURE........ xi ABSTRACT........, xv I INTRODUCTION.... e e o~ II. LITERATUhE SUREVY DESCRIPTION OF EXISTING METHODS FOR THE DETERMINATION OF INSITU PERMEABILITY, TNSITU COMPRESSIBILITY AND RESERVOIR LIMITS............... L Ao Well Tests in Infinite Reservoirs... Be Well Tests in Finite Reservoirs..... 7 C. Effect of Linear Faults on Pressure Behavior l Behavior......... 9 D. Interference of Several Wells Pumping in a Common Aquifer............ 10 E. Skin Effects....... 11 F. Other Topics.........., 11l III. DETERMINATION OF ERROR IN NEGLECTING THE NONLINEAR TERM IN THE PARTIAL DIFFERENTIAL EQUATION DESCRIBING THE FLOW OF SLIGHTLY COMPRESSTIBLE FLUIDS IN POROUS MEDIA..... 13 A. Derivation of Non-Linear Differential Equation...... *. 13 B. Solution of Non-Linear Differential Equation for Radial Flow, Constant Terminal Rate Case....... o 18 1, Explicit Numerical Method..... 21 20. Implicit Numerical Method....., 26 C. Solution of Non-Linear Differential Equation for Linear Flow, Constant Terminal Rate Case.......... 3 iii

TABLE OF CONTENTS (CONT'D) Page 1, Explicit Numerical Method. *.. 36 2, Implicit Numerical Method.. 40 D. Analysis of Results and Application to Reservoir Problems...... 45 IV. EFFECT OF NEIGHBORING GAS FIELD (OR LINE OF CONSTANT PRESSURE) ON INSITU PERMEABILITY AND COMPRESSIBILITY MEASUREMENTS IN AQUIFERS FOR CONSTANT RATE PUMP TESTS 53 A. Prediction of Pressure Behavior During DrawdowJn or Build-Up Pump Tests..... 53 1, Pressure Behavior of Observation Well During Drawdown and Build-Up Tests......... 55 2. Pressure Behavior of Pumping Well During Drawdown and Build-Up Tests.. 62 B. Evaluation of Error in the Measurement of Insitu Permeability and Insitu Compressibility.,..... 68 C. Description of Graphical Method for Locating External Line of Constant Pressure (or Gas-Water Interface)... 73 D. Example Problems........ 76 V. EFFECT OF LE1AKAGE INTO THE PERMEABLE STRATA THROUGH THE CONFINING CAP AND BOTTOM ROCK ON THE INSITU PERMEABILITY AND INSITU COMPRESSIBILITY MEASUREMENTS IN AQUIFERS OR OIL FIEIDS.............., 9. 81 A. Prediction of Pressure Behavior During Drawdown Tests p.... 82 B. Prediction of Pressure Behavior During Build-Up Tests........, 84 C. Evaluation of Error in the Measurement of Insitu Permeability and Insitu Compressibility. I...., *. 88 D. Example Problems.......... 92 VI, INTERPRETATION OF FIELD WELL TEST DATA.... 98 A. Field A..... b O. O..., 98 iv

TABLE OF CONTENTS (CONT'D) Page 1. Approximate Equations Describing Gas Flow in Porous Media for Constant Rate Tests..... e 98 2. Analysis of Pressure Drawdown Data 102 B. Field B.......... 105 1. Drawdown Test..... 105 2. Build-Up Test............ 114 C. Field C. i...... 114 1. Drawdown Test..... 121 2. Build-Up Test....... *... 124 VII. SUMMARY, CONCLUSIONS, AND RESULTS........ 126 VIII. RECOMMENDATIONS FOR FUTURE WORK.... 130 B IBLIOGRAPHY.e...... 132 APPEND ICES......0 4 * o *. *. * * * 0 ~ i ~ o e ~ ~ I ~ e ~ ~ e ~~V

LIST OF TAB LES Table Pagef IV-1 Dimensionless Pressure Drop PD(RD, TD) for Radial System with Constant Pressure line......... 60 VI-1 Pressure Data for Flow Test on Gas d^Eell in Field A................ 104 VI-2 Field B - Pressure Drawdown Data.... 108 VI-3 Field B - Pressure Build-Up Data.... 109 VI-4 Field C - Pressure Drawdown Data..... 118 VI-5 Field C - Pressure Build-Up Data.,... 119

LIST OF FIGURES Figure Pa ly III-1 Effect of Non-Linear Term on Dimensionless Pressure Drop versus Dimensionless Time for Radial System, Constant Terminal Rate Case.. *. *. 27 III-2 Dimensionless Pressure Drop versus Dimensionless Time for Linear System, Constant Terminal Rate Case. e * *.* 0 39 III-3 Percent Error in Dimensionless Pressure Drop for Radial System, Constant Terminal Rate 6........ 46 III-4 Percent Error in Dimensionless Pressure Drop for Linear System, Constant Terminal Rate Case......., a a. 51 IV-1 Pumping Well and Observation Well near a Line of Constant Pressure... 54 IV-2 Mathematical Model and Location of Image Well...5.... 4 IV-3 Dimensionless Pressure Drop for Observation Well in Radial System with Constant External Pressure Line..,. *.. 0 IV-4 Steady State Dimensionless Pressure Drop for Radial Systems with Constant External Pressure Line.... * 1* e v I *,* e IV-< Pressure Build-Up for Observation Well in Infinite Radial System.. * * * 0 63 IV-6 Pressure Build-Up for Observation Well in Radial System with External Line of Constant Pressure, RD = 6. 0. * 64 IV-7 Dimensionless Pressure Drop for Pumping Well in Radial System with Constant External Pressure Line.. 0 0 66 IV-8 Pressure Build-Up for Pumping Well in Radial System with Constant External Pressure Line, RD = 100.... o.. 0 67 vii

LIST OF FIGURES (CONT'D) F igure Page IV-9 Determination of the Minimum Error in the Measurement of Insitu Permeability Resulting from an External Line of Constant Pressure........ * 72 IV-10 Determination of Error in Measurement of Insitu Compressibility Resulting from External Line of Constant Pressure.. 75 IV-11 Illustration of Graphical Method for Determining Location of External Line of Constant Pressure or Gas-Water Interface.*. I........, 77 IV-12 Location of Wells for Problem IV-1 ~ ~. 79 V-1 Dimensionless Pressure Drop for Radial System with Leakage through Caprock and/or Bottom Rock.... * *....... 85 V-2 Steady State Dimensionless Pressure Drop for Radial Systems with Leakage through Confining Caprock and/or Bottom Rock.. 87 V-3 Pressure Build-Up for Observation Well in Radial System with Leakage r = 0.2.. 89 B V-4 Ratio of Observed Permeability to Actual Permeability versus r for Radial System with Leakage.. * 91 V-5 Ratio of Observed Compressibility to Actual Permeability versus r measured at Inflection Point for RadialBSystem with Leakage...e....*., a..... 93 VI-1 Well Pressure versus Time for Flow Test on Gas Well in Field A....... 106 VI-2 Location of Wells in Field B.... 107 VI-3 Pressure Curves for Drawdown Test on Field B.......... 1 10 VI-4 Pressure Curves for Build-Up Test on Field B ~..........,. 115 viii

LIST OF FIGURES (CONT'D) Figure Page VI-5 Location of Wells in Field C... * 117 VI-6 Pressure Curves for Drawdown Test on Field C............ *... 122 VI-7 Pressure Curves for Build-Up Test on Field C e12 ix

LIST OF APPENDICES Appendix Page A Computer Program for Numerical Explicit Method used in Determining the Error in Neglecting the Non-Linear Term in the Partial Differential Equation Describing Radial Flow of a Slightly Compressible Fluid in Porous Media........ 140 B Computer Program for Numerical Implicit Method used in Determining the Error in Neglecting the Non-Linear Term in the Partial Differential Equation Describing Radial Flow of a Slightly Compressible Fluid in Porous Media........ 0 146 C Computer Program for Numerical Explicit Method used in Determining the Error in Neglecting the Non-Linear Term in the Partial Differential Equation Describing Linear Flow of a Slightly Compressible Fluid in Porous Media....... 0 15 D Computer Program for Numerical Implicit Method used in Determining the Error in Neglecting the Non-Linear Term in the Partial Differential Equation Describing Linear Flow of a Slightly Compressible Fluid in Porous Media....... 160

NOMENC LATURE Engineering units are underlined. A Cross sectional area normal to flowT, cm', (eet' a Scaling factor defined by Equation (III-55) B Term defined by Equation (V-4a) Ci Term defined by Equation (III-66) c Compressibility, vol/(vol)(atm), vol/(vol)(psi c (obs) Observed compressibility, vol/(vol)(atm), vol/(vol) (psi) Di Term defined by Equation (III-66) Ei Exponential integral, Ei (-x) = e dl h Reservoir height, feet h' Caprock thickness, feet h It Bottom rock thickness, feet Io Modified Bessel function, first kind, zeroth ordex Jo Bessel function, first kind, zeroth order J1 Bessel function, first kind, first order K Permeability, darcys, millidarcys K (obs) Observed permeability, millidarcys K' Permeability of caprock, millidarcys K" Permeability of bottom rock, millidarcys Ko Modified Bessel function, second kind, zeroth order Ki~j Term defined by Equation (III-71) L Distance between pumping well and linear fault, feet

Q. L, Distance between pumping well and observation well, centimeters, feet ye) La Distance between observation well and image well (See Figure IV-2), centimeters, feet Logarithm base e log Logarithm base 10 M Dimensionless coefficient, defined by Equation (III-34) for radial system and Equation (III-93) for linear system m Slope of drawdown or build-up curve, psi/ycle n Number of homogeneous layers in porous media Pb Pressure base for gas measurement PD Dimensionless pressure (PD)gas Dimensionless pressure for gas flow Pt Dimensionless pressure drop for well in 'infinite radial system (Table 10-5, Katz et al (50)) p Pressure, atmospheres, psia pf Flowing bottom hole pressure at shut-in, psia Po Initial pressure, atm, psia q, Q Flow rate, reservoir conditions, cubic centimeters/second, barrels/day QG Gas flow rate, SCF/day R Dimensionless radius, R - r 4 rw Dimensionless length ratio, RD = - r Distance from center of pumping well to point of pressure measurement, centimeters, feet re Radius of circular reservoir, feet rw Radius of pumping well, feet S Storage coefficient, S - + ch, feet/psi xii

Se Skin effect s Direction of flow T Gas temperature, OR T Transmissibilitv, T = Kh//, (millidarcy)(feet)/ centi.poise Tb Temperature base for gas measurement, OR TD Dimensionless time defined by Equation (IV-13) To Dimensionless time of drawdown test, T o- 000633K+t. t Time, seconds, days tD Dimensionless time defined by Equation (III-33) to Duration of drawdown test, days At Time since cessation of drawdown test, days XU Well function, a = -I VxvyVz Fluid velocity in x, y, z direction, respectively w Variable defined by Equation (III-36) x Space coordinate Xi Term defined by Equation (III-70) Xc Linear reservoir characteristic length, feet Yi Term defined by Equation (III-68) y Space coordinate Zi Term defined by Equation (III-68) z Space coordinate z Gas compressibility factor z Average gas compressibility factor xiii

Viscosity, centi oise,2Z Average viscosity, centipoise Porosity of porous media, fraction Roots of J1 ( ) = O given by Equation II-7a) p Density 7 Del operator, V =+ ' j ~? k Laplacian operator, V7 Force gradient, i -g$ + xiv

ABSTRACT The purpose of this study was to investigate the relationship between unsteady state well performance and insitu characteristics of reservoirs. Certain insitu conditions often encountered in reservoirs were postulated and their effect on the unsteady state performance were obtained by calculations, The error in neglecting the non-linear term in the differential equation describing the flow of a slightly compressible liquid was evaluated~ This information is important in interpreting reservoir drawdown and build-up pump test data. A mathematical expression describing the pressure behavior in a reservoir was derived for the case where a gas field or a line of constant pressure is located near the vicinity of the pump test in an aquifer or oil field. The magnitude of the error in the evaluation of the insitu permeability and insitu compressibility was determined0 A graphical method for locating the gas-water interface from drawdown tests on water wells located near a gas field is described, The pressure behavior of a reservoir when leakage is occurring through the caprock, described by M. S. Hantush, was expressed in engineering units and extended to the case of leakage through both the cap and bottom rock and for pressure build-up tests in addition to drawdown tests. The magnitude of the error in the xv

measurement of insitu permeability and insitu compressibility was evaluated. The pressure behavior of an aquifer when leakage is occurring was observed to be very similar to the pressure behavior of an aquifer located near a gas field. The magnitude of the error in neglecting the non-linear term in the differential equation describing the flow of a slightly compressible fluid in a porous media was evaluated by numerical means. The effect of neglecting this term was found to be negligible in all field data investigated, An equation giving the conditions when neglecting this term is not justified are presented. Analyses of well test data from three fields for the insitu characteristics are presented. xvi

I, INTRODUC TI ON Before the optimum depletion schedule for an oil or gas reservoir can be determined, it is necessary to obtain numerical values for the reservoir characteristics. These characteristics 'include the reservoir permeability, effective compressibility, porosity, and fluid viscosity. In addition the nature of the reservoir limits, the amount of gas or oil contained in the field must be known before the optimum number of wells, well spacing, and production schedule can be determined, Storage of natural gas in water sands has made it necessary to evaluate the characteristics of an aquifer prior to gas injection-production history. In order to evaluate the suitability of an aquifer for gas storage, it is necessary to determine if a suitable caprock exist and if there is suff icient permeability for the required gas deliverabilityo It is reasonable therefore that considerable research and money have been spent developing methods for evaluating reservoir and fluid characteristics and obtaining field data, Fluid characteristics can be determined by obtaining a sample of the fluid from the reservoir and evaluating its properties in a laboratory,

-2 -Determination of the reservoir rock properties can also be examined in laboratories, but it is not known how to average rock permeability in order to obtain the reservoir effective permeability. Thus it is necessary to perform well tests in the field in order to evaluate the effective permeability and compressibility, Unsteady-state flow behavior must be analyzed since steady state conditions exist only prior to fluid flow tests in the reservoir. A pseudo-steady state can exist in a limited reservoir and is defined as the state when the pressure change with respect to time is constant and equal at all points in the reservoir, Available unsteady state flow equations are used to calculate the following information: 1. Insitu permeability and insitu compressibility in infinite aquifers of constant thickness 2, Insitu permeability and insitu compressibility in finite circular aquifers of constant thickne ss 3, Location of linear faults 4., Skin effects (reduced permeability within few feet of pumping well) The equations and methods used to determine these properties are presented in the next section. Unfortunately, the performance of many reservoirs cannot be described by the equations presented in the next section. This failure may be due to neglecting important

parameters or failure of the mathematical equations to describe the flow, Effects of heterogenities in reservoirs may contribute to the tabnormal" behavior, The validity of neglecting the non-linear term in the partial differential equation describing the flow of a slightly compressible fluid was evaluated. It was found that neglecting this term is usually justified. The errors in the insitu permeability and insitu compressibility calculated from well test data if an infinite aquifer of uniform thickness is assumed are determined for the following cases: 1i A gas field or an external line of constant pressure is located near the test wells 2. Water is leaking through the cap and/or bottom rock into the aquifer. Field well test data are analyzed in Section VI to demonstrate the application of available methods for determining insitu reservoir permeability and insitu reservoir compressibility.

II. LITERATURE SURVEY - DESCRIPTION OF EXISTING METHODS FOR THE DETERMINATION OF INSITU PERMEABILITY, INSITU COMPRESSIBILITY, AND RESERVOIR LIMITS Drawdown and build-up well test results are used to estimate the insitu properties and geometry of underground reservoirs. These properties include insitu permeability and insitu compressibility as well as location of reservoir boundaries and reservoir limits (hydrocarbon volume). In addition, the effect of interference from neighboring hydrocarbon fields on a common aquifer can be predicted, Descriptions of several methods used in interpreting well tests are presented in this section. A, Well Tests in Infinite Reservoirs The pressure behavior of a well in an infinite reservoir during a constant rate drawdown pump test presented by Horner (35) and Theis (84) is given in engineering terms by f= o + Ei ()-~C (iir-') xh 4 (os. -o 633) Kt where: c compressibility, vol/(vol)(psi) Ei = exponential integral, Ei(-x) = -. e"du h reservoir thickness, feet K permeability, millidarcys

p = pressure, psia q = pumping rate, reservoir conditions, bbl/day r = distance from center of pumping well to point of pressure measurement, feet t = time, days o/ = viscosity, centipoise = porosity, fraction Equation (II-1), which assumes the pumping well radius is infinitesimal, was originally developed by Lord Kelvin and is known as the continuous point source solution. Values of the exponential integral are available in tables and graphs (19) (28) (43). If the value of the argument 4 -,oo33)Kg is less than.01, Equation (II-1) (19) (35) (51) (92) can be approximated by p=- ' <h- l62(o e (A 0633t + 0.351/3 (II2) Thus if the well pressure is plotted versus log time, the insitu permeability is calculated from the slope by r = 162. 6e (II-3) and the insitu permeability from Equation (II-2) where m = pressure drop, psi/cycle. An example calculation of the insitu permeability and insitu compressibility is given by Katz et al (51). Equations (II-l) and (II-2) are approximate solutions which are valid for K greater than

-6 -1000. The pressure behavior for lower values of the argument is given by Van Everdigen and Hurst (86) and Chatas (12) The pressure behavior of an infinite reservoir during a build-up test, obtained by superimposing a negative and equal production rate on Equation (II-1), (35) (51) (92) is aeo + 70.6 ~. Ei/- 4(0 3)(A). (II-) - E;(- co. o0633) KKt i where: Et - time since cessation of pump test, days to duration of pump test, days If O'c r12 If 410.oo0633) Kt is less than 0.01, Equation (II-3) can be approximated by A) = -Ib, _ 1K6, (t (IIh) Several examples of the determination of the insitu permeability from a build-up test are available in the literature (4) (35) (51) (71). If the pressure is plotted versus log1o0 -at the insitu permeability is calculated from the slope by K = 062.6 --- ~ as(II-6) &rnh~

-7 -The cumulative water influx for a constant terminal pressure drawdown is given by Van Everdigen and Hurst (86) Chatas (12), Katz et al (50) (51), and Jacob and Lohman (42)~ Other methods of analysis, additional information and case studies for build-up and drawdown tests are given by Ammann (2), Arps (4), Bruce (9), Collins and Kolodzie (16), Dolan et al (20), Driscoll (21), Hazebroek et al (62), Maier (57), Matthews et al (58), Matthews and Stegemeier (59), Miller et al (62), Pirson and Pirson (72), Pitzer et al (73), Schrenkel (79), and Thomas (85). Analyses of well tests and pressure behavior in gas fields are presented by Accord (1), Aronofsky and Jenkins (3), Bruce et al (8), Carter (10), Carter et al (11), Cornell and Katz (17), Cornell (18), Dykstra (22), Jones et al (45), Jones, L. G. (46, 47), Jones, P. (48, 49), Katz et al (50), Kidder (53), Layton (51+), McMahon (61), Pottman et al (74) Smith (80), and Swift and Kiel (82)0 B. Well Tests in Finite Reservoirs The point source solution to the flow of a fluid into a single well of finite radius, developed by Muskat (66) is Kh 4 re 2 re - t (0 06O33 Kta (II-7) 2(0. o00633Kt 'T C~" A&C 5 T2 (Aft)

-8 -where: = roots of J1(X) 0= (II-7a).o = Bessel function, first kind, ozeroth order J, = Bessel function, first kind, first order re = radius of circular reservoir, feet rw = radius of pumping well, feet The corresponding apprioximate solution given by Horner (35) is S = op e; (O.(- (006333) K t ) (II-8) '-E < t4(o.oo633fKt ) 4(o.oo(,33) K,_ exp 4(o o633t J'je The last two terms of Equation (II-8) account for the effect of the external boundary. Thus Equations (II-6) or (II-7) are used to predict the pressure behavior of a reservoir during a drawdown 1PU test. The pressure behavior during a build-u test is again found by superposition. If to is the total pumping time and At the time after cessation of pumping, the approximate point source solution (35) is 0.434 Ei (,.0.0 0633Kt. (II-9) 40o o0o,063)Kt, exp ( (o.00633KVto I

-9 -Graphs of the last two terms in Equations (II-8) and (II-9) are given by Horner (35). C. Effect of Linear Faults on Pressure Behavior The method of images (35) is used to describe pressure behavior of a well producing near a linear faulto Thus if the distance from the production well to the fault is L, the pressure behavior for a drawdown test is given by +ow = < O KM3 Ei E- 4(633 Ei(- o, 00633oKt, (II-10) and for a build-up test by -Ir = + 7(0.Ei (-{0.00633)/K(.6+t) A 003 6) 33 tIto+) E; (-.4 oo,(,633) Kito - E;(- o0. o0633f ) Pumping time and the distance L are usually large enough to allow Equation (II-11) over the first part of the build-up curve to be approximated by 162- ' + At Kh ', - O. 4 34 E~i (- o.0033 K to)}(II12) For large shut-in times, the pressure behavior can be calculated from

-10 -~PVw = fe h C~81.t,,~~ ( ~to At )(II-13) Thus when the pressure is plotted versus log10( tt), the slope over the latter portion is twice the initial (35). D. Interference of Several Wells Pumping in a Common Aquifer The pressure behavior of wells producing on a common aquifer can be obtained by superposition of the separate effects. The pressure in an aquifer where other wells are producing is given by Kh E ( 4(0.00633)Kt (.oo6s3(4)Kt:; I where: q - production rate of given well, bbl/day qi - production rate interfering well, bbl/day Li distance from given well to "i" interfering well, feet n = number of interfering wells rw radius of given well, feet Additional procedures for predicting the effect of interference are given by Coats et al (14), Hurst (39), Mortada (63) (64), Parson (70), Stevens and Thodos (81), and Warren (90).

-11 -E. Skin Effects The resistance to flow caused by a reduced permeability within a few feet of the pumping well is known as the "skin effect" and may be calculated by the following equation (88) (92) Se = lt+n 3 -1 where: S. = skin effect Plhr = pressure at shut-in time of one hour, psia Pf flowing bottom hole pressure at shut-in, psia m slope per cycle of straight-line portion of build-up curve, psia/cycle Application of the skin effects are discussed by Arps (4), Brons and Marting (6), Hurst (37), Johnson et al (44), and Van Everdingen (88). F. Other Topics The effect of leakage on the performance of aquifers is discussed by Hantush (25) (28) (29) (31), Hantush and Jacob (30), Jacob (41), Walton (89) and Witherspoon et al (93). Reservoirs with different permeabilities are analyzed by Henson et al (33), Katz (52), Lefkovits et al (55), Loucks and Guerrero (56), Mueller (65), and Warren (91).

-12 -Pressure behavior of partially penetrating wells was investigated by Hantush (27) and Nisle (69). Water movement and pressure behavior in oil and gas storage reservoirs are described by Coats (13), Coats et al (14), Hutchinson and Sikora (40), Katz et al (50) (51), Katz (52), and Rzepczynski (77). Compressibilities of reservoir rock have been measured by Fatt (23) and its effect on permeability by McLatchie et al (60). Numerical methods were used to solve two phase flow by Nielsen (68). Counter current gravity segregation was investigated by Briggs (5). In summary, available solutions to flow of fluids in homogeneous reservoirs of constant thickness used in calculating insitu characteristics include: lo Drawdown tests in infinite or finite reservoirs - insitu permeability and compressibility 2. Build-up tests in infinite or finite reservoirs - insitu permeability 3, Drawdown tests in infinite reservoirs with leakage - insitu permeability 4. Drawdown tests - reservoir hydrocarbon volume (reservoir limit test described by Jones (48) (49)), Although one well is sufficient for calculating the insitu properties, two or more wells increase the reliability, The dissertation extends these tests to include: 1, Build-up tests in infinite aquifer with leakage - insitu permeability 2. Drawdown tests in aquifers for locating gas fields or lines of constant pressure in vicinity of test wells.

