Understanding waterflood performance with modern analytical techniques

Journal of Petroleum Science and Engineering 81 (2012) 100–111

Contents lists available at SciVerse ScienceDirect

Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol

Understanding waterflood performance with modern analytical techniques Omer Izgec Chevron Energy Technology Company, 1500 Louisiana Street, 77002, Houston, TX, United States

a r t i c l e

i n f o

Article history: Received 27 December 2010 Accepted 30 November 2011 Available online 23 December 2011 Keywords: waterflood analytical models

a b s t r a c t Application of reservoir simulation models to understand and forecast reservoir performance is the norm. In contrast, analytical tools provide required simplicity while capturing the events occurring at smaller time scales, which are ordinarily sacrificed in numerical simulations to keep the run time reasonable. This study illustrates the use of some recently developed analytical tools with high-frequency production data to understand field-wide and well-based flood performance. Also, modified definitions of coupled capacitance–resistance-aquifer model and reciprocal productivity index are introduced. Dynamic flow and storage capacity curve concept is developed using production data only. The use of these curves for characterizing the type of interwell and reservoir connectivity is illustrated with two case studies. The methods presented are based on fundamental-reservoir engineering concepts; therefore, they are versatile and easy to use. Synthetic examples illustrate and verify the methods used in the study. Case studies demonstrate the power of integrating analytical models for better understanding of evolving waterfloods. © 2011 Elsevier B.V. All rights reserved.

1. Introduction With the ongoing advances in computer technology, reservoir simulation becomes one of the preferred tools for reservoir management. The appeal of a reservoir simulation model stems from providing a single platform where different types of data, such as geological, geophysical, reservoir fluid and rock properties, etc., are integrated in a physical model and used for performance predictions. When properly calibrated and verified by alternative methods, the reservoir simulation prediction can be reasonably accurate. Despite offering many advantages, reservoir-simulation models have limitations. For instance, for a full-field study, both large timesteps and grid-block sizes, often used for practical turnaround time for a single run, causes loss of temporal and spatial details. Reservoir simulators most of the time fail to account for dynamic and finer scale well conditions (damage, plugging, fracturing, etc.). These lead to inaccurate prediction of performance and optimistic forecasts, especially for secondary and tertiary recoveries. In recent years, integration of surveillance data with full-field simulation studies has gained attention. Case studies of Haddad et al. (2004), Maschio et al. (2009), and more recently of Kim et al. (2010) showed the obvious benefits of data integration. Formal procedures for conditioning the static reservoir description with pressure-transient data were reported by Landa et al. (2000) and Kamal et al. (2005). These studies show the promise of merging analytical methods with full-field simulations. Some recent advances in analytical modeling chronicle that the amount of information that

E-mail address: [email protected]. 0920-4105/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.petrol.2011.11.007

can be learned from production data is vast. For instance, following Yousef et al.'s (2006) work, Sayarpour et al. (2009a, 2009b) introduced the capacitance–resistance models (CRMs) for dynamic evaluation of waterfloods. This simple, easy-to-use tool proved to be very powerful in field applications (Sayarpour et al. 2009b). While capacitance–resistance models resolve interactions between wells, Izgec and Kabir (2009a) showed that the modified-Hall integral (MHI) can accurately resolve dynamic events occurring around injectors, such as plugging and fracturing. In a later publication Izgec and Kabir (2009b), the same authors showed that channel size and fracture conductivity may be estimated using surveillance data. Recently, Kabir and Boundy (2010) showed that reciprocal-productivity index (RPI) can be used to assess the dynamic support (injection or aquifer) acting on a producer. This study shows that a great deal of insight can be gained from simple analytical tools. Furthermore, the array of information learned from analytical tools can guide an engineer in making changes to the flow model that are physically consistent. Specifically, we demonstrate the use of (1) coupled capacitance–resistance/aquifer model to assess the nonuniform strength of the aquifer, interwell connectivity and preferential flow of injectant, and reservoir-drive indices, (2) CRM-derived flow- and storage-capacity curves to identify the thief zones and flow geometry, (3) MHI to evaluate the performance of the injectors, (4) modified RPI to identify time-varying injection/ aquifer support. 2. Understanding aquifer and injection support Natural water encroachment with reservoir depletion is common in many oil and gas reservoirs. Understanding this reservoir/aquifer

O. Izgec / Journal of Petroleum Science and Engineering 81 (2012) 100–111

interaction is crucial to reservoir management for optimal hydrocarbon recovery. An aquifer may be substantially larger than the reservoir containing hydrocarbons, leading to infinite-acting or strong water-drive behavior; or may be small enough to have negligible effect on reservoir performance. Therefore, the importance of quantification of global and local aquifer strengths (i.e., felt at individual wells) cannot be overemphasized in strong and moderate waterdrive reservoirs. In water-drive reservoirs, aquifer influx experienced by each well is a function of reservoir heterogeneity and operating variables. Direct application of analytical aquifer models, being applicable to the entire reservoir, is nondiscriminating at the well scale. In a recent publication, Izgec and Kabir (2010) introduced a coupled approach for modeling differential aquifer influx received at each well. In their study, the combined use of CRM and an analytic aquifer model underpins the proposed approach. The CRM relies on signal-processing techniques in which injection rates are treated as input signals and production rates are the reservoir response or output signals. Interference between wells and response delay constitutes the unknown system parameters. Yousef et al. (2006) defined CRM as nonlinear multivariate-regression analysis tools, which account for compressibility and fluid flow in the reservoir. As detailed by Izgec and Kabir (2010) CRM is the solution to continuity equation for flux; so it is based on fundamental reservoir engineering concepts. In this study, a modified and simplified version of the Izgec– Kabir approach is proposed. Coupling the Carter–Tracy aquifer model with tank model representation of CRM (Sayarpour et al. 2009a) gives: qF ðt n Þ ¼ qF ðt n−1 Þe

Δt

−τn F


 Δt − n þ 1−e τF

!  BΔpðt Dn Þ−W e ðt Dn−1 ÞpD ′ jn n × I F þ f aj pD ðt Dn Þ−t Dn−1 pD ′ jn

ð1Þ

where subscript “n” refers to the present time step and “n − 1” the previous. And pD and tD are the classical dimensionless definitions of pressure and time. τ is the time constant (signal travel time), and I is the total filed injection rate. Izgec and Kabir (2010) showed that Brigham's closed-form equations (1997) can be used to calculate the pD and tD easily and accurately; Appendix A details the modified approach.

