Pressure Buildup Analysis With Wellbore Phase Redistribution

Transcription

1 Pressure Buildup Analysis With Wellbore Phase Redistribution Walter B. Fair Jr., SPE, Shell Oil Co. Abstract This paper presents an analysis of the effects of well bore phase redistribution on pressure buildup tests. Wellbore phase redistribution is shown to be a well bore storage effect and is incorporated mathematically into a new solution of the diffusivity equation. Dimensionless pressure solutions based on an infinite radial reservoir are presented for typecurve matching to analyze pressure buildup tests influenced by well bore phase redistribution, and example analyses of actual field data are included. The parameters that affect phase redistribution and gas humping are documented also. This information permits analysis of many anomalous pressure buildup tests which previously could not be analyzed quantitatively. Introduction Pressure buildup tests and other types of transient pressure tests have been used for many years to evaluate reservoir fluid flow characteristics and well completion efficiency. The basic theory and equations for the analysis of these tests are well documented.! M~ny factors that influence the pressure response in transient flow conditions have been investigated - i.e., the effects of reservoir boundaries, heterogeneities, and fractures, wellbore storage of fluids, and various types of well impairments, skin effects, and completion practices. However, little information concerning the effects of the redistribution of gas and liquid phases in the well bore has been presented. The phenomenon of well bore phase redistribution occurs in a well which is shut in with gas and liquid flowing simultaneously in the tubing. As shown by Stegemier and Matthews, 2 when such a well is shut in at the surface, gravity effects cause the liquid to fall /81/ $00.25 Copyright 1981 Society of Petroleum Engineers of AIME APRIL 1981 and the gas to rise to the surface. Because of the relative incompressibility of the liquid and the inability of the gas to expand in a closed system, this redistribution of phases causes a net increase in the well bore pressure. When this phenomenon occurs in a pressure buildup test, the increased pressure in the well bore is relieved through the formation, and equilibrium between the well bore and the adjacent formation will be attained eventually. However, at early times the pressure may increase above the formation pressure, causing an anomalous hump in the buildup pressure which cannot be analyzed with conventional techniques. In less severe cases, the well bore pressure may not rise sufficiently to attain a maximum buildup pressure. General analyses of well bore phase redistribution have been presented by Stegemeier and Matthews 2 and by Pitzer et al. 3 Both of these investigations documented the association of the pressure buildup hump with phase redistribution and indicated that the size of the hump was correlated with the amount of gas flowing in the tubing. Stege meier and Matthews also noted an apparent correlation between estimated gas rise velocity and the time at which the hump occurred. Earlougher! also noted (on the basis of the shape of the log t:..p vs. log t:..t plot of buildup test data) that phase redistribution seems to be related to the problem of well bore storage. Other authors have recognized the significance of well bore phase redistribution; however, no complete analysis of the phenomenon has been presented and general methods for analyzing buildup data influenced by phase redistribution in the wellbore have not been available. Mathematical Analysis of Phase Redistribution If we consider a well where well bore phase 259