III. DETERMINATION OF ERROR IN NEGLECTING THE NON-LINEAR TERM IN THE PARTIAL' DIFFERENTIAL EQUATION DESCRIBING THE FLCJ OF SLIGHTLY COMPRESSIBLE FLUIDS IN POROUS MEDIA The purpose of. this section is to define the conditions under which one is justified in neglecting the nonlinear term in the differential equation describing the flow of slightly compressible liquids in porous media, A graph is presented showing the reservoir pressure behavior as a function of dimensionless time when the term is not neglected, A second graph gives the maximum values of dimensionless time for various values of the dimensionless coefficient of the non-linear term beyond which neglecting this term is not justified. Example problems illustrate how one may determine if neglecting the non-linear term is justified. Katz et al (50) and Rowan and Clegg (76) noted that when a single equation of state is used in the derivation of the differential equation for flow of a slightly compressible liquid in a porous media, a non-linear term appears. Many authors have used two equations of state in deriving the differential equation for the flow of a slightly compressible liquid and thus avoided the term in their erroneous derivation~ A, Derivation of Non-Linear Differential Eauation The differential equation describing the flow of a slightly compressible liquid is obtained by combining the following three equations:

1. The equation of continuity 2. Darcys law 3. Equation of state for slightly compressible liquids The continuity equation is obtained by writing a mass balance on a differential element showing the net change in the mass flow rate in and out of the element is equal to the rate of change of the mass in the element. In Cartesian coordinates, the continuity equation is rx + + + + = ~ + > (III-l) where: t = time x,y,z - space coordinates vx = fluid velocity in x direction vy = fluid velocity in y direction vz - fluid velocity in z direction =< - fluid density t - time = porosity of the porous media Darcys law is an experimental observation that the velocity of a homogeneous fluid flowing through a porous media is proportional to the force potential i and inversely proportional to the viscosity. V A= ' e Vi (III-2)

-15 -where: i-gz +. K = permeability As pressure / = viscosity o = density If the effect of gravity is neglected, Equation (III-2) can be written K,,J Vs _ _ s (III-3) where s is the direction of flow. Writing Equation (III-3) for each component in Cartesian coordinates yields Vx = A. (III-4) \/Y /4 ( /I-5) V - ~ 4 (III-6) The equation of state can be derived from the definition of compressibility I V _ I (II7 If the compressibility is independent of pressure, then integrating Equation (III-7) gives ~( \b) An / = o I_ n

The equation of state is obtained by exponentiating Equation (11I-8) and rearranging e P-~ e -P.f~ (III-9) Expanding Equation (III-1) yields + ~V), V, +/o ' =v t + Vs + e=0 (III-1l) Applying the chain rule to Equation (III-10) gives t~P ax VX + v. K ax + ay VY + / ( +~' ~ v. +/: ~ v. _a v. ~ Differentiating Equation (III-9) gives c e ('~'.) (III-12) Substituting Equations (III-4), (III-5), (III-6), and (III-12) in Equation (III-11) and cancelling 1o e in each term gives cK K t _ cK2/ K - _ (III-13) /,, y A,, - -a.

-17 -Thus rearranging Equation (III-13) gives the differential equation describing the flow of a slightly compressible fluid in a porous media e + + 9+ C (( )+ ()+ (XlC ) ) (III-14) For linear flow, Equation (III-14) reduces to e4 C a =&c. d0 (I II 1) For radial flow, Equation (III-14) is given by ez~ t e +c(~ a= c A (III-16) Neglecting the non-linear terms in Equations (III-13), (II-14), and (III-15) yields diffusivity equations that most authors use in solving for the flow of a slightly compressible liquid in a porous media: 1. General 17e #= /t e (III-17) 2. Linear flow I X2 =,f (III-18) 3. Radial flow b'~L~+ t e = 4 f (III-19)

-18 -B. Solution of Non-Linear Differential Eguation for Radial Flow, Constant Terminal Rate Case In addition to the differential equation derived for radial flow in part A Po + rt + C( ) ~"-a= ' (III-16) one initial condition ( - 0) = (111-20) and two boundary conditions '_r. | ~= t K t a 0 (III-21) Ads (OO, t ) _ -o (111-22) are necessary to define the problem. The terms in these equations are defined below: A = area perpendicular to flow, square centimeters K permeability, darcys = pressure, atmospheres o = initial pressure, atmospheres Q = rate of production, cc/sec r = radius, centimeters

-19 -rw = radius of well, centimeters t = time, seconds, = viscosity, centipoise = porosity, fraction Substitution of the dimensionless quantities R — = (III-23) p = 2 Tr(SOT-) (III-24) K t to=~~~~~ ^+cd (IT1 25) into Equations (III-16), (III-20), (III-21), and (III-22) yield the following dimensionless equations PD ( R, O) = O (11-27) R()I P, (, t) = 0 ('II-29) respectively.

-20 -Define the dimensionless coefficient M Q= _ c (III-30) 2 fh I and substituting in Equation (III-26) yields a R R ( + ) R ) t ( III-31) Thus the pressure can be calculated at any time by -~ P o 2t hQ K Po (111-32) In Engineering field units, the dimensionless quantities defined by Equations (III-25) and (III-30) can be calculated from _ o, o063 K (III-33) M. =- i.2. c 4.. (,2-3) M (III-34) where: c = compressibility, vol/(vol) (psi) h = thickness of porous media, feet K = permeability, millidarcys q = flow rate, bbl/day r - radius of well, feet

-21 -t = time, days - porosity, fraction A = viscosity, centipoise Thus the reservoir pressure in psia is calculated by ~- <1b141.2: p ( I- Io where: po = initial pressure, psia p = pressure, psia PD = dimensionless pressure The values for dimensionless pressure, PD, as a function of dimensionless time, tD, are solved in this section. No analytic solution to the differential Equation (III-31) is known. Thus the Equation (III-31) along with the initial and boundary conditions, Equations (III-27), (III-28), and (III-29) were solved by replacing them by finite difference equations and solving the difference equations numerically on the IBM 7090. Partial differential equations may be solved by an explicit or an implicit method. These methods are currently being used to solve reservoir problems for which analytical solutions are not available (8) (15) (33) (34) (65) (68) (75) (93)0 1. Explicit Numerical Method Equation (III-31) was solved numerically by the following method. Transform R using

-22 -w _ ( - e (III-36) so that the limits of R, I - R - ~ is transformed to the limits of w, O w I. The values of each term in Equation (III-31) in terms of w are derived below. Differentiating Equation (III-36) gives d.w = e-(R-I) = 1- w (111-37) d"R Thus* - - =3P Jw =,i ( I-_w) (III-38) and 6=2P w (L- w )J Rw C W w 3 —2 (I-w) (III-39) Solving Equation (III-36) for R gives R = I- A (I-w) (iII+40) Substitution of Equations (III-38), (III-39), and (III-4O) in (III-31) gives *Note that the subscript D is dropped in this section to avoid confusion with other subscripts to be introduced later.

-23 -(I-s)z b2p II:-W P ( —W).QP + M-(IW ) gJ( a J (iii4l) which reduces to (I~w)L 1- -(WV- (I,,W) ~P + M (Zw)2(4ZP) = ( "2) +M(;-,,)'( ~ ' -to Equation (III-42) is solved in the region Osw I ( III43) and 0- t, -_ (IIIL-~) by representing the region by a grid w = i nw (III,45) and t= - jt tD (III-46) where A w and A tD are the increments of distance and time respectively. Thus P; j defines the value of the pressure at the point P

A finite difference representation of Equation (III-42) is (/.w)2 (.,+,, -2P P, P j) _[l J -v>(I-1w) 3 ()..)7 i- iAW P.., Solving Equation (111-47) for Pi j, gives an explicit expression for P;,J+, that is, P;, j, is equal to known quantities. P(A (( A w )L,. { (,2 P4 + jJ) ~im l 2[1 ) (lwj(+,,jj-m P, -j + (,) -J8 -VI ( -- I,'-.,. ' R_, J Pt,;, An analysis of the stability of the numerical An analysis of the stability of the numerical Equation (III-48) (15) (34) requires that -2 nto < 2 (III-49) or on rearranging at0 < w) (oIII0) is a necessary condition for a stable solution. Thus the maximum size of the time step is limited by the choice of A W.R

-255 -The boundary conditions defined by Equation (IIIT28) in terms of w is WP= (III51) Equation (III-l1) can be expressed in numerical form by using a three point Lagrangian fit for P and evaluating the derivative at w = o. Thus - 3 P.,, 1+ 4 P.,, - P..;, = I (III-52) 2aw is the numerical expression for Equation (III-51). The remaining boundary condition, Equation (III-29) in terms of w is P= - = O (III-53) The initial condition, Equation (III-27) is eP, O (ITI-54+) The MAD (Michigan Algorithm Decoder) program used to solve Equations (III-48), (III-52), (III-53), and (III-5+) is given in Appendix A. This program was used to evaluate the values of dimensionless pressure for values of dimensions less time from 0 to OO1. The results are presented in Figure (III-1). Values of dimensionless pressure for values of dimensionless time greater than 0.01 were determined by an implicit method described in the next part.

-26 -2. Implicit Numerical Method The solution of Equation (III-31) by an implicit numerical method is developed below, Equation (III-31) is transformed to the w plane by w =- t- '' -R-) (III-55) Note that a scaling factor "a" is introduced in this transformation. Hence a( -)P (pII-56) and aR, = (I-( W)t ' -~(w) (III-5 7) Solving Equation (III-55) for R gives The differential equation in the w plane is obtained by substituting Equations (III-56), (III-57), and (III-58) into Equation (III-31). I2( I-W), a' P -at ( - # W, ~ i(-W) P (III-59) + M oLA (I -W)!p)2 p AW t

10 oOO'ce) w U/) Uf) w a- III ~ ~ ~ ~ ~ ~ DMESOLS =" M-,,t U) Z I w.011.001.01.1. 10 ~100 DIMENSIONLESS TIME, tD Figure III-1. Effect of Non-Linear Term on Dimensionless Pressure Drop versus Dimensionless Time for Radial System, Constant Terminal Rate Case.

-28 -where 0 _ w I (III6O) Expressing Equation (III-60) in difference form gives 2 ( 2Aw (III-61) a.2 (I- f ~ __._______ _ _ PI_ _1 ~a -W) 2 aw (i M, _ = P(I4).-, Replace the non-linear term in Equation (III-61) by ( +lJ*I - 1) P= (4P ' J+I-R-', ) (111I-62) where..,.j.,? - P;,,., is estimated from previous time steps or previous iterations and corrected until PI'I+ - - P -, (III-63) Substituting Equation (III-63) in Equation (III-61) and rearranging gives

-29 -Sp i (I-i;w)2 (I-;, ),, ( - i Lw) p, _2_ (_-iw)_ I (4w) + 2 w (q- - v(l-;w))2aw -M (I ' w)' * 4 (A)2 -,.,J~', )A'f-t. Multiply Equation (III-6+) by (Zw) to obtain the implicit numerical equation for the differential equation,, w(-ijw) - Aw (i-iiw) ) P+8,,.) 2( ( - -,-))) + f-i (i w )+ -d 3_t(j 6w = — {( P jw)2 + At,.. ( -; w) ( w) AW r 8 ) _; R + w 2 4 J at A

-30 -Equation (III-65) is solved by the following procedure. Assume Pj.,, j. = G j+C; +D; (III-66) Substitution of Equation (III-66) into Equation (III-65) yields J ( Z ) +,,j+, ( Y;) + C; P;JJI + Di(X;)= K,; (II1-67) where; = (- w) aw (I-;4 ), w( -;Aw) - _ 2( ~.-.,~(2 -;i w )) wc V} j #* (iii-68) _ ML (t_; a~l (s * - P _+ pi Y; -2 (I -; w)2 (.w (IIi-69) -u t <D _ /'4 (I-;4w)a (~ *;+lJ'l - P,2;) (III-70) i, =; 71)

-31 -Solving Equation (III-67) for P;j.,, gives % J f Z.-; + K'< D; X; (II72) '+ I',j+,l c; + Y C; X;i y; Comparison of Equation (III-72) and Equation(III-68) yields -Z'; cj+1 = I ~~~~~(III-73),+, - C; X, + Y; and D;, - (III.74,) C; Xi + y; Hence the recursion relationships are obtained for C; and D; which will allow the calculation of C; and D; i = 2..., imax provided CI and DI can be found. The boundary condition, Equation (III-28) in the w plane is given by.~v |? = (III75) aw w..0 OL Using the same procedure as in the derivation of Equation (III-52), Equation (III-75) can be approximated by a 3nPo.;, d+ 4 P,:+ - Pa.j+ (h-c76) aaw. and hence

-32 -pw_3PeJ., = -'; (11III-77) Evaluating Equation (III-67) for i = 1 gives P,;,,j (z,) * FP1J,, (Y,) + P,J.,I (X,)= (I11-78) s ince oOJ, S CI P J +D1 (III-79) Substituting Equation (III-77) into Equation (III-78) yields ( 2 _ -3 Poa + 4 P., )J, ++(Y,) (III-80) +PO, jl XI) ( K,, or,,;(X,- 3Z, )+RJ,, (4,Z-, Y,) j -,z, (iii-81) Solving Equation (III-81) for Po,+i yields Pp j4Z +Y.) 3Z (+ I Iz 82) " x, -+-Z,82) Comparison of Equation (III-82) with Equation (III-65) shows that - 4 Z, - Y3 (11183) C - (I-$3)

-33 -and Kgj -(Z w Z~a w A)(IIN-8Y) X, - 3 Z, All values of Ci can now be calculated from Equations (III-73), (III-74), (III-83), and (III-84). The last equation in the matrix has the form piX on J ' P I= I 1(1J l Ctj +-8 The boundary condition for i - imax is p^ j+= C (11iii86) The resulting tri-diagonal matrix can be solved by the following procedure: 1. Set P, = 2. Set p p 3. Calculate C, and D, 4. CalculateC;,D; for i = 2,..., imax ~5. Solve for P;, from Equation (III-64) 6. Test P,+,jl ' -P;, *', _ i, 7. If this condition is not satisfied, set Ptj+ P- i,iL for i = O.., " 8. Repeat steps 2 through 7 9. Proceed to next time step The MAD program used to solve the equations in this part is given in Appendix B. The values of dimensionless

pressure, P0, evaluated for values of dimensionless time, tD, from 0.01 to 100, are presented in Figure (III-1). C. Solution of Non-Linear Differential Equation for Linear Flow, Constant Terminal Rate The differential Equation (III-15) describing the linear flow of a slightly compressible liquid __ _ -, 4&= (III-15) along with the initial condition -P (Xj O): o0 (III-87) and two boundary conditions 'x-o - A _- o (III-88) ~ ( * i,~ " ) "-Ho (III-89) define the constant terminal rate case in infinite linear porous media. Substituting the dimensionless quantities X = x (III-90) PO QX (I111-91)

-35 -K 3 tD = KA c x (III-92) M X.- C E4 (II193) into Equations (III-26), (III-27), (III-28), and (III-29) gives the dimensionless equations to be evaluated, respectively. aX+M - a = (III-9~4) Po (X, O) - 0 X o (IIIO95) ax = -I (III-96) X O PD (~ t0) = O (III-97) Thus the pressure at any time is given by e=~ _- A K Po (III-98) The dimensionless terms defined by Equations (III-92) and (III-93) in engineering field units are:

-36 -t = 0.00633 Kt (III-99) and M = _ 887.6 cAx. (III-100) AK where: A = cross sectional area normal to flOw, (feet) c = compressibility, vol/(vol)(psi) K = permeability, millidarcys q flow rate, bbl/day t = t ime, days Xc = reservoir characteristic length (arbitrary), feet * = porosity, fraction )A4a viscosity, centipoise Thus the reservoir pressure in psi can be calculated from b=,b_ 887.0 6 Do f.xe p0 (III-101) As in previous case for radial flow, the linear flow case will be solved by numerical means using both implicit and explicit methods. 1. Explicit Numerical Method Equation (III-94) is transformed by = - e - (III-102)

-37 -so that limits 0 X = e are transformed to 0 = w - 1. The values of each term in Equation (III-94) are given by ~P _ 1P dw AP x. w W) (III 103) and a2P )P Substituting Equations (II-103) and (III-lOL) in Equation (III-9~) yields (IW)2 P (-w) P +M(I-w) ( P) = P (III-105) Equation (III-105) can be represented in difference form by (L2.w)I-1 P:P (III1o06) +M ( I-iw)e{.,;.=.. _, { 2, } /\ twhere i and j refer to the mess points of the distance and time increments respectively. Rearranging Equation (III-106) and solving for P;,;,, gives JI I,.PtJ (P..-w). "(. i MW) (Pi j AW). -(/-,~,,)~ a-,,J- P,, ) + M (1- ),P)-" )2}

the difference equation in implicit form. Again an analysis of the stability (15) (34) requires that at0 < z (\w)P (III-108) The boundary condition given by Equation (III-96) is _ P| = I (III-109) which may be expressed in numerical form by _.+AU~Z +4p,,2*L --- -J (III-110) The remaining boundary condition is P @ = 0 (IlI-ll ) The initial condition is given by P; = 0 (III-112) Equations (III-107), (III-110)), (III-111, and (III-112) were solved on the IBM 7090 computer for imax - 800 The MAD program is presented in Appendix C. Dimensionless pressure was evaluated for values of dimensionless time from 0 to 0,01. Values of dimensionless pressure for values of dimensionless time greater than 0.01 were found by an implicit method as discussed in the next part. These results are shown in Figure (III-2).

.000' 0 Cr: -~~~~~~~~~~ - U) UL) - I w.01.001.01 1 1 10 100 DIMENSIONLESS TIME, tD Figure 111-2. Dimensionless Pressure Drop versus Dimensionless Time for Linear System, Constant Terminal Rate Case

-40 -2. Implicit Numerical Method The non-linear differential Equation (III-92) is transformed by w = - e-.x (-II-113) as follows.* bP -P Jw = R (- w) P iB.te ( }_,V)2 dw -as (g-w) JiW(III-115) X W w X ex 2 w Substituting Equations (III-114) and (III-115) into Equation (III-94) yields b-(,-W)L P - af(l-~,) ~ (III-116) +Maa (I-w) (w = aOP w I A difference equation approximation of Equation (III-14) is given by (111-117) The subscript D s dropped as in the preous derivations

Again the non-linear term in Equation (III-15) is replaced as in Equation (III-61) by ',j* )~ -+ P )( P l-, j ) (11162) where (PP*, - ) is estimated from previous time steps or previous iterations. Substituting of Equation (III-62) into Equation (III-117), rearranging and multiplying by ( Aw) gives J~ - ) Z 4 ( ' i /\w) (- A ' +I4J..il 5(I};Aw)' 2 *(+,, J+ -)} =- p,( "6) The same procedure is used to solve Equation (111-118) as was used to solve Equation (III-65). Assume P3.,,., = C; P;,, + Di (111-119) Substituting Equation (III-119) in Equation (III-118) gives P;.,,, (Z;) + P;@,,J (Y;) (. III-120) +(C;,4,, +D;)Xi = Ki,j where Sz Equ t I -i;,w) ( P-;aw - )s + ) X;

Y; = -2. (1-;,) _ ' (III122) K; = - Pg; ( " a (III-124) Solving Equation (III-120) for P;., yields -p- Z; z p 4- Z \@K;, j -D;X; (III-125) J~+j -C1; '+' Y, C' X + Y; If Equation (III-125) is compared to Equation (III-119), it is obvious that C; + = _ Z. (III-126) C';Xi +y; and D K, j -D; X; (III-127) +j = C X; + Y; Equations (III-126) and (III-127) are the recursion relationships used to calculate the values of Ci and Dio. The boundary condition, Equation (III-88) in the w plane is a x I a x- (III-128) and hence iP ( 2 aI. = - - (III-129) AW iOWIO a

The remaining boundary condition, Equation (III-97), in the w plane is P ( I, tb) =o (III-130) The initial condition corresponding to Equation ( III-95) is P(w,0)= 0 O'w I (1II-131) The differential form -of the Lagrangian three point formula evaluated at the initial point is used to approximate Equation (III-129). ____;,, +4_,_,__-P2__;+_1 _ I (111-132) Rearranging Equation (III-132) gives Pzj j+ 2 /\ 3 Poj + 4 PI~, (III-133) Evaluating Equation (III-120) at i 1 gives P2,;+i (Z,) Pj+, (Y,)+ PO,j(X,) = IK,, (III-134) Substituting Equation (III-133) irn Equation (III-134) gives (2, _ 3 Po., + 4 P,.,)Z + P) (Y) ( III-13 ) + Po,+, (X,) = K,,j

-44 -or Po j., (X, - 3Z,) R.., (4Z,.Y)= 2K,, - 2,.wZ (III-136) Solving Equation (III-136) for P.j.,, yields P P {- 4,,Y, 2 E, -4WZ,/OL (III-137) DX,*I XI-Z, X, - 3z, Comparison of Equation (III-137) with Equation (III-119) shows that C, ~ - 4Z, -Y, (II-138) Xi - 32Z and D,: X - Z (III-139) X, -3 Z, All values of Ci can now be calculated from Equations (III-126), (III-127), (III-36), and (III-139). The last equation in the matrix has the form Since The tri-diagonal matrix can now be solved, and the values of P found for i = 09 1i 2,..., imax. The same procedure as outlined on page 33 for the radial model is used in this case.