101

In the original study, CRMIP (injector–producer control volume, see Sayarpour et al. 2009a) was used to quantify nonuniform aquifer strength acting at each producer. In a typical heterogeneous system, the pressure-drop in each producer's drainage area, created by changing production rates, leads to differential aquifer influx. Previously, the aquifer parameters for a region were treated equally. These parameters include porosity, permeability, and thickness feeding each well. Therefore, for a given region the aquifer parameters remain the same; however, each individual well experiences differential influx because its drainage radius is evaluated at each timestep. In this study, instead of connecting an aquifer to each well, we start with a single aquifer attached to a tank, and then estimate the unknown parameters of the tank–aquifer system. Once the instantaneous aquifer influx to field is calculated, we use CRMIP formulation and let CRM find the aquifer influx acting on each well. This modification makes procedure simpler, and computationally less expensive. First, the concept is illustrated and verified with a synthetic model and later two field applications are provided. 2.1. Proof of concept We verify the notion of the coupled modeling approach with a synthetic example, wherein an aquifer is attached to a reservoir with 4 injectors and 9 producers, as shown in Fig. 1. The simulated model response is considered to be the “truth” and the injection/production data of this model is fed into the CRM-aquifer or CRMA model. Streamline simulations revealed the magnitude of aquifer influx at each well. Fig. 1 shows the synthetic model heterogeneity. Simulation input parameters are given in Table 1. An analytical aquifer model was attached to the reservoir model, having a thickness of 60 ft, permeability of 100 md, compressibility of 10 −5 1/psi, encroachment angle of 360, and porosity of 0.2. Variable production rates and corresponding flowing-bottomhole pressure in each well generated the necessary perturbation for CRMA. The corresponding streamlines at each timestep allowed cumulative water influx calculations for each well. Put simply, we used the unique feature of streamlines to generate allocations between the injector–producer and aquifer–producer pairs. Simulation-derived production data is considered as “field response” and fed into the CRM as input. Once again, the promise of CRMA is to capture interwell connectivity and estimate the aquifer strength acting on each well using production data only. Fig. 2 shows the production match quality achieved by

Fig. 1. Synthetic model used for verification of CRMA approach: permeability field (left) and corresponding streamlines.

102

O. Izgec / Journal of Petroleum Science and Engineering 81 (2012) 100–111

Table 1 Model parameters. Grid Grid size, ft Datum, ft Pinitial, psi Øavg kx,avg, mDa ky,avg, mDa kz,avg, mDa λx/L λy/L λz/h Sor M a

101 × 101 × 10 21, 21, 5 3300 1485 0.2 120 120 12 5 1 0.30 0.25 1

μo (@datum), cp Rs, scf/rb Bo, rb/stb Bw, rb/stb ρo, °API cf, 1/psi co, 1/psi cw, 1/psi Pb, psi WOC, ft GOC, ft Swir

2 200 1.15 1.01 56 3.0E-06 5.0E-06 1.0E-06 250 4000 2000 0.2

Permeability is log-normally distributed.

CRMA method and Fig. 3 shows the crossplot of streamline- and CRMA-derived aquifer allocation factors. Almost perfect accuracy of the estimated allocation factors confirms the robustness of the new CRMA approach. True water influx (simulated) and corresponding CRMA estimation is provided in Fig. 2, as well. The approach was also able to estimate the designed aquifer properties correctly. Next, this approach is applied to an offshore reservoir experiencing moderate to strong aquifer influx. 2.2. XC1 reservoir application The XC1 field is an early Cretaceous, multi-Darcy channelized sandstone reservoir, as shown in Fig. 4. In spite of a fairly homogenous distribution of the petrophysical properties, some of the producers do not respond to injection. This observation indicates that local heterogeneities were not completely understood. Initial reservoir and bubble point pressures are 1350 and 1205 psi, respectively. The solution gas–oil ratio (GOR) is 130 scf/stb, and kv/kh is ~0.2. Oil is highly viscous (~ 140 cp) and has ~20° API gravity. The adverse

mobility ratio has been a serious issue in the XC1 field because wells had to be shut-in or abandoned because of their high water cuts. The field was entirely developed with horizontal producers and injectors. It was concluded that the injection water slumps down, under-runs the oil column and then cones up to producers. This process inevitably leads to significant by-passing of hydrocarbons. In spite of the unfavorable mobility ratio, the XC1 reservoir seems to respond to injection at some parts of the reservoir. Fig. 5 makes this point clear by showing the good correlation (R 2 = 0.83) between fluid production and water injection. We will come back to this observation later in the discussion part. The reservoir is also attached to a moderately strong aquifer, which helps support reservoir pressure. Because water injection started very early in field's development, the real size of the aquifer was not completely understood. From well performance and observed shut-in reservoir pressures, one can observe that the aquifer strength acting at each producer is different. To quantify the real size and connectivity of the aquifer, good surveillance data is needed. Difficulty arises when one attempts to quantify the size and connectivity of strong aquifers if water injection starts in early life of the field development. This makes it impossible to distinguish between the effect of injected water and natural influx. Indeed, this is the case observed in XC1 reservoir. Because low injection efficiency/bigger aquifer and high injection efficiency/smaller aquifer combinations can honor the material balance, it is possible to generate equiprobable realizations. To explore this uncertainty, we decided to investigate three realizations with 65%, 75%, and 85% injection efficiencies. These three cases produced exactly the same quality of history match for the 620 day time window. For brevity, we present the history match for only the 75% injection efficiency case in Fig. 6. In this figure, symbols show the real field data and lines represent CRMA-derived production estimation. A previous material balance study showed that ~75% injection efficiency is more realistic; hence, this case is selected for more detailed analysis. Based on this injection efficiency, the

20

21

Data

Water Influx, MrB/D

CRMA

qT, MrB/D

17 14 11 8

CRMA

Data

18 15 12 9 6 3 0

5 0

100

200

300

400

500

0

100

time, days 1,500

200

300

400

500

time, days

P01 CRMA

2,500

Data

P08 CRMA

Data

qT, RB/D

qT, RB/D

2,000 1,000

500

1,500 1,000 500

0

0 0

100

200

300

400

500

0

100

time, days Fig. 2. Production history match obtained by CRMA.

