Reservoir characterization using dynamic capacitance–resistance model with application to shutin and horizontal wells
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Abstract
Capacitance–resistance model (CRM) is a nonlinear signal processing approach that provides information about interwell communication and reservoir heterogeneity. Several forms of CRM have been introduced; however, they would deliver erroneous model parameters if production history involves shutin period. To address this issue, this study presents a dynamic capacitance–resistance model (DCRMP), a comprehensive formulation that is capable of handling multiple shutin periods in different producers. CRM model parameters are representative of the geological information. Accordingly, two geologically identical synthetic examples are used to validate DCRMP; one including shutin periods in historical production data of some producers and the other one with all continuously operating wells. Obtaining the same model parameters and the high quality of fitting in both cases proved the reliability of DCRMP, which allows the utilization of historical data to characterize the reservoir behavior in real cases. Investigation of uncertainty on the fitted model parameters was also performed to demonstrate that confidence intervals are affected mostly by two aspects; permeability distribution and interwell distance. It is shown that though the confidence intervals in the heterogeneous fields are relatively higher than the homogeneous examples, higher permeability and lower producer–injector distance reduce the uncertainty of model parameters in both cases. This study also applies the proposed model in reservoirs with horizontal wells and further examines the impact of well direction and length of the productive interval on the connectivities between wells.
Keywords
Capacitanceresistance model Reservoir characterization History matching Waterflood Shutin well Horizontal wellAbbreviations
 BHP
Bottomhole pressure
 CMG
Computer modeling group Ltd.
 CRM
Capacitance–resistance model
 CWI
Cumulative water injection
 DCRMP
Dynamic capacitance–resistance model
 ICRM
Integrated capacitance–resistance model
 IMEX
Implicit–explicit black oil simulator
 IWC
Interwell connectivity
 MLR
Multivariate linear regression
 MSHW
Multisegmented horizontal well
List of symbols
 L, F, t
Mean length, force, and time, respectively
 \(c_{\text{t}}\)
Total compressibility (L^{2}/F)
 \(f_{ij}\)
Interwell connectivity between injector \(i\) and producer \(j\), dimensionless
 \(I\)
Water injection rate (L^{3}/t)
 \(J\)
Productivity index (L^{5}/F − t)
 \(N_{\text{p}}\)
Cumulative liquid production rate (L^{3}/t)
 \(n_{\text{P}}\)
Total number of producers
 \(n_{\text{T}}\)
Total number of historic time periods
 \(n_{\text{I}}\)
Total number of injectors
 \(P_{\text{wf}}\)
Well bottomhole pressure (F/L^{2})
 \(q\)
Total liquid production rate (L^{3}/t)
 \(R^{2}\)
Correlation coefficient
 \(\tau\)
Time constant, t
Introduction
Waterflooding is known as the most frequently used secondary recovery method due to its proven success ratio, application ease, and cost efficiency (Gözel 2015). Recovery efficiency of a waterflood is highly depended on the sweep efficiency and the ratio of oil–water viscosity (Craig 1971; Gözel 2015). Several studies were conducted to propose a way of estimating reserves, recovery rates, and flood life, which, as Thakur and Satter (1998) states, are the most important goals of a waterflooded reservoir management. Volumetric, empirical and classical methods, performance curve analysis, and numerical reservoir simulation constitute the five common methods used to characterize the waterflood performance.
The capability of interwell connectivities (IWCs) to infer reservoir’s geological properties prompted some to present several methods regarding this issue. Heffer et al. (1997) used Spearman rank correlations to detect the relationship between injection–production well pairs and infer them to the geomechanical features of the reservoir. Jansen and Kelkar (1997) investigated the dependence of injection/production rates and pressure on the location of active wells in a waterflooded reservoir. Pizarro (1998) also utilized the Spearman rank method to compare observed data with numerical simulation results and reported the advantages and drawbacks. Soeriawinata and Kelkar (1999) presented a superpositionbased approach to resolve the effect of multiple injectors on a single producer by using crosscorrelation between the summations of the injection rates with the production rate.
Alejandro and Lake (2002) developed a robust multivariate linear regression (MLR) technique to calculate the connectivity and diffusivity filter (time lag) between injection–production well pairs and estimate the total liquid (oil and water) production of wells, simply using injection and total production rates in waterflood systems. They analyzed the interaction between wells such that water injection and total production, respectively, are regarded as stimulus and response in a reservoir system. Gentil (2005) explained the physical meaning of IWC and examined the relationship between transmissibility and heterogeneity. Dinh and Tiab (2008) extended MLR’s application and established a relationship between IWCs and bottomhole pressure (BHP) in injection and production wells. Although using MLR was a major breakthrough toward estimating IWC within a short time in a practical way, it suffered from some important limitations such as the assumption of constant BHP during the simulation.
Capacitance–resistance model
Yousef et al. (2006) introduced capacitance–resistance model (CRM), a nonlinear datadriven model to estimate the IWCs between production and injection wells within various conditions accurately. CRM considers the effect of capacitance (compressibility) and resistance (transmissibility), which correspond to two parameters, respectively: The degree of fluid storage (time constant, \(\tau\)) and the degree of connectivity (weight coefficient, \(f\)) between wells. By considering injection rates as input data and production rates as output, the CRM is derived based on the total fluid mass balance in the control volume. In addition to synthetic examples, Yousef et al. (2006) validated this approach by applying to real fields.

