We compare the Kurganov-Tadmor (KT) two-dimensional second order semi-discrete central scheme in dimension by dimension formulation with a new two-dimensional approach introduced here and applied in numerical simulations for two-phase, two-dimensional flows in heterogeneous formations. This semi-discrete central scheme is based on the ideas of Rusanov's method using a more precise information about the local speeds of wave propagation computed at each Riemann Problem in two-space dimensions. We find the KT dimension by dimension has a much simpler mathematical description than the genuinely two-dimensional one with a little more numerical diffusion, particularly in the presence of viscous fingers. Unfortunately, as one can see, the KT with the dimension by dimension approach might produce incorrect boundary behavior in a typical geometry used in the study of porous media flows: the quarter of a five spot. This problem has been corrected by the authors with the new semi-discrete scheme proposed here. We conclude with numerical examples of two-dimensional, two-phase flow associated with two distinct flooding problems: a two-dimensional flow in a rectangular heterogeneous reservoir (called slab geometry) and a two-dimensional flow in a 5-spot geometry homogeneous reservoir.
Simulation of multiphase porous media flows with minimizing movement and finite volume schemes
Inria, Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé（法国国家信息与自动化研究所，里尔大学，法国国家科学研究中心）；Inria, Project team MOKAPLAN and Mathematics department, Université de Liège（法国国家信息与自动化研究所，列日大学）；Department of Mathematics and Statistics, McGill University（麦吉尔大学数学与统计系）；IECL Université de Lorraine（洛林大学计算数学与复杂系统研究所）；GFM Universidade de Lisboa（里斯本大学）
In this article we formulate and implement a computational multiphase periporomechanics model for unguided fracturing in unsaturated porous media. The same governing equation for the solid phase applies on and off cracks. Crack formation in this framework is autonomous, requiring no prior estimates of crack topology. As a new contribution, an energy-based criterion for arbitrary crack formation is formulated using the peridynamic effective force state for unsaturated porous media. Unsaturated fluid flow in the fracture space is modeled in a simplified way in line with the nonlocal formulation …
An Adaptive Subdomain Coupling Approach in Domain Decomposition for Multiphase Porous Media Flow
SKLMS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；College of Computer Science and Technology, National University of Defense Technology（国防科技大学计算机学院）
Multiphase flow is a critical process in a wide range of applications, including carbon sequestration, contaminant remediation, and groundwater management. Typically, this process is modeled by a nonlinear system of partial differential equations derived by considering the mass conservation of each phase (e.g., oil, water), along with constitutive laws for the relationship of phase velocity to phase pressure. In this study, we develop and study efficient solution algorithms for solving the algebraic systems of equations derived from a fully coupled and time-implicit treatment of models of mult…
For both isothermal and thermal petroleum reservoir simulation, the Constrained Pressure Residual (CPR) method is the industry-standard preconditioner. This method is a two-stage process involving the solution of a restricted pressure system. While initially designed for the isothermal case, CPR is also the standard for thermal cases. However, its treatment of the energy conservation equation does not incorporate heat diffusion, which is often dominant in thermal cases. In this paper, we present an extension of CPR: the Constrained Pressure-Temperature Residual (CPTR) method, where a restricte…
We present efficient deep learning techniques for approximating flow and transport equations for both single phase and two-phase flow problems. The proposed methods take advantages of the sparsity structures in the underlying discrete systems and can be served as efficient alternatives to the system solvers at the full order. In particular, for the flow problem, we design a network with convolutional and locally connected layers to perform model reductions. Moreover, we employ a custom loss function to impose local mass conservation constraints. This helps to preserve the physical property of …
In fractured natural formations, the equations governing fluid flow and geomechanics are strongly coupled. Hydrodynamical properties depend on the mechanical configuration, and they are therefore difficult to accurately resolve using uncoupled methods. In recent years, significant research has focused on discretization strategies for these coupled systems, particularly in the presence of complicated fracture network geometries. In this work, we explore a finite-volume discretization for the multiphase flow equations coupled with a finite-element scheme for the mechanical equations. Fractures a…
In this study, we develop a probabilistic approach to map the parametric uncertainty to the output state uncertainty in first-order hyperbolic conservation laws. We analyze this problem for nonlinear immiscible two-phase transport in heterogeneous porous media in the presence of a stochastic velocity field. The uncertainty in the velocity field can arise from the incomplete description of either porosity field, injection flux, or both. The uncertainty in the total-velocity field leads to the spatiotemporal uncertainty in the saturation field. Given information about the spatial/temporal statis…
