THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Chemical and Metallurgical Engineering Second Annual Report ENGINEERING STUDIES ON MOVEMENT OF WATER IN CONTACT WITH NATURAL GAS (Project N031) Donald L. Katz M. Rasin Tek Maurice C. Miller Stanley C. Jones ORA Project 02935 under contract with: The American Gas Association New York, New York administered through; OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR October 1961

Members of the Supervising Committee For American Gas Association Research Project N031* J. Ao Vary, Chairman Michigan Consolidated Gas Company O0 Co Davis Natural Gas Storage Company of Illinois B. B. Gibbs United Gas Corporation E. Vo Martinson Northern Natural Gas Company CO E. Stout Columbia Gas System Service Corporation S. Jo Cunningham American Gas Association Ro Mo Duke, Secretary American Gas Association *A subcommittee of the AGA Pipeline Research Committee, Ro Ho Crowe, Chairmano

TABLE OF CONTENTS Page LIST OF TABLES v LIST OF FIGURES vii ABSTRACT xi INTRODUCTICN 1 PART I. CALCULATION PROCEDURES DEVELOPED, EXTENDED, AND IMPROVED DURING THE SECOND YEAR A. ANALYSIS OF RESERVOIR PERFORMANCE BASED ON FIELD DATA 7 1. Development of Equations 7 2. Summary of Working Equations and Stepwise Calculation Procedure 11 B. LINEAR FLOW MODEL 14 1. Constant-Pressure Case 15 2, Constant-Rate Case 16 3. Variable-Pressure Solution 17 C. EQUATIONS COMMON TO THE VARIOUS METHODS 18 1. Calculation of Bottom-Hole Pressure, Psia 18 2. Calculation of Gas in Place 19 3. Calculation of Pore-Volume Ratio 20 4. Calculation of Dimensionless Time 20 D. SUMMARY OF WORKING EQUATIONS FOR RADIAL AND THICK SAND MODELS 21 1. Radial Flow Model 21 2. Thick Sand Model 24 PART II. FIELD CASE STUDIES FIELD A 33 1. Geology 33 2. Field Data 33 3. Calculations 33 4. Results 41 iii

TABLE OF CONTENTS (Concluded) Page FIELD B 49 1. Geology 49 2o Field Data 49 35 Calculations 53 4o Results 57 FIELD C 63 1l Geology 63 2, Field Data 63 35 Calculations 63 4o Results 71 FIELD D 77 lo Geology 77 2. Field Data 77 35 Calculation on Field D: Example of Resistance-Function Method 80 FIELD E 95 1, Geology 95 2. Field Data 96 30 Results 96 FIELD F 111 1o Geology 111 2. Field Data 111 3. Results 111 FIELD G 123 1. Geology 123 2. Field Data 123 30 Results 123 FIELD H 135 1, Geology 135 2, Field Data 135 35 Results 135 ACKNOWLEDGMENT 147 REFERENCES 149 iv

LIST OF TABLES Table Page I. Data Required and Used for Field Studies on Various Models. 2 II. Dimensionless Pressure Drop vs. Dimensionless Time for Thick Sand Model 25 III. Parameters M and A for the Thick Sand Model 25 IV. Definition and Units of Parameters used in the Various Mathematical Models and in the Resistance Function 32 V. Parameters used for Fit and Prediction of Field Data 146 A-1. Gas Field A Data 35 A-2. Gas Inventory and Pressure History for Field A 36 A-3. Calculated Values for Field A using Radial Model 38 B-l. Storage Field B Data 51 B-2. Gas Inventory and Pressure History for Field B 52 B-3. Calculated Values for Field B using Thick Sand Model 55 C-1. Storage Field C Data 65 C-2, Gas Inventory and Pressure History for Field C 66 C-'3 Calculated Values for Field C using Linear Model 68 D-l1 Storage Field D Data 78 D-2. Gas Inventory and Pressure History for Field D 79 E-1. Storage Field E Data 97 E-2. Gas Inventory and Pressure History for Field E 98 v

LIST OF TABLES (Concluded) Table Page F-l, Storage Field F Data 112 F-2o Gas Inventory and Pressure History for Field F 113 G- 1 Gas Field G Data 124 G-2. Gas Inventory and Pressure History for Field G 125 H-1. Storage Field H Data 136 H-2. Gas Inventory and Pressure History for Field H 137 vi

LIST OF FIGURES Figure Page A-l. Areal sketch showing boundary of Field A. 34 A-2. Comparison of Field A predicted pressures, assuming no water movement with observed pressures. 42 A-3. Comparison of Field A pressures predicted using radial model (infinite aquifer) with observed pressures. 43 A-4. Comparison of Field A volume ratio predicted using radial model (infinite aquifer) with volume ratios calculated from field data. 44 A-5. Comparison of Field A pressures predicted using resistance function with observed pressures. 45 A-6. Comparison of Field A volume ratio predicted using resistance function with volume ratios calculated from field data. 46 A-7. Resistance function vs. time for Field A. 47 B-1. Areal sketch showing boundary of Field B. 50 B-2. Comparison of Field B predicted pressures, assuming no water movement with observed pressures. 58 B-3. Comparison of Field B pressures predicted using thick sand model (infinite aquifer) with observed pressures. 59 B-4. Comparison of Field B pressures predicted using radial model (infinite aquifer) with observed pressures. 60 B-5. Comparison of Field B volume ratio predicted using radial model (infinite aquifer) with volume ratios calculated from field data. 61 C-l. Areal sketch showing the boundary of Field C. 64 C-2. Comparison of Field C predicted pressures, assuming no water movement with observed pressures. 75 vii

LIST OF FIGURES (Continued) Figure Page C-3. Comparison of Field C pressures predicted using linear model (infinite aquifer) with observed pressures. 74 C-4. Comparison of Field C volume ratio predicted using linear model with volume ratios calculated from field data. 75 D-l. Areal sketch showing boundary of Field D. 81 D-2. Comparison of Field D predicted pressures, assuming no water movement with observed pressures. 89 D-3. Comparison of Field D pressures predicted using radial model (infinite aquifer) with observed pressures. 90 D-4. Comparison of Field D volume ratios predicted using infinite radial model with volume ratios calculated from field data. 91 D-5. Comparison of Field D pressures predicted using resistance function with observed pressures. 92 D-6. Comparison of Field D volume ratios predicted using resistance function with volume ratios calculated from field data, 95 D-7. Resistance function vs. time for Field D. 94 E-l. Areal sketch showing boundary of gas Field E, 100 E-2. Comparison of Field E predicted pressures assuming no water movement with observed pressures for an initial pore volume of 440 million cubic feet. 101 E-3. Comparison of Field E predicted pressures assuming no water movement with observed pressures for an initial pore volume of 520 million cubic feet. 102 E-4. Comparison of Field E pressures predicted using thick sand model (infinite aquifer) with observed pressures. 103 E-5. Comparison of Field E pressures predicted using radial model (infinite aquifer) with observed pressures. 104 viii

LIST OF FIGURES (Continued) Figure Page E-6, Comparison of Field E pressures predicted using finite radial model (R=9) with observed pressures. 105 E-7. Comparison of Field E volume ratio predicted using finite radial model (R=9) with volume ratios calculated from field data. 106 E-8o Comparison of Field E pressures predicted using resistance function with observed pressures. 107 E-9. Comparison of Field E volume ratio predicted using resistance function with volume ratios calculated from field data. 108 E-10, Resistance function vs. time for Field E. 109 F-l, Areal sketch showing boundary of Field F. 115 F-2. Comparison of Field F predicted pressures, assuming no water movement with observed pressures. 116 F-3. Comparison of Field F pressures predicted using radial model (infinite aquifer) with observed pressures, 117 F-4, Comparison of Field F pressures predicted using finite radial model (R=10) with observed pressures, 118 F-5. Comparison of Field F volume ratio predicted using finite radial model (R=10) with volume ratios calculated from field data. 119 F-6. Comparison of Field F pressures predicted using resistance function with observed pressures. 120 F-7. Comparison of Field F volume ratios predicted using resistance function with volume ratios calculated from field data. 121 F-8. Resistance function vs. time for Field F. 122 G-l. Areal sketch showing boundary of Field. G. 127 ix

LIST OF FIGURES (Concluded) Figure Page G-2. Comparison of Field G predicted pressures assuming no water movement with observed pressures. 128 G-3. Comparison of Field G pressures predicted using radial model (infinite aquifer) with observed pressures. 129 G-4. Comparison of Field G volume ratio predicted using radial model (infinite aquifer) with volume ratios calculated from field data. 130 G-5. Comparison of Field G pressures predicted using resistance function with observed pressures. 131 G-6. Comparison of Field G volume ratio predicted using resistance function with volume ratios calculated from field data. 132 G-7. Resistance function vs. time for Field G. 133 H-1. Areal sketch showing boundary of Field H. 139 H-2. Comparison of Field H predicted pressures assuming no water movement with observed pressures. 140 H-3. Comparison of Field H pressures predicted using radial model (infinite aquifer) with observed pressures. 141 H-4. Comparison of Field H volume ratio predicted using radial model (infinite aquifer) with observed pressures. 142 H-5. Comparison of Field H pressures predicted using resistance function with observed pressures and those calculated assuming no water movement. 143 H-6. Comparison of Field H volume ratio predicted using resistance function with volume ratios calculated from field data. 144 H-7. Resistance function vs. time for Field H. 145 x

ABSTRACT The purpose of American Gas Association Project N031, "Engineering Studies on Movement of Water in Contact with Natural Gas," begun at The University of Michigan in May, 1959, was to investigate the behavior of aquifers adjacent to gas storage reservoirs. The equations and procedures developed during the first year (see that year's progress report) were applied during this second year to typical cases of aquifer water-movement problems for which direct data from fields were available. This work permitted a quantitative evaluation of predictive and simulative accuracy of the equations and methods developed as well as their practical applicability to various cases. xi

INTRODUCTION When water drive is present, volumetric studies of gas production or storage reservoirs, simulation of inventory gas-pressure behavior, optimization of production-pressure schedules, development of gas storage bubbles in aquifers, and of many other reservoir engineering problems of practical significance must take into account the nature and extent of the water movement at reservoir boundaries. A broad investigation of various problems encountered in the movement of water in contact with natural gas was initiated by the American Gas Association as Project N031 in May, 1959. The first year of the research project was devoted to defining various geometric models and problems encountered in the field and developing solutions and computational procedures permitting the prediction of water movement for various cases. This work was reported in the first annual report issued in September, 1960* The models of gas fields and surrounding aquifers studied during the first year included: 1. horizontal radial flow; 2. radial flow with thickness parameters; 3. hemispherical flow; 4. reservoirs bounded by elliptic geometry; and 5. multilayered sands. In addition to the above, where the geometry is the controlling factor, a generalized method for analyzing the reservoir performance from direct field data on that particular reservoir was developed and reported as a special computational procedure to be applied to actual field case studies, To check out, test, and evaluate the equations and procedures developed for predicting the water movement, a program based on study of analysis of direct field data was initiated early during the second year of the projecto Several gas storage and producing companies were contacted and visited with assistance from AGA and API. As a result, field data from several companies operating gas fields subject to water drive have been received, screened, and selected for evaluation of various theories and methods developed during the first year. Table I below indicates the data required and used for field studies on various models, The field case studies completed during the past year included eight fields analyzed by digital computational procedures and mathematical models developed during the first year on the project. These fields are located in Michigan, Kansas, Illinois, New York, and the Gulf of Mexico, 1

TABLE I DATA REQUIRED AND USED FOR FIELD STUDIES ON VARIOUS MODELS Aquifer Characteristics Type of formation Lateral extent Average thickness, variation range Average porosity, variation range Average permeability (core data), variation range Permeability (in situ) from well tests Formation and liquid compressibility (if known) Vicsocity of liquid Density of liquid in aquifer-undisturbed Initial pressure or water level Depth of the aquifer sand Data on Producing or Storage Fields a. Field geometry Location: state, county, township Type of formation Nature of structure, contour map with water level, cross sections Number and location of gas wells Number and location of observation wells b. Gas sand formation and fluid properties Porosity Permeability Reservoir temperature Gas gravity Reservoir depth Baseconditions for gas measurement Gas compressibility factor data c. Data on production-pressure history of gas reservoir Initial discovery pressure with datum Reservoir formation or well head pressures (weekly or monthly time increments) Inventory (cumulative) gas vs. time 2

Radial, Linear, and Thick Sand models and the Resistance Function method have been used to predict, compare, and evaluate the equations developed for simulating the actual reservoir behavior observed. To ascertain the nature and relative degree of effectiveness of water movement, the performance predicted for each case has been compared with performance computed on the basis of no water movement. These field studies are described in detail in this reporto It may be noticed that four sections precede the major section of this report entitled "Field Case Studies." These sections include; Analysis of Reservoir Performance Based on Field Data (The Resistance Function Method), The Linear Flow Model, Equations Common to the Various Methods, and A Summary of Working Equations for Radial and Thick Sand Models. 5

PART I. CALCULATION PROCEDURES DEVELOPED, EXTENDED, AND IMPROVED DURING THE SECOND YEAR

As work on field studies progressed during the second year of the project it became necessary to develop some additional calculation procedures. The first major phase of this development concerned the resistance function method, which was improved in its practical applicability and predictive accuracy since last year's report. The second phase saw the development of the Linear Modelo In addition, a list of equations common to most of the methods and a summary of the calculation procedures for the Radial and Thick Sand Models are included in this section, A, ANALYSIS OF RESERVOIR PERFORMANCE BASED ON FIELD DATA A calculation procedure for predicting performance in water-drive fields without the usual assumptions of simplified aquifer geometry, homogeneity, and isotrophy is presented here, A characteristic curve called the "resistance function" is calculated from field-performance data. Its extrapolation allows prediction of future water movement and pressure behavior when a production-injection schedule is given or assumed, A previous weakness of the method due to its sensitivity to small errors in pressure data has been eliminated by "fitting" the radial model for an infinite aquifer to the first few data points. The best fit of this mathematical model yields the initial portion of the resistance function, The remainder of the curve (which is less sensitive to these errors) is computed from an algebraic rearrangement of the superposition equation using field performance data directly. Calculation of the resistance function is unstable. However, suitable damping based on theoretical restrictions on the shape of the curve restores stability at the expense of a slight loss of accuracy. In its present form, the method is suitable for processing on a digital computer without intermediate manual operations. 1o DEVELOPMENT OF EQUATIONS Solutions to the partial differential equations for pressure drop at the gas-water interface of an aquifer as a function of time and water influx rate for various field geometries may all be expressed in the general form: AP = Klewf(tD) (1) 7

This equation gives the pressure drop for a constant rate of influx, ew. Both K1 and the function f(tD) are determined by the physical properties and geometry of the system and are different for each aquifer shape. They are "known" for the simplified geometries when the aquifer properties are given. For the radial model which is used in determining the initial portion of the resistance curve: K1 = 25.l5t/(Khf) (la) 0.00633 KAt tDn = K2 n = 0 —— 006 (lb) p 2crb where n indicates the nth time increment, each of which has a duration of At days. The function f(tD) has been solved by Van Everdingen and Hurst and is presented in tabular form. l,24 See Table IV, page 32 for nomenclature. For the case of a nonhomogeneous anisotropic aquifer with an irregular geometry, we assume that some function of time,Klf(tD), exists. The problem is to find it. Equation (1) may be modified by the superposition principle to account for a nonconstant water influx rate: t f(tD) AP = K1 ew(tD-T) dT (2) where tD is "dimensionless time" and r is a dummy variable of integration. Equation (2) may be approximated by an equivalent finite difference form: n AP = K1 ew(n+lj)AtD jAtD ( j=l Any desired degree of accuracy can be obtained by reducing the size of the dimensionless time increment, AtD. The function f(tD) may be replaced by another function g(t), where t is actual, rather than dimensionless time. If K1 is combined with this new function inside the summation sign and At is taken as unity then Eq. (3) becomes: n APn = X ewn+l.j'AZj (4) j=l 8

where AZn ZnZnl = KlAgn and An - Po-Pn As used in Eq. (4) ewn is the average water influx rate during the nth time interval. This rate is equal to the decrease in the volume of the gas bubble during the interval: Vn-l-Vn 1 a VO Pbn- a VO Pb Gpn T ewn = + b - b - + j At At Pnl /p+b ) / P /-+b b (5) The constants a and b arise from a linear approximation of the variation of the gas compressibility factor with pressure, z = a+bP, We can write Eq. (5) more compactly by defining a new variable, Sn: Vo PbT Sn = - G p (6) At( -+b) AtTb n Po where Gp is the cumulative gas produced, SCF at pressure and temperature bases Pb and Tb, respectively. Equation (5) now becomes: ew = (a/Pn.- + b)Snl - (a/Pn + b)Sn (5a) Our problem now is to calculate Zn as a function of time. Toward this goal the first and last terms of the summation of Eq. (4) are removed from under the summation sign and the equation is solved for ewn: n-l ~n = --- ( * Po - P n - ewZ' ^n —j ~ ~~ (A) ewn - AZ Pn ew AZn ew n+l-j AZ (7) j=2 Now ewn is eliminated between Eqs. (5a) and (7) and the resulting equation is solved for AZn: n-1 AZn = AZ + + AZPO - + ) + b)Snl AZ L w~ j=2 (8) - When Z is plotted vs. time the curve must pass through the origin, ioeo, Zo = 0; and it must have a positive slope which must either decrease or remain constant with increasing time. Therefore, theoretically, AZn cannot be greater than AZn-l nor less than, zero. However, small errors in the pressure data will cause violation of these rules, Since the calculation of AZn depends upon all the previous AZ's, these errors propagate themselves, resulting in an unstable computation. Thus it is mandatory to place restraints 9

