Engineering Library APPLICATIONS OF UNSTEADY STATE GAS FLOW CALCULATIONS by John D. Janicek and Donald L. Katz University of Michigan ABSTRACT The unsteady state differential equation for flow of gas from resetvoirs has been solved approximately for relatively large values of time by using a constant diffusivity. The solution gives the flowing sand-face pressure as a function of time for constant flow rate from an infinite reservoir. A parallel development starting with the basic differential equation gives a series of relationships from an assumption that the rate of pressure decline is constant with respect to radius within the effective radius of drainage but changes as the effective radius of drainage increases. These relationships for known reservoir constants and Darcy flow include: 1. Rate of pressure decline of a flowing well as a function of flow rate and effective radius of drainage. 2. Pressure distribution in the reservoir for a given flow rate and effective radius of drainage. 3. The relationship between the effective radius of drainage where gas begins to flow toward the well bore and the Darcy apparent radius of drainage in the steady state Darcy equation. 4. The effective radius of drainage as a function of time. Presented at Research Conference on "Flow of Natural Gas from Reservoirs", University of Michigan, June 30 - July 1, 1955.

-2 - Except for a mathematical constant, this development results in the same time-dependence of the sand-face pressure for relatively large values of time as the approximate mathematical solution mentioned above, The relationship for time dependence of sand-face pressure in an infinite reservoir is used to predict the effect of the superposition of a series of successive flow transients on the sand-face pressure. Pressure build-up curves, apparent stabilization pressures, the exponent of backpressure curves for normal and reversed sequence flows, time variation of the performance coefficient and the isochronal back-pressure test are discussed in light of the above relationship with superposition. At this point in the paper the deviations from Darcy's law at high flow rates are incorporated into the equation for time dependence of sandface pressure. A test procedure is suggested for obtaining the performance characteristics of gas wells, including time as a variable. Also, a procedure is given for calculating the performance characteristics of a gas well from core data including time as a variable. APPLICATIONS OF UNSTEADY STATE GAS FLOW CALCULATIONS The back-pressure curve is widely used for predicting the deliverability of a gas well. A well is opened to a flow rate and the well head pressure is allowed to stabilize to a slow rate of decline, if possible. The flow rate is then changed and the well again allowed to stabilize. The data are plotted as the difference in the squares of the closed-in pressure ( pf) and the flowing sand-face pressure (p ) versus the flow rate, generally yielding a straight line on log-log paper of slope 1/n and intercept c. In many cases, however, the intercept c tends to decline as the time allowed for stabilization is increased and the value of n sometimes depends on whether or not the flow rates were changed in increasing order or in decreasing order,

-3 - In order to better understand the reactions to be expected of a reservoir, an idealized Darcy flow reservoir is studied first. The usual steady state Darcy laminar flow formula may be derived by setting the rate of pressure decline equal to zero, which in many cases may be applied with good accuracy to reservoir in a "pseudo"- steady state where the pressure drops very slowly. The effect of time as a variable may be approximated by deriving all the results from the simple assumption that the rate of pressure decline is a constant for all radii within a "radius of drainage". Although the rate of decline is assumed constant independent of radius, it is allowed to change as the radius of drainage moves out into the formation. The results are used to derive a back-pressure equation including the effects of deviations from Darcy's law with a procedure for evaluating the necessary constants from either well test data or from core data. It is shown that the prediction of deliverability while the radius of drainage is moving is different from the condition when it becomes stationary after reaching the boundary of the reservoir or because of well interference. Methods of handling these two regimes are indicated. In this paper steady decline is defined as the condition when the radius of drainage becomes constant and the pressure at the radius of drainage declines at essentially the same rate as the equalized average formation pressure within the effective radius of drainage. DIFFERENTIAL EQUATION The ideal gas reservoir under consideration is one of uniform permeability, porosity, and thickness. The gas is at a constant temperature with uniform viscosity, molecular weight, and compressibility factor. Flow is radial and where a bounded reservoir is under consideration, it is assumed to have a circular boundary with the well bore in the center.

-4 - A material balance around a differential volume yields the equation of continuity, in one dimension,.(p = _ 0V)= (1) ~ x It Although Darcy's law for flow of gases through porous media may not apply at high flow rates, it is used in a simplified system and discussion is given later upon the significance of the deviations: v = - - (2) ax The equation of state is given by pM (3)M P = T (3)R T Combining the above equations for a radial system: 22 2 2 3 P 1 P 2p.0 ap p 1 p (4 p Dr r ar k 3.t ' t where Y = k p/p. 0 is commonly called the diffusivity. Equation (4) may be written in the dimensionless form according to the scheme of Bruce, Peaceman, Rachford, and Rice(2) for radial flow. 2 22 2 _p 1 o P P 1 P (5) + - = = - (5) 3R2 R 3 R O 2P Q~ For the most part, except for certain constants, the use of capital letters indicates dimensionless variables and lower case letters imply the use of consistent units unless the units are marked as a subscript.

-5 - METHOD OF FINITE DIFFERENCES Electronic computers have been used to find numerical solutions of Equation (5) for the case of constant rate of production. Cornell and (4) Katz applied the Schmidt method for solving the equation for generalized boundary and initial conditions. In order to do so, the non-linear coefficient (1/P) in Equation (5) was set equal to (1/P ) for the particular case in average order to remove the non-linearity. This is the same as using an average diffusivity Y in Equation (4). The non-linearity in the basic differential equation may be taken into account with the graphical procedure, if desired, as illustrated with the following one-dimensional example. Equation (5) written as a finite difference equation in one -dimension becomes 2 2 2 2 PA P = AP (6 ( A X) A. Consider a general profile as shown in Figure 1 at some given time. and center attention on three general points 1, 2, and 3 which are separated by 2 2 2 the distance AX. Let PI, P2, P3 be points on the profile at time Q, and 2 2 let P be the new value of P2 at the time 0 + AG. Substituting these 2, AG 2 values into Equation (6) we have 2 2 2A. 2 2 2 P2, AO P P X) (P3 + P1 2 P) (7) Now let A (8) (AX) 4

-6 - Then 2 2 2 2 2 P +P 3 P -P P = P2 (P2 - - ) (9) Equation (9) shows that each increment in pressure decline as found by the (4 (9) usual Schmidt procedure must be decreased by a factor equal to P2, which is different for each point in the profile. In the usual procedure A ( 10) 2,avg ( X)2 4 and 2 2 P + P 2 2 2 1 3 2 ~2,AQ = 2 (11) Therefore 2 2 2 P 2I+ 3 12P +P 2, AQ 2 ( 12} Graphically, the point given by Equation (12) is given by the intersection of 2 2 line joining P1 and P3 with the vertical line at "2". The correction may be applied by similar triangles by means of a ruler marked from zero to one. 2 The zero mark is placed at P on the curve and "one" is placed horizontally 2 2 even with the point (P1 + P3)/2. A distance equal to P2 is marked off on the ruler, and this point is projected horizontally to the point P directly 2 2, AG 2 below P2. Although the construction lines are shown in Figure 1, most of them are unnecessary to be actually drawn. The correction for the non-linearity may be applied to a radial flow system in the same way as described above. The procedure is similar to (4)(9) the one dimensional case except that the radial distance is plotted on a logarithmic scale with the lines separated by AR. The uncorrected new

-7 - 2 2 2 value of P is found at the intersection of a line from P to P at the vertical 2 line "2". This is corrected by multiplying the change in P2 by the pressure at that point using the method of similar triangles. The main advantage in the graphical procedure is the generality of the boundary and initial conditions which can be handled. Disadvantages lie in the fact that a large scale must be used in order for the finite difference approximation to be accurate and that the method does not provide an easy method for determining the characteristic constants of the gas reservoir. APPROXIMATE.AN.ALLYTICAL SOLUTION Consider the case of constant rate of production from an infinite reservoir. It has been shown that the effect of the non-linearity is to decrease the rate of pressure drop in proportion to the pressure. The correct solution to the non-linear differential equation should lie between the solutions of the corresponding linear equations using constant values for the non-linear coefficient (1/P) in Equation (5) (or the diffusivity y' in Equation (4)) equal to the highest and lowest values in the problem at hand, namely, P = P = 1. If P S c is the particular constant pressure used, the solution to the linear equation at the sand-face for large values of time and beginning from uniform shut-in conditions is given by 22 Q I 8 P P Q 8 PD PfP - P = - = - n 2 (13) 4 R 4 R s s or 2 2 (-P - p) = 1635 r:~ T5R qMCD log psia k.h. n -d 4ft -4 5.93 X 10 k, p — cps. t (13a) L C 0 r 2 p s, ft

