SPE 56487 Analysis and Interpretation of Well Test Performance at Arun Field, Indonesia T. Marhaendrajana, Texas A&M U., and N.J. Kaczorowski, Mobil Oil (Indonesia), and T.A. Blasingame, Texas A&M U. Copyright 1999, Society of Petroleum Engineers Inc. below the dew point (p =4,200 psia), these data are quite dew This paper was prepared for presentation at the 1999 SPE Annual Technical Conference and valuable as an aid for estimating well deliverability and for Exhibition held in Houston, Texas, 3–6 October 1999. assessing the need for well stimulation. Well test analyses are This paper was selected for presentation by an SPE Program Committee following review of also useful for general reservoir management activities such as information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to injection balancing and optimizing the performance of single correction by the author(s). The material, as presented, does not necessarily reflect any position and multiple wells. of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum For the analysis of well test data, we use the single-phase gas Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial pseudopressure approach–as this method does not require know- purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; ledge of relative permeability data and it is much more conven- illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX ient (and practical). The use of the single-phase pseudopressure 75083-3836, U.S.A., fax 01-972-952-9435. approach is justified by the concept that most of the reservoir is (or at least until recently, was) at pressures above the dew point. Abstract Further, any multiphase pseudopressure approach will require This paper presents a comprehensive field case history of the knowledge of a representative relative permeability data set, as analysis and interpretation of all of the available well test data well as a pressure-saturation function that is used to relate from the giant Arun Gas Field (Sumatra, Indonesia). Arun Field pressure and mobility (i.e., relative permeability). has estimated recoverable reserves on the order of 18-20 TCF The combined approach of using the single-phase gas and has 110 wells—78 producers, 11 injectors, 4 observation pseudopressure coupled with a homogeneous reservoir model for wells, and 17 abandoned. Approximately 100 well tests have the analysis well performance in gas condensate reservoir has been performed, and the analysis and interpretation of these data been documented.1-4 This particular approach yields an accurate suggests strong existence of condensate accumulation near the estimate of kh (the permeability-thickness product)–however, the wellbore as well as a "regional pressure decline" effect caused estimated skin factor is much higher than the actual (near-well) by competing production wells. skin factor. This "high skin" phenomenon occurs because there We demonstrate and discuss the analysis/interpretation tech- is an accumulation of condensate liquid in the near-well region niques for wells that exhibit condensate banking and non-Darcy that behaves like second, lower permeability "reservoir" around flow. We found that the 2-zone radial composite reservoir the wellbore. To resolve this problem, a 2-zone radial composite model is effective for diagnosing the effects of condensate bank- reservoir model is used, where the inner zone represents the ing at Arun Field. "condensate bank," and the outer zone represents the "dry gas We also demonstrate the development and application of a reservoir." The application of the 2-zone radial composite new solution for the analysis and interpretation for wells that reservoir model for the analysis of well test data from a gas exhibit "well interference" effects. Our new solution treats the condensate reservoir was demonstrated by Raghavan, et al,3 and "interference effect" as a regional pressure decline and this then by Yadavalli and Jones.4 proposed solution accurately represents the well interference We use the 2-zone radial composite reservoir model in our effects observed in many of the well tests obtained at Arun analysis if we observe a distinct inner zone (due to con-densate Field. Furthermore, this solution is shown to provide a much banking) in the well test data. This approach provides us with better interpretation of well test data from a multiwell reservoir estimates of: system than the existing approaches–i.e., assumption of no-flow (cid:1) Effective permeability to gas in both the "condensate or constant pressure boundaries, data truncation, etc. bank," and in the "dry gas" reservoir, (cid:1) Mechanical skin factor, Introduction (cid:1) Radial extent of condensate banking, The objective for performing well test analysis at Arun Field is In some (often many) cases, wellbore storage masks the inner to provide information regarding flow properties (e.g. permea- region and affects our ability to obtain a complete suite of re- bility, near-well skin factor, and the non-Darcy flow coefficient) sults. In such cases, we only obtain an estimate of the total skin as well as the radial extent of condensate banking around the factor. For the purpose of well stimulation (our objective is to wellbore. As the reservoir pressure at Arun Field has declined 2 T. Marhaendrajana, N.J. Kaczorowski, and T.A. Blasingame SPE 56487 maximize well productivity ), knowledge of the total skin factor We have provided an extensive validation of this new solu- is useful. tion using numerical simulation for the case of square reservoir Another phenomenon that we have observed at Arun Field is with nine wells. For purposes of demonstration (not actual the so-called "well interference" effect–which potentially leads application), all wells are produced from the same starting time us to misinterpret well test data. In fact, many of the well test and produce at the same constant rate. The dimensionless pres- cases taken at Arun Field show this behavior–this well inter- sure and pressure derivative functions from both the analytical ference effect tends to obscure the radial flow response, and and numerical solutions are plotted versus the dimensionless hence, influence our analysis and interpretation efforts. time function in Fig. 2. The open circles are the results from The first attempt to provide a generalized approach for the numerical simulation and the lines are the results from the new analysis of