on the otherwise direct calculation of the resistance function. The obvious upper limit, whenever AZn from Eq. (8) is greater than AZn-i, is to set AZn equal to AZnlo A lower boundary is suggested by a limiting condition of the curve corresponding to the radial model for an infinite aquifer. At large times the slope of this curve is proportional to the natural log of time: Zn = A~ n n + B thus AZn = Zn Znl = A n ) and AZn-1 = A In n-l) therefore In (mn) AZn = AZn-1 n- (9) ien (n-2) n-2 The first term of a series expansion of the In terms is (n-2)/(n-1); thus the lower limit for AZn is: AZn = AZnl(- ) (9a) i.e,, if AZn as calculated from Eq. (8) is less than the value calculated from Eq. (9a), then AZn is set equal to the latter value. A moment's consideration will reveal that the above restrictions are too severe. For example, let us assume that we are generating a portion of the Z curve which should actually be a straight line, but due to an error in the pressure data, one AZ is less than the previous values. If the upper limit, as stated above, were applied, all the subsequent AZ's would have smaller values and the Z-curve could never "recover" The same argument applies to the lower boundary in addition to the fact that this particular limit does not necessarily apply to all geometrieso A factor, c, rectifies the situation: (1) whenever AZn > (l+E) )AZn-l AZn = (l+E)AZnl (lOa) n-2 n-2 (2) or whenever AZn < (l-E)AZnl(_), AZn = (le) AZn-i( )o (10ob) An e = 0,02 has been found to work satisfactorily. It should be pointed out 10

that condition (lOa) above gives rise to the possibility of an upward-curving resistance function which violates the rule that the second derivative with respect to time, d2Z/dt2, be zero or negative. Whenever a positive second derivative occurred in the application of this method to field data, a straight line, tangent to the high points of the Z-curve, was drawn producing a new "smoothed" curve. Extrapolation and predictions were based on this new curve. A programming modification is planned which will automatically produce tangents to the Z-curve, continuously back-correcting it, Equations (5a) and (7) are combined and solved for Pn: n Pn2 + AZ1 (- + Sn- - bSn + AZ - P - aSnAZ = 0 Let n [1-i ^ Bn = AZl p1 + b)Sn- - 1bSn + j ewnlj Z P - ~~j=2 and Cn = aSAAZI Therefore P = B + Cn - Bn1) The water influx during the nth time interval is calculated using the smoothed AZn value and the corresponding calculated pressure, Pn: n ewnO =Pn' ewn+lj ~ AZ (12) j=2 2. SUMMARY OF WORKING EQUATIONS AND STEPWISE CALCULATION PROCEDURE (1) Calculate K1 and K2 from best estimate of reservoir field properties K1 = 25.15lt/(Khf) K2 = 0.006328 KAt/(dlcri) 11

(2) Calculate all Sn values to be used from an estimate of the initial pore volume, VO, and cumulative gas production data, Gn Sn = Vo/[(a/Po + b)At] - [PbT/(TAt)] Gp = C - C2~ " n = 1,2Y.oo9 L C1 C2 Gpn (3) Compute dimensionless time values corresponding to the first few data points tDn = K2 n n, = 1,2,..., M (4) Using linear interpolation and values from tables 1' calculate dimensionless pressure values: (tDn -tD ) Pn = Pj=l + - (Pj- Pjl) (tDj-tDj-) where j and j-l refer to the table values of p(tD) and tD nearest to and respectively greater and less than tDno (5) Calculate Ap1... ApM APn = Pn - Pn-l (6) Compute Pn and ewn alternatively, starting with P1: Pn = B Cn - B where Bn = T[(a/Pnl- + b)Sn-l - bSn]KiAp - Po + K1 j ewn+lj Apj j=2 J and Cn = aSnKlApi n ewn = (Po - Pn - K1 ewn+lj Apjj/KAp n = j=2 / n = 1,2,o.., M 12

(7) Find the percentage deviation between calculated pressures and selected pressure data X Pcalci-Pdatai % deviation = 100 Pdatai where Pi's are selected from P1..oPM, but do not necessarily include all the values. (8) Change the value of K1 and repeat steps (6) and (7). Repeat until the minimum deviation between calculated and data pressures is obtained. (9) Compute AZoo. AZM AZn = Kloptimum Pn (10) Find AZM+1 n-l AZn = Po - P - wn+lj AZj + [(a/P + b) Sn - (a/Pn-l + b)Snl]AZ j=2 The asterisk denotes pressure data. Pressures without it are calculated. (11) If AZ++1 does not fall within the following limits set it is equal to the lower value if below or to the higher limit if above: n-2 (l-) (n-l) AZn-1 < AZn < (1+E) o AZn_1 A value of C = 0.02 has been found to work satisfactorily. (12) Calculate PM+l using the corrected value of AZM+1 P = I4B2 + Cn - Bn where n Bn = - [(a/Pn_1 + b)Sn - n - Sn]Z P + ewn AZ 2 - wn+l-j A j=2 13

and Cn = aSnAZi (13) Compute ewM+l n ewn = P - Pn - ewn+lj. AZ /AZ j=2 (14) Repeat steps (11), (12), and (13), for n = M+2...L. (15) "Observed" and calculated pore volume ratios may be calculated for n=l... L n o\ +a SnAt Vo obs n Vo n (V)cac 1 - (At ew VO Vo c alc V j=1 (16) If a finite aquifer is indicated (as has been the case in all the present field studies), i.e., if a plot of Zn vs. n approaches a straight line as n approaches L, extrapolation is obtained by setting AZL+l i..AZN all equal to AZL. Otherwise AZn+1 = AZn for n = L... N. (17) Calculate SL+l.oo SN (see step 2) using projected gas production schedule. (18) Predict pressures and water influx rates using the extrapolated AZn values in steps (12) and (13). B, LINEAR FLOW MODEL The partial differential equation governing one-dimensional, unsteadystate flow of a compressible liquid through porous media is 14

a2p = LP ~ E (1) ax2 K at Definition of the new variables P = P- (2) 9 = Kt/(tc) (3) where po is initial aquifer pressure, allows (1) to be written 82P _ (4) ax aQ The solutions to Eq. (4) for constant boundary pressure and constant water influx rate across the boundary are obtained as follows. 1. CONSTANT-PRESSURE CASE Here the initial and boundary conditions for solution of Eq. (4) are P(x,O) = 0 (5) P(oo~,) = O (6) P(O,Q) = Ap (7) If P(x,s) denotes the Laplace Transform of P(x,G) with respect 9, then transformation of Eq. (4) yields P - sP = 0 (8) dx2 for which the solution satisfying Eqs. (6) and (7) is - Ap - )s x P = - e (9) The rate of water influx into the reservoir (at x = O) is KA 6p KA 6P q = KA~) = KAP) = q(G) ax/x=O 1 axx=o 15

where A is the cross-sectional area of the linear flow system. The Laplace Transform of q is KA o P q(s) = ax=o (10) t The cumulative water influx into the reservoir is We = qdt or *J o We = fqd (11) The Laplace Transform of We is We We 2 1- Clc KA 1 oP K pN e K = K s xx=o12) Inserting ) from Eq. (9) into Eq. (11) yields ax/x=o W = -OcA sN (13) and taking the inverse transform of this expression yields We = -2 cA(po-p) AN (14) which is the constant-pressure-case solution of Eq. (4). Thus the cumulative water influx We is proportional to the square root of time. 2. CONSTANT-RATE CASE The initial and boundary conditions for the constant-rate case are P(x,O) = 0 (15) P(oo,O) = o (16) a)x, = (17) ax /x=o AK where q is the constant rate of water influx into the reservoir. 16

The solution to Eq. (8) for these conditions is P(x,s) = - -- 1- e (18) AK S3/2 and the inverse transformation of this function is x2 P(x,-) = - - e 4 AK The reservoir pressure at x = 0 is thus given as Po - p 2 (19) VNf AK When field units are used, = o.o-o628 Kt t in days, c vol/vol psi, ~ is a fraction, q is in ft3/day, A in ft2, p in cp, K in md, and Eq. (19) becomes Po- P = 178 q \ a (20) which is the constant-rate-case solution of (4). 3. VARIABLE-PRESSURE SOLUTION When the reservoir pressure is not constant but rather a specified function of time, the superposition principle can be applied to (14) to obtain i=j -1 We. = 1 XcA j AP E (j-i) (21) X e j = i=O where Wej = cumulative water influx at time jAt, ft3 O. o006328K AG = transformed time increment, --- - At APi = Pi-lPi+l; P-1 = Po Pi = reservoir pressure at time iAt, psia At = time increment, days. 17

Equation (21) gives comulative water influx for any known or specified pressure schedule, Po, pi, P2,.... 4. VARIABLE-RATE CASE Application of the superposition principle to the constant-rate-case solution, (20), yields i=j-1 ^Po - Pj =- LAKAt i=O where pj = reservoir boundary pressure at time jAt, psia At = time increment, days AVi = 2Vi-Vi+l-Vi_1; AVo = Vo-VI Vi = reservoir pore volume at time iAt, ft3 Equation (22) gives reservoir pressure pj whenever the schedule of porevolume variation is known or specified. Co EQUATIONS COMMON TO THE VARIOUS METHODS Equations used to calculate terms common in several programs are included below. These calculations concern: (1) bottom-hole pressures when wellhead pressures are given; (2) inventory gas in place (lb-moles) from initial pore volume and production-injection history; (3) pore-volume ratiosfrom calculated and observed pressures; (4) dimensionless time; and (5) prediction of pressure when no water movement is assumed. 1. CALCULATION OF BOTTOM-HOLE PRESSURE, PSIA Field-pressure data are usually expressed in psig or psia (wellhead). The pressure in the reservoir is determined by the following procedure. If the data are given as psig (wellhead), add 1407 to obtain psia (wellhead) and calculate bottom-hole pressure by 18

Mh/(l44 zRT) Pbhp = Pwh e where Pbhp = bottom-hole pressure, psia Pwh = wellhead pressure, psia M = molecular weight of gas = gas gravity x 29 h = depth gas well below surface, feet R = 10.73 (psia)(ft)3(lb-mole)1(oR)-1 T = reservoir temperature, ~R z = gas compressibility factor 2. CALCULATION OF GAS IN PLACE The number of pound moles initially in a gas reservoir, no, is calculated by PoVo PoVo Vo no = = (2) zRT (a+bPo)RT (P+b)RT where no = initial gas present in field, pound-moles Vo = initial pore volume, ft3 a,b = constants in correlation z = a + bP The cumulative gas in place, nj, for any time step "j" is given by Pbase Gp. nj = - b s RTbase where 19

Pbae = base pressure for measurement of gas, psia Tbase = base temperature for measurement of gas, ~R Gp. = cumulative gas produced (gas injected is defined as negative production), standard cubic feet j = "j" time step 3. CALCULATION OF PORE-VOLUME RATIO The ratio of the pore volume at any time step "j" to the initial pore volume is determined as follows: V. znjRT (a+bPj)njRT.............. ---- u.-~ ~(4) Vo PjVo PjVo where Pj may be either an observed or calculated pressure. 4. CALCULATION OF DIMENSIONLESS TIME Dimensionless time, tD, and the incremental dimensionless time, AtD, used in the radial, thick sand, and hemispherical models are given in terms of field units by tD 00o633 Kt ( tD cr (5) and AtD =O. 00633 KAt 6).tD (6) Micrab where tD = dimensionless time AtD = incremental dimensionless time K = permeability, millidarcys t = time, days At = time per increment, days j = viscosity, centipoise 20

= porosity, fraction c = compressibility of aquifer rock and water, vol/vol psi rb = radius of gas reservoir or inner radius of aquifer, feet Note that tj = jAt (7) and tDj = jAtD (8) 5. PREDICTION OF PRESSURE ASSUMING NO WATER MOVEMENT Pressures assuming no water movement are calculated from the gas law, p = Vo (a+bPj)nRT () Vo Vo Solving (9) for Pj yields anjRT ( p = - (10) Vo-bnj RT D. SUMMARY OF WORKING EQUATIONS FOR RADIAL AND THICK SAND MODELS The developments of the Radial Flow Model and Thick Sand Model were presented in our 1960 annual report. The Radial Flow Model has been modified to correct for the expansion and contraction of water under the gas bubble. A brief development of the working equations in their present form is given below, 1. RADIAL FLOW MODEL Water influx for the constant terminal pressure case is calculated by superposing the effects of each constant time step, giving i=j-1 Wej = icrihf APi Q(.j-l) At + Th^cr.f1(Po - Pj) (1) i=O 21

where APi = Pi-i - Pi+l and -1 = Po (2) The first term on the right-hand side of Eq. (1) gives the water influx into the reservoir boundary; the second term represents the compression or expansion of water under the gas reservoir. The dimensionless cumulative influx Q(tD is tabulated in the papers by Van Everdingen and Hurst,4 Chatas,l and the Handbook of Natural Gas Engineering2 [Tables (10-4) -(10-8), p. 426]. The dimensionless time is given by tD ~.00o633 Kt tD -r 2 -- (3) ipcrb In Eqs. (1), (2), and (3) above, the terms in field units are given by = fractional porosity c = formation and water compressibility, vol/vol psi h = formation thickness, ft K = formation permeability, millidarcys t = actual time, days p. = water viscosity, centipoise rb = inner aquifer radius, ft f = fraction of complete circle represented by reservoir fl = fraction of cylinder rbgh of water under gas reservoir We. = cumulative water influx at time jAt A volumetric balance gives Vj = Vo - Wej (4) 22

where Vo is the initial pore volume and Vj the pore volume at the j time step. Vj may be calculated using the gas law v -= - (5) Pj and the gas compressibility constant z; by z = a+bPj (6) Combining (1), (2), (4), (5), and (6) we obtain j-2 Ja+. njRT = V o - ch^crbf APiQj_ J i=O - th~crbf Pj_2Q1 - rh/crbfl Po (7) + Pj(rxhcr f Q1 + Trhcrbfl) Solving (7) for Pj yields Pj = C + 2C1 + C2 (8) where j-2 Tthcr2 [f C APi Qj-i+fQl Pj-2+flPo]+bnjRT-V i=O C1 =.iO (9) 2 chcrbf Ql+2rchbcr-bf and anjRT 23 25

2. THICK SAND MODEL The superposition principle applied to the equation governing the reservoir pressure drop for the thick sand model, constant-terminal-rate case, results in i=j-1 Po - Pj = 50~2 X (11) P0 - PJ -e - -'A P(j-i) (ii) Krb R i=O where Pj = reservoir pressure at time t = jAt, psia Aei = (e)i+l - (e)i (e) i = average rate of water influx during the time increment from (i-l)At to iAt = Vil-Vi, ft3/day Pj-i = the value of P at dimensionless time tD = (j-i)AtD At = time increment chosen for representation of ew as a step function of time, days AtD = (.00633 KAt)/biLcr, dimensionless time increment The dimensionless pressure drop, Pj, is listed as a function of dimensionless time in Table II for tD less than 10, For dimensionless time tD greater than 10, 7 can be calculated from the equation P = A + ~ In tD (12) where the value of the constant A is dependent upon M, as given in Table III. M is defined by M = h/(rbKR) (13) where KR is the ratio of vertical to horizontal permeability in aquifer (dimensionless) Substituting the equivalent for Aei, (2Vi-Vi+l-Vi-l)/At, in Eq. (11), combining this with the equation of state for gases, and solving for Pj, one obtains: Pj = C1 + lC - C2 (14) 24

TABLE II DIMENSIONLESS PRESSURE DROP VS. DIMENSIONLESS TIME FOR THICK SAND MODEL tD = dimensionless time = Kt/jcr-b M = h/(rbKR) tD P M =.0 M =.1 M = 3 M =.5 M =.7 M =.9 M = 1.0.1 1.433.741.329.287.285.285.285.2 2.675 1.362.536.412.383.377.377.3 3.649 1.849.699.509.454.436.436.4 4.464 2.257.835.591.512.483.483.5 5.166 2.608.952.661.562.522.521 1 7.693 3.871 1.373.914.743.663.648 2 10o590 5.319 1.856 1.204.950.824.792 3 12.442 6.245 2.164 1.389 1,082.927.885 4 135789 6.919 2.389 1.523 1.178 1.001.952 5 14.848 7.448 2.565 1.629 1.254 1.060 1.005 6 15.722 7.885 2.711 1.717 1.316 1.109 1.049 7 16.465 8.257 2.835 1.791 1.369 1.150 1.086 9 17.684 8.866 3.038 1.913 1.456 1,218 1.147 10 18.198 9.123 3.123 1.964 1.493 1.246 1.173 TABLE III PARAMETERS M AND A FOR THE THICK SAND MODEL M A.05 6.5644.10 3.3065.30 1.1846.50 0.8010.70 0.6622.90 o.6000 1.00 0.5911 25

where`j -2 C1 = - 1 ( -50.21 ) AVi ji + (2Vj_1 - Vj_2) - i RTnjb - Po Kr, Atf i=o Kt^~~~~ ( ~lI~(14a) and C2 =( -52 RTnja (4b) Krb'/r Atf and AVi = 2Vi - Vi+l Vi-l (14c) where Vi = reservoir pore volume at time t = iAt, ft.3 If tD is less than 10 and the value for M is not given in:the table, dimensionless pressure, P, can be calculated by a LaGrangian polynomial fit of three table values (Table II) whose values of M are nearest to the actual M: (M-M2) (M-M3) (M-M1) (M-M3) (M-M1) (M-M2) P =1 -......p +...P3,M1-M2) (M1-M3) (M2-'M1) (M2-M3) (M3_M1} (Ms-M2) Equation (15) is applied twice, the first time for tD (Table II) less than the actual value, and then for tD greater. Finally the dimensionless pressure drop corresponding to the required M and tD is obtained by linear interpolation. As an example, suppose that M = 0.65, and tD = 3.5~ First table values of P corresponding to tD = 3 and M =.5,,7, and.9; 1.389, 1.082, and 0.927, respectively, are used in Eqo (15) Thus: M = o.65 M1 = 0.5, P1 = 1.389 M2 = 0.7, P2 = 1.082 Ms3 = 0o9, PS = 0.927 26

and P for M = 0,65 and tD = 3 is found from Eq. (15). Then the same equation is used to solve for P at tD = 4: M = o.65 M1 = 05, Pi = 1.523 M2 = 0.7, P2 = 1.178 M3 = 0.9, P3 = 1.001 The P for tD = 3.5 is, by linear interpolation, midway between the two values calculated using Eq. (15). 27