200

300

time, days

400

500

O. Izgec / Journal of Petroleum Science and Engineering 81 (2012) 100–111

1.00

aquifer allocation factors

qres, normalized

Calculated (CRMA)

0.20

0.15

0.10

0.05 0.05

0.10

0.15

103

0.20

Observed (streamline simulation)

0.80 0.60 0.40 0.20 0.00 0.00

y = 0.9624x R² = 0.8267 0.20

0.40

0.60

0.80

1.00

qinj, normalized

Fig. 3. Cross plot of streamline simulator- and CRMA-derived aquifer allocation factors.

Fig. 5. Good correlation between field injection rate and liquid production rate indicates that reservoir responds to the injection.

average aquifer influx was estimated at 41,000 STB/D. After estimating the aquifer size, connectivity and instantaneous aquifer influx to the field, the aforementioned method was used to find the aquifer strength acting on each well. For simplicity, we divided the field into four injection areas. These injection areas are not dictated by geological features or rock/fluid characteristics; injectors and surrounding producers were simply grouped based on their spatial location. Fig. 7 shows the percent aquifer influx acting on each injection area calculated by CRMA. This analysis serves for three purposes: (1) understanding the size and connectivity of the aquifer, (2) understanding aquifer influx acting on different areas of the reservoir, and (3) as a precursor to history matching with a numerical reservoir simulator. Calculated differential aquifer influx is to be used for history matching of the flow model of XC1 reservoir. This example clearly illustrates how simple analytical tools can help in the understanding of the individual well performances, areal heterogeneity, and differential aquifer influx acting on different areas of the reservoir, and can aid in history matching studies. 2.3. XC2 reservoir application The XC2 field comprises two sands whose sand-to-sand connection is not fully understood. A moderate aquifer supports the XC2_1 sand whereas XC2_2 has no connection to this aquifer. A number of now-abandoned producing wells were completed both in XC2_1

and XC2_2 sands, particularly LP1 which was drilled right through the water underlying the XC2_1, as Fig. 8 shows. Because of the pressure differential between the XC2_1 and XC2_2 sands, water rises in LP1 to reach XC2_2. A packer was set in the upper part of LP1 to prevent water from rising any further, but the asset team thinks that some amount of water rises from the aquifer to the XC2_2 sand, thereby creating a “dump flood”. This phenomenon is thought to bring pressure support to the producers in this sand. These two sands may also be connected through sand-to-sand contacts, and cross-flow effects between these two reservoirs are not yet fully understood. Our first objective was to quantify the aquifer influx, meaning the dump-flood rate, acting on the XC2_2 sand, which has no direct connection to the aquifer. The LI1 injector is completed in both sands; and we also want to allocate injections between the sands. The liquid rate match obtained by the CRMA analysis for the LP1 drilled in XC2_1, and LP2 well located in XC2_2 sands are provided in Fig. 9. The high-quality match conveys the capability of the method. Based on CRMA-derived allocation factors, results show that ~5% (~1375 RB/D) of the LI1 injection supports XC2_1 sand. The dumpflood support was estimated at ~2800 RB/D. This example showed the promise of material-balance-based CRMA approach to (1) quantify aquifer strength acting on each

Fig. 4. 3D view of channelized XC1 reservoir (oil saturation shown).

104

O. Izgec / Journal of Petroleum Science and Engineering 81 (2012) 100–111

Field Liquid Production

Field Oil Production 1 Field

Field 0.8

CRM

0.6 0.4 0.2 0

Normalized Oil Rate

Normalized Liquid Rate

1

CRM

0.8 0.6 0.4 0.2 0

0

200

400

600

800

Time, days

0

200

400

600

800

Time, days

Fig. 6. Cumulative liquid match and cross plot of daily rates obtained from CRMA against field data.

sand, and (2) to understand the transmissibility between the reservoirs. Uncertainty about the effect of dump flood is also addressed through this study. Using this tool in a probabilistic fashion would help understand and manage the waterflood more efficiently. 3. Understanding reservoir-drive mechanisms With advances in computing technology, high-resolution numerical simulators have become the preferred approach for reservoir engineering studies. Pletcher (2002) provides a comprehensive review of the available material-balance techniques, which were considered to be the norm before the advent of numerical simulators. Today, material-balance studies seldom precede a full-field numerical modeling, presumably because material balance is implicit in simulation. Nonetheless, Kabir et al. (2008) provide an excellent example of how simple analytical material-balance studies provide valuable information at a fraction of time needed for detailed numerical modeling. In their study, they showed energy indices plot as one of the key lessons learned from simple material-balance modeling. In this study, we show that energy indices can be also obtained from a CRM approach. The fundamental CRM equation is given in Eq. (1). With a simple algebra and ignoring the effect of bottomhole pressure changes, this equation can be written as

 Δt Δt − n n − n þ 1−e τF IF þ 1−e τF ! ′ BΔpðt Dn Þ−W e ðt Dn−1 ÞpD jn : pD ðt Dn Þ−t Dn−1 pD ′ jn

qF ðt n Þ ¼ qF ðt n−1 Þe 

Δt

−τn F

ð2Þ

In Eq. (2), the first, second and third terms represent effects of the expansion drive, injection support, and aquifer influx, respectively. These components can be calculated at each timestep and then can be normalized to be plotted as an energy indices graph; Fig. 10 presents such a plot for the aforementioned “proof of concept” example. This plot shows that contribution of the aquifer is decreasing in time, replaced by water injection. The steady trend of the “volumetric term” is the indication of the perfect voidage replacement (steady average pressure) in the field. The results are in perfect agreement with the streamline-simulation-derived reservoir drive indices. Next, use of this concept is demonstrated in a real field. 3.1. XC1 reservoir application The approach was applied to XC1 reservoir for which the reservoir details were provided previously. In XC1, despite changes in relative volumes of aquifer influx over 13 years of history, the aquifer has been an important contributor of the reservoir energy. This point is illustrated by the CRMA-derived reservoir energy indices, as shown in Fig. 11, where the calculated instantaneous aquifer influx is also presented. It is worthy to mention that a longer period of field history is studied compared to previous examples of XC1 where only 650 days of the field history was analyzed. As of December 2009, the aquifer influx amounts to ~ 35% of the total voidage from the standpoint of overall performance (Fig. 11). For brevity, we only show field-wide reservoir energy indices; however, the method could easily be applied at the well level which may provide important insights into the poorly supported areas for remedial action. 4. Understanding interwell connectivity, swept volume and channelized flow Interwell connectivity is a key issue in any field development planning, especially for success of secondary and tertiary recovery methods. Fluid PVT properties, geochemical fingerprinting, tracer

Fig. 7. Nonuniform aquifer strength acting on different parts of the XC1 reservoir.

