Type Curves for Finite Radial and Linear Gas-Flow Systems: Constant-Terminal-Pressure Case

Transcription

1 Type Curves for Finite Radial and Linear Gas-Flow Systems: Constant-Terminal-Pressure Case Robert D. Carter, SPE, Amoco Production Co. Abstract This paper presents gas-pfpduction-rate results in type curve form for finite radiai llnd linear flow systems produced at a constant terminal (bottomhole) pressure. These results can be used in the analysis of ll"tual gas and oil rate/time data to estimate reservoir size and to infer reservoir shape. The type curves are based on dimensionless variables that are a general~zed form of those presented previously. 1,2 In addition, an approximate drawdown parameter is presented. Example applications that demonstrate the applicability of the type curves to a variety of reservoir configurations are given. The Appendix contains derivations of the dimeq~ionless variables and the drawdown parameter. Introduction The gas-bearing rock in some low-permeability gas fields consists of sandstone lenses of uncertain but limited size. In such fields, the reservoir area and volume drained by individual wells cannot be inferred frotn well spacing. Moreover, good reserve estimates using plots of I'/z vs. cumulative production are often not possible because of the difficulty of obtaining reservoir pressure from buildup tests. Therefore, reserve estimation techniques that use performance data, such as production rate as a function of time, are needed. Although this problem has been recognized, the techniques proposed in the past for application to gas reservoirs have been mostly empirical. The present work offers a method that is consistent with the basic theory of gas flow in porous media for analyzing production data to estimate reserves. This method will also provide some inference about reservoir shape. Type Curves Basic Assumptions. Six basic assumptions are made in generating the type curves. 1. The flow geometry is radial; therefore, the reservoir either is circular and is produced by a concentrically located well of finite radius or is a sector of a circle produced by the corresponding sector of the well (Fig. 1). In the limit as (re/rw)=r-4i, the flow regime becomes a linear one. 2. Permeability, porosity, and thickness are constant throughout the reservoir. 3. The pressure at the well radius (usually corresponding to the bottomhole flowing pressure [BHFPD is held constant. Copyright 1985 Society of Petroleum Engineers OCTOBER The initial reservoir pressure is constant (independent of position). 5. Non-Darcy flow is neglected. 6. The flowing fluid is either a gas with viscosity and compressibility that vary with pressure or an oil with a constant viscosity/compressibility product. Definition~. The type curves are based on specially defined dimensionless time (td), dimensionless rate (qd), a flow geometry parameter (71), and a drawdown parameter (A). These variables are defined by the following equations, which are derived in the Appendix. I424qT(lIB I) qd=... (1) akh[m(pi)-m(pwj)] _ X 10-4 X 24kt 2 td - 2 (XI.(2) CPP.(Pi)C g(pi)r w 2 71= (R -I) (::)... (3) 2 A= P.(Pi)C g(pi) X 2 Results. The type curves for rate as a function of time are presented in Fig. 2. A finite-difference radial-gas-flow simulator was used to generate the data for constructing the type curves. Two flow periods can be identified. The infinite-acting (or transient) period is that period before which the curves become concave downward. The transient period ends at td values ranging from about 0.15 to about 1.0, depending on the value of 71 that characterizes the curve. The curves are concave downward during the late-time or depletion period. Notice that the primary characterizing parameter during the infinite-acting period is 71, and A is the characterizing parameter for late-time behavior (td>i). The curves for 71=1.234 (linear flow) are straight lines with a negative half-slope during the infiniteacting period. As indicated in the Appendix, the expressions for q D and t D when used in conjunction with the parameter