2 redistribution occurs, it is apparent that well bore storage also must occur. If the well bore could not store fluids of finite compressibility, the phase redistribution process could either (1) physically not occur or (2) be associated with a zero pressure increase. It is also interesting to note that the techniques presented by Stegemeier and Matthews 2 and Pitzer et al. 3 for minimizing wellbore phase redistribution also minimize well bore storage effects. For a well where well bore storage occurs the effects of the storage can be described by Eq. 1.4 The effect of the changing sand-face flow rate on the wellbore pressure also can be obtained from Eq. 2.!bi.. = 1- CD dpwd... (1) q dt D dpwd _ 1!bi (1 - )..... (2) dt D CD q To describe the effect of wellbore phase redistribution, note that not all of the pressure change in the wellbore can be attributed to well bore storage flow rate effects, since some of the pressure change is caused by phase redistribution. Thus, Eq. 2 can be modified by adding a term describing the pressure change caused by phase redistribution, as in Eq. 3, which also can be rearranged to show the sand-face flow rate dependency in Eq. 4. dpwd 1!bi.. dp<f;d -- = - (1- ) (3) dt D CD q dt D!bi.. = 1-CD ( dpwd _ dp<f;d )... (4) q dt D dt D Eq. 4 also can be written in the form of Eq. 1 by defining a pseudowellbore-storage coefficient given in Eq. 5. CD = C D (1- dp<f;d / dpwd )... (5) e dt D dt D In this form, it is apparent that wellbore phase redistribution is a form of well bore storage, since when dp<f;d -- ~O, C ed $,C D, dt D which implies that the effect of phase redistribution always will cause an apparent lowering of the wellbore storage coefficient given by Eq. 5. In addition, when dp<f;d dpwd -->--, dt D dt D the pseudostorage coefficient becomes negative, indicating a reversal in the direction of flow. When this occurs, a pressure buildup test becomes more like a pressure falloff test, and the gas hump results. By considering the physical process of phase redistribution, certain properties of the phase redistribution pressure function P <f;d can be inferred, even though few published data are available to determine the functional form. If the gas and liquid phases in the well bore before shut-in behave as a 260 homogeneous fluid (i.e., the well is not "heading"), the required pressure function must have a value of zero at shut-in (time zero). Also, at long times, the phase redistribution must stop so that its derivative with respect to time must approach zero. If it is specified further that no gas enters solution in the liquid phase, then it can be shown that the pressure function must increase monotonically to its maximum value. These conditions are described by Eq. 6. limp<f; =0. t-o.... (6a) lim P<f; = C<f;' a constant.... (6b) t-oo lim dp<f; =0. t-oo dt... (6c) Furthermore, in considering the effect of gas bubbles or slugs rising through a column of liquid, note that when the first gas bubble or slug reaches the surface after shut-in, the pressure in the well bore must increase by some amount. This pressure increase causes a decrease in the volume and an increase in the density of all other gas bubbles or slugs. Both of these effects cause a decrease in the rise velocity of all the remaining gas, so that the rate of pressure change therefore must decrease. The same argument can be made for gas bubbles or slugs reaching the surface at later times. In addition, since gas bubbles and slugs in an actual well bore may be of widely varying initial sizes, their rise velocities will be distributed over a considerable range. Therefore, it is expected that the phaseredistribution pressure initially would rise quickly and then slowly approach its maximum value C</>. This observation leads to the exponential function III Eq. 7, which satisfies the constraints of Eq. 6. Also, the one available set of unpublished laboratory data on the phase-redistribution pressure seems to confirm the following functional representation. * P<f; =C<f;(1-e- t10i )..... (7) In Eq. 7, the parameter C<f; represents the maximum phase redistribution pressure change and a represents the time at which about of the total change has occurred. An estimate of C</> can be obtained by noting that the gas in the well bore will rise to the surface with the total gas volume remaining constant; this is caused by the assumed incompressibility of the liquid in the wellbore. C<f; can be estimated by Eq. 8 when (1) the gas/oil ratio in the well bore is assumed constant, (2) temperature effects are neglected, (3) the liquid is assumed incompressible and the gas ideal and weightless, and (4) a linear increase in well bore pressure with depth applies. A more general method for estimating C<f; can be derived from Appendix 1 in Ref. 2. C<f; = Pgej-Pwhj..... (8) In(!!EL ) Pwhj 'Personal communication from G.L. Stegemeier, Shell Oil Co., March SOCIETY OF PETROLEUM ENGINEERS JOURNAL