CRMT (control volume is the whole field),

CRMP (each producer has a drainage volume),

CRMIP (a control volume for each injector–producer pair).
The objective function (Eq. 7) should be minimized associated with Eqs. 5 or 6 (in case the BHPs are constant) by considering the constraints in Eq. 4. This leads to an MLR analysis in which any local minimum found by Eq. 7 is the global minimum. In previous nonlinear CRMs, as number of model parameters increase or extreme fluctuations present in injection rates, finding global minimum is hard and may lead to erroneous solution by being stuck in unrepresentative local minimum. Using the ICRM, the linear regression provides a unique set of model parameters representing the global minimum and reduces the computation time. Salehian and Soleimani (2018) improved the matching performance of ICRM by employing two consecutive objective functions for both monthly and cumulative liquid production match. Recently, there have been several efforts to characterize layered reservoirs with different types of CRM along with their application into conventional and smart reservoirs (Mamghaderi and Pourafshary 2013; Prakasa et al. 2017; Salehian et al. 2018; Temizel et al. 2018; Zhang et al. 2015, 2017). Nevertheless, there is still lack of information in the application of CRM in shutin and/or horizontal wells. To address these issues, this paper modifies the classic CRMP presented by Sayarpour (2008) and extends its application to more realistic reservoir and well conditions.
Weber et al. (2009) stated that shutin periods present a problem for CRM, as it cannot distinguish the zero rate of production due to the shutin or abandonment in response to possible operational reasons (i.e., extremely low permeable zone, barriers around well, formation damage, etc.). Hence, using these models in reservoirs in which some production wells are shutin for a specified period or abandonment would result in underestimated connectivities, as optimization skews model parameters downward to account for zero production in given time steps. Kaviani et al. (2012) and Soroush et al. (2014) addressed this issue by modifying the history matching window. Altaheini et al. (2016) presented a modified injection rate as an extra expression to previously proposed models. More recently, however, as de Holanda et al. (2018) explains, it is still necessary to develop a comprehensive approach to address CRM’s issue with shutin periods in production history.
The reservoir models developed in previous studies about CRM only consider the application of the proposed model in vertical wells. To the best of our knowledge, proposed forms of CRM have not yet been tested in reservoirs with horizontal wells. Therefore, application of CRM (DCRMP in this study) in horizontal wells becomes necessary to certify that CRM successfully characterizes the reservoir dynamic behavior regardless of well configuration.
This study addresses these two issues (i.e., shutin periods in production history and application of CRM in history matching of horizontal wells) by presenting a modified model, dynamic capacitance–resistance Model (DCRMP), based on mathematical and physical derivations. We then validate the new model through a heterogeneous field including temporary shutin periods in different producers. Thereafter, we apply the proposed model for characterizing the reservoirs including horizontal wells to show the ability of CRM in waterflood characterization regardless of the type of well, and to illuminate the impact of well configuration and its direction (if horizontal wells is used) on model parameters. We also analyze the confidence interval of obtained DCRMP parameters in both heterogeneous and homogeneous examples to understand their relationship with the physics of the reservoir.
Mathematical derivation of dynamic capacitance–resistance model (DCRMP)