The MAD program was used to evaluate values of dimensionless pressure, PD, for several values of the coefficient M, for values of dimensionless time, tD, from 0.01 to 100. These results are presented in Figure (III-2). D, Analysis of Results and Aplication to Reservoir Problems The percent error in the measurement of the pressure drop for a constant rate pump test in a radial flow system is shown in Figure (III-3). The value of dimensionless time for which the error is less than one percent may be approximated by to ( 1 erro7) 0.00 ~ (III-142) Ma The following two example problems show how to determine if neglecting the non-linear term in the differential equation describing the flow of a slightly compressible fluid in a porous media is justified. Example Problem III-1 Consider an aquifer with the following physical properties and dimensions: Compressibility, c = 7 x 10-6vol/(vol)(psi) Thickness, h - 160 feet Permeability, K 200 millidarcys Porosity, * =.21 Viscosity, / - 1 centipoise

12 101 Z) Irl LU o r s:-I0 it t~ 4 U o~~~~~rr P for Radlal system O

If a well with a one foot diameter, completed in the aquifer, is pumped at a constant rate of 300 bbl per day for three days, determine if one is justified in neglecting the non-linear term. Solution The coefficient M is calculated from Equation (III-34) Mt -1+412 Ca' W ' 7.2)fxtilZiOi)( () (3009.27 xo-6 1' - ~~(60o)(2z00) and the value of dimensionless time at the end of three days by Equation (III-30) _ 0o0633 K t _ (0.Oo633)(16o)(3) - 8.3 K/06 to - A C ~ (I)(0,21)(87xlO') (OS)Z The value of dimensionless time for which a one percent error in the calculation of pressure drop is made in neglecting the non-linear term is calculated from Equation (III-1L2) 0.0 oool 0001 1.16 xOTi tM. (27 /o -) a 1' 6 x O Neglecting the non-linear term results in less than a one percent error in the calculation of the pressure drop. Thus one is justified in neglecting the non-linear term..and using solutions based on the diffusivity Equation (,11-19)0

-~~8 -Example Problem III-2 A natural gas storage field is situated on a large aquifer. The gas field and aquifer have the following dimensions and physical properties: Compressibility, c = 7 x 10-6 vol/(vol) (psi) Thickness, h = 20 feet Permeability, k = 200 millidarcys Porosity, 4 = 0.18 Water Viscosity,,. = 0.8 centipoise (reservoir conditions) Radius of gas field, rb= 3000 feet If the gas field is produced for ninety days at a rate which causes a water influx rate of 20 000 bbl/day, determine if neglecting the non-linear term in the solution of the flow equation is justified. Solution Again the coefficient M is calculated from Equation (III-34) M= 141.2 I4.2)(71)o-6)(Oa)(20o0o) ooo396 I~ ~~ (200) (20) The value of dimensionless time at the end of 90 days is to = 0.00633 4t _ (.0 o0633)(200)(90) _ D,. $ c A,2 (o. 8)(O.8)( 7 xl o-6)(3, 06)

Equation (III-142) is used to determine the value of dimensionless time above which an error greater than one percent would be made to E d0.009 0.001 o= 2 = (396 -4) 6370 Thus the solutions based on the solution of the diffusivity equation can be used to calculate pressure drop. The error in pressure measurement resulting from neglecting the non-linear term for a constant rate pump test in a linear flow system is shown in Figure (III-4), Again the value of dimensionless time for which the error is less than one percent may be approximated by to ( IZ error) ~- 0.001 (III-142) Note that this is the same expression found in the radial case. An example problem demonstrates how to determine if the non-linear term may be neglected in the linear flow case. Example Problem III-3 A gas field is situated on a large aquifer. The gas field and aquifer are located between two parallel faults so that the linear equations can be used to predict the pressure. The aquifer has the following dimensions and physical properties: Cross sectional area to flow, A = 500,000 square feet Compressibility, - 7 x 10-6 vol/(vol) (psi)

-5O0 -Permeability, K = 50 millidarcys Porosity, O. 10 Water Viscosity, H/ = 0.7 centipoise Gas is injected into the field for thirty days causing a water efflux rate of 10,000 bbl/day. Justify neglecting the non-linear term in the differential equation used in determining the pressure behavior in the aquifer. Use 1,000 feet as the characteristic length. Solution The value of M, calculated from Equation (III-98), is M = 887.63 CX. _ (887.6r)(Txlo-6')(o.7)(10, ooo)(000oo) HA (50)(50,oo00o) i.73 xO 10 Dimensionless time at the end of thirty days is given by t- O.OO, 33 t. - (O.OO633 (o5) (30) DCC. C Xc o(0o7)(o I )(7x 1o-)(10') = 119.4 Equation (III-142) is used to determine the value of dimensionless time when the error reaches one percent 0.001 0.00I 3_o tbi 70 error) Mo =.74 l-3) 330 Since the value of dimensionless time at the end of thirty days is less than the value which results in a one

12 I-. Lu ~1O I0 a.. LUJ 0 I~1 7' a: O3 6 -a. a: 4 o ir~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ r 0 DIMENSIONLESS TIME, tD1 D.~~~~~~~~~~~~~~~~~ Fig~~ T1..4 Ter cen tl Errt r Ca e D h e son ies Pressure Drop for Linear Syste m, C t n Rate Case.~ ~ ~ ~ ~ ~ ~~7Costn

percent error in the calculation of pressure drop, solutions based on the diffusivity Equation (III-18) are valid. The same conclusion results for any selected value for the characteristic length.

IV. EFFECT OF NEIGHBORING GAS FIELD (OR LINE OF CONSTANT PRESSURE) ON INSITU PERMEABILITY AND COMPRESSIBILITY MEASUREMENTS IN AQUIFERS FOR CONSTAiNT RATE PUMP TESTS Field data on pressure behavior of wells have been observed to deviate considerably from the behavior predicted by the methods described in Section II. Failure of the mathematical model to describe the reservoir is partially responsible for these deviations. The presence of a gas field, an outcropping of the porous media on the bottom of a lake or stream, or a sudden increase in the permeability are a few conditions which may cause the deviation. Effect of these conditions on the measurement of the insitu permeability and compressibility will be evaluated in this section. Furthermore, the unsteady state pressure behavior of wells during constant rate pump tests will be described for these situations. A. Prediction of Pressure Behavior During Drawdown or BuildPump Tests Field pump test data from aquifers can be analyzed to determine if a gas field (or line of constant pressure) is located near a pumping or observation well if the theoretical pressure performance is known as a function of the distance between the pumping and observation wells in addition to the distance between the wells and the gas field. These relationships are developed below0 -53 -

O OBSERVATION WELL * PUMPING WELL CONSTANT PRESSURE LINE Figure IV-1. Pumping 'Well and Observation Well near a Line of Constant Pressure. Y OBSERVATION WELL PUMPING WELL MAGE WELL -Xo + Xo X Figure IV-2. Mathematical Model and Location of Image SWell.

Pressure Behavior of Obser-vation Well During Drawdown and Build-up Tests Figure (IV-1) shows a sketch of a pumping well and an observation well near a line of constant pressure. The mathematical equations which describe the pressure behavior of an observation well during a drawdown pump test is given by the diffusivity equation Ox2/+ 2. =C <~ (IV-1) the initial condition - (x, Y~o) = -o (IV-2) and the boundary conditions ~ +o ( X, ye, t) = (IV-3) Ye ~~:,h(-Xo;~ t ) = 27 K $(IV-5) + (so) D, t) - 2K (IV-6) If the distance between the pumping well and the observation well is greater than 30 times the well radius, Mortada (63) showed that the point source solution presented

by Horner (35) is valid. Thus the solution to Equations (IV-1) through (IV-6) is given by the addition of a point source at x - -xo, y = 0 and a point sink at x xo, y O 0 (See Figure IV-2). (IV-7) where: c = compressibility, vol/(vol)(atm) Q = production rate, cc/sec J/ = viscosity, centipoise K = permeability, darcys h = thickness, centimeters Ei exponential integral, Ei(-x)=-/,e d p - pressure, atmospheres Po = initial pressure, atmospheres, = distance between pumping well and observation well, centimeters P = distance between observation well and image well (See Figure IV-2), cent imeters = porosity, fraction Replacing the terms in Equation (IV-7) by the following dimensionless quantities p 2 TrK/(p-fY (IV-8)

-57 -TD= + C KtQ (IV-9) RD = R. (IV-10) yields PD 2 - TTD- o) (TVi) If field units are used, the expression for the dimensionless pressure (Equation IV-8) is given by PD /141 2kh (f.'-t') (IV-12) where: h = reservoir thickness, feet K = permeability, millidarcys p = pressure, psia Po = initial pressure, psia q = production rate, bbl/day A viscosity, centipoise and the expression for dimensionless time becomes T7 = 00633 K -t (IV-13) where: = porosity, fraction c = compressibility, vol/(vol)(psi)

-58 -t = time, days, = distance between pumping and observation well, feet The distances in Equation (IV-10) must be expressed in the same units. Numerical values of dimensionless pressure drop, PD, as a function of dimensionless time, TD, are presented in Figure (IV-3) and Table C(V-1) for values of RD between 1 and 16 and TD less than 1,000. Steady state values for the dimensionless pressure drop, PD, as a function of dimensionless distance ratio, RD, are presented in Figure (IV-W). The pressure behavior for a build-up pump test is obtained by superimposing a negative flow rate on the time after the pumping well is shut-in. Thus ri -~ c~ M C, ak C _ - So c - 4 K (to t)) E (4K (o+ (IV-I14) E; (t + E4- K 't or if dimensionless pressures PD, and field units are used 4( 00633) -E 4' 3 E; (,. )a(33)a.t, )

3.5 3.0 2.5 0 cr_ Wc 2. 12/1p = 0 a2. 1.5 nW 12p =. I z 00 roo 1000 DIMENSIONLESS TIME, TD FiGU0c IV-3. D1Irn3nsinless Pressu'e Drop for Observation Ie11 in Radial SyIteri with Constant Extgrnal Pressur,3 Line.

TABLE 1I32 -)L...._._ __-_- DIMENSIONLESS PRESSURE DROP, P, (RD,tD ) FOR RADIAL SYSTEM WITH LINE OF CONSTANT PRESSURE DIM. TIME DIMENSIONLESS PRESSURE DROP, to RD=l.l RC)l.25 RD 1.5 RC)2 RD 2.5 R03 RD4 RD 6 RD 8. R0O R0Z12 RD16.01.0000.0000.0000.000C.0000.0000.0000.u00O.0000.0000.0000.COOO.02.0CO0.0000.0000.0000.0000.0000.0000.0000.0000.0000.0000-.0000.03.0000.0000.0000.0000.0OCC.00 00.0000.0000.0000.0000.oo.0000.04.OLOI.0001.0001.0001.ocro1.0001.0001.0001.0001.0001.0CO1.0001.05.0004.0005.0006.0006.0006.0006.0006.0006.0006.0006.0006.0006.06.0010.0014.0015.0015.015.0015.0015.0015.0015.0015.0015.0015.07.0019.0029.0032.OC32.0032.0032.OA32.0032.0032.0C32.0032.0032.05.0031.0049.0055.0056.0056.0056.0056.OC56.0056.0056.0C56.0056.09.0045.0074.0085.0087.0087.0087.0087..087.0087.0087.0887 70u81.15.00 1 0103.0121.0125.0125.0125'.0125.0125.0-125.0125.0125.0125.15.0153.0284.0367.0392.0393.0393.0393.0393.0393.0393.0393.0393.20.0243.0473.0652.0729.0134.0735.0735.0735.0735.0735.0735.0735.25.0316.0640.0923.1078.1096.1097.1C97.iC97.1097.1091.1097.1097.30.0380.0787.1171.1420.1458.1462.1463.1463.1463.1463.1463.1463.40.0479.1020.1586.2037.2140.2158.2161.2161.2161.2161.2161.2161.50.0549.1192.1903.2554.2743.2789.2799.2799.2799.2799.2799.2799.60.0602.1323.2157.2984.3268.3351.3376.3376.337-6.3376.3376.3376.70.0644.1425.2358.3343'.3722.3851.3898.3901.3901.3901.-3901.3901.80.0676.1507.2520.3644.4117.4295.4373.4378.4378.4378.4318.4378.90.0702.1574.2656.3902.4462.4693.4806.4817.4817.4817.4817.4817 1.00.0723.1630.2769.4124.4764.5047.5203.5221.5221.5221.5221.5221 2.00.0833.1909.3351.5322.6514.7225.7876.8111..8121.8121.8121.8121 2.50.0852.1965.3476.5602.6953.7813.8683.9084.9113.9114.9114.9114 3.00.0868.2009.3566.58-C2.7270.8245.9301.9881.9942.9946.9946.9946 3.50.0881.2037.3632.5949.7508.8513.9789 1.0546 1.0650 1.0659 1.0660 1.0660 4.00.0890.2063.3683.6063 71693.8832 1.0187 1.1110 1.1266 1.1283 1.1284 1.1284 5.00.0902.2095.37-51.6226.7961.9213 1.0786 1.2016 1.2289 1.2334 1.2339 1.2339 6.00.0910.2117.3803.6336.8147.9479 1.1218 1.2710 1.3110 1.3195 1.3209 1.3210. 7.00.0916.2133.3838.6417.8282.9616 1.1544 1.3258 - 1.3786 1.3920 1.3947 1.3952 8.00..0921.2145.3864.6477.8385 1.9828 1.799 1.3702 1.4353 1.4542. 1.4587 1.4591 9.00.0924.2155.3885.6527'.8467.9948 1.2003 1.4072 1.4835 1.5083 1.5151 T3169 10.09.0927.2162.3902.6568.8533 1.0044 1.2171 1.4382 1.5251 1.5558 1.5652 1.5681 12.00.0931.2174.3927.6627.8636- 1.0193 1.2429 1.4872 1.5928 1.6355 1.6508 1.6569 14.00,.0935.2182.3945.6670.8706 1.0302 1.2618 1.5243 1.6459 1.6999 1.7216 1.7320 16.00.0937.2188.3958.6702.8764 1.0384 1.2765 1.5534 1.6889 1.7529 1.7812 1.7967 18.00.0939.2193.3969.6727.88C-8 1.0445 1.2879 1.5767 1.7240 1.7975 1.8322 1.8534 20.Oo.0940.2197.3977.6747.8842 1.0498 1.2973 1.5960 1.7533 1.8352 1.8763 1.9036 40.00.0947 -.2214.4016.6838.9001 1.0740 1.3406 1.6883 1.9009 2.0359 2.1220 2.2089 60.00.0949.2220.4029.6869.9054 1.0821 1.3556 1.7215 1.9565 2.1161 2.2266 2.3552 80.00.0950.2223.4035.6885.90-81 1.0862 1.3632 1.7383 1.9858 2.1593 2.2845 2.4418 100.00.0950.2224.4039.6894.9098 1.0887 1.3677 1.7490 2.0038 2.1862 2.3211 2.4985 200.00.0952.2228.4047.6913.9130 1.0936 1.3110 1.7701 2.0409 2.2422 2.3994 2.6252 400.OC.0952.2230.4051.6922.9147 1.0961 1..3816 1.7809 2.0600 2.2721 2.4412 2.6960 600.00.0953.2230.4o52.6925.9152 1.0969 1.3832 1.7845 2.0664 2.2822 2.4556 2.7204 800.00.0953.2231.4053.6927.9155 1.0974 1.3840 1.7863 2.0696 2.2872 2.4628 2.7335 1C00.0S.0953.2231 --.4C53.6928.9156 1.0976 1.3844 1.7874 2.0716 2.2903 2.4672 2.7412

-61 -4.8 4.4 -0 4.0 -0 3.6 -en) 2.8 -a/ 2.4 U) L) 0 UI 1.6 z 1.2 0.8 0.4 ~I 0 100 LENGTH RATIO, RD =L2/LI Figure IV-4. Steady State Dimensionless Pressure Drop for Radial Systems with Constant External Pressure Line.

The values for the exponential integral can be replaced by an equivalent form (19). i; (-x') -= x + 0 5 772 + (-) x (IV-16) If the shut-in time is large so that / ( O.oo0633) K Z / less than 0.01, and the value of 3z is very large compared to,, then I I -t PO 2 A ( a&t 2 ' C 'tThus when a well in an infinite radial system is pumped, a straight line is obtained as illustrated in Figure (IV-5). The effect of the presence of a line of constant pressure in the vicinity of the pumping and observation:Liells is shown in Figure (IV-6) for RD = 6. Here a slope of 1.1515 is not obtained as in Figure (IV-5), Thus if ~2 is not very large compared to Q., substituting Equation (IV-16) in Equation (IV-15) gives for large shut-in time PD - - (IV-18) Hence an asymtope value of 0 is obtained for long pump tests as shown in Figure (IV-6). 2. Pressure Behavior of Pumping Well During Drawdown and Build-up Tests The point source solution can not be used to describe the pressure behavior of a pumping well during a

0 0.2 0.4 0.6 0.8 0 T0o 0.00633 Kto/p cr2 o I.Ow:r 1.2 03 w 1.4 n,'t~l -I I a. %T010 cn 1.6 w - 1.8 z 0 (n 2.0 z w 2.2 2.4 -,,00 2.6 2.8,,,~~~ I,,,,, I,,, I,?, 11 I,,,,. I,-,, I 2.8 10,000 1000 100 10 to+ At at Figure IV-5. Pressure Build-Up for Observation Well in Infinite Radial System.

0 0.2 s0.4 T 0.00633 Kt0/ p cr2 a. 0 W- 0.6 C cr 0.8 w 1.0 1.2 U) U) V) i.4 z 0 z w 1.8 2.0 10,000 1000 100 10 to+ At At Figure IV-6. Pressure Build-UP for Observation Well in Radial System with External Line of Constant Pressure, RD = 6.

drawdown test for values of dimensionless time below 1000o The dimensionless pressure drop, PD for a pumping well is given by D = Pt (it) + 2 # i(- 4 ) (IV-19) where: ~D = o0. 0 0633 K t (IV-20) RD = r Pt = dimensionless pressure drop for well in infinite radial system (Table 10-5, Katz et al (50), Chatas (12), Van Everdigen and Hurst (86)) The engineering units are listed below Equations (IV-12) and IV-13). For values of dimensionless time above 1000, Equation (IV-11) can be used to predict the ideal pressure behavior at the well. Values of dimensionless pressure, PD, versus dimensionless time, tD, are given in Figure (IV-7). The pressure performance of the pumping well during a build-up test near a line of constant pressure will be similar to that of a well in an infinite radial system if the pumping time is short. The build-up curve for a pumping well where RD = 100 is shown in Figure (IV-8).

5 ~~~~~~0~~~~~~~~~~~~~~~~~~~~F I00 0 1.~~~~~10 100 1000 10,000~ ~ 100,0 00 Iz, D TE TD xterna~l Pressure Line.

0.5 _ R= 2/rW = 100 0. To -0.00633 Kto/p crW2 irY 1.0 r A ~0 ~ ~ ~ ~ z 2.5 3.0 3.5 10,000.1,000 100 10 to + At Figure IV-8. Pressure Build-Up for Pumping Well in Radial System with Constant External Pressure Line, RD = 100.

-68 -B, Evaluation of Error in the Measurement of Insitu Permeability and Insitu compressibility The insitu permeability and insitu compressibility are obtained for infinite radial reservoirs by plotting pressure versus loglo pumping time. The insitu permeability can be obtained from the slope of this plot. (51) (92) rn=-16 2.6 4A (IV-21) whe re: m = slope of pressure vs loglo time, psia/cycle q = pumping rate, bbl/day K = permeability, millidarcys h - reservoir thickness, feet, = viscosity, centipoise Solving Equation (IV-21) for K yields;,~ - /62. i,>(IV-22) The insitu compressibility is obtained by evaluating the expression for the pressure drop _ = _ 70. 006 33) t, (IV-23) for the compressibility, The terms in Equation (IV-23) are defined below Equations (IV-12) and (IV-13), Thus solving Equation (IV-23) for c gives (o. 00633) K t E - o (IV-24) ~u b Ei 0pO 6 (IV-24

-69 -Unfortunately, there may be considerable error in the determination of the insitu permeability and insitu compressibility if there is an external line of constant terminal pressure in the vicinity of the pumping and observation wells. The magnitude of this error will be shown below. The numerical values for the slope of a plot of dimensionless pressure drop, PD, versus dimensionless time, TD, when an external line of constant pressure is present can be obtained by writing Equation (IV-11) in integral form PD= 4f a.e~ -— Uf2 Li e d. (IV-25) D4T0~~~~~L 4To and differentiating with respect to log10 TD = a, - 2. 3 e4T (4-T (IV-26) The error due to the external line of constant pressure is given by the last term in Equation (IV-26). The error in the measurement of the insitu permeability is a minimum if the maximum value of the slope from Equation (IV-26) is used. The value of dimensionless time, TD, when the slope, m, is a maximum for a given value of the dimensionless length ratio, RD, is obtained by differentiating Equation (IV-26) with respect to loglo TD setting the resulting expression equal to zero, and solving for TD. Thus

3)feRK 4 TD R- e TD 0 (IV-27) a/~L, To~) - To e-.e and TD RD -J (IV-28) 4 o where a- = logarithm base e The minimum error in the measurement of insitu permeability for a given value of the dimensionless length ratio, RD, is obtained by the following steps: 1. Use Equation (IV-28) to find TD. 2. Find the slope for this value of TD from equation (IV-26). 3. Calculate the minimum error in K from equation (IV-29). ___o__) I e + eQ (IV-29) where K (obs) = observed permeability, millidarcys. A plot of the ratio of observed permeability to the true permeability, K (o(s) versus the dimensionless length ratio, RD, is shown in Figure (IV-9). Note that the observed permeability

-71 -is always greater than the true permeability. Thus if K equals 1,3, the observed permeability is 30 percent higher than the true permeability, The error in the measurement of insitu compressibility which corresponds to the error in the measurement of insitu permeability is obtained by the following analysis. If an external boundary of constant pressure is present, the dimensionless pressure drop is given by Equation (IV-ll ) PD =-f9 Ei(- E (IV- l) In the absence of an external boundary (R= - ), the dimensionless pressure drop is given by EP = 2 E, (- 4T )i (IV-30) Thus if no external line of constant pressure is assumed when there is such a line, the error in the measurement of insitu compressibility is as follows, Equation (IV-30) is assumed to give the correct pressure drop when the actual pressure drop is given by Equation (IV-11), thus Ei (- 4T (obs)) E i ( T ) E ) (IT Solving Equation (IV-31) for -4Tl gives 4Td As) 4 r E( 4TD) ( )1D (IV-32) 4 Tr% (o b,) 4 n4To

2.27 2.1 2. 1.9 1.8 1.71 mw6~ 1.5 1.4 1.3 1.2 2 3 4 5 6 7 8 910 20 40 100 L2/LI Figure IV-9. Determination of the Minimum Error in the Measurement of Insitu Permeability Resulting from an. External Line of Constant Pressure.

-73 -Expand TD (obs) yield ing -:(0)C)T EK LF 4'T ) E i i(- )3 (IV-33) Solving Equation (IV-33) for c ) gives C c(bs) - K (o 6s) (V ) Co= (. o)(T) 4i { E ) Ei (4T R ) (IV_3) where c(obs) is the measured value of the insitu compressibilityO The ratio of observed compressibility to the true compressibility, c (L, versus the dimensionless length ratio, RD, is shown in Figure (IV-10). As in the case for the observed permeability, the observed compressibility is always greater than the actual compressibility. C. Description of Graphical Method for Locating External Line of Constant Pressure (or Gas-Water Interface) The location of the external line of constant pressure (or the gas-water interface if a large gas field is located in the vicinity of the well test) can be determined graphically if the following information is available: 1. The true effective permeability (Obtain from pressure data from the pumping well or core data) 2, The observed permeability at an observation well at a known distance from the pumping well (Obtain from pump test)

-74. 3. The observed permeability at a second observation well at a known distance from the pumping well (Obtain from pump test) The procedure for locating the external line of constant pressure or the gas-water interface is as follows. For the purpose of discussion, assume the wells are located as shown in Figure (IV-ll). 1. Calculate the value of K(obs) for observaK tion well No. 1. 2. Using the value of K(obs) from step 1, read K the value of RD from Figure (IV-9). 3. If the distance from the pumping well to observation well No. 1 is ta" feet, then the distance from observation well No. 1 to the image well with respect to observation well No. 1 is (RDa) feet. Draw a circle of radius (RDa) around observation well No. 1. 4. Determine the locus of all points located midway between pumping well and the circle of radius (RDa) (Shown as locus No. 1 in Figure (IV-11)). 5. Calculate the value of K(obs) for observation well No. 2. C Using the value of Kobs) from step 5, read the value of RD for observation well No. 2.

2.4 2.2 2.0 1.8 -C,, 1.6 1.4 1.2 1.0 2 3 4 6 8 10 20 40 6080100 L2 /LI Figure IV-1O. Determination of Error in Measurement of Tnsitu Compressibility Resulting from External Line of Constant Pressure.

-76 -7. If the distance from the pumping well to observation well No. 2 is "b" feet, the distance from observation well No. 2 to the new image well is (RDb). Draw a circle of (RDb) around observation well No, 2o 8. Determine the locus of all points located midway between the pumping well and the circle of radius RDb. (Shown as locus Noo 2 in Figure (IV-ll), 9. The location of the external line of constant pressure (or the gas-water interface) is given by the intersection of locus No, 1 and locus No. 2. (This is shown as points A and B in Figure (IV-11).) 10. If the exact location is required, i.e., whether the external line of constant pressure is located at point A or point B, then, pump either observation well No. 1 or observation well No. 2, determine the K(obs) at the K other observation well and repeat steps No, 1 through No. 4, Locus No, 3, obtained by this method, will intercept either point A or point B. D, Example Problems Several example problems are presented to illustrate the application of the material in this section.