Fig. 8. LP1 dump-flood mechanism.

O. Izgec / Journal of Petroleum Science and Engineering 81 (2012) 100–111

LP1 field

1.00

LP1 CRM

Normalized liquid rate

Normalized liquid rate

1.00

0.75

0.50

0.25

0.00

105

LP2 field

LP2 CRM

0.75

0.50

0.25

0.00 0

100

200

300

400

0

100

200

Time, days

300

400

Time, days

Fig. 9. Production match obtained with CRMA for LP1 and LP2 producers.

4.1. XC1 reservoir application The CRMA-derived connectivity matrix of Area2 (Fig. 7) in XC1 reservoir is visualized in Fig. 12. There is only one injector (UI1) in the selected area; however, the CRM analysis was done for the whole reservoir by simultaneously matching production history of all producers. In this map, size of an arrow signifies the degree of communication intensity between an injector and a producer. Although most of the producers in Area2 are in good communication with the injector, because of the nature of the oil, a polymer flooding pilot is considered as a future development option. Success of a polymer flood is dictated by how the reservoir is connected and the existence of the local heterogeneities. The CRMA-derived connectivity map aids in understanding the degree of connectivity between the well pairs, as shown in Fig. 12. Apparently, the UP3 producer is not in communication with the UI1 injector. Perhaps more alarmingly, only 43% of the UI1 injection is shared with the neighboring producers; the rest of the injection supports other producers in the field. In contrast, field data reveal that most efficient producers are located in this area. This point is made clear in Table 2, where cumulative production of each area is normalized by the cumulative production of the field. Injectors in Area2 accounted for 28% of the total field injection, while 41% of the total production is attributed to this area. This imbalance indicates existence of external support received from the aquifer and/or other injectors. Strong aquifer support acting on this area was previously made clear in Fig. 7. This analysis not only aids in understanding the interactions of producers with the injector in this area, but also quantifies the communication of the area with the rest of the reservoir/aquifer system. These are valuable information for designing a potential EOR project.

4.2. XC2 reservoir application Flow capacity–storage capacity diagrams have appeared in the reservoir-engineering literature for decades (Schmalz and Rahme, 1950; Lake, 1989). Also known as F–C curves, they were originally derived for 2D, vertical cross section, noncommunicating, layered reservoirs. F–C is a simple plot of cumulative flow capacity, F or kh, vs. cumulative storage capacity, C or Øh. It is also possible to generate F–C curves from production data using CRM. The CRM-derived curves are referred as F–Φ curves to emphasize the differences between the static, 2D F–C curves and these dynamic estimates of flow heterogeneity. CRM-derived curves are obtained from dynamic data and they are likely to better reflect the flow path distribution and geological features in the reservoir. This information is extremely important because the F–Φ curve discloses information about both degree and type of the communication (through channel, highpermeability streak, or matrix) between an injector–producer pair. Interwell connectivity coefficient (fij) of CRM readily represents the flow capacity between wells. The swept volume between an injector and a producer can be easily calculated from the CRM parameters. In CRMIP (injector–producer control volume, Appendix A) formulation fij, τij and Jij are fitting parameters. Pore volume associated with the producer j and injector i can be calculated by the product of τij and Jij. An F–Φ plot for an injector can be generated by Fi ¼

φi ¼

∑m j¼1 f ij

ð3Þ

N

p ∑j¼1 f ij

∑m j¼1 τ ij J ij N

p ∑j¼1 τ ij J ij

:

ð4Þ

0.75

Reservoir Energy, %

testing, and transient-pressure testing are the classical approaches to discern reservoir connectivity. Maybe the most well-known methods are interference and pulse tests. In these tests one well sends the signal and the other one receives it. But for fields where there are multiple producers and injectors, these tests are difficult to conduct because the signal can be distorted by production responses of other wells. The CRM offers the promise of rapid evaluation of waterflood performance by discerning the connectivity between well pairs. In this study, CRMA and modified-RPI is used in two reservoirs. CRMA, in this work, is not only used for mapping interwell connectivity but also to estimate the type of the connection (matrix, or channel/ high-perm streak). For this purpose, we introduced CRM-generated flow capacity (F)–storage capacity (Φ) curves. One of the key objectives of this analysis is to understand well connectivity so that appropriate measures, such as pattern realignment can follow for optimal flood performance. The secondary objective is to condition the earth model with the results of CRM. While CRM aids these purposes, we introduce a modified version of RPI to assess the effect of voidage-replacement ratio or VRR on sweep in a heavy-oil flood.

Volumetric

Aquifer

Secondary

0.50

0.25

0.00 0

100

200

300

400

time, days Fig. 10. Reservoir drive indices of the synthetic example.

500

106

O. Izgec / Journal of Petroleum Science and Engineering 81 (2012) 100–111

Aquifer Influx Aquifer influx, MRB/D

100

75

50

25

0 10/1995

12/1999

1/2004

2/2008

Date Fig. 11. Reservoir drive indices of XC1 reservoir.

The F–Φ curves around each injector are generated for the XC2 reservoir. Two of the curves are given in Fig. 13. The 45° line shows perfect displacement. In this ideal case, 20% of the flow comes from 20% of the sweep volume and next 20% comes from the next 20% of the sweep volume, etc. The connectivity matrix obtained from the CRM analysis shows 11% of the LI1 injection supports the LP3 producer. The F–Φ curve shows that this communication is through very-low pore-volume (Φ). This can be seen in Fig. 13. Further, the time constant (τ, the signal travel time) between this pair is very small compared to the other pairs. This shows that the communication is through a high permeability streak or a channel. Horizontal permeability field of the XC2 reservoir is given in Fig. 14. This figure shows that there are some geological features (circled) which may lead to CRM-observed communication. Although the current reservoir simulation model captures some of the local trends, these features might be more pronounced, which are apparently underestimated in the current earth model. Another interesting connection is between the LI2 and LP7 pair. The F–Φ plot around this injector is also given in Fig. 13. Because Φ (swept volume) is relatively large compared to F (allocation), the LP7 requires more injection from LI2. Apparently, the LP4 producer receives preferential energy from the LI2 injector. This behavior may be driven by the local heterogeneities, which are mildly captured in the current flow model, as shown in Fig. 15. The high permeability region at the bottom of the figure (circled with black) may contribute to preferential flow through LP4. On the other hand, low permeability region (circled with orange) may be the reason for the limited connection between LI2 and LP7. Overall, the degree of heterogeneity may have been underestimated in the current model. In these examples, we showed how CRM/F–Φ plot derived results can be used for improved understanding of local heterogeneities. A

