Oil production strategies traditionally attempt to combine and balance complex geophysical, petrophysical, thermodynamic and economic factors to determine an optimal method to recover hydrocarbons from a given reservoir. Reservoir simulators have traditionally been too large and run times too long to allow for rigorous solution in conjunction with an optimization algorithm. It has also proven very difficult to marry an optimizer with the large set of nonlinear partial differential equations required for accurate reservoir simulation. A simple capacitance-resistive model that characterizes the connectivity between injection and production wells can determine an injection scheme that maximizes the value of the reservoir asset. Model parameters are identified using linear and nonlinear regression. The model is then used together with a nonlinear optimization algorithm to compute a set of future injection rates which maximize discounted net profit. Research previously conducted has shown that this simple dynamic model provides an excellent match to historic data. Based on a number of simulated and two actual fields, the optimal injection schemes based on the capacitance-resistive model yield a predicted increase in hydrocarbon recovery of up to 60% over the extrapolated historic decline. An advantage of using a simple model is its ability to describe large scale systems in a straightforward way with computation times that are short to moderate. However, applying the capacitance-resistive model to large reservoirs with many wells presents several new challenges. Reservoirs with hundreds of wells have longer production histories that often represent a variety of different reservoir conditions. New wells are created, wells are shut in for varying periods of time and production wells are converted to injection wells. Additionally, history matching large reservoirs by nonlinear regression is more likely to produce parameters that are statistically insignificant, resulting in a parameter dense model that does not accurately reflect the physical properties of the reservoir. Several modeling techniques and heuristics are presented that provide a simple, accurate reservoir model that can be used to optimize the value of the reservoir over future time periods. 1.0 Practical Aspects of Capacitance-Resistive Modeling Traditional reservoir simulators use detailed material balances to calculate reservoir pressures and oil saturations at multiple points in the reservoir. Finite difference equations valid for each grid-block require estimation of various parameters (compressibilities, permeabilities, porosities, etc.) throughout the reservoir. If these parameters are considered to be time-varying, (as they often are), they must be updated as the simulation progresses. This can be a daunting task for large simulations that can contain millions of grid-blocks. In spite of the difficulty, there have been several attempts to optimize oil production using traditional models (Barnes, 1990; Fang and Lo, 1996; Kosimidis et al., 2004 and 2005; Wang et al., 2002). Albertoni and Lake (2003) first proposed an inter-well connectivity model as an attempt to simplify the reservoir to a system of inputs (injection wells) and outputs (production wells). Subsequent versions of the capacitance-resistive model (CRM) expanded the number of model parameters and examined their physical meaning (Albertoni, 2003; Yousef, 2005; Yousef et al., 2006). Several attempts have been made to use the CRM to maximize predicted oil production (Liang et al., 2007; Sayarpour et al., 2007; Sayarpour, 2008). To solve for the necessary model parameters, the model requires only historic injection rates and total production rates - data that are typically already measured and collected. The CRM does not require a priori estimation of physical reservoir properties. However, history matching the model to historic data provides valuable information about the reservoir.

fax 01-972-952-9435. AbstractThis paper presents a methodology to optimize the value of oil produced by adjusting the water injection rates in a mature field.The reservoir is considered to be a multi-well input-output system with the injection rates as the inputs and production rates as the outputs. A simple capacitance model based on control volume of a given producer is fitted to total injection/production data obtained from a three dimensional reservoir simulator. A power law relationship models the instantaneous water-oil ratio as a function of the cumulative water injected. An optimization problem for oil production is formulated to optimize future performance, where oil production rates and water injection rates are optimized simultaneously based on the selling price of the oil produced and the cost of water injected or the disposal cost of water produced.The simulation results show that the simple capacitance model satisfactorily captures the long-time behavior between injectors and producers. The solution of the constrained optimization problem for oil production using nonlinear programming can maximize the future economic return of the reservoir asset. Various patterns of optimal injection are followed under different revenue objectives. The optimal injection rates depend on the cost of water injected/disposed, which leads to some wells being shut in while other wells operate at their maximum values.The proposed optimization method can be used for realtime production control and is appropriate for simultaneous optimization of well rates in a field where injectors are shut-in frequently.

Finite difference approximations to partial derivatives are generally based on Taylor series, which are polynomial expressions for the unknown variable as a function of the grid locations. In many problems, approximate analytical solutions are known that incorporate the physics of the process. It is proposed that such expressions be used to derive finite difference equations. Increased accuracy is anticipated, particularly when the solutions are highly non-linear, singular, or discontinuous.Reservoir simulation is such a problem. Flow in petroleum reservoirs results from injection and productions from wells, which are relatively small sources and sinks. Near singularities in the pressure around the wells result. The immiscibility of the fluids causes an oil bank to form in front of displacing water, and near discontinuities in the saturations occur. This paper investigates the utility and accuracy of finite difference equations for reservoir pressures based on two new functional forms: ln(r) and 1/r, where r is the distance to the well. The ln(r) form is based on pressures from line sources, and thus is effective at representing straight line wells. The 1/r form is based on pressures from point sources. The sum of many points represent more complex wells. Both are found to greatly increase the accuracy of the simulated reservoir pressures relative to solutions based on the polynomial approach.

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