Use of POD in control of flow through porous media

Abstract

During the design of development concepts for the exploitation of oil and gas reservoirs, frequent use is made of numerical simulation of the flow of multi-phase fluids through porous rock. Recently, increased attention has been paid to systematic optimization of well positions and operating parameters (rates, pressures) over the life of the reservoir. Here we consider optimization of the displacement of oil towards production wells through the injection of water in other wells. Model-based optimal control of this “water flooding” process generally involves multiple simulations, which makes it into a time-consuming process. A potential way to address this issue is through the use of proper orthogonal decomposition (POD), We addressed the scope to speed up optimization of water-flooding a heterogeneous reservoir with multiple injectors and producers. We used an adjoint-based optimal control methodology that requires multiple passes of forward simulation of the reservoir model and backward simulation of an adjoint system of equations. We developed a nested approach in which POD was first used to reduce the state space dimensions of both the forward model and the adjoint system. After obtaining an optimized injection and production strategy using the reduced-order system, we verified the results using the original, high-order model. If necessary, we repeated the optimization cycle using new reduced-order systems based on snapshots from the verification run We tested the methodology on a reservoir model with 882 states (441 pressures, 441 saturations) and an adjoint model of 882 states (Lagrange multipliers). We obtained reduced-order models with 35-43 states only. The reduction in computing time was 52%.