A simple model to infer interwell connectivity only from well-rate fluctuations in waterfloods Ximing Liang PII: DOI: Reference:
S0920-4105(09)00171-5 doi: 10.1016/j.petrol.2009.08.016 PETROL 1740
To appear in:
Journal of Petroleum Science and Engineering
Received date: Accepted date:
9 October 2007 2 August 2009
Please cite this article as: Liang, Ximing, A simple model to infer interwell connectivity only from well-rate fluctuations in waterfloods, Journal of Petroleum Science and Engineering (2009), doi: 10.1016/j.petrol.2009.08.016
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ACCEPTED MANUSCRIPT A simple model to infer interwell connectivity only from well-rate fluctuations in waterfloods Ximing LIANG School of Information Science & Engineering, Central South University, Changsha, Hunan 410083, China; Fax: +86-731-8830700; Email:
[email protected]
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Abstract
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This paper presents a procedure that systematically accounts for the interactions between wells in a reservoir. The reservoir is considered to be an input-output system with the injection rates as the input and production rates as the output. A simple capacitance model is developed to infer the interwell connectivity only from the injection/production data fluctuations on a reservoir. A commercial simulator on a synthetic field with five injectors and four producers is used to test the approach. The simulation results show that the simple capacitance model satisfactorily captures the long time dependent behavior between injectors and producers. Keywords: Capacitance model; Interwell connectivity; Data fluctuations; Simulation
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1. Introduction
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Managing production of an oil reservoir to maximize the future economic return of the asset is very important. The techniques to analyze past performance and then to predict the future vary from an educated guess to very complex numerical approximations. Most models rely on fitting or matching historical data. In petroleum fields, oil production is often constrained by the reservoir conditions, flow characteristics of the pipeline network, fluid-handling capacity of
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surface facilities, safety and economic considerations, or a combination of these considerations (Kosmidis et al. 2004; Wang, 2002). This requires simultaneous consideration of the interactions between the reservoir, the wells, and the surface facilities. Production and injection rates are the most abundant data available in any injection project. One can analyze these data to obtain valuable and useful information about interwell connectivity. The resulting information may be used to optimize subsequent oil recovery by changing injection patterns, assigning priorities in operations, recompletion of wells, and in-fill drilling. Several methods have been used to compare the performance of a producing well with that of the surrounding injectors. Heffer et al. (1995) used Spearman rank correlations to relate injector-producer pairs and associated these relations with geomechanics. Refunjol (1996) also used Spearman analysis to determine preferential flow trends in a reservoir and related injection wells with their adjacent producers and used time lags to find an extreme coefficient value. Sant’Anna Pizarro (1998) validated the Spearman rank technique with numerical simulation and pointed out its advantages and limitations. Panda and Chopra (1998) used artificial neural networks to determine the interaction between injector-producer pairs. Soeriawinata and Kelkar (1999), who also used Spearman rank analysis, suggested a statistical approach to relate injection wells and their adjacent producing wells. They applied superposition to introduce concepts of constructive and destructive interference. See also the works of Araque-Martinez (1993) and Barros-Griffiths (1998). Albertoni and Lake (2003) indicated interwell connectivity by a linear model with coefficients estimated by multiple linear regressions. The linear model weights quantitatively indicate the communication between a producer and the injectors in a
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waterflood. Filters were adopted to account for time lags between producer and injector. Gentil (2005) explored the physical meaning of the weights and proposed a new way to interpret them. These insights are used to better understand the underlying assumptions of the model used by Albertoni and Lake and to construct a procedure for incorporating production data into geostatistical permeability distribution models. Ali et al. (2005) used a more complicated model that includes capacitance (compressibility) as well as resistive (transmissibility) effects. Two coefficients are determined for each injector-producer pair: one parameter (weight) quantifies the connectivity and the other (time constant) quantifies the degree of fluid storage between the wells. In this work, as in Ali et al. (2005), the reservoir is considered as a system that converts inputs (injection rates) into outputs (production rates). However, compared with Ali et al. (2005), a simpler capacitance model is considered, where weights are calculated to indicate the connectivity between each injector-producer pair and time constants are determined to indicate the degree of fluid storage around each producer. The model has been applied to numerically simulated data (Eclipse model) on a synthetic field with five injectors and four producers. The results show that the simple capacitance model can successfully capture the attenuation and time lag in the field studied.