The abovementioned DCRMP model eliminates shutin wells at each time step and automatically considers active wells when calculating liquid production rate of each producer. This formulation of dynamic interwell connectivity supports the extra contribution from injection wells (i.e., higher \(f_{ij}^{'} \left( {t_{k} } \right)\)) toward active producers, when some other producers are shutin in a particular time step. The impact of horizontal wells will also be studied through changes in \(f_{ij}\) and \(\tau_{j}\) relative to the cases where all the wells were vertical.
Results and discussions
In this section, DCRMP is evaluated for two synthetic reservoirs, identical in all aspects (geological properties, water injection rate history, wells’ configuration, and locations), but different in production history, as in the first case all wells operate continuously and the second case includes multiple shutin periods in different producers. In each case, we apply DCRMP to characterize the system and match the history of waterflood performance. We focus on the quality of fitting and accuracy of characterization parameters by comparing DCRMP to the classical CRMP. After validating the new model, we evaluate the capability of CRMs in fully active horizontal wells as well as multisegmented horizontal wells. In addition, we compare the confidence intervals on estimated parameters within heterogeneous and homogeneous examples.
In this work, commercial reservoir simulator CMG (2017) is used to simulate all synthetic cases, while authors specified all geological information and injection history. Then, a computer code in Python was developed based on the Levenberg–Marquardt algorithm to fit the data.
All synthetic reservoirs contain two phases: water and oil. The oil gravity (API˚) is 35, porosity is 21%, formation volume factor is equal to 1.012 bbl/STB, and fluid compressibility is 2.85E−6 l/psi. We assess the performance of DCRMP based on the accuracy of model parameters and quality of match in two different production scenarios. In case I, producers operate continually, while multiple shutin periods are implemented in production history of wells in case II.
After validation of DCRMP in heterogeneous case I and II, we apply it to four homogeneous synthetic reservoirs with horizontal wells to investigate the influence of well direction on IWCs. The production history of horizontal wells has been also involved with shutin periods. We also consider a reservoir in which the horizontal producer is a multisegmented well, that is, only some intervals of the well are producing, to study the effect of productive length on CRM outcomes. Finally, we discuss the dependence of model parameters on each other by analyzing the correlation coefficient values.
Validation of DCRMP
Case I: Heterogeneous field without shutin periods
Estimated model parameters by DCRMP in case I
\(\tau_{j}\)  I1  I2  I3  I4  I5  

P1  43.25  0.295  0.283  0.210  0.152  0.124 
P2  41.31  0.298  0.121  0.207  0.260  0.137 
P3  43.94  0.157  0.285  0.242  0.192  0.291 
P4  38.71  0.250  0.312  0.342  0.396  0.448 
Confidence intervals in heterogeneous case i
Uncertainty assessment on model parameters is vital to evaluate the reliability of history matching. In spite of traditional numerical and analytical reservoir models, CRMs are easy to use for statistical analysis. Several approaches such as generating ensembles of history matching results, clustered computing techniques, and Monte Carlo simulations are proposed to study the uncertainty (Landa et al. 2005; Sayarpour 2008). Kim (2011) calculated the confidence intervals for model parameters by both nonlinear and linear regression using Weber (2009)’s method in which time constants are not regression parameters, but constants. In this research, we utilized confidence intervals and correlation coefficients to infer uncertainty on the fitted model parameters. The Ftest method was used for calculating confidence intervals to compare our null model, which is the best fit for model parameters, with an alternate model, where one of the parameters is fixed to a specific value.
95% confidence intervals of model parameters calculated by DCRMP in case I
\(f_{ij}\)  I1  I2  I3  I4  I5  \(\tau_{j}\) 