-8 - where ~ is Euler's constant ( = 0. 5772...), D = e, and Q is a dimensionless flow rate. The maximum error expected by using this equation as a solution of the non-linear differential equation corresponds to using the limiting values of P giving c' 2 Q 1 Maximum error in P = - In (14) s 4 P s 2 Thus, for P = 0.85 and Q = 0. 02 the error in P would be less than one s s tenth percent. This flow rate corresponds to almost 1, 000 MCFD from a reservoir 10 feet thick, permeability of 100 md, 1000 psia pressure, and temperature of 100'F. Equation(13) is represented graphically in Figure 2. An approximate derivation of Equation (13) will be given using the concept of approximating unsteady state conditions by a succession of steady decline conditions. These steps in the derivation will yield some insight into the relationship between steady and unsteady states, the meaning of "radius of drainage", the rate of propagation of disturbances, and the relationship between finite and infinite reservoirs. SUCCESSION OF STEADY DECLINE RESERVOIRS Consider a bounded or finite reservoir producing at a constant rate. When the well is first opened the rate of pressure drop is very large, but after some time the rate of decline may become relatively small. If as a first approximation, the rate of pressure decline is set equal to zero then integration of Equation (5) yields the usual Darcy radial laminar flow formula 2 2 2 Q a Pf - = - In - (15) 4 R s

where R and R are respectively the dimensionless Darcy apparent radius a S of drainage and sand-face radius, both of which are assumed fixed for the moment and at constant pressures. If the Darcy apparent radius of drainage, R a, is given then the corresponding sand-face pressure P can be calculated for any flow rate Q, This Darcy pressure distribution is illustrated in Figure 3. One unsatisfactory characteristic of this equation is that it gives no information concerning the variation of the Darcy apparent radius of drainage with time. Strictly speaking, the derivation assumes that gas is entering and leaving the system at the same rate in order to maintain constant pressures throughout the reservoir. However, Muskat and also MacRoberts. have used the Darcy pressure distribution together with a material balance to describe the change in the Darcy radius of drainage with time for certain boundary conditions, In order to obtain an improvement on the Darcy formula for treating unsteady state problems, the concept of a steady decline reservoir is introduced. The usual definition of steady state implies no pressure variation with time. This condition should theoretically never be realized in a producing reservoir because of the material balance requirements. A single well in a bounded reservoir is defined as being in a condition of steady decline after the effect of the original drawdown has been felt in all parts of the reservoir and every point in the reservoir declines in pressure at the same rate as the average equalized pressure of the reservoir. It is recognized that this idealized approximation may not hold as well near the well bore as for points further out in the formation but it is felt that this approximation is a step forward from the Darcy condition of zero pressure decline and yields much semi-quantitative information which should be applicable to a large number of cases. This condition is also intuitively more satisfactory in that it satisfies the material balance for the total reservoir, in consistent units

-10 - r r b b <qP z R T zR T g g r r and average g _ (17) B t ir h M (r - r ) b s Equation 17 states that for a reservoir of given properties and a given flow rate the rate of pressure decline depends only on the square of the boundary radius rb. At this point the pressure gradient must be zero since there is no flow across this boundary. It seems plausible to assume that after a well is opened that the region near the well bore is affected first and that this region becomes larger with time. As a first approximation, it would seem that for each instant of time there should be a point at which gas has just begun to flow toward the well bore. At this point, where the net flow approaches zero as a limit, the pressure gradient is zero. This radius is very analogous to the bounding radius of a finite reservoir and this radius is called the effective radius of drainage, R, to distinguish it from the Darcy or apparent radius of drainage, R. These radii are illustrated in Figure 3 in dimensionless form. Since the effective radius of drainage has the same properties as a boundary radius, it follows that the average rate of pressure decline within this radius should be given by Equation 17 with r substituted for rb. If at e b any instant of time the reservoir within the effective radius of drainage acts like a bounded reservoir in steady decline, then unsteady state decline may be pictured as a sequence of steady decline reservoirs bounded at r until r e e reaches the real boundary rb and becomes stationary. This is illustrated in Figure 4. Equation 17 states that when r is very small then the rate of decline is very large, but as r becomes exceedingly large then the rate of decline becomes exceedingly small, explaining the apparent stabilization of the sand-face pressure some time after initiating flow. With these

-11 - approximations, it will be shown! that the variation of the effective radius of drainage can be calculated as a function of time. The method will consist of setting the rate of pressure decline in Equation (5) equal to a constant and applying appropriate boundary conditions at R and R. It should be recognized e s at the outset that although mathematical reasoning is used to derive results based upon approximations relying on physical intuition, that the results cannot be expected to be mathematically rigorous in every respect. One inconsistency which will be pointed out is the failure of the approximation of the rate of pressure decline being independent of radius to hold well near the well bore for very steep pressure gradients. Another is that a reservoir obeying the differential Equation (5) should theoretically feel any disturbance at every point immediately upon opening the well. In the latter case, however, it is felt that physical intuition is as reliable as the mathematical description of the system. The idea of approximating an unsteady condition by a succession of steady conditions was suggested by Muskat. RATE OF PRESSURE DECLINE, d dO The partial differential equation (5) may be transformed into the following ordinary differential equation if the rate of pressure decline is considered constant, using dimensionless variables dP 1 d P 2: 2 (18) dO R d(ln R) The boundary conditions which must be satisfied are 2 d.P Q d(ln) at R=R (19)

-12 - 2 dP d(ln r) 0 at R= R.e (20) These conditions follow from the fact that at the boundaries the flow rate is constant or steady and thus the profile must have the slope indicated by the radial laminar flow formula written for a differential section. The first integration of Equation.(18), considering the left hand member constant, together with the substitution of the boundary conditions given by Equations (19) and (20) gives not only the pressure gradient as a function of R but also permits the evaluation of the rate of pressure decline as a function of the effective radius of drainage. dP Q 2 2 dQ R - R e s (21) or dp -psia = 0. 375 d thr z TOR qMCFD h 0 (r - r) ft e s ft2 (22) For a reservoir of z = 0.9, T = 6000R, h = 40 ft, 0 = 0.24, q = 5000 MCFD, the rate of pressure drop will be about 0. 1 psia per hour when the effective radius of drainage has become about 1027 feet. Equation (22) is represented graphically in Figure 5. This rate of decline should hold fairly well near the sand-face if the pressure gradient is not too steep. PRESSURE DISTRIBUTION The pressure gradient obtained from the first integration is 2 dP Q d(lnR ) 2 2 2 R - R ( 1 - ) (1-2.2 R - R e s (23)

-13 - Integration of Equation (23) gives the pressure profile which may be written in the following two forms, using Pf as the pressure at the effective radius of drainage and P as the pressure at the sand-face s p -p.= - {l 2 Z ) (ln -- ) - 4 R - R R _ e s.2 2 Q 'R 2 2 2 s R p -Ps ( 1. + R2 2 ) (ln 4 RR )R e s s R - R e R - R e s _ 2 2 R - R s 2 2 R - R e s (24) (25) Adding the above equations, the relationship between the pressure fective radius of drainage and the pressure at the sand-face of the is obtained: at the efwell bore (26) 2. 2 2 2' RR 2 2 Q Rs Re sPP = - 1 + 2 ) (ln 1 4 R R R _ e s s ) -S -.4 where & = 1 according to this derivation. We shall later use - = Euler's constant = 0. 5772 instead of unity while the radius of drainage is moving in order for our final results to agree with Equation 13, and then use & ~ = 1 after the effective radius of drainage reaches the outer boundary. Pf (= pf,/pf = 1 ordinarily) is used here rather than unity as the pressure at the effective radius of drainage since Equations (24), (25), and (26) are valid even after the effective radius of drainage reaches the outer boundary of the reservoir, after which the pressure, pf,, at the boundary also declines with time. Figure 6 gives a generalized graphical representation of the pressure distribution derived from Equation (24) for a steady decline 2 2 reservoir when r /r > 100 and for g = 1. To obtain the pressure distribution while the effectie s but ion while the effective radius of drainage is moving, it is suggested to