-77 -\ \ OBSERVATION 1 /I \ WELL(2)IO / \\ \-_ WELL 2 Figure IV-11. Illustration of Graphical Method for Determining Location of External Line of Constant Pressure or Gas-W:ater Interface.

-78 -Problem IV-1 The distance between an observation well and pumping well in an aquifer is 200 feet, The pumping well is located 300 feet from the edge of a large gas field. The locations of the wells in reference to the gas field are shown in Figure (IV-12). If the true value for the insitu permeability is 100 millidarcys, determine the insitu permeability that would be measured at the observation well, Solution The value of Ll and L2 are shown in Figure (IV-12). L1 200 feet L2- 824 feet Calculate the ratio L2 _ 824 4.12 L1 200 and read the value of K(obs) from Figure (IV-9) K K = 1.21 Thus the value of the observed permeability is K(obs) = 1.24 K = 1.24 (100) 124 millidarcys Problem IV-2 If the true value of the insitu compressibility for the aquifer described in Problem (IV-1) is 7 x 10-6 vol/(vol)(psi), determine the value of insitu compressibility that would be measured at the observation well.

AQUIFER GAS FIELD OB S ERVAT I ON WELL 400' it 400 PUMPING WELL IMAGE WELL GAS WATER INTERFACE Figure IV-12. Location of Wells for Problem IV-1.

-80 -Solution The value of L2 * 4.12 is obtained from ProbL1 lem (IV-1). Read the value of c(obS) from Figure (IV-10). cobs) = 1.28 The observed value of the insitu compressibility is c(obs) = 128c - 1.28 (7 x 10-6) c(obs) = 8.96 x 10-6 vol/(vol) (psi)

V. EFFECT OF LEAKAGE INTO THE PERMEABLE STRATA THROUGH THE CONFININING CAP AND BOTTOM ROCK ON THE INSITU PERMEABILITY AND INSITU COMPRESSIBILITY MEASUREMENTS IN AQUIFERS OR OIL FIELDS Leakage through the confining cap and bottom rock and its effect on the pressure behavior of well tests and the resulting insitu permeability and insitu compressibility is evaluated in this section. The equations describing leakage are presented in engineering units so that comparison can be made with previous work in analysis of reservoir performance. Furthermore, the effect of leakage is shown to be similar to the effect of an external line of constant pressure discussed in the previous chapter. Thus it may be difficult or even impossible to differentiate between the two effects by a single well test. Pressure behavior for both drawdown pump tests and build-up pump tests are presented, Example problems at the end of the section illustrate the application of this material to field data. Hantush (28) (31) and Hantush and Jacob (25' examined the problem of leakage during pump tests in underground aquifers. Their results are presented in terms of transmissibility, coefficients of storage, and well functions; terms used by hydrologists, These results are transformed into engineering terms such as dimensionless pressure drop, dimensionless time, permeability, compressibility, viscosity, and porosity so that their work can be -81 -

-82 -compared with previous accomplishments in the field. The superposition principle is used to extend these results to build-up pump tests. A. Prediction of Pressure Behavior During Drawdown Tests The differential equation describing the flow of slightly compressible fluids in a porous media with leakage (25) is given by -e r e -1Z 04'3-= C. e (V-1) 2 +B 0", OoO633 t The initial condition is 0+ (~, ~) =p (V-2) and the boundary conditions are <) (o t) =t o (V-3) and r 14- 4 (V-4) where: B = KKh/(-h, +- ), feet (V-4a) c compressibility, vol/(vol)(psia) h - thickness of permeable zone, feet h' = thickness of caprock, feet h" - thickness of bottom rock, feet K = permeability, millidarcys

-83 -K' = permeability of caprock, millidarcys K' = permeability of bottom rock, millidarcys p - pressure, psia Po = initial pressure, psia q = flow rate, bbl/day r = distance from center of pumping well to point of pressure measurement, feet t = time, days = porosity, fraction / = viscosity, centipoise The solution to Equations (V-1) through (V-4) is given by Hantush and Jacob (25) as tq - A3 = 2 KO - 1O( )7E; ( 4 3 2)7 ( 28-.4/A h + exp (- 5 72 + +, (-I -o ( ) (V-5) ncl.*~2)/z 4 8 where: u = well function, u = 4 Io = Modified Bessel function, first kind, zeroth order Ko - Modified Bessel function, second kind, zeroth order tD = 0o00633Kt//A icr2

-84 -Equation (V-5) wrritten in terms of engineering un its is P,~ 4 ( B )-2 I( i) LE r tD )h +2T exp ( 0. 5 7 T 2 (4 -- ) k ( 44 ( wz (V-6) 6 to (n+)!32 4 4to where p 141.2 (-~o) lh (V-6a) The dimensionless pressure drop, PD, is showaJn as a function of dimensionless time, tD, with parameters of /( ) in Figure (V-I). Note that the general pressure behavior is similer to that observed with the external line of constant terminal pressure, Figure (IV-3). Extensive tables of a versus (-B-) are given by Hantush (28). For large values of dimensionless time, tD, a steady state pressure drop is obtained. This pressure drop in dimensionless terms is shown in Figure (V-2) as a function of ( B). B. Prediction of Pressure Behavior During Build-up Tests The pressure response for build-u well tests is obtained by superimposing a negative flow rate of the same

3.5 I I I III 3.0 -0 2.5 r/B = 0. I D2.a: | / / r/B = 0.2 a: -I~~~~~ I ~~~r/B=0.25 u) 1.5 U) w L r/B =0.3| z zl.0- r/B=O.' w 0.5 -Q01 2. 1.01.1 I I0 100 00 DIMENSIONLESS TIME, tD Figure V-1. Dimensionless Pressure Drop for Radial System With Leakage tlhrough Caprock and/or Bottom Rock.

-86 -magnitude on the drawdown test after the pumping well is shut-in. Thus the pressure behavior during build-up can be predicted by PD = 4KO (j) -2 I(-j[E i ( 8 )J +2 exp( -r r2 to ) 0 S 7 72 (4to) + [;Ei (- 4 —).7 - 4 + II ( B. I)/ r _ _ _ _ _o __4 to 4 B2t4 4 D 4 ((v-7) 4 Ko (B) + 2 lo (5)[Ei ( - r2 t/ ) r - 2!exp ( t o 5 7 7 (4at(4t +- Ei ( +, ) -O )cn o, r ( )M~- m + l)!, -t~ 16(',) (n+2 )!2 |4B/t where: 0.00633 KA (vt-8) tD = (V-8) tD = 0.00633 K t (V-9) 6t = time since well shut-in, days t = time since pump test started, days

4.8 4.4 0 4.0 0C 0 c 3.6 w 3.2 C,, C) w 2.8 a. 2.4 1rJ oJ 112.0 z 0 z 1.2 -0.8 0.4 0 0.1 I I IC r/ B Figurs V-2. Steady State Dimensionless Pressure Drop for Radial Systems -wit. Leakage through Confining Caorock and/or Bottom Rock.

-88 -The effect of leakage in build-up tests is shown in Figure (V-3) for-r = 0.2. Comparison of Figure (V-3) with Figure (IV-6) shows that the effect of leakage on the pressure drop is very similar to the effect of an external line of constant pressure. Thus additional observation wells are necessary to differentiate between the two effects. C. Evaluation of Error in the Measurement of Insitu Permeability and Ins itu C0mpressibit Standard methods used to obtain insitu permeability and insitu compressibility are described in Section II. The numerical value of the slope if leakage is occurring from a plot of dimensionless pressure drop versus dimensionless time is obtained by expressing Equation (V-6) in integral form (28) Po =I 2 i exp (-a 4 8 ) Do V-10) and differentiating with respect to log10 tD m - d =__ 2. 303 -'- (V-ll) M JQ oltD-) 2 e. If B = oO (no leakage), then the slope is 2.303 4 m = 3 e 4te (V-12) which is the expression for the slope of radial flow in an infinite, radial, porous media.

0 0.2 -o.4 - a. To =0 00633 Kto/pU~cr2 = 0 nT3 a 0.6 -w 0.8 -U)3 I.0 - U)1.2 - U) -J 1.4 -z 0 \ o0 1.6 -Z — 1.8 -2.0 -I0,000 I000 I00 I 0 to + At AAt iguro V-.Presur Buld-p fr Obervtio Wel i RaialSystm wtheakge, 0.2r Figure ~~~V-3. Pressure Build-Up for Observation Well in Radial System with Leakage, 02

-90 -The minimum error in the measurement of insitu permeability is obtained by using the slope at the point of inflection as shown by Figure (V-1). The value of dimensionless pressure at this point is tD 2( (V-13) Substitution of Equation (V-13) in Equation (V-ll) gives the slope at the point of inflection _2. 303 r 2 3 03 (V-14) Thus the ratio of the observed insitu permeability to the true insitu permeability at the point of inflection is given by K(obs) e (I -1 e'The values of Mob) versus B obtained at the point of inflection are shown in Figure (V-4). The pressure drawdown at the inflection point defined by Equation (V-13) is PD = (V-16) If no leakage is occurring, the pressure at the inflection point is a e i,- r I_ / (V-17)

3.0 2.8 2.6 2.4 2.2 K 2.0 - 1.8 1.6 1.4 1.2 0.01 0.02 0.03 0.04 0.06 0.08 0.1 0.2 0.3 0.4 0.6 0.8 r/ B Figuro V-4. Ratio of Observed Permeability to Actual Permeability vrsuq ~ for Radial System with Leakage.

-92 -Setting Equation (V-16) equal to Equation (V-17) gives K2 )(v-18) 2 E i (4tD (OS) Solving for t (obs) gives 4-tb(obs)= E i {Ko (i) (V-19) and hence K(ob5) t( - Crons) ___ = ~ 4[-Ko(0 (V-20) Solving Equation (V-20) for C(obs) gives Thus at the point of inflection the ratio of the observed compressibility to the actual compressibility is given by c(obs) _ 2e F-iE;f-K P(B (V-22) Values of the ratio of observed compressibility to the actual compressibility B using the slope obtained at the point of inflection in a plot of dimensionless pressure drop versus loglo time is shown in Figure (Va-). D. Example Problems Two examples are given to illustrate the application of methods developed in this section.

3.6 3.. Zi.. 3. ~2.8 2. C 013S C -2. 2.2 2. A 1.6 1.4 1.2 0.01 0.02 0.03 0.04 0.06 0.08 0.1 02 03 0406 08 r/ B.... Figure V-5. Ratio of Observed Compressibility to Actuml Permeability versus Z Measured at Inflection Pointffor Radial system with Leakage. B

-94. Problem V-1 A well completed. in an aquifer is pumped at a rate of 40 gal/min. The aquifer is 20 feet thick and has a permeability of 200 millidarcys. The caprock is 5 feet thick and has a permeability of 0.001 millidarcys. The bottom rock is 10 feet thick and has a permeability of 0.5 millidarcys. If the pressure is measured at an observation well 141 feet from the pumping well, determine the value of insitu permeability and insitu compressibility if the maximum slope of the drawdown curve is used and the pressure drop at the point of inflection is used to calculate the insitu compressibility. Assume the true compressibility of the formation is 8 x 10-6 vol/(vol)(psi). Solution Calculate the value of B using Equation (V-4a) ~/=/'' ~ I"7(200) 282 fee(2) \ K hi j (+)_ (o0O. t 0 282 feet Thus r =14 - 282 - Use Figure (V-4) or Equation (V-15) to find K(obs) for -L= 0.5 K(o(s) K(obs) = 1.65 (200) K(obs) = 330 millidarcys, the observed permeability

-95 -Use Figure (V-5) to find ( for - =r 0, c(os)- - 92 C(obs) = 1,92 (8 x 10-6) C(obs)= 1.536 x 10-5 vol/(vol)(psi) This example shows that if the bottom rock is permeable, the error in the insitu permeability and insitu compressibility can be large. Problem V-2 An 92 foot thick aquifer is being considered for storage of natural gas. Core data shows that a 20 foot caprock with a permeability of 0.02 millidarcys lies above the aquifer. The core data also showed a 10 foot, tight sandstone below the aquifer has a permeability of 1 millidarcy. A well was pumped at 10 gal/min. Pressure data obtained from an observation well 100 feet from the pumping well was used to measure the insitu permeability, If the measured value of the insitu permeability is 165 millidarcys, what is the true permeability of the aquifer? Solution The true value of the permeability must be found by trial and error. Trial No, 1 Assume 05' Use Figure (V-4) or Equation (V-15) to find K(6S) K 1.65

-96 -Thus 5 _= r 1 - 200 0.5 -,0.5 _2= 40) 000 and K-'6s) - 5 = oo00 m;ll;daIrcys Check value of K using Equation (V-4a) KJ K h'K. "oooo Ke 40, ~ ( K0" _ 40 000/ 0. I2 ~92,~ ~ 10o,/ 44 P;- lildwcys Since the two values of K do not agree, it is necessary to assume a new value of - and repeat the calculations. Trial No. 2 Assume B =0.3 Use Figure (V-4) or Equation (V-15) to find lob K (obs) _35 KThus Thus 8 - o.3 = 333 0.332 03 ooo 52=~,,, oo0

and K = K<ob4) _ 1635 = 122 2IM;M11;igrys /', 3S' - 1,35 This value is checked using Equation (V-4a) kh = 111,000 (., + )I, o0 /ooo 0 2 _ 122 m; ll dcrcy5 Since the two values of K agree, K = 122 millidarcys is the correct value for the permeability.

VI. INTERPRETATION OF FIELD WELL TEST DATA The pressure behavior of one gas field and two aquifers are analyzed in this section. Replacement of p by p2 in the equations for flow of slightly compressible fluids to describe the flow of gas when the pressure drawdown is less than ten percent is justified. Analysis of well drawdown and build-up tests in the two aquifers will illustrate many problems involved in the interpretation of reservoir pressure data. A. Field.A Pressure data from a drawdown test on a gas well completed in the deep Frio trend of Southwest Texas was presented by Accord (1). These data are analyzed using a procedure similar to the procedure for analyzing the flov of slightly compressible liquids. Since the pressure drop was less than 10 percent of the initial pressure, the equations for a slightly compressible fluid can be used if p is replaced by p2 (17) (50) (92). Justification of this procedure is given below: 1. Approximate Equations Describing Gas Flow in Porous Media for Constant Rate Tests Darcys law for the radial flow of gas in porous media is given by (VI-1) -98 -

-99 -where: A = cross sectional area normal to flow, square centimeters q = gas flow rate, flow conditions, cc/sec K = permeability, darcys p = pressure, atmospheres r = radial distance, centimeters >/ = viscosity, centipoise The gas law can be used to convert the gas flow rate to standard conditions. Thus Gr ( (VI-2) where: Pb = pressure base for gas measurement, atmospheres QG - flow rate at standard condition, cc/sec T = temperature at flow conditions, OK Tb = temperature base for gas measurement, K z gas compressibility factor, flow condition The area normal to flow at a given radius is A = 2nh (VI-3) where: h - reservoir thickness, centimeters Substituting Equations (VI-2) and (VI-3) in Equation (VI-l) and rearranging gives

-100 -QG' = ' K T."d' (VI-4) which simplifies to.e, =.e,~,Z_"TQ6: (VI-5) The b'oundary condition at the well bore is given by C:1T I (VI-6) where RD = -W (VI-7) The diffusivity equation for the flow of gas in a porous media, derived by Katz et al (50), is RD < (VI-8) The initial condition is CD( RD, o) = ~ o, Ro a o (VI-9) or -'P' ( )D, A O):~)= Ro _ O (VI-10) The remaining boundary condition is,. So — ^. 'SO >0 (VI-11) *0- ~

-101 -or ~,o _ s =,~2 tA 0 (VI-12) Define dimensionless pressure and dimensionless time for the flow of gas as follows (P) = (VI-13) _t j 2 K1o.~ ( t )945 P E2 i (VI-14) Substituting Equations (VI-13) and (VI-14) in Equations (VI-8), (VI-6), (VI-10), and (VI-12) yield respectively aPP RS dR D 9(,R, R (VI-16) RD -~ c~ PC (RD, C) = O (VI-17) ( PD)fXS = O (ti)e ~ O (VI-18)

-102 -Equations (VI-15), (VI-16), (VI-17), and (VI-18) are identical to the equations of flow of a slightly compressible fluid except that p is replaced by p2. If pt is plotted versus logl0t and engineering units are used, the slope of the drawdown curve is given by (50) = _ 0, 4E24. M z T Oqd (VI-19) hkt where: h = reservoir thickness, feet K = permeability, millidarcys m - slope, (psi)2 /cycle Po = initial pressure, psia QG = gas flow rate, SCF/day, for Tb 600F, P = 14.7 psia T = reservoir temperature, ~R z average gas compressibility,c= average gas viscosity, centipoise The error in the application of this method is due to the change in p given in Equation (VI-14). 2. Analysis of Pressure Drawdown Data The pressure drawdown data and values of pg are given in Table (VI-1). Note that the pressure drop in this test is less than 10 percent of the initial pressure. Other reservoir data needed to analyze the pump test data are:

-103 -Average viscosity, /A 0.0362 centipoise Average gas compressibility factor, z 1.3 Reservoir temperature, T = 729 0R Reservoir thickness, h =8 feet The gas flow rate for the pressure drawdown test is 600 SCF/day. The slope over the initial drawdown of a plot of p2 versus loglot in Figure (VI-1) is used to calculate the insitu permeability. The sharp changes in the slope over the latter history reflects the effect of reservoir faults as discussed in Section II. Thus using m =-1,300,000 psi2/cycle the insitu permeability to gas is calculated by rearranging Equation (VI-19) S = -1424 zz2 T o (VI-20) K= — 1424 (oo 362)(,1) (72) (600) =.d 8 (- i 300, oo) =2 It is interesting to note that when the question for a slightly compressible fluid was applied to this data directly by Accord (1), a value of 3.0 millidarcys was obtained for the insitu permeability to gas.

TABLE VIll PRESSURE DATA FOR FLXOW T EST ON GAS WELL IN FIELD A (ACCORD (1)) Flow Time Pressure, psia (Pressure)e (psia) Hours (well-bottom) (well-bottom) 0.000 9340 87.24 x 106 0.117 9122 83.21 o. 250 9085 82.54. o00 9059 82.07 0.750 9049 81.88 1 9038 81.69 2 9018 81.32 3 8997 80*95 4 8976 80.57 5 8960 80.28 6 8950 80.10 7 8934L 79.82 8 8924 79.64 9 8914 79.46 10 8908 79.35 11 8903 79.26 12 8893 79.09 14 8867 78.62 16 8846 78.25 18 8830 77.97 20 8815 77.70 22 8794 77.33 24 8778 77.05 28 8745 76.48 32 8690 75.52 36, 8648 74.79 40 8625 74.39 44 8590 73.79 48 8560 73.27 56 8500 72.25 64 8460 71.57 72 8431 71.o8

-105 -B. Field B Drawdown and build-up test data from an aquifer located in St. Peter Sandstone in the Illinois Basin are presented below. Values for the physical properties of the sandstone obtained from core data are porosity 14o.5 percent, water viscosity 1 centipoise, sand thickness 164 feet, and permeability 168 millidarcys. Well (B-l) located in the aquifer was pumped at a rate of 1028 barrels per day. Pressures were observed at the Pumping Well (B-1), three Observation Wells (B-2) (B-3) B-6) completed in the aquifer, and two Observation Wells (B-4) (B-5) completed in the first permeable sand above the aquifer. Locations of the wells are shown in Figure (VI-2) and pressure measurements are listed in Table (VI-2) for the drawdown test and in Table (VI-3) for the build-up test. The distance between the Pumped Well (B-1) and the observation wells are: Well (B-2) - 1021 feet Well (B-3) - 1760 feet Well (B-4) - 1232 feet Well (B-5) - 1320 feet Well (B-6) - 6600 feet 1. Drawdown Test A plot of the pressure change versus loglotime for the drawdown test is shown in Figure (VI-3). The

84 i 1 83 82 81 80 cn a. 79 z -78 -77 76 75 74 73 72 711I L 0.1 1 10 100 TIME, HOURS Figure VI-1. Well Pressure versus Time for Flow Test on Gas Well in Field A.

-107 -(B-6) (B-3) 1760' (B-4) (B-2) i 1232' 1021' (B-5) (B -I) 1320' 0 DENOTES WELL Figure VI-2. Location of Wells in Field B.

-108 -TABLE VI-2 FIELD B - PRESSURE DRAWDOJWN DATA Pumping Pressure Drop, Feet of Water Time, Hours Well B-1 Well B-2 Well B-3 Well B-4 Well B-5 Well B-6 O O 0 0 0 0 0 0.017 60 0.033 80 0.050 95 0.067 120 0.o83 135 0.100 14+6 0.117 151 0.113 159 0. 150 164 o.162 169 0o.50 183 -0.o333 198 0.05 0. 20 201 0.09 0.01 0.50 202 0.20 0.03 o.58 0. l3 o.og09 0.67 0.60 0.13 O. 75 204 0.77. 19. 02 0.82 1.05 0.25 0.02 0.91 1.20 0.34 0.02 1.00 205 1.50 0.o 2 0.04 1.08 1.73 0.52 1.16 1.93 0.61 1.25 20+4?.05 0.66 0.0+ 1.33 230 0.79 1.42 245 o. 88 1.50 206 2.61 0.98. 0.07 0.01 1.58 2.79 1.67 2.95 1.75 208 3.05 1.26 0.12 0.01 1.82 3.20 1.91 3.35 2.00 210 3.50 1.4 8 o. 1 o0.01 2.25 3.80 2.50 210 +.16 1.91 0.19 0.01 2.75 +.4+5 3.0 210 4.70 2.31 O. +6 0.01 3.5 5.05 2.65 0.55 0.01 210 5.52 2.93 0.72 0.02 5 210 6.15 3.03 1.15 0o.10.6 212 6.55 3.78 1.37 0. 12 7 208 7.00 4.1i 1. 92 0.14 8 205 7.30 4.+1 2.22 0.241 9 208 7.:5 4.67 2.62 0.31 10 208 7.85 1. 87 2.84 0.36 11 7.85 4.87 0.35 12 209 8.30 5.31 3.44 0.62 0.10 14 210 8.70 5.64 3.96. 75 c. 10 16 210 9.00 5.9+4 4.29 0.89 C.12 18 212 9.30 6.22 1.55 1.05 c.18 20 211 9.60 6.48 4. 81 1.35 0.25 23 10.13 6.92 1.35 0.31 241 210 10.05 6.92 5.28 1.64 0.35 30 205 10.50 7.41 5.87 2.39 c..45 35 205 10.89 7.73 6.26 2.85 c.67 209 11.05 7.93 6.54 3.40 c. 66 +5 203 11.40 8.28 6.78 3.80 O. 88 47 11.40 8.28 6.78 3.80 0.76 50 210 12.00 8.53 7.13.33 1.00 55 202 12.10 8.68 7.27 4.80 1.05 59 12.25 8.93 7.47 5.1.4 0.96 60 200 12.25 8.93 7.47 5.11 1.24 65 9.18 1.25 70 200 12.30 9.18 7.73 5.99 1.26 71 12.30 7.73 5.99 1.16 72 205 12.18 9.32 7.87 6.14 1.30

-109 -TABLE VI-3 FIELD B- PRESSURE BUILD-UP DATA Shut-In Pressure Drop, Feet of Water Time, Hours Well B-1 Well B-2 Well B-3 Well B-4 Well B-5 Well B-6 0.050 1441 30 12.18 9.3'3 0.100 721 31 12.18 9.33 0.162 445 341 12.18 9.34 6.17 0.250 291 34 12.18 " 34 7.87 6.17 0.333 219 31 12.18 9.34 6.20 0.42 172 31 12.18 9.33 6.19 0.50 145 29 12.15 9.31 6.19 0.58 125 28 12.05 9.24 /7.87 6.19 0.67 119 27 11.83 9.20 7.87 6.19 0.75 97 26 11.60 9.13 7.87 6.21 0.82 89 25 11.40 9.07 7.87 6.23 0.91 80 24 11.10 8.9.9 7.87 6.23 1.00 73 22 10.87 8.91 7.87 6.24 1.25 59 21 10.55 8.57 7.87 6.25 1.39 1.5 49 20 10.01 8.43 7.85 6.26 1.42 1.75 42 9.57 8.21 7.84 6.28 1.42 2.0 37 9.27 7.98 7.77 6.31 1.42 2.5 29.8 8.69 7.58 7.7C 6.34 1.42 3.0 25.0 8.15 7.23 7.60 6.36 3.5 21.6 8.08 6.95 7.47 6.37 4 19.0 7.48 6.63 7.37 6.38 15.4 6.82 6.28 7.01 6.46 1.42 13.0 6.43 5.88 6.72 6.48 1.39 7 11.3 6.05 5.53 6.35 6.39 1.39 8 10.0 5.60 5.31 6.o05 6.47 1.39 9 9.0 5.42 5.03 5.81 6.49 1.39 10 8.2 5.25 4.98 5.62 6.49 1.39 11 7.6 4.49 1.26 12 7.0 4.94 4.67 5.14 6.52 1.47 14 6.2 4.65 4.45 4.78 6.48 1.55 16 5.5 4.37 4.17 4.47 6.45 1.52 18 5.0 9.55 4.10 3.98 '4.19 6.40 1.52 20 4.6 3.95 3.81 3.99 6.34 1.51 23 4.1 9.03 3.814 1.36 2+ 4.0 3.79 3.65 3.73 6.22 1.59 30 3.4 3.40 3.30 3.39 6.05 1.55 35 3.0 3.13 3.06 3.,11 5.89 40 2.8 8.99 2.95 2.85 2.89 5.64 1.50 45 2.6 8.80 2.82 2.77 2.83 5.47 1.49 47 2.54 2.84 2.58 2.77 1.41 50 2.44 8.66 2.84 2.82 2.77 5.14 1.60 55 2.31 8.50 2.37 9.56 2.64 5.14 1.52 59 2.22 2.13 60 2.20 8.42 2.39 1.43 2.42 4.96 1.46 65 2.11 8.35 2.37 2.141 2.42 4.76 1.44 70 2.03 8.25 2.35 2.28 2.31 4. L"2 1.43 71 2.02 2.13 83 1.87 2.03 95 1.76 3.78

0 -1 -- en 3 ^ = Well (B- 3) ~ \ am | ~ z 0 0 w 0 = Well (8 - 2) cl) H (1 - = Well (B- 3) w - P = fll -T n- -- = Well (B-5) (n a= Well (B-6) 1~~~~~0 -10 1130.1 1 10 100 TIME, HOURS Figure VI-3. Pressure Curves for Drawdown Test on Field B.