well test data from multiwell reservoir systems was analytical solution. We note that these solutions are identical– presented by Onur et.al.5 The application of their method is which validates our new analytical solution, at least concept- limited as it assumes that all of the wells in question produce at ually, as a mechanism to analyze well performance behavior in a the same time and that the pseudosteady-state flow condition is multiwell reservoir system. achieved prior to shut-in. Arun Field has been produced for over 20 years and currently in "blowdown" mode–we can assume that Regional Pressure Decline Model. To perform an analysis of the reservoir is at currently at pseudosteady-state flow condi- well test data in a multiwell reservoir system, we must account tions). for the "well interference" effects caused by offset production Drawdown and buildup tests induce local transient effects– wells. Our approach is to consider this interference as a which are controlled by the reservoir and near-well permeability, "Regional Pressure Decline," where this pressure drop acts the skin factor, and the non-Darcy flow coefficient–but not the uniformly in the area of investigation. reservoir volume. Most of the well tests performed at Arun Our "well interference" model assumes that all of the wells in Field are relatively short (< 5 hours producing time), and the the reservoir are at pseudosteady-state flow conditions at the pseudosteady-state flow condition is not established in the area time the "focus" well is shut-in. After shut-in, the reservoir in of investigation given such short production times. the vicinity of the focus well will experience a significant To address this issue we have developed a new method for pressure transient effect, but the offset wells remain on the analysis of well test data from a well in multiwell reservoir production. In simple terms, we assume that any rate change at where we treat the "well interference" effect as a "Regional the focus well (including a drawdown/buildup sequence) has Pressure Decline." We note numerous cases of pressure buildup little effect on the offset wells. However, these offset producing tests taken at Arun Field where the pressure actually declines at wells will eventually have a profound effect on the well late times during the test–indicating communication with the performance at the focus well–including, as we noted from surrounding wells that are still on production. observations at Arun Field, the flattening and decline of a Our new approach employs a straight-line graphical analysis pressure buildup trend taken at the focus well. of data on a Cartesian plot (D t (dD p/dD t ) versus D t2/D t )–where The analytical solution for the focus well in this particular e e e scenario is: (the details are provided in Appendix B) this yields a direct estimation of permeability. We have called this the "well interference" plot. The multiwell model used to pwD(tDA)= pD,1([xwD,1+e],[ywD,1+e],tDA,xwD,1,ywD,1) derived this approach also allows us to use type curve and +2p t (a –1).......................................................(2) DA D suesminiglo tgh ea nnaelyws is"–wwehlle rine t"ecrfoerrreenccteed"" mdaotdae fl upnrcotipoonsse da rein g ethnee rnaetexdt where: a D= VqpcBt ddtp = VqpcBt a 1 1 section. Where: e =r / 2 and r =r / A. p (x ,y ,t ,x ,y ) wD wD w D,1 D D DA wD,1 wD,1 New Technique for the Analysis of Well Test Data in a is the constant rate, single well solution for well 1 (i.e., the focus Multiwell Reservoir System well) and is given by Eq. A-4. Multiwell Model. In our new solution we assume a bounded We have shown in Appendix C that the early-time pressure homogeneous rectangular reservoir, with an arbitrary number of buildup equation (in [p -p (D t=0)] format) derived from Eq. 2 ws wf wells, positioned at arbitrary locations (as shown in Fig. 1). We can be written as: also assume a single compressible fluid phase, but we allow any 1 4 A p (D t )+2p (a –1)D t = ln D t +s..........(3) rate or pressure condition to be imposed at the well. However, sD DA D DA 2 eg DAer2 w our efforts in this work are focused on the constant flowrate Where D t is given by: case. The general analytical solution for the constant rate case is DAe D t t given by: D tDAe= D t DA+ptDA .......................................................(4) DA pDA p (x ,y ,t )= D D D DA The derivative formulation of Eq. 3 is given by: iSn=we1ll qD,ipD,i(xD,yD,[tDA–tsDA,i],xwD,i,ywD,i)u(tDA–tsDA,i)..(1) D tDAeddD tpsD = 12 –2p (a D–1)DD tt2DA ...............................(5) DAe DAe The derivation of Eq. 1 is given in Appendix A. This solution is Eq. 5 suggests that a Cartesian plot of D t (dp /dD t ) vs. valid for all flow regimes (e.g., transient and boundary-domi- D t2 /D t will form a straight-line trend. FuDrAtehermsDore, DseAme ilog nated flow) and its computation is extremely rapid. DA DAe plot of p +2p D t (a -1) versus D t forms a straight line, sD DA D DAe SPE 56487 Analysis and Interpretation of Well Test Performance at Arun Field, Indonesia 3 where this trend can be used to estimate formation permeability We propose Eq. 6 as a new plotting function for the pressure and skin factor. derivative function–where this relation specifically accounts for To validate Eqs. 3 and 5, we use our new analytical solution the effects of well interference. (Eq. 1) for the case of a homogeneous, rectangular reservoir In Fig. 6 we plot the various pressure derivative functions consisting of nine wells with a uniform well spacing (see inset versus shut-in time or effective shut-in time. The solid lines schematic diagram in Fig. 3). Our focus well is the center well denote the single well approach (including the well interference in this configuration. In this validation scenario, all wells except effects (there is no correction)) while the symbols denote the the focus well are put on production at the same time and at the multiwell approach (correcting for the well interference effect as same constant flowrate–are produced long enough to achieve suggested by Eq. 6). We find that the multiwell approach yields pseudosteady-state flow conditions. a more clear (and longer) 0.5 line for the pressure derivative Once this occurs, the focus well is then put on production for function. various producing times to simulate various case of pressure In Fig. 7, we show that when we plot p +2p D t (a -1) sD DA D drawdown (tpDA = 