2 ~ ~ a=1 a<1 a=1 10 III ~ = ~ = ~ = ~ = 1.01 ~ = ~.. = 1.. = _ qo = 1424qT(I/B,) okh[m(p;) - m(pwf)) to _ x x 24 kt <u,)2 ~(",)c (P,)'w Fig. 1-Flow system basis for type curves. Fig. 2-Radial-linear gas-reservoir type curves. are a generalized form of the definitions presented earlier. 1,2 Also, the Appendix shows that the two sets of definitions tend to coincide as R increases. Each value of 11 corresponds to a value of R. Fig. 3 is a plot of 11 as a function of R. As R increases, 11 approaches a value of 1. For values of R greater than 30, 11 ~ 1. In an earlier study, 2 constant-terminal-pressure results for liquid flow are extended to a variety of well/reservoir configurations for the late-time period during which the rate follows an exponential decline. The connection between this earlier work and the results presented in the present paper is discussed in the Appendix. The approximate correlating parameter (A) accounts for the changing value of JlC g during depletion. A A-value of unity corresponds to liquid flow (viscosity/compressibility product is constant); a A-value of around 0.5 corresponds to maximum gas-reservoir drawdown. A derivation and discussion of A appears in the Appendix. The rate/time data appearing as a log-log plot in Fig. 2 is shown as a semilog plot in Fig. 4. These curves show that only when A= 1 is the behavior consistent with exponential decline (straight line on Fig. 4 coordinates). For values of A < 1, the curves are concave upward, with the most pronounced curvature at the lowest value of A. Thus, in cases in which A < 1, a straight -line extrapolation of sernilog rate/time data may yield performance predictions that are too conservative. In the strictest sense, the type curves apply only to radial flow in a circular reservoir toward a concentric well (or in a sector of a circular reservoir spanning 21l"a radians). Nevertheless, they may be used with good approximation to match late-time data for many well/reservoir systems of nonradial shape. Tests of the type curves using simulator-generated data for a variety of well/fracture/ LINEAR CASE For Coordinates For Use In Oeterrnini'.g (1] = 1.234) Reservoir Area Or Volume From Type Curve Match Values Of The Product q 0 to '1 = (ailb,) (;) (R2-1) f- A. 1 f-- f-- A. = F= ~:==' A. = F= 1424qT(lIB, ) qo = F= okh[m(p;) - m(pwf)) ~= to _ x 10-4 x 24 ;t (a,)2 ~= f-- "'1'(PI)C g (PI)r w 1::::= l,-'-'-~"""""'~'"'-':-~~~=:;::;:;:r:::::====~ RE/RW = R Fig. 3-Type curve parameter ll(r) \ '. r- o Fig. 4-Semilogradial-linear gas-reservoir type curves. to 720 SOCIETY OF PETROLEUM ENGINEERS JOURNAL

3 ~ = ~ ~ ~ = ~ = 1.01 ~ = ~ = ~ ~ ~ = 1.01 ~ = l. - 1 l l qT(1/B, ),qo - ~kh[m(p,) - m(pw,ll i to _ x 10 4 X24r (a,)2 IW<Pi)C g (p,)r w l. 1 l. = l qT(1/B, ), qo; ~kh[m(pi) - m(pw,ll " _ x 10'4 x24kt ( )' o qlp(~)cg(pi)rw2 « I j lullllull 0.D to Fig. 5-Rate behavior of fractured and unfractured wells. Fig. 6-Example 1 type-curve match. reservoir configurations have confirmed the usefulness of these curves. Fig. 5 shows the type-curve matches for fractured and unfractured wells in small, irregularly shaped reservoirs, illustrating the difference in characteristic behavior caused by the fracture. In this case, the fracture causes the rate/time behavior to approach that of a linear flow system (R-+ 1 and 1J = 1.234); the unfractured case is matched by the type curve for 1J = (large R). In these cases, the irregular external boundary apparently does not cause the rate/time behavior to differ significantly from that of the type-curve system. The type curves cannot be used directly to analyze the behavior of wells connected to two or more independent- 1y acting reservoir segments of differing size, shape, and/or (k/</». In such cases, the curves can be used in decomposition or trial-and-error history-matching procedures, which are beyond the scope of this paper. For multiple reservoir-segment cases, the early rate data plotted on