P1  0.0467  0.1110  0.0515  0.0929  0.1525  0.0735 
P2  0.0452  0.1648  0.0518  0.0529  0.1449  0.0727 
P3  0.0870  0.1127  0.0457  0.0770  0.0691  0.0583 
P4  0.0574  0.0898  0.0322  0.0397  0.0567  0.0330 
Case II: Heterogeneous field with shutin periods
In case II, it is assumed that production history involves four shutin periods that each of them lasts for around 6 months. P1 and P4 experience one shutin period and two periods are assumed in production history of P2 (i.e., P3 operates continuously). The same geology (Fig. 3) and injection rates (Fig. 4) as of case I are used in case II. We aim to validate DCRMP based on two facts: first, model parameters must be independent of production scenario and second, the quality of liquid production history matching should be acceptable.
As mentioned in the previous section, conventional CRMs such as CRMP and ICRM may not provide completely satisfactory characterizations of model parameters when some producers are abandoned or temporarily shutin. The inability of those models to distinguish shutin periods lead to unrealistic and underestimated connectivities due to the wrong interpretation of zero production rates as an indicator of low permeability around the well. This brings the idea of using an improved CRM modification (DCRMP) based on the producerbased representation of CRM, called CRMP, which was presented by Sayarpour (2008). That is to say, the DCRMP is insensitive to the number and length of shutin periods as it eliminates the connectivity of shutin producers at each time step to avoid illogical results. Hence, in the real cases where some wells might be shutin by the operator for a while, DCRMP would be a good choice to characterize the reservoir and forecast the waterflood performance. In this paper, DCRMP is applied only to characterize the synthetic cases without any production optimization.
Estimated model parameters by DCRMP in case II
\(\tau_{j}\)  I1  I2  I3  I4  I5  

P1  43.25  0.295  0.283  0.210  0.152  0.124 
P2  41.32  0.298  0.121  0.207  0.260  0.137 
P3  43.95  0.157  0.285  0.242  0.192  0.291 
P4  38.71  0.250  0.312  0.342  0.396  0.448 
Waterflood history matching in horizontal producers
In this section, we apply DCRMP to a synthetic homogeneous reservoir including a horizontal well. The reason for choosing a homogeneous reservoir is being able to detect the effect of well configuration in the absence of any other heterogeneity (e.g., permeability, porosity, etc.). Second, we apply our model to the same reservoir, where the horizontal producer is a multisegmented horizontal well, to investigate the influence of productive length on model parameters. Finally, the impact of the direction of a horizontal well is examined by changing horizontal producer’s direction. Note that all reservoir and fluid properties, as well as injection history of studied cases, are identical. We acknowledge that previous forms of CRM may be useable in reservoirs with horizontal wells. However, to our knowledge, the application of a CR model in horizontal wells has not yet been investigated in the literature.
Case III: Basic homogeneous field
Average reservoir and fluid properties in case III and later
Number of grid blocks  100 × 100 × 5 
Size (ft^{3})  2000 × 2000 × 100 
Horizontal permeability \(K_{\text{h}}\) (md)  100 
Vertical permeability \(K_{\text{v}}\) (md)  20 
Porosity (%)  22 
Producer bottomhole pressure constraint (psi)  2000 
Reservoir temperature (°F)  158 
Oil density (API˚)  35 
Formation volume factor (bbl/STB)  1.01182 
Fluid compressibility (l/psi)  2.85E−6 
Formation compressibility (l/psi)  1E−5 
Estimated DCRMP parameters for case III
\(\tau_{j}\)  I1  I2  I3  I4  I5  

P1  33.873  0.291  0.338  0.253  0.192  0.168 
P2  35.592  0.322  0.191  0.269  0.325  0.155 
P3  29.722  0.176  0.271  0.232  0.208  0.348 
P4  32.634  0.210  0.200  0.246  0.276  0.330 
Confidence intervals in homogeneous case III
95% confidence intervals of model parameters calculated by DCRMP in homogeneous case III
\(f_{ij}\)  I1  I2  I3  I4  I5  \(\tau_{j}\) 