-14 - use Figure 6 or Equation (24) at distances relatively far from the well bore where the approximation of ~ = 1 and steady decline s.hbuld be more accurate, and to use Equation (25) with the value of p from Figure 2 near the well bore to take into account the substitution of = 0. 57722 while r is 2 2 e moving. This is equivalent to correcting the values of (Pf - P )/Q read from Figure 6 for radii near the well bore by adding a value of (1 - 0. 57722)/4 = 0. 105695. This should be quite accurate since Equation (25) reduces to the expected Darcy distribution near the sand-face for a given value of Ps. Figure 6 may be used directly after the effective radius of drainage becomes constant and steady decline begins. The figure shows that for a steady decline reservoir the difference in the squares of the outer boundary 2 2 pressure and the sand-face pressure, (pf, - p ), remains constant with time. Figure 7 shows example profiles before and after the effective radius of drainage reaches the outer boundary. THE RATIO R /R e a To find the relationship between the effective radius of drainage, R, where fluid just begins to flow, and the apparent radius of drainage, R, used a in the Darcy laminar flow formula equate Equations (15) and (26) under conditions where the effective radius of drainage is much larger than the sandface radius. We find that 2 R e in 2- = (27) R a Let us define R r -2 -a a 2 D = - = = e 2 (28) R r e e

-15 - Thus, it may be seen that when R >> R then R is directly proportional to R. If & = 1, then D = 0.606 and if S = 0.5772, then D = 0.749. The usual laminar flow formula, Equation (15), can be rewritten in the following form under these conditions: 2 2 Q D R 2 2 Q D Re P - =- ln - (29) f s. 4 R s Note that Equation (29) holds even after R reaches the outer boundary and -e steady decline begins, stating that the difference in squares of the outer boundary pressure and the sand-face pressure is constant during steady decline. The declining boundary pressure Pf should be used when R = 1. Using a value of & = 1 during steady decline, Equation (28) states that the apparent radius of drainage in the usual laminar flow formula stabilizes to about sixtenths of the outer boundary radius. The variation of R /R with R is illusa e e trated schematically in Figure 8. No attempt is made to predict the ratio in the transition regions at either end of the curve. TIME DEPENDENCE OF THE SAND-FACE PRESSURE P: s INFINITE RESERVOIR The time dependence of the sand-face pressure can be found by eliminating R between Equation (21), applied at the sand-face, and Equation e (29) and integrating. Using Equation (29) rather than Equation (26) in effect neglects the time required for Re to become much greater than R. The integration can be carried out directly in terms of error functions which may then be expanded by semi-convergent series to give a result similar to Equation 13, under certain conditions. 2 2 2 Q 8 P D P F Pf - P = -in (- - -- + - ) 30 f s 5 4 R F F s s s

-16 - where F = 1 - ( 2)+ (1)(3) (Q z) - (I)(3)(5) ( +. (31) 8 P 8 P 8 P The subscript of F indicates what pressure to insert. Equation (30) reduces to Equation 13 for large values of time if F - 1 and P is used instead of P o This result may be obtained more directly by carrying out the integration using an average pressure P such that 2 dP 1 dP d O 2 P d (3 C The integration yields 2 2 2 Q 8 P D P f P - In ( - + 1) (33) f s 4 R s Equation (33) reduces directly to Equation (13) for large values of time, provided we use & = - In D =0. 5772 instead of the i = 1 derived in Equation (26). Figure 2 gives the graphical representation for large values of time, 2 2 According to Equation 26, time should be large enough so that R /R 2 2 2 e s = 8 0 P /R > 100 for 1% accuracy in Pf - P. The equality follows from c s f s. Equations (13) and (29). TIME DEPENDENCE OF THE EFFECTIVE RADIUS OF DRAINAGE, R e Comparison of Equations (13) and (29) yields the effective radius of drainage as a function of time for large values of time R 8 GP (34) e c This is the same as the fraction of the area bounded by rb which has felt the disturbance, since the dimensionless radius of the outer boundary Rb = 1. The area drained at a given time is essentially independent of the flow rate

-17 - and depends only upon the properties of the reservoir and fluid. For a given reservoir the area of drainage is dependent only on time so long as the pressure level is essentially constant. The rate of propagation of the effective radius of drainage is seen to depend essentially on the diffusivity constant 6 of the basic Equation (4) and the radius of drainage or time dre 4 X = - in consistent units (35) dt r e and -3 and 056 x 10 k p 2 md c, psia. r t (36> e, ft g 0 hr cp A similar, though not identical, result has been derived by MacRoberts( using a Darcy pressure distribution and constant sand-face pressure. Equation (36) is shown graphically in Figure 9. For a reservoir of 500 md, 2000 psia, 0. 02 cp. viscosity, 0. 24 porosity, the example for which the effective radius of drainage was 1027 ft. for a rate of decline of 0. 1 psia per hour, the time required to reach this decline rate is about 4. 67 hours. For a reservoir of 1 md, 450 psia, 0.015 cp viscosity, 0. 12 porosity, the effective radius of drainage will be only about 80 feet after one day. Note from Equation (35) that the rate of increase of the effective radius is inversely proportional to the magnitude of the radius already attained. TIME DEPENDENCE OF SAND-FACE PRESSURE, P5' FINITE RESERVOIR When a well is opened at a constant rate of production the unsteady state condition may be described by a succession of steady decline reservoirs bounded at the effective radius of drainage. This succession continues until the effective radius of drainage reaches the real boundary of the reservoir, after which steady decline ensues. In this analysis, it is assumed that the movement of the effective radius of drainage occurs as if the reservoir were

infinite until the outer boundary is reached, whereupon steady decline abruptly begins. Actually, the transition will probably begin gradually as the effective radius of drainage approaches the real outer boundary. For large values of time Equation (13) may be used to estimate the sand-face pressure as a function of time. The time required for the effective radius of drainage to reach the edge of the reservoir may be estimated from Equation (34) or Equation (36)' After steady decline begins the pressure at the outer boundary, Pf also begins to decline. If Qb is defined as the time required for R to reach the b' e outer boundary, then using Equation (21), Pf is obtained as a function of ~, for R e R e s 1 - Pf = Q (Q - 0b) (37) where -b (38) b =8P c If P b is the sand-face pressure when R just reaches the outer boundary then sb e using Equations (29) and (37) to get P as a function of ~ during steady decline: 2 2 2 2 2 P - = 1 - P = 2 Q (- b) + Q (~ - ) (39) sb s f b b where 2 2 Q D P = 1 --- In - (40) 4 R S 2 2 Pf declines linearly with time but P declines directly proportional to Pr. A simplified procedure for computing the sandface pressure would be to use Figure 2, allowing the boundary pressure pf, to vary with time and using a constant effective radius of drainage equal to the actual boundary radius. The problem is then reduced to using Figure 2 both before and after the effective radius of drainage reaches the boundary, provided pf, is used after L i the radius of drainage reaches the known maximum. The time, tb9 is required for r to reach the outer boundary. Figure 8 or Equation (36) in the e following form gives the time, tb:

-19-.cp b t =....p. (41) hr 1.056 X 10 k p md c, psia After the time tb, Figure 10 or Equation (22), in the following form may be used to compute pf,: 0.375 z T Pf - = 2 thr - tb ) qMCFD (42) hft r hr It is suggested to use D = r /r = 0.749 while the effective radius of drainage, r, is moving, and to use D = 0. 606 when r becomes stationary at the bounde e ary rb. It is emphasized that time may be used as a variable in Figure 2 only for times less than tbe It would seem that P could be found as a function of Q directly by s applying rate of pressure decline, Equation (21), directly at the sand-face. It seems, however, that this rate of decline, which is the same as the average rate of decline of the reservoir within the effective radius of drainage is more accurate when applied to points relatively distant from the sand-face where the pressure gradient is small and the pressure is nearer to that of the average reservoir pressure. Calculation of the decline of P by means of Equation (21) s is actually slightly inconsistent with the method given above according to 2 2 Equation (29) since it is assumed that P1 and P decline together, or 2 2 dP dP d3 dO (43) from which dP Pf dP = ' (44) dO P d ( s It is thus seen that the two methods of computing steady pressure decline at the sand-face become identical when flow rate is low enough that Pf/P is fs approximately unity. That inconsistent results can be derived merely emphasizes the fact that the original assumptions are not exactly consistent with the basic differential equation.