10-5, 10-4, 10-3, 10-2 respectively—where the versus D tDAe on a semilog scale we obtain a much better semilog dimensionless time in this comparison is based on total reservoir straight line trend than simply plotting p versus D t . This sD DAe area). All of the other producing wells are left on production observation (as well as the general multiwell result (Eq. 6)), and the focus well is shut-in for a buildup test sequence. The proves that we must take into account the effect of other pro- case of single well in a reservoir of equal volume is also ducing wells in the analysis of pressure buildup test data in a considered for comparison and discussion. multiwell reservoir system. The pressure derivative responses for the focus well (for In Appendix C we provide Eqs. 3 and 5 in terms of field various producing times, t ) are plotted for the pressure buildup units. These results are given as: p cases using D t (dp /dD t ) versus D t as shown in Fig. 3. qBm DA sD DA DA p +m D t= p +162.6 · This plotting function was originally proposed by Onur et.al5 as ws c wf,D t=0 kh a means for estimating reservoir volume from a pressure buildup k test. The data for the multiwell case are denoted by symbols and log(D t )+log –3.22751+0.8686s ............(7) e mf cr2 the data for the single well case are given by the solid lines. t w We note that the data from the single well case approaches dp qBm D t2 zero as shut-in time increases–this is a distinct characteristic of a D t ws =70.6 –m ...............................................(8) edD t kh cD t "no-flow" boundary. On the other hand, the data from the multi- e e well case decrease linearly to negative values as shut-in time where the "effective time" function is given by t D t increases–i.e., there is no "approaching zero" behavior. This D t = p e (t +D t) behavior is an indication of "well interference," i.e., the produc- p ing wells dominate the behavior of the reservoir system, and and the slope term is given by hence, the behavior of the "focus" well. m =0.041665dp q qtot–1 ...........................................(9) On Figs. 4 and 5 we provide plots of the D t (dp /dD t ) c dt qtot q DAe sD DAe function versus D t2 /D t as suggested by Eq. 5. These figures For the purpose of type curve matching using standard sin- DA DAe (Fig. 4 is in semilog format, and Fig. 5 is in Cartesian format) gle-well type curves, we use the "corrected" pressure and pres- show that the data for the multiwell cases follow a straight line sure derivative functions from Eqs. 7 and 8. The functions are: predicted by Eq. 5 (this is shown most clearly on Fig. 5). The Pressure function: slope of this line is proportional to the rate of change of the p +m D t........................................................................(10) average reservoir pressure at the time of shut-in for the focus ws c Pressure derivative function: well. dp D t2 For all cases, we note the general horizontal pressure deriva- D t ws +m ..............................................................(11) tive behavior (i.e., the 0.5 line) as prescribed by Agarwal6 for edD te cD te transient radial flow. The effect of the closed boundary (i.e., the pressure derivative function decaying to zero) is only apparent Analysis Procedures for Multiwell Reservoirs at very late times. Figs. 4 and 5 can be used as diagnostic tools To analyze pressure buildup tests taken in multiwell systems, we to identify the effect of well interference on pressure buildup test recommend the following procedures: behavior in a multiwell reservoir system. Step 1 Plot D t (dp /dD t ) versus D t2/D t on a Cartesian scale. e ws e e From the straight-line trend we obtain the slope m and c Analysis Relations for Multiwell Reservoirs intercept b . We calculate permeability using the inter- c We would like to develop a method for the analysis of pressure cept term as: buildup test data from a well in a multiwell reservoir system that qBm exhibits "well interference" effects. With that objective stated, k=70.6 ..........................................................(12) b h we immediately note that Eq. 5 can be rearranged to obtain: c Step 2 The Horner plot [(p +m D t) versus log((t +D t)/D t)] can D t dpsD +2p (a –1) D t2DA = 1...................................(6) also be used to estiwmsatec formation propperties. From DAedD tDAe D D tDAe 2 the straight-line trend observed on the Horner plot, we 4 T. Marhaendrajana, N.J. Kaczorowski, and T.A. Blasingame SPE 56487 obtain the slope m as well as the intercept term, corresponding flow regimes (and reservoir features) en- sl (p +m D t) . Permeability is estimated using: countered during a test. ws c D t=1hr qBm Horner Plot–We also use a "Horner" semilog plot of the k=162.6 ........................................................(13) m h shut-in pseudopressure function, p , in order to provide sl pws And the skin factor is calculated using: the "conventional" semilog analysis of a particular data set. We typically provide the semilog trends for the (p +m D t) – p s=1.1513 ws c D t=1hr wf,D t=0 "inner" and "outer" regions corresponding to the con- m sl densate and dry gas portions of the reservoir (respectively), for cases where both characteristics are t p k –1.1513 log +log –3.22751 ......(14) observed. t +1 fm cr2 p t w Muskat Plot–The "Muskat" plot is a relatively new Step 3 In order to use standard single-well type curves for approach for establishing/confirming boundary-domi- type curve matching, we must make the appropriate nated flow behavior during a pressure buildup test "corrections" given by Eqs. 10 and 11. These relations sequence.7 We use a Cartesian plot of the shut-in are: pseudopressure function, p , versus the base derivative pws function, dp /D t . This plot provides an extrapolation Pressure function: pws a to the average reservoir pressure (or in this case, pseudo- p +m D t.............................................................(10) ws c pressure), based on the principle that the pseudopressure Pressure derivative function: and pseudopressure derivative functions are represented D t dpws +m D t2 ...................................................