P1  0.0282  0.0617  0.0254  0.0421  0.0701  0.0439 
P2  0.0275  0.1039  0.0255  0.0257  0.0708  0.0420 
P3  0.0477  0.0699  0.0260  0.0396  0.0389  0.0381 
P4  0.0423  0.0889  0.0265  0.0304  0.0384  0.0371 
Case IV: Horizontal well in North direction
The only difference between case IV and case III (basic reservoir) is the existence of one horizontal producer in P4’s location which is drilled toward the North in the middle of formation (i.e., 50 ft from the top of the reservoir). The length of the horizontal section is 200 ft and it is fully perforated. Other reservoir and well properties, injection history and fitting window are identical to case III (Table 4).
Estimated DCRMP parameters for case IV
\(\tau_{j}\)  I1  I2  I3  I4  I5  

P1  31.189  0.272  0.304  0.247  0.175  0.172 
P2  33.637  0.303  0.153  0.268  0.309  0.158 
P3  27.377  0.158  0.231  0.233  0.187  0.351 
P4  33.732  0.246  0.203  0.376  0.283  0.373 
Case V: Multisegmented horizontal well
Estimated DCRMP parameters for case V
\(\tau_{j}\)  I1  I2  I3  I4  I5  

P1  30.040  0.301  0.331  0.249  0.208  0.188 
P2  33.414  0.326  0.173  0.269  0.358  0.186 
P3  25.283  0.186  0.260  0.225  0.230  0.386 
P4  30.233  0.183  0.167  0.242  0.196  0.312 
Case VI: Horizontal well in West direction
Estimated model parameters by DCRMP in case VI
\(\tau_{j}\)  I1  I2  I3  I4  I5  

P1  28.436  0.281  0.321  0.245  0.166  0.177 
P2  30.394  0.322  0.162  0.256  0.262  0.169 
P3  24.143  0.174  0.255  0.223  0.182  0.361 
P4  27.348  0.248  0.189  0.263  0.393  0.344 
As observed in previous cases, time constants, nevertheless, seem to be insensitive to the direction of the horizontal producer. It is worth mentioning that even though obtained model parameters are similar for different well configurations, they are not exactly same. Nevertheless, all of them can offer an acceptable match of geological information.
Conclusions
The main objectives of this study were to develop the modified version of CRM to characterize waterflooded reservoirs including multiple shutin periods in production history and utilize the proposed model in reservoirs with horizontal wells. We applied the new model DCRMP to several heterogeneous synthetic cases and validated its consistency with the imposed geological information. We report that the DCRMP is an effective tool to obtain realistic insight into the reservoir characteristics and production performance when production data includes shutin periods. The application of DCRMP in future prediction and production forecast is recommended as a potential future work.
The analysis of confidence intervals on estimated model parameters showed that the higher permeability and lower interwell distance affects the uncertainty positively. We obtained lower uncertainty for the connectivities of injector–producer pairs in the higher permeability regions as well as those pairs with lower interwell distance. Results also demonstrated higher uncertainty on model parameters in heterogeneous examples in comparison with the similar homogeneous case.
This paper presents informative facts about application of CRMs in waterflooded reservoirs containing horizontal wells. We validated that DCRMP can match the production rates perfectly. It was also shown that using a horizontal well instead of a vertical well improves the connectivities of the well, especially with those injectors on the same path of the horizontal producer. In addition, application of DCRMP to a reservoir with multisegmented horizontal well indicated that a fully productive horizontal well receives larger connectivity values with other injectors compared to a multisegmented (partially productive) one. Nevertheless, no big differences were observed between the time constants of vertical and horizontal wells with different characteristics, which mean that the type of well configuration does not affect time constants remarkably.
Notes
Acknowledgements
Authors deeply appreciate Computer Modeling Group (CMG) for donating the license of CMG 2017.
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