MULTI-WELL RESERVOIR: MAXIMUM EFFECTIVE DRAINAGE AREA PER WELL As a first approximation assume that each well can be treated independently of all the other wells. The problem then is reduced to finding the maximum effective drainage area to be associated with each well. Unsteady state calculations could then be made as if the reservoir were infinite until the effective radius of drainage reaches the maximum effective drainage radius associated with the well, after which steady decline calculations could be made as for a reservoir bounded at the maximum radius of drainage. One method of obtaining the area to use might be to take the geometrical area associated with each well converted into the equivalent circle. If S is the distance between wells and r is the maximum effective radius of drainage, then em r = (45) em Another method is as follows. Define steady decline in a multi-well reservoir as the condition where the average rate of decline of each well is the same as that of every other well or of the reservoir as a whole. Assume that the entire reservoir has the same properties and that the total reservoir area is known to be of equivalent radius rb. Then basing the dimensionless radius on this value of rb and using Equation 21, we have at steady decline, neglecting the sand-face radius Q. dP 2 -^- = Q = -- (46) R dQ em. or 2 Q. r q R - = e = - = (47) em.. Q b q where "i" refers to the individual well and Q is the total reservoir production rate. The fraction of the total reservoir drained by a single well in steady

-21 - decline is equal to the fraction of the total reservoir production produced by that well. A good approximation might be to consider the geometrical area of a group of wells surrounding the well in question and then proportion the drainage area according to the flow rates. This method of proportioning areas was suggested by a paper by Matthews SUPERPOSITION OF SUCCESSIVE FLOW TRANSIENTS: INFINITE RESERVOIR Superposition is a method of describing the net reaction of a system to a series of transients or impulses by adding together or superimposing the effects of the individual transients or impulses as if they acted independently of each other( 8(14. This principle may be applied to systems governed by linear differential equations. We have shown that the solution of our non-linear differential equation for a constant rate transient in an infinite reservoir may be approximated by the solution to a linear differential equation, Equation (13), within the maximum error given by Equation (14). The case of a series of constant rate transients will be considered in which the well is opened to a constant flow rate for a period of time, then changed to a different constant flow rate for another period of time, and so on. The net impulse on the system with each change in flow rate is the net difference in flow rates, and each impulse causes a corresponding net change in the pressure observed at the sand-face. The total pressure drop at the sand-face may be calculated by adding the change in pressure due to each impulse acting through its respective net impulse time. For example, the total decrease in the square of the pressure at time 2 due to the effects of two successive flows Q1' Q2', beginning respectively at times zero and Q1 is as follows, using Equation (13) with P = 1 2 2( 2 2 Q1 8 D 2 2 Q1 8 D (2 - 1) (4 P-P -- In +in - - (48) f s2 2 4 R 4 R s s

-22 - or 2 2 Q2 2 1 8 D 02 (.Q1) 8 D(~ -0 ) P P - P = — ln + in f s 22 4 2 s s Q+ 8 D2 (2 - O1) + -- in 2 -- (49) 4 R s or 2 2 Q2 8 D (2 - ) Q Q P P iP = - n + -(50) f s2 4 R 4 0 - 0 s 2 1 Note from Equation (50) that the effect of the first flow period on the last becomes negligible when the total flow time becomes very large compared with the first interval. When the second term becomes negligible the equation reduces to the form of Equation (13) which states that the pressure at the sandface could be calculated as if the well were opened from shut-in conditions at the rate Q2 for its net time of flow (02 - 1). The physical meaning of superposition may be illustrated graphically by means of Figure 11. The total pressure drop given by Equation (48) may be further expanded into the three components given by Equation (49). The first term represents a production of rate Q1 for the total time Q0 from the time the well was first opened, represented by ABD in Figure 11; the second term represents an injection of rate (-Q1) for the time (2 - 01 ) from the time the flow rate was changed at time Q1' represented by FH in the figure; the third term represents a production of rate Q for the time (0Z - 0Q ) from the time the flow rate was changed, represented by FG in the figure. In other words, after the flow rate has been changed to Q2 the system acts as if it were still producing Q1 from the first flow which is cancelled by an injection of amount (-Q1) to give a net flow of Q2. The algebraic sum of the three curves ABD, FH, and FG gives the net drawdown for the system, ABC. Superposition states that the net effect of a complicated series of changes in a single reservoir can be computed by taking the sum of the effects

-23 - on a number of independent reservoirs acting over the same period of time, each beginning in the same equalized condition. In our case: a single reservoir producing Q1 until time 0., at which time injection of Q1 begins in another independent equalized reservoir to cancel the continued production of Q, and a production of Q2 begins in a third reservoir. Note that once a reservoir is brought into the picture it must remain for the rest of the time, its net production being cancelled by another independent reservoir if necessary. It is seen that eventually the effects of injection and production of Q will tend to cancel, curves FH and ABD, and the net effect will then be essentially Q2 through its time period. Thus, FG and BC approach each other. BUILD-UP CURVES Superposition may be applied directly to the extrapolation of build-up curves obtained upon shutting in a well after a known period of flow, provided there is no well interference so that an infinite reservoir may be assumed. If a well producing Q1 is shut in at time l'. then the net impulse is (-Q1) and the pressure at time ((0 - I1) from shut in is, using -c = 1, 2 2 2 2 Q1 8 D 02 (-Q1) 8 D (G2- 1) P -P 2 -=n + In 2 (51) 4 R 4 R s s 4 In C (52) Thus, the square of the sand-face pressure versus the ratio of the total time to the time from shut-in plotted on semi-log paper should extrapolate as a straight line to the original shut-in pressure at unity on.the time ratio scale as illustrated in Figure 12. If the time of the first flow, Q1' is long with respect to the build-up time, (Q2 -.1) or if the sand-face pressure was changing very slowly prior to shutting the well in, the first term in Equation (51) may be considered 2 2 as approximately equal to Pf - P so that Equation (51) becomes: f sl

-24 - (p2 -p ) 1.635 L z T 5.93 X 10 p (t -t (sZ s si i 2 cp ~R d Pc,psia 2 1 psia ' _1hr. — =...... --.log 2.. 2 k 1000 m, C 0 r,MCFD hft k md cp 0 s,ft:(53) =A log B (t2 - tl)hr (54) where (t - t ) is the time measured after shutting in the well. Houpeurt(8) has 2 k hr suggested using Equation (53) for finding the permeability if all the other properties in the constant A are known. It is seen that if the left hand.- member of Equation (53) or (54) is plotted versus the logarithm of time in hours, then the slope is A, from which the permeability may be evaluated. In addition, however, the constant B may also be evaluated from the intercept of such a plot at t = 1. Indeed, if a reservoir follows Darcy's law it would seem that this hr would be a very good method for testing a well to evaluate the constants of the reservoir, A and B, from build-up pressures taken as a function of time after shutting in a well which has been producing at a constant rate for a long period of time. The sand-face pressure would be declining at a very slow rate just prior to shutting in. This technique is illustrated in Figure 13. APPARENT STABILIZATION PRESSURE If a well is opened to a flow Q1 for a period of time and then the flow rate is increased to Q2, the sand-face pressure may appear to stabilize at a different value for the second flow period from that obtained by opening the well to the flow rate Q beginning from equilibrium shut-in conditions and 2 flowing for a.time equal to the second. flow period, as shown in Figure 14. If 1 is the time of the flow Q, and 0 is the total time for the successive 1 2 2 1 2 flow test, then P f - P is given by Equation (50), f s2

-25 - For the single flow test for a time @ 2 2 2 Q2 8 D Pc P- P ln 2 (55) f s2 2 4 R s If the single flow test is allowed the same time to stabilize 0a, as, that of the successive flow test from. the time of opening to the second. flow, 2 - "1, it is seen by comparison of Equations (50) and (55) that the successive flow test will "stabilize" at a lower pressure than that of the single flow test, the difference in the square of the sand-face pressures for the two cases being the second term in Equation (50). This effect is confirmed by Haymaker, Binckley, and Burgess ) If the time for the second flow is large with respect to the first, then the second term in Equation (50) becomes small and the effect of the first flow period becomes small. Note that the effect of the first flow period depends not only on the time of the transient but also the magnitude of the flow rate. Thus, the "stabilized" pressures obtained on a successive flow test will depend not only on the time intervals chosen but also the magnitudes of the flow rates of all the preceding flow periods. Thus, the position and slope of a back pressure test using successive flows may depend not only on the time for each flow period but also on the magnitudes of the flow rates and the sequence used. This analysis assumes that the effective radius of drainage does not reach the extremity of reservoir during the test. )EXPONENT OF A NORMAL SEQUENCE BACK-PRESSURE PERFORMANCE CURVE In the normal sequence back-pressure test the well is opened to successively larger flow rates and the apparent stabilized bottom hole sand-face pressure obtained for each flow rate. A plot of the difference in the squares of the reservoir equalized formation pressure and the bottom hole sand-face pressure versus the flow rate usually gives a straight line of slope l/n on log log paper, represented by an equation of the form