(11) by a single term exponential function. edD t cD t e e "Well Interference" Plot–The "well interference" plot is a new approach for verifying the "regional pressure Analysis of Pressure Buildup Test from Arun Field decline" behavior associated with producing wells in a In this section we discuss our analysis and interpretation of multiwell reservoir. This approach uses a Cartesian plot selected pressure buildup cases taken from the Arun Gas Field in of the pseudopressure derivative (D p ') function versus Sumatra, Indonesia. We provide a wide range of examples, the shut-in pseudotime group (D t2/D t p). The slope of the where the examples shown exhibit some type of "well interfer- a ae resulting straight line (if a straight line exists) is used to ence" effects, as well as the effect of condensate banking "calibrate" the multiwell pressure and pressure derivative (several cases). correction functions (Eqs. 10 and 11, respectively). Recall that we have analyzed and interpreted approximately 100 well tests from the Arun Field. Our goal is to identify cases that Well C-I-18 (A-096) [Test Date: 28 September 1992]. The clearly illustrate certain types of behavior–in particular: corresponding figures showing our analysis for this case are provided in Figs. 8 to 11. Well C-I-18 (A-096) was completed (cid:1) Effects of non-Darcy flow (not as prevalent). in June 1991 and had an initial pressure of approximately 3173 (cid:1) Condensate banking (2-zone radial composite model). psia at the time of completion. The results for this case are sum- (cid:1) "Well interference"/boundary effects. marized in Table 1. The following well test cases are presented: Test Summary Plot–Fig. 8 clearly shows the condensate (cid:1) Well C-I-18 (A-096) [Test Date: 28 Sep 1992] banking phenomena, as the pseudopressure derivative (cid:1) Well C-II-15 (A-040) [Test Date: 26 May1993] exhibits 2 distinct horizontal trends. The raw derivative (cid:1) Well C-IV-11 (A-084) [Test Date: 05 Jan 1992] data appear show a reservoir boundary, while the cor- (cid:1) Well C-IV-11 (A-084) [Test Date: 04 May1992] rected derivative data show a continuance of the infinite- (cid:1) Well C-IV-16 (A-051) [Test Date: 16 Mar 1993] acting reservoir behavior (which is a result of our new (cid:1) Well C-IV-16 (A-051) [Test Date: 16 Sep 1993] multiwell correction function). Horner Plot–Fig. 9 verifies the condensate banking with Orientation for the Analysis/Interpretation Sequence: a semilog straight line for the condensate bank as well as Our analysis and interpretation of each well test case centers on the dry gas portion of the reservoir. This plot also shows the following plot sequence an apparent reservoir boundary at late times. Test Summary Plot–We use a log-log plot of the Muskat Plot–The Muskat plot provided in Fig. 10 shows pseudopressure drop (D p ) and pseudopressure derivative p a reasonable straight-line trend at late times (to the left of (D p ') functions versus the effective shut-in pseudotime p the plot). However, this plot also shows a deviation function (D t ). This plot includes the analysis results ae from the expected linear trend at very late times, where and the simulated test performance using these analysis this behavior prompts us to suggest that the nearby results. The primary value of this plot is the visual- producing wells have caused a specific interference ization of the pseudopressure derivative function, and the effect. SPE 56487 Analysis and Interpretation of Well Test Performance at Arun Field, Indonesia 5 An argument could be made that the derivative function "Well Interference" Plot–In Fig. 19 we observe a rela- itself is the cause–as the derivative algorithm can skew tively consistent linear trend in the data, and conclude the derivative values near the end-points. While plaus- that well interference efforts are a possible mechanism. ible, we suggest that the nearby producing wells are the However, this trend only approaches zero, and does not most likely cause of the "well interference" effects. actually extend to negative values, which is one criteria "Well Interference" Plot–Fig. 11 exhibits a slightly scat- associated with the well interference model. tered, but clearly linear trend. This observation validates Well C-IV-11 (A-084) [Test Date: 04 May 1992]. This is an- our previous suggestion that deviation of the derivative other well test performed on Well C-IV-11 (A-084) on 4 May function at late times is due to well interference from 1992 and the results for this case are summarized in Table 4. surrounding production wells. Our new multiwell The purpose of this test was to evaluate the effectiveness of an reservoir model uniquely predicts this behavior. acid fracturing treatment performed in January 1992. Well C-II-15 (A-040) [Test Date: 26 May 1993]. Well C-II-15 Test Summary Plot–Comparing the log-log summary (A-040) was completed in January 1981 and had an initial pres- plot for this case (Fig. 20) with that of the previous test sure of approximately 6444 psia at the time of completion. The (Fig. 16), we immediately note a significantly smaller results for this case are summarized in Table 2. "hump" in the derivative function, suggesting that the Test Summary Plot–In Fig. 12, the pseudopressure deri- well has been substantially stimulated. We note that the vative function clearly shows the condensate banking flowrate prior to this test is to the pre-stimulation test is phenomena, as well as a closed reservoir boundary (raw about half that of the present test, but the maximum data). The corrected derivative data (although a bit pseudopressure change at late times for the pre-stimula- scattered at very late times) suggests the continuation of tion test is almost twice that for the present test. the infinite-acting radial flow regime. From Fig. 20 we conclude that the effect of condensate Horner Plot–In Fig. 13 the shut-in pseudopressure banking is fairly modest, with only a slight "tail" in the function clearly indicates the presence of condensate pseudopressure derivative function. A boundary feature banking (note the two semilog straight-line trends). The is apparent at very late times and the raw data again sug- condensate bank feature appears to dominate most of the gest that this is a closed boundary feature, while the well performance behavior. "corrected" data indicate infinite-acting radial flow. Muskat Plot–The Muskat plot in Fig. 14 shows a rela- Horner Plot–We note two apparent semilog straight line tively good linear trend at late times and tends to confirm trends on the Horner plot shown in Fig. 21. Both trends the presence of a "closed boundary" feature. were constructed using the