the type curve graph can fall above and have a steeper slope than the curve for linear flow (1J= 1.234). The type curves can be applied to constant-terminalpressure-rate data from a single-phase liquid system, such as an oil reservoir above the bubblepoint. This can be a useful addition to other pressure-transient results because the reservoir size of a slightly compressible liquid reservoir often can be estimated from the production of a relatively small fraction of the original fluid in place. Use of the Type Curves to Estimate Gas or Oil Content Matching Procedure for Gas Wells. To analyze gas well data, use the following type-curve-matching procedure. 1. Using Eq. 4, compute A for the system and data set of interest. This computation requires information that is normally available. Data items needed to determine A are initial reservoir pressure, Pi; BHFP, P wf; and curves or tables of viscosity, z-factor, and m(p) as functions of pressure covering the pressure interval from P wf to Pi' Although in principle it is possible to determine A from the type-curve match itself, this is not recommended. This parameter should be determined before type-curve matching is done, and the type-curve match should honor the calculated value of A. For undersaturated oil reservoirs, A=1. OCTOBER Plot rate q (MscflD or MMscflD) as a function of time (t) in days using the same log-log scale as the type curves. (Although qd is expressed in Mscf/D, either Mscf/D or MMscflD may be used in this plot.) "Smooth" rate data should be used. If actual rate values are erratic or fluctuate, it may be best to obtain averaged values of rate by obtaining the slope of straight lines drawn through adjacent points spaced at regular intervals on the plot of cumulative production vs. time. This plot should be made on tracing paper or on a transparency so that it can be laid over the type curves for matching. 3. If the computed value of A for the data to be matched is not the same as one of the values for which a type curve is shown, the needed curve can be obtained by interpolation and graphical construction. Graphical quadratic interpolation is recommended.. 4. Match the rate data to a type curve corresponding to the computed value of A. The type~curve-matching process is described by Earlougher. 3 From this match, values of q D and t D corresponding to specific values for q and t are obtained. Also, a value for 1J should be obtained from the match. It is strongly emphasized that latetime data points are to be matched in preference to earlytime data points because matching some early rate data often will be impossible. 5. An estimate for the gas that would be recoverable by reducing the average reservoir pressure from its original value to Pwt is then given by where the units for gas quantity (Mscf or MMscf) are determined from the units for gas production rate (Mscf/D or MMscflD). Original gas content can be determined as (~) G(pi)=AG z I (6) (~)i- (~)Wf 721

4 TABLE 1-DAT A FOR EXAMPLE PROBLEM 1 Pressure (psia) ,201 1,801 2,401 3,001 3,601 4,201 4,801 5,401 Viscosity (cp) z-factor are q=1 MMscf/D [ x 10 3 std m 3 Id], qd=0.605, 71=1.045, t=l,ooo days, and td=1.1. With Eq. 5, 1 X 1,000 x =2,860 MMscf. 0.55xO.604xl.l With Eq. 6, Time (days) ,600 2,000 3,000 5,000 10,000 Production Rate (MMscf/O) 'For this problem, Pi = 5,400 psia; P wi = 500 psia; T = 726 R; h = 50 It; <I> = 0.035; s w = 0.5, A = 0.55 Matching Procedure for Oil Wells. The type curves for A= 1 can be applied to oil reservoirs above the bubblepoint. The procedure is similar to that described for gas wells. Production rate in STB/D is plotted vs. time (in days) on the same log-log scale as the type curves. After a match is obtained on one of the curves for A = 1, original oil in place tnay be computed with 4 It is clear from Eqs. 5 and 7 that determining reservoir gas or oil content does not require knowledge of the values of parameters appearing in the equations defining q D and t D. Such values would be needed only if