-111 -pressure drops in Wells (13-4) and (B-5) show that there is communication between the St. Peter sandstone and the next sandstone above the St. Peter. The change in slope of the drawdown curve in Wells (B-2) and (B-4+) indicates that there is pressure communication in the vicinity of these wells. See Figure (VI-2). Data from Wells (B-2) and (B-3) were used to determine the insitu permeability and insitu compressibility. The slope of the drawdown curve for Wells (B-2) and (B-3) are -6.5 and -5.0 feet of water/cycle, respectively or -2.82 and -2.17 psi/cycle respectively. The insitu permeability calculated from Equation (II-6), is K =- 161.6 2 i 6.. tK 16265tu (II-6) Thus for Well (B-2) -~, 6 (104) ()= 361 rI1;,Carcy-s - 2.82 (164) and for Well (B-3) K= -162. 2 ( o2.s)() = 470,; II;dg~ic.ys - 2, /7 ('164) The insitu com iressibil is obtained by solving Equation (II-1) for c

-112 -A drawdown of -6.5 feet of water (-2.82 psi) at 6 hours (0.25 days) for Well (B-2) was used to calculate the insitu compressibility for Wsell (C-2). C = C. 0 6 x /0 vol /(VI)(P,;) Using the drawdown of -9.3 feet of water (-4.03 psi) at 72 hours (3 days), for Well (B-3) the insitu compressibility is given by C= -4.(o.oo633)(470) (3) E 4.03 (o0 145)( 1) ( 1760)z - (70 6)(102 (470) C/64) o= 6*35 x /0' vol/(vol)(ps;) This value is checked using the drawdowrn -4.14 feet of water (-1.79 psi) at 7 hours (0.292 days) for Well (B-3) S = - 4 (o oo633)+47o)(o 2q2) E-) - I. 7E' t L (-70)1 16 4 J c = 7.1 7 /(l)(p4)

These results for the insitu compressibility are not only inconsistent, they are less than the compressibility of the water itself, These data strongly indicate that in addition to leakage, the formation is heterogeneous and can not be represented by a single pseudo homogeneous porous media. The pressure drawdown for a layered reservoir can be calculated from +p = + 70. 6> i + (.oo0633Kt (VI-22) where: hi = height of the ith layer, feet Ki - permeability of the ith layer, mill idarcys n = number of distinct homogeneous layers qi = flow rate from the ith layer, bbl/day If the heterogeneities in a reservoir are known, then the pressure behavior can be determined. Unfortunately the converse is not true, since there may be an infinite number of combinations of stratifications, leakage, faults, etc. which will give essentially identical pressure behavior~ Furthermore the flow may not be radial and. forcing the data to fit this model may lead to large errors in the determination of the insitu properties. Applications of hemispherical model and thick sand model (bottom water drive) to reservoir well test data are given by Katz et al (51).

Barometric pressures were not recorded during this test. Failure to correct the pressure data for barometric changes can cause serious error in interpreting pump test data when the pressure drawdown is only a few feet of water, 2. Build-up Test The pressure change versus log10 ( +t t ) is shown in Figure (VI-4) for the build-up test. The slope of the curves for Wells (B-2) and (B-3) is m = -6 ft. of water/cycle m = -2.6 psi/cycle The insitu permeability calculated from Equation (II-6) is K- - h K — (-.6,) ( 16 4) K = 393 ',;llic;dacys C. Field C Drawdown and build-up test data from a second aquifer in the Illinois Basin are analyzed. Location of the pumping Well (C-l) and three observation Wells (C-2) (C-3) (C-4) in the Mt. Simon formation are shown in Figure (VI-5l).

-1 /11 o=Well(B-2). cr -2 = Well (B-3) -- o13 =Well (B-4') = Well (B-5) Well (B-6) LL 0 -4 ur aII I -1 -81 / -12 -0 0 0 0000 -13 1000 100 10 1 TIME, HOURS Figure VI-4. Pressure Curves for Build-Up Test on Field B.

-116 -Well (C-1 was pumped at a rate of 2740 barrels per day (80 gallons per minute). Pressure observations corrected for changes in barometric pressure, at the Pumping Well (C-1) and the three Observation Wells (C-2), (C-3), and (C-4) are recorded in Table (VI-4). The pressure behavior, corrected for changes in barometric pressure, is given in Table (VI-5) for the build-up test. The horizontal distances between the Pumping Well (C-1) and the observation wells are Well (C-2) - 4611 feet Well (C-3) - 2869 feet Well (C-41) - 6934 feet The distances between the pumping well and the observation wells when the vertical displacement of the formation is considered is Well (C-2) - 4612 feet Well (C-3) - 2880 feet Well (C-4) - 6934 feet The aquifer is several thousand feet thick and contains several non-continuous shale streaks, Core data show the permeability of the sandstone varies from a few millidarcys to several darcys. Thus the meaning of insitu permeability and insitu compressibility is questionable in such a heterogeneous formation. Pump test actually measure the transmissibility, T, and storage coefficient, S, where these terms are defined by

-117 -(C-3) 2869' ~~69~~~~~2~~ \4611' (C-4) (C-2) e DENOTES WELL Figure VI-5. Location of Wells in Field C.

TABLE VI-1+ FIELD C — PRESSURE DRAWDOWN DATA Pumping Pressure Drop, Feet of Water Time, Hours (Corrected for Barometric Pressures) Well C-1 Well C-2 Well C-3 Well 'C-+ 0.0 0 0.1 398 0.2 561 0.3 634+ o0:+ 706. 5 746 0. 75 7941 1 811 2 81+7 847 847 4.25 847 0.03 0.01 5 81+7 0.03 6 81+7 0.02 0.05 7 81+7 0.04 0.10 8 81+7 0.09 0.16 9 81+7 0.11+ 0.22 10 81+7. 18 0.29 11 81+7 0.27 0.35 12 81+7 0.32 0.39 13 81+7 0.39 0.43 [14' 81+7 0.441+1+ 0. 41+ 15 81+7 0.48 0.45 17 847 0. 5 0.45 18 847 0o. 59 0.6 20 842 0.67 0.49 23 0. 87 060 25 837 0.96 0.65 27 1.08 0.70 30 816 1.16 '0.72 32 1.25 0.78 0.11 35 852 1.49 0.95 0.16 4+0 883 1.68 1.03 0.19 1+5 841+ 1.79 1.01 0.28 50 837 1.98 1.13 0.37 52 2.00 1:16 0o41 55 837 2.11 1.19 0.44 57 832 2.25 1.28 0.53 65 832 2.41 1.16 0.72 70 848 2.62 1.1+47 0.71+ 75 837 2.66 1.5.5 0.82 80 840 2.75 1.56 85 90 842 2.92 1.68 91 114 837 3.24 1.88 1.17 126 3.39 1.99 1.28 138 835 2.,0 1.41 150 837 3.4+5 2.o 1. 43 162 819 3.53 2.12 1.46 174 826 3.65 2.20 1.53 186 826 3.76?.28 1.59 198 821 3.91 _.42 1.59 210 821 3.92 2.41 1.72 222 4+.o01 2.55 1.84 231+ 3.91 2.43 1.83 2416.02 2.56 1.92 258 778 4.0o5 2.55 1.83 261+ 773 1+.16 2.64 1.93 270 1+.17 2.63. 1.93 282 1.21 2.61+ 1.92 288 4.30 2.73 2.02 291+ 4.37 2.71 1.99 306 4.33 2.75 2.02 312 792 1.42 2.92 2.17 318 4.1+2 2.83 2.12 330 4.50 2.93 2.23 336 4.53 2.99 2.23 342 4.50 2.93 2.19 351 773 1+.62 3.03 2.28 360 778 1.65 3.06 2.31 366 776 4.71 3.03 2. 27 378 778 4.66 3.09 2.33 381+ 789 1+.65 3.07 2.31 390 789 4.72 2.98 2.32 1+02 778 4.69 3.12 2.33 1+08 778 4.66 3.09 2.30 1+11+ 778 4.66 3.09 2.30 1+26 778 4.75 3.19 2.37 1+50 789 3.25 2.:1 51o 789 5.03 3.30 '.61 528 791+ 39 2.61+ 750 771 5.30 73 2.79 870 779 5.65 1o 3.18 990 834 5.66 ~.03 3.02 1152 798 5.75 1+.25 3.27

-119 -TABLE VI-5 FIELD C - PRESSURE BUILD-UP DATA Pressure Drop Shut-in Feet of Water (corrected) Time, Hours Well C-1 Well C-2 Well C-3 Well C-4 6 193 5.67 4,15 3.25 18 65 5'.o8 3.73 3,17 24 49 4.76 3.54 30 39.4 4.56 3 52 3.11 42 28*5 4.11 3.34 2,95

-120 -T = Kh//, millidarcy feet/centipoise S = +ch, feet/psia Except for the inclusion of viscosity, these terms are not new and have been used by hydrologists for decades (28) (42) (84). The pressure for a drawdown test in terms of transmissibility, T, and storage coefficient, S, is calculated by e + 70 Ei - 40oo63T} (VI-23) The storage coefficient, S, can be obtained by solving Equation (VI-23) for S. Thus =4(.o00633)Tt Ei{-2 (<-.i)T (VI-24) Comparison of Equations (II-1) and (VI-23) shows that the transmissibility, T, can be calculated from the slope of the drawdown or build-up test by = o _ _ 162.6 S (VI-25) The expression for dimensionless time in engineering units can be written as t -. o G33 T (VI-26) D I~s rp. -

-121 -1. Drawdown Test Pressure change versus log time is plotted in Figure (VI-6) for the drawdown test. Data from each of the three observation wells are used to determine the insitu transmissibility, T, and storage coefficient, SO Equation (VI-25) is used to calculate the insitu transmissibility, and Equation (VI-24) is used to calculate the storage coefficient. The slopes for the drawdown test shown in Figure (VI-6) are Well (C-2), m = -3.05 ft water/cycle = -1.34 psi/cycle Well (C-3), m = -2.5 ft water/cycle = -1.08 psi/cycle Well (C-L), m = -2.2 ft water/cycle = -0.95 psi/cycle The corresponding values for the insitu transmissibility given by Equation (VI-25) T = _ '62 6 + (VI-25) are calculated for each of the three observation wells. Well (C-2), T = 162.6(2740) 332,000 milli-1,3 darcy feet/centipoise Well (C-3), T = -162.6(270 412,000 milli-1.08 darcy feet/centipoise Well (C-4), T = -162.6(2740) = 468,000 milli-0.99 darcy feet/centipoise The average insitu transmissibility for the three wells is T = 404,000 millidarcy feet/centipoise

0IO 0:: w LI. 0 i —2 I..w z 0 -3 w o=Well (C-2) o~~~~~~~~~ "= Well (C-:3) w o = Well (C-4) a: -4 a.. 0.~~~~~~~~~~~~~~~~~~~~~~~~ -6 4 6 8 10 20 40 60 80 100 200 400 600800o1000 TIME, HOUR — Figure VI-6. Pressure Curves for Drawdown Test on Field C.

-123 -Since the value for the aquifer thickness, h, is questionable, solving the expression for the transmissibility Kh for the insitu permeability is not recommended for this field. The values for the storage coefficient, S, are calculated by Equation (VI-24).(0o,0633)Tt E-,{3-J)T tI for each of the three observation wells. Values for the drawdown at 720 hours (30 days) are read from Figure (VI-6). Well (C-2) S = +(O.00633)(332,000) (30s)E, (-S,) (,433) (332 4 Ood0 (4~612)2 E1 1 70.6 (Z740 s = o o C/2 -e-t tent; po;Sc / ps; a well (c-3) S= 4 (0, oo63 3)(4-122 o O) (30) E; -3. 7 (O+33)(412, ooo)7 o880) 70.,.( o740) S- 0.00072 feet ce-nfpoise / p.;a Well (0-4) 5 = 4 (0. o63 3) (46e, oo) (30) E -2 q.(o,433)(468 Coo) (65934)2 ' L 70.6 (2740) 5- = O00021 feet cen't'poi /pSIaw

The average value for the storage coefficient, S, for Field C is Save O=.OO35 (feet)(centipoise)/psia The expression for the storage coefficient S =/v 4 ch should not be solved for the insitu compressibility since the effective value for the aquifer thickness, h, is not known for Field C. 2. Build-up Test The build-up curves for Wells (C-2), (C-3), and (C-4) are shown in Figure (VI-7). The curves are extrapolated to. +z4t to =At 1 in order to obtain the slope. The slope for these curves is -2.8 ft. of water/cycle or -1.21 psi/cycle. The insitu transmissibility, given by Equation (VI-25) is T = -162.6(2740) -1.21 T = 368,000 millidarcy feet/centipoise This value is within the range of values obtained for the insitu transmissibility from the drawdown test.

0 w-1w/ 1-3 o = Well(C-2) 0 o =Well (C-4) 0 z-5 to+(t to+At Figure VI-7. Pressure Curves for Build-Up Test on Field C.

VII. SUMMARY, CONCLUS IONS, AND RESULTS Several problems involving the unsteady state behavior of fluids in underground strata were investigated in this dissertation. These problems include: 1. Evaluation of the error in neglecting the non-linear term in the partial differential equation describing the flow of a slightly compressible liquid in a porous media, 2. Determination of mathematical expressions describing the unsteady state pressure behavior for radial flow in a reservoir during drawdown and build-up pump tests when gas fields or lines of constant pressure are located in the vicinity of the well o 3. Determination of mathematical expressions in engineering units describing the unsteady state pressure behavior for radial flow in a reservoir during drawdown tests when leakage is occurring through the confining cap and bottom rock and extended these results to build-up tests, 4. Illustration of many difficulties encountered when actual field data are analyzed. -126 -

-0127 -The following results and conclusions were obtained in this study. 1o The effect of neglecting the non-linear term in evaluating the unsteady state pressure behavior is negligible for both radial and linear systems when tD < 0001 M '2 where: tD =,0o00633 M' t, M =t ~I for radial flow and -887.64C~;) for linear flow AKA - cross sectional area normal to flow, (feet) c = compressibility, vol/(vol)(psi) h = thickness of porous media, feet K permeability, millidarcys q flow rate, bbl/day r - radius of well, feet t = time, days xc = reservoir characteristic length, feet = porosity, fraction / = viscosity, centipoise 2. The pressure behavior, during a drawdown test when a gas field is located nearby is given by

-128-.K A,o i (o.o00633 Kt-E i (- b.0o633 Kt) where: Ei = exponential integral R, distance from pumping well to point of distance measurement, feet )l distance from point of pressure measurement to image of pumping well, feet The observed insitu permeability and insitu compressibility are larger than their true values and the magnitude of these errors are given in Figures (IV-9) and (IV-10). 3. The pressure behavior during a build-up test when a gas field is located in the vicinity of the test is ~0 ii - El 006 33) K(ttt) E (o 0633)X where: to = length of drawdown test, days At = time since cessation of pumping, days

-129 -4. The pressure performance versus dimensionless time when leakage is occurring through the cap and bottom rock was evaluated for drawdown and build-up tests. These results are presented in Figures (V-l) and (V-3). The complex expressions for the pressure behavior are given by Equation (V-6) for the drawdown and Equation (V-7) for the build-up. The error in the measurement of insitu permeability and insitu compressibility is shown in Figures (V-4) and (V-5). 5. Two fields characteristics, transmissibility, T, and storage coefficient, S, should be evaluated for reservoirs in which it is difficult or impossible to obtain the reservoir thickness. These characteristics defined by and S= =ch can be used to predict the reservoir performance. 6. A graphical method for locating an external line of constant pressure or a gas-water interface is described.

VIII. RECOMMENDATIONS FOR FUTURE WORK Methods have been presented in this dissertation for predicting the pressure behavior of constant rate pump tests when (1) a gas field is near or a line of constant pressure is present and (2) leakage is occurring through the cap and bottom rock. The usefulness of this work needs to be substantiated by field data and extended to the ase of constant pressure well tests. Specific problems and tests which would enhance our understanding of unsteady state reservoir behavior are given below. 1. Drawdown and build-up well tests should be conducted near the edge of a gas field to evaluate the usefulness of the material presented in Section IV on pressure behavior near a gas field. These tests could be performed near the edge of a gas storage field during the few months preceding or after the gas withdrawal season. It is further recommended that both constant rate and constant pressure pump tests be performed. 2. The pressure behavior for constant pressure drawdown and build-up tests should be determined for the case when a gas field is located in the vicinity of the test site. -130 -

-131 -It probably will be necessary to solve the partial differential equation, boundary, and initial conditions by numerical methods, 3. The pressure behavior for constant pressure drawdown and build-up tests need to be evaluated when leakage is occurring through the cap and bottom rock. Again numerical methods will probably be necessary to obtain the desired mathematical solutions0

BIBLIOGRAPHY 1. Accord, H. K., Jr., A Study of Pressure Drawdown Analysis in Oil and Gas Reservoirs, M.S. Thesis, University of Houston (1961). 2. Ammann, C. B., "Case Histories of Analyses of Characteristics of Reservoir Rock from DrillStem Tests", Jour. Pet. Tech. 12, 27, (May, 1960). 3. Aronofsky, J. S. and Jenkins, R., "A Simplified Analysis of Unsteady Radial Gas Flow", Trans. AIME 201, 149 (1954). 4. Arps, J. J., "How Well Completion Damage Can Be Determined Graphically, World Oil 140, No. 5 225 (1955). 5. Briggs, J. E., Countercurrent Gravity regation in Porous Media, Ph.D. Thesis, University of Michigan, (1963). 6. Brons, F. and Marting, V. E., 'The Effect of Restricted Fluid Entry on Well Productivity", Trans. AIME, 222, 1-172, (1961). 7. Brons, F. and Miller, W. C., "A Simple Method for Correcting Spot Pressure Readings", Trans. AIME 222, 1-803, (1961). 8. Bruce, G. H. Peaceman, D. W., Rachford, H. H., Jro. and Rice, Jo. D., "Calculations of Unsteady-State Gas Flow Through Porous Media", Trans. AIME, 1_8A 79, (1953). 9. Bruce, W. A., "Pressure Prediction for Oil Reservoirs", Trans. AIME, 1!, 73, (1943). 10. Carter, R, D., "Solutions of Unsteady-State Radial Gas Flow", Jour. Pet. Tech,. 14, 549, (May, 1962). 11. Carter, R. D., Miller, S. C., Riley, H. G., "Determination of Stabilized Gas Well Performance from Short Flow Tests", presented at the 37th Annual Fall Meeting, The Society of Petroleum Engineers of AIME, Los Angeles, California, (October 7-19, 1962),o -132 -

-133 -12. Chatas, A. T., "A Practical Treatment of Non-Steady State Flow Problems in Reservoir Systems"l, Petro. Engr. (May, June, and September, 1953). 13. Coats, K. H., Prediction of Gas Storage Reservoir Behavior, Ph.D. Thesis, University of Michigan, (1959). 14. Coats, K. H., Tek, M. R., and KatzD. L., "Method for Predicting the Behavior of Mutually Interfering Gas Reservoirs Adjacent to a Common Aquifer", Trans. AIME 216, 247, (1959). 15. Coats, K. H., "Selected Topics in Numerical Analysis"' Notes prepared for course in Chemical Engineering Department, University of Michigan, 16, Collins, R. E. and Kolodzie, P. A., Jr., "Effect of Interference on Pressure Build-Up -- A Method for Calculating Porosity-Thickness Product", presented at the 34th Annual Fall Meeting of the Society of Petroleum Engineers of the American Institute of Mining, Metallurgical and Petroleum Engineers in Dallas, (October 4-7, 1959). 17. Cornell, D. and Katz, D. L., "Pressure Gradients in Natural Gas Reservoirs", Trans. AIME, 198, 61 (1953). 18. Cornell, D. "Applying van Everdingen and Hurst Solutions to Natural Gas Flow Problems", World Oil, 134, (February, 1956). 19. Craft, B. C. and Hawkins, M. F., Jr., Applied Petroleum Reservoir Engineerin, Prentice Hall, Englewood Cliffs, N. Ji, (1959). 20. Dolan, Jo P., Einarsen, C. A., and Hill, G. A,, "Special Applications of Drill-Stem Test Pressure Data"' Transm AIME, 210 318, (1957). 21. Driscoll, V. J., "Use of Well Interference and Build-Up Data for Early Quantitative Determination of Reserves, Permeability and Water Influx", presented at the 37th Annual Fall Meeting of the Society of Petroleum Engineers of AIVIE, in Los Angeles, October 7-10, 19620 22. Dykstra, H., "Calculated Pressure Build-Up for a LowPermeability Gas-Condensate Well", Trans. AIHE, 222 1-1131, (1961), 23o Fatt, I., "Pore Volume Compressibilities of Sandstone Reservoir Rocks", Trans. AIME 213, 362, (1958)

24. Gladfelter, R. E., Tracy, G. W., and Wilsey, Lo E., Selecting Wells Which Will Respond to Production Stimulation Treatment", 0il and Gas Journal, 54, No. 3, 126, (1955). 25. Hantush, M. S. and Jacob, C. E., "Non-Steady Radial Flow in an Infinite Leaky Aquifer", Trans. AGU, 3, 95, (1955). 26. Hantush, M. S. and Jacob, C. E., "Steady Three-Dimensional Flow to a Well in a Two-Layered Aquifer", Trans. AGU, 3k, 286, (1955). 27. Hantush, M S.,"Nonsteady Flow to a Well Partially Penetrating an Infinite Leaky Aquifer", Proc. Irai A.. So__c., 10, (1956). 28. Hantush, M. S., "Analysis of Data from Pumping Tests in Leaky Aquifers", Trans AGU, 37Z, 702, (1956). 29. Hantush, M. S., "NQnsteady Flow to Flowing Wells in Leaky Aquifers", Jour. Geophys. Res. 4, 1043 (1959). 30. Hantush, M. S.. and Jacob, Co E., "Flow to an Eccentric Well in a Leaky Circular Aquifer", Jour. Geophys Res., A, 3425, (1960). 31. Hantush M. S. "Modification of the Theory of Leaky Aquifers", our. Geophys. Res., 6, 3713, (1960). 32. Hazebroek, P., Rainbow, H., and Matthew, C. S., "Pressure Fall-Off in Water Injection Wells" Trans. AIME, 21, 250, (1958). 33. Henson, W. L,, Wearden, P. L., and Rice, J. D., "A Numerical Solution to the Unsteady-State PartialWater-Drive Reservoir Performance Problem", Trans. AIME, 222, II-184, (1961). 34. Hildebrand, F. B., Introduction to Numerical Analysis, McGraw.-Hill, New YY.ork (19 35. Horner, D. R. "Pressure Build-Up in Wells" Proc. Third Worla Petroleum Congress, Sec. II, 503, E. J. Brill, Leiden (19 51). 36. Hovanessian, S. A., "Pressure Studies in Bounded Reservoirs", Trans. AIME, 222, II-223, (1961).

