Mixed Volume Element-Characteristic Fractional Step Difference Method for Contamination from Nuclear Waste Disposal

A positive semi-definite problem of three-dimensional incompressible contamination treatment from nuclear waste in porous media is discussed in this paper. The mathematical model is defined by a nonlinear initial-boundary system consisting of partial differential equations. Four important equations (an elliptic equation, two convection-diffusion equations and a heat conductor equation) determine the physical features. Considering the physical natures and computational efficiency, the authors introduce the conservative mixed finite volume element, upwind approximation and multistep difference to solve this system. The pressure and Darcy velocity are computed by a mixed finite volume element. The concentrations and temperature are solved by a combination of upwind approximation, multistep difference and mixed finite volume element. A multistep difference is used for approximating the partial derivative with respect to time. Mixed finite volume element and upwind differences are given for solving the convection-diffusions equations. Numerical dispersion and nonphysical oscillations could be eliminated, and the computational efficiency is improved by using a large time step. Furthermore, a conservative law is preserved and error estimates in $L^2$-norm is obtained. Finally, two numerical experiments are given to show the efficiency and possible applications.

Mixed finite volume element-upwind mixed volume element of compressible two-phase displacement and its numerical analysis

Research on the scientific and engineering problems of porous media has drawn increasing attention in recent years. Digital core analysis technology has been rapidly developed in many fields, such as hydrocarbon exploration and development, hydrology, medicine, materials and subsurface geofluids. In summary, science and engineering research in porous media is a complex problem involving multiple fields. In order to encourage communication and collaboration in porous media research using digital core technology in different industries, the 5th International Conference on Digital Core Analysis & the Workshop on Multiscale Numerical and Experimental Approaches for Multiphysics Problems in Porous Media was held in Qingdao from April 18 to 20, 2021. The workshop was jointly organized by the China InterPore Chapter, the Research Center of Multiphase Flow in Porous Media at the China University of Petroleum (East China) and the University of Aberdeen with financial support from the National Sciences Foundation of China and the British Council. Due to the current pandemic, a hybrid meeting was held (participants in China met in Qingdao, while other participants joined the meeting online), attracting more than 150 participants from around the world, and the latest multi-scale simulation and experimental methods to study multi-field coupling problems in complex porous media were presented. Cited as: Yang, Y., Zhou, Y., Blunt, M. J., Yao, J., Cai, J. Advances in multiscale numerical and experimental approaches for multiphysics problems in porous media. Advances in Geo-Energy Research, 2021, 5(3): 233-238, doi: 10.46690/ager.2021.03.01

A combined mixed and discontinuous Galerkin method for compressible miscible displacement problem in porous media

A miscible displacement of one compressible fluid by another in a porous medium is governed by a nonlinear parabolic system. A new mixed finite element method, in which the mixed element system is symmetric positive definite and the flux equation is separated from pressure equation, is introduced to solve the pressure equation of parabolic type, and a standard Galerkin method is used to treat the convection‐diffusion equation of concentration of one of the fluids. The convergence of the approximate solution with an optimal accuracy in L2‐norm is proved. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 229–249, 2001

The focus of this work is the numerical approximation of differential problems involving partial differential equations (PDEs) of elliptic nature, in the context of modelling and simulation in uid dynamics in porous media. The dissertation aims to contribute with an implementation of a multiscale multigrid algorithm, recently introduced in the literature, designed for solving ill-conditioned large linear systems of equations derived from those classes of PDEs, typically associated with Poisson problems of pressure-velocity with boundary conditions typical of ow in porous media. The Poisson problem discussed here is identied from the coupled convectiondiffusion transport system counterpart of PDEs, with dominated convection, and by a linearization by means the use of an operator splitting approach based on the physics of the underlying model. The methodology used for the discretization of the Poisson elliptic problem is mixed hybrid nite elements. The numerical solution of the resulting linear system will be addressed by a multiscale multigrid preconditioned conjugate gradient (PCG) method. We combine both methodologies in order to capture the distinct wavelengths associated with the different wavelengths from the assosiated self-adjoint Poisson operator, strongly inuenced by the heterogeneity of the geological properties of the porous media, in particular to the absolute permeability, which in turn might exhibit very large uctuations of orders of magnitude. Numerical experiments in applications of uid dynamics problems in porous media are presented and discussed for a verication of the results obtained by direct numerical simulations with the multiscale multigrid algorithm under consideration.