estimations of permeability, reservoir areal extent, or shape were to be attempted. Applications of the Type Curves Example Calculation 1. This example performance data set was obtained from a simulator run in which a fractured well and its drainage area were matched. The geometry was nonradial. Input data and resulting output are given in Table 1. The calculated value of A for this problem is 0.55; therefore, the type curves for a A value of 0.55 can be used directly from Fig. 2. The early portion of the rate/time data before about 20 days cannot be used with the type curves of Fig. 2. This reinforces the statement that the late-time data points are always to be matched in preference to early-time data. Fig. 6 shows the type-curve match of the data. (The rate scale in MMscf/D and the time scale are not shown.) Match points 722 =3,176 MMscf. From simulator results, the correct value for G(p i) is 3,087 MMscf [88.4 x 10 6 std m 3 ]. Therefore, the error in the type curve determination is 3,176-3,087 Error(%)= xl00=2.9%. 3,087 To obtain a definitive match, it is necessary to have some data extending into the portion of the type curves that are concave downward (td ;;;0.7 to td ;;;2, depending on the value of 71). The drainage area, S, for the well can be estimated with the following equation: G(Pi) S=------~-~ Pi) 520x he/> ( -- Zi 14.7T... (8) Substituting values frotn Table 1 and type-curve-match results where applicable yields 3,176 S= ( )-( X-l-0--~6--)' (50)[(0.07)(1-0.5)] -' x726 or S=7.19x10 6 sq ft, or S=165 acres. Input data for the drainage area dimensions are for a drainage area of 160 acres [64.75 hal. SOCIETY OF PETROLEUM ENGINEERS JOURNAL

5 AG = G(Pi) - G(Pwd (BCF) CASE CONFIGURATION FROM TYPE CURVE ACTUAL ERROR CJ % 2 CO % 3 OJ % 4 CJ % 5 I-I % % Infinite 7 Conductivity % Fracture " '---" %... ~--800' c=well % Fig. 7-Test application result summary. Once a type-curve match is obtained, a prediction of future performance is generated by simply converting q D - t D rate/time pairs along the matched type curve. Example Calculation 2. A well has been on production for 300 days. All the rate data fallon a negative half-slope line. The current producing rate is 20 Mscf/D [0.57 X 10 3 std m 3 /d]. The computed value for A is OCTOBER 1985 A definitive type-curve match cannot be made for this well because no data correspond to the concave downward portion of the type curve. However, a lower bound estimate for G(Pi)-G(Pwj) can be made. All the data will fall on the early portion of the type curve for 1]=1.234 and A=0.55. If the last data point is assumed to be matched at the end of the negative halfslope straight line on this curve, the following match 723

6 points are obtained: q=20 Mscf/D [0.57 x lo3 std m 3 /d] qd=0.68, 1]=1.234, t=3oo days, td =0.6, and A=0.55; therefore, a lower bound value for 20x300x G(p i) - G(p wj) = X x-0.-6 =32,995 Mscf. Because the well has produced 12 MMscf [343.6xlO 3 std m 3 ], gas remaining to be produced (to reduce the average reservoir pressure to Pfj) is at least = MMscf [944.8 x lo x lo3 =601.2 x lo3 std m 3 ]. Additional Simulator Tests of the Type Curves Fig. 7 presents a summary of nine tests of the type curves. in which the performance data for each test were generated with a gas reservoir simulator. Of these, six well/reservoir systems were symmetrical, and three were asymmetrical. The predicted well performances were type-curve matched to obtain estimates of fig. Fig. 7 presents a comparison of the computed and actual values of G(Pi)-G(Pwj). They are in good agreement. These results suggest that the type curves should be applicable to a wide variety of well/reservoir configurations. Summary 1. Log-log type curves have been developed for ptoduction rate as a function of time for flow in bounded radial and linear gas and liquid systems having constant pressure at the outflow face. The type curves are based on definitions of dimensionless rate, q D, and time, t D, that are generalizations of those presented earlier. The new, generalized, dimensionless quantities permit representation of solutions for R approaching unity. Hence, flow in linear systems may also be represented. 