We implement a computational periporomechanics model for simulating localized failure in unsaturated porous media. The coupled periporomechanics model is based on the peridynamic state concept and the effective force state concept. The coupled governing equations are integral-differential equations without assuming the continuity of solid displacement and fluid pressures. The fluid flow and effective force states are determined by nonlocal fluid pressure and deformation gradients through the recently formulated multiphase constitutive correspondence principle. The coupled peri-poromechanics is…
In subsurface multiphase flow simulations, poor nonlinear solver performance is a significant runtime sink. The system of fully implicit mass balance equations is highly nonlinear and often difficult to solve for the nonlinear solver, generally Newton(-Raphson). Strong nonlinearities can cause Newton iterations to converge very slowly. This frequently results in time step cuts, leading to computationally expensive simulations. Much literature has looked into how to improve the nonlinear solver through enhancements or safeguarding updates. In this work, we take a different approach; we aim to i…
We present a computational framework for dimension reduction and surrogate modeling to accelerate uncertainty quantification in computationally intensive models with high-dimensional inputs and function-valued outputs. Our driving application is multiphase flow in saturated-unsaturated porous media in the context of radioactive waste storage. For fast input dimension reduction, we utilize an approximate global sensitivity measure, for function-value outputs, motivated by ideas from the active subspace methods. The proposed approach does not require expensive gradient computations. We generate …
The numerical approach has been performed to study the Bingham fluid flow through an oscillatory porous plate with Ion-Slip and Hall current. Initially, at time; t = 0 both the fluid and the upper plate are at rest. At time; t > 0 the upper plate begins to oscillate in its own plane while the lower plate is stationary. The lower plate temperature is constant while the upper plate temperature has oscillated. A uniform magnetic field is applied perpendicular to the plates. To obtain the dimensionless equations from the governing non-linear partial differential equations, the usual transformation…
In this article we formulate a stable computational nonlocal poromechanics model for dynamic analysis of saturated porous media. As a novelty, the stabilization formulation eliminates zero-energy modes associated with the original multiphase correspondence constitutive models in the coupled nonlocal poromechanics model. The two-phase stabilization scheme is formulated based on an energy method that incorporates inhomogeneous solid deformation and fluid flow. In this method, the nonlocal formulations of skeleton strain energy and fluid flow dissipation energy equate to their local formulations.…
This work proposes an original preconditioner that couples the Constrained Pressure Residual (CPR) method with block preconditioning for the efficient solution of the linearized systems of equations arising from fully implicit multiphase flow models. This preconditioner, denoted as Block CPR (BCPR), is specifically designed for Lagrange multipliers-based flow models, such as those generated by Mixed Hybrid Finite Element (MHFE) approximations. An original MHFE-based formulation of the two-phase flow model is taken as a reference for the development of the BCPR preconditioner, in which the set …
Many subsurface engineering applications involve tight-coupling between fluid flow, solid deformation, fracturing, and similar processes. To better understand the complex interplay of different governing equations, and therefore design efficient and safe operations, numerical simulations are widely used. Given the relatively long time-scales of interest, fully-implicit time-stepping schemes are often necessary to avoid time-step stability restrictions. A major computational bottleneck for these methods, however, is the linear solver. These systems are extremely large and ill-conditioned. Becau…
A posteriori error estimates and adaptivity for locally conservative methods. Inexpensive implementation and evaluation, polytopal meshes, iterative linearization and algebraic solvers, and applications to complex porous media flows
Robust and accurate fully implicit finite-volume schemes applied to Darcy-scale multiphase flow and transport in porous media are highly desirable. Recently, a smooth approximation of the saturation-dependent flux coefficients based on Implicit Hybrid Upwinding (IHU) has been proposed to improve the nonlinear convergence in fully implicit simulations with buoyancy. Here, we design a truly multidimensional extension of this approach that retains the simplicity and robustness of IHU while reducing the sensitivity of the results to the orientation of the computational (Cartesian) grid. This is ac…