2. Model
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The balanced capacitance model (BCM) and the unbalanced capacitance model (UCM) are two different approaches in describing interwell connectivity. Both use the total production (oil+water+gas) rates (in reservoir volumes/time) and the injection rates (in reservoir volumes/time) for every well in a waterflood as input data. They are based on material balance (oil, water) and do not depend on the well locations. The capacitance model is a total mass balance with compressibility. In a real waterflood, there are multiple producers and injectors acting simultaneously and more than one injector usually influences the total production rate at a given producer. We consider a reservoir with m injectors and n producers. The governing material balance at reservoir conditions can be described by the following differential equations:
where ct is the total compressibility of the volume; Vp is the original pore volume p (t) j is the average pressure in the volume drained by of the drainage volume; producer j ; i k (t) is the injection rate in injector k (1,2,...,m) and q j (t) is the total production rate in producer j (1,2, ..., n) . The first term on the right side of Equation (1) is the total water flowing to producer j from all m different injectors. These equations state that at any time the net rate of mass depletion from the drainage volume can be accounted for by a change in average pressure. Equation (1) is based on the assumptions that the total compressibility of a reservoir is small and constant and there is no fluid transfer out of or into the volume. To obtain a description that is based entirely on rates, we use the following linear
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productivity model
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where p wfj (t) and J j are the flowing bottomhole pressure and productivity index of the producer j , respectively. Equation (2) holds only for stabilized flow and is unlikely to be accurate in circumstances in which rates are constantly changing. Its appropriateness can only be established by numerical simulation and application. However, the linear productivity model (2) (and its analogous alternative definitions) is almost universally applied in describing well performance in practice. Eliminating the average pressure in Equations (1) and (2), we can obtain a model to describe the relationship among production rate, injection rates and bottomhole flowing pressure:
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In this work, we assume a constant flowing bottomhole pressure for simplicity. By defining
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as the time constant of the volume drained by producer j , we obtain the following simple capacitance model
This model provides one time constant
for each producer j and one weight
for each injector-producer ( k j ) pair. The reservoir is considered to be an input-output system with the injection rates
as its input and the production rates
as its output. From equation (4), we can take the outputs
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as the state variables and obtain its state space formula as follows.
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where the state vector
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input vector
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system matrix
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and control matrix
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With
From the solution to Equation (5):
one can see that the total production rate can be decomposed into two components. The first term on the right of Equation (6) is the response of its initial production rate, which accounts for primary production associated with the total production. The second component is the contribution from the injection rates of all injectors. Thus, under the condition of constant flowing bottomhole pressure, model (5) incorporates the effects of primary production and multiple injectors. The m× n weights
and n time constants
can be determined by minimizing the squared errors between measured historical production rates and those generated by the following discrete
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Where
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is the length of the sampling interval over which the injection rate is held constant. To ensure conservation of mass, we enforce additional constraints (8) and (9) on the weights by their physical meaning.
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A two-stage optimization procedure allows us to determine the optimum values of Akj and τj . For a given set of τ j ’s, a constrained multivariate linear regression under constraints (8) and (9) is used to determine the λkj ’s, as Equation (7) is linear in λkj when the τj are known. The optimum set of λkj is obtained after iterating on the τj . Actually, one can also directly use a standard nonlinear constrained optimization method to obtain the optimum solution of λkj and τj . The m × n weights generated from minimization provide a quantitative expression of the connectivity between each injector-producer pair; the larger the weight, the greater the connectivity. The n time constants are direct measures of the rate of dissipation around each producer; the larger the time constant, the larger the dissipation rate.
