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Register a new userModeling flow in porous media with double porosity/permeability: A stabilized mixed formulation, error analysis, and numerical solutions
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Added by
Kalyana Babu Nakshatrala
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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The flow of incompressible fluids through porous media plays a crucial role in many technological applications such as enhanced oil recovery and geological carbon-dioxide sequestration. The flow within numerous natural and synthetic porous materials that contain multiple scales of pores cannot be adequately described by the classical Darcy equations. It is for this reason that mathematical models for fluid flow in media with multiple scales of pores have been proposed in the literature. However, these models are analytically intractable for realistic problems. In this paper, a stabilized mixed four-field finite element formulation is presented to study the flow of an incompressible fluid in porous media exhibiting double porosity/permeability. The stabilization terms and the stabilization parameters are derived in a mathematically and thermodynamically consistent manner, and the computationally convenient equal-order interpolation of all the field variables is shown to be stable. A systematic error analysis is performed on the resulting stabilized weak formulation. Representative problems, patch tests and numerical convergence analyses are performed to illustrate the performance and convergence behavior of the proposed mixed formulation in the discrete setting. The accuracy of numerical solutions is assessed using the mathematical properties satisfied by the solutions of this double porosity/permeability model. Moreover, it is shown that the proposed framework can perform well under transient conditions and that it can capture well-known instabilities such as viscous fingering.
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Modeling flow through porous media with multiple pore-networks has now become an active area of research due to recent technological endeavors like geological carbon sequestration and recovery of hydrocarbons from tight rock formations. Herein, we consider the double porosity/permeability (DPP) model, which describes the flow of a single-phase incompressible fluid through a porous medium exhibiting two dominant pore-networks with a possibility of mass transfer across them. We present a stable mixed discontinuous Galerkin (DG) formulation for the DPP model. The formulation enjoys several attractive features. These include: (i) Equal-order interpolation for all the field variables (which is computationally the most convenient) is stable under the proposed formulation. (ii) The stabilization terms are residual-based, and the stabilization parameters do not contain any mesh-dependent parameters. (iii) The formulation is theoretically shown to be consistent, stable, and hence convergent. (iv) The formulation supports non-conforming discretizations and distorted meshes. (v) The DG formulation has improved element-wise (local) mass balance compared to the corresponding continuous formulation. (vi) The proposed formulation can capture physical instabilities in coupled flow and transport problems under the DPP model.
The objective of this paper is twofold. First, we propose two composable block solver methodologies to solve the discrete systems that arise from finite element discretizations of the double porosity/permeability (DPP) model. The DPP model, which is a four-field mathematical model, describes the flow of a single-phase incompressible fluid in a porous medium with two distinct pore-networks and with a possibility of mass transfer between them. Using the composable solvers feature available in PETSc and the finite element libraries available under the Firedrake Project, we illustrate two different ways by which one can effectively precondition these large systems of equations. Second, we employ the recently developed performance model called the Time-Accuracy-Size (TAS) spectrum to demonstrate that the proposed composable block solvers are scalable in both the parallel and algorithmic sense. Moreover, we utilize this spectrum analysis to compare the performance of three different finite element discretizations (classical mixed formulation with H(div) elements, stabilized continuous Galerkin mixed formulation, and stabilized discontinuous Galerkin mixed formulation) for the DPP model. Our performance spectrum analysis demonstrates that the composable block solvers are fine choices for any of these three finite element discretizations. Sample computer codes are provided to illustrate how one can easily implement the proposed block solver methodologies through PETSc command line options.
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As groundwater is an essential nutrition and irrigation resource, its pollution may lead to catastrophic consequences. Therefore, accurate modeling of the pollution of the soil and groundwater aquifer is highly important. As a model, we consider a density-driven groundwater flow problem with uncertain porosity and permeability. This problem may arise in geothermal reservoir simulation, natural saline-disposal basins, modeling of contaminant plumes, and subsurface flow. This strongly nonlinear time-dependent problem describes the convection of the two-phase flow. This liquid streams under the gravity force, building so-called fingers. The accurate numerical solution requires fine spatial resolution with an unstructured mesh and, therefore, high computational resources. Consequently, we run the parallelized simulation toolbox myug with the geometric multigrid solver on Shaheen II supercomputer. The parallelization is done in physical and stochastic spaces. Additionally, we demonstrate how the myug toolbox can be run in a black-box fashion for testing different scenarios in the density-driven flow. As a benchmark, we solve the Elder-like problem in a 3D domain. For approximations in the stochastic space, we use the generalized polynomial chaos expansion. We compute the mean, variance, and exceedance probabilities of the mass fraction. As a reference solution, we use the solution, obtained from the quasi-Monte Carlo method.
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