2. The type curves are also based on a drawdown parameter, A, that permits good approximate representation of real gas flow with a single set of curves. 3. When the type curves are plotted in semilog form (log qd vs. td on a linear scale), all the curves are concave upward except for the liquid flow case (A= 1). Thus ithas been shown that straight-line semilog extrapolations of gas-well rate data may predict rates and recovery that are too low. 4. The type curves may be used with good approximation to describe the late-time behavior of many gaswell/reservoir systems that are neither radial nor linear and in fact may be asymmetrical. 5. For many well/reservoir systems, the rateitime data can be used in a simple type-curve matching and calculation procedure to estimate original gas in place. However, data are required over that portion of the producing history corresponding to a t D value of at least 0.7 to 2 for reliable estiml,ltes. If only data corresponding to values of t D less than this range are available, it may be possible to obtain only a lower-bound estimate of gas in place. Nomenclature 724 A = area of flow system in the plane normal to the direction of flow, sq ft [m 2 ] B j = coefficient of the jth term in Eq. A -15 as defined by Eq. A-16 C e = effective compressibility, psi- I [kpa -I] (as defined in Ref. 5) C g (p) = gas compressibility, defined by Eq. A-4, psi- I [kpa-l] Co = compressibility of oil, psi -I [kpa -I ] GDR,G DL = dimensionless cumulative production expressed as a fraction of the gas that would be recovered by reducing the average reservoir pressure from Pi to Pwj G(p) = reservoir gas content corresponding to average reservoir pressure p, Mscf or MMscf [std m 3 ] h = pay thickness, ft [m] J n = Bessel function of first kind of order n k = effective permeability, md L = length of linear system parallel to the direction of flow, ft [m] m(p) = real gas pseudopressure ot real gas flow potential, defined by Eq. A-2, psi2/cp [(kpa)2/(pa's)] P = pressure, psia [kpa] Pi = initial reservoir pressure, psia [kpa] Pwj = bottornhole flowing pressure, psia [kpa] P' = pressure variable of integration, psia [kpa] q g = gas production rate, Mscf/D [std m 3 /d] q D = dimensionless rate used in the type curves, defined by Eq. A-18 qdr = dimensionless time defined by Eq. A-lO qo = oil production rate, STBID [stock-tank m 3 /d] r e = external radius of type curve reservoir region, ft [m] r w = internal radius of type curve reservoir region, ft [m] R = radius ratio r e/r w s w = water saturation, fraction S = well drainage area, acres [hal t = time, days [d] t D = dimensionless time used in the type curves, defined by Eq. 2 t DL = dimensionless time as defined by Eq. A-12 tdr = dimensionless time defined by Eq. A-9 T = reservoir temperature, OR [K] Y n = Bessel function of second kind of order n z(p) = gas deviation (or supercompressibility) factor, dimensionless a j = time coefficient in the jth term in Eq. A-15 as defined by Eq. A-17 1] = type curve parameter characterizing effect of radius ratio R, dimensionless, defined by Eq. 3 SOCIETY OF PETROLEUM ENGINEERS JOURNAL

7 A = type curve parameter used to characterize gas well drawdown, dimensionless, defined by Eq. 4 Il = gas viscosity, cp [Pa' s] IlC g = mean value of viscosity/compressibility product over the pressure interval Pi to Pwf, cp/psi [Pa' s/lcpa], defined by Eq. A-5 or A-6 (J = fraction of 27r radians defining the concentric ring sector constituting the type curve approximation to the reservoir shape, dimensionless cp = hydrocarbon porosity (for a gas reservoir, usually porosity times [1-s w)), fraction References 1. Fetkovich, M.J.: "Decline Curve Analysis Using Type Curves," J. Pet. Tech. (June 1980) Ehlig-Economides, C.A. and Ramey, H.J. Jr.: "Pressure Buildup for Wells Produced at Constant Pressure," Soc. Pet. Eng. J. (Feb. 