permeability values estimated for the "inner" (condensate) and "outer" (dry gas) "Well Interference" Plot–In Fig. 15 we note a reasonably regions. These trends appear to be reasonable, and well defined linear trend, although there is considerable should be considered accurate. scatter at very late times. Using this trend, we find that the corrected functions in Fig. 12 (the log-log plot) do Muskat Plot–In Fig. 22 we present the Muskat plot for suggest infinite-acting radial flow behavior. this case and we note a very well-defined linear trend at late times, the extrapolated average reservoir pseudo- Well C-IV-11 (A-084) [Test Date: 05 Jan 1992]. Well C-IV- pressure (p ) is 1882.8 psia. For the pre-stimulation 11 (A-084) was completed in Arun Field in August 1990 and p had an initial pressure of approximately 3835 psia at the time of case, we obtained a p estimate of 1920.0 psia (Fig. 18). p completion. The results for this case are summarized in Table 3. "Well Interference" Plot–Fig. 23 provides the "well Test Summary Plot–Fig. 16 provides a log-log plot of interference" plot for this case. Comparing the trend on the pseudopressure functions where the effect of conden- this plot with the pre-stimulation case (Fig. 19), we note sate banking is not obvious, but a large wellbore storage/ a remarkable similarity in the estimated slopes of the skin factor "hump" is observed in the pseudopressure data–confirming that the character of the regional pres- derivative function. As is typical at Arun Field, there sure distribution has not changed significantly. does appear to be the influence of a closed reservoir Well C-IV-16 (A-051) [Test Date: 16 Mar 1993]. Well C-IV- boundary at very late times (uncorrected data). 16 (A-051) was completed in Arun Field in March 1985 and had Horner Plot–The Horner plot shown in Fig. 17 gives a an initial pressure of approximately 5818 psia at the time of the response that one would expect from a well in an in- original completion. This well was "sidetracked" and finite-acting homogeneous reservoir. There are no ob- recompleted in mid-1989. The results for this case are provided vious/apparent features resembling condensate banking in Table 5. or reservoir boundaries. Test Summary Plot– In Fig. 24 we present the summary Muskat Plot–The Muskat plot in Fig. 18 exhibits a fairly log-log plot and we note a fairly well defined boundary well-defined linear trend at late times, confirming the feature. However, there are no obvious signs of conden- presence of a "closed boundary" feature. 6 T. Marhaendrajana, N.J. Kaczorowski, and T.A. Blasingame SPE 56487 sate banking–the estimated skin factor is low (0.62) and reservoir pseudopressure (p ) for this case is 1634.1 p an excellent match of the data is obtained without using psia, while for the pre-stimulation case, we obtained a p the 2-zone radial composite reservoir model. p estimate of 1788.3 psia (Fig. 26). Horner Plot–The Horner plot in Fig. 25 shows the "Well Interference" Plot–In Fig. 31 we present the "well "classic" character of a pressure buildup test performed interference" plot for this case and comparing the linear on a well in an infinite-acting homogeneous reservoir, trend on this plot with the trend for the pre-stimulation with only wellbore storage and skin effects present. case (Fig. 27), we find that the estimated slopes have Only a very slight deviation of the pseudopressure trend changed considerably. However, given the somewhat is seen at late times, where this behavior is presumed to ill-defined nature of the trends in both cases, we can only be a boundary effect. conclude qualitatively that the regional pressure distribu- Muskat Plot–In Fig. 26 we present the Muskat plot for tion has changed. this case and we note an excellent straight-line trend at Summary and Conclusions late times (as would be expected). However, we also In summary, we have developed a rigorous and coherent note that the very last data deviate systematically from approach for the analysis of well test data taken from multiwell this trend, suggesting the possibility of external in- reservoir systems. Using the appropriate (dry gas) pseudopres- fluences–i.e., the well interference effect. sure and pseudotime transformations, as well as the 2-zone "Well Interference" Plot–In Fig. 27 we immediately radial composite reservoir model and the non-Darcy flow model, observe a new feature, the systematic deviation of the we have effectively analyzed all of the well test data taken from derivative trend. In fact, it appears that two linear trends Arun Field. could be constructed. We have elected to use the "ear- We have also provided new insight into the effects of well lier" trend as we have more confidence in these data interference in large multiwell reservoirs. The most innovative (which are not influenced by "endpoint" effects in the aspect of this work is the development of the new multiwell derivative algorithm). However, either trend could be solution and corresponding analysis procedures. The most used, and perhaps an "average" trend should be used. practical aspect of this work is the demonstration/validation of The point of this exercise is that we clearly observe the these multiwell analysis techniques for the majority of wells at effects of well interference, but we have no single ex- Arun Field. planation for this behavior. For example, the drainage The following conclusions are derived from this work: pattern may be non-uniform, and/or wells beyond the immediate vicinity could be affecting the pressure distri- (cid:1) The new "multiwell" solution has been successfully de- bution in the area of the focus well. rived and applied for the analysis of well test data taken from a multiwell reservoir system. Well C-IV-16 (A-051) [Test Date: 16 Sep 1993]. This is an- (cid:1) The appearance of "boundary" effects in pressure build- other well test performed on Well C-IV-16 (A-051) on 16 Sep- up test data taken in multiwell reservoirs can be cor- tember 1993 and the results for this case are summarized in rected using our new approach. Care must be taken so Table 6. This well test was performed to evaluate the effective- as not to correct a true "closed boundary" effect. ness of an acid fracturing treatment performed in May 1993. (cid:1) The 2-zone radial composite reservoir model has been Test Summary Plot–From the early time data in Fig. 28 shown to be representative for the analysis