The Use of Capacitance-Resistive Models for Rapid Estimation of Waterflood Performance

Abstract

Abstract The capacitance-resistive model (CRM) offers the promise of rapid evaluation of waterflood performance. This semianalytical modeling approach is a generalized nonlinear multivariate regression technique that is rooted in signal processing. Put simply, a rate variation at an injector introduces a signal, with the corresponding response felt at one or more producers. CRM uses production and injection rate data and bottomhole pressure, if available, to calibrate the model against a specific reservoir. Thereafter, the model is used for predictions. We focused on three different control volumes for CRMs: the volume of the entire field, the drainage volume of each producer, and a drainage volume between each injector/producer pair. Unlike the numerical simulation approach, the CRMs use only production/injection data to predict performance, which provides simplicity and speed of calculation. Once the CRM is calibrated with historical production/injection data, we use an optimization technique to maximize the amount of oil produced by reallocating water injection rates. To verify CRM predictions, the models were tested against numerical flow-simulation results. Two case studies showed that the CRMs are able to successfully history match, and maximize the amount of oil produced by just reallocating water injection. This study introduces analytical solutions to the fundamental differential equations of the capacitance model based on superposition in time. In so doing, this approach adds flexibility, simplicity, and computational speed to the work presented previously. Introduction The CRM relies upon signal-processing techniques in which injection rates are treated as input signals and production rates are the reservoir response or output signals. Inter-well connectivity as well as response delay constitutes the unknown system parameters. Therefore, the model parameters of the reservoir reflect the connectivity between each injector/producer pair based on the historical injection and production data. Thereafter, performance predictions can be made with the fitted model parameters. In this regard, CRM may also be viewed as a nonlinear multivariate regression analysis tool, which accounts for compressibility and fluid flow in the reservoir (Yousef et al. 2006). Unlike the grid-based numerical-simulation approach, CRM models the reservoir flow behavior in accord with interactions (connectivity) between well pairs. Using injection/production data, Albertoni and Lake (2003) used a linear multivariate regression technique with diffusivity filters to predict the total fluid production of a well based on injection rates. In continuation of Albertoni's work (2002), Gentil (2005) explained the physical meaning of multivariate-regression-analysis constants by expressing the connectivity constant as a sole function of transmissibility. Yousef et al. (2006) showed the improved capability of extracting reservoir properties from injection and production data by introducing the capacitance model. The capacitance model considers the effects of compressibility, pore volume, and productivity index in nonlinear multivariate regression by introducing a time constant to characterize the time delay of the injection signal at the producers. Therefore, connectivity indices and time constants can reflect reservoir and fluid properties between injectors and producers. In this study, we introduce analytical solutions for fundamental differential equation of the capacitance model based on superposition in time. We present these solutions for three different reservoir-control volumes:volume of the entire field,drainage volume of each producer, anddrainage volume between each injector/producer pair. These analytical solutions facilitate CRM's application for rapid assessment at different levels of a field study, from a single well, to a group of wells, and to an entire field. CRM's analytical solutions in conjunction with the physical meaning of its weights, its capability to discern reservoir properties, its flexibility in taking timesteps, its simplicity, and speed are major advantages over those presented previously. We present applications of CRM to synthetic and real reservoirs by combining its results with an empirical oil-fractional flow model. This fractional-flow approach, introduced by Gentil (2005) and developed by Liang et al. (2007), allows the maximization of oil production rates by reallocating water amongst the injectors.