PART II. FIELD CASE STUDIES

The eight fields studied individually and in detail by methods developed during the first year will be referred in the following as Fields A, B, C, etc., without specific mention of the names, location, and without designating the particular operating company. The descriptions of each of these fields include all the available data on geology, geometry, and past production-pressure history. The mathematical procedures and specific calculations are shown in detail once for every mathematical model. A summary of parameters used in the application of the models and resistance function to these field studies is presented in Table IV. A summary of the values of these parameters is presented at the end of the report in Table V (see page 140. It may be noted that two K1 values are listed for each model in Table V. The first value is estimated from physical parameters of the field and aquifer and the second value is that which gave the best prediction of the reservoir behavior. 31

r-{ 0 ed o H -)0 - -P Fzl H *H H pi Pi P4 CO C) CC Eij oj \9 ca o Ij a ~> M s C z H ~.P >i UZg~~~~~ Q I s<1 Us I - I o r'r" C\ ] 0). Iz I? v r is;^ r:- rd o co.i 1H C D~" r d ~ W I I ~ r | ~r.- O U1 b ~ $ X W m0 -I r CO C - H LIi 4 O a) o Qo a a) m ~) oI F,.4 a 0 ) Hc rd 0 ( 0Wi ~ ~ ~ 4 P a-) = -P,- C Od 0~^ (U-s~~~~~~~) - -r- -, a 0 pj(- 1 m m, r-I., c4 4-' d Ea P |s l | $ g = h | g | H p-'H CHCH32 H I ~o0 <I. eel kP 0 - 0p O a);j aiC -H Hfe*^*^~~~~~~~~~~~H *HEaC) R-4 - oE a)-a a) *H H H $ - C 4 d w 0 H a) a) AH PH EH'a H C fH r U) H^ a) a) s< 5 H a) C) a) 03 4')d *m H< 52

FIELD A Field A, one of two large gas reservoirs located under the Gulf of Mexico off the coast of Louisiana, was discovered in 1951. The two fields we located on opposite sides of a fault running through a structural dome and have a 400-foot vertical displacement with no pressure or aquifer communication across the fault. The discovery pressure of Field A has declined from 3745 psia (bottom hole) to 3175 psia after producing over 172 billion standard cubic feet of gas. To date, this field has only been produced and its location makes conversion to storage operation doubtful. Figure A-1 shows a plan sketch of Field A. 1. GEOLOGY Field A is a structural dome bounded by a fault, located 7400 ft below the Gulf of Mexico. The pay is fine- to medium-grained, moderately cemented sandstone separated b several shale lenses. The shale has numerous breaks and cannot be correlated from well to well. The location of this field makes drilling and coring operations very expensive; thus the available geological information is limited. 2. FIELD DATA The reservoir and fluid properties of Field A are given in Table A-1 and the production-pressure history in Table A-2. The initial pore volume was found by comparing the predicted and observed pressure for several selected initial pore volumes. The best estimate for the initial pore volume using both the radial model and resistance function method is 3850 million cubic feet. 3. CALCULATIONS (a) Application of Radial Model to Field A.-The data needed for the application of the Radial Model to Field A are taken from Tables A-1 and A-2 and listed below. 33

I WATER 7Z G AS / ha / AWATER WATER Fig. A-1. Areal sketch showing boundary of Field A. 54

TABLE A-1 GAS FIELD A DATA Aquifer Characteristics Type of formation Sandstone Porosity 29% Permeability 1474 millidarcys Thickness 170 ft Depth to top of aquifer 7929 ft Viscosity of liquid in aquifer 0.5 centipoise Compressibility of brine and aquifer 7x106 vol/vol-psi Data on Gas Field a. Field geometry Location Gulf of Mexico Type of formation Sandstone Nature of structure Structural dome b. Gas-sand formation and fluid properties Gas compressibility correlation, z = a + bP Constant a 0.776 Constant b 4.17x10-5 psi 1 Depth to top of sand 7800 ft Reservoir temperature 640~F Reservoir thickness 129 ft Base conditions for gas measurements Pb = 14.7 psia Tb = 520~R c. History of gas reservoir Discovery date 1951 Initial discovery pressure 3745 psia (bottom hole) Initial pore volume 3850x10 cu ft 35

TABLE A-2 GAS INVENTORY AND PRESSURE HISTORY FOR FIELD A Pressure Base = 14.7 psia Temperature Base = 520~R Cumulative Gas Production P Wl B m, P, Well Bottom, psia Date MMSCF at End of Month at End of Month 1951 Feb 3745 Aug 2,680 3731 1952 Feb 7,254 3708 Aug 12,342 3688 1953 Feb 17,497 3670 Aug 23,162 3645 1954 Feb 30,898 3621 Aug 39,909 3595 1955 Feb 49,474 3555 Aug 60,471 3508 1956 Feb 74,126 3470 Aug 88,523 3430 1957 Feb 103,237 3383 Aug 117,024 3324 1958 Feb 135,775 3265 Aug 154,123 3220 1959 172,591 3175 36

i =.29 hgas = 129 ft haquifer = 170 ft c = 7x10-6 vol/vol psi K = 1474 millidarcys = 0.5 cp a = 0.776 b = 4.17x10-5(psi)At = 182.5 days f =.425 fi = (170-129/170) = 0.24 Tr = 640~R Vo = 3850x106 ft3 r2 = 77x106 ft2 AtD is calculated from 0.00633 KAt 0.00oo633(1474(182.5) AtD = ------- = --------------------- = 22 H/crb2 (0.5) (.29) (7X10 6) (77X106) and tD for each time step is determined by tD(l) = j(At) = 1(22) = 22 tD(2) = 2(22) = 44 tD(3) = 3(22) = 66, etc. The values of QiAtD for an infinite radial model corresponding to the above tD values are interpolated from Table 10-4 in the Handbook of Natural Gas Engineering and are listed in Table A-3. The initial number of lb moles, no, is calculated from Vo 389oxlo6 no = PRT(Tp b) (10.73)(640) (376 +0.-0000417) = 2.252x109 lb moles 37

0 O H 0o o co -o o o H O - 0 L\ 0 o0 0 H-I H 0 o -- rt4 o o o o0 \ c r-I Cf F- -I -_: H Co - 00 2 0 c0 ON o o O c CO Co C o N o oN H 0 0 \ ON DO\ ON ON O o o o o ON N ON ON O O ro i 0\ 0 0 0\ 0\ 0 0\ 0\ 0 00 P\ \ 00c i0) Ea o V rd c -0O cO 0 t — 0 C 0 CO ON 0 N. L N \ n L-C \1-D 0 0O 0 ON O N\ C- \0 N\ -H - K\N \OKN \ \0C CO 0 H-4 0 ON\' OP- 0 \ 0\ O \ 0\ O\ C co t- \irt- Kl i- -') 0\ \ 0\ 0\ 0 \ 0\ 0\ 0\ 0\ 0\ 0\ 0 \ 0) 0 0 0.0 - O o 0 C- O C 0 10 00 o dS Po 01 H 0 0 0 H 0-0 CO H ON C- CC\O C- H C cE o M i rco t-l_ o —t t j 0\ rc \ C- o C0 - o C- o c o 0-o A PR i -' t ^- t —O -o 0o0 U\ [c \ -o - Co j j ra)12 Id n 3d i o 1- 1H ON o0 \ C0j- t - -t c r c- o CH \o IH\C C N co o coH0 Ko- \ r-O O C CO [CO nCO r O - CO Hl K4 f- C O \C Ea-P -- t- \C \0 D \-) L) 0n -n -- - 0 C\0 rO C 0) ON\ O O [C\ C C N ON N O N [C [C pC\ [ co\ co C-= / - \o 0)H0 0 F~1Xt ~ ~ ~ o co co co co c 0 01 co co Co H H 0H \ co HO coo O0 Co 0) -0 - 0 H O O O-1 o HI0 O oo 0 ON 0-0 [C 0 CO ON C^ H C O oN H C- 0 0 0 CO0 C O. -c o 0 C H coo N- 0 t C- C \ H, \ H H8 H O CQ -H ~ j C~j C~j ~ H r — i o 0\ QIr\o co c o

and the cumulated gas in place for each succeeding time step from Pbase Gp. nj = no.... RTbase n, = 2.252x109 - (14.7)(2.68x109) 2.245xl09 lb moles (10.73)(520) n2 = 2.252x109 - (14.7)(7.254x109) 2.233x109 lb moles (10.73)(520) The remaining values for the gas inventory are tabulated in Table A-5.3 The pressure can now be predicted from Eqs. (17), (18), and (19) for j greater than 1. For j = 0, 1, Po(calc) = Po(initial) = 3745 psia Pj(calc) = C1 + NcF+ C2 C- = K1(fQPj_-2+f Po) +bnjRT-Vo 2TK1( fQl+f1) anjRT iKl(f Ql+f1) where K1 = hocrb. The value of K1 was varied until the absolute deviation observed and calculated pressures was a minimum. Calculations for K1 = 3750, the optimum value, are given below. P1 = C1 + vC- + C2 C1 = T(3750) {(.425)(13.17) (3745) +.24(3745) + (4.17xlO-5) (2.245x109) (10. 7) (640) - 3850xl06/ [(2r) (3750) {(. 425) (13. 17) +.24)] _ (0.776)(2.245xl09) (10. 75) (420) C2 x(3750){(.425)(13.17)+.24) P1 = 3732.0 psia

P2 = C1 + NvC + C2 C = fK l(fAPoQ2+fQlPo+flPol+bnjRT-Vo 2TK1(fQi+fl) anj RT C2 = =tK1(fQl+f1) APo = PO - P1 = 3745 - 3732 = 13 C1 = ir(3750) [.425(153)(22.4) +.425(13.17)(3745) +.24(3745)] + 4.17xlO-5(2. 33x1l0) (10735) (640) -3850xl6 / [2n(37.50) ((.425) (13,17) +.24)] 2^ =.776(2.233) (10073(640) [ T(3750) (. 425) (13o17) +t(3750) (24) ] P2 = 3711 3 psia This procedure is repeated until all pressures are calculated (see Table A-3). The pore volume ratio is predicted by Vj fa+bPj V- = Pj J njRT/Vo V, _ _j.776+4.17x105(53732) V_ =..7.76+4o7yx0-5(3732) (2.245xlOs)(10.7)((640)/385Oxl06 = 0,998 VO 3732 The remaining pore volumes, calculated for the predicted and observed pressure, for each time step with the respective gas volume (lb mole), are also tabulated in Table A-3. The pressures, assuming no water movement, were calculated from Eq. (10). 40

an RT Pj (no water movement) = -- - Vo-bnjRT p _ (.776)(2,245x109) (0.73) (640) 7509 P1 3730.9 3850x106-(4.17xlO-5)(2.245x109)(1073) (640) A summary of all the numbers calculated in processing Field A data by the radial model is given in Table A-3. 4. RESULTS The performance predicted for a producing reservoir in which water movement is not considered shows lower pressures than actually observed (Fig, A-2). Application of the radial model, the closest approximation of the reservoir shape (Fig. A-l), with an infinite aquifer to the field data gave a predicted performance almost identical to that observed (Fig. A-3). This agreement is also reflected in the volume ratios calculated from these pressures (Fig. A-4). As all the predicted pressures are in good agreement with the observed, the aquifer is either infinite or the boundary of the aquifer has not had time to influence the pressure history. The resistance function method (see Field D for example calculation (Fig. A-5) also gave close agreement between the calculated and observed pressures. This agreement was also reflected in the corresponding pore volume ratios (Fig. A-6). The slope of the resistance curve of Field A becomes constant toward the end of the history, indicating that the reservoir if of finite extent. However, the effect of the boundary is just beginning to be felt, and the deviation from infinite aquifer behavior is not significant. The resistance curve for Field A is shown in Fig. A-7. 41

3800 3600 - m\ \ a. 3400 I. > 3200 i la POINTS CALCULATED FROM FIELD DATA W - o POINTS CALCULATED ASSUMING NO WATER MOVEMENT ESTIMATED INITIAL PORE VOLUME OF 3000 GAS SAND, Ve = 3850 MMCF 2800 I I I I I I I I I 1 0 1 2 3 4 5 6 7 8 TIME, YEARS Fig. A-2. Comparison of Field A predicted pressures, assuming no water movement with observed pressures. 42

.3400 a. A POINTS CALCULATED FROM FIELD DATA a: O POINTS CALCULATED USING RADIAL MODEL _ WITH INFINITE AQUIFER > 3200 U ESTIMATED INITIAL PORE VOLUME OF GAS W SAND, Vo = 3850 MMCF K/"+crb2 19 (md)(cpfl(psi)(ft) " h+crb = 3750 (f t?/( psi ) 3000 2800 i, I II I I I 0 1 2 3 4 5 6 7 8 TIME, YE A R S Fig. A-3. Comparison of Field A pressures predicted using radial model (infinite aquifer) with observed pressures. 43

1.0 -I. < I\ 2.3 3'.96 -- 0 o.95 P POINTS CALCULATED FROM FIELD DATA n 0 POINTS CALCULATED USING RADIAL MODEL m_ WITH INFINITE AQUIFER W 94 ESTIMATED INITIAL PORE VOLUME OF GAS.94 SAND, Vo = 3850 MMCF K//Jicrb = 19 (md)lcpr'(psi)(.ft).93- h+crb = 3750 (ft)5/(psi).92.91 L I I, I,I l I I I I I I I 0 I 2 3 4 5 6 7 8 T I ME, YEARS Fig. A-4. Comparison of Field A volume ratio predicted using radial model (infinite aquifer) with volume ratios calculated from field data. 44

3800 3600 -- - 3400 - POINTS LCULATED FRO FIELD DATA 0POINTS CALCULTED I E TA POINTS CALCULATED USING RESISTANCE FUNCTION ESTIMATED INITIAL PORE VOLUME OF GAS SAND, Vo =3850 MMCF 25.15 u/(khft ) = 5.77 x 10'(psi)(ft)J(six months)I 0.006328 kAt/(u+crbL) = 10.88 (six months)-l 3000 2800 I I I I 1 0 I 2 3 4 5 6 7 8 TIME, YEARS Fig. A-5. Comparison of Field A pressures predicted using resistance function with observed pressures. 45

1.0.98 k-.96 0 a a: A POINTS CALCULATED FROM FIELD DATA o 0 POINTS CALCULATED USING RESISTANCE 0: FUNCTION w.94 <0n ESTIMATED INITIAL PORE VOLUME OF GAS W SAND, Ve = 3850 MMCF ).1 25.15 u /(khfat) 5.77 x IO6( psi) (f t3(six months) 0.006328 kAt/((pcrbt) 10.88(six monthsf).92.9o 0 1 2 3 4 5 6 7 8 T I ME, Y E A RS Fig. A-6. Comparison of Field A volume ratio predicted using resistance function with volume ratios calculated from field data. 46

110 I I'I I 100 90 80 70 < 60_ ESTIMATED INITIAL PORE VOLUME OF GAS SAND, Vo = 3850 MMCF - 50 N 40 30 20 10 0 1 2 3 4 5 6 7 TIME, YEARS Fig. A-7. Resistance function vs. time for Field A. 47

FIELD B Field B, a northern Illinois storage reservoir, was formed by injecting gas into a sandstone aquifer initially containing only water. Development of the reservoir for gas storage was initiated in early 1957 and gas injection started on Nov. 30, 1957. By December 5, 1960, 16,339,900 MCF of gas had been injected. A plan sketch, Fig. B-l, shows a horizontal view of this field. 1. GEOLOGY The storage sand, encounted at an average depth of 2450 feet below the surface, is 2500 feet thick. This sand consists of alternate layers of fine- to very-coarse-grained sandstone with several randomly spaced, noncontinuous, thin, green shale laminations. Imprevious layers of dolomite and shale form the cap rock. The lower two layers of dolomite and shale in this cap rock are more than twenty and forty feet thick, respectively. No faults are known to exist in the immediate vicinity of the aquifer. Although geological information indicates an "infinite" aquifer, this cannot be ascertained without analysis of field-performance data. 2. FIELD DATA A summary of the reservoir and aquifer properties is given in Table B-l and the production-pressure history is tabulated in Table B-2. Although the configuration of the aquifer and gas bubble does not conform to any of the present mathematical models, the field's performance can be approximated by the radial or thick sand model. The mathematical models used impose the restriction that the radius of the gas-water interface may not change with time. Hence one is forced with an awkward situation for an "aquifer" storage field, i.e., one in which the initial gas bubble radius is zero. It is necessary to create some pseudo initial conditions or assign a later time in the field's history, when the gas bubble has grown to a finite size, as the starting point for analysis. The latter route was chosen for field B.'49

WATER G^X S WATER... WATER V/'/ GAS Fig. B-1. Areal sketch showing boundary of Field B. 50