3 While a is not determined as easily, it is known that it will depend mainly on those factors which control the gas bubble or slug rise time in a well. Finally, to keep the dimensionless quantities consistent, the dimensionless phase-redistribution pressure function is defined in Eq. 9. where khp P D = qb~ C _ kh C D qb~' kl ID= 2 ~Ctr w ' and ka ad= 2 ~Ctr w In SPE preferred SI units, replace with x 10-6 and with 3.6 x Determination of Dimensionless Wellbore Pressures To obtain dimensionless pressure solutions for use in the analysis of pressure buildup tests, it is necessary to incorporate the effects of well bore phase redistribution into the diffusivity equation. For radial flow in an infinite, homogeneous, isotropic reservoir of a fluid of small compressibility, this problem is stated in dimensionless variables as follows. The diffusivity equation is a 2 pd 1 apd apd =-... (10) ard rd ard aid The boundary conditions are PD (rd'o) =0... (11) lim PD(rD,I D ) =0... (12) rd- ex> _ ( a PD ) =1_cD(dPWD _ dp D). ard rd=1 did did... (13) P D = [PD _s(a PD )] _... (14) ward rd-i Several authors 5 have shown that this problem also can be written as a convolution integral to account for well bore storage. This approach leads to Eq.15. APRIL I'lXI. [dpwd(td) _ dp D(tD) ]1... (15) did did j Eq. 15 can be solved for (PwD)' the Laplace transform of the desired pressure function. This results in Eq. 16. (s denotes the Laplace transform variable.) [s (PD) +S][1 + CDS 2 (P D)] (p D) =. w s[1 +CDS(S (PD) +S)]... (16) Note that the solution still is entirely general, since no constraints have been placed on either P D or P D' except that these functions exist and are Laplace transformable. Thus, if P D represents any type of reservoir condition, the required pressure solutions for those conditions can be determined in principle. This statement also applies to the phase redistribution pressure. In this work, Eq. 9 is used for the phase redistribution effect. Its Laplace transform is: C D C D (P D) = (17) S S+ 1IaD The required expression for (p D) has been presented by Van Everdingen and Hurst 4 as Eq. 18, where Ko and Klare modified Bessel functions. Ko(Ys) (PD) = s312 KI (Ys)'... (18) It also has been shown that at long times this simplifies to the line source solution in Eq. 19, since YsKI (Ys)-1 whens-oor/d-oo. 1 (PD) = -Ko(Ys)... (19) s A further long-time approximation for (P D) can be obtained by noting that as ID -oo,s-o, and Ko(Ys) - Ys - [In( 2 +1'], where 1'= denotes Euler's constant. This gives Eq. 20. (PD) = - ~ [In(~) +1']... (20) Combining the definition of (P D) in Eq. 17 with the various forms of (P D) from Eqs. 18, 19, and 20 yields required expressions for (PwD) as follows. Cylindrical Source Well. [ Ko(Ys) ] [ 2 ( 1 1)] ski (Ys) +S 1 + CDC Ds ~ - s+ 1Ia D { 1 + K (Ys) ]J CDs [ 0 Ys +S ski ( s)... (21) 261

4 Line Source Well. C \(s[1 + CDs[KO (Vs) +SlJ)... (22) Line Source Well Long-Time Approximation. Fig. 1 - Type curves with phase redistribution (CaD = 20, CD = 100). "wo 'OO-~-----,----:=cr-----,-----, , [ Vs J [S-ln(-) -1'][1 + CDC<I>Ds (- - )] 2 s s+ liad \ (s[1 + CDs[S-ln( ~) -I'll)..... (23) The long-time approximating form of the PwD function can be derived from Eqs. 21,22, or 23 by noting that C~o ]() Co 1000 O>,'-;-o'----''~, ----"-':-'--,-, ---",,'----'-,"'-----',,' Fig. 2 - Type curves with phase redistribution (CaD = 20, CD = 1,000). S2(~- 1 )-0 as s-o(t D - oo). s s+ liad Thus, these equations reduce to the well bore storage equation given by Agarwal et al. 5 which further approaches Eq. 24. PwD """PD +S.... (24) The short-time approximation also can be obtained from Eq. 21 by noting that the well bore storage factor obtained by letting C<I>D = 0 reduces to 1. (PwD ) = -2-'... (25) storage s CD OliO", ---"~'----'-;"'--,-, --':,,'-----","'-----',,' Fig. 3 - Type curves with phase redistribution (CaD = 20, CD = 10,000). "wa '00, ,, , -, , O\O'-;-,---,,~,---,-':-O.--"-----':"'-----","'------'''' Fig. 4 - Type curves with phase redistribution (CaD = 100, CD = 1,000). 262 Also, sinces 2 [lis- [li(s+ 1)/(aD)J]-I/aD at large s,. (PwD) must approach 1 C<I>D. (PwD) """ '... (26) CDs ads and P wd approaches td PWD=-C ' ad where 1 1 C<I>D -= (27) CaD CD ad Note that Eq. 27 indicates that a representation very similar to well bore storage will exist at short times. This is consistent with Earlougher's 1 earlier comments. To obtain dimensionless pressures for use in analyzing pressure buildup tests with wellbore phase redistribution, Eqs. 21, 22, or 23 must be inverted. Since these expressions are too complicated for analytical inversion, the inverse Laplace transforms SOCIETY OF PETROLEUM ENGINEERS JOURNAL

