Application Of Adjoint-Based Optimal Control To Gas Reservoir With A Memory Effect

Abstract

Summary In recent years researchers in oil-gas industry have established that the contribution of memory is significant for the modeling of fluid flow in unconventional reservoirs. According to modern works, the memory effect appears due to the contrast between high permeable fractures and nanoporous matrix leading to the gap of fluid velocities at the interface between these media. Also in the homogenization procedure from micro to macro scale the nonlocality in time (in other words, the memory) reflects the delay of fluid pressure and density between subdomains with different properties of pore space geometry. Mathematically, a memory-based fluid flow model can be described by the system of integro-differential equations. Despite the fact that a large number of journal articles are devoted to numerical methods for the forward solution of such equations, the problems of optimization and optimal control of these systems are actual and insufficiently studied. We consider the one-dimensional model of gas filtration and diffusion as a model with memory. The system includes a partial differential equation for filtration in fractures and weakly singular Volterra integral equation of the second kind, which describes the diffusion of gas from blocks with closed nanopores. Numerical simulation, obtained using a Navot-trapezoidal algorithm, shows that the effect of memory influences on the distribution and the time evolution of pressure and density in comparison with the classical double porosity model. The pressure-constrained maximization of discounted cumulative gas production was chosen as a basic optimization problem. The appearance of memory in the model makes the standard adjoint-based approach not applicable since it was developed only for conventional systems of partial differential equations. The novel adjoint model for media with memory was obtained from the necessary conditions of optimality using the classical theory of calculus of variations and efficiently applied to production optimization problem. In conclusion we compare optimal control scenarios for the model with memory and for the classical double porosity model. Analysis has shown the importance of memory accounting in reservoir optimization problems.