UP4, 10% UP5, 15% UP3 no support

UP6, 12%

UP7, 6%

Fig. 12. Connectivity (gain) map of Area2, and% gains.

formal procedure to integrate these results to a flow model has not been developed as yet. Integration of CRM-derived properties is a daunting task. This difficulty arises from dynamic nature of connectivity, as opposed to the static permeability field, defined in a flow model. Another simple method we used was a modified version of RPI proposed by Kumar (1977). In the original formulation, dimensionless wellbore pressure (PDw) is plotted against dimensionless time based on the well drainage area (tDA) defined by   kh p −pwf 141:3qμB i

ð5Þ

0:000264kt : ∅μct A

ð6Þ

pDw ¼ t DA ¼

Kumar suggests plotting Cartesian plot of pDw vs. tDA to assess the nature of the boundary condition being, partial water drive, partial fluid injection, constant pressure and excessive voidage replacement. Such a plot is provided in Fig. A-1 for variety of values of water drive and/or fluid injection strength (f). For qualitative use, this plot can be simplified as a Cartesian graph of (pi − pwf)/q vs.. t. Recently, Kabir and Boundy (2010) demonstrated the use of this plot to discern temporal variation of the energy support acting on a well. In an RPI plot: (1) positive slope indicates weak injection/aquifer support, (2) negative slope means excessive voidage, and (3) a plateau indicates voidage balance. Here, we propose that plotting injection rate, total production (water, oil, and gas) rate RPI and oil rate RPI on the same graph can reveal significant information about displacement efficiency. Such a plot is provided for the LP4 well of XC2 reservoir in Fig. 16. In this plot we recognize two periods as marked in the graph. In period 1, decreases in injection rate lead to positive slope of “conventional” RPI, which is plotted for the total liquid rate. This signature indicates the need for more energy support, as explained above. In contrast, the oil-RPI shows negative slope, which is an indication of improved oilproduction response. Here, reduced water injection causes more uniform displacement of oil and, hence, less viscous fingering or preferential flow of water. In period 2, an increase in injection rate brings about a negative slope in “conventional” RPI. This signature is an indication of excessive

Table 2 Normalized production and injection by area.

Produced Injected

Area1

Area2

Area3

0.24 0.31

0.41 0.28

0.35 0.45

O. Izgec / Journal of Petroleum Science and Engineering 81 (2012) 100–111

107

1.00

1.00 LP7

LP7

LP6

0.75

0.75

LP2 LP4

F

F

LP5

0.50

0.50 LP9

0.25

LP1

LP2

0.25

LP4

LP8

0.00 0.00

LI2

LI1

LP3

0.25

0.50

0.75

1.00

0.00 0.00

Φ

0.25

0.50

0.75

1.00

Φ Fig. 13. F–Φ diagram for LI1 and LI2 injectors.

voidage. In this region, the oil-RPI exhibits a positive slope which means decreasing oil production. This case illustrates that in highmobility-ratio waterfloods, optimum voidage replacement ratio (VRR) may be below unity. We concluded that the injection water slumps down, under-run the oil column, thereby facilitating water cresting at the producers. This process promotes significant bypassing of oil. Several simulation cases were run to find the optimal depletion strategy for this reservoir. Based on many sensitivity runs, the asset team found out that XC2 reservoir should be maintained at 85% voidage to maintain the reservoir pressure and improve oil recovery. Replacing voidage by more than 85% results in poor sweep; hence, rapid increase in water cut. This point is also made clear in Fig. 17. Circled area on this plot shows the favorable response of field water–oil-ratio (WOR) to reduced water injection. As illustrated, the conventional RPI can provide valuable insights in displacement efficiency. Future work should focus on finding the optimum VRR from this plot in a quantitative fashion. 5. Understanding plugging and fracturing around injectors

the classical Hall plot with improved diagnostic capabilities (Izgec and Kabir, 2009a). MHI enables a field engineer's fast evaluation of injector performance and identifies the time periods where plugging, fracturing, and recompletion take place. Proposed methods involve plotting the Hall integral, analytical and numerical derivatives and injection data together. Potential signatures can be interpreted as: (1) no separation of Hall integral and derivatives is indicative of continued matrix injection, (2) downward separation of derivative from integral indicated formation fracturing, and (3) upward separation means formation plugging. Detailed discussion can be found in Izgec and Kabir (2009a and 2009b). We analyzed two wells with this method. The UI2 plot (Fig. 18) demonstrates matrix injection signature up to an injection volume of 1 MM Bbls, corresponding with a period of about 2 months. Thereafter a short plugging period, matrix injection and long fracturing effect is clear with some small cycles of plugging. Previous work performed also demonstrated that injection pressure for UI2 injector has for all but very early time been above the calculated fracture pressure. A steady decline in UI3 injectivity performance can be observed throughout all three years of operating life, as shown in Fig. 19. The

Performance of the XC1 field injectors were evaluated with another analytical tool, the modified-Hall Integral (MHI). This method builds on

LP4 LP7 LP3

LI2 LI1

mD Fig. 14. Horizontal permeability field of XC2 reservoir. CRM results show that the geological trends emphasized in circle may be more pronounced (underestimated in the current earth model).

Fig. 15. Horizontal permeability field of XC2 reservoir. CRM results show that the geological trends emphasized in circle may be more pronounced (underestimated in the current earth model). Low permeability region around LP7 and high permeability channel connecting LI2 and LP4 is mildly captured in the current earth model.

O. Izgec / Journal of Petroleum Science and Engineering 81 (2012) 100–111

54 1

50

0.8

2

46 42

0.6

38

qinj, MSTB/D

Reciprocal-PI, D/STB

1

34 0.4 220

recip_pi_o+w

recip_pi_o

230

240

LI2 injection

250

30 260

Hall Integral/DHI, Thousand psi-D

108

300

Derivative Hall Integral Num. Derivative

200

100

0 0.E+00

time, days

2.E+06

4.E+06

6.E+06

Cumulative Injection, STB

Fig. 16. Combined use of injection rate (LI2), reciprocal productivity index for liquid and oil (LP4) show effect of injection rate on waterflood efficiency.