5 were calculated numerically using an inversion technique presented by Stedhfest 6 adapted for use on the TI-59 programmable calculator. The type curves shown in Figs. 1 through 6 indicate that the pressure functions may show a tendency toward a damped oscillation. According to Stehfest, 6 such an oscillation may render the numerical technique useless unless certain conditions on wavelength of oscillation are met. However, it can be shown that the functions obtained in this work do not oscillate, since the Laplace transform can be written as the sum of three terms. Two of the terms represent monotonic functions, while the inverse transform of the remaining term has a single maximum. Thus, Stehfest's criteria of functional "smoothness" is met on each term and, by virtue of the linearity property of the Laplace transform and of the numerical technique, it is valid to use the numerical method with these functions. Eq. 21 was programmed and inverted for several values of the well bore-storage coefficient CD and skin factor S. Results and a comparison with data previously reported by Agarwal et al. 5 are shown in Table 1. The excellent agreement indicates that the numerical technique is well suited to the calculator precision. Eq. 22 also was programmed and inverted for several values of CD and S, again with close agreement to the Agarwal et al. 5 results for the line C. O 100 C O -l0.000 Fig. 5 - Type curves with phase redistribution (CaD = 100, Co = 10,000). C. O 1000 Co TABLE 1 - COMPARISON OF CALCULATED Pwo WITH REF. 5 (CYLINDRICAL SOURCE WELL, C<t>o = 0) to Pwo Co =100,S= , , , ,000, ,000, Pwo, Ref Fig. 6 - Type curves with phase redistribution (CaD = 1,000, Co = 10,000). Co = 10,000, S = , , , ,000, ,000, CD = 100, S=O , , , ,000, ,000, Fig. 7 - Comparision of dimensionless pressures with and without phase redistribution. CD = 10,000, S = , , , ,000, ,000, <\PRIL

6 TABLE 2 - COMPARISON OF CALCULATED PwD WITH REF. 5 (LINE SOURCE WELL, C",D = 0) to PwO CD = 100, S=O , , , ,000, ,000, Pwo, Ref Co = 10,000, S=O , , , ,000, ,000, CD = 100, S=O , , , ,000, ,000, CD = 10,000, S = , , , ,000, ,000, TABLE 3- COMPARISON OF PwD CALCULATIONS (C"'D =0) Pwo, Pwo, Cylinder Line Pwo, to Source Source Approx. Co =100,S= , , , ,000, ,000, Co =10,000,S= , , , ,000, ,000, CD =100,S= , , , ,000, ,000, CD = 10,000, S = , , , ,000, ,000, TABLE 4 - COMPARISON OF PwD CALCULATIONS (5 = 0) Pwo, Pwo, Cylinder Line Pwo, to Source Source Approx. CD = 100, C",o = 10, ad = 1, , , , ,000, ,000, CD = 10,000, C",o =100, ad = 1, , , , ,000, ,000, CD = 100, C",o = 10, ad = , , , ,000, ,000, CD = 10,000, C",o =100, ad = , , , ,000, ,000, SOCIETY OF PETROLEUM ENGINEERS JOURNAL

7 source well solution, as shown in Table 2. Finally, Eq. 23 was inverted and, again, the results shown in Tables 3 and 4 are in close agreement with previous results. Eq. 23 therefore was used in the remainder of this study. To facilitate use of the dimensionless pressures in buildup analysis, CaD from Eq. 27 was used as a variable rather than CiD' which is more difficult to determine. Results of the inversion are shown in Tables 5 and 6, and log PwD vs. log td type curves are presented in Figs. 1 through 6. Accuracy is ±O A comparison of the phase-redistribution type curves with well bore-storage type curves is shown in Fig. 7. Note that at long times the new curves coincide with the storage curves, while at early times the apparent wellbore storage effect is obvious. At intermediate times, the phase redistribution effect causes the curves to trend away from the apparent storage behavior to the true storage behavior. At large values of CcpD' the "gas hump" is apparent, while at small values of C cpd the phase redistribution effect is much diminished. A potential problem in pressure data interpretation also is shown at intermediate values of CcpD' since the curve for CcpD = JO resembles the storage curve with CD = 100 while the true CD is 1,000. An attempt to type-curve match phase-redistribution data to a storage curve TABLE 5 - DIMENSIONLESS WELLBORE PRESSURES WITH PHASE REDISTRIBUTION to ,000 2,000 5,000 10,000 20,000 50, , , ,000 1,000,000 2,000,000 5,000,000 10,000,000 CaD = 20, Co = 100 S=O S=10 S= C",o = CaD = 20, Co = 1,000 S=O S=10 S= Cpo = CaD = 20, Co = 10,000 S=O S=10 S= Cpo = ,000 2,000 5,000 10,000 20,000 50, , , ,000 1,000,000 2,000,000 5,000,000 10,000, C",o = C",o = C",o = ,000 2,000 5,000 10,000 20,000 50, , , ,000 1,000,000 2,000,000 5,000,000 10,000,000 C",o = C",o = C",o = APRIL