37* Hurst, W. "Establishment of the Skin-Effect and Its Impediment to Fluid Flow Into a Well Bore", Petro Engr., (October, 1953). 38, Hurst, W", "The Simplification of the Material Balance Formula by the Laplace Transformation", Trans. AIME,9 2_1, 292, (1958). 39. Hurst, W., "Interference Between Oil Fields", Trans. AIME 219, 175, (1960)0. 40. Hutchenson, T. S. and Sikora, V. J., "A Generalized Water-Drive Analysis", Trans. AIME, 216, 169, (1959), 41. Jacob, C. E,, "Flow in a Leaky Artesian Aquifer'"Trans. Am. Geophys. Union, 27, 198-205, (1946)o 42. Jacob, C. E. and Lohman, S. W., "Nonsteady Flow to a Well of Constant Drawdown in an Extensive Aquifer", Trans. Am. Geophys. Union, 3,, 559-569, (1952). 43. Jahnke, E. and Emde, F., Tables of Functions, Dover Publications, New York (..i9W). 44, Johnson C. R., Greenkorn, R. A., and Widner G, Wo, "A Variable-Rate Procedure for Appraising Wellbore Damage in Waterflood Input Wells", Jour. Pet. Tech, 2, 85-89, (Jan., 1963). 45, Jones, Ke R.,Pearson, W. C., and Riley, H. G,, "Some Practical Aspects of Unsteady-State Gas Flow Related to Gas-Well Performance", Jour. Pet Tech, j, 41, (Jan,, 1963). 46. Jones, L. G,, "An Approximate Method for Computing Nonsteady-State Flow of Gas in Porous Media", Trans. AIME, II-264, (1961). 47, Jones, L, G,, "Reservoir Reserve Tests", Jour. Pet. Tech., l, 333, (March, 1963). 48. Jones, P., "Reservoir Limited Test", The Oil and Gas Journal,,9 No. 59, p. 184, (1956). 49, Jones, P., "Reservoir Limit Test on Gas Wells", Jour. Pet. Tech., 14, 613, (June, 1962). 50, Katz, D. L,, Cornell, D., Kobayashi, R., Poettmann, F. H., Vary, J. A., Elenbaas, J. Ro, Weinaug, Co F,, Handbook of Natural Gas Engineering, McGraw-Hill, New York (1959).

-136 -51. Katz, D. L,, Tek, M. R., Coats, K. H. Katz, M. L., Jones, S. C., and Miller, M. C., Movemenrt of Underground Water in Contact with Natural Gas, American Gas Assoc., New York, (1963). 52. Katz, M. L., Fluid Flow and Heat Transfer in Stratified System, Ph.D. Thesis: University of Michigan. (19.60). 53. Kidder R. E., "Unsteady Flow of Gas Through a SemiInfinite Porous Medium", Journal Appl. Mech. 24L 329, (Sept., 1957). 54. Layton, D. D., "Predicting Reservoir Performance by Applying an Unsteady State Solution", presented at the Gas Techmology Symposium sponsored by LouArk, East Texas, and Mississippi Sections of the American Institute of Mining, Metallurgical, and Petroleum Engirieers in Shreveport, April 17-18, 1958. 55. Lefkovits, H. C., Hazebroek, P, Allen, P. P, and Matthews, C. S.. "A Study of the Behavior of Bounded Reservoirs Com)osed of Stratified Layers", Trans. AIME, 222, II-[L3, (1961)..56, Loucks, T. L. and Guerrero, E. T., "Pressure Drop in a Composite Reservoir", Trans AIME, 222, II-170, (1961). 57. Maier, L. F, "Recent Developments in the Interpretation and Application of DST Data", Jour. Pet. Tech., 1213-1222, (November, 1962). 58. Matthews, C. S., Brons, F., and Hazebroek, P., "A Method for Determination of Average Pressure in a Bounded Rese:cvoir", Trans. AIMiEB 201, 182, (1954), 59. Matthews C. S. aiad Stegemeier, G. L., "A Study of Anomalous Pres sure Build-Up Behavior", Trans. AIME, 213, 44, (1958). 60. MicLatchie, A. S., Hemstock, R. A., and Young, J. 1., "The Effective! Compressibility of Reservoir Rock and Its Effects on Permeability", Trans. AIME, 213 386, (1958). 61. McMahon, J. J., "'Determination of Gas Well Stabilization Factors From Surface Flow Tests and Build-Up Tests", presented at the 36th Annual Fall Meeting of the Society of Petroleum Engineers of AIME in Dallas, October 8-11, 1961.

-137 -62. Miller, C. C., Dyes, 'A. B*, and Hutchinson, C, A., Jr. "The Estimation of Permeability and Reservoir Pressure from Bottom-Hole Pressure Build-Up Characteristics", Trans. AIME, 1, 91, (1950). 63, Mortada, 1M, "A Practical Method for Treating Oilfield Interference in Water-Drive Reservoirs", Trans. AIMEI 204, 217 (1955). 64. Mortada, M., "Oilfield Interference in Aquifers of Non-Uniform Properties", Trans. AIME, 2 412, (1960). 65. Mueller, T Do, "Transient Response of Nonhomogeneous Aquifers", Soc. vetr. ingr. Jour., 2, No, 1 (1962). 66, Muskat, M., The Flow of Homogeneous Fluids Throug Porous Media, McGraw-Hill, New York, (1937). 67. Nielsen, R, G., "How to Calculate Unsteady State Flow", The Oil and Gas Journal, (July 26, 1951+). 68. Nielsen, R. L., On the Flow of Two Immiscible Incompressible Fluids in Porous Media, Ph.D. Thesis, University of Michigan, (1962). 69. Nisle, R. G., "The Effect of Partial Penetration on Pressure Build-Up in Oil Wells", Trans. AIIME, 2_, 85 (1958). 70. Parson, R. L., "Discussion on Prediction of Approximate Time of Interference Between Adjacent WIells", Trans AIME. 21.6, 436, (1959). 71, Perrine, R. L,, "Analysis of Pressure Build-Up Curves", API Drill, and Prod. Pra. c 482, (1956). 72. Pirson, R. S. and Pirson, S, J., "An Extension of the Pollard Analysis Method of Well Pressure Build-Up and Drawdown Tests", presented at the 36th Annual Fall Meeting of the Society of Petroleum Engineers of the AIME in Dallas, October 8-11, 1961, 73. Pitzer, S C, Rice, Jo D,, Thomas, C, E., "A Comparison of Theoretical Pressure Build-Up Curves with Field Curves Obtained from Bottom-Hole Shut-In Tests", Trans. AIME, 216, 416, (1959). 74, Poettmann, F. H., Schilson, Ro E., "Calculation of the Stabilized Performance Coefficient of Loir Permeability Natural Gas Wells", Trans. AIME, 216, 240, (1959),

-138 -75. Rachford, H. H., Jr., Taylor, R. D., and Dyke, P. M., "Application of Numerical Methods to Predict Recovery From Thin Oil Columns", Trans. AIME, 213, 193, (1958). 76. Rowan, G. and Clegg, M. W., "An Approximate Method for Transient Radial Flow", Soc. Petr E nrJour., 2 No. 3, (1962). 77. Rzepczynski, W. M., Katz, D. L., Tek, M. R., and Coats, K. H., "The Mt. Simon Gas Storage Reservoir in the Herscher Field", paper presented at Research Confrerence on Underground Storage of Natural Gas, University of Michigan, (July, 1959). 78. Scheidegger, A. E., The Physics of Flowr Through Porous Media, MacMillan Co., New York (19-9) 79. Schrenkel, J., "Field Examples of Barrier Detection with Unsteady-State Techniques", presented at the 36th Annual Fall Meeting of the Society of Petroleum Engineers of AIME in Dallas, October 8-11, 1961. 80. Smith, R. V., "Unsteady-State Gas Flow into Gas Wells", Trans. AIE, 222, I-1151, (1961). 81. Stevens W. F. and Thodos, G., "Prediction of Approximate Time of Interference Between Adjacent Wells", Trans. AIME, 216, 433, (1959). 82* Swift, G. W. and Kiel,, G.,, "The Prediction of GasWell Performance Including the Effect of Non-Darcy Flow", Jour. Pet. Tech., 14, 791 (July, 1962). 83. Tek, M. R., Coats, K. H. and Katz, D. L. "The Effect of Turbulence on Flow of Natural Gas Through Porous Reservoirs", Jour. Pet. Tech., 14, 799 (1962), 84. Theis, C. V., "The Relation Between the Lowering of the Piezometric Surface and the Rate and Duration of Discharge of a Well Using Ground-Water Storage", Trans. Am. Geophys. Union, 16, 519 (1935). 85. Thomas, G. B., "Analysis of Pressure Build-Up Data", Trans. AIME, 198, 125, (1953). 86. van Everdingen, A, F. and Hurst, W., "The Application of the Laplace Transformation to Flow Problems in Reservoirs", Trans. AIME, 186, 305, (1949).

-139 -87. van Everdingen, A. F,, Timmerman, E. H. and McMahon, J. J., "Application of the Material Balance Equation to a Partial Water-Drive Reservoir", Trans. AIME, 59, 1, (1953). 88o van Everdingen, A F., "The Skin Effect and Its Influence on the Productive Capacity of a Well", Trans. AIME, 1 171, (1953). 89. Walton, Wo C. Leaky Artesian Aquifer Conditions in Illinois, RI 39, Illinois State Water Survey, Urbana, Illinois, (1960). 90, Warren, J. E. and Hartsock J. H, "Well Interference"' Trans. AIME, 219, 393, (1960). 91. Warren, J, E. and Price, H. S., "Flow in Heterogeneous Porous Media", Trans. AIME, 222, II-153, (1961), 92. Warren, J. E.: Private Communication (1963). 93. Witherspoon, P. A., Mueller, T. D., and Donovan, R, WO, "Evaluation of Underground Gas Storage Conditions in Aquifers Through Investigations of Groundiwater Hydrology", Jour, Pet. Tech, 14, (May, 1962).

APPEND IX A COMPUTER PROGRAM FOR NUMERICAL EXPLICIT METHOD USED IN DETERMINIING THE ERROR IN NEGLECTING THE NON-LINEAR TERM IN THE PARTIAL DIFFERENTIAL EQUATION DESCRIBING RADIAL- FLOW OF A SLIGHTLY COMPRESSIBLE FLUID IN POROUS MEDIA A description of the IBM 7090 computer program used to determine the error in neglecting the non-linear term in the partial differential equation describing the flow of a slightly compressible fluid for radial flow is presented in this appendix. This appendix includes the MAD (Michigan Algorithm Decoder) program, the program nomenclature, the flow diagram, a list of information required, and an example problem illustrating the use of the program. This program was used to calculate values of dimensionless pressures for values of dimensionless time below 0.01. The following information is required in the computer program: I. The number of length increments. (Eighty length increments were found to be satisfactory,w = 1 2, The value of the time increments. (A/t 0.00005 is an acceptable value of At for Aw 1=d-0 Stability requirements demand that (At)/(A'W) is less than 0.5.) 3. The number of time increments to be evaluated. 4. The number of M coefficients to be evaluated, (The value of M is defined by Equation (III-34)). -140 -

-141 -[. The numerical values of the M coefficient to be evaluated. The computer program uses the information to calculate values of dimensionless pressure versus dimensionless time for each value of the dimensionless coefficient M. Deck Assembly 1, IBM center control cards 2. Radial, Explicit Numerical Program (MAD or Binary Deck) 3. Input Data The input data is read in "simplified input format". For Example, IMAX = 80, DELT = 0.00005, MMAX = 2, JMAX = 49, M(1) O, 9 1* Definitions of the terms used in the simplified input are given below in the nomenclature. Flow Sheet The instruction J - a b, J> c in the flow sheet means that the set of calculations is calculated for J = a and repeated in increments of b until the condition J > c is satisfied, For J3 a, b, J >c refers to iterative calculations for a given box, Through d, J a, b, J > refers to iterative calculationris through "circle d". The Flow Sheet is presented at end of Appendix,

Nomenclature Used in IBM Program and Flow Sheet ADIM Term used to dimension P(I,J) vector A At/ (w)2 DELT At DELW Aw DT At/(F) F Scaling coefficient "a". In this dissertation, F = 1.0 I Space index IMAX Number of space increments J Time index JMAX Number of time increments K1 Term used in program, K1 =l-iLAw M Dimensionless coefficient defined by Equation (III-34+) MMAX Number of dimensionless coefficients P Dimens ionless pressure R Counter used in IBM program X2 Term defined by X2 = P+,,j - Pi-,,j X3 Term defined by X3 = (X2)2 X4 Absolute value of X2 Z Value of dimensionless time at a given time step, Z= jAt

---- -— L --- —LL — B --- —---------- L- ----------- - _ _ —. ---- 00o M.Co MILLER Q203N 002 045- 000.._ eo. t. $COMPILE MAD, EXECUTE, DUMP, PUNCH OBJECt.PRINT OBJECT R EVALUATION OF NON-LINEAR TERM IN NON-LINEAR, R SECOND ORDER. PARTI A_! D [IFQF..ERfNJ.-....I.. --- —------. MAD I rIgrum - R FOR FLOW OF SLIGHTLY;OMPRESSIBLE LIQUIDS.................... RLiEJJ_ ADJ_ OEL...... --- —------------------------------- R RSOLVED BY DIFFERENCE EQUATIONS,EXPLICIT FORM R. ---..........__DMENIQN_ P(l L -— __ --- —- ---------------------------— ___ INTEGER I,J,K,L,R,N,IMAX,MMAX,JMAX VECTOR VALUES ADIM=2,101,100 F=1. S4 READ DATAIMAX,DELT,MMMAX.JMAX DELW=1./IMAX A=DELT/DELW/DELW DT =DELT/F/F ___________ --- —--— _ PRINT R__-ESULT_DLTIMAXMMAXX AM LI _.__ P(IMAX. 1)=0. THROUGH 56,FOR R=1,1,R.G.MMAX. THROUGH S1,FOR I=0,1,iI.GIMAX 'Z=O. J=O PRINT COMMENT l$EVALUATION OF NON-LINEAR TERM IN NON-LINEAR, lSECOND' ORDER, PARTIAL DIFFERENTIAL EQUATION FOR FLOW OF SLIGH 2TLY$. ---- -----------— PRINT COMMNT $0COMPRESS IBLE LIQ UISI A _N__IIT _ AD.IALJ_AQPEL_ 1$ P_ R._ I NT R, DESU LT,DE ELWAF__________________ PRINT COMMENT $0 N TIME PRESSURES S5 THROUGH S2,FOR I=1,1,I.Go(IMAX-1) X2=P I+1,0)-P(I-1,0) X4=:ABS,(X2) WHENEVER X4.L. 1.E-10, X2=0. X3=X2*X2 Kl=(1,-I*DELW) P(I, 1)=P I,0)+A*(Kl*Kl*.(P( I+1,0-2*P( I,O)+P( I-1O') 1-DELW/2*(K1-K/1/ (F -ELOG.(K1)))*X2 2+M (R) *K1*K1/4.*X3) WHENEVER P( I, 1).L.1.E-20,P (I,1)=0 P(O, 1)=(2.*DELW/F+4.*P(1, l')-P(2, 1>)/3. J=J+1 Z=Z+DT............A____!___APRINT. FORMAT R...SLT1....J, Z_ VECTOR VALUES RSLTl=$I4,F9.6,F12,7*$ ~ --- —------- WHENEVER_ J.G.JMAXTRANSFIRJQ__56 ______ ------- THROUGH.S.3, FOR I=0,1,I.G.IMAX S3 P(I,Q)=P(I,1) TRANSFER TO S5 S6 CONTINUE TRANSFER TO S4 END OF PROGRAM. IMAX=80, DELT=.00005, MMAX=2, JMAX= 49,.M(1)=0.,1.* ____ -... _- - ------------ ----— _ _ ---1 —Pg- — r-m —rin L:uL.. LEOSALUAL HT LY_O E___Q=L l~E9r_ E~0NQ _ PQEB~____TIAL _ O R.FLQW _St GHTY....... _ COMPRESSIBLE LIQUIDS IN INFINITE-.RADIAL MODEL ___: _ — _________..._______M..__.QQ________ _- 0QQQEQ5. oELw.012500 A.320O0; ---- -------— _ F. — = —____l.QQOQDQ______ __________ - - ____ — _______-_______'__-_ --- - - N TIME PRESSURE 1.000050.0083333 - 2..Ol00100.-1- 80___ 3.000150.0142833 4 __.000200..0163656___ 5.000250.0181936 6.000300.0198440 7.000350.0213605 8.000400.0227713 9.000450.0240960 10.000500.0253485 11.000550 ',.0265395 12.000600.0276771 13.000650.0287680 14.000700.0298173 15.000750.0308294 16.000800.0318081 17.000850.0327563 18.000900.0336766 19.000950.0345715 20.001000.0354427 21.001050'.0362922 22.0__01100.0371213 23.001150.0379315 24.001200.0387240

-144 -25.001250.0394998 26.001300.0402600 27.001350.0410055 28.001400.0417369 74.001450.0424552 30.001500.0431609 31.001550.0438547 32.001600.0445371 33.001650.0452086 34.001700.0458699 35.001750.0465212 36.001800.0471630 37.001850.0477958 38.001900.0484198 39.0-01950.0490354 40.002000.0496430 4i.002050..0502428 42.002100.0508351 43.002150.0514202 44.002200.0519983.45.002250....0525697 46.002300.0531345 47.002350.-0536931 48.0oo02400oo.o0542455 49.002450.0547920 50.002500.0553328 - ----- ----------------- -- -IBM -.... -— Pmrogr-am-t....Print. Out..... EVALUAT -— NLEL EE-&-N EI FOR FLOW OF SLIGHTLY COMPRESSIBLE LIQUIDS IN INFINITE RADIAL nODEL.........._.._. ______ -. ~.__________M(2) = 1.0000001.DELT * 5.0 OE-059, DELW =.012500, - A =.320CD; F * 1.000000 N TIME PRESSURE 1.000050.0083333 -2a.... f,l8aP_910_Q... _ ~.......AN --- — 3.000150.0143028.000200.___....0163989. 5.000250.0182415 6.000300.0199070' 7.000350.0214391 8.000400.0228659 * 9.000450.0242067 10.000500.0254755 11.000550.0.2668306 -- 12.000600.0278372 13.000650.0289448 14.000700.0300110 15.000750.0310400 16.000800.0320356.. -— 000850..-033b0008 18.000900.0339383 19.000950.0348503 20.001000.0357387 21.001050.036605 22..001100..037451.7 23.001150.0382792 24.001200.0390890 25..001250.0398822 26.001300.0406597 27.001350.0414225 28.001400.0421714 29.001450..042907.1 30.001500.0436302 31.001550.0443414 32.001600.0450413 33.001650...0457303 ---34.001700.0464090,o i~~~i- -------— ~,l —i --- — *~~35.001750.0470779 36.001800.0477372 37.001850.0483875 38.001900.0490290 39.001950.0496622 40.002000.0502873 41.002050.0509046 42.002100.0515145 43.002150.0521171 44.002200.0527128 45.002250-`...05301V7 --- —46.002300.0538842 47..002350_-. —..015-46-3.. ---48.002400.0550303 49.002450.0555944 50.002500.0561528..............................................

FLOW DIAGRAM () IREAD DATA DELW = I./IMAX I PRINT RESULT START) L -~ ~IMAX,.DELT,M, A = DELT/DELW/DELW DELT, IMAX, NMAX,= FOR T I~1,1,>PIMAX,- )=0 NMAX, JMAX DT= DE LT/F/F JMAX, A, M(1)... M(MMAX) PRINT COMMENT THROUGH S6 FOR I=0,1, I>IMAX Z =. EVALUATION OF NON- LINEAR TERM IN NON- LINEAR, R= 1, I, R> MMAX / P(I,O) = 0.0 J =O. SECOND ORDER, PARTIAL DIFFERENTIAL EQUATION M(R /)FOR FLOW OF SLIGHTLY COMPRESSIBLE LIQUIDS IN INFINITE RADIAL MODEL FOR I= 1,1, I > (IMAX-1) I P X2 = P(I+1,0)- P(I-1,0) P X4 = ABS. (X2) WHENEVER X4.L. 1.E-O10, X2=0. PRINT RESULTS PRINT COMMENT X3= XX2 x X2 M (R), DELT, DELW, A, F I IN TIME PRESSURE KI = (I.- x DELW) ~.) DETELWA, P(I, 1) ' P(I,O)+Ax(KIxKi x(P(I+ 1,0)-2. x P(I,O) + P(I-I,O))DELW/2. x (K1 -K 1/(F- ELOG.(K1))) x X2 + M(R) x K1 x K1/4. x X3) WHENEVER P(I,1).L. 1.E-20, P(I,)= O. CONTIN UE P(O,1) = ( 2. x DELW/F+4. x P( 1, )- P(2,1))/ 3. (~ J= =+, PRINT JMAX FOR I=1,,,,. Z= Z+ DT J,2, P(O, I) I>IMAX P(I,O)= P(I,1),P,, S

APPEND IX B COMPUTER PROGRAM FOR NUMERICAL IMPLICIT METHOD USED IN DETERMINING THE ERROR IN NEGLECTING THE NON-LINEAR TERM IN THE PARTIAL DIFFERENTIAL EQUATION DESCRIBING RADIAL FLOW IN A SLIGHTLY COMPRESSIBLE FLUID IN POROUS MEDIA An implicit numerical method was used to calculate values of dimensionless pressure for values of dimensionless time above OOl, A description of the IBM 7090 computer program used in this calculation to determine the error in neglecting the non-linear term in the partial differential equation describing the flow of a slightly compressible fluid for radial flow is presented below, As in the previous appendix, the MAD (Michigan Algorithm Decoder) program, the program nomenclature, the flow sheet, a list of required information, and an example problem illustrating the use of the program are given. The following information is required in the computer programs 1. The number of length increments. (Eighty length increments were found to be satisfactory, Aw -,) 2, The number of time increments to be evaluated, 3, The numerical value for each time increment, 4, The number of M coefficients to be evaluated, (The value of M is defined by Equation (III-34) 50 The numerical values of the M coefficients to be evaluatedo 6, The allowable error between the assumed and calculated pre s sure s o - 1 46 -

i17 -70 The maximum number of iterations allowed for a given time step. The above information is used by the computer program to calculate values of dimensionless pressure versus dimensionless time for each value of the dimensionless coefficient M. Deck Assembly 1. IBM center control cards 20 Radial, Implicit Numerical Program (MAD or Binary Deck) 3. Input Data The input data is read in the "simplified input format". For example, IMAX = 80, JMAX = 51, MMAX = 2, M(1) - 0., 1., B 209 MAXDIF 19E-6, TIME(1) 0.0001, etc.* Definition of these terms used in the simplified input are given in'?the nomenclature below. Flow Sheet The instruction J = a, b, J >c in the flow sheet means that the set of calculations is calculated for J = a and repeated in increments of b until the condition J >c is satisfied, For J. a, b, J >c refers to iterative calculations for a given box. Through d, J m a, b, J >c refers to iterative calculations through "circle d"o The Flow Sheet is presented at end of Appendix.