and inter- we note a feature in the pseudopressure derivative func- pretation of well test data from Arun Field (most wells tion that appears to suggest fracture stimulation (or a exhibit radial composite reservoir behavior). least a significant improvement in near-well communica- (cid:1) The effect of non-Darcy flow on pressure buildup test tion). This feature is not apparent in the previous test analysis seems to be minor for the wells in Arun Field. (see Fig. 24). The "raw" pressure derivative function in Although not a focus of the present study, our analysis Fig. 28 shows the apparent effect of a closed boundary at of the pressure drawdown (flow test) data appear to be late times, while the "corrected" data suggest radial flow much more affected by non-Darcy flow effects. in an infinite-acting homogeneous reservoir. Nomenclature Horner Plot–In Fig. 29 we have the Horner plot for this A =Area, ft2 case, where this particular plot does not provide any B =Formation volume factor, RB/MSCF evidence of flow impediments, and suggests (at very late c =Total compressibility, psi-1 times) that a boundary has been encountered. t D =Non-Darcy Flow Coefficient, (MSCF/D)-1 Muskat Plot–The Muskat plot for this case is shown in h =Thickness, ft Fig. 30 and we note a well-defined linear trend at late k =Formation permeability, md times, although we observe (as in the pre-stimulation p =Pressure or pseudopressure, psia test), that the last few points deviate systematically from q =Sandface flow rate, MSCF/D the straight-line trend. The extrapolated average r =Wellbore radius, ft w SPE 56487 Analysis and Interpretation of Well Test Performance at Arun Field, Indonesia 7 s =Near-well skin factor, dimensionless ¶ 2p ¶ 2p Snwell D + D +2p q d (x –x ,y –y )u(t –t ) t =Time, hr ¶ x2 ¶ y2 D,i D wD,i D wD,i DA sDA,i D D i=1 V =Pore volume ¶ p p = D .....................................................(A-2) x =x-coordinate from origin, ft ¶ t DA y =y-coordinate from origin, ft where: x =x-coordinate of well from origin, ft 2p kh(p–p(x,y,t)) w p = i y =y-coordinate of well from origin, ft D q Bm w ref m =Fluid viscosity, cp kt f =Porosity, fraction tDA= fm cA t x x = Acknowledgements D A y We acknowledge the permission to publish field data provided y = D A by Pertamina and Mobil Oil Indonesia. We also acknowledge In Field units, these dimensionless variables are: financial and technical support provided by Mobil E&P Tech- nology Company (MEPTEC). p = 1 kh(pi–p(x,y,t)) D 141.2 q Bm ref And finally, we acknowledge the technical, and computing kt t =0.0002637 support services provided by the Harold Vance Department of DA fm cA t Petroleum Engineering at Texas A&M University. Eq. A-2 is solved analytically subject to the assumption of a no- flow condition at all reservoir boundaries (i.e., this is a sealed References ("no flow") boundary system). Following the work by Marhaen- 1. Jones, J.R., Vo, D.T., and Raghavan, R.: "Interpretation of drajana and Blasingame7 (1997), the solution of Eq. A-2 is: Pressure Buildup Responses in Gas Condensate Wells," SPEFE p (x ,y ,t )= (March 1989). D D D DA 2. Thompson, L.G., Niu, J.G., and Reynolds, A.C.: "Well Testing for Snwell q p (x ,y ,t ,x ,y )u(t –t )..........(A-3) Gas Condensate Reservoirs," paper SPE 25371 presented at the D,i D,i D D DA wD,i wD,i DA sDA,i i=1 SPE Asia Pacific Oil & Gas Conference & Exhibition held in Where p is the dimensionless pressure for the case of a single Singapore, 8-10 February 1993 D,i well in bounded reservoir produced at a constant rate. 3. Raghavan, R., Chu, W.C., Jones, J.R.: "Practical Consideration in the Analysis of Gas Condensate Well Tests," paper SPE 30576 From Ref. 7, the solution for a single well in a bounded rec- presented at the SPE Annual Technical Conference & Exhibition tangular reservoir produced at a constant rate is given as: held in Dallas, U.S.A., 22-25 October, 1995. p (x ,y ,t ,x ,y )= 4. Yadavalli, S.K. and Jones, J.R.: "Interpretation of Pressure Tran- D,i D D DA wD,i wD,i sient Data from Hydraulically Fractured Gas Condensate," paper S ¥ S ¥ (x +x +2nx )2+(y +y +2my )2 SPE 36556 presented at the 1996 SPE Annual Technical 1 E D wD,i eD D wD,i eD Conference and Exhibition held in Denver, Colorado, U.S.A., 6-9 2m=–¥ n=–¥ 1 4tDA October 1996. 5. Onur, M., Serra, K.V., and Reynolds, A.C.: "Analysis of Pressure (x –x +2nx )2+(y +y +2my )2 Buildup Data Obtained at a Well located in a Multiwell System," +E D wD,i eD D wD,i eD 1 4t SPEFE (March 1991) 101-110. DA 6. Agarwal, R.G.: "A New Method to Account for Producing Time Effects When Drawdown Type Curves Are Used to Analyze (x +x +2nx )2+(y –y +2my )2 D wD,i eD D wD,i eD Pressure Buildup and Other Test Data," paper SPE 9289 presented +E1 4t DA at the 1980 SPE Annual Technical Conference and Exhibition, Dallas, Sept. 21-24. 7. Marhaendrajana, T. and Blasingame, T.A.: "Rigorous and Semi- (xD–xwD,i+2nxeD)2+(yD–ywD,i+2myeD)2 +E Rigorous Approaches for the Evaluation of Average Reservoir 1 4t DA Pressure from Pressure Transient Tests," paper SPE 38725 presented at the 1997 SPE Annual Technical Conference and .....................................................................................(A-4) Exhibition, San Antonio, TX, 5-8 October 1997. Although Eq. A-4 appears tedious, it is actually quite simple to compute and is remarkable efficient (i.e., fast). When applying Appendix A(cid:190)(cid:190)(cid:190)(cid:190) Multiwell Solution Eq. A-4, we use a convergence criteria of 1x10-20. The mathematical model that describes the pressure behavior in a bounded rectangular reservoir with multiple wells producing at Appendix B(cid:190)(cid:190)(cid:190)(cid:190) Regional Pressure Decline Model an arbitrary constant rate (Fig.1) is given by (Darcy units): ¶ 2p ¶ 2p Snwell qB fm c ¶ p In this Appendix we provide the development of the "regional + – i d (x–x ,y–y )u(t–t )= t pressure decline" model, where in this particular case, we utilize ¶ x2 ¶ y2 Ah(k/m) w,i w,i s,i k ¶ t i=1 a complete multiwell solution, where this solution assumes that .....................................................................................(A-1) the surrounding production wells are at pseudosteady-state flow In dimensionless variables, Eq.A-1 is written as: conditions. 