8 If the producing time is large compared to the shut-in time M D, then td+md=::t D, so that Eq. 28 sim- plifies to could give a reasonable type-curve match, but any estimates of reservoir parameters might be greatly in error. Fortunately, this problem can be resolved by comparing the estimates of the true and apparent storage constants as in the following examples. kh Ww(tD+tJ.tD ) -Pwj1= (tj.td ) qbp. Analysis of Pressure Buildup Tests Generally, to analyze pressure buildup test data, the... (29) superposition principle is applied to any dimensionless pressure functions yielding kh Thus, a log (pw -Pwj) vs. log (tj.t) plot of the buildup data can be type-curve matched according to W (t +tj.t ) -P 11 = standard procedures, providing that P wd (t D + tj.t D) qbp. w D D w PwD (td) -PwD (td + tj.td ) +PwD (tj.td ).. (28) =:: PwD (t D)' If this assumption is not valid, the more general superposition in Eq. 28 must be used; TABLE 6 - DIMENSIONLESS WELLBORE PRESSURES WITH PHASE REDISTRIBUTION CaD =100, CD =1,000 CaD = 100, CD = 10,000 CaD = 1,000, CD = 10,000 td S=O S=10 S=20 S=O S=10 S=20 S=O S=10 S=20 Cq,D =1 Cq,D =1 Cpo = , , , , , , , , , ,000, ,000, ,000, ,000, C",D =10 C"'D =10 C"'D = , , , , , , , , , ,000, ,000, ,000, ,000, Cq,D = 100 Cq,D =100 C<p.D = , , , , , , , , , ,000, ,000, ,000, ,000, SOCIETY OF PETROLEUM ENGINEERS JOURNAL

9 TABLE 7 - PRESSURE BUILDUP DATA FOR FIELD EXAMPLE 1 ", (PSI).r-~~-r~~~,---~-,~~~~~~-, q = 212 BID (33.71 m 3 /d) J1. = 4cp(4x10-3 Pa s) B = 1.1 RB/STB(1.1 res m 3 /stock tank m 3 ) h = 10 ft (3.05 m) Aw = bbl/ft ( m 2 ) <J> = 0.28 c t = 60x10-6 psi- 1 (8.70x10-6 kpa- 1 ) r w 2 = sq ft ( m 2 ) PI = psi/ft (7.46 kpa/m), measured '-MATCHPOINT >,10.' oil loo.''wo'lol. t TYPE CURVE ~OR CaD 400 S- 0 Co 750 Cl2lo- 10 Fig. 8 - Pressure buildup data, Example 1. t,.t (hours) o / Jw psi (kpa) 296 (2041) 449 (3096) 520 (3585) 574 (3958) 597 (4116) 588 (4054) 576 (3971) 576 (3971) 576 (3971) 578 (3985) 578 (3985) however, since there is apparently no drawdown equivalent to w ellbore phase redistribution in a buildup test, the appropriate PwD functions must be used for PwD (t D) and PwD (t D + t:.t D). An example of such superposition would be the case of a short drawdown with storage followed by a buildup with phase redistribution. The superposition of this rate history would be as in Eq. 30, where phase redistribution effects are in the third term only. kh qbp. [Pw (td +t:.td ) -Pwjl = PwD (td'cd,s) -PwD (td + t:.td,cd,s) + PwD (t:.td,cd,s,p<j>d)'... (30) In SPE preferred SI units, 1/141.2 is replaced by 7.27r x 10-6 in Eqs. 28 through 30. The examples provided in the following section illustrate the analysis of bottomhole pressure-buildup surveys which are influenced by well bore phase redistribution; however, note that not all surveys are analyzed as easily. The tests documented here are taken in gas-lifted oil wells in southern Louisiana, and one factor which makes these tests amenable to analysis is that little free gas enters the wellbore after shut-in. Most of the gas in the tubing string which contributes to the phase redistribution process originates from the annulus through the gas-lift valves. Thus, the true wellbore storage coefficient CD is controlled by the rising liquid level of the inflowing fluid which remains essentially constant. In other cases, it is not obvious that the density of the inflowing fluid remains constant, since the flowing gas 1 oil ratio may change as the well is shut in; this would cause a changing storage coefficient. In addition, the compression of the gas near the surface may not be accounted for correctly, which also will cause a variable wellbore storage. Although APRIL 1981 at long times these affects will be negligible, at short times they will cause the pressure buildup to deviate from the unit slope log P wd vs. log t:.t D line. In wells which flow significant quantities of free gas, this effect may be pronounced. 