TABLE B-1 STORAGE FIELD B DATA Aquifer Characteristics Type of formation Sandstone Porosity 14.8% Permeability 100 millidarcys Thickness 2500 ft Depth to top of aquifer 2490 ft Viscosity of liquid in aquifer 1 centipoise Compressibility of brine and formation 8x10-6 vol/vol-psi Data on Storage Field a. Field geometry Location Northern Illinois Type of formation Sandstone Nature of structure Structural dome b. Gas-sand formation and fluid properties Gas compressibility correlation, z = a + bP Constant a 0.98 Constant b -0.000083 Depth to top of sand 2450 ft Reservoir temperature 530~R Reservoir thickness 40 ft Base conditions for mas measurements Pb = 14.65 psia Tb = 520~R c. History of gas reservoir Date storage operations started November 30, 1957* Initial discovery pressure 1097.4 psia* (well bottom) Initial pore volume 110x106 cu ft *March 26, 1960, used as initial time in this study. 51

0 pQ c\o Ot>- -l- 0 \ 0 n\t-0- s 0L-\n O\ L0- 0 O ONO OJ CO CO t* oo S O O ooOH C\COCO \ C O 0 Co 0 c cat CU j C\CU C Cj C'1 CJ CJ CN C0 H t r r r- O O O 0O0 0 O r P-1 rl r- HH r-i r-I I H H H Hr- rH Pd,.-1 H 0n HGOU\C-j0\ LC\CxJn K(\ t —z tH- 0t C-z(I — 100\ r H cO JL\ ~rfA) W H \ C\jOJ 0 - d- t co co CO H (- 0 \JC Hr-AC\ M- H C0 0 OO H IJS S oo i OCOCOOO H K oPH -P pc0 0 H r r-lr-HlHOHHH HHH - Hr-AC H H H HHH Hr- HH H HHH S I 0 - D C!- I Pm g I 1 0~~~~~~~~~~~~~0 C P E-! Cf)C pq d 3 01\ COa 7! C-C zico OD-rCCcO \ \ -:J- m-0 0 rH n OOLC —c( M _::i- r\C-j 05 ~) 0 (\JK>\(0C00 z0\ t0- \ — r — O -n\ \ n n HLC \ LD \, \OD r\ - L\\ \O N\ D * p cOH r H-ri - -J C!C H C OJ C\J C C\J CJ CK\ C\J l C\ Cj C jt CJ!j Hr- r- Hr-l li l r i - 1111 DII I1 O I P), 0 CO C 1 CO 0)D~ CM O0\\ -C\0 O- r-ACO -r-C L( C\E i\\ON\ 0"0 O\0 Kl 0 K l'\ t - l; 09 *rlC MC H M-r-lC r- ICA NCM CCr-U M C\ \ j- 1 4 LCC rHC P co;J ~ 11 ^ 1 ~1 1 (D H 52

By March 26, 1960, after more than nine billion SCF of gas had been injected, the field was shut in for about a month and the pressure stabilized to a value near the discovery pressure of the aquifer. This date was selected as the initial time, tD. The "initial" pore volume was calculated from the quantity of gas injected and the field pressure. 3. CALCULATIONS (a) Application of Thick Sand Model to Field B.-The data required for the application of the Thick Sand Model to Field B are taken from Tables B-l and B-2 and are listed below. I =.148 has = 40 ft haquifer = 2500 ft c = 8x10-6 vol/vol-psi K = 100 millidarcys. = 1 centipoise a = o.98 b = -0.000083 psi-' At = 7 days Tr 530~R Vo = 110x106 ft3 r2= 5.9x106 ft2 b Kr = 1 M =.963 M1 =.7 M2 =.9 M3 = 1.0 The incremental delta time is calculated from 0.00633 KAt AtD = 2 0.00633(100)(7). =.64 (1)(.l48)(8xlo-6)(5.9xlO6) and tD for each time by (tD)j = jAt tD(l) = 1(.64) =.64 tD(2) = 2(,64) = 1.28 tD(3) = 3(,64) = 1.92 53

The values of dimensionless pressure P. are calculated by Eq. (15), page, when tD was less than 10, and Eq. 12), page, when tD was larger than 10. An example calculation is shown for P1 using 0.7, 0.9, 1.0 for Ml, M2, M3, respectively. - = (.963-.9)(.963-i.o) (.616) (.7-.9) (.7-10) (.963-.7)(.963 -1.0) + (.56o0) (.9-.7) (.9-1.0) (.963-.7)(.963-.9) (55 + 0-.7) (..50..9) =.55622 The remaining values for P. are given in Table B-3. The cumulative gas in place is found by: nO % _V, VO - 0 a-+b) RT 110x106 (98 _0.000083) (10.73) (530) 1097o 4 = 2,382x107 lb moles and Pbase Gpe n. = n0 - ------ - ~ no RTbase nl = 2.382x107 (14.7)(-204x106) (10.73)(520) = 2.436x107 lb moles 54

E M 08. -O -\t- - (N — 0 O\ \ if\ODAn t —-o-n cgO O N cO O\ 0 No ON O nNoN -O - O c C o\O N oH oN Co o5 IK. _- NO _N O _- ONN H0 -IO (-N NO t- O\ NO n o N OO Lr\ \O nO 005 O O\ O\ 0'\ O \ ON ONOO O O HHHH04 C0440 \ c\HN b- K'\ tNo \ H- t —O'- H — t — N c O (- (H+0PA 5co 6I 40 cOCO 0\r-l O I0iC\ \ \ Cr\^O i-l -(-t \tl 00P - 0 0 CM ^FQOU c no KlO\ cQ n \ O r- 0 t'\ nn \r0f'IO \ t \ r \0 \ a r-N o o W cu r\ In\9- f1 ~jO\ C in \ M Cy D~,O [' —- _['-O\ L rl\Lrl\ n n aad (R C! 0 o o Cy\ O\ tc- ),o 0Os l- ^^^ Ol t.- ON n \0 \,o -I \ 4 r ^ t \^o r o cu o u co U-\ t — t ci c \O \, o KM co \^ o \, n - \, cu E XD E b- o ioi tO A c6 c( \Z co' o~ c6 M( cy Sya t A c \lo-tAci A ( cAAo y (sA ON No'NO O HN ( 7 N -O NON0 O \HN-O'OHO Q a -H Ca or^*~o to 1, o -00 ONOO ON\O -O O N o H -ON In o H ONH -:tO\-O c < CH \ o t- 04 o No\ o co co t- N- o N noCON Cu co \otO cu 0 (- 5-OrNHd p\o H t^CO 04NONO\N7O NO \NOi:-cN OtNO-N \O0o'otN t-O 0 coNooo No ONONO O \ NOh\ ioCU ONOCO HN N H4 NO - NONNNOC N NON NNON N N NO 0 NO N N4co i - 888 rQi I I i i i i i' i i I — I I i I 0i i i i i i i i i -i 4 i i r-4i -Iri +0 o o CiK t-; t l Cm in\ n 0 in 0\ *r 0\ In 2 \ \ t- cu CQ w Ko \ e- n 0 N \co \ocqot<- i- Cr\ 0o\ O\ cOi ON it G\ _ tO- K D~ \ \0 \o "~ C C OO~ ) c4\ ON (G\ 0 J rtr oc CM % t~ \_ o O \\ t- O,O - ON ON 0 0 — d- cu C — cm 0' cu rl K - rI ( 0 -- ON o\ c co W H'U - ~ OHOOO OOH COH CO CO O NO N N N H rO NO rHO O OcJ O rcH - ONiONO H H 0 NO t-N NOOOO NON t-O —t A A —— n \O\O\O \NO t-NN t O, N4 0t 0' -NO\ NON LA 55 d (m N \ ONH \0 H.\-.t- co 4 HNONO -\o N tN005\- t-0 C(3O H- ON N KO, Oa, C\oO (NO OCNNN 5-O NO — \05-N n\ NO -N -- N oNpO 04 0 0 0 0 0 0 0 0 0 0 4 04 04 NO N N No NO NO N O NO NO N NO rNO NO NO NO NO 4 NO NO N A N NO rO N N N NO N N O NO NO NO E OOO00N 0 0 N N-O4 —KNNNN -400 NO(OHONb (N

n 2.582x10 - (14.7)(-548xl06) n2 = 2.382x107 - k4'{/k-~4 zv / (10.73) (520) = 2.474x107 lb moles The values for n3 through n50 are listed in Table B-3. The value for K3 50.2i K3 -- Krb Nr Atf was varied until the sum of the absolute differences between the predicted and observed pressures was a minimum. The value found for K3 was 0.00007591. The pressures can now be calculated from Eq. (14), page P (predicted) = Po(observed) = 1097.4 1 Pi = C1 + ICI - C2 C1 = - - (K3) [VP1 - P1RTnlb} - Po 2 = - (0.00007591) {( llOxl0) (55622) 2 -.55622(10.73)(530)(2.436x107) (-.000083) ) - 1097,4 C2 = -K3P1RTnla - (0.00007591)(.55622) (10 73)(530) (2.436xl7) (.98) P1 = 1115.7 The pressures assuming no water movement are determined by: 56

p an *RT VO-bnjRT P, (.98) (2.436x10)(10 73) (530) llx06+ooo 000083(2.436x107) (lo 73) 53(0) = 1119.7 The pore volume ratio is determined by V(/Vo (a+bPj)njRT PjVo V/Vo (calc) = (.98-,000083(1115~7) (2.456x107) (10.73)(530) (1115.7) (ll0x 06) 1.0039 The remaining values are tabulated in Table B-3. (b) Application of Radial and Resistance-Function Method.-Examples of the Application of the Radial Model and Resistance-Function method are given in Field A and Field D, respectively. 4. RESULTS Calculations based on the following were performed for Field B: 1. no water movement; 2. thick sand model, infinite aquifer; and 3. radial model, infinite aquifero The comparisons between predicted and observed pressure performance for these cases are plotted in Figs. B-2-B-4, respectively. The predictions obtained using the models are all considerably better than the prediction based on the assumption of no water movement. The thick sand model gave the best prediction, but was only slightly better than that of the radial model. However, the correspondence between the predicted and observed performance leaves much to be desired. The volume ratios predicted using the thick sand model are shown in Fig. B-5. 57

I I. I, I' I' I 1800 1600 a. /~ ^ POINTS CALCULATED FROM FIELD DATA bJ It O POINTS CALCULATED Mj / ASSUMING NO WATER gn MOVEMENT c 1400 a 1400 J ESTIMATED INITIAL PORE VOLUME OF GAS ~ P SAND, Von 110 MMCF () 1200 1000 0 10 20 30' I TIME, WEE KS FIg. B-2. Comparison of Field B predicted pressures, assuming no water movement with observed pressures. 58

I' I' I 1 I' I 1800 A POINTS CALCULATED FROM FIELD DATA o POINTS CALCULATED USING THICK SAND MODEL WITH INFINITE AQUI FER 1< 600 ESTIMATED INITIAL PORE VOLUME _T ~ OF GAS SAND, V = 110 MMCF K/pucrb = 14.4 (md)(cp) (psi)(ft)f K r */Krbir t t+ =7.6 x lO(cp)(md) (ft)t(doys q)b 0_ a 1400 hi hi 1200 1000 0 10 20 30 40 50 TIME,WEEKS Fig. B-3. Comparison of Field B pressures predicted using thick sand model (infinite aquifer) with observed pressures. 59

1800 A POINTS CALCULATED FROM FIELD DATA o POINTS CALCULATED USING RADIAL MODEL WITH INFINITE AQUIFER ESTIMATED INITIAL PORE VOLUME OF GAS SAND, V = IIO0 MMCF < 600 L K//ucr *~14.4(md)(cpl)(psi)(ft)' h+crb 5 4506 (ft'(psi)D. 1400 hi 1200 - - o000. 0 I0 20 30 40 50 TIME, WEEKS Fig. B-4. Comparison of Field B pressures predicted using radial model (infinite aquifer) with observed pressures. 60

3.0 I I I I I A POINTS CALCULATED FROM FIELD DATA * POINTS CALCULATED USING RADIAL )2' MODEL WITH INFINITE AQUIFER ~2.5 >. *ESTIMATED INITIAL PORE VOLUME o OF GAS SAND, Vo = 110 MMCF K/u+crb = 14.4 (md)(cpT'(psl)(tt)' 2. 0 h+crb = 4506 (ftf (psi)' 1.0 - - > 0.51 U) 0 I I I I 1 I, I 0 10 20 30 40 50 TIME, WEEKS Fig. B-5. Comparison of Field B volume ratio predicted using radial model (infinite aquifer) with volume ratios calculated from field data,.6l

FIELD C Field C, a western New York storage reservoir discovered in 1934, produced gas for eight years prior to conversion to gas-storage operation. The reservoir was nearly depleted during the initial eight years when 3,582 million standard cubic feet were produced accompanied by a wellhead pressure decline from an initial value of 2124 to 247 psia. 1. GEOLOGY This field, 4555 feet below the surface, consists of weakly cemented sandstone. Extensive coring operations were unsuccessful because the cementing material was not strong enough to hold the sand grains together. The rock is composed of fairly coarse to medium-grained quartz sandstone, open in texture, with small amounts of secondary silica and calcite as cementing material. The sandstone was deposited during a rapidly westward advancing sea; the old shore deposits were reworked during temporary interruptions of this encroachment. This sand is found throughout central and western New York in lenticular beds. The structure of this field is a small structural dome, elongated east and west, and bracketed by two paralleled reverse faults to the north and south. The presence of these faults suggests that the linear model would best approximate this reservoir, An areal sketch of the gas and water boundary is shown in Fig. C-lo 2. FIELD DATA The reservoir and fluid properties are summarized in Table C-l and the production-pressure history tabulated in Table C-2. 35 CALCULATIONS (a) Application of Linear Model to Field C,-The location of Field C between two parallel faults suggested that the Linear Model would be the most suitable choice for analyzing the data from this field. The following data from Tables C-l and C-2 were used: 65

WATER WAT ___ WATE R V/2 G AS Fig. C-1. Areal sketch showing the boundary of Field C. 64

TABLE C-1 STORAGE FIELD C DATA Aquifer Characteristics Type of formation Sandstone Porosity 8% Permeability 50 millidarcys Thickness 14 ft Depth to top of aquifer 4555 ft Viscosity of liquid in aquifer 1 centipoise Compressibility of brine and formation 7x10-6 vol/vol-psi Data on Storage Field a. Field geometry Location Western New York Type of formation Sandstone Nature of structure Structural dome b. Gas-sand formation and fluid properties Gas compressibility correlation, z = a + bP Constant a 0.983 Constant b -0.000081 Reservoir temperature 582~R Reservoir thickness 5 ft Base conditions for gas measurements Pb = 14.75 psia Tb = 520~R c. History of gas reservoir Discovery date 1934 Initial discovery pressure 1900 psig (wellhead) Initial pore volume 24x10 cu ft Date converted to gas storage 1942