8 T. Marhaendrajana, N.J. Kaczorowski, and T.A. Blasingame SPE 56487 This condition presumes that any rate change (e.g., a pressure Appendix C(cid:190)(cid:190)(cid:190)(cid:190) Development of Pressure Buildup drawdown/buildup sequence) at the focus well will have very Analysis Method in Multiwell System little effect on the surrounding production wells. Therefore, a For a pressure buildup test performed on the focus well (i.e., pressure drawdown or pressure buildup test will cause transient well 1) after a period of constant rate production in the focus flow conditions only in the vicinity of the focus well–not in the well (with the surrounding production wells at pseudosteady- entire reservoir. Given the short period of a well test compared state flow conditions) the pressure at well 1 (i.e., the focus well) to the entire production history of the reservoir, this local is given by: transient phenomena is a reasonable and logical assumption. p (t )= p ([x +e],[y +e],t ,x ,y ) wD DA D,1 wD,1 wD,1 DA wD,1 wD,1 Our new "regional pressure decline" model is written as fol- – p ([x +e],[y +e],t –t ,x ,y ) D,1 wD,1 wD,1 DA pDA wD,1 wD,1 lows (Darcy units): ¶ 2p ¶ 2p q B Snwell qB +2p tDA(a D–1)............................................(C-1) + – 1 d (x–x ,y–y )– i From Eq. C-1, we can write the pressure buildup equation for ¶ x2 ¶ y2 Ah(k/m ) w,1 w,1 Ah(k/m ) i=2 well 1 (in [p-p ] format) as: fm c ¶ p i ws = k t ¶ t psD(D tDA)= pwD(tpDA+D tDA)–pwD(D tDA) .....................................................................................(B-1) – p ([x +e],[y +e],D t ,x ,y ) D,1 wD,1 wD,1 DA wD,1 wD,1 Writing Eq. B-1 in terms of dimensionless variables, we have +2p (a –1)(t +D t ).............................(C-2) ¶ 2pD + ¶ 2pD +2pq d (x –x ,y –y )+2p Snwell q We would like to usDe the [ppwDs-Apwf(D Dt=A0)] format for our pressure ¶ xD2 ¶ yD2 D,1 D wD,1 D wD,1 i=2 D,i buildup formulation, therefore we proceed as follows: = ¶ pD [pws–pwf(D t=0)]=[pi–pwf(D t=0)]–[pi–pws]............(C-3) ¶ tDA Hence, .....................................................................................(B-2) p (D t )= p (t )– p (D t ).............................(C-4) sD DA wD pDA sD DA For the case of a no-flow outer boundary, the solution of Eq. B-2 Substituting Eqs. B-7 and C-2 into Eq. C-4, and rearranging, we is given as: obtain: Snwell p (D t )= p (x ,y ,t )= p (x ,y ,t ,x ,y )+2p t q sD DA D D D DA D,1 D D DA wD,1 wD,1 DAi=2 D,i pD,1([xwD,1+e],[ywD,1+e],tpDA,xwD,1,ywD,1) .....................................................................................(B-3) +2p (a –1)t D pDA Where pD,1(xD,yD,tDA,xwD,1,ywD,1) is the solution for the case of – pD,1([xwD,1+e],[ywD,1+e],tDA+tpDA,xwD,1,ywD,1) a single well in a bounded rectangular reservoir producing at a + p ([x +e],[y +e],D t ,x ,y ) constant rate. This solution is given by Eq. A-4. Eq. B-3 is only D,1 wD,1 wD,1 DA wD,1 wD,1 –2p (a –1)(t +D t )....................................(C-5) strictly valid in the vicinity if the focus well (i.e., well 1) and is D pDA DA used solely to model the pressure-time performance at the focus Cancelling the similar terms, Eq. C-5 reduces to the following: well. p (D t )= p ([x +e],[y +e],t ,x ,y ) sD DA D,1 wD,1 wD,1 pDA wD,1 wD,1 From material balance, we have: – p ([x +e],[y +e],t +t ,x ,y ) dp = qtotB..................................................................(B-4) + pD,1([xwD,1+e],[ywD,1 +e],DDtA ,xpDA ,wyD,1 )wD,1 dt V c D,1 wD,1 wD,1 DA wD,1 wD,1 p t –2p (a –1)D t ..........................................(C-6) Defining a new parameter, the "well interference" coefficient, D DA a , we obtain Using Eq. A-4 for the p variable, we have D D,1 a = Vpct dp = Vpct a ................................................(B-5) pD,1([xwD,1+e],[ywD,1+e],tDA,xwD,1,ywD,1)= D q B dt q B 1 1 Substituting Eq. B-4 into Eq. B-3, and using the definition given 1 S ¥ S ¥ E (2xwD+e +2nxeD)2+(2ywD+e +2myeD)2 by Eq. B-5, we obtain: 2m=–¥ n=–¥ 1 4tDA p (x ,y ,t )= p (x ,y ,t ,x ,y )+2p t (a –1) D D D DA D,1 D D DA wD,1 wD,1 DA D .....................................................................................(B-6) +E (e +2nxeD)2+(2ywD+e +2myeD)2 1 4t The first term on right-hand-side of Eq. B-6 is the pressure DA response caused by well 1 (i.e., the focus well), and the second (2x +e +2nx )2+(e +2my )2 term is the pressure response due to the other active producing +E wD eD eD 1 4t wells in the reservoir system. DA Evaluating the pressure response at the focus well, we have: (e +2nx )2+(e +2my )2 p (t )= p ([x +e],[y +e],t ,x ,y ) +E eD eD ................(C-7) wD DA D,1 wD,1 wD,1 DA wD,1 wD,1 1 4t DA +2p t (a –1)..................................................(B-7) DA D The early time (i.e., small D t) approximation for Eq. C-7 is: where e =r / 2 and r =r / A. wD wD w SPE 56487 Analysis and Interpretation of Well Test Performance at Arun Field, Indonesia 9 2e2 Multiplying both sides of Eq. C-15 by 141.2qBm /kh and substi- 1 p ([x +e],[y +e],t ,x ,y )= E D,1 wD,1 wD,1 DA wD,1 wD,1 2 1 4t tuting the material balance relation, dp/dt=5.615q B/fhAc, we DA tot t have .....................................................................................(C-8) dp q q Inserting the definition of e into Eq. C-8, we obtain pws+0.041665 dt qtot qtot–1 D t= p ([x +e],[y +e],t ,x ,y )= 1E 1 rw2 p +162.6qBm · D,1 wD,1 wD,1 DA wD,1 wD,1 2 1 4tDA A wf,D t=0 kh .....................................................................................(C-9) t D t p k Substituting Eq. C-9 into Eq. C-6, we have log +log –3.22751+0.8686s (t +D t) fm cr2 p t w p (D t )= 1E 1 rw2 – 1E 1 rw2 sD DA 2 1 4tpDA A 2 1 4(tpDA+D tDA) A ...................................................................................(C-16) To determine coefficient of D t on the left-hand-side of Eq. C-16, + 1E 1 rw2 –2p (a –1)D t .................(C-10) we proceed by differentiating Eq. C-14 with respect to D t , 2 1 4D t A D DA DA DA which gives us the following result Ugrsailn tge rtmhes, lwoeg awrirtihtem Eicq .a Cpp-1ro0x aism:ation for the Exponential Inte- D t dpsD = 1 tpDA –2p D t (a –1).......(C-17) DAdD t 2 (t +D t ) DA D psD(D tDA)= 12ln 4tepgDArA2 – 12ln 4(tpDAe+g D tDA)rA2 Multiplying DbAoth sidesp DbAy (tpDDAA+D tDA), we obtain w w t pDA + 12ln 4DetgDArA2 –2p(a D–1)D tDA................(C-11) (tpDAt+D tDA)D tDAddD ptsD = 12 –2p (a D–1)(tpDAt+D tDA)D tDA w pDA DA pDA Collecting the logarithm terms in Eq. C-11, we have ...................................................................................