1981) Earlougher, Robert C. Jr.: Advances in Well Test Analysis, Monograph Series, SPE, Richardson, TX (1977) 5, Craft, B.C. and Hawkins, M.F.: Applied Petroleum Reservoir Engineering, Prentice-Hall Publishers, Inc., Englewood Cliffs, NJ (1959) AI-Hussainy, R., Ramey, H.J. Jr., and Crawford, P.B.: "The Flow of Real Gases Through Porous Media," J. Pet. Tech. (May 1966) van Everdingen, A.F. and Hurst, W.: "The Application of the Laplace Transformation to Flow Problems in Reservoirs," Trans., AIME (1949) 186, Kidder, R.E.: "Unsteady Flow of Gas Through a Semi-Infinite Porous Medium," J. Applied Mechanics (Sept. 1957) APPENDIX Derivation of Type Curve Variables and Parameters and Validation of the Type Curves The type curves were generated from numerical solutions of unsteady-state radial gas-flow and exact liquid flow solutions. The definitions for q D, t D, and 1/ were derived by use of exact liquid flow solutions. These variable definitions, therefore, are rigorously correct for liquid flow (A= 1). From the numerical gas flow solutions, it was found that they were also valid for gas flow (A < 1). The following provides derivations of A, q D, t D, and 1/. The basic assumptions are given in the text. Derivation of A. A parameter A that approximately correlates the drawdown behavior of different constant-terminalpressure gas flow systems is derived. This parameter makes possible the concept and creation of generalized type curves that can be used (with good approximation) with any natural gas composition, initial reservoir pressure, and outflow face (BHFP) pressure. By appropriately modifying the viscosity and z factor (as functions of pressure), it is also possible to determine values of A that account for confining-pressure effects on permeability and porosity, assuming that confining pressure is a function only of local reservoir pressure. First, recall that the time rate of change of gas content at a point in a gas reservoir (for isothermal flow) is pro- OCTOBER 1985 portional to o(p/z)/ot. If Darcy's equation holds, the partial differential equation describing the flow of gas through a porous medium is L[k,m(p)] = vo(p/z)/ot. The specific form of the operator L depends on the coordinate system used and other considerations. By use of the chain rule, it can be shown that o(~) = d(~) om where ot dm ot P p'dp' m(p) =2 I Pc Il(p ')Z(p ') It also can be shown that d(~) 1 ~=2[Il(P)Cg(p)], where... (A-I)... (A-2)... (A-3) Cg=;(l-~ :)... (A-4) A result equivalent to Eq. A-3 was given by AI-Hussainy et al. 5 Now determine a mean value for the product Il(p)c (p) over the interval (Pi, Pwf)' Replacing the derivative on the left side of Eq. A-3 with a finite-difference quotient and solving for the viscosity/compressibility product, one obtains or - [(~)i- (~)wj p.c =2 g [m(pi)-m(pwf)] Pi p'dp' ) Il(p ')z(p ') Pwf,... (A-5).... (A-6) Eq. A-6 is almost the same as Eq. 34 of AI-Hussainy et al. 5 The distinction is that Eq. A-6 is in terms of fixed pressure values Pi and Pwf' whereas AI-Hussainy et al. were concerned with an average pressure p, which is a function of time. Finally, define a parameter (A) as A= Il(p i)c g(pi),... (A-7) Il C g 725

8 TABLE A-1-REAL GAS PROPERTIES USED TO TEST THE I.. CONCEPT Pressure Viscosity m{p) cg{p) {psi a) (cp) z-factor (psi 2 /cp) (psi -1) x x10-2 1, x x , x x , x x10-3 2, x x10-3 3, x x , x x10-3 4, x x10-3 4, x xlO- 3 5, x x , x x , x x 10-4 ~ = ~ = ~ = ~ = 1.01 ~= II ; ; o. 1 ). = 1 ) _.. ). =.55 REAL GAS FLOW SIMULATOR RESU LTS (~ ) XA =.55 (Cases 1 and 4) ea.75 (Cases 2,5. and 6) AA =.99 (Case 3) 1424qT(lIB, ) qo - ~kh(m(p,) m(pw')] '0_2.634,,0 ',24k' ( 4,)' f{i~(p.)cg (P.)'w Fig. A-1-Results validating the I.. concept. TABLE A-2-PRESSURE TRAVERSES USED TO TEST THE I.. CONCEPT Case Pi (psia) Pwt (psia) 6,000 1,636 6,000 3,547 6,000 5,910 4, ,000 2,173 2, A or, substituting Eq. A-5 into Eq. A-7, then dimensionless cumulative production is expressed as and is the gas produced to time t expressed as a fraction of the gas