TABLE C-2 GAS INVENTORY AND PRESSURE HISTORY FOR FIELD G Pressure Base 14.75 psia Temperature Base = 520~R Cumulative P, Wellhead, Cumulative P, Wellhead, Cumulative P, Wellhead, Cumulative P, Wellhead, Date Gas Produc- psig, at End Date Gas Produc- psig, at End Date Gas Produc- psig, at End Date Gas Produc- psig, at End tion, MMSCF of Month tion, MMSCF of Month tion. MMSCF of Month tion, MMSCF of Month Initial 1900 1934 Nov 2.0 -- Dec 33.4 1935 Jan 79.8 1941 Jan 3432 -- 1947 Jan 3576 680 1953 Jan 2975 Feb 99.9 -- Feb 3481 Feb 3617 - Feb 3014 Mar 11.3 -- Mar 3528 -- Mar 3642 607 Mar 3064 1440 Apr 124.6 1855 Apr 3562 217 Apr 3642 - Apr 3062 May 138.6 -May 3580 212 May 3642 630 May 3004 June 140.1 1825 June 3580 226 June 3659 651 June 2927 July 143.7 1825 July 3580 238 July 3599 693 July 2847 Aug 143.7 1820 Aug 3580 253 Aug 3583 728 Aug 2769 Sept 149.5 1850 Sept 3580 265 Sept 3576 - Sept 1776 Oct 152.5 1840 Oct 3580 -- Oct 3552 -- Oct 2661 Nov 152.5 1830 Nov 3580 -- Nov 3545 - Nov 2658 Dec 182.2 1825 Dec 3580 305 Dec 3550 -- Dec 2701 1936 Jan 211.5 1825 1942 Jan 3580 -- 1948 Jan 3568 -- 1954 Jan 2780 Feb 244.7 Feb 3580 -- Feb 3591 -- Feb 2792 Mar 260.2 1825 Mar 3580 -- Mar 3618 - Mar 2798 1629 Apr 262.4 1825 Apr 3581 330 Apr 3633 1015 Apr 2794 May 262.4 1825 May 3582 350 May 3633 - May 2753 June 262.4 -- June 3582 370 June 3609 - June 2691 July 262.4 -- July 3574 385 July 3585 -- July 2617 Aug 280.2 -- Aug 3537 423 Aug 3562 - Aug 2545 Sept 280.2 -- Sept 3480 477 Sept 3536 - Sept 2479 1877 Oct 280.2 1770 Oct 3456 530 Oct 3531 -- Oct 2437 -- Nov 297.3 1770 Nov 3456 -- Nov 3511 -- Nov 2483 -- Dec 318.8 1740 Dec 3458 548 Dec 3492 - Dec 2638 1937 Jan 319.8 1943 Jan 3469 -- 1949 Jan 3492 -- 1955 Jan 2764 Feb 320.3 -- Feb 3488 530 Feb 3492 - Feb 2845 Mar 320.3 -Mar 3504 -- Mar 3492 -- Mar 2903 Apr 320.3 -Apr 3504 530 Apr 3492 -- Apr 2865 1527 May 320.3 -May 3504 535 May 3492 - May 2810 -- June 320.3 -June 3475 560 June 3492 - June 2744 July 320.3 -July 3460 589 July 3446 - July 2677 Aug 320.3 1750 Aug 3436 616 Aug 3381 - Aug 2614 Sept 320.3 -- Sept 3399 643 Sept 3308 - Sept 2556 1804 Oct 321.3 1745 Oct 3359 670 Oct 3235 1290 Oct 2495 Nov 321.3 -- Nov 3359 680 Nov 3217 - Nov 2567 Dec 373.2 1725 Dec 3389 -- Dec 3226 - Dec 2792 1938 Jan 384.5 1720 1944 Jan 3414 645 1950 Jan 3267 -- 1956 Jan 2909 Feb 384.5 1725 Feb 3439 -- Feb 3324 - Feb 5073 Mar 386.8 1725 Mar 3462 610 Mar 3360 1110 Mar 3186 1049 Apr 386.8 1725 Apr 3470 -- Apr 3392 -- Apr 3145 May 386.8 1725 May 3461 -May 3337 - May 3081 -- June 386.8 1730 June 3429 650 June 3283 - June 3016 July 386.8 1725 July 3392 688 July 3229 -- July 2938 Aug 386.8 -- Aug 3361 720 Aug 3173 -- Aug 2866 Sept 392.0 1725 Sept 3342 -- Sept 3117 1420 Sept 2811 1625 Oct 401.0 1725 Oct 3324 -- Oct 3074 Nov 432.4 1715 Nov 3324 770 Nov 3088 Dec 458.8 1685 Dec 3370 -- Dec 3176 1939 Jan 611.1 -- 1945 Jan 3424 -- 1951 Jan 3246 Feb 878.5 - Feb 3458 Feb 3319 Mar 1192 Mar 3458 665 Mar 3332 1180 Apr 1424 1250 Apr 3458 680 Apr 3286 May 1614 1180 May 3458 685 May 3223 June 1794 1085 June 3458 700 June 3152 July 2026 990 July 3458 July 3078 Aug 2230 910 Aug 3458 -- Aug 3008 Sept 2280 905 Sept 3458- Sept 2945 Oct 2432 - Oct 3458 720 Oct 2889 Nov 2608 -- Nov 3458 730 Nov 2899 1595 Dec 2791 -- Dec 3476 -- Dec 2978 1940 Jan 2944 1946 Jan 3496 -- 1952 Jan 3067 Feb 3059 -- Feb 3502 680 Feb 3108 Mar 3161 - Mar 3502 -- Mar 3152 1336 Apr 3221 Apr 3502 -- Apr 3161 -- May 3247 -- May 3502 675 May 3115 June 3253 385 June 3502 695 June 3054 July 3256 385 July 3502 710 Jdly 2989 Aug 3256 395 Aug 3502 -- Aug 2928 Sept 3257 400 Sept 3503 -- Sept 2874 1650 Oct 3276 395 Oct 3504 -- Oct 2854 Nov 3328 -- Nov 3504 -- Nov 2835 -- Dec 3581 360 Dec 3540 -- Dec 2920 66

, =.08 hgas = 5 ft haquifer = 14 ft c = 7x10-6 vol/vol psi K = 50 millidarcys = 1 centipoise a = 0.983 b = -0.000081 psi-1 At = 30.45 days/time interval Tr = 582~R Vo = 24.0x106 ft3 Gas gravity = 0.577 Gas well depth = 4,600 ft Since wellhead pressures, psig, were reported, the bottom hole pressures were calculated by: Mh/144 zRT Pbhp Pwh e { ((29) (. 577) )}(4,600) / ((144) (10. 73) (582) } PO = (1900 + 14.7)e = 2124.3 ((29)(.577() (4,600)/{(144)(10.753)(582) P6 = (1855 + 14.7)e = 2073.4 The remaining values for the first ten points are summarized in Table C-3. As in the previous example of calculations, the cumulative gas in place was calculated by: Vo O a (p-+b) RT 24.0xl06 (98-. oooo081) (10.73) (582) 2124.3 = 1.008x107 lb moles 67

Id Crl ( 0 H CO CO H wO W ~ o o q CQ CQa Q 0 o tN -o- H 0 H H 0 0 o 0 0 0 N oo \ CO C- -0 Lr\n C-t- C *li 0 0\ 0\ \ 0\ \ o \ -Ul D r O 0 0\ 0 0 0 0 0 0 0 a) r ( E Q+3PH ( 1 1 I ON NN ON r(N o IQ( Q O | I c I IoC- i K\ K( \ (a) A I 0 0 0 0 (2S ( O O OC J ON CO I-4 r1 o o - S UL ) CI C 0 C C UO aU) rdat, ~ 0H oU 4 r.-n K'X _3 CQ rd S fi k h eli El O..d.. Yi pa;) r1l - 00 t - ) 0 n \ t H l 0 0 N N NO O O ON o o o r-J Ca- a) cn *rl O r' " U 0 1 O N ( - - H C- C fr, Pd U *H X K c O n O O O O H Cco') N O O 0 0 C 0 C O o o C t — - O HO HO H r O /- r1 *rl g o &p r-C) ^ o H U CQU 0\ O K co K c (d o a r-f z z rco rq ri r~ / * ~I (H H0 0 O C- I C"O CU O H CO (N CM U 0 - C - O0' o. o - n co o o- ~o N, O r -,,-I CO - - 6O 8 0 El 68

and n = - Pbase Gp nj n R Tbase n = 1.008xl07 (14-.7) (2,016,000) (10.73)(520) n- = 1.007x107 lb moles n2 = 1.008x107 -14.7( 3,440,000) (10,73)(520) = 9.987xlO lb moles The remaining values of cumulative gas in place are given in Table C-3. The pressure assuming no water movement is given by: p. = an 1RT Vo-bnj RT P = (.98) (1.007xl07) (10.73) (582) P1 ='..................... (24x106) -(.o000081) (007xl7) (10.73) (582) = 2123.4 The pressures were predicted using the water influx equation for the constant-terminal-pressure case, i=j-1 OcA NQ V - Wej = cJ E Pi I K1 i=O From a mass balance, Vj = Vo - We where Vj = reservoir volume at time step j, ft3 the gas low zjnjRT a+bPj Vj = -- = --- njRT Pj Pj 69

and an expansion of the water influx equation, j-2 a+bP = G F AG njRT Vo cA i- P AP - +_ + cA i=O Let = A" 0.006335 KAt K4 = OcA 7f = 0cA Solving the above equation for Pj yields P = C1 + C + C where j-2 K4 ( APi T?+Pj-2 +bnj RT-Vo i=0 J C1 2K4 anjRT C2 = JK4 j = Pi-l'i+l; P1 P The value for K4 was varied until the difference between the observed pressures and predicted pressures was a minimum. This value was K4 = 689. PO(calc) = Po(observered) = 2124.3 psia P1 = C1 + 1/C + + C2 C1 K4Po+bnjRT-Vo 2K4 689(21244. 3) -. 000081(1. 007x107) (10.73) (583) -24x106 2(689) 70

anlRT C2 = --- K4 o 983 (1.007x107) (1073) (582) (689) P1 = 2123.4 P2 = C1 + IC + C2 APo = P_1 - P1 2124.3 - 2123.4 = 0.9 psi K4 (APo I2 +Po) +bn2RT-Vo C1 2K4 689(0.9 2 +2124.3 -.00008(9.987xl06)(10.73) (582) -24.0x106/(2(689) ) an2RT C2 K4 o.983(9.987x06) (10 73) (582) 689 P2 = 2109.7 See Table C-3 for the remaining values predicted using the Linear Model. 4. RESULTS Nearly all the virgin gas of Field C was produced prior to the date of storage conversion. This quantity of gas, converted to the initial reservoir 71

temperature and pressure, was used as a lower limit for estimating the initial pore volume. This limit (Vo = 23.8 million cubic feet) was close to the value which gave the minimum deviation between calculated and observed pressures, 24.0 million cubic feet. Pressures calculated, assuming no water movement, are considerably lower than the observed pressures after the initial drawdown of Field C (Fig. C-2). Considerable improvement is obtained when the rate of water movement is determined by use of the linear model (Fig. C-3). A comparison of the reservoir volume ratios calculated from these pressures is given in-Fig. C-4. Note that at one point about 85% of the volume originally occupied by the gas had been replaced by water. The resistance-function method in its present form did not give a satisfactory prediction for this field. Future work is planned to overcome the difficulties encountered, 72

4 0 4 0 I 0 ~~~~~~~~ — u; z o <..joCP >" ^~~~~~~~~~~~~~~~~~ 4 N ~ 40 Q N ^ U. ^< s 3 — ~*- < ^ 2 ^ ^ _ - 0 " — 0 0 ~ Q Q -'46" I 4 < < o 0^^^~ MM~~ M~~<o 0 4- ~ cd *D — ^' I II Iw w 9z.pq 2- I.z O Co W, —_ o v y> 0Oy s 4 1 M - 7~C_'a w~~~~~~~~- 00 0 --- 0 0 CY U

I | W I | | | | | T | | I | I' o 1D N ~ X >,0_ O o 0w w Z N ^ ^ > -0 -0 --. 3 1 0 ) I - 9 N- 20 i 4 0,.., ~,, o x -^^^^. ^ i lx i^ 1 - H _ gZ —,-6jo E: rd I I I I I I I I I I I I Iz I I — I C.<1 o N0 O" o d C~v O 00r 0D P> CM 0 00 - 00 (C 2o 0 O rd H -H 0 0 0'. 0 VISd'3aifSS3bld I1OAd3S3l 7^ k Hr bF Cril Q _ IO O~~~~~~~~~~~~~~~~~

% At CC * L CLTo ^ow X - #>,d 0 4 0 o a U.' H o o w -, I l - - 1 zw n E r< W 4 C D O 0w 4I~1~~~~~~~~~~~~~~~~~~~^ 0 - o a. o w Dp -J Z /' 754~4 0 Z%~ CH rd ^^ ~ 00 Z.. E 4 _r0f8 ^ H0 H: — I — -- ~ x ~~~rO-^4 0:-a o 0 z" *A /A'OOl~d z^ 0 -IO A d lOad 3S,4 70 -- -p. 00 N'6~ 75 ~ ~. ~ ~ 0

FIELD D Field D, a Kansas gas reservoir, was recently converted to storage operations. The first well in the field was completed in July, 1918, with an open flow capacity of 10,000 MCF per day from a sand. 1340 ft below the surface. The initial wellhead shut-in pressure was 520 psig, and the reserve originally contained 24,264,356 MCF (14.65 psia base) of gas. Prior to conversion to gas storage in May, 1958, water had encroached the depleted areas of the field and had successively drowned out the lower-lying propertieso The ability to move water out of the storage area and to inject gas in quantities sufficient to make this field commercially acceptable has been established. 1, GEOLOGY The gas-bearing sand that comprises Field C is an anticlinal structure with closure exhibited to the limits of the gas field. This structure appears to be of sufficient magnitude to have held the original volume in place. The sand body grades up-dip to the northeast into shaley sands and shales which effectively seal off up-dip migration of the gas; while to the south, the sand extends a considerable distance beyond the southern limit of the field, Thick shale beds overlie this storage sand, This sand rests directly upon the Mississippi Lime throughout most of the field, In a few wells a thin shale bed was found between the storage sand and Mississippi Lime, The storage sand is uniform in texture and composed of round, medium-sized grains, 2. FIELD DATA The reservoir and aquifer properties are given in Table D-1 and the production-pressure history is tabulated in Table D-2. The initial pore volume of the gas sand was determined from the production history prior to the conversion of the field to storage operation. Since the field had been shut in for several years, the initial pore volume used in this study is the volume at the time the field was converted to storage operation and not the original volume when the field was discovered in 1918. This value, Vo = 30 MMCF, also gave the best results when the mathematical models and resistance-function method were applied to the field data, 77

TABLE D-1 STORAGE FIELD D DATA Aquifer Characteristics Type of formation Sandstone and limestone Porosity 30* Permeability 500 millidarcys Thickness 10 ft Depth to top of aquifer 1364 ft Viscosity of liquid in aquifer 1 centipoise Compressibility of brine and formation 7x10 vol/vol-psi Data on Storage Field A. Field geometry Location Kansas Type of formation Sandstone Nature of structure Anticlinal b. Gas-sand formation and fluid properties Gas compressibility correlation, z = a + bP Constant a 0.9975 Constant b -1.25x10-4 Reservoir temperature 560~R Reservoir depth 1554 ft Base conditions for gas measurements Pb = 14.65 psia Tb = 5200R c. History of gas reservoir Discovery date July 1918 Initial discovery pressure 408 psia (wellhead) Initial pore volume 30x106 cu ft Date converted to gas storage 1958 78

TABLE D-2 GAS INVENTORY AND PRESSURE HISTORY FOR FIELD D Pressure Base = 1465 psia Temperature Base = 520~R Cumulative Gas Production, P, Wellhead, psia, Date MMSCF ~~~Date MN~M'~SCF ~at 23rd of Month at 23rd of Month Initial 408 1958 May - 757 560 June -1319 572 July -1877 589 Aug -2506 602 Sept -3140 610 Oct -3763 614 Nov -4349 613 Dec -4481 585 1959 Jan -4417 552 Feb -4229 527 Mar -3936 503 Apr -4317 534 May -5505 601 June -6531 627 July -7131 627 Aug -7664 626 Sept -8176 626 Oct -8628 626 Nov -8658 601 Dec -7571 536 1960 Jan -6126 466 Feb -4582 396 Mar -3353 353 Apr -5261 533 May -7751 614 June -8873 626 July -9561 627 79

An areal sketch of the gas-water interface, Fig. D-l, and the field properties, Table D-l, indicate that the radial model should be applicable for analyzing water movement in this thin, pie-shaped reservoir. 3. CALCULATION ON FIELD D; EXAMPLE OF RESISTANCE-FUNCTION METHOD Find the resistance function for Field D. It has been decided to fit the first ten data points with the radial model, constant-rate case. The following data are available for Field D. n P(well bottom psia) Gp(MMSCF) 1 592.1 -757 2 604.5 -1319 3 622.1 -1877 4 635.5 -2506 5 643.8 -3140 6 647.9 -3763 7 646.9 -4349 8 618.0 -4481 9 583.9 -4417 10 558.1 -4229 11 53353 -3969 12 565.3 -4317 13 634.5 -5505 14 661.4 -6531 15 661.4 -7131 16 660.3 -7664 17 660.3 -8176 18 660.3 -8628 19 634.5 -8658 20 567.4 -7571 21 495.1 -6126 22 422.9 -4582 23 378.6 -3353 24 564.3 -5261 25 647.9 -7751 26 660.3 -8873 27 661,4 -9561 Po = 435.3 psia Vo = 30.0 MMCF h = 10 ft i = 0.30 [ = 1.0 cp K = 500 md f = 0.25 80

VERY LOW PERMEABILITY WATER I | WATER mI GAS Fig. D-1. Areal sketch showing boundary of Field D. 81

Pb = 14.65 psia Tb 520~R c = 7x10-6 vol/vol psia T = 560~R a = 0.9975 b = -.000125 psi- At = 30.45 days/time interval. (1) Calculate K1 and K2: K 25.15 25 15x.000 Khf 500x10x1.0 0 006328 KAt 0. 006328 KAtithf K2 = ocr2 rbc VO b ~cVo 0.00628(500) (30.45) (3.1416) (10) (1.0) (1. 0) (7x0-6) (30x106) 14.413 (2) Calculate SO... S27: Sn = V0 - Ob Gp (a+b)At \Tbt n Po =o3 50xl06_ (14.65x560) (r 9975 -.000125)30.45 \5wx350.45 435.3 = 4.547x108 - 0.5181(0) = 4.547x10o S1 = 4.547xlo8 - (0.5181)(-757xlO6) = 8.4697x10 etc. (3) Calculate dimensionless time, tD1... tDlo tD = K2' n 82

tD = (14.4135)(1) = 14.413 tD = (14.413)(2) = 28.826 etc. (4) Find corresponding dimensionless pressure values from tablesl24 using linear interpolation. tDn-tDj l Pn = Pjl-1 + --- (Pj - Pj-) tDj-tDj-1 14413-10000 (1.8294-1.6509) 15.000-10.000 = 1.8084 P2 = 2.1251 p3 = 2.3167, etc. (5) Compute Ap1.. APlo: APn = Pn- PnApi = 1.8084-0 = 1.8084 Ap2 =125180 = 1251-18084 = 0.167 Ap3 = 2.3167-2.1251 = 0.1916, etc. (6) Compute P1, ewl; P2, eW2,.. Plo, ewlo(see below). (7) Find percentage deviation between calculated and "observed" pressures. (8) Change the value of K1 and repeat steps (6) and (7). For purposes of this example calculation the "optimum" value only of K1 will be used to illustrate steps (6) and (7). Its value, 19983 x 10-4, was found by trial and error using a modified half interval method. Starting with the value of K1 calculated in step (1), the optimum value was obtained after 15 trials. 85