Fig. 18. MHI plot for injector UI2 shows plugging and fracturing cycles in early time and continues fracturing in late time.

Hall derivative response suggests that this is coincident with plugging behavior. UI3 is a converted production well, which was originally gravel-packed. Gravel-packed wells tend to be having a negative outcome on injectivity. There is no evidence of fracturing having occurred over the life of this well. In part, this may be due to the larger estimated frictional pressure-drop occurring in the smaller tubing size as the operating wellhead pressure is similar to other fractured injection wells in the reservoir. Alternately, it may be too early in the well's life for fracturing to have occurred based on trends observed in other wells. Another likely cause of plugging in the injectors is owing to oil and/or solids content of the injected fluid. In some cases, injected contaminant levels can be seen to have exceeded the design specification; however, a limited data set is now available. Identifying the plugging and fracturing behaviors may be particularly important when considering the merits and behavior of future polymer EOR schemes in the field. MHI method provides a rapid evaluation of the event occurring around injectors using high-frequency, real-time surveillance data.

frequency production data can be better used by employing analytical models where — in most cases — computational time is not a concern. Also, the array of information learned from analytical methods can guide engineers to condition the simulation model to dynamic data. As discussed by Kabir and Boundy (2010), analytical methods can capture the events occurring at small timesteps (hours to days), which cannot be captured with conventional reservoir simulation practices where the larger time steps (months) are frequently required. Of course, ease of use and analytical techniques' ability to process high-frequency data, and granularity of the solution comes with a price. Analytical methods, by their nature, are simplified and may honor less physics. On the other hand reservoir simulation is a unique platform in that variety of data are merged with a cost of intensive labor and computation time. The methods used here can be considered as bridge between conventional diagnostics and detailed numerical methods. Conventional waterflood plots, such as the semilog plot of WOR vs.. cumulative oil production, cartesian gas–oil-ratio (GOR) vs.. time, and annotated well history plots can provide information about the efficiency of the waterflood. Although these plots are very easy to generate, they are all pure diagnostics and are not able resolve the root cause of the poor waterflood efficiency. In this context, more granular methods, such as the CRM can be used to resolve the complex interactions between the well pairs. Such studies can shed light on the root cause of the poor waterflood response of a specific producer. In the recent years, interwell connectivity methods gain considerable attention, as testified by the contributions of Albertoni (2002), Yousef et al. (2006), Dinh and Tiab (2008), Kaviani (2009), and Sayarpour et al. (2009a and 2009b). In most cases (except Kaviani

6. Discussion This study attempts to illustrate the use of some modern analytical and semianalytical tools for improved understanding of waterfloods. The use of permanent downhole gauges (PDG) has become popular in recent years. PDGs provide a continuous record of bottomhole pressure and flow rate measurements. These continuous measurements may also provide important information on changing reservoir properties, and events occurring in small time intervals. High-

WOR

80

10.0

40

qinj, MRB/D

100.0

0

1.0 0

20

40

Oil Cum., MMRB Fig. 17. Reduced water injection caused decrease in WOR. This might be because of high viscosity of crude oil (147 cp) which leads to immediate increase in water cut in high injection rates due to poor mobility ratio.

Hall Integral/DHI, Thousand psi-D

120

900

Derivative Hall Integral Num. Derivative

600

300

0 0.E+00

5.E+06

1.E+07

Cumulative Injection, STB Fig. 19. MHI plot for injector UI3 shows continues plugging.

O. Izgec / Journal of Petroleum Science and Engineering 81 (2012) 100–111

10

1

Liquid rate

Inj

7. Conclusions

Well Count

4

Following conclusions are drawn from this study: Well Count

Rate, MRB/D

8 6

2 0 12/01/96

0 01/09/01

02/17/05

109

03/28/09

Time Fig. 20. Liquid production, injection rates and well count in XC2 reservoir.

et al.'s work) importance of the effect of well count on an interwell connectivity mapping study was not discussed. Introducing a new well or shutting an existing well brings about redistribution of the connectivities between well pairs. Therefore, the periods that are going to be analyzed should be selected such that the well count remains constant. In Fig. 5, a cross plot of field-wide injection and liquid production is given for an initial assessment of the XC2 producers' response to injector. In Fig. 20 instantaneous injection and production rates are plotted together with the number of active producers for the same field. Interestingly, the producer well count and field-wide injection give similar signatures that are reflective of field production. In such a case, distinguishing the effect of drilling campaign and changing field-wide injection rate on field production becomes difficult. Effects of changing well count can be modeled with the approach provided by Sayarpour et al. (2009a) for a tank model only. Strictly speaking, if the complex interactions between well pairs are being investigated, time periods where the well count remains constant should be identified and studied. Dynamic data integration and condition earth models with the high-frequency analysis techniques (rate transient analysis, pressure transient analysis) have been studied previously. In this context, attempts made in probabilistic history matching and forecasting (Landa et al. 2005) with various geological realizations may be constrained considerably by the information gained from a suite of analytical tools. Although some illustrative examples are shown here, a formal procedure is not introduced to integrate CRM results in model building. Variable skin that can be calculated from MHI method can be incorporated into the full-field simulation models. Yet, contrary to common belief that current reservoir simulators are full-physics, the industry still struggles to correctly predict the waterflood performance. This is not just because of our lack of ability to resolve finescale heterogeneities but is also caused by the lack of physics introduced into the current reservoir simulators. Geomechanics (fracturing, subsidence), and pore-scale phenomenon (plugging) are such areas which are most of the time not included in simulation practices. In this regard, analytical tools can aid history matching efforts. The value of using analytical models employing high-frequency data can be summarized as follows: (1) Understanding the waterflood performance at different scales: reservoir, group of wells, and individual wells. (2) Integrating learnings from a suite of analytical tools to condition the flow model. (3) Understanding the events occurring in small time scales. (4) Understanding the evolving nature of the flood (changing skin, injection support, aquifer influx). (5) Corroborating reservoir behavior with various tools for consistency and quality assurance.