-26 - 2 2 n q = c (p p) (56) where c is a constant which may be termed the performance coefficient or productivity index. The principle of superposition will be applied to determine whether or not the effect of using a successive flow test could influence the value of nh obtained from the test even in a reservoir obeying Darcy's law. For the purpose of illustration of method and ease of mathematical manipulation, the conditions of the test will be chosen such that the duration of each flow period is the same and the flow rates are increased in the ratio of 1: 2:3...:k where k denotes the number of the flow period. The end of the successive time intervals may be denoted 1: 201 301:.k0l, and the flow rates Q1: Q...:Q = Qi: 2Q1....:kQ. At the end of the first flow period we may write 2 2 2 Q1 8 D. G Pf- P = —n -- (57) f sl 2 4 R s At the end of the third 2 2 1 8 D (3Q1) (2Q1 - Qi) 8 D (201) P-P = -- +in in f s3 2 + 2 s3 R 4 R s s 2 (3Q1 - 2Q ) 8 D @ +,Q Q — in D 20 (58) 4 R s 2 3Q 8D0 Q Q 2 2 1 1 P P = -- in - 2 + n2 in 3 (59) s3 4 2 4 4 s Q 8 D 2 1 1 = -- in I — + - ln (3:) (60) 4 R 3_ s___

The results may be generalized for the kth flow period 22 k 8 D_ 2 2 Note that Equation (61) states that the value of (Pf - P ) obtained will be the same as if the test were a single flow test, given by the first term, plus an increase due to the preceding flow periods, given by the second term. For a series of single flow tests in a Darcy flow reservoir, the slope and the value of n should be unity and the dimensionless performance coefficient, C, should be the reciprocal of 2 1 8D ~ - In 4 R s provided the same interval is used for each test beginning from equalized shut-in conditions. Such a test is called isochronal because of the equal flow periods from shut-in. It is seen from Equation (61) that on a plot of 2 2 log (Pf - P ) versus log Q that as one proceeds from the lowest flow rate, corresponding to. the first test in a successive flow test, to the highest flow 2 2 rate that an increasing amount must be added to (Pf - P ) of the isochronal x s flow test in order to correspond to the successive flow test. This has the effect of raising each point to a higher value as one proceeds to the right and thus the slope is increased above unity and the value of n decreased below unity. If Equation (56) is used to approximate Equation 61 then the value of n should be as follows, using the fact that Qk/QI = k, and evaluating n using.the first and kth flow periods log k n _. '- -,, -, - -,, -, -.-,, -, 8 D Q 1 In( ---2 ) +2- ln(k') R k log k 2 (62) 8 D D ln 2.2 R s

-28 - It is indeed a striking result that Equation (62) gives the values of n as substantially the same function of time over a wide range of time for a wide range of values of k. Figure 15 shows that the value of n tends to approach unity provided that the intervals of flow are sufficiently large, thereby diminishing the effect of the use of successive flowso. It should be noted that the values of n obtained in this analysis was independent of the absolute magnitude of the flow rates used because of the special manner in which the ratios of the flow rates were chosen. A far more complicated function for n may be expected for arbitrary rates of flow and arbitrary flow intervals. This example illustrates the possibility of obtaining values of n less than unity for a normal sequence back-pressure test in a Darcy flow reservoir. EXPONENT OF A REVERSE SEQUENCE BACK-PRESSURE CURVE The principle of superposition will now be used to investigate the slope of the back-pressure curve for a reverse sequence test in which the well is opened to the largest flow rate first followed by successively smaller flow rates. The conditions of the test to be analyzed are that the flow intervals will again be the same for each flow rate and the comparative magnitudes of flow will be successively Q /Q2 = Q /Q3 Q 3/ = = Q /Q 2. Applying superposition for the third flow, we can obtain: 2 2 8D0 382D Q 2 P -P =- In ( R ) +l _ Iln- + - In (63) 3s 3 2 = - 21n- + in 2n2 (64) 4R 2 s The results may be generalized for the kth flow interval

-29 - m=k=l 2 2 2k m fork > 1 Equation (65) states that the values of(Pf - P ) obtained in the reverse sequence test will be higher than that of an isochronal test by the amount indicated by the second term, the amount increasing with each successive flow. But since the sequence of the test is from high flow rates to low flow rates the deviation increases as we proceed from right to left on the usual log log plot of the back-pressure curve. This tends to decrease the slope and thus raise the value of n above unity. The value of n may be shown to be as follows, using the fact k-1 that Q1/Qk = 2, and using the first and kth flow periods to evaluate n. k-l n = 8 D 2 (66) 8 D ~ ln 2 k-l Rs log 2 - 2 m=kl-i 8 DO - m+ 1 n ( — ) + Z2 nR m=l m It is again striking that Equation 66 gives the values of n as substantially the same function of time over a wide range of time for a wide range of values for k. Figure 16 shows that the values of n tends to decrease toward unity if the interval chosen for each flow period is sufficiently large, thereby diminishing the effect of the use of a successive flow test. It is emphasized again that other choices of flow rates and time intervals may give a far more complicated functional relationship for n. This example illustrates the possibility of obtaining values of n greater than unity for a:reverse sequence test in a Darcy flow reservoir.

-30 - TIME VARIATION OF THE PERFORMANCE COEFFICIENT The equation of the normal sequence back-pressure curve in dimensionless form may be written as follows: 2 2 Q =C (Pf - P ) (67) 2 2 From Equation (67) it follows that C is the value of Q when (Pf - P ) = 1, which, in dimensionless form, corresponds to the absolute open flow condition, For the normal sequence test illustration given by Equation (61) the 2 2 value of C may be obtained by setting (P - P) = 1.o Let the value of k for which this is true be k', in which case Q = C = k' Q1 Q is the value of the first flow rate, bearing in mind that in this example the actual test flow rates follow the pattern Q:Q Q2 = 1: 2:..: k. Then from Equation (61) Q, 8 D 1 1 = - n 2 + - in (k' '.) (68) 4 R k' s or C 8 D2 Q C 1 = - In - + — in ( — ) (69) 4 R s C Q1 Equation (69) is plotted in Figure 17 as essentially C as a function of time with Q1 as a parameter. For a series of single flow tests giving an isochronal performance curve the second term vanishes and C declines with an increase in the flow period used as shown by the dotted line. If a successive flow test is used, the value of C depends not only on time but also on the absolute magnitude of the flow rates used. But again if large values of time are used for the flow intervals, the decline of C approaches that for the isochronal test and the effect of both the choice of the magnitude of the flow rates and the preceding transients becomes small.

ISOCHRONAL AND OTHER TEST PROCEDURES Compare the form of the usual back-pressure curve equation with the unsteady state flow equation for constant production rate for a Darcy flow reservoir, 2 2 Q C (Pf - P) (70) 4 2 2 Q 2 ( (Pf P ) (71) 8 D D In -2 R s 2 Pf rather than unity in the dimensionless equations has been used above since the actual reservoir pressure used in these equations may be different from that used in originally evaluating the dimensionless groups. Equation (7.1) should describe an isochronal test in which each constant flow. rate is run for the time 0 9 the well then being shut in to equalized conditions before each flow period. Comparison of the two above equations show that for an isochronal test of a Daficy flow reservoir n = 1 (72) 4 C = —. 42 (73) 8 D ~ In 2 R s The isochronal back-pressure test data of Cullendar(5I indicate, however, that n is generally less than unity. Although unsteady state calculations indicate some deviation of n from unity when a successive flow test is used, no such deviation is predicted for an isoc.hronal test. The major assumptions used in this investigation are (1) the use of constant values for reservoir