Nomenclature used in IBM Program and Flow Sheet A TIME (I)/(Aw)2 AA C(I) * P(I) B Maximum number of iterations for a given time step BB C (I) X (I) C (I) Terms defined by Equations (III-73) and (III-83) V (I) Terms defined by Equations (III-74) and (III-84) DD l-i Aw DELW Length increment, Aw DP Calculated Pressure Drop DPSTAR Assumed Pressure Drop F Scaling factor "at' defined by Equation (III-55) I Space index I3 Counter used in IBM program IMAX Number of space increments J Time index JMAX Number of time increments K (I) Terms defined by Equation (III-71) K1 K2 K39,K4 Terms used in IBM program L Counter used in IBM program M Dimensionless coefficient defined by Equation (III-3)+) MAXDIF Allowable difference between assumed and calculated pressure drop MMAX Number of dimensionless coefficients N - 1 P Dimensionless pressure

POLID Dimensionless pressure from previous time increment R I+ 1 TIME Dimensionless time X (I) Terms defined by Equation (III-70) XT TIME (J) / F / F Y (I) Terms defined by Equation (III-69) Z (I) Terms defined by Equation (III-68)

IBM.. Center_ Contr_ Cord M.C. MILLER. Q203N 002 045 000 -----— C- — M —L4 —E ---— N --- —----------— 002 —......045- ~ 000 --- —----------------------------- SCOMPILE MAD, EXECUTE, DUMP PUNCH OBJECPRINT OBJECT MAD Pro ram R EVAtLUATION -or NN- LINEAR TERM IN NC- -t-INEAReN -.-. --- —._ --- —---- -- --- RSECOND ORDER, PARTIAL DIFFERENTIAL EQUATION --------—. --- —— R-FOR FLOW — OF —E H..tG - E —(MR —iQU I.DS ------------------- RIN INFINITE RADIAL MODEL...... ---. -.-. ---R —... --- - --------- RSOLVED BY DIFFERENCE EQUATIONS, IMPLICIT FORM -R. -.. --- —--- _... EXECUTE FTRAP. -------. --- —---- D I MENSI oN —P —)) -M 5Ot+D4-5-+-trX(5-0 tOO.Y (500),Z ( 500 ) PSTAR 00 --- 1),K( 500),DP( 500) DPSTAR (500),C(500) POLD( 500),TIME 200) - ---------- -. - INTEGER IJRL, N-I --- - N)4A-X-,-NAMMMAX-,ALMAXI1I2B131 M12: ----- -- F-1. -READ —DATAY-IMA-X'yd4A'X-M -,MAX-t - IMA'XBT ME ( 1 ) T' I-MMME(-MA-X't - DELW=1~/IMAX --.THROUGH - S14 -, FOR L-a,1L —GiLr6 MMAX(- '... THROUGH S1,FOR I=0tl, I.GIMAX S1 P(I )00. -- ~A=TIME(1 )/DELW/DELW..- PRINT —COMMENT$1EVALU4A-T-4PN OF-'F-ON-LI- EAR 'TERM IN NON-LINEARr-S ---1ECOND ORDER,PARTIAL DIFFERENTIAL EQUATION FOR FLOW OF SLIGHTL 2YS -........PRINT COMMENT $ COMPRESSIBLE LIQUIDS IN INFINITE RADI'AL MODEL I$ -- -.. PRINT COMMENT SOSOLVED BY DIFFERENCE EQUATI'ONStIMPLICIT FORM'$ ---.- -------—.PR-I NT RE-SbfLTS-4 5 —t --- —--- E-S LW,-. —MAXDIFB,IMAXF. ' PRINT COMMENT $0 N TIME PRESSURE BN P( 1IMAX-1)$ - JO ~I31. S2 THROUGH S3,FOR I=01,ItIG.IMAX.-.POLt: ), P (t - -..... -''l'.. S3 PSTAR(I ) P ( I 512 THROUGH S4,FOR Il.tIt-,It.-G.IMAX-1 -- DD=1 -I*DELW' KI=DD*DD K2=DD*DELW/2...-....K3..)../4).l. -----— ' - ' ---. (-PSTAR-4t+1- t-PSTAR(I-1)) - K4=K2/(F-ELOG. (DD)) Z I )=K1-K2+K3+K4 Y(I)=-2,*K1-l./A X(I)-Ki+K2-K3-K4 -. S4 K(I)=-POLD( I )'/A-................ —...-,-(Ht-f —4'+*Z ('tt-+Y (t-]:'-'~t{f —i —r4-rt-]=t') ' ' -~:D( 1)=(K(.1.)-2 *DELW/F - — 1 ' - -- *Zlt)-)-/Xtl)-3-.*.Z.)) ' THROUGH S6,FOR I.=ll9I.G.-IMAX-1 BB=C (.I)*X(I) S6 'D(R)=(K(I)-D(I)*X(I))/.(BB +Y(I)-) P(IMAX )=0.. -... - THROUGH S7,FOR I= IMAX.-1,ILel AA=C(I)*P(I)' 57 P(I-1 )=AA +D(I) WH4ENEVER -M4Lt- wE-rO-t. —TRAN 6F'-ER-T-O 510... RTEST R -.. THROU.i0- SeFO- -- { —I {- G-I-MA#X —t-..... - - -.. R=I+l *.NI-I-.......... -...-.... DP(I)=PP(R )-P(N DPSTAR(I ) PSTAR4-R ----...-PST-AR-4N ----)-. WHENEVERABS.(DP(I)-DPPSTAR(I) )G.MAXDIFTRANSFER TO S9 S8 CONTINUE --... - TRANSFER TO S10 9 - - I313+1 ---- --- WHENEVER I3.G.B,TRANSFER TO 515 THROUGH S11,FOR- lt0-i.-,4G.4-MAX --- —. --- — --. — ' Sl1 PSTAR(I )=P(I ) TRANSFER TO S512 --- S10 J=J+1 -----.T- T t E -J t-/ fF /..................F... WHENEVER XT G.* 1000., TRANSFER TO. 55 —. -. ---- — PR-NT -FORMAT ---RSL-T1 -,X,-P+t-,P- -tP I-MAX-1-) — - - VECTOR VALUES RSLTl1SI4,F12'4,F12.7,tI5,1PE207*$ ----------......- —...... T RAN $FR — -TO — 53 --- —------------------------. —.-.- —. -. ''.''. S5 PRINT FORMAT RSLT2,JXT. P(O),I3,P(IMAX-1) VECTOR VALUES RSLT2= -4,1PE2.3,F -2.7,I5vtPE20S7*S ----. ---S13 WHENEVER J.E.JMAX,TRANSFER TO 514 -..-....-.. —. --- —-A=(' TI-ME —J-I-) —'M"E-( --- -" ------ - E' D —'-... ~.. ----..13=1 -------------—.. ---T-RAN-fER —FO --- —----------------------------._ --- 515 PRINT COMMENT $1 TOO MANY INTERATIONS IN PSTAR CALCULATIONS PRINT RCSULT5 P(O)...P' IMAX,rPTAR'O.V ~,PSfTARf-.M rX-) ---t - -- S14 CONTINUE ---------- -----— ND-OF-PRGGR-AM —

$DATA~-11 -IMAXB809 JMAX=51,MMAX=2, M(1)=O.l-'-T ---..'. Data Input B=20, MAXDIF=1.E- 69 - — TI'ME(-)1 *.0001l00015, 90002,.0003,.0O04-r.0005*0006, O0007,.0008,.0009, --- *001,.0015,.002,.003,,004,.005,.006,.007,.008,.009. - - —.O..01,.015,.02,.03,. 04,. 05,.06,07,.08,,09,. — -- --.1,.15, 2,.3,.4,.5,,6,.7,.8,9,1.,1.5,2.,3.,4.,5.6. 7.,8.9. 10. -..... ---.- - 15e,20.,30.,40,50.,60.,70.,80.,90, 100.,150.,200.,300.,400.,500.,600., ----- --—.. ---- -. —. --- —- —. --- — 700.,800.,900,,1000.,1500.,2000.,3000.,4000.,5000.,6000.,7000.,8000,9000.,10000.* - -. — -------- -... ___~~~ __ --- ___ --- —-----— IBM Program PrinQ ui _EVALUqIQNOF NON-LINEAR TERM IN NON-LINEARt SECOND ORDER9PARTIAL DIFFERENTIAL EQUATION FOR FLOW OF SLIGHTLY COMPRESSIBLE LIQUIDS IN INFINITE RADIALMODEL SOLVED BY DIFFERENCE EQUATIONS,IMPLICIT FORM M( 1).000000, DELW -.012500, AXIF - 10.0000OOE-07, IMAX = 80, F = 1.000000 N TIME PRESSURE BN PIIMAX-1) 1....0001_..01330i1 1I.OOOOOOOE_ 00Q - -_________ 2.0001.0154438 1.0000000E 00 3.0002.0173404 1.OOOOOOOE 00 4.0003.0205250 1.000000E 00 5.0004.0233026 1.0000000E 00 6.0005.0257895 1.0000000E 00 7.0006.0280575 1.0000000E 00 8.0007.0301540 1.0000000E 00 9.0008.0321117 1.0000000E 00 10.0009.0339541 1.0000000E 00 11.0010'.0356991.1 OOOOOOOE 00 12.0015.0429614 1.0000000E 00 13.0020.0492553 1.0000000E 00 14.0030.0596028 1 000000E 00 15.0040.0685006 1.0000000E 00 16.0050.0763816 1.OOOOOOOE 00 17.0060.0835065 1.00000OOE 00: 18.0070.0900438 1.OOOOOOOE 00 19.0080.0961090 1 3.4972836E-38 20.0090.1017849 1 2.5337114E-37 21.0100.1071333 1 1.4480600E-36 22.0150.1290299 1 7.2678235E-24 23.0200.1477552 1 1.4805471E-22 24.0300.1779687 1 6.1525390E-18 25.0400.2035498 1 9.2093271E-17 26.0500.2259100 1 7.3227713E-16 27.0600.2458918 '1 4.0990127E-15 29.0800.2807147 1 6.6971315E-14 30.0900.2961888 1 2.1592918E-13 31.1000.3106538 1 6.2244555E-13....32.10. --- —--— s.. ----~4 —5fso....... 33.2000.4163586 1 3. 5444978E-08 34.3000.4912657 1 1.5180347E-06 35.4000.5527991 1 7.7374445E-06 36.5000.6051975 1 2.4491473E-05 37.6000.6509620 1 6.0029113E-05 -— 38..700.-6168 r1 1. --- —-- - - --..2458737E —04 39.8000.7284333 1 2.2967137E-04.-.-.......90-006-... 7616. 3..87-25?-779-E ---0.4... 41 1.0000.7928471 1 6.0905688E-04 4Z 1. 5000.9098961 I 4.0091186E-03 43 2.0000 1.0036693 1 9.7457930E-03 44 --- —--—;. 0 --— T T;T41-6T0 ---— T --- —----- 2.-63T87-0Z - - 45 4.0000 1.2499011 1 4.8928528E-02 ---- 46 --- —- —..;U0- ----..3T47b63 T ---— T194T --- —-- 47 6.0000 1.4131096 1 9.3642391E-02 48 7.000U -1.417Z550 1 1.1485905E-01 49 8.0000 1.5331367 1 1.3464326E-01 '- 5 ---0 --- —.0 -;-. 58728T ---— Tz8 —2- 20....51 10.0000 1.6257329 _1 1.6940413E-01

_IBM o__Pra m___riinLt Out EVALUATION OF NON-LINEAR TERM IN NON-LINEAR, SECOND ORDER,PARTIAL DIFFERENTIAL EQUATION FOR FLOW OF SLIGHTLY COMPRESSIBLE LIQUIDS IN INFINITE RADIAL MODEL SULVED BY DIFFERENCE EQUATIONS,IMPLICIT FORM M(2). -.000000, DEL.. 012500, MAXDIF 0 10 O- 0.000000E-i0'7,' '-'..'-. 2 IMAX = 80, F = 1.000000 N TIME PRESSURE BN P(IMAX-1) 1.0001.0133231 3.OOOOOOOE 00 2.0001.0154783 3.OOOOOOOE 00 3.0002.0173884 ' 2 OOOOOOOE 00 4.0003.0206026 3.O000000E 00 5.0004.0234113 3.0000000E 00 6.0005.0259301 3.OOOOOOOE 00 7.0006.0282309 2.0000000E 00 8.0007.0303605 2.OOOOOOOE 00 9.0008.0323517 2.OOOOOOOE 00 10.0009.0342279 2.0000000E 00 11.0010.0360069 2.OOOOOOOE 00 12.0015.0434415 3.0000000E 00 13.0020.0499087 3.00000OOE 00' 1.4 -.0030.0606041 3 0000000E 00 15.0040.0698504 3.OOOOOOOE 00 16.0050.0780802 3.0000000E 00 17.0060.0855540 3.000000E 00 18.0070.0924401 3.0000000E 00 19.0080..0988541 3.3.5976035E-38 20.0090. 1048786 3 2.6073725E-37 21.0100.1105753 3' 1.4907704E-36 22.0150.1342009 3 7.7363950E-24 23.0200.1546447 3 1.5785186E-22 24..0300.1882595 3 6.7221543E-18 25.0400.2172161 3 1.0092352E-16 26 ~.0500.2429319 3 8.0487968E-16 27.0600.2662525 3 4.5187569E-15 28.0700.2877217 3 2.0006314E-14 29..0800..3077101.3 7.4267524E-14 30.0900.3264824 3 2.4016695E-13 31.1000.3442339 '3 6.4438660E-13 32 '.'' —..l.-. ---I ---.;ii T. --- —- 4. --- —-- ----— 53-&3-6M3TOIE. —'. 33.2000.4819923 4 4.4000771E-08 34.. 300 -.;-88068- 4 2264578- 06" 35.4000.6803159 4 1.0607364E-05 6' — ''""'...5.-00d.7630929 -4 34'86E — 37.6000.8389783 4 8.6384336E-05 38.7000.9096145. --- —- 1.8362337E-04 39.8000.9761125 4 3.4670'87E-04 40 '.9000 1.0392574 -.4 ---- ' 5.9883187E-04 41 1.0000 1.0996277 4 9.6483438E-04..42..5000 1. 362 5 1 6 6'8963E — 43 2.0000 1.6004583 6 2.2088459E-02 44 3.0 000 T-.2;02T73..T9. —. -....-.' 8. 8''2'024909E-02 '45 4.0000 2.4276088 8 1.7708651E-01 46 5.0000 ' 2.8113426 9' 3.1038776E-01 47 6.0000 3.1851252 9 4.8506430E-01 48 7.0000.S 52-2 — 9 7.0424624E-01 49 8.0000 3.9161483 9 9.7126147E-01 50 9.0000 4.277163 — 10 -. - 1.2903'542E 00 51 10.0000 4.6369280 10 1.6683324E 00

FLOW DIAGRAM '-AX, aMAX, M(])... M(MMAX), MMAX, B, TIME (D)... T(MMX) L a 1, 1, L >MMA PRINT COMMENT EVALUATION OF NON- LINEAR, SECOND ORDER, PARTIAL FOR J *O. 1, I>IMAX _ DIFFERENTIAL EOUATION FOR FLOW OF SLIGHTLY K2 * DO DEA TIME(M)/DELW/LDELW t! P(l) *0.0, A s TIME(I)/ DELW/DELW COMPRESSIBLE LIOUIDS IN INFINITE RADIAL MODEL SOLVED BY DIFFERENCE EOUATION,IMPCT FORM PRINT RESULTS PRINT COMMENT FOR,1,MAX PSTAR3M ) P P(DR FOR I -1, 1, I >(IMAX) DD ~ (1.- I xDELW) K1 -DD xDD K2 ~ DD x DELW/2. C (1) ~ -(4.x Z(1) + Y(1))/(X(t) -3. x Z ()) K3S (M(L)/4.) x K1 x ( PSTAR(I+I)- PSTAR(I- 1)) K48 K2/(F-ELOG.(DD)) D(1)~(K(1)-2. x DELW/FxZ(I))/(X(1)-3. x Z(1)) Z(l) ~ K1- K2 + K3 * K4 Y(l) a -2.x K1-1./A X(l) ~ KI +K2 - K3 - K4 K(l) ~ - POLD (I)/A FOR I ~, 1, I<IMAX-I R -I+I FOR I 1,-1, I<l BB C (D ) X(I) PT I MAX) G. AAA CM(I) x p (T C(R) - -Z(I)/(BB+ Y(D)) P(I-1) ~ AA+ D(I) D(R)" (K(]) - D(l) x X(I))/(BB + Y(I)) FOR 1=1,1, I>(IMAX-I) R sI*l D(]) P(R) - P(N) DPSTAR (I) PSTAR (R)- PSTAR ( N WHENEVER.ABS.(DP(I)-DPSTAR(I)).G. MAXDIF,TRANSFER TO S9 CONTINUE -PSTAR () - p() +I PRINT XT.- TIME (J)/F/F J, XT, P(O), 13, P(]MAX- ) CON TIN UE

APPEND IX C COMPUTER PROGRAM FOR NUMERICAL EXPLICIT METHOD USED IN DETERMINING THE ERROR IN NEGLECTING THE NON-LINEAR TERM IN THE PARTIAL DIFFERENTIAL EQUATION DESCRIBING LINEAR FLOW OF A SLIGHTLY COMPRESSIBLE FLUID IN POROUS MEDIA A description of the IBM 7090 computer program used to determine the error in neglecting the non-linear term in the partial differential equation describing the flow of a slightly compressible fluid for linear flow is presented below. Although this program is very similar to the method used for radial flow described in Appendix A, this appendix is enclosed for completeness. The MAD (Michigan Algorithm Decoder) program, the program nomenclature, the flow diagram, a list of required information, and an example problem illustrating the use of the program are given. The following information is required in the computer program~ 1. The number of length increments. (Eighty length increments were found to be satisfactory9 Aw 8-~ ) 2. The value of the time increment. (At - 0,00005 is an acceptable value of At for Aw =- 0 Stability requirements demand that (At)7(Aw)2 is less than 005.) 3. The number of time increments to be evaluated. 4-. The number of M coefficients to be evaluated. (The value of M is defined by Equation (III-100).) 51, The numerical values of the M coefficients to be eva 3,~e~zluated oC -,,.._

-J15 -This information is used by the computer to calculate values of dimensionless pressure versus dimensionless time for each value of the dimensionless coefficient Mo Deck Assembly 1. IBM center control cards 2. Linear, Explicit Numerical Program (MAD or Binary Deck) 3. Input Data The input data is read in "simplified input format"o For example, IMAX t 80, DELT = 0.00005', MMAX = 2, JMAX = 49, M(1) = 0., l o* Definitions of the terms used in this input are given below in the nomenclature. Flow Sheet The instruction J = a, b, J >c in the flow sheet means that the set of calculations is calculated for J S a and repeated in increments of b until the condition J > c is satisfied. For J = a b, J >c refers to iterative calculations for a given box. Through d, J = a, b, J >c refers to iterative calculations through "circle d". The Flow Sheet is presented at end of Appendix. Nomenclature used in IBM Program and Flow Sheet ADIM Term used to dimension P(I,J) vector A At/(Aw)z DELT At

DELW Aw DT A t/(F) F Scaling coefficient "a". In this dissertation, F 1!.0. I Space index IMAX Number of space increments, A w IJ Time index JMAX Number of time increments K1 Term used in program, K1 1-i Aw M Dimensionless coefficient defined by Equation (III-100) MMAX Number of dimensionless coefficients P Dimensionless Pressure R Counter used in IBM program X2 Term defined by X2 = P +,j-Pi-, X3 Term defined by X3 = (X2)l X4 Absolute value of X2 Z Value of dimensionless time at given time step, Z = jAt

. --- —--------- - ---- ------------------ A -- lBG CenterCnritrol Cards. M.C. MILLER' Q203N 002 045 000 -------— M- I-ER — -- ----- ----— 3 --- —-------- - - 2 ----45 O- 04 — - A- Program -- SCOMPILE MAD, EXECUTE, DUMPt PUNCH OBJECTPRINT OBJECT MProgr R EVALUA-TION-OF- NOI LIICAR TER M1 NO- -INEAR ~ - - --.-. --- —. R SECOND ORDER, PARTIAL DIFFERENTIAL EQUATION --------------- —. —.R.-FOR —F-4OW OF -SLIG4T.TL-Y —4OMPRE&SB4-E —L- tQUIDS R IN INFINITE LINEAR MODEL. --- *..... R....... RSOLVED BY DIFFERENCE EQUATIONSEXPLICIT FORM DIMENSION P( 8100,ADIM),M(10)........... —. --- —.I N T EGER I 9 J 9 K-vL,R rN- AAM*X-,MMA*w-AJMX --- - - VECTOR VALUES ADIMs29101.100............ FI1 '. S4 READ DATA9IMAXDELT.MMMAXoJMAX _-___ _ ---. — — ELW. / IMAX — -—.. - I ---- A=DELT/DELW/DELW DT =DELT/F/F ' - PRINT RESULTS DELTIMAX,MMAX,JMAX, AM(1)'*.MIMMAX' P(IMAX, 1)O. THROUGH S6,FOR RtlilR.GoMMAX THROUGH SIrfOR-4 —m-1t,'4 -M-..- 44 -- Si' P( I, o')OO Z=O. J=O - PRINT. COMMENT SlEVALUATION OF NON-LINEAR TERM IN NON-LINEA* -- ISECOND ORDER, PARTIAL DIFFERENTIAL EQUATION FOR FLOW OF SLIGH PRINT COMMENT $OCOMPRESSIBLE LIQUIDS IN INFINITE LINEAR MODEL 1is PRINT RESULTS M(R),DELTDELWAF. PRINT COMMENT $0 *N ' TIME PRESSURES S5 THROUGH S2,FOR Il1,1itl G(IMAX-1) X2PiI+1q-0)-P(I-1,0). X4=oABSo(X2) WHENEVER X4.L. 1.E-1lO X2-0O X3=X2*X2 K11(1-I#*DELW.. P(I 1)=P(I, 0)+A*(Kl*Kl*i(P(I+1,0)-2.*P(IPO)+P(I-10 )). 1-DELW2**K1*X2... 2+M(R)*Kl*Kl/4*X3l) WHENEVER P(I 1),L.l.E-209PIIl)-w0o. 52 CONTINUE P(O, 1)-1(2*DELW/F+4.e*P,1 l)-P(2v 1))/3. J.J+1 ZZ+OT.. PRINT FORMAT RSLTlJZP(Ol,1) VECTOR VALUES RSLTl-S1I4F9g69F12.7*S WHENEVER J;G.JMAXTRANSFER TO S6 THROUGH S3, FOR' I=0,1,Ie.G.IMAX S3 P(I,O0)P(II,) TRANSFER TO 55 S6 CONTINUE TRANSFER TO S4 END OF PROGRAM IMDTAX=80, ~"Data Input IMAX=80, DELT=.00005, 'MMAX=2, JMAX= 49, M(1)O,1* Data Input -__ --- --- -------- --- - -------- IBM_-P —amintL ut.__EVALUAION OF NON-LINEAR TERM IN NON-L INEAR SECOND ORDER1 PARTIAL DIFFERENTIAL EQUATION FOR FLOW OF SLIGHTLY. COMPRESSIBLE LIQUIDS IN INFINITE- LINEAR MODEL _........... --- —-------— AL = --- —------- DELT = 5.000000E-O 05t DELW =.012500, A. 3200Q -E -..03. -300................ E_ -—........ ----_QO _ Q ---------- -------—. --- —-...............................:....................... N TIME PRESSURE 1.000050.0083333......_OQD1O_-Q.._OQII.8225 3.000150.0143279 4.Q_0__Q_0 -— _.O.Q16433-6_ 5.000250.0182852 6.000300.0199593 7.000350.0214997 8.000400.0229345 9.000450.0242831 10___ 0.___.Q50__.255596 11.000550.0267747 12.000600.0279364 13.000650.0290513 14.007Q00.0301248 15.000750.0311610 16.000800.0321638 17.000850.0331361 18.000900.0340806 19.000950.0349995 2. 0010 ___ 0_.358949 21.001050.0367684 ___ Q D___O4 ILQ....___..ObZLZ..... 23 —.001150.0384560 24.001200.0392726