(1) In reservoirs experiencing moderate to strong aquifer influx, quantification of the aquifer's real strength becomes infeasible if the injection starts in field's early life. Lack of adequate surveillance data preventing establishing the aquifer's size precipitates this issue. (2) An array of information derived from simple analytical tools can give insights into dynamic reservoir events occurring at different levels — field, injector/producer pairs, and single well. (3) Information obtained from a well-level CRM study can be used to condition reservoir simulation models. Similarly, MHI methods provide information about plugging and fracturing. This information can also be incorporated into the flow model — as variable skin — for better waterflood performance predictions. (4) Combined used of F–Φ diagram and CRM can reveal important information about local heterogeneities. (5) Combined use of a suite of analytical (or semianalytical) tools can provide consistency and quality assurance in understanding reservoir behavior. Nomenclature A drainage area, ft 2 q flow rate, B/D t time, D τ time constant, 1/D I injection rate, B/D fij rate allocation or producer gain B aquifer constant, ft 3/psi p pressure, psi pi initial reservoir pressure, psi pe average reservoir pressure, psi pD dimensionless pressure pDW dimensionless wellbore pressure tD dimensionless time tDA dimensionless time based on well drainage area rD dimensionless radius re drainage radius, ft rw wellbore radius, ft We cumulative aquifer influx, STB Wi cumulative injection, STB IH Hall integral, psi-D DHI analytical derivative of Hall integral, psi-D/RB DHI, numeric numerical derivative of Hall integral, psi-D/RB h formation thickness, ft kx permeability in x-direction, mD ky permeability in y-direction, mD kz permeability in z-direction, mD Rs solution gas/oil ration, SCF/RB Bo oil formation volume factor, RB/STB Bw water formation volume factor, RB/STB cf formation compressibility, 1/psi co oil compressibility, 1/psi cw water compressibility, 1/psi ct total compressibility, 1/psi Fi flow capacity of injector i Φi storage capacity of injector i Jij productivity index, RB/D/psi μ viscosity, cp s* pseudo skin Acknowledgment The author is indebted to Chevron Management for permission to publish this study. Interpretation and opinions presented are solely

110

O. Izgec / Journal of Petroleum Science and Engineering 81 (2012) 100–111

those of the author. The author benefitted from discussion on analytical tools with C. Shah Kabir and Morteza Sayarpour. Colleagues Bulent Izgec and G. Michael Shook provided insightful feedback.

Brigham's aquifer influx functions

Appendix A. Overview of the methods used

W e ðt Dn Þ ¼ W e ðt Dn−1 Þ þ

13

f=0 f=0.25

11

f=0.5 9 f=0.75

pDw

Carter–Tracy aquifer model is given by

7

f=1 f=1.25

5

f=1.5 3

f=1.75

! BΔpðt Dn Þ−W e ðt Dn−1 ÞpD ′ jn ðt Dn −t Dn−1 Þ pD ðt Dn Þ−t Dn−1 pD ′ jn

ðA  6Þ

where subscript “n” refers to the present timestep and “n − 1” the previous. This model requires calculation of dimensionless pressure. Brigham (1997) presented semi-analytic solutions for both unsteady-state and pseudo steady-state aquifer influx. For the appropriate treatment of these equations, for all practical purposes, the solutions switch immediately from one to the other. These solutions apply to different boundary conditions, such as constant-rate and constant-pressure inner boundary and with infinite, constant pressure, and no-flow at the outer boundary. These relations are given as

f=2.0

1 0.00

0.25

0.50

0.75

1.00

tDA Fig. A-1. Cartesian plot of Kumar's function.

Capacitance–resistance model CRM can be developed for different control volumes (Sayarpour et al., 2009a). Derivation of the CRM for an injector–producer control volume (CRMIP) is provided here. For a control volume between and injector and producer, we can develop the CRMIP governing differential equation from mass conservation. Liang et al. (2007) presented the governing differential equation for this capacitance model by dqij ðt Þ 1 dpwf ;j 1 þ qij ðt Þ ¼ f ij ii ðt Þ−J ij τ ij τij dt dt

pD ¼

2πkhðp−pi Þ qμ

ðA  7Þ

tD ¼

kt ∅μct r 2e

ðA  8Þ

rD ¼

ra re

ðA  9Þ

8 1 3 2 2 > > >
pD ¼ 1:1237t D −0:4326t D  1:945 t D ≤0:7 > > t ≤0:328 ð r −1 Þ 1 1:024 > D D > t > 0:7 lnt D þ 0:80907 þ pD ¼ > D > 0:729 < 2 ðt D þ 0:4Þ : pD ¼ 4 2 > lnr −1 r 3r > D 2t D > D >
 > þ t D > 0:328ðrD −1Þ1:945 pD ¼
D −  > 2 > 2 2 2 > > r −1 4 r −1 : r −1 D D D ðA  10Þ

ðA  1Þ Reciprocal productivity index

where τij is time constant,
 ct V p τij ¼ J ij

ðA  2Þ

and pore volume, Vp, total compressibility, ct, and productivity index, J, are associated with the control volume between producer j and injector i; the fij term, connectivity, represents the steady-state fraction of the rate of injector i flowing toward producer j. qij ðt Þ : f ij ¼ ii ðt Þ

For a well which reached to steady-state flow in the center of a square reservoir following relation can be written (Kumar, 1977): pDw ¼

  pDw ¼ pi −pwf

−

qij ðt n Þ ¼ qij ðt 0 Þe t

−τn

þe

ij

t

t n −t 0 τij

δ

∫t n0 eτij

−

þ f ij ii ðt n Þ−e

t n −t 0 τ ij


 diij dpwf ;j −J ij dδ: dδ dδ

t DA ¼ # ii ðt 0 Þ ðA  4Þ

Assuming fixed injection rate i(Δtn) = Iin and a linear BHP variation during the time interval Δtn and replacing ii(tn) and ii(t0) we can write   qij ðt n Þ ¼ qij ðt n−1 Þe

−

Δt n τ ij

 !

þ

−

1−e

ðA  11Þ

kh 141:2qμB

0:000264kt ∅μct A

ðA  12Þ

ðA  13Þ

Kumar suggests plotting Cartesian plot of pDw vs. tDA to assess the nature of the boundary condition being, partial water drive, partial fluid injection, constant pressure and excessive voidage replacement. Such a plot is provided in Fig. A-1 for varying values of f (water drive and/or fluid injection strength). In an RPI plot: (1) positive slope indicates weak injection/aquifer support, (2) negative slope means excessive voidage, and (3) a plateau indicates voidage balance. Modified Hall integral

Δt n τ ij

  Δpwf ;j ðt n Þ :  f ij I i ðt n Þ−J ij τ ij Δt n

for t DA ≥ðt s ÞDA

where pDW and tDA are given by

ðA  3Þ

Solution for Eq. (A-1) can be written as     "

  1 4A  f  ln þ 2πð1−f Þt DA 4 2 γC A r2w

ðA  5Þ

Izgec and Kabir (2009a and 2009b) presented the modified Hall integral (MHI) formulation. Three curves are used in their formulation.