-32 - characteristics, (2) neglecting the non-linearity of the basic differential equation, which depends primarily on the validity of (3) Darcy's. law. For any given test, it would seem that use of average reservoir characteristics would not be too bad an assumption provided that the pressure drop was not too great or the flow rates too high. Within the error given by Equation (14) the use of a linear approximation to the non-linear differential equation should be all right. Houpeurt(8) has investigated the effect of changes in some properties on the value of n and has found them insufficient to explain large deviations of n from -unity. He concludes that although Darcy's law may be valid for describing the flow of petroleum in porous media, it does not adequately describe the flow of gases because of the significant changes in kinetic energy due to the expansion and contraction of the gas in the irregular porous medium (4) described as turbulent flow by Cornell and Katz. Houpeurt derives the following steady flow formula.for flow between two fixed. pressures at r and r a s in a radial sy stem, in consistent units 2 2 2 2 LzRTq ra zRg Tq Pf -- P ln 2 + - 2 (74) Ps 2h2 2 wT h k M r l6 T h k r M s s where ', zeta, is a characteristic of the porous medium. This is the same as the usual Darcy formula with the addition of the second term propor-. (6) tional to the square of the flow rate. Green and Duwez and also Cornell and Katz(4) have used the quadratic form for expressing the pressure drop to take into account the deviations from Darcy's law 2 2 2 2 lzRTq r2 3 z RTq Pf - = - in + - (75) s 2 2..2 2 i hk M r 2 r h r M s s where p is a characteristic of the porous medium. Comparison of equations (74) and (75) shows that T = 8k. P (76)

-33 - It is seen therefore that the results of the above researches are the same except for the description of a constant of the porous medium. Cornell and (4) Katz correlate P with an electrical resistivity factor F, related to the deviation of the flow path from a straight line, and a constant k of the medium. If we make the rough approximations, F = 4/0 and k = 0. 0283 (k d/0) 3 derived from the work of Cornell and Katz, then we have = 3.89 X 10 F 5 — 5(77): 5/4 3i/4 2 kmd md which is plotted in Figure 18. P can also be obtained by a flow test on a core. Houpeurt illustrates the approximation of Equation (74) by an equation of the form of the usual back pressure curve, Equation 56, provided that the range of flow rates is not too great. Using Equation 34 converted to consistent units, giving the time variation of the effective radius of drainage, r, and e Equation (28) for r /r we have for the performance coefficient a e n n 1 2 Tr h k M 1 2 Tr hkM c = r = D (7 uLzR Tln — j L z R TlnL g 2g.2 s s where \ and n are numbers required to force the fit. For the purposes of establishing constants by means of a back-pressure test Equation (75) and its exponential approximation may be written as follows, using Equations (28) and (36) to substitute time for the effective radius of drainage as a variable 2 2 f - Ppsia2 2 1000 = A CFD log B thr + C MCFD 2 2 e 2 = qMC log + C F (79) r s

34 - 11 n n 2 22 2 2 1P ' s p - P Ps si 1 1ns(af $ asipsia qc - "(80) - log B thL 1000 J CL 1000 lj The reservoir characteristics equivalent to A, B, and C are given in the nornenclature. Equation (79) shows that A is the slope of a plot of(pf - p /1000 q versus log t and the intercept is log (Cq + A log B) at thr = 1. This data may be obtained. from pressure data taken as a function of time at a single flow rate. The time must be long enough so that the line from which A is evaluated repre2 2 2 2 2 sents times such that (B t )/D = r /r > 100 for 1% accuracy in (pf - p51 hr e s according to Equation 26. Note that although the deviation termin Equation (79) does not depend on time, it was derived with the understanding that r a> r.....a S and therefore the time must be large enough for this to be true. The constant C may be obtained by a series of isochronal drawdowns for several flow rates. 2 2 A plot of (pf - p)/q versus q has a slope equal to C provided the plot represents the same time from equalized shut-in conditions for all data points. The values of A and C then may be used with the intercept of the first plot (Cq + A log B) to obtain the value of B. With the value of B, a check should then~be made to be sure that the constants were determined from data for Bt which >100. If this is not true then one should not expect the derived Dc for large values of time. Isochronal data from at least two flow rates may-be used to evaluate n in Equation (80), q 2 logn = (81) 2 2 (Pf ps I bearing in mind that the pressures in the above equation must represent the same time from equalized shut-in.conditions for both flow rates. The other constants maybe evaluated from -data from a single drawdown in which the

pressure data is recorded as a function of time. Using the value of n obtained above the value of the performance coefficient q C = Pf- Ps) ooo1000 may be evaluated for each data point. Then. since 1/n 2z 2 1 (Pf - f psia2 ( ) = i/n - = ( A) log B thr (82) c 1000 qMCFD a plot of (1/c) versus log thr will have a slope of (X A) and intercept AAiog B at t - 1. hr A method of testing using a pressure build-up curve might be developed for determining the reservoir constants as suggested for A and B by Equations (53) and (54) and by Figure 13 for a Darcy flow reservoir. Equation (79) may be used even after the radius of drainage has stopped. moving, if properly interpreted. Use of time as a variable assumes that the effective radius of drainage is still moving outward to its maximum value, discussed in the section pertaining to multiwell reservoirs, and that pf is the constant formation pressure at the effective radius of drainage. The maximum radius of drainage r may be computed by, say, Equations (45) or (47), and the time, tb, required to reach this maximum may be computed from Equation (34) or (38) given as follows in terms of the empirical constants 2 2 D r t em (83) hr B.r After the time tb Equation (79) may still be used, in the form using the constant radius of drainage r and substituting pf, for pf, where pf, is the declining pressure at the outer boundary. This pressure may be predicted from Equation (22) or (37) in the following form:

2 2 (Pf - Ppsia 1000 A B f, psia 2 2 2. qMCFD (t - tb)hr 4. 606 D (r -r ) p b hr em s c, psia Equation (84) states that the pressure at the maximum effective radius of drainage declines at the same rate as the average equalized pressure of the reservoir within the maximum radius of drainage. The use of Equation (79) after steady decline begins follows from Equation (26) which states that after the radius of drainage becomes fixed the difference in squares of the pressures at the outer boundary and at the well bore becomes constant. The problem then is to compute the decline of pf, from Equation 84, then calculate p from pf by means of Equation (79). EXAMPLE BACK-PRESSURE CURVE FROM TEST DATA 2 2 Figure 19 shows a plot of (pf - ps)/q versus q for a series of 3 hour if s drawdowns from the isochronal test data for Gas Well 3 taken by M. Ho (5) 7 Cullender. The slope of this line gives the value of C = 1.32 X 10. Fig2 2 ure 20 gives a plot of (pf - p )/q versus log t for a 214 hour drawdown in which the flow rate varied from 1229 MCFD at 1 hour to 1156 MCFD at 214 hours. The slope of this plot gives the value of A = 0.0108 and the intercept to be 0. 0046 = A log B + C q. Using the value of A and C calculated above and a value of 1175 MCFD for q, we obtain a value of B = 2. 577. v Thexreforee, data should have-been usedfor times greater than about 22 hours in order for B thr /D > 100. Although annual 72 hour test data were available they were not used because of the long period between tests. The use of the three hour tests is questionable not only as to short time interval but also because they were taken 7 years after the long drawdown. The data were used for purpose of illustration. The back pressure equation in quadratic form may then be expressed as follows:

-37 - 2 2 (Pf - s)'psia' -7 2 000- = 0. 0108 qMCD log 2. 577 th + 1.32X 107 q (85) with pressure in psia, flow rate in MCFD, and time in hours. Equation.(85) is plotted-in Figure 22. The effective radius of drainage is given by 2 r' B - = - th = 4.59 th (86) 2 2 hr hr r D s In ten days the effective radius of drainage would be about 33 r if the resers, ft voir obeyed Darcy's law and had uniform properties. Because of acidization or non-uniform properties r should be looked upon as some equivalent sandface radius which may be many times the actual sand-face radius. Using the usual form of the back-pressure curve, tile value of n may be approximated from two three-hour drawdowns at different flow rates by means of Equation (81). For flow rates of 965 MCFD and 4, 318 MCFD the 2 2 2 values of (pf - p ) were respectively 10, 190 and 48, 960 (psia), from which the value of n = 0.955. The values of n for all combinations of the threehour runs ranged from 0. 948 to 0.961 with an average value of 0.956. Using 1/n the long drawdown, the value of (1/c) was calculated for each point by means of Equation (82) and plotted versus log thr in Figure 21. The slope gives the value of ~?A = 0. 00767. The intercept gives the value of A log B. 0. 0035. Using the value of -AA above, t'ea V"aiuie.of B = 2:. 86 is.obtaained which agrees fairly well with. the B. = 2. 577 from the other method. Using the value of A from the first method ' =0..7_tL isobtained. The backpressure equation in exponential form is then 0.956 2 2 0.956 f Pspsia qMCFD 0 0.00767 log 2.86 th 1000 2 (87)

