Nonlinear Finite Volume Methods for Simulating Flow and Transport in Fractured Porous Media

Abstract

Control volume based numerical methods are one of the most widely used methods for modeling flow and transport phenomena in porous media. Conventionally, the linear Two-Point Flux Approximation (TPFA) method is routinely used to compute the transmissibility for cell-to-cell connections in the computational domain because of its simplicity, efficiency and robustness. The TPFA method, however, is only consistent for the so-called K-orthogonal grids. In practice, the reservoir grids are hardly K-orthogonal as they are often dictated by complex geological features. To address this issue, linear Multi-Point Flux Approximation (MPFA) methods have been proposed which can rigorously accommodate general non-orthogonal grids populated by full anisotropic permeability tensors at the cost of sacrificing the monotonicity properties for highly anisotropic reservoirs. In recent years, there has been a growing interest in a class of nonlinear finite volume methods for modeling flow in challenging reservoirs because of the consistency and monotonicity properties they enjoy. In this work, we have developed nonlinear finite volume methods that utilize the concept of the harmonic averaging point with improved robustness. We gave a new derivation of the harmonic averaging point to better understand its behavior. Based on the new derivation, a correction algorithm is proposed to identify those 'ill-placed' harmonic averaging points and modify their locations accordingly by solving an associated minimization problem. The resulting nonlinear finite volume methods are either positivity-preserving (nonlinear TPFA method) or extremum-preserving (nonlinear MPFA method). Compared to existing methods, they can be applied to solve more challenging problems. We tested the performance of the nonlinear TPFA method for modeling highly compressible flow in anisotropic reservoirs in a fully implicit manner. The results show that the nonlinear TPFA method can accurately capture the anisotropic effect and is more robust than the classical MPFA-O method when the permeability anisotropy is strong. Underground geological formations are almost always fractured to some extent. Fractures of multiple scales introduce significant heterogeneity and anisotropy to the reservoir. To model flow and transport in these fractured reservoirs in a robust manner, we developed a Discrete Fracture-Matrix (DFM) model based on the nonlinear finite volume methods in 2D space. One challenge faced by any DFM model is the accurate and efficient treatment of fractures. We made a simple modification to the star-delta transformation so that it can be used for modeling both conductive and blocking fracture intersections. The nonlinear finite volume methods are used to calculate transmissibility for matrix-matrix connections while the linear TPFA method is used to couple the matrix and fracture together. Numerical convergence tests were conducted to assess the effect of the linear TPFA coupling on the accuracy of the DFM models when there is strong anisotropy in the rock matrix. We then extended the DFM models to 3D spatial dimensions. The 3D DFM models are more complex compared to their 2D counterparts mainly because of possible intrinsic permeability anisotropy inside the fracture plane itself. The concept of the harmonic averaging point was first generalized so that one harmonic averaging point can be defined at each fracture intersection. The nonlinear TPFA method is then used to compute transmissibility for fracture-fracture connections in addition to matrix-matrix connections. New nonlinear fluxes were derived for fracture intersections to avoid including the intermediate control volumes at fracture intersections in the final discretized systems. Finally, to model flow and transport in densely fractured reservoirs, we introduced a modeling workflow that partitions the fractures into small-scale and large-scale fractures based on the fracture length. A hybrid numerical upscaling approach is proposed to upscale the small-scale fractures together with the hosting matrix into a single anisotropic continuum while the large-scale fractures are modeled explicitly using the Embedded DFM (EDFM) model.

Date of Award	May 2021
Original language	American English