Numerical Simulation of Multiphase Non-Darcy Flows: Generalized Approach

Abstract

A set of different numerical algorithms for non-Darcy flow models is developed and compared to each other in order to estimate functionality of algorithms and their potential of embedding into existing reservoir simulation software. In addition, a question of using such updated software to study an applicability of various non-Darcy flow models for unconventional reservoirs is discussed. The approaches are based on generalization of a linear Darcy law in which a flow equation is modified by nonlinear expressions of a flow rate and other reservoir values, so various formulations of non-Darcy flows from different research papers can be described as particular cases of such a general formula. Next, this generalized flow equation is applied to the modified black-oil equations, but an exclusion of a flow rate as unknown is impossible due to properties of the generalization. A finite volume discretization and Newton linearization are performed, and several techniques of computationally efficient solution are observed. A prototype of reservoir simulation program based on obtained mathematical model is constructed. Several numerical experiments are performed in order to verify numerical solutions and applied algorithms. Convergence rates of calculations by different approaches to non-Darcy flows are studied. The most significant finding is an existence of common approaches to exclude discretized and linearized flow equations at each iteration of nonlinear solver. This is important due to a presence of different non-Darcy models derived from different prerequisites (such as Forchheimer quadratic law and power law for non-Newtonian fluid) which can be studied through general algorithm as a research framework. Equally important is that the developed approaches are practically efficient and could be implemented in previously developed software without significant rearrangement of their code and algorithms in order to immediately gain practically useful simulations of non-Darcy flows or to explore their applicability, which is still an issue to resolve. The novelty of the considered approaches is in ability to embed non-Darcy flow models into present reservoir simulation software keeping most of existing algorithms and data structures implemented. Taking into account that the algorithms are based on a generalized form of non-Darcy flows, it is possible to calculate a wide range of models preserving computational complexity.