Pn ='Bn +Cn - Bn where n Bn = )+ b Snl - bS K apl - Po + KZ 1 * 2 e2 n+l-j and Cn = aSnKlApl B1 = 1 T0(j997 -.000125) 4.5475xlo0 + 000125x8.4697xlo8 1.983x10-4x1.8084 - 455.5 + 1.983x10-4(0), Ci = (0.9 ) (84975) (8.4697xlO8) (1.804 8084) P1 = J/B + C1 - B1 = 573.0 psia n ewn (Po - Pn - K1 ewn+lj * Apj)/(KApl) j=2 ew1 = (435.3 - 573.0 - 1.983xl0-4(0))/(1.983xl0-4xl.8084) = -383.349 MCF/day Similar calculations yield the following results. 84

pn(data), psia Pn(calc), psia ewn MCF/day AZnxlO,psia/(ft3/day) 1 592.1 573.0 -383.349 35.900 2 604.5 598.3 -386.669 6.272 3 622.1 613.4 -387.422 3.775 4 63555 631.4 -407.816 2.740 5 643.8 643.7 -415.172 2.131 6 647.9 651.6 -414.745 1.735 7 646.9 654.9 -405.681 1.461 8 618.0 623.8 -306.600 1.126 9 583.9 588.6 -213.300 1.126 10 558.1 553.7 -130.147 1,126 The percentage deviation is found by dividing the sum of the absolute values of the differences between columns 2 and 3 by the sum of column 2 and multiplying by 100: 10 100 / Pn(calc) -Pn(data) % deviation = n=2 0.82% 10 L Pn(data) n=2 (9) Compute AZ1... AZlo: AZn = K1 (optimum). APn AZ1 = (1.983x10 4) (1.8084) = 3.585x10-4 psi/(ft3/day) AZ2 = (1.983x10-4)(0,3167) = 0.629x10-4, etc. (10) Find AZ,, n-1 AZn = po - zj ewn+l + [(a/Ps + b)Sn - (a/Pn-l + b)Sn-liAZ /ewl j=2 J AZ,, = 435 55.3 - 5 - [(-130.15xl03) (6.273x10-5) + (-213.30xl03) (3.776x10-5)+ + (-386.70x103)(1.127x10-5)] + (.9975 -.000125)25.112x108-( 9975 1\555.5'5535 7 -. 000125) 26.459x10J 3.590x10- 4/(583538x103) = 1.398x10-4 85

Note that the asterisk on Pn denotes data rather than calculated pressure. (11) Set AZ1l so that it satisfies the following: (l-e) (n-l) AZn-l < AZn < (1+e) AZn-1 (1-.02) (1-) (1.126x10-5) AZll (l+.02) (1.126xC0-5) 11-1.991x10-5 s AZ11< 1.147x10-5 From the above equation, AZll is too high. Therefore it is set equal to (l. 2) (AZo) = 1,147xlO-5 (12) Calculate P11: Pn= nB+ + Cn - Bn where n Bn = i f[(a/Pn-l + b)Snl - bSn]AZi - Po+ w ewn _~ AZJ 2 - j=2 and Cn = aSnAZi B11 = 553 7.000125) 26.459x108 +.00015x25.x2511 3.950x104 -435.3 + [(-130.14xl03) (6.273x10-5) +...+(-383.38x103) (1.147xlO-) Cl = (.9975)(25.112x8) (35.590xl0-4) P11 = 521.0 psia. 86

(13) Calculate ewll: n ewn = - - n ewn = (PO - Pn - ewn+l-j AZj)/AZ4 j=2 435.3-521.0-[(-130.15x103)(6.273x10-5)+...+(-383.4x10 )(1.147x10 5) ] ewl 1 3.950xio-4 -58.236x103 ft3/day. (14) Repetition of steps 10 through 13 yield the following results: n Pn(data), psia Pn(calc), psia ewn MCF/day AZnxlO, psi/(f3/day) 11 533.3 521.0 - 58.236 1.147 12 565.3 540.7 -134.548 1.157 13 634.5 613.5 -334.801 1,187 14 661.4 654.1 -410.144 1.218 15 661,4 659.3 -385.553 1.248 16 660.3 660.9 -366.489 1.157 17 660.3 662.2 -3535845 1,066 18 660.3 660.9 -338.341 1.005 19 634.5 639.4 -271.592 0.944 20 567.4 566.3 - 73.268 0.944 21 495.1 483.1 123.458 0,974 22 422.9 398.2 306.706 1.005 23 378.6 332.3 426;509 1.005 24 564.3 471.7 - 22.646 10035 25 647.9 613 6 -379.582 1.035 26 660.3 650.9 -401.579 1,066 27 661.4 661.1 -337.397 1o096 The resistance curve is required by the problem in the sum of the AZ's, Zn = Zn1 + AZn Z1 = 35.900x10-5 psi/(ft3/day) Za = 35.900x10-5 + 6.273x10-5 = 42.173x10-5, etc. These points are plotted in Fig. D47. Note that the units for Z have been changed in Fig. D-7 to psi/(MMCF/month), 87

Pressure calculations based on the assumption of no water movement are thousands of pounds in error as shown in Fig. D-2. This is a good example of a field for which it is imperative to take water movement into account. Application of both the radial model and the resistance function to Field D resulted in good agreement between observed and calculated pressures for its entire history (Figs. D-3 and D-5). Volume ratios calculated from the predicted pressures are shown in Figs. D-4 and D-6 for the two methods, respectively. Note that the percentage error is necessarily the same for a volume ratio prediction as for corresponding pressure prediction. The straight line exhibited by the latter portion of the resistance curve (Fig. D-7) indicates that theaquiferE outer boundary is influencing the field's pressure behavior, i.e., that the aquifer is finite. However, due to the limited data, the extent of the aquifer cannot be accurately determined since a nearly identical prediction was obtained using the infinite radial model. Although the effect of the aquifer boundary is small now, its importance will progressively increase with time. 88

350 - I I I I I I A POINTS CALCULATED FROM FIELD DATA 0 POINTS CALCULATED ASSUMING NO WATER MOVEMENT ESTIMATED INITIAL PORE VOLUME OF 3000 GAS SAND,Ve =30MMCF 2500 02000 00. 0 89. 1500 - - 0 4 8 12 16 20 24 28 TIME, MONTHS Fig. D-2. Comparison of Field D predicted pressures, assuming no water movement with observed pressures. 89

750 700 60 w 500 0. A POINTS CALCULATED FROM FIELD DATA b o POINTS CALCULATED USING RADIAL MODEL 450 WITH INFINITE AQUIFER ESTIMATED INITIAL PORE VOLUME OF GAS SAND, P =30 MMCF 400L k/pcrb= lo (md)(cp) (psi)(ftr) \ h4crb2= 8150 (ft)(psi) \ 350 - 0 4 8 12 16 20 24 28 TIME, MONTHS Fig. D-3. Comparison of Field D pressures predicted using radial model (infinite aquifer) with observed pressures. 90

8 0 / z | ~/ ^JA POINTS CALCULATED FROM FIELD DATA I' S / O POINTS CALCULATED USING RADIAL "t 3 _ < MODEL WITH INFINITE AQUIFER,{~ ~ ESTIMATED INITIAL PORE VOLUME OF GAS,jp~ ~ SAND, \6= 30 MMCF 2L k/+crb 10a IO (md)(cp) f(pl)(ft) hcr e=8150 (ftP(psi)'I TIM E, MONTHS Fig. D-4. Comparison of Field D volume ratios predicted using infinite radial model with volume ratios calculated from field data. 91

700 650 60 0. w | A POINTS CALCULATED FROM FIELD DATA \ r L45| o POINTS CALCULATED USING RESISTANCE \ FUNCTION ESTIMATED INITIAL PORE VOLUME OF 1 GAS SAND, V. 30 MMC 400 25.15 u/( khfat) 6.52 x 10 psl)(ft) (month) l 60 - 0.006328 ka/,j+crbT) *14.41 (month )1 \ 0 4 8 12 16 20 24 28 TIME, MONTHS Fig. D-5. Comparison of Field D pressures predicted using resistance function with observed pressures. 92

8 7 5' / 4 > I A; POINTS CALCULATED FROM FIELD DATA 0 POINTS CALCULATED USING RESISTANCE > | / FUNCTION W 3 co 3 ESTIMATED INITIAL PORE VOLUME OF W GAS SAND, V, - 30 MMCF 25.15 l/( khf&At) = 6.52 x 6(psui)(ft3( month) 0.006328 kah/(Tcrbz) = 14.41 (month)-' 4 8 12 16 20 24 21 TIME, MONTHS Fig. D-6. Comparison of Field D volume ratios predicted using resistance function with volume ratios calculated from field data. 93

20 15 ESTIMATED INITIAL PORE VOLUME OF GAS SAND, U E X V,= 30 MMCF 2 10 N 5 - 0, X 0 4 8 12 16 20 24 28 32 36 TIME, MONTHS Fig. D-7. Resistance function vs. time for Field D. 94

FIELD E Field E was discovered in Michigan in 1941 and produced for six years before being converted to gas-storage operation in 1947. During this period, 8,026,657 MCF of gas were produced accompanied by a pressure decline of 4355 psia to 267 psia (wellhead). Recovery of pressure over short periods of high production indicated the gas-water interface was mobile. This field is one of the several gas storage fields which supplies gas to Michigan consumers during periods of high gas demand. Gas is replaced during the summer months, thereby enabling the gas pipeline to operate during periods of low demand. 1. GEOLOGY Field E is one of several Michigan gas reservoirs found in the Michigan stray sand. This sand is a series of disconnected lenses located at random rather than a continuous layer of sandstone. For this reason, the formation has been designated as stray sand. The sand was deposited during the geologic time known as the "Michigan" and is an integral part of the Michigan formation. The Michigan stray sand is an ancient sand bar deposited in a shallow sea at the beginning of the Michigan time. During this period, the level of the lake changed frequently and some folding of the rocks caused formation of long, narrow, irregular, gentle arches with wide troughs forming shoals in the shallow sea. Erosion of a land mass located nearby, along with waves and currents, resulted in the deposition of sand lenses, along with mud and ooze, which became discontinuous shale streaks. The ridges located near the surface caught the sand as it rolled along tbe sea floor forming the sand bars which later became structural domes of which Field E is one, The sea floor was sinking with respect to the land mass during this period so that eventually sand was no longer deposited from the nearby shore, and a thick layer of mud and ooze from farther inland was deposited on the sand bar. Water was later squeezed out by the weight of overlying beds compacting the mud and calcareous ooze into impermeable layers of shale and limestone, while compacting the sand bar into a sandstone. The layers of shale and limestone are several hundred feet thick and form a good cap rock for the sandstone bar. Underlying the stray sand is the Marshall formation, a porous and permeable sandstone like the stray. Although several shale lenses are found between the stray and Marshall formations, no consistent thickness of sealing shale is found between the two structures. In many places water in the Marshall is free to migrate into-the-stray if the pressure within the stray is reduced. 95

The invasion of the Marshall water into the stray helps maintain the pressure in the stray. Providing that the movement of this water is controlled and is not allowed to come into a production well, thereby waterlogging the well, the water is beneficial. 2. FIELD DATA A summary of reservoir and fluid properties is given in Table E-l and the production-pressure history is tabulated in Table E-2. A sketch showing the boundary of storage field is shown in Fig. E-l. 3. RESULTS Predicted pressures assuming no water movement are compared with the observed field pressure history in Fig. E-2. The deviation between these two curves is considerable during the first ten years' operation but decreases after gas storage begins. This improvement may be attributed in part to the cycling of the pressures about the discovery pressure. It must be realized that the degree of apparent water movement is related to the assumed initial gas-in-place. The pressure behavior of a large reservoir with no water movement is similar to that of a smaller reservoir with water drive. Figure E-3 shows that with a suitable choice of Vo for Field E the pressures calculated, assuming no water movement, compare reasonably well with the observed pressure history. Pressures predicted using the thick sand model with an infinite aquifer are in good agreement with the observed pressures during the production period as shown in Fig. E-4. This agreement was not maintained during the storage cycles where the predicted pressures are higher than the observed. The results from the radial model with an infinite aquifer (Fig. E-5) are nearly identical with those from the thick sand model. The high prediction during the latter portion of the field historyindicate that the aquifer may be finite rather than of infinite extent. Therefore the finite radial model with several values of R, the ratio of the aquifer radius to the gas bubble radius was tried. Good agreement for the entire history was obtained for R = 9. This is shown in Fig. E-6. The variation with time of the reservoir volume ratio, V/Vo (the present gas bubble volume divided by the initial gas volume), calculated from field data and also the radial model (R = 9), is shown in Fig. E-7. The prediction of Field E by the resistance function method also gave close agreement between predicted and observed pressures (Fig. E-8) and 96

TABLE E-l STORAGE FIELD E DATA Aquifer Characteristics Type of formation Marshall sandstone Porosity 20o Permeability 200 millidarcys Thickness 100 ft Depth to top of aquifer 1120 ft Viscosity of liquid in aquifer 1 centipoise Compressibility of brine and formation 7x10-6 vol/vol-psi Data on Storage Field a. Field geometry Location Michigan Type of formation Michigan stray sandstone Nature of structure Structural dome b. Gas-sand formation and fluid properties Gas compressibility correlation, z = a + bP Constant a 0.998 Constant b -0.00016 psi-1 Reservoir temperature 530~R Reservoir thickness 20 ft Base conditions for gas measurements Pb = 15.025 psia Tb = 520~R c History of gas reservoir Discovery date 1941 Initial discovery pressure 453 psia (wellhead) Initial pore volume (440 to 470)xlO cu ft Date converted to gas storage 1947 97

TABLE E-2 GAS INVENTORY AND PRESSURE HISTORY FOR FIELD E Pressure Base = 15.025 psia Temperature Base = 520~R Cumulative P, Wellhead, Cumulative P, Wellhead, Cumulative P, Wellhead, Cumulative P, Wellhead, Date Gas Produc- psia, at End Date Gas Produc- psia, at End Date Gas Produc- psia, at End Date Gas Produc- psia, at End tion, MMSCF of Month tion, MMSCF of Month tion, MMSCF of Month tion, MMSCF of Month Initial 455 1941 Sept 152 Oct 305 Nov 505 Dec 784 1942 Jan 1044 434 1947 Jan 8016 255 1952 Jan - 671 507 1957 Jan 2495 407 Feb 1281 425 Feb 8018 252 Feb 345 478 Feb 3774 372 Mar 1581 417 Mar 8020 251 Mar 501 468 Mar 3258 337 Apr 1708 412 Apr 8022 251 Apr - 17 485 Apr 2243 404 May 1814 409 May 8023 258 May -1994 540 May 778 442 June 2001 406 June 8024 260 June -4172 597 June -1085 491 July 2200 404 July 8025 264 July -6595 657 July -4039 562 Aug 2402 - Aug 8026 267 Aug -7470 676 Aug -7200 638 Sept 2680 -Sept 8006 266 Sept -7839 682 Sept -7255 649 Oct 2964 - Oct 7957 268 Oct -7199 659 Oct -6987 631 Nov 3238 376 Nov 7899 267 Nov -6361 636 Nov -6050 610 Dec 3388 374 Dec 7883 267 Dec -5096 602 Dec -3844 557 1943 Jan 3542 - 1948 Jan 7873 270 1953 Jan -2586 536 1958 Jan -1587 487 Feb 3671 364 Feb 7830 269 Feb - 770 488 Feb 274 435 Mar 3805 364 Mar 7767 273 Mar 356 452 Mar 1975 390 Apr 3976 351 Apr 7497 288 Apr 450 455 Apr 1575 406 May 4188 - May 7420 291 May - 129 469 May -1473 501 June 4373 343 June 7317 294 June - 347 480 June -4395 581 July 4511 340 July 7101 302 July -2974 548 July -5208 596 Aug 4653 333 Aug 6921 308 Aug -5286 609 Aug -5966 599 Sept 4803 330 Sept 6838 311 Sept -6247 631 Sept -6335 615 Oct 4991 328 Oct 6785 312 Oct -6857 638 Oct -6228 608 Nov 5172 326 Nov 6757 313 Nov -6018 620 Nov -5634 589 Dec 5306 323 Dec 6755 314 Dec -4001 564 Dec -1543 483 1944 Jan 5417 319 1949 Jan 6800 312 1954 Jan -1174 491 1959 Jan 1856 389 Feb 5534 513 Feb 6901 308 Feb 362 448 Feb 4136 316 Mar 5663 304 Mar 6941 307 Mar 1257 422 Mar 4643 292 Apr 5732 308 Apr 6968 306 Apr 937 422 Apr 3359 361 May 5842 309 May 6953 307 May - 236 465 May 1136 423 June 5950 305 June 6931 309 June -1725 509 June -1019 485 July 6023 295 July 6900 310 July -3170 546 July -2896 562 Aug 6032 300 Aug 6863 312 Aug -5015 592 Aug -6096 613 Sept 6047 303 Sept 6836 313 Sept -6079 618 Sept -7040 632 Oct 6077 305 Oct 6878 311 Oct -6007 615 Oct -7001 618 Nov 6115 298 Nov 6954 307 Nov -4428 573 Nov -4014 547 Dec 6138 - Dec 7055 304 Dec - 929 482 Dec - 964 464 1945 Jan 6139 295 1950 Jan 7117 302 1955 Jan 1633 411 1960 Jan 1120 400 Feb 6140 - Feb 7189 298 Feb 3347 357 Feb 2830 356 Mar 6141 - Mar 7273 295 Mar 4290 348 Mar 4178 313 Apr 6142 - Apr 7353 293 Apr 4363 322 Apr 4655 295 May 6142 - May 7219 299 May 4403 344 May 1993 388 June 6142 - June 7041 307 June 1237 425 June 377 447 July 6142 - July 6933 311 July -2199 512 July -1659 499 Aug 6142 309 Aug 6898 316 Aug -4927 580 Aug -3071 539 Sept 6180 315 Sept 4774 375 Sept -5523 596 Sept -6197 619 Oct 6349 - Oct 2336 451 Oct -5520 605 Oct -6850 625 Nov 6510 294 Nov 2143 462 Nov -2686 535 Nov -5512 565 Dec 6679 - Dec 2552 449 Dec 748 446 Dec -2547 470 1946 Jan 6850 287 1951 Jan 4539 382 1956 Jan 2915 390 Feb 7001 - Feb 5349 353 Feb 5160 328 Mar 7149 - Mar 5565 364 Mar 6196 290 Apr 7291 274 Apr 4328 384 Apr 6087 292 May 7438 270 May 2587 439 May 5738 291 June 7573 268 June - 360 519 June 4470 July 7690 263 July -1817 558 July 2697 395 Aug 7822 262 Aug -3885 614 Aug 581 449 Sept 7927 254 Sept -5072 637 Sept -1096 517 Oct 8007 254 Oct -5689 649 Oct -1222 508 Nov 8011 256 Nov -3975 599 Nov -1918 513 Dec 8014 258 Dec -2917 565 Dec - 499 482 98

volume ratios (Fig. E-9). It should be noted that the best results were obtained with VO = 470 MMCF rather than 440 MMCF used in the above model studies, The initial portion of the resistance curve was generated by the infinite radial model using field data from the first five years. The remainder of the curve was produced directly from the observed field data without the use of any model. However, the curve was concave upward in this portion violating one of the conditions of the resistance functions Thus a smoothed Z-curve was drawn as a straight line passing through the final point of the unsmoothed curve and tangent to it at a previous time step as shown in Fig. E-10, satisfying all restrictions imposed by the method. 99