(C-18) t · D t Using the following identity in terms of D tDA and D tDAe, we have p (D t )= 1ln 4 pDA DA A –2p (a –1)t dp t D t dp dD t sD DA 2 eg (t +D t )r2 D DA D t sD = pDA DA sD DA pDA DA w DAedD t (t +D t ) dD t dD t DAe pDA DA DA DAe ...................................................................................(C-12) 2 t D t dp (t +D t ) Rearranging Eq. C-12 we obtain: = pDA DA sD pDA DA (t +D t ) dD t t t · D t pDA DA DA pDA psD(D tDA)+2p (a D–1)D tDA= 12ln e4g (tppDDAA+D tDDAA)rAw2 = (tpDAt+D tDA)D tDAddD ptsD ..................(C-19) pDA DA 1 4 A = 2ln eg D tDAer2 ..........(C-13) Substituting Eq. C-19 in Eq. C-18, we obtain The D tDAe function is very similar to the "effewctive" shut-in time D tDAeddD tpsD = 12 –2p (a D–1)(tpDAt+D tDA)D tDA proposed by Agarwal.6 The difference being that D t is DAe pDA DAe or finally, we have defined using dimensionless times based on the total drainage area, A. D t dpsD = 1 –2p (a –1) D t2DA t · D t DAedD t 2 D D t D t = pDA DA DAe DAe DAe (t +D t ) ...................................................................................(C-20) pDA DA This is an intermediate result, when we reduce these relations to Eq. C-20 suggests that a plot of D tDAe(dpsD/dD tDAe) versus the field units, we will use the Agarwal effective shut-in time (D t ). D t2DA/D tDAe group will form a straight-line trend. Rearranging e Eq. C-20 further, we obtain Including the near-well skin factor, Eq. C-13 becomes: psD(D tDA)+2p(a D–1)D tDA= 12ln e4g D tDAerA2 +s......(C-14) D tDAeddD tpDsDAe +2p (a D–1)DD ttD2DAAe = 12 ........................(C-21) w Substituting the definition of the appropriate dimensionless Equation C-14 suggests that plot of psD+2p D tDA(a D-1) versus variables (in terms of field units) into Eq. C-20, we have log(D tDAe) will form a straight line. Substituting the definition of dp qBm dp q q D t2 the dimensionless variables (in terms of field units) into Eq. C- D tedD wts =70.6 kh –0.041665 dt qtot qtot–1 D t 14, we obtain e e 1411.2 kh(pwsq–Bpmwf,D t=0) +2p 0.00fm02c6A37kD t qqtot–1 = From... .E..q.... .C...-.2..2..,. .w...e.. .c..a.n.. .d..e..f.i.n..e.. .t.h..e.. .s..l.o..p..e. .t..e.r..m...,. .m...c.,. .a..s.......(C-22) t dp q q 1ln 4 0.0002637k tpD t +s...............(C-15) mc=0.041665 dt qtot qtot–1 ..................................(C-23) 2 eg fm cr2 (t +D t) Therefore, we can now write Eq. C-22 as follows t w p 10 T. Marhaendrajana, N.J. Kaczorowski, and T.A. Blasingame SPE 56487 dp qBm D t2 For the purpose of performing type curve analysis using the D t ws =70.6 –m ......................................(C-24) edD t kh cD t standard single-well type curves, we must use the corrected e e Equation C-24 suggests that plot of D t (dp /dD t ) versus D t2/D t pressure and pressure derivative functions which are derived e ws e e from Eq. C-24 and Eq. C-26. These "correction" functions are: will form a straight-line trend with a slope m and an intercept c 70.6qBm /(kh)—where we define the intercept term as b . We Pressure function: c can calculate formation permeability using pws+mcD t.................................................................(C-29) qBm Pressure derivative function: k=70.6 ..............................................................(C-25) bch D t dpws +m D t2 .......................................................(C-30) Recalling Eq. C-16, we have edD t cD t dp q q e e p +0.041665 tot–1 D t= ws dt qtot q TABLE 1(cid:190)(cid:190)(cid:190)(cid:190) SUMMARY OF ANALYSIS FOR WELL C-I-18 (A-096) qBm p +162.6 · [TEST DATE: 28 SEPTEMBER 1992] wf,D t=0 kh Cartesian Horner Log-log Muskat t D t Parameter Plot Plot Plot Plot p k log +log –3.22751+0.8686s Inner zone permeability --- 7.36 7.36 --- (tp+D t) fm ctrw2 k1, md Outer zone permeability 15.3 15.3 15.3 --- ...................................................................................(C-16) k , md 2 Substituting Eq. C-23 into Eq. C-16 gives us: Near-well skin factor, s --- 0.68 0.129 --- p +m D t= p +162.6qBm · Total skin factor, st --- 4.55 --- --- ws c wf,D t=0 kh Non-Darcy flow coeff. --- --- 5x10-6 --- D, (MSCF/D)-1 k log(D t )+log –3.22751+0.8686s ...(C-26) Inner zone radius, r, ft --- --- 19 --- e mf cr2 i t w Average reservoir pseudo- --- --- --- 1148.6 pressure p, psia where effective time is given by p t D t TABLE 2(cid:190)(cid:190)(cid:190)(cid:190) SUMMARY OF ANALYSIS FOR WELL C-II-15 (A-040) D te = (t p+D t) [TEST DATE: 26 MAY 1993] p Cartesian Horner Log-log Muskat Eq. C-26 suggests that a plot of (p +m D t) vs. log(D t ) will yield ws c e Parameter Plot Plot Plot Plot a straight line from which we can determine permeability (from Inner zone permeability --- 7.20 7.20 --- the slope term) and skin factor (from the intercept term). The k , md 1 coefficient m is obtained from a Cartesian plot of D t (dp /dD t ) c e ws e Outer zone permeability 61.4 61.4 61.4 --- versus D t2/D te. k2, md Using this approach, one can construct an appropriate semilog Near-well skin factor, s --- -0.138 -0.707 --- plot for a well undergoing a pressure buildup test in a multiwell Total skin factor, st --- 28.1 --- --- reservoir system. In fact, the p +m D t data function can also be Non-Darcy flow coeff. --- --- 1.6x10-5 --- ws c D, (MSCF/D)-1 used in the conventional Horner plot format [i.e., p +m D t ws c Inner zone radius, r, ft --- --- 21 --- versus log((t +D t)/D t)]. i p Average reservoir pseudo- --- --- --- 1132.8 The formation permeability and near-well skin factor are cal- pressure p, psia p culated using the following relations (respectively): qBm TABLE 3(cid:190)(cid:190)(cid:190)(cid:190) SUMMARY OF ANALYSIS FOR WELL C-IV-11 (A-084) k=162.6 ............................................................(C-27) [TEST DATE: 5 JANUARY 1992] m h sl Cartesian Horner Log-log Muskat (pws+mcD t)D t=1hr– pwf,D t=0 Parameter Plot Plot Plot Plot s=1.1513 m Inner zone permeability --- --- --- --- sl k , md 1 t Outer zone permeability 6.04 6.04 6.04 --- p k –1.1513 log +log –3.22751 k , md t +1 fm cr2 2 p t w Near-well skin factor, s --- --- --- --- ........................................................................................(C-28) Total skin factor, st --- 33.5 33.5 --- where m is the slope of the straight-line trend on a semilog plot Non-Darcy flow coeff. --- --- --- --- sl D, (MSCF/D)-1 [i.e., (p +m D t) versus log(D t ) (Agarwal format) or p +m D t ws c e ws c Inner zone radius, r, ft --- --- --- --- versus log((t +D t)/D t) (Horner Format)]. i p Average reservoir pseudo- --- --- --- 1920 pressure p, psia p