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3. Numerical Results
To show the utility of our model, we applied it to three groups of numerically simulated data (Eclipse) on a synthetic field (see Albertoni, 2003; Yousef, 2005, 2006, for the details). The results of these applications are presented and discussed below. The Synfield consists of 5 injectors and 4 producers and the location of wells is shown in Figure 1. It has only vertical wells and its characteristics are similar to an actual reservoir. The Synfield dimensions and the grid size are 31 × 31 × 5 and 40× 40×6 ft, respectively. The distance between injector and producer is 800 ft. The oil-water mobility ratio is equal to one and the oil, water and rock compressibility are 5×10-6, 1×10-6 and 1 × 10-6 psi-1, respectively. The oil and water compressibility is independent of saturation. The flowing bottomhole pressure is constant. The Synfield is a single-layered anisotropic reservoir and we set its permeability in the following three different cases. Case 1: permeability is 5 md everywhere; Case 2: permeability in horizontal direction is three times of that in vertical direction; Case 3: permeability in vertical direction is three times of that in horizontal direction. The same injection data as Yousef (2005, 2006) are used in all three cases. These data, as shown in Figures 2a-e, were randomly selected from different wells in a real field and proportionally modified to be in agreement with the Synfield injectivity. The numerical simulation extends for 100 months (approximately 3000 days), which represent a history of 100 data points of production. The corresponding total production rate, oil production rate and water production rate for four producers in the
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three different cases are shown in Figures 3a-d, 4a-d and 5a-d, respectively. Using an active set truncated-Newton algorithm (Liang, 2005), the numerical values of the weights and time constants in the three different cases are obtained from our simple capacitance model and are shown in Tables 1a-b, 2a-b and 3a-b, respectively. In Figure 6a-c, arrows or cones that start from injector k and point to producer j represent the weights λkj . The larger the arrow is, the larger the value of the weight and the greater the connectivity between the two wells. These figures reveal different characteristics of the medium between each injector and producer pair. The λ’s are larger for near well pairs (e.g. λ11 , λ12 ) than for more distant well pairs (e.g. λ13, λ14) corresponding to greater connectivity between closer well pairs. This also indicates that λ’s manifest distinct characteristic of the medium. Another important characteristic is the symmetry in λ’s. The weights obtained in Case 1 are symmetric in horizontal, vertical and diagonal direction as expected from a homogeneous reservoir. As expected from a homogeneous reservoir, the weights obtained in Case 2 are symmetric in horizontal and vertical direction, respectively, but they are asymmetric in diagonal direction and those in horizontal direction are larger than those in vertical direction. We have the same observation from Case 3 as from Case 2. The weights obtained in Case 3 are symmetric in horizontal and vertical direction, respectively, and they are asymmetric in diagonal direction and those in vertical direction are larger than those in horizontal direction. The symmetry in λ’s was pointed out and it was concluded that the λ’s do not depend on injection rates (Albertoni and Lake, 2003; Yousef, 2005, 2006). Here, we further examine this observation using our simple capacitance model and confirm that λ’s do not depend on injection rates and they only depend on the reservoir properties and the relative location of the well. That is, our simple capacitance model describes the injection-production behavior and the weights quantify the connectivity between wells appropriately. Figures 7a-d, 8a-d and 9a-d show comparisons between the modeled total production rate using the simple capacitance model and the total production rate observed in the numerical simulation for all producers in the three different cases, respectively. The coefficient of determination R2 (Jensen et al. 2003) on the total production rate matching is shown in every figure. The matches of total production rate for all producers in the three different cases yielded R2 >0.998, which means the simple capacitance model can match the numerical simulation data very well. As a consequence, we conclude the simple capacitance model successfully captures the attenuation and time lag in the field studied.
4. Conclusions In this work, we present a procedure that systematically takes into account the interactions of an integrated oil and water production system. The reservoir is considered to be an input-output system with the injection rates as its input and production rates as its output. A simple capacitance model is sufficient to predict dynamic behaviors of injectors and producers in a reservoir. The proposed model was applied to three groups of numerically simulated data (Eclipse) on a synthetic field with 5 injectors and 4 producers. The results show that the proposed capacitance model matches all numerical simulation data very well for the long time behavior for each producer, and can successfully capture the true attenuation and time lag behavior between injectors and producers.