I i WATER //// GAS WATER / WATER Fig. E-l. Areal sketch showing boundary of gas Field E. 100

0 o -. so - r0 0 CH > 2-' — C 0 0 OX 2W HODO W o o 0 o j. 2 O - ~ tO g L I t J.0 0 Z0k~ 0 101 c o i f, 0 o m ~10A~13S3~ U

0 0 OD? 0 40d0J - 0 (ft K~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~lul P^ ^~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~b0Q OLC aCH *H 0 2 rd S IL o o ~o~y ^ L i-j -'ij - 10 -N~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Aft~~~~~~~~~~~~ br) Cc ~~~~~~~~~~~~~I 4 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~w; ~CH ~~~~2~~~ o rd o rrt 4 O F-l~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~OQ 1w 20 -ClO 0 W0(0 z 0~~~~~~~~~~~~~~~~~~- I L C L cco o~~~~~~~~~~~~~~~~~~~~~ N co P wz WoW z 0 CL IL W U) 1E~ ~ ~ ~~~~~0 0 UO cr P~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~L

0 c\j T~~~~~~~~~~~ OD CH ~l 2 ~ fl (d U) 0 U)& rA rA U) 1rd oc ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~U) ~: o w U Id Pi - ul 5 2 ^ ^h.^~~~~~~~~~~~~~ % U) U) CO U) y -- 0 2 =)- -. Z U. 0 o 0 o m ~ 3' 9\0 w - a: IC 06S. - U^ |i U. 1 0 CHr-. - -~ U^ ~C L U 0 co IA Wrd 1-IL 0 E'o 1*.C0 n Id 0 -Q z~~~~~~~~~~~~ OIL CH 0 r!:l0U o- ~- U )o *** ^ ^art4 W Wz 41 s x jp 1-~ar M- 41 4~~^ 1I - 44 UU -' 4 5 0 ID'3 S 1 105 Qu) 1-~~~~~~~~~~~~~~~~ N'U ktU v,~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~H' 0 t~~~~' (0 IA 44' ~~~o = COIC VIJ'3IlS3I c iO d3 W Y W~~~~~~~~~~~~~10

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^ ~~~I I'^' I' -' —<1 - co 1-1 ^1.r-i (( rd 0 F3 0 z " C'q^^rf^~~~~- 0 l d > () -.r~! s^ I ~ji/^ -'l ~ 4106Ic j 0 IL Z 00(~w ~~~~~~~~~~~~0 0~~~~~~~o o I4 ti It 0kw 06 0<0 O 0 U) 0 40 II E0 Ir o a, C o o -- -' -: 0 O. lo6 O O cn ~~~~~~~~~~~~~~~~~~~~~~~~ T- 9` ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ b. -

0- 0 Id ~ _ _.. ^> 0 < z o - i<D? D. 4) r-I X F (D X dw n Cq - 0 10'" 0 rd >- ). - -S " <o ^.... P 0 - 0 W W O 0 < ~~~o - U)^%., 0 Ea w E Qil u. z'0 0- - <0. flL t o HI~ 0, ~. 10, z.O U ) CL 0IL U. N 0M <1 o II ~ I r1 is u) g., ~~4~~~~~lc': 1'S i~c~l,::1 1 = ~3 n'Cf F~~~10

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0 00 I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ — -------- -0\(U LL~~~~~~~~~~~~~~~~~~~~~~~~~1 0 \.r8N 0 0 \ \ C ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~o Zf I 4~~~~~~~ 4; 4 (3~~~~~~~~~~~~~. O Ow~~~~~~~~~~~~~o ~ CU IIA C,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~,0 0 ~ ~ ~ ~ ~ ~~~~~~ 0 0 0 0o^ (0 0 W =) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~CHa 3 I \ \ S \ I ^~~~~~~~~~~~~~~~~ -~ ~ ~ ~ ~ ~ J NOIN dlC t:1\ I d -'Z 2 I \ )o 0 10 =t 10 M 0.~~~~~~0 O~~~~~~~~~~~~~~( o 4 r I-I 4 2) 0 Ia~~~~~~~~~~~~~~~~~~~~~c H. N OYW /Dfl ~ ~ I 3d ISd'Z 109

FIELD F Field F, another Michigan storage reservoir discovered in 1933, produced 6.3 billion standard until 1941 when the field was converted to gas storage. During this period, the wellhead pressure dropped from 534 to 301 psia. 1. GEOLOGY Field F is one of the several Michigan gas reservoirs located in the Michigan Stray Formation. The geology of this formation is dicussed in the geology section of Field E. An areal sketch of this field is given in Fig. F-1. 2,. FIELD DATA Pertinent aquifer and reservoir properties of Field F are listed in Table F-l and the production-pressure history is tabulated in Table F-2. 3. RESULTS The shape of this reservoir, as shown in Fig. F-l, indicates the use of either the radial or thick sand model, Since the lateral extent of the reservoir is large compared to the aquifer thickness, as Table F-l indicates, the radial model was selected for analyzing thel.performance of this field. The pressure variation with time for Field F calculated assuming no water movement is illustrated in Fig. F-2. Figure F-3 shows the pressure prediction utilizing the infinite radial model. Good agreement is obtained for the first fifteen-year pressure history, but the high prediction for the remainder indicates that the aquifer is finite and that the effect of the outer boundary is being felt. Therefore, several different values of R, the ratio of the outer to the inner radius of the aquifer, were investigated. The best prediction was obtained with R = 10, This is shown in Fig. F-4. The corresponding volume ratios calculated from these pressures are presented in Fig4 F-5: It may be noted that some of the wild fluctuations in the volume ratios calculated from field data are probably due to errors in these data. 111

TABLE F- STORAGE FIELD F DATA Aquifer Characteristics Type of formation Marshall sandstone Porosity 17Yo Permeability 200 millidarcys Thickness 175 ft Depth to top of aquifer 1383 ft Viscosity of liquid in aquifer 1 centipoise Compressibility of brine and formation 7x106 vol/vol-psi Data on Storage Field a. Field geometry Location Michigan Type of formation Michigan stray sandstone Nature of structure Structural dome b. Gas-sand formation and fluid properties Gas compressibility correlation, z = a + bP Constant a 0.997 Constant b -1.60x10-4 Reservoir temperature 523.5~R Reservoir thickness 12.5 ft Base conditions for gas measurements Pb = 14.755 psia Tb = 520~R c. History of gas reservoir Discovery date 1933 Initial discovery pressure 554.4 psia (wellhead) Initial pore volume (270 to 290)xlO cu ft Date converted to gas storage 1941 112

TABLE F-2 GAS INVENTORY AND PRESSURE HISTORY FOR FIELD F Pressure Base = 14.755 psia Temperature Base = 520~R Cumulative P, Wellhead, Cumulative P, Wellhead, Cumulative P, Wellhead, Cumulative P, Wellhead, Date Gas Proauc- psia, at End Date Gas Produc- psia, at End Date Gas Produc- psia, at End Date Gas Produc- psia, at End tion, MMSCF of Month tion, MMSCF of Mnnth tion, MMSCF of Month tion, MMSCF of Month Initial 534*4 1940 Jan 5867.984 313.4 1947 Jan 2642.459 467.7 1954 Jan -1043.168 561.3 1933 Feb 104.929 - Feb 5939.891 308.4 Feb 3152.013 453.7 Feb - 974.214 567.5 Mar 209.858 - Mar 6013.479 307.4 Mar 35535.242 423.7 Mar 556.919 496.5 Apr 314.785 - Apr 6064.457 507.4 Apr 3621.498 423.7 Apr 1432.722 473,4 May 315.550 - May 6098.6352 305.4 May 3555.473 451.7 May 1744.804 466.6 June 316.315 - June 6111.949 302.4 June 3225.241 455.7 June 1255.518 495t5 July 317.080 - July 6131.614 302.4 July 2791.329 480.7 July 170.067 544.8 Aug 317.844 - Aug 6149.656 -Aug 2550.513 502.7 Aug - 798.647 578.1 Sept 18.609 - Sept 6164.948 3504.4 Sept 2009.456 518.7 Sept -2257.816 630.4 Oct 319.574 - Oct 6174.521 303.4 Oct 1654.184 533.9 Oct -3402.225 664.0 Nov 320.139 - Nov 6196.179 -Nov 1686.022 530.2 Nov -3460.456 657.6 Dec 320.903 - Dec 6219.892 302.4 Dec 1838.973 517.5 Dec -3297.078 645.6 1954 Jan 322.127 - 1941 Jan 6262.580 302.4 1948 Jan 2161.579 498.4 1955 Jan -2582.489 o09.8 Feb 33557.626 - Feb 6300.297 302.4 Feb 2409.510 486.9 Feb -2829.258 625.1 Mar 354.145 - Mar 6329.195 301.4 Mar 2608.787 476.1 Mar - 13.847 501.4 Apr 370.664 - Apr 6308.537 - Apr 3677.556 433.9 Apr -1594.550 586.6 May 387.183 - May 6261.992 310.7 May 3413.051 452.3 May 432.867 492.8 June 403.701 - June 6200.383 319.7 June 2494.919 493.1 June 3697.062 337.2 July 420.220 - July 6130.776 326.7 July 2791.329 521.9 July 4026.329 338.0 Aug 436.739 - Aug 6075.184 333.7 Aug 1091.343 546.2 Aug 2367.983 464.6 Sept 4535.258 - Sept 6292.098 5533.7 Sept 510.431 566.6 Sept 107.554 554.9 Oct 469.777 - Oct 5830.934 345.7 Oct 574.937 558.6 Oct -2845.212 657.5 Nov 494.453 - Nov 5662.757 356.7 Nov 679.526 551.2 Nov -2540.963 616.5 Dec 510.972 - Dec 5427.396 372.7 Dec 1909.645 495.2 Dec -1755.802 584.2 1935 Jan 615.424 504.4 1942 Jan 5257.793 381.7 1949 Jan 2981.478 449.3 1956 Jan -2379.787 627.7 Feb 715.767 - Feb 5098.0536 390.7 Feb 3472.562 434.7 Feb -1592.806 619.6 Mar 797.450 - Mar 4913.288 399.7 Mar 4015.939 412.5 Mar 2076.743 401.6 Apr 866.470 - Apr 4750.371 409.7 Apr 3778.954 433.3 Apr 3586.334 44.4 May 918.187 - May 4660.273 414.7 May 3536.310 447.0 May 1625.561 493.3 June 951.410 - June 4484.523 422.7 June 2861.761 479.9 June 1369.218 485.6 July 978.500 - July 4292.7853 452.7 July 1998.312 512.2 July 2893.320 411.7 Aug 1005.908 492.4 Aug 4089.309 441.7 Aug 1364.590 530.7 Aug 3525.462 384.9 Sept 1045.904 - Sept 3828.507 453.7 Sept 1572.336 512.8 Sept 1430.628 524.7 Oct 1104.493 - Oct 3581.775 464.7 Oct 1305.202 531.4 Oct -2423.435 660.9 Nov 1188.949 - Nov 3366.603 473.7 Nov 1321.770 518.9 Nov -3161.480 659.6 Dec 1322.952 - Dec 3330.217 472.7 Dec 1925.154 499.9 Dec -3205.054 651.2 1936 Jan 1483.052 - 1943 Jan 3330.343 471.7 1950 Jan 1357.585 522.6 1957 Jan -1846.768 597.0 Feb 1650.409 - Feb 3330.343 470.7 Feb 3357.435 415.9 Feb -2401.685 608.6 Mar 1781.119 - Mar 3330.343 468.7 Mar 4503.945 380.1 Mar 2041.525 388.7 Apr 1890.450 - Apr 3224.534 472.7 Apr 4476.422 395.3 Apr 2769.028 396.4 May 1954.170 454.4 May 3040.791 481.7 May 3041.999 481.4 May 3239.111 381.0 June 2007.105 - June 2859.421 487.7 June 2034.069 518.7 June 3241.617 389.2 July 2052.912 - July 2662.789 495.7 July - 296.136 605.0 July 3244.106 393.1 Aug 2111.521 - Aug 2481.5373 501.7 Aug -1183.771 627.8 Aug 3246.651 396.7 Sept 2171.565 - Sept 2324.826 507.7 Sept -1503.596 632.4 Sept 768.024 533.5 Oct 2271.022 442.4 Oct 2183.701 512.7 Oct -1540.579 628.2 Oct -3093.726 676.6 Nov 2406.754 - Nov 2067.094 518.7 Nov -1033.297 600.8 Nov -3244.760 655.0 Dec 2551.897 - Dec 20435.586 519.7 Dec 500.759 537.8 Dec -3269.195 646.9 1937 Jan 2714.541 - 1944 Jan 2058.547 515.7 1951 Jan - 632.275 589.2 1958 Jan -3263.054 642.3 Feb 2862.128 421.4 Feb 2125.648 511.7 Feb - 518.660 575.6 Feb -1974.590 597.4 Mar 3005.467 414.4 Mar 2168.632 505.7 Mar 198.425 554.0 Mar -1012.380 554.2 Apr 2117.671 -Apr 2170.573 505.7 Apr 609.924 544.9 Apr - 862.323 544.2 May 3196.288 406.4 May 2042.566 510.7 May 497.678 544.8 May 393.776 496.3 June 3256.293 - June 1891.869 516.7 June 95.930 562.8 June 1603.653 447.4 July 3305.921 July 1778.604 521.7 July 27.213 565.5 July 305.738 530.9 Aug 3556.825 406.4 Aug 1778.707 516.7 Aug -2408.755 662.1 Aug - 910.586 569.6 Sept 3423.842 - Sept 1784.541 515.7 Sept -2727.598 661.6 Sept -1655.590 589.9 Oct 3528.041 - Oct 1789.301 517.2 Oct -2783.145 656.9 Oct -4287.813 688.3 Nov 3657.543 - Nov 1812.641 510.7 Nov -2781.608 653.6 Nov -3332.657 611.7 Dec 3827.245 - Dec 1951.831 503.7 Dec -1655.208 594.8 Dec -2214.954 598.9 1938 Jan 4004.603 - 1945 Jan 2201.755 494.7 1952 Jan -2950.770 652.7 1959 Jan -2609.600 617.1 Feb 4155.721 567.4 Feb 23508.576 488.7 Feb -1594.951 595.1 Feb -3565.431 683.8 Mar 4289.708 - Mar 2215.053 496.7 Mar 1070.417 489.6 Mar - 23.423 488.8 Apr 4391.712 - Apr 2096.284 502.7 Apr 1978.102 462.5 Apr 1522.745 432.6 May 4466.455 367.4 May 2103.501 498.7 May 3239.083 407.8 May 3067.589 555.9 June 4518.079 - June 2013.352 511.2 June 3542.360 403.5 June 3569.368 347.0 July 4558.851 - July 1869.960 512.7 July 3542.382 411.7 July 3421.890 368.9 Aug 4600.734 - Aug 1724.106 520.7 Aug 2113.384 490.5 Aug 2143.210 449.3 Sept 4662.932 - Sept 1607.629 522.7 Sept - 70.716 589.7 Sept -1057.390 601.3 Oct 4728.946 - Oct 1536.188 524.7 Oct -1469.734 623.7 Oct -3891.675 690.5 Nov 4824.940 - Nov 1580.732 520.7 Nov -1835.956 624.9 Nov -2921.076 618.8 Dec 4945.975 - Dec 1884.749 505.7 Dec 1883.665 621.1 Dec -1426.579 542.9 1939 Jan 5077.915 - 1946 Jan 2212.740 488.7 1953 Jan -1838.443 615.6 1960 Jan -3939.747 656.8 Feb 5202.412 - Feb 22545.273 474.7 Feb -1824.777 612.9 Feb -2459.391 568.7 Mar 5327.239 - Mar 2548.177 477.7 Mar - 649.691 553.9 Mar 1004.883 438.2 Apr 5424.021 - Apr 2564.924 477.7 Apr 1262.722 482.4 Apr 882.228 454.1 May 5475.182 - May 2492.619 483.7 May 1342.722 493.2 May 3240.918 341.9 June 5509.761 324.4 June 2299.415 496.7 June - 451.860 587.7 June 3243.159 353.0 July 5537.029 322.4 July 2087.799 509.7 July 452.260 566.8 July 1195.468 478.5 Aug 5568.150 - Aug 1904.855 517.7 Aug -1262.961 600.1 Aug - 999.058 578.8 Sept 5607.510 Sept 1766.045 521.7 Sept -2629.077 650.9 Sept -3164.583 647.2 Oct 5660.082 321.4 Oct 1696.954 522.7 Oct -3361.325 663.9 Oct -3672.215 646.2 Nov 5718.031 - Nov 1814.805 512.7 Nov -3382.980 660.4 Nov -3698.866 638.5 Dec 5782.258 315.4 Dec 2175.093 492.7 Dec -2530.894 618.8 Dec 113