1-3 In addition, the problems of buildup analysis without phase redistribution also exist in the analysis of data with phase redistribution. Multiple stringers, mechanical problems, and other effects may make analysis difficult, if not impossible. Another observation of interest in the analysis of buildup surveys is a discrepancy between storage constants calculated from pressure data and from well completion data. I have found that these estimates of the storage constant rarely agree. The type curves presented in this work offer one possible explanation for this common discrepancy, since at small times it is apparent that the phase redistribution effect greatly controls the apparent storage constant CaD calculated from the pressure data. Several types of storage behavior are observed in practice. 1. When Cad ::= CD' the well exhibits a true storage behavior. 2. When CaD < CD' the buildup usually is controlled by phase redistribution. 3. When CaD >C D, mechanical problems, multiple lay:~rs, or an enlarged wellbore usually can explain the discrepancy. Note that these observations are based on experience and apply only to pressure tests in unfractured reservoirs where the storage is caused by a rising fluid level. Note also that the correct value for the fluid density or gradient must be used in calculating the true storage constant, since an error in the gradient will cause directly a corresponding error in the calculated storage constant. For this reason, it is recommended that the fluid gradient under flowing and static conditions be measured in conjunction 267

10 TABLE 8 - PRESSURE BUILDUP DATA FOR FIELD EXAMPLE 2... " "00'.,-----, ,-----, , , IPSI),=~--f~-+---~I j-~ j OOOOOQ(3) T~PfCLJRVf for ldt) loci) 1()() ) L---"-"-i'-Cjl!O 10 "wo l8"" t."w 'I) 1]0""... ' 1HR loops' Fig. 9 - Pressure buildup data, Example 2. q = 14BID(2.23m 3 /d) Ii = 4 cp (5 x 10-4 Pa s) B = 1.05 RB/STB (1.05 res m 3 /stock tank m 3 ) h = 20 ft (6.10 m) Aw = bbl/ft ( m 2 ) <I> = 0.28 Ct = 150x10-6 psi- 1 (2.176x10-5 kpa- 1 ) r w 2 = ft2 ( m 2 ) PI = psi/ft (9.50 kpa/m), measured M (hours) o C4Jw psi (kpa) 102 (703) 190 (1310) 254 (1751) 278 (1917) 306 (2110) 292 (2013) 284 (1958) 273 (1882) 276 (1903) 276 (1903) 276 (1903) 278 (1917) 281 (1937) with the buildup data. In gas-lifted oil wells, this gradient must be measured below the point of gas-lift gas entry. Any differences in the flowing and static gradients which cannot be attributed to frictional effects generally will give an indication of the flow of free gas from the reservoir. The gradients used in the following examples were measured in conjunction with the pressure surveys. Note also that the type curves presented are not meant to replace semilog analysis methods or the use of previously published type curves. If it is possible to analyze well test data using semilog methods, greater accuracy will be obtained in nearly all cases, mainly due to the similarity of the shape of the type curves which makes type-curve matching difficult. When such simple analyses are not possible, however, the type curves presented in this work may permit approximate analysis which would not be possible otherwise. Example Analyses Example 1 is an actual set of pressure buildup data measured in a gas-lifted oil well in southeast Louisiana. The basic data are shown in Table 7 and a log-log plot of the pressure data is shown in Fig. 8. From the data plot in Fig. 8, a point on the unit slope straight line is estimated to be!::t.p = 153 psi (loss kpa) at!::t.t = 0.1 hour. The wellbore storage coefficient is calculated as in Eq. 30 and the apparent storage coefficient as in Eq. 31. The gradient used in Eq. 30 is calculated from flowing- and static-pressure surveys measured in conjunction with the buildup test. Aw C= - = bbllpsi PI ( m 3 IkPa)..... (30) 268 qb!::t.t C a = 2-- = bbllpsi 4!::t.p ( m 3 IkPa)..... (31) Since C a < C, we can conclude that phase redistribution effects are significant. These values yield CD =752 and CaD =407. The data then are matched to the type curves for CaD =400 and CD =750 as indicated, with a match