or 2 2 l; 046 (Pf Pspsia = 0.00767 q log 2.86 thr (88) 1.000 where flow.rate is in MCFD, time inhours, and pressure in psia. This equationis plotted in Figure 22..To suggest the effect of using a successive flow. test for this well, ass-ume a-test as discussed in conjunction with Figure 15. If a three-hour successive flow.test is- used,.then from Equation (86) r /r = 13.76, and so 222 ra/r = 7.73. Assuming that the dotted portion of the curve is approximately correct then the slope -of the back-.pres sure'curve would be about 0. 80. This neglects deviations from Darcy's law. EXAMPLE BACK-PRESSURE-CURVE,FROM.CORE DATA The back-pressure curve, Equation (75),: may be written in the following form using time instead of the apparent radius of drainage as a variable, 2 2 -4 2 1.65~.To 5'~ 9'3 X 10 (Pf -s~psia cp R.MCD kmdPc, psia thr log 2 1000 h k r ft md cp S ft -16 2 1.109 X 10 16 - T T q MCFD + 2 (89) h r ft sft (6) (4) Green and Duwez and-also Cornell and Katz give a method of measuring P by a laboratory test. For the purposes of this example,::Equation 77 or Figure 18 is used to estimate P from the permeability and porosity. Consider the following. characteristics:

pf = 3000 psia T = 140~F = 600~R k = 50 md = 0.015 cp 0 = 0.12 z = 0.9 h = 20 ft M = 17.4 (0.6 gravity) r = 0.5 ft (3 = 203 X 10 ft s The equation of the back-pressure curve then becomes 2 2 (PfBP) psia hr X - Pa = 0.01324 qMCFD log (197,'000 th) + 1.037 X 10 qM MC00 MCFD 1000 (90) The three-hour isochronal back-pressure curve is plotted in Figure 23. CONC LUSIONS The following procedure may be used for predicting the performance of a Darcy flow reservoir for a single drawdown from shut-in conditions provided that the reservoir constants are known. The two lumped constants, A and B, from flow, tests or core data, the original formation pressure, pf, the equivalent sand-face radius, r, and the outer boundary radius, rb, are sufficient to evaluate all the groups used in. the derived equations and charts. In a multiwell reservoir, Equations (45) or (47) may be used for computing the maximum effective radius of drainage, r, which may be used instead of em r of the entire reservoir for the boundary radius to be associated with a particular well. Figure 2, which uses D - 0. 749 in the dimensional tonstant, may then be used to predict the sand-face pressure, Ps, from the flow rate, q, and either time, t, or the effective radius of drainage, r, until the time, e tb, when the maximum radius of drainage is reached. The time, tb, may be predicted from the maximum effective radius of drainage, r, by means of........ ' e ra "

Figure 9. This figure may also be used to predict the effective radius of drainage, r, for any other time less than tb. For any given. value of r the pressure profile or distribution may be computed directly from Figure 6 for points distant from the well bore but. a correction of 0...105695 should be 2 2 added to values of (pf - p )/Q read for radii near the well bore to correct for using = 0. 57722 instead of JI = 1 while r is moving. The rate of pressure decline may be found from Figure 5. This rate of decline should apply at the sand-face provided the pressure gradient is not. too steep. After the time tb the effective radius of drainage is assumed fixed. at r. The b e em pressure, pf, I at the maximum..radius of drainage may be computed as a function of time from Figure 10. Figure 6 may then be used to calculate p s from the known value of r /r by using pf, instead of pf in the figure. The S e figure may also be used to compute the entire pressure profile directly even near the well bore without making any corrections since & = 1 for a steady decline reservoir with constant re. Figure 2 may be used instead of Figure 6 to get the sand-face pressure during steady decline provided the abscissa is used in terms of radius of drainage, a value of D = 0. 606 is used, and pf, is substituted for pf. The constants A, B, and C.for use in the quadratic form of the backpressure curve, Equation (79), or ( A A), B, and n for use in the exponential form, Equation (80), may be obtained from flow tests or from core data. These equations give the sand-face pressure as a.function of flow rate, -time, and the formation pressure, pf or pf,, at the effective radius of drainage. The movement of the effective radius of drainage may be predicted from Figure 9 and pf, may be predicted from Figure 10. The principle of superposition may be used to find the net effect of a series of successive constant flow rate transients. Alternately, the graphical procedure of finite differences may be used to predict the effect of complicated boundary conditions in a Darcy flow reservoir if the constants A, B r, rb, and. pf are known.

1=141 - NOMNC~TLAT IRE Sy bol1 ~zR T R a65P A empirical constant 2.303 200 T hCaMh kOR 4D Dk p ~ Use appropriate q in emiiclcnsat2 consistent units or P 0 rNCFD with A and C. 5e93 x klO dk p cp r~mt c~si hrfor D 0.749 C'ft Esft C also dimrensionless perforrriance coe.Iffiient; of Equation (70). c usual perfr.mrance coefficient6, Equation (5) D ra/S =,0749 for r e ( rem.; 0,606 for r r F series factor defined by Equation (31). F also resistivity -factor.h thickness of the reservoir k p ei ea bil.yI k also an index ihdicLating the niunaber of a succ~essive flow test k2 reservoir constant connected with deviations from Darcy's "Law. ln natural logarithm, base. e., log coIMMon logarithm, base 10. O~verall length of a one-dimensional flow system. H molecular weight n exponent in the usual form of the back-pressure equation p dimensional pressure Pf9$Pf respectively,t the original equalized formation pressure and the formation pressure at-the maximuma radius-of drainage, rem9 which may have dropped below the original pressureppfa

P dimensionles.s pressures, pD/p1 P unty o p-#pfwhichever is appropriate. 2 pz R T Q dimensi~onless f low rate-I-.. 2kh Iff, 2, h P f kin f 9~ r R 'R g S t T 4000 A 2.3 03 PF fraction of the gas within the boundary rb which has been removed, dimensional radius' dimensionless radius r/rb gas constant =. 1545 using consistent units in f t-# ass-second System. distance between wells for square spacing., dimensional time absolute temperature, OR v Velocity x linear dimensional distance X dimensionless linear distance x/L. z compressibility factor reservoir constant ac counting for deviations from Darcy's law used by Green anM Duwez and Cornell and Katz, dimensions of length1 (Gamma) diffusivity constant 2,836 x lO f 2nsiar (Delta): %iler's constant -W0.5772 also -In D2 in which case it may be either 0.5772 or unity.* 0'porosity (lambda). constant to force quadratic form of back-pressure curve into usu al exponential forrm P viscosity p density (zeta) reservoir constant for deviations from Darcy's law, used by Houpeurt, dimensions of length.

9 dimensionless' time 13x104md ~sia t Pf r2 B ~ D2 e 2 thr 8 C rb Bsubserit a apparent radius or drainage indicated by Darcy steady flow formula b actual boundary radius usually, buit the maximum effective radius of~ drainage may be substituted for it in a multiwell'reaervoirs also, indicates that re has just reached rb or rem C average, constant pressure used to eliminate non-linearity in the basic differential1 equation. e effrective radius of drainage em maximum effeactive radius or drainage f formnation pressure at the etffective radius of drainage indieates that formation pressure at the effective radius or drainage, re has dropped below the original pressure after re becomes constant, I~vk running index maximum s sand-facea. 1.0 Aronofsky, JS.,ad Jenkins, E., "~A Simplified Analysis or Unsteady.Radial Gas- lw" Tras. A.I.M.ELi 4 (l94)4. 2. Bruce, G0H.., -Peacoman, D.,W,., Rachfforct, sr,, HJJ,, and Rice, J.,D,., to-calculations of Unsteady-State Gas Flow Through -Porous Media, J.Pet,,Tech.,,, 79 (March 19S3). 3, CarslawpH,US,., -and Jaeger, J.C., Conductio'n or Heat in Solids, Oxford University Press,, p. 2.83,v 1947.6 L4. Cornell, D',, and Katz, D.L., "~Pressure Gradients in Natural Gas Reservoirs'~, J,?Pet,,Tech'.. 198 61 (March 1953).

