Physics-based preconditioners for flow in fractured porous media

Abstract

Discrete fracture models are an attractive alternative to upscaled models for flow in fractured media, as they provide a more accurate representation of the flow characteristics. A major challenge in discrete fracture simulation is to overcome the large computational cost associated with resolving the individual fractures in large-scale simulations. In this work, two characteristics of the fractured porous media are utilized to construct efficient preconditioners for the discretized flow equations. First, the preconditioners are tailored to the fracture geometry and presumed flow properties so that the dominant features are well represented there. This assures good scalability of the preconditioners in terms of problem size and permeability contrast. For fracture dominated problems, numerical examples show that such geometric preconditioners are comparable or preferable when compared to state-of-the-art algebraic multigrid preconditioners. The robustness of the physics-based preconditioner for less favorable fracture conditions is further demonstrated by a systematic degradation of the fracture hierarchy. Second, the preconditioners are physics preserving in the sense that conservative fluxes can be computed even for an inexact pressure solutions. This facilitates a scheme where accuracy in the linear solver can be traded for efficiency by terminating the iterative solvers based on error estimates, and without sacrificing basic physical modeling principles. With the combination of these two properties a novel preconditioner is obtained which bridges the gap between multiscale approximations and iterative linear solvers.

Heterogeneity preserving upscaling for heat transport in fractured geothermal reservoirs

“…Equation 10 can be solved either via a direct solver or by using an appropriate iterative solver, see, e.g., [32]. Similarly, after discretizing Eq.…”

“…If the coupling between flow and transport is assumed to be weak, a single fine scale pressure solve may be sufficient, in which case the pressure solve will not constitute a large part of the overall simulation cost. Alternatively, iterative solvers or multiscale methods tailored for fractured media can be applied to effectively produce conservative approximations to the velocity field on the fine scale, see for example [32,33,35]. This would in particular increase the computational efficiency in the case of compressible flow, where the pressure equation needs to be solved several times.…”

Algebraic dynamic multilevel method for embedded discrete fracture model (F-ADM)

“…ADM extends the applicability of the multiscale methods to fully-implicit (stable) simulations, allows for crossing the scales for all unknowns with a multilevel dynamic mesh, does not require reconstruction of conservative flux field, employs the basis functions which are computed only at the beginning of the simulation, and does not rely on any smoothing iterative procedure. The development of such a dynamic multilevel scheme for fractured media has not yet been addressed, despite their high importance in the geo-scientific community and extensive literature for fine-scale consistent discrete representations [22][23][24][25][26][27][28][29][30][31][32]. Such a dynamic multilevel approach allows for capturing explicit fractures at their relevant resolution while maintaining the scalability of the simulation for real-field applications.…”

Fully Coupled Nonlinear Fluid Flow and Poroelasticity in Arbitrarily Fractured Porous Media: A Hybrid‐Dimensional Computational Model

“…This model can be reduced to a discrete fracture model, which is appropriate when the system is dominated by highly conductive fractures, or the porous matrix is nearly impermeable (e.g., Erhel et al, ; Hyman et al, ). The model can also be modified by using empirical (Unsal et al, ) or averaged (Sandve et al, ) matrix flow for beneficiary trade‐off between accuracy and efficiency. Since fractures often occur at multiple length scales, the discrete fracture‐matrix model becomes computationally demanding if all fractures are to be resolved.…”