-158 -25.001250.0400725 26.001300.0408568 27.001350.0416263 28.001400.0423818 29.001450.0431241 30.001500.0438539 31.001550.0445717 32.001600.0452782 33.001650.0459738 34.001700.0466590 35.001750.0473344 36.001800.0480002 37.001850.0486570 38.001900.0493050 39.001950.0499446 40.002000.0.505762 41.002050.0511999 42.002100.0518162 43.002150.0524253 44.002200.0530273 45.002250.0536226 46.002300.0542114 47.002350.0547939 48.002400.0553702 49.002450.0559407.-._______ _ _ ___IL~s2___I.MP..mrinLu. 50.002500.0565053.....~ALUArlQ -1N_ R MRTR. _IXQ.~R~E~.. i8-P AL i~F-~LLAL -XU&tQN~ FOR FLQH OF SLIGHTLY....... COMPRESSIBLE LIQUIDS IN INFINITE LINEAR MODEL.MI... -..... -_-.. _ _0..__ --- —___-.__ELa =,.012500,@ A....32000u,...................._..-..,0___2Q9 _ _ __ ___ __ _ _. _... N TIME PRESSURE 1.000050.0083333.-.-.2 Q....Q10_Q....11.. I829... 3.000150.0143475 -4...... Q2QQ __-_-_ —._01_6467_0_ 5.000250.0183333 6.000300.0200228 7.000350.0215790 8.000400.0230299 9.000450.0243950 10.000500.0256881 11.000550.0269199 12.000600.0280986 13.000650.0292306 14.000700.0303211 15.000750.0313747 16.000800.0323947 17.000850.0333844 18.000900.0343464 19.000950.0352829 20.001000.0361959 21.001050.0370871 22.001100.0379580 23.001150.0388100 24.001200.0396444 25'.001250.0404621 26.001300.0412643 27.001350.0420516 28.001400.0428251 29..001450.0435853 30.001500.0443330 31.001550.0450688. 32.00160.0.0457933 34.001700.0472102..35.0'017i50.0479036 36.001800.0485875 37.001850.0492624 38.001900.0499285.39..-f... '..0505863 40.002000.0512360.00205'i-.....=-05 i~.0518779 42.002100.0525124 43.002150.0531396 44.002200.0537599 ------ 45 — 0 —O2Y50 ---. ---- 054'"Y4 ---- 46;002300.0549804 4... 5T ---.02.... --- —-; —5.oSsYs - 48.002400.0561757 49.002450.0567644 50.002500.0573474.............................................

FLOW DIAGRAM RE AD DATA DE LW = 1. / IMAX P RINNT R ESUULT I MAX, DE LT, M, A = DELT/ DELW/ DELW DELT, I MAX, NMAX, PMX NMAX, JMAX DT==DE LT//F/F JMAX, A, M(l)... M (MMAX) PRINT COMMENT THROUGH S6 ~ --- —-el FOR I=O,1, I >IMAX Z =O. EVALUATION OF NON-LINEAR TERM IN NON-LINEAR, 2 R = 1, 1, R >MMA Pt IO) = 0.0 J = O. SEC.OND.ORDER, PARTIAL DIFFERENTIAL EQUATION L ~~FOR FLOW OF SLIGHTLY COMPRESSIBLE LIQUIDS IN INF'INITE LINEAR MODEL S FOR I 1, 1, I >(IMAX -1) ~ X2 = P(I+1,O)- P(I-1,O) X4 = ABS. (X2) PRINT RESULTS PRINT COMMENT WHENEVER X4.L..E-bX2=O. M (R),DELT, DE LW, A,F N TIME PRESSURE X3= X2 x X2 K =(- I x DELW) P(, 1)= P(I,O)+Ax(KI xKl x(P(I+1,O)-2. x P(I,O) + PO-1,0))DELW/2. x KI x X2 +M(R) x Ki x K1/4. x X3) CONTINUE N(O, 1 ) ( (2. x DELW/ F + 4.' x P(I1, I))- P( 2,11))/3. PRINT FOR I= 1, 1, J=J+1 >JMAX J 2, P(O, 1) I>IMAX Z = Z DT PI ) 1

APPEND IX D COMPUTER PROGRAM FOR NUMERICAL IMPLICIT METHOD USED IN DETERMINING THE ERROR IN NEGLECTING THE NON-LINEAR TERM IN THE PARTIAL DIFFERENTIAL EQUATION DESCRIBING LINEAR FLOW OF A SLIGHTLY COMPRESSIBLE FLUID IN POROUS MEDIA An explicit numerical method was used to calculate values of dimensionless pressure for values of dimensionless time above 0,01 Although this computer program is very similar to Appendix B it is included for completeness. The MAD (Michigan Algorithm Decoder) program, the program nomenclature, the flow sheet, a list of required information, and an example problem illustrating the use of the IBM 7090 program are described below. The following information is required in the computer program: 1. The number of length increments. (Eighty length increments were found to be satisfactory, Aw =. ) 2. The number of time increments to be evaluated. 3. The numerical value for each time increment. 4+. The number of M coefficients to be evaluated. (The value of M is defined by Equation (III-100).) 50 The numerical values of the M coefficients to be evaluated, 6. The allowable error between the assumed and calculated pressures. 7. The maximum number of iterations allowed for a given time step. The computer program uses this information to calculate the numerical values of dimensionless time versus dimensionless

for each value of the dimensionless coefficient M. Deck Assembly 1. IBM center control cards 2. Linear, Implicit Numerical Program (MAD or Binary Deck) 3o Input Data The input data is read in the "simplified input format". For example IMAX = 80 JMAX 51, MMAX - 2 M(1) = 0.I 1.9 B - 20, MAXDIF - 1.E-6, TIME(1) = 0.0001, etc.* Flow Sheet The instruction J e a, b, J >c in the flow sheet means that the set of calculations is calculated for J S a and repeated in increments of b until the condition J >c is satisfied:, For J = a, b, J >c refers to iterative calculations for a given box. Through d, J = a, b, J >c refers to iterative calculations through "circle d", The Flow Sheet is presented at end of Appendix, Nomenclature used in IBM Program and Flow Sheet A TIME(I)/(Aw) AA C(I) * X (I) B Maximum number of iterations for a given time step BB C(I) * X(I) C(I) Terms defined by Equations (III-126) and (III-138) D(I) Terms defined by Equations (III-127) and (III-139)

wi62,DD l - i.Aw DELW Length increment, Aw DP Calculated pressure drop DPSTAR Assumed pressure drop F Scaling factor "a" defined by Equation (III-113) I Space index I3 Counter used in IBM program IMAX Number of space increments J Time index JMAX Number of time increment s K(I) Terms defined by Equation (III-124) KlK2,K3 Terms used in IBM program L Counter used in IBM program M Dimensionless coefficient defined by Equation (III-100) MAXDIF Allowable difference between assumed and calculated pressure drop MMAX Number of dimensionless coefficients N I 1 P Dimensionless pressure POLD Dimensionless pressure from previous time increment PSTAR Assumed value for dimensionless pressure R I+ 1 TIME Dimensionless time X(I) Terms defined by Equation (III-123) XT TIME (J) /F/F Y(I) Terms defined by Equation (III-122) Z(I) Terms defined by Equation (111-121)

-163 -IBM Center Control Cards M.C. MILLER Q203N 002 045 000 M.C. MILLER Q203N 002 045 000 $COMPILE MAD, EXECUTE. DUMP, PUNCH OBJECT,PRINT OBJECT MAD Program R EVALUATION OF NON-LINEAR TERM IN NON-LINEAR, RSECOND ORDER9 PARTIAL DIFFERENTIAL EQUATION RFOR FLOW OF SLIGHTLY COMPRESSIBLE LIQUIDS RIN INFINITE LINEAR MODEL R RSOLVED BY DIFFERENCE EQUATIONS, IMPLICIT FORM R EXECUTE FTRAP. DIMENSION P(500),.M(500),D(500),X(500),Y(500)tZ(500),PSTAR(500. 1),K(500),DP(500),DPSTAR(500),C(5001)POLD(500),TIME(200) INTEGER IiJ,RL, NtIMAXtJMAX,MMAXALMAX,I1l,12B,I3,M1 F=1. READ DATAtIMAXJMAXtM(1).e.-M(MAX),MMAXB,TTIME(Il),,TIME(MAX) 'DELW=1./IMAX THROUGH S14, FOR'L=ll1,LoG.MMAX THROUGH S1,FOR I=O,1,I.GIIMAX S1 P(I )=0.0..A=TIME(1)/DELW/DELWW PRINT COMMENT$1EVALUATION OF NON-LINEAR TERM IN NON-LINEAR, S lECOND ORDERPARTIAL D'IFFERENTIAL EQUATION FOR FLOW OF SLIGHTL 2Y$. PRINT COMMENT $ COMPRESSIBLE LIQUIDS IN INFINITE LINEAR MODEL 15 PRINT COMMENT O'SOLVED BY DIFFERENCE EQUATIONS,IMPLICIT' FORM$ PRINT RESULTS M(L), DELW, MAXDIFB,IMAX,F PRINT COMMENT $0 N TIME PRESSURE 8N P1 IIMAX-1)$ J=O I3=1 S2 THROUGH S3,FOR I=O,1,I.G.IMAX POLD(I) = P(I) S3 PSTAR(I )=P(I ) S12 THROUGH 54,FOR I=11i,Io.GIMAX-1 DD=(1.-I*DELW) K1=DD*DD K2=DD*DELW/2. K3=(M(L)/4.)*K1 *(PSTAR(I+1 )-PSTAR(I-1)) Z(I)=K1-K2+K3 Y(I)=-2.*K1-1./A 'X(I)=K1+K2-K3 S4 K(I)=-POLD(I)./A C(1)=-(4.*Z(1)+Y(l())/(X(1)-3*Z(1)) D(1)=(K(1)-2.*DELW/F 1 *Z(1))'/(X(1)-3o*Z(1) THROUGH S6,FOR I=1,1,IoG.IMAX-1 R=I+1 BB=C(I)*X(I) C(R)=-Z(I)/(BB +Y(I)) S6 D(R)=(K(I)-D(I)*X(I))/(BB +Y(I)) P(IMAX )=0. THROUGH 57,FOR I= IMAX,-1I.L.1 AA=C(I)*P(I) 57 P(I-1 )=AA +D(I) WHENEVER M(L) oE.Oo, TRANSFER TO S10 RTEST R THROUGH S8,FOR I=1,1,IoGo(IMAX-1) R=I+1 N=I.-1 DP(I)=P(R )-P(N ) DPSTAR(I)=PSTAR(R )-PSTAR(N ) WHENEVER.ABS.(DP(I)-DPSTAR(I)).G.MAXDIF,TRANSFER TO 59 58 CONTINUE TRANSFER TO S10 59 13=I3+1 WHENEVER I3.G.B,TRANSFER TO 514 THROUGH S11,FOR I=O,1,IGIMAX Sil PSTAR(I )=P(I ) TRANSFER TO 512 SIO J-J+1 XT=TIME(J)/F/F WHENEVER XT.0. 1000., TRANSFER TO 55 PRINT FORMAT RSLT1,J,XT,P(O),I3,P(IMAX-1) VECTOR VALUES RSLT1=$I4,F12.4,F1277,I5, 1PE20.7*$ TRANSFER TO 513 S5 PRINT FORMAT RSLT2,J.XT,P(0),13,P(IMAX-1) VECTOR VALUES RSLT2=$I4,1PE12.3,F12-7, 15,1PE207*$ 513 - WHENEVER JeE.JMAXITRANSFER TO 514 A=(TIME(J+1)-TIME(J))/DELW/DELW 13=1 TRANSFER TO S2 S14 CONTINUE END OF PROGRAM

-164 -$DATA Data Input IMAX80 JMAX=51,MMAX=2, M(1):0.o,1. 8=20, MAXDIF=1.E- 6, TIME(1)=.0001.00015,.0002, 0003,.0004,.00059,.0006.00079,.00089.0009, 0001,.00159,002,0039,004,.005,.006,.007,008,0099.01,.0159,02,.03,.049.059.06.07,9.08,.099.1,.15,.2,.3,.4,.5,.6,.7,.8,.9,1.,1.5,2.,3.,4.,5..6.,7,,8.,9.,10., 15.,20.,30.,40.,50.,60.,70.,80.,90.,100.,150.,200.,300.,400.,500.,600., 700.,800e,900.,lOO1,1500.,2000.,OO.30.40040.,5000.,6000.,7000.,8000., 9000.,.10000.* -EVACUATI~I~~~BM APF~ru.-rr~~_nogram-PrinLuLm POutm ---'.EVALUATIONOF NON-LINEAR TERM IN NON-LINEARt._SECOND ORDER UPARNIAL DIFFERENTIAL EQUATION FOR FLOW OF SLIGHTLY COMPRESSIBLE LIQUIDS IN INFINITE LINEAR MODEL SOLVED BY DIFFERENCE EQUATIONS.IMPLICIT FORM M(1) =.000000, DELW =.012500. MAXDIF = 10.OOOOOOE-07O B 2 IMAX = 80, F = 1.000000 N TIME PRESSURE BN P(IMAX-1).1.0001.0133468 1.OOOOOOOE 00 0. 2.0001.015512.7.1.0000000E 00 3.0002.0174321 1.0000000E 00. 4.0003.0206648 1.0000000E.00. 5.0004.0234902 1.0000000E 00 6.0005.0260250 1.000008E 00 7.0006.0283411 1.0000000E 00 — 8.000.0304857 1 OOOOOOOE 00. 9.0008.0324915 1.0000000E00 10.0009.0343821 1.OOOOOOOE 00 11.0010 '0361753 1.0000000E 00 12.0015.0436779 I.0000000E O 13.0020.0502114 1.0000000E 00.1...4......o0030....0610352 1.........-0000000E 00.. 15.0040.0704066 1.0000000E 00 16.0050.0787589 1.ooo0000000ooE 00 17.0060.0863525 1.0000000E 00 18.0070.0933564 T 1.1088234E-38 19.0080.0998863 1 9.2418500E-38.20.0090.-1060-250.1. 6.0052935E-37.. 21.0100.1118342 1 3.3806798E-36...22.0-150..1359989..1. 1.7167136E-23 23.0200.1569611 1 3.49959458-22 24.0300.1915496 1 1.47005758-17 25.0400.2214234 1.2.20325448-16 26.0500.2480053 — 1 - 1.75409858-15 27.0600.2721461 1 9.8308289E-15 28..0700.2943941 1.. 4.3449056E8-14 29.0800.3151238 1 1.61007648-13 30.0900.3346030 1 5.1974005E-13 31.1000.3530298 1 1.49999708-12 32.1500.42946129 1 1.12883.04E-08 33.2000.4960114 1 8.89803508-08 34.3000.6055173 1 3.9127982E-06 35.4000.7000678 1 2.0126939E-05 36.5000.7841834 T 6.4224688E-05 37.6000.8605650 1 1.5861864E-04 38 ------- ' --- 1O_ Y____.9 1-15f ------- f --- —----— 3 A-6297-0 38.7000.9309516 1 3.3162597E-04 39.8000.9965303 1 6.1570762E-04 ' ~' ' ' ~~~-~' —^ --- —------ 40.9000 1.0581490 1 1.0454163E-03 41 1.0000. 1.1164367 1 1,6554139E-03....42 — 1l.5000 1.3586604 1 1T. 162544_2ET 0_2 43 2.0000 1.5686593 1 2.9132885E-02 44 3. 0000.. —f.;91497-47...... —. —. 8.7942625E-02........ 45 4.0000 2.2138803 1 1.6229092E-01 46 5.0000 24794863 - -1 2.4530167E-01 47 6.0000 2.7200514 1 3.3206388E-01.48 7.0000 2.9407316. 1 4.1933646E-01 49 8.000o 3.1449153 1 5.0509205E-01 50....9.0000 3...3334933-8..1 i.....5.8812507E-O01 51 10.0000 3.5124638.1 6.6776519.-01

-165 -M-..Proram Print Out EVALUATION..OF NON-LINEAR_ TERM IN NON-INEAR._SECOND_RER _PARTIAL DIFFERENTIAL. EQUATION. FOR FLOW OF SLIGHTLY. COMPRESSIBLE LIQUIDS IN INFINITE LINEAR MODEL SOLVED BY DIFFERENCE EQUATIONSOIMPLICIT FORM M(2) = 1.000000,. ELW..012500, MAXDIF - 10.OOOOOOE-07 B 2C IMAX o 80, F - 1.000000 N TINE PRESSURE BN P(IMAX-1) 1_..._t ~. _ ~ _..____ ~.Q00 1:... sQ13366_9-.~~___3 -.0 _Q0_Q 0JEQ00 2.0001.0155474 3.0000000E 00._3.0002.0174812 2,OOOOOOOE 00 4.0003.0207432 3.0OOOOOOOE 00 5.0004.0236002 3.OOOOOOOE 00 6.0005.0261676 3.OOOOOOOE 00 7.0006.0285169 2.OOOOOOOE 00 8.00074.0306954 2.OO00000E 00 9.0008.0327355 2.0000000E 00.00 -~~ - ------------ ~ —~ --- — bb 'dbd_ _0 10.0009.0346607 2.0000000E 0 11.0010.0364887 2.0000000E 00 12'.0015.0441688 3.0000000E 00 13.0020.0508817 3.0000000e OO0 ~ ---~ --- —----------------------- -' --- —14.0030.0620681 3.OOOOOOOE 00 15.0040.0718052 3.000O000E 00 — 16.0050.0805253 3 0000 00 —E- 00 17.0060.0884888 3.Ooo00ooooE 00 18.0070.0958643 3 1.1404954E-38 19.0080.1027672 3.9.5095649E-38 20.0090.1092804 3 6.1818725E-37.21.0100.1154652 3 3.4815888E-.6 22.0150.1415240 3 1.8302682E-23 23.0200.1643947 3 3.7373441E-22 24.0300.2028455 3 1.6111119E-17 25...0400.2366256..3.. 2.4224841A-16 26.0500.2671563 3 1.9348174E-15 27.0600.2952857 3 1.0878372E-14 28.0700.3215597 3 4.8233255E-14 29.0800.3463504 3 1.793f378E-13 30.0900.3699235 3 5.8071772E-13 31.1000.3924754 3 1.6814848E-12 32.1500.4902508 4 1.4069004E-08 33.2000.5783575 4 1.1261847E-07 36.3000.7330062 5 5.4064626E-06 35.4000.8742882 5 2.8750841E-05 36.5000 1.0066097 5 9.4679370E-05 37.6000 1.1325689 5 2.4133995E-04 38.7000 1.2538115 5 5.2107086E-04 39.8000 1.3714461 5 9.9980067E-04 40.9003 1.4862493 55 1.7556938E-03 41 1.0000 1.5987870 5 2.8774391E-03 42........ 1.5000 02.133220-f23 8... 2;8702732E -02 43 2.0000 2.6523208 8 8.6282647E-02 44 3.0000 3.6680869 11 3.9015816E-01 45 4.0000 4.6748586 13 9.5703807E-01 46 5.0000 5.6780592 13 1.8492870E 00 47 6.0000 6.6798893 14 3.1628510E 00

UNIVERSITY OF MICHIGAN.3 9015 03483 8543 FLOW DIAGRAM DD'. I /""~,..oE. MAX, JMAX, M()... M(MMAX), MMAX,, TIME (D)... T (MAX) L 1, 1, L> MMAX PRINT COMMENT EVALUATION OF NON-LINEAR, SECOND ORDER, PARTIAL FOR l 0, 1, I >iMAX A DIFFERENTIAL EQUATION FOR FLOW OF SLIGHTLY P (I~) 80~.: COMPRESSIBLE LIOUIDS IN INFINITE LINEAR MODE SOLVED BY DIFFERENCE EOUATION, IMPLICT FORM PRINT RESULTS PRINT COMMENT J-O FOR 1-O, I, I>AX M(L), DELW, MAXDIF,. IMAX, f N TIMF PRESSURE BN P(IMAX-I) 13- POLD() P(l) PSTAR (I) P(1) FOR a1, 1, I>(IMAX) DD (I.- I x DELW) KI~ RDD x DD K2 - DD x DELW/2. C(I) a-(4. x Z()*+ Y(I))/(X(i)-3. x Z(!)) K3- (M(L)/4.) KI R ( PSTAR(+I)-PSTAR (I- I)) Z(I) ~ KI - K2 * K3 D(l ) (K(1) - 2. x DELW/F x Z(I))/(X(I)- 3. x Z(0)) Y() ~ -2. x KI-. /A X(I) ~ KI + K2 - K3 K(I) ~ - POLD(I )/A FOR I 1, I, I >IMAX -I R a I + I FOR I 1,-I, I < I BBaC(l) X(l ) P(MAX AA * C(1) x p(l) F C(R) ~ -Z()/(88 + Y(l)) P(l- I ) ~ AA + D(I) D(R) ~ (K() - D() x X(Z))/(88+ Y(I)) FOR I1,1I, I>(IMAX-I) R -1+l N 1-ID(l) = P(R) - P(N) DPSTAR () ~ PSTAR (R) - PSTAR (N) WHENEVER,ABS.(DP(I) - DPSTAR(I)).G. MAXDIF, TRANSFER TO S9 CONTINUE }~-.4~~~~~INUF FOR I0, 1, I IMAX *I3>8~:.PSTAR (]) ~P() XT a TIME M/F/F J3~~"~~, XT, P(O), 13, P(IMAX-) PRINT A.(TIME(J+I)- TIME(J)}/DELW/DELW J: XT, P(O), 13, P(OMAX - I)13 CONTI NUE