point chosen as C<I>D = 10, S = 0, t D = 6,800 at!::t.t = 1 hour, and PwD = 1.02 at!::t.pw = 100 psi (689 kpa). From the standard definitions of td and PwD' the permeability is calculated as follows. FrompwD match: k= 134 md. From td match: k= 144 md. Example 2 consists of data taken in a well in southeast Louisiana producing at low rates and high water cuts from a shaly sand. Pressure buildup data is given in Table 8 and the log!::t.pw vs. log!::t.t plot is shown in Fig. 9. From the static and flowing surveys taken in conjunction with the pressure buildup, C is calculated as shown and C a also is estimated from an extrapolation of the short-time data. Aw 3 C= - = bbllpsi ( m IkPa). PI qb!::t.t C a = -- = bbllpsi 24!::t.p ( m 3 IkPa). Since C a < C, phase redistribution effects are believed to be significant, so CD and CaD are calculated to be SOCIETY OF PETROLEUM ENGINEERS JOURNAL

11 CD = 115 and CaD = 16. The data are matched on. the type curves for CD = 100, CaD = 20 and the best match is estimated as shown in Fig. 9 to be Cq,D = 10, S = 5, PwD = 2.80 at Llp w = 100 psi (689 kpa), and td = 320 at Llt= 1 hour. From the standard definitions of P wd and t D' the permeability is calculated as follows. FromPwD match: k= 1.45 md. From td match: k=2.16 md. Fig. 9 also seems to indicate that the last two data points may be close to the semi log straight line. Using semilog analysis techniques, the permeability and skin are estimated to be 1.46 md and 3.4, respectively. Since the true straight line may not have been reached and only two points are used to determine the semi log straight line, these estimates are in adequate agreement with the estimates obtained from type-curve matching. Summary and Conclusions It has been recognized for some time that wellbore phase redistribution can cause anomalous pressure buildup behavior in oil and gas wells. General aspects of the phenomenon have been presented previously2,3; however, no technique for the analysis of such tests has been available. The work presented in this paper provides an analysis of the well bore phase redistribution problem and, with an assumed behavior based upon physical arguments, provides a general method for the analysis and description of such anomalous pressure buildup tests. It has been shown that the well bore phase redistribution problem is a complex well bore storage phenomenon, and mathematical methods previously applied to well bore storage problems have been extended to solve this more general problem. In the analysis of buildup surveys, I have found that the observed storage constant often does not agree with that calculated from the well completion properties. One possible explanation for this observation lies in the apparent storage observed to be associated with phase redistribution. Even though a hump may not be observed, phase redistribution effects may cause an inobvious distortion in the data plot. Analysis of such data by other type curve techniques may yield totally meaningless results. In view of this, it is recommended that the true and apparent storage coefficients always be calculated and checked for consistency before proceeding with detailed analysis of a buildup survey. The main assumption of this work is the exponential form used in representing the phase redistribution pressure function. I have found that this form apparently represents phase redistribution in a gas-lifted oil well very well; however, no meaningful experimental data are available to substantiate this completely. Such data would be useful either in verifying this function or in proposing a new function for the phase redistribution pressure. This data could be collected by measuring the pressure in a well shut in simultaneously at the surface and at the bottom of the tubing string using APRIL 1981 equipment described in Ref. 3. Laboratory experiments also could measure this pressure. This data and further analytical work is definitely needed to determine the range of well conditions over which the assumed form is applicable and to extend the basic technique to other conditions. In this work, only positive values of the skin effect factor have been considered. It would pose no major problem to calculate dimensionless well bore pressures for negative skin factors by the technique described by Agarwal, et