Acknowledgments This research is supported by National Key Basic Research and Development
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Program of China (2002CB312203). I would like to thank the China Scholarship Council and Dr. Thomas F. Edgar for providing the opportunity to visit UT-Austin. I appreciate the cooperation of Dr. Thomas F. Edgar and Dr. Larry W. Lake in oil production optimization study and for their contribution to scientific discussions. I am very grateful to Daniel B. Weber for the simulation on Eclipse and the numerically simulated data.
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References Albertoni, A., Lake, L.W., 2003. Inferring connectivity only from well-rate fluctuations in waterfloods. SPE Reservoir Evaluation and Engineering Journal, 6, 6-16. Araque-Martinez, A.N., 1993. Estimation of autocorrelation and its use in sweep efficiency calculation. M. S. Thesis, The University of Texas at Austin. Barros-Griffiths, I., 1998. The extreme Spearman rank correlation coefficient in the characterization of the north buck draw field, M. S. Thesis, The University of Texas at Austin. De Sant’Anna Pizarro, J.O., 1998. Estimating injectivity and lateral autocorrelation in heterogeneous media. Ph. D. Thesis, The University of Texas at Austin. Gentil, P.H., 2005. The use of multilinear regression models in patterned waterfloods: physical meaning of the regression coefficients. M.S. Thesis, The University of Texas at Austin. Heffer, K.J., Fox, R.J., McGill, C.A., Koutsabeloulis, N.C., 1995. Novel techniques show links between reservoir flow directionality, earth stress, fault structure and geomechanical changes in mature waterfloods. SPE 30711, 91-98. Jensen, J.L., Lake, L.W., Corbett, P.W.M., Goggin, D.J., 2003. Statistics for Petroleum Engineers and Geoscientist. Elsevier Science, Amsterdam, The Netherlands, 199p. Kosmidis, V.D., Perkins, J.D., Pistikopoulos, E.N., 2004. Optimization of well oil rate allocations in petroleum fields. Industrial and Engineering Chemistry Research, 43, 3513-3527. Liang, X.M., 2005. Active set truncated-Newton algorithm for simultaneous optimization of distillation column. Journal of Central South University and Technology (English Edition), 12(1), 93-96. Panda, M.N., Chopra, A.K., 1998. An integrated approach to estimate well interactions. SPE 39563, 517-530. Refunjol, B.T., 1996. Reservoir characterization of north buck draw field based on tracer response and production/injection analysis. M. S. Thesis, The University of Texas at Austin. Soeriawinata, T., Kelkar, M., 1999. Reservoir management using production data. SPE 52224, 1-6. Wang, P., Litvak, M.L., Aziz, K., 2002. Optimization of production operations in petroleum fields. SPE 77658, 1-12. Yousef, A.A., Gentil, P.H., Jensen, J.L., Lake, L.W., 2005. A capacitance model to infer interwell connectivity form production and injection rate fluctuations. SPE 95322, 1-19. Yousef, A.A., 2006. Investigating statistical techniques to infer interwell connectivity from production and injection rate fluctuations. PhD dissertation, The University of Texas at Austin.
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Figure 7d. Model match for producer P04 in Case 1
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Figure 7a. Model match for producer P01 in Case 1
Figure 8a. Model match for producer P01 in Case 2
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Figure 7b. Model match for producer P02 in Case 1
Figure 7c. Model match for producer P03 in Case 1
Figure 8b. Model match for producer P02 in Case 2
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Figure 9b. Model match for producer P02 in Case 3
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Figure 8c. Model match for producer P03 in Case 2
Figure 9c. Model match for producer P03 in Case 3
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Figure 8d. Model match for producer P04 in Case 2
Figure 9a. Model match for producer P01 in Case 3
Figure 9d. Model match for producer P04 in Case 3