recoverable by reducing the average reservoir pressure from Pi to Pwj If dimensionless time and rate, respectively, are defined for the linear system by x 10-4 x 24kt cpp-(p i)c g (p i)l 2... (A-12) which is Eq. 4 of the text. A value for A of unity corresponds to liquid flow (or gas flow in the case where (Pi -Pwj)/Pi 4, 1). For an ideal gas, taken herein as one with constant viscosity and z factor, Eq. A-8 reduces to and then (211')(1,424)LT q '--,... (A-13) Ak[m(p i) - m(p wf)] For a producing well, O<Pwj<Pi' Therefore, with substitution of the limiting values zero and P i for P wf into the above expression, it is found that 0.5 < A < 1.0 for an ideal gas. Some Properties of 1 If dimensionless time for the radial flow system is given as tdr = 2.634x 10-4 x 24kt,... (A-9) cpp-(pi)c g(pi)r~ and dimensionless rate is given as 1,424qT qdr=,... (A-10) akh[m(p i) - m(p wj)] 726 GDL=AJ o rtdl qdldtljl,... (A-14) where G DL is the gas recovered at time t expressed as a fraction of the amount recoverable by reducing the average pressure from Pi to P wj. Thus A is a factor that relates dimensionless gas recovery to the integral over dimensionless time of dimensionless rate. Consider two flow systems that have the same flow geometry but different gas compositions. If the pressure traverses (Pi, Pwj) for the two systems are selected so that each system has the same value of A (the pressure traverses will not be the same in general), then the rate/time curves expressed in dimensionless form will be in very close (but not exact) agreement. Thus it is possible to construct rate/time depletion type curves for gasflow systems based on one gas composition that can be applied with ~cceptable accuracy to systems having different gas compositions. SOCIETY OF PETROLEUM ENGINEERS JOURNAL

9 The greatest disagreement between two systems will usually occur at later times when prediction accuracy is less important. At very late times when the average reservoir pressure is approaching Pwl' the final rate/time behavior should approach an exponential decay curve determined by Il(Pwf)c g(pwf) This time portion of the rate behavior may be of less interest because often it will occur near the point of abandonment. Two systems with different gas compositions will not usually have the same value for Il(p w/)c g (p wi) even though they have a common value of A. For convenience, the type curves were generated with an ideal gas system and then compared and tested with simulator results for a real gas system whose properties are given in Table A-I. Several comparison cases were run in which specified values of A were obtained by the use of different pressure traverses with the real gas system of Table A-I. The pressure traverses and corresponding values of A are in Table A-2. Values for A were computed using Eq. A-8. Notice that more than one pressure traverse can yield a given value for A. The real gas simulator results are shown plotted as points on the type curves in Fig. A-I. These results are for re/rw=r= 1,000, but similar agreement was obtained also for flow in a linear system. Derivation of the Dimensionless Type-Curve Variables qd, td, and 11 The type curves are based upon radial-linear flow systems. The exact liquid (A= 1) flow solution for cumulative production as a function of time for radial systems is given by Eq. VII-to of Ref. 6. The exact solution for production rate as a function of time can be obtained by the use of the first derivative with respect to time of this equation. The result is 00 qdr(tdr)= ~ Bje-a.