at. 5 Note, however, that such an approach places a great emphasis on the accuracy of the various functions used at small times and these functions are inherently more difficult to evaluate with great precision. In the phase redistribution problem, such an approach would require the evaluation of the dimensionless pressures at extremely small dimensionless times, dimensionless storage coefficients, and dimensionless phase redistribution time parameters. Although the numerical work presented here is based upon an infinite, homogeneous, radial reservoir model, the basic concepts are much more general. In particular, it is possible to apply the techniques used in this study to other reservoir models and thereby to obtain techniques for the analysis of data in fractured systems as well as other practical situations. N omencjature A = w B= C= cross-sectional area of the well bore, bbl/ft (m 2 ) formation volume factors, RSB/STB (res m 3 Istock-tank m 3 ) well bore storage coefficient, bbl/psi (m 3 /kpa) apparent storage coefficient, bbl/psi (m 3 /kpa) apparent dimensionless storage coefficient, C a C D= 2 a 27r(j>c thr w = ( C q,d + _1 ) _ 1 ad CD (CaD = 2p(j>~~r w 2) C ed = effective dimensionless storage coefficient defined in Eq. 5 c ( compressibility, psi - 1 (kpa - 1 ) CD dimensionless well bore storage coefficient, ( CD = C ) 2p(j>c(hr w 2 phase redistribution pressure parameter, psi (kpa) 269

12 (PD) (PwD) (P<I>D) P Pgej Pw Pwhj PwD PD P<I> P<I>D dimensionless phase redistribution pressure parameter, khc<i> C<I>D = qbjl _ 7.27rx 1O-6khC<I» ( C<I>D - ---q-b-jl--:r:... reservoir thickness, ft (m) reservoir permeability, md modified Bessel function of second kind of order n Laplace transform of P D Laplace transform of P wd Laplace transform of P<I>D pressure, psi (kpa) flowing pressure at point of gas entry, psi (kpa) well bore pressure, psi (kpa) flowing wellhead pressure, psi (kpa) dimensionless well bore pressure dimensionless pressure phase redistribution pressure, psi (kpa) dimensionless phase redistribution pressure, _ P<I>D - khp<i> qbjl _ 7.27rx 1O-6khP<I» ( P<I>DqBJl q = flow rate, BID (m 3 Id) qsj = sand-face flow rate, BID (m 3 Id) r = radius, ft (m) r w well bore radius, ft (m) r D = dimensionless radius, r D = r / r w s = Laplace transform variable S = skin factor t time, hours t D = dimensionless time, 0.OO0264kt t D = 2 Jlctr w _ 3.6x 1O-6kt) ( td - 2 Jlctr w phase redistribution time parameter, hours Jl Pj 7 = dimensionless phase-redistribution time parameter, 0.OO0264ka ad= 2 Jlctr w _ 3.6x 1O-6ka) ( fxd- 2 Jlctr w Eulers constant,,,(= fluid viscosity, cp (Pa s) fluid density, psilft (kpa/m) dummy variable of integration porosity (fraction) pressure difference, psi (kpa) shut-in time, hours dimensionless shut-in time Acknowledgment The author thanks the management of Shell Oil Co. for permission to publish this paper. References I. Earlougher, R.C. Jr.,: Advances in Well Test Analysis, SPE Monograph Series, Dallas (1977) Stegemeier, G.L. and Matthews, C.S.: "A Study of Anomalous Pressure Build-Up Behavior," Trans., AI ME (1958) 213, Pitzer, S.C., Rice, J.D., and Thomas, C.E. : "A Comparison of Theoretical Pressure Build-Up Curves with Field Curves Obtained from Bottom-Hole Shut-In Tests," Trans., AIME, 216, Van Everdingen, A.F. and Hurst, W.: "The Application of the Laplace Transformation to Flow Problems in Reservoirs," Trans., AIME (1949) 186, Agarwal, R.G., Rafi AI-Hussainy, and Ramey, H.J. Jr.: "An Investigation of Well Bore Storage and Skin Effect in Unsteady Liquid Flow: l. Analytical Treatment," Soc. Pet. Eng. J. (Sept. 1970) ; Trans., AIME, Stehfest, H.: "Algorithm Numerical Inversion of Laplace Transforms," Communications oj the A em (1970) 13, Handbook oj Mathematical Functions with Formulas, Graphs and Mathematical Tables, Abramowitz, M. and Stegun, l.a. (eds)., Nat!. Bureau of Standards Applied Mathematics Series 55, U.S. Dept. of Commerce (1972). SI Metric Conversion Factor psi x E+OO kpa SPEJ Original manuscript received in the Society of Petroleum Engineers office July 16,1979. Paper accepted for publication Feb. 21,1980. Revised manuscript received Nov. 10, Paper (SPE 8206) first presented at the SPE 54th Annual Technical Conference and Exhibition, held in Las Vegas, Sept , SOCIETY OF PETROLEUM ENGINEERS JOURNAL