Cullender, M.i,, "The.1;4,ochronal Performance Method of Determining the Flow Characteristics of Gas Wells", Paper No, 330-G, A.1OXIJ4E. Dallas, Texas (1953). 6. Green,, L.,9 ad Duwez,. P,,'"Fluid Flow Throug Porous Metals", J. Appi, 7. Haymaker, F.R., Binckley, C,W.,v and Burgess, F,.,. "Method of Establishing a $ta'bilized Back'-Pressure yure for Gas Wells Producing from Reservoirs of Extremely Lw Permeability.', Petroleum Trans,s A,oI,M.eE.,$ 18-6, 77 (1949). 8. Houpeurtp A,, "Analogy of Radial Circular Transient Flow of Gazes in Porous Media".'Revue de LvInastituit Francais du Petrole et Anales des Combustibles Liquides, -8, o, 4, p.* 129, ino,, p.9 193 n, ~ 6, P. 248, (1953)e 9,8 Jakob, M., Heat Transfer, Vol. 1, John Wiley & Sos eowYork, (1950). 0.s Je'nkins, R. and Aronof sky, J.S.,, "Unst'eady'Radial Flow of Gas Through Porous Media", J.Appl.Mech.,' 20, No.o 2,9 210 (June 1953)). 1. Mac-goberts,, D,,T., "Effects of Tra n.sie'nt Conditions in Gas -Reservoirs" Trans., AJ.IMO,E., 186, 36 (1949). 2, Matth-ews, Bronz, ad Hazebroekj, "Method for Determination of Average Pr'essure in a Bounded'Reservoirl", Petroleum Trans., A.I.'M.E., 201,0 182 (1954)0 3. Mskat, M., Flow of Homogeneou.S Fluids,* NcGraw4Iill Book Compan-y, (19P7)* L. Van Everdingen, AF., and Hurst, W., "The Application of the Laplace Transformation to Flow Problems in Reservoirs," Petroleum Trans., A.I.M.E,, 186,p 305 (1949),

2 P I I I _____________________________ I I Ax -J I 2 3 x Figure 1. Solution of Non -Linear Differentia'l Equation by Graphical Method of Finite Differences 2 P f 2 P 2 P sl 2 P s2 2 ~s3 Ral a2 a3 R =lI 0 0 1 0 m.2 P I R log R R S log R. Figure 3. Darcy Pressure Distributions. Figure 4. Pressure Distri-. butions for a Succession of Steady Decline Re servoirs

Figure 2. Sand Face Pressure as a Function of Time or Radius of Drainage for Constant Rate of Production 5 U) U) a X 0 > Cr 0 N Eq. j 00 II Ni U) 2 2 2 2 8 D P 9k p D r r D c- = 5. 93 X 10-4 md Pc, psia hr D e a R2 0 2 r 2 r 2 S cp s S S s s tt

I0 U) $:4 4p IoC~I I Igo I 02 z T.R qMF h 0 (r 2-r 2) 2 ft e s ft Figure 5. Rate of Pressure De-cline Within the Effective Radius of Drainage, r, as a Function of r eand the Flow Rate, eq* N "-4 ul N.z P U 5 H E: Po 5 400 N N p4 pL4 R/R = r/r e e Figure 6. Generalized Pressure Distribution for r 2/r 2 >100. e s

1.0 0.8 0.6 e 0.4 0.2 0 I 10 R/R ff 100 200 Figure 7. Example Dimensionless Pressure Distributions for r /r = 200 and Q = 0 2. b s 0.749 0.606 l-~ - rl %%. N r /r e b I Figure 8. Ratio of the Darcy Apparent Radius of Drainage to the Effective Radius of Drainage, r /r. a e

44 I0 103 105 10' 109 10 kmd Pc, psia - -~0 ' hr cp Figure 9. Effective Radius of Drainage, r e, as a Function of Time, t, for an Infinite Reservoir.

I I 11 E0-4 1013 10 10 1 (t -tb)hr Figure 10. Boundary Pressure, pf,' as a Function of the Total Time, t,, and the Time, tbs Required for re to Reach the Boundary., rb* (Use r in This e m Figure for Multi-Well Reservoirs),.e

2 ps 2 P S 2 Pf - -T I I Dimensionless time, 0 Figure 11. Effect of Superposition of Two Successive Flow Periods on the Sand-Face Pressure of an Infinite Darcy Flow Reservoir ABD —Flow of Q 1 Only FH — Injection of Q1 Only FG — Production of Q2 Only ABC —Net Superimposed Drawdown r-. I t t2 -- (log scale) 2 1 Figure 12. Extrapolation of Build-up Pressure Data to the Original Formation Pressure of a Darcy Flow Reservoir (N N., (TJ.ax N NP r-4 N l 04.Q U W V 0 0 0 1 - I SLOPE A / I I RCEPT I LOG B I I p2 / I / /INTE =A I I a.1 10 (t2 - tlhr 100 Dinsion s time Dimrensionless time O (log scale.) Figure 13. Use of Build-up Pressure Data to Evaluate Constants of a Darcy Flow Reservoir. Figure 14. Comparison of Single Flow and Successive Flow Apparent Stabilization Pressures ABC —Successive flow test: Q1' Q2 DE — Single flow test: 0.

0.6 _ _ _ _ I We %F I I 0 2 8 D P 0 ci1 2 =D R a 2 zre r_ 2 r S S Figure 15. Effect of Superposition on Exponent, n, of a Normal Sequence Successive Flow Back Pressure Curve 'for Test Conditions: Equal Flow Periods, 0 andFlow Rates Increasing as Q:Q:Q:Q4 1:2:3:4 I. 11 6.1I I t F 0, IO2 ci1 2 R 5 I 05 I 06 t10' 2 2 2 rel al 2 2 r r 5 5 Figure 16. Effect of Superposition on the Exponent, n, of a Reverse Sequence Successive Flow Back Pressure Curve for Test Conditions: Equal Flow Periods, 0, and Flow Rates Decreasing as.: Q 1 Q2 Q Q2 Q3 Q

loll a s4 k i K& Io 10 -4 4st 9 S 10 8 -C I U 0 N 0. 10 0.20 - 0.30 - I 10, % % % I& 11. m I I 2 2 8D c 2 el al....=D --- - R ~ y S 8 8 Figure 17. Effect of Superposition on the Dimensionless Performance Coefficient; C, of a Normal Sequence Successive Flow Back Pressure Curve for the Test Conditions: Equal flow periods, 01, and flow rates increasing asQ1 QQ3:Q = 1:2:3:4 1o7' 1( _-I I 10 102 103 permeability, k, in md. Figure 18. Correlation of P with Permeability k, and porosity, 0., for correction of Devi — ations from Darcy's Law.

(d *0 CA N Q44 i4 Q U 0 0 0.0115 0.01'10 0.0105 -4 1 a 0 2000 4000 Figure 19. Graph for Evaluation of Constant C in the Quadratic Form of the Back Pressure Equation N Cd Ca Q U o* 0 0 0 N (M.rd *~CA P4 1 N 4 (^ %tor Q X 4 U 0 0 0 r-~ -i P- j (a) 10 thr thr Figure 20. Graph for Evaluation of Constants A and B in the Quadratic Form of the Back Pressure Equation Figure 21. Graph for Evaluation of Constants {(A) and B in the Exponential Form of the Back Pressure Equation

'2 4 " NM 0 104 105 10 10 qMCFD Figure 22. 72 Hour Isochronal Pressure Curves Computed Using Constants Derived from Test Data. Dotted Curve from Exponential Equation and Solid Curve 4 fr om Quadratic Equation (pf = 441 )sia) 10 103 0a o / 'qMCFD Figure 23. Three-hour Isochronal Back Pressure Curve Computed from Core Data

IE RATA APPLICATIONS OF UNSTEADY STATE GAS FLOW CALCULATIONS John D. Janicek and Donald L. Katz 1. Page 7, 'Equation (13a), for '; rCad repd 2. Pagie 10, line 12 from botVoma, for"Figure 3" read "Figure 4". 3. Page 12, Equation (20), f or " diP2 0 "read "dv P8 0 " &(In r) d(lu R) 4. Page 21, line 5 from top, for "Matthews" read "Matthews, Bronz, and Hazebreek."f 5.. Page 31, line 5 from bottom, for "Cullendar" read "Cullender." 6.Page 40, line 5 from top, for " pf p)/" reavd"(P~. r2/P 7. Page 41, in definition of D, for"D -r/r "read "D -r/e 8., Figure, 1?, abscissa should-read In lD2~e r - In r&I R2 2 r2

UNIVERSITY OF MICHIGAN 3 9015 03024 6725