The pressure and volume ratio predictions obtained from the resistance function method are presented in Figs, F-6 and F-7, respectivelyo The shape of the resistance function itself (Fig. F-8) indicates that the aquifer surrounding Field F is of finite extent, confirming the conclusion of the model studyo 114

WATE R /GA0 WATER I Z WATER AGAS Fig. F-l. Areal sketch showing boundary of Field F. 115

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FIELD G Field G is a large gas reservoir under the Gulf of Mexico off the coast of Louisiana, As a result of this gas field's isolation, it is rather doubtful that this field will be used for gas storage. 1, GEOLOGY As Field G is located in the same sand formation as Field A, the properties of this field consist of fine to medium-grained moderately cemented sandstone with numerous discontinuous shale lenses. 2. FIELD DATA The reservoir and fluid properties for Field G are summarized in Table G-l and the production-pressure history, in Table G-2, The initial pore volume was determined by comparing the predicted and observed pressures for several initial pore volumes. The best estimate for the initial pore volume using the radial model is 950 MMCF while the resistance function's best estimate is 1000 MMCF. Note that since this has never been used for storage, only primary production data are available. 3. RESULTS The performance predicted for this reservoir in which the correction for water movement is ignored shows lower pressures than are observed (Fig. G-2)., The radial model, the best representation of Field G based on the reservoir geometry (Fig. G-l), was selected for determining the degree of water movement. With the exception of the last few points, the predicted pressures are in good agreement with the observed pressures as shown in Fig. G-35 The difference between the predicted and observed performance in the last year of the field history may be due to confinement of the aquifer. However, not enough data are available to investigate this possibility with a geometric model. The reservoir volume ratios calculated from this prediction is presented in Fig. G-4. 123

TABLE G- GAS FIELD G DATA Aquifer Characteristics Type of formation Sandstone Porosity 29% Permeability 1474 millicarcys Thickness 100 ft Depth to top of aquifer 7500 ft Viscosity of liquid in aquifer 0.5 centipoise Compressibility of brine and formation 7x10-6 vol/vol-psi Data on Gas Field a. Field geometry Location Gulf of Mexico Type of formation Sandstone Nature of structure Structural dome b. Gas-sand formation and fluid properties Gas compressibility correlation, z = a + bP Constant a 0.776 Constant b 4.17x105 Depth to top of sand 7400 ft Reservoir temperature 640~R Reservoir thickness 100 ft Base conditions for gas measurements Pb = 14.7 psia Tb = 520~R c. History of gas reservoir Discovery date 1951 Initial discovery pressure 3485 psia (well bottom) Initial pore volume 950x10 cu ft 124

TABLE G-2 GAS INVENTORY AND PRESSURE HISTORY FOR FIELD G Pressure Base = 14.7 psia Temperature Base = 520~R Cumulative Gas Production Well Bottom sia PD Well Bottom, psia, Date MMSCF at End of Monh at End of Month at End of Month 1951 Feb 3485 Aug 747 3458 1952 Feb 2047 3437 Aug 3593 3410 1953 Feb 5146 3397 Aug 6994 3370 1954 Feb 8771 3357 Aug 10620 3335 1955 Feb 12448 3318 Aug 14250 3303 1956 Feb 16100 3282 Aug 18113 3257 1957 Feb 20790 3215 Aug 23352 3170 1958 Feb 21289. 3157 Aug 28318 3113 1959 Feb 29689 3105 125

Application of the resistance function method to the data from Field G resulted in better agreement between the calculated and observed behavior than was predicted by the radial model, as is shown in Figs. G-5 and G-6. The fact that the slope of the resistance curve (see Fig. G-7) decreases for the entire history shows that the reservoir is either infinite or that the effect of the boundary has not been felt. 126

____WATER g GAS WATER WATER Fig. G-1. Areal sketch showing boundary of Field G. 127

3500 3400 -- - 3 $300.I 3200 o 0 3100 \ co I lA POINTS CALCULATED FROM FIELD DATA 3000 \ POINTS CALCULATED ASSUMING NO WATER MOVEMENT ESTIMATED INITIAL PORE VOLUME OF GAS SAND, V = 950 MMCF 2900 2800 I, I,, I,, I,, [ j 1 0 1 2 3 4 5 6 7 8 TIME, YEARS Fig. G-2. Comparison of Field G predicted pressures assuming no water movement with observed pressures. 128

3500 I I I 3400 o c. (0 O POINTS CALCULATED USING RADIAL MODEL WITH INFINITE AQUIFER 3200 — ESTIMATED INITIAL PORE VOLUME OF GAS SAND, Ve = 950 MMCF K/utcrb'= 76.5 (md)(cp)'(psi)(ft) hcrb2= 293 (ftt (psi) 3100 0 2 4 6 8 10 12 14 16 TIME, YEARS Fig. G-3. Comparison of Field G pressures predicted using radial model (infinite aquifer) with observed pressures. 129

I —.9. 0.97.95-:! A/ POINTS CALCULATED FROM FIELD DATA w, O POINTS CALCULATED USING RADIAL MODEL'.940 WITH INFINITE AQUIFER ESTIMATED INITIAL PORE VOLUME OF GAS SAND, Vo = 950 MMCF.93' K/P/crb = 76.5 (md)(cpfl(psi)(ft)' hgcrbs 293 (ft)(psi)'.92.91.90 0 2 4 6 8 10 12 14 16 TIME, YEARS Fig. G-4. Comparison of Field G volume ratio predicted using radial model (infinite aquifer) with volume ratios calculated from field data. 130

3500 3400 3300 - - It I a. 0: m u 3200 A POINTS CALCULATED FROM FIELD U) DATA O POINTS CALCULATED USING RESISTANCE FUNCTION ESTIMATED INITIAL PORE VOLUME OF GAS SAND, V0= 1000 MMCF ~3100 25.15 P//(khfAt) 1.545 x IO'( psi)(tt)3(six months)l 3100. 0.06328 kAt/i=crb) - 43.9 (six months)1 3000ooo I I I I I I I L I I O I 2 3 4 5 6 7 8 TIME, YEARS Fig. G-5. Comparison of Field G pressures predicted using resistance function with observed pressures. 131

1.0 POINTS CALCULATED FROM FIELD DATA.98. w FUNCTION ESTIMATED INITIAL PORE VOLUME OF GAS SAND, % 1000 MMCF 25.15p/(khfAt) = 1.545 x 10 (psl)(ft) (six monthsf' _.92 t0.00632-8 k&t/(jutct-)= 43.9 (six months)'.90 0 1 2 3 4 5 6 7 8 TIM E, Y EA RS Fig. G-6. Comparison of Field G volume ratio predicted using resistance function with volume ratios calculated from field data. 132

400 I I 300 L / cL 200 / ESTIMATED INITIAL PORE VOLUME CL i / OF GAS SAND, V. = 1000 MMCF OL I I I I 100 0 2 3 4 5 6 7 8 TIME, YEARS Fig. G-7. Resistance function vs. time for Field G. 133

FIELD H Field H is a large gas field discovered in Michigan in August, 1956, with an original void space of 900 million cubic feet and discovery pressure of 1346 psia (well bottom). To date, 111 storage wells and 23 observation wells have been drilled. Before the field was converted to storage operation in March, 1960, more than 16 billion standard cubic feet had been produced. The acceptibility of this field for storage operation has been in question since the permeability is extremely low (2.4 millidarcys). The answer will be ascertained after operating a few years. As the available data are rather limited, this study should be considered as preliminary to a more complete investigation to be presented next year after more complete information is available. In addition, corrections due to interference resulting from a neighboring gas field on the same aquifer will be presented. An areal sketch of this field is shown in Fig. H-l. 1. GEOLOGY In contrast to the previous fields investigated, the composition of Field H is dolomite instead of sandstone. This dolomite is very "tight" with an estimated permeability of only 2.4 millidarcys. This formation is designated as the A2 dolomite in the Salina Formation of the Silurian Age. 2. FIELD DATA The gas and aquifer properties are summarized in Table H-l and the gas inventory and pressure history is listed in Table H-2. The value for the initial pore volume which gave the best agreement between observed and predicted pressures for both the radial model and resistence function method was 900 million cubic feet. 3. RESULTS A comparison of the observed and predicted pressures (Figs. H-2, and H-3) when the assumption of no water movement or water movement in a radial gas bubble, show the predicted pressure are not in agreement with those ob135

TABLE H-1 STORAGE FIELD H DATA Aquifer Characteristics Type of formation Dolomite Porosity 11.26% Permeability 2.4 millidarcys Thickness 19.7 ft Depth to top of aquifer 2662 ft Viscosity of liquid in aquifer 1 centipoise Compressibility of brine and aquifer 7xlO6 vol/vol-psi Data on Storage Field a. Field Geometry Location Michigan Type of formation Dolomite Nature of structure Anticlinal b. Gas-sand formation and fluid properties Gas compressibility correlation, z = z + bP Constant a 1.0 Constant b 1.9437x10-4 Depth to top of sand 2642 ft Reservoir temperature 532~R Reservoir thickness 19.7 ft Base conditions for gas measurements Pb = 14.73 psia Tb = 520~R c. History of gas reservoir Discovery date 1957 Initial discovery pressure 1746 psia (well bottom) Initial pore volume 900x10 cu ft Date converted to gas storage 1960 136

TABLE H-2 GAS INVENTORY AND PRESSURE HISTORY FOR FIELD H Pressure Base = 14.73 psia Temperature Base = 520~R Cumulative Gas Production, P, Well Bottom, psia, Date MMSCF at End of Month 1957 Dec -- 1346 1958 Jan 457 Feb 1009 Mar 1491 Apr 1713 May 1813 - June 1840 1252 July 1854 - Aug 1946 Sept 2098 Oct 2423 Nov 2870 Dec 3714 1959 Jan 4615 Feb 5370 Mar 5812 Apr 5925 May 6025 1186 June 6191 July 6749 Aug 7753 Sept 8360 Oct 9235 Nov 10720 Dec 12340 1960 Jan 14370 Feb 16130 993 Mar 16110 Apr 14540 May 11820 June 9139 July 6036 Aug 2925 Sept 1242 1415 137

served. The application of the resistance function method (Fig. H-5) improved the prediction considerably, but the disagreement is still large for some points. Part of the disagreement can be rationalized when one considers the pressure gradients within the gas bubble itself. In the previous fields investigated, the permeability of the gas formation is several orders of magnitude larger, thereby allowing use of an average gas pressure for the entire fieldo As a result of the low permeability in this field, pressure gradients of several hundred pounds within the gas bubble itself have been found. When these gradients are considered, the discrepancy between observed and predicted pressure is not so surprising. The calculation of pore volume ratios (Figs. H-4 and H-6) show only a slight water movement is present, the void space being reduced by less than 5% prior to gas injection. The shape of the resistance curve, plotted in Fig. H-7, indicates that the effect of the aquifer boundary has not been felt, 1538

WATER G/ / 5^/ / WATER WATERWATER F GAbAS Fig. H-l. Areal sketch showing boundary of Field H. 139

1400 1350 1300 c) IL 1250 A uO POINTS CACUAED ASSUMING NO I 200 1150 (I) A POINTS CALCULATED FROM FIELD 1100- DATA ESTIMATED INITIAL PORE VOLUME 1050 OF GAS SAND, V. x 900 MMCF 1000 950 0 4 8 12 16 20 24 28 32 TIME, MONTHS Fig. H-2. Comparison of Field H predicted pressures assuming no water movement with observed pressures. 140

1400 1300 cn P01 N TS CALCULATED FROM? rc o POINTS CALCULATED USING RADIAL 11001 MODEL WITH INFINITE AQUIFER ESTIMATED INITIAL PORE VOLUME OF GAS SAND, V. = 900 MMCF K //u~rbZ = 0.022 60 (md)(cpJ'l(pei)(tt)'~' h~rblt= 25,630 {ft.)/{psi)-I I I I I I 0 co A POINTS CALCULATED FROM 0 4 8 12 16 20 24 28 32 0 POINTS CALCULATED USING RADIAL MODEL WITH INFINITE AQUIFERONTHS Fig. H-3. Comparison of Field H pressures predicted using radial 141 ESTIMATED INITIAL PORE VOLUME OF GAS SAND, Ve = 900 MMCF K/+u~cra=:0.02260 (md)(cpf(p*i)(ftt \ h+crbIL 25,630 (fD/ plI)-' 1000 0 4 8 12 16 20 24 28 32 TIME, MONTHS Fig. H-5. Comparison of Field H pressures predicted using radial 141

98- 0.97 - 0 A W.95- A POINTS CALCULATED FROM FIELD DATA O POINTS CALCULATED USING RADIAL MODEL WITH INFINITE AQUIFER.9 4 ESTIMATED INITIAL PORE VOLUME OF GAS SAND, Ve =900 MMCF K/lu+crb =O.02260(md)(cp)l(psi)(ft)-.93 h+crb=25,630 (tt)3/(psi)-I.92.9, I I I I I, I I 0 4 8 12 16 20 24 28 32 TIME, MON THS Fig. H-4. Comparison of Field H volume ratio predicted using radial model (infinite aquifer) with observed pressures. 142

1400 1350 - 1300 1250 C,) 0.I w 10 w w A POINTS CALCULATED FROM FIELD cc 1100 DATA O POINTS CALCULATED USING RADIAL MODEL WITH INFINITE AQUIFER, RATE CASE + POINTS CALCULATED ASSUMING NO 0 4 8 12 16 20 24 28 32 WATER MOVEMENT ESTIMATED INITIAL PORE VOLUME ILOO OF GAS SAND, Vb = 900 MMCF 25.15j/(Khf~t) =3785 xleO'(psi)(tt)lmonth) 0.006328 KAt/(pucrlt) =.004543 (month)'1 95O 0 4 8 12 16 20 24 28 32 Fig H-5. Comparison of Field H pressures predicted using resistance function 143

1.0 A.9:~.98 --- 0,.97 - |d25.15 /(Khfat)-3785 x\Io(pei)(ftj5month).96A POINTS CALCULATED FROM FIELD DATA 0.91 > I0 POINTS CALCULATED USING RADIAL. ~ MODEL WITH INFINITE AQUIFER,MN u) RATE CASE with volume rats ESTIMATED INITIAL PORE VOLUME OF GAS SAND, Vo= 900 M MCF 25.15,/(Khft) 3785 xlO'(psi)(ft)(month).9 -- 0.006328 KAt/(f1*crl)=.004543 (month)'.92 0 4 8 12 16 20 24 28 32 TIME, MO'NTHS Fig. H-6. Comparison of Field H volume ratio predicted using resistance function with volume ratios calculated from field data. 144

140 120 100 z 0 80 2-L,_j X~~ESTIMATED INITIAL PORE VOLUME 60ow / OF GAS SAND, Ve 900 MMCF ( 60 N 40 20 0 4 12 16 20 24 28 32 TIME, MONTHS Fig. H-7. Resistance function vs. time for Field H. 145

K^ coC CO L CC \ 010 o'~~~~~~~~~~~~0, H H H H | - 0 0 H | C.-, (1W ) I I I I I - (DO L I WI\ O O c LO OC C - HgO Q L) _- \O —L Lco H0 ( CO H o4 4 41l l Hl l l t 4 K H-4 O O zi-O l\ I O l O j — P 0-P O - 0 O —] 0 r-t cO CO'1 i 1 Ok'c\ Or\~ —!.C \IDr-A O'\ rO_ ~ rd 1 1 0 (1 a) U a) ^ o cQ - r O 00 0l0 -. \ 00 0000 000 00 0 HI FZI ^ccI V\W\ H. H I-C\K\ - ta) — ONGO- LO 00 \ I ^ a, C\Lr - - c Od ",. H \ C \1O O H H- 0 0n, coC 10 O C C0 0\0 0 rd S!-) *H () H I -l m S E-IlI p ^ d O o L\ L\ CQ \K _j -_I 0 o - *n 0 0 0' c1o 3 P-<^3 Pr 3~ 8 8 88 8 8 8 o 8 8 c DCH ro o CD *rH I~~~~~~~~~~~- iH C ~ ~ ~cO rd -d-;|r;; j i |c CI I I cI C ^ CI Pi C rd ( c d a)C c do co U)o co co r rd *H pq pq C F=4F

ACKNOWLEDGMENT The authors wish to gratefully acknowledge the encouragement, interest and many constructive suggestions from the members of the supervising committee and participating gas and oil companies for Project N031o The funds for the research program were made available by the American Gas Association under the auspices of its Pipeline Research Committee. 147

REFERENCES lo Chatas, A. T., "A Practical Treatment of Nonsteady-State Flow Problems in Reservoir Systems," Pet. Engr., May, 19535 2, Katz, D. L., et al,, Handbook of Natural Gas Engineering, McGraw-Hill Book Co., Inc., New York, 1959. 35 Katz, D, Lo, Tek, M, R, Coats, K. H., and Katz, M. Lo, Engineering Studies on Movement of Water in Contact with Natural Gas, American Gas Association Project N031, Annual Report, The University of Michigan, September, 1960o 4. Van Everdingen, Ao F,, and Hurst, W., "The Application of the Laplace Transformation to Flow Problems in Reservoirs," Trans. AIME, 186, 305 (1949). 149