<abstract><p>In this paper, a two-grid mixed finite volume element (MFVE) algorithm is presented for the nonlinear time fractional reaction-diffusion equations, where the Caputo fractional derivative is approximated by the classical $ L1 $-formula. The coarse and fine grids (containing the primal and dual grids) are constructed for the space domain, then a nonlinear MFVE scheme on the coarse grid and a linearized MFVE scheme on the fine grid are given. By using the Browder fixed point theorem and the matrix theory, the existence and uniqueness for the nonlinear and linearized MFVE schemes are obtained, respectively. Furthermore, the stability results and optimal error estimates are derived in detailed. Finally, some numerical results are given to verify the feasibility and effectiveness of the proposed algorithm.</p></abstract>

In this paper we consider adaptive numerical simulation of miscible and immiscible displacement problems in porous media, which are modeled by single and two phase flow equations. Using the IMPES formulation of the two phase flow equation both problems can be treated in the same numerical framework. We discretise the equations by an operator splitting technique where the flow equation is approximated by Raviart–Thomas mixed finite elements and the saturation or concentration equation by vertex centered finite volume methods. Using a posteriori error estimates for both approximation schemes we deduce an adaptive solution algorithm for the system of equations and show the applicability in several examples.

In this paper we are interested in two phase flow problems in porous media. We use a Dual Mesh Method to discretize this problem with finite volume schemes. In a simplified case (elliptic - hyperbolic system) we prove the convergence of approximate solutions to the exact solutions. We use the Dual Mesh Method in physically complex problems (heterogeneous cases with non constant total mobility). We validate numerically the Dual Mesh Method on practical examples by computing error estimates for different test-cases.

Numerical modelling of mass transport problems in porous media: A review

In this article, we establish a new mixed finite element procedure, in which the mixed element system is symmetric positive definite, to solve the second‐order hyperbolic equations. The convergence of the mixed element methods with continuous‐ and discrete‐time scheme is proved. And the corresponding error estimates are given. Finally some numerical results are presented. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

Theoretical investigation in applied oil and gas recovery problems is based on mathematical models for multiphase fluid flows in porous media. The base model takes into account the continuity equations and Darcy’s laws for each of the phases, as well as the algebraic expression for the sum of the saturations. The equation for pressure is used for numerical implementation of these problems. In the present paper, we indicate the basic conservation properties for differential models, and discuss the necessity of their fulfillment at the discrete level. We construct explicit-implicit consistent approximations for multicomponent filtration problems in porous media and build splitting schemes for compressible media.

In this article, an approach for the efficient numerical solution of multi-species reactive transport problems in porous media is described. The objective of this approach is to reformulate the given system of partial and ordinary differential equations (PDEs, ODEs) and algebraic equations (AEs), describing local equilibrium, in such a way that the couplings and nonlinearities are concentrated in a rather small number of equations, leading to the decoupling of some linear partial differential equations from the nonlinear system. Thus, the system is handled in the spirit of a global implicit approach (one step method) avoiding operator splitting techniques, solved by Newton’s method as the basic algorithmic ingredient. The reduction of the problem size helps to limit the large computational costs of numerical simulations of such problems. If the model contains equilibrium precipitation-dissolution reactions of minerals, then these are considered as complementarity conditions and rewritten as semismooth equations, and the whole nonlinear system is solved by the semismooth Newton method.

In this work we introduce and analyze a mixed virtual element method for the two-dimensional nonlinear Brinkman model of porous media flow with non-homogeneous Dirichlet boundary conditions. For the continuous formulation we consider a dual-mixed approach in which the main unknowns are given by the gradient of the velocity and the pseudostress, whereas the velocity itself and the pressure are computed via simple postprocessing formulae. In addition, because of analysis reasons we add a redundant term arising from the constitutive equation relating the pseudostress and the velocity, so that the well-posedness of the resulting augmented formulation is established by using known results from nonlinear functional analysis. Then, we introduce the main features of the mixed virtual element method, which employs an explicit piecewise polynomial subspace and a virtual element subspace for approximating the aforementioned main unknowns, respectively. In turn, the associated computable discrete nonlinear operator is defined in terms of the $$\mathbb {L}^2$$ -orthogonal projector onto a suitable space of polynomials, which allows the explicit integration of the terms involving deviatoric tensors that appear in the original setting. Next, we show the well-posedness of the discrete scheme and derive the associated a priori error estimates for the virtual element solution as well as for the fully computable projection of it. Furthermore, we also introduce a second element-by-element postprocessing formula for the pseudostress, which yields an optimally convergent approximation of this unknown with respect to the broken $$\mathbb {H}(\mathbf {div})$$ -norm. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are presented.

A multiple-relaxation-time lattice Boltzmann model for convection heat transfer in porous media