}tDR, where and j=1 and a j is the jth root of (A-IS) sionless variables can be obtained by normalizing on the basis of the first-term coefficients B I and a I. Thus qdr 2 qd=--, td=altdr BI Notice that B I and a I in this case are obtained for radial geometry with Eqs. A-I6 and A-17. It is also possible to use the exact solution for linear liquid flow to define a q D and a t D in terms of B I and a I for linear flow; qd(td) for radiall10w and qd(td) for linear flow will coincide for large t D. In fact, it can be shown that based on radial flow is equal to qd(td) based on linear flow. In terms of qd and td, Eq. A-IS becomes At large values of t D, the terms under the summation sign will become small relative to e -t D. Therefore, all the type curves for different specific values of R will tend to coincide for large values of t D. The parameter 1/ is a function of R and is used to compute drainage area gas content. It is obtained by noting that X (Jhcf>r~ll(p i)c g (p i)[ m(p i) - m(p wi)] or, by use of the definitions of q D and t D,.6.G=~ (~) (R2-1) at. qdtd A 2 BI Thus, 1/ is defined by T.. (A-I9) (A-20) n= (R2-I) (at),./... (A-2I) 2 BI and Eq. A-20 may be written as The values of aj are real and distinct. Also, al <a2 <a3. For the sake of compactness, it is desirable to develop a coordinate system for areal-extent (reservoir limit) type curves that bring the curves together at large values of time. In view of the foregoing equations, it is clear that at late times the first term on the right side of Eq. A-IS will predominate. Therefore, the needed system of dimen- OCTOBER G= ~..... (A-22) qdtda Original gas in place is given by... (A-23) 727

10 The parameter 'f/ is shown as a function of R in Fig. 3. Notice that 'f/-+ 1 as R increases. Fetkovich I and Eh1ig-Economides and Ramey2 have shown in effect that, when R is large (i.e., greater than 30), and 0(1 == (R22~ 1) + (In R-~)... (A-24) BI == 1+ (In R-~).... (A-25) Therefore, for R>30, Validation of the q D -t D -11 System for A. < 1 Numerical solutions were obtained for constant-terminalpressure radial ideal gas flow for values of A of 1, 0.75, and When these results were plotted in terms of qd and t D, they formed distinct curves at late times determined by A-values only. At early times, the curves tend to be defined by 'f/ (or equivalently by R). As 'f/ approaches its limiting value as R approaches unity, early-time separation of the curves for different A values is manifested. The validity of the early-time behavior of the curves as R approaches unity (linear flow) was confirmed by a study utilizing Kidder's expression of the semi-infinite gas flow problem with constant terminal pressure. 7 All of these characteristics can be observed in the basic log-log type curves shown in Fig. 2. Gas Flow Behavior as R Approaches Unity (Linear Flow) As R-+ 1, it can be shown that ( R2-1) 0(1. 0(1 limn../ (radial) =lim --- ~=~. R--+I R--+I 2 Blradial Bilinear Thus a limiting value for 'f/ as R-+ 1 is available. Because 11" 0( =-and B =2 1 linear 2 1 linear ' then. (~)2 'f/linear= hm 'f/radial = --- = R--+I 2 This value is shown for 'f/(r= 1) in Fig. 3. This value permits a smooth continuation to R = 1 of the function 1/(R). Relation Between the Type Curves and the Results of Ehlig-Economides and Ramey. Ehlig-Economides and Ramey2 show that late-time rate behavior for liquid flow in a variety of reservoir shapes can be expressed by an equation involving an exponential in time. This equation includes a shape factor to account for flow geometry. When this equation is normalized with the same procedure as described in this Appendix for these type curves, the result is equivalent to that obtained with the late-time portion of the Fig. 2 type curves with a value of 'f/ of unity. The exponential equation result of Ref. 2 relies on the assumption that the Laplace transform of a specific latetime approximation to the pressure solution for constant rate can be used in Eq. VI-32 of Ref. 6. This will not always be possible for every configuration. SI Metric Conversion Factors ft x 3.048* E-Ol psi -1 x E-04 psi x E+OO OR x 5/9 scf x E-02 *Conversion factor is exact. m Pa- I kpa K std m 3 SPEJ Original manuscript received in the Society of Petroleum Engineers office May 24,1984. Paper accepted for publication Feb Revised manuscript received Feb Paper (SPE 12917) first presented at the 1984 SPE Rocky Mountain Regional Meeting held in Casper May SOCIETY OF PETROLEUM ENGINEERS JOURNAL