O. Izgec / Journal of Petroleum Science and Engineering 81 (2012) 100–111

Starting with pseudo steady-state equation: 
  141:2iw Bμ r ln e −0:5 þ s : pwf −pe ¼ kh rw

ðA  14Þ

Integrating both sides with respect to time 
    141:2W i Bμ r ln e −0:5 þ s : ∫ pwf −pe dt ¼ kh rw

ðA  15Þ

The original Hall plot is generated by plotting the integral term against cumulative injection, Wi. Using the procedure that authors provided, analytical derivative can be written as: 
 r DHI ¼ α 1 W i ln e þ s rw

ðA  16Þ

where ∝1 ¼

141:2Bμ : kh

ðA  17Þ

One can also easily take the numeric derivative of the Hall integral by use of the following expression

DHI;numeric ¼

  d∫ pwf −pe dt dlnðW i Þ

nþ1

≅

n

IH −IH lnðW i Þnþ1 − lnðW i Þn

ðA  18Þ

where   IH ¼ ∫ pwf −pe dt:

ðA  19Þ

Three plots generated from above relations serve for the necessary diagnostics of the plugging, fracturing and channelized flow by: (1) no separation of Hall integral and derivatives is indicative of continued matrix injection, (2) downward separation of derivative from integral indicated formation fracturing, and (3) upward separation means formation plugging. Detailed discussion can be found in Izgec and Kabir (2009a and 2009b). References Albertoni, A. 2002. Inferring interwell connectivity only from well-rate fluctuations in waterfloods. MSc. Thesis. The University of Texas at Austin, TX, US.

111

Brigham, W.E., 1997. Water influx and its effect on oil recovery: Part 1. aquifer flow. Technical Report, SUPRI TR 103, Contract No. DEFG22-93BC14994. US DOE/ Stanford University, Stanford, CA, USA. Dinh, A., Tiab, D., 2008. Inferring interwell connectivity from well bottomhole-pressure fluctuations in waterfloods. SPEREE 11 (5), 874–881. Haddad, S., Proano, E., Patel, Y., 2004. A method to diagnose depletion, skin, kh, and drive mechanism effects using reservoir monitoring data. Paper SPE 90032 presented at the SPE Annual Technical Conference and Exhibition, Houston, 26–29 September. Izgec, B., Kabir, C.S., 2009a. Real-time performance analysis of water-injection wells. SPEREE 12 (1), 116–123. Izgec, B., Kabir, C.S., 2009b. Identification and characterization of high-conductive layers in waterfloods. Paper SPE 123930 Presented at the SPE Annual Technical Conference and Exhibition, New Orleans, 4–7 October. Izgec, O., Kabir, C.S., 2010. Quantifying nonuniform aquifer strength at individual wells. SPEREE 13 (2), 296–305. Kabir, C.S., Al-Khayat, N.I., Choudhary, M.K., 2008. Lessons learned from energy models: Iraq's South Rumalia case study. SPEREE 11 (4), 759–767. Kabir, C.S., Boundy, F., 2010. Analytical tools aid understanding of history-matching effort in a fractured reservoir. Paper SPE 135039 Presented at the SPE Annual Technical Conference and Exhibition, Florence, Italy, 20–22 September. Kamal, M.M., Pan, Y., Landa, J.L., Thomas, O.O., 2005. Numerical well testing: a method to use transient testing results in reservoir simulation. Paper SPE 95905-MS Presented at the SPE Annual Technical Conference and Exhibition, 9–12 October, Dallas, Texas, USA. Kaviani, D. 2009. Interwell connectivity evaluation from well rate fluctuations: a waterflooding management tool. Ph.D. Dissertation, Texas A&M University, College Station, TX. Kim, J.U., Datta-Gupta, A., Jimenez, E.A., Detlef, H., 2010. A dual scale approach to production data integration into high resolution geologic models. JPSE 71, 147–159. Kumar, A., 1977. Strength of water drive or fluid injection from transient well test data. JPT 29 (11), 1497–1508. Lake, L.W., 1989. Enhanced Oil Recovery. Prentice-Hall, Englewood Cliffs, New Jersey. Landa, J.L., Kamal, M.M., Jenkins, C.D., Horne, R.N., 2000. Reservoir characterization constrained to well test data: a field example. SPEREE 3 (4), 325–334. Landa, J., Kalia, R.K., Nakano, A., Nomura, K., Vashishta, P., 2005. History-matching and associated forecast uncertainty analysis — practical approaches using cluster computing. Paper IPTC 10751 Presented at the International Petroleum Technology Conference, Doha, Qatar, 21–23 November. Liang, X., Hunan, C., Weber, D.B., Edgar, T.F., Lake, L.W., Sayarpour, M., Al-yousef, A., 2007. Optimization of oil production based on a capacitance model of production and injection rates. Paper SPE 107713 presented at the Hydrocarbon Economics and Evaluation Symposium, 1–3 April, Dallas, Texas, U.S.A. Maschio, C., et al., 2009. A methodology to reduce uncertainty constrained to observed data. SPEREE 12 (1), 167–180. Pletcher, J.L., 2002. Improvements to reservoir material-balance methods. SPEREE 1, 49–59. Sayarpour, M., Zuluaga, E., Kabir, C.S., Lake, L.W., 2009a. The use of capacitance–resistance models for rapid estimation of waterflood performance and optimization. JPSE 69 (2009), 227–238. Sayarpour, M., Kabir, C.S., Lake, L.W., 2009b. Field applications of capacitance– resistance models in waterfloods. SPEREE 12 (6), 853–864. Schmalz, J.P., Rahme, H.D., 1950. The variation of waterflood performance with variation in permeability profile. Prod. Monthly 15 (9), 9–12. Yousef, A.A., Jensen, J.L., Lake, L.W., 2006. A capacitance model to infer interwell connectivity from production and injection rate fluctuations. SPEREE 9 (6), 630–646.