Problems In Porous Media Research Articles

Modeling gas breakthrough and flow phenomena through engineered barrier systems using a discrete fracture approach

Compacted clays are being considered to build engineered barrier systems (EBS) intended for the safe isolation of high-level nuclear waste (HLW) and spent nuclear fuel (SNF). The corrosion of the metallic canister containing the HLW/SNF will lead to the generation and buildup of the gas pressure in the more internal part of the clay buffer. This phenomenon would eventually trigger the formation and propagation of fractures in the clay barrier, jeopardizing its safety functions. In this work we propose to use the fragmentation technique (MFT) to model evolving fractures in clays triggered by gas pressurization. The MFT has been successfully used to model the formation of fractures in concrete, drying cracks in soil, hydraulic and thermo-fractures in rocks. In this work, we extend the MFT to deal with multiphase fluid flow in deformable porous media, and we upgraded a fully coupled computer finite element code using the extended technique. The proposed approach is first verified against analytical solutions and is then applied to model gas breakthrough experiments in clays. A very satisfactory performance of the method is observed in all the analyses, showing the potential of the MFT to tackle multiphase flow problems in deformable porous media with evolving discontinuities.

Mathematical modelling of free convection in a square cavity filled with a bidisperse porous medium for large values of Rayleigh number

"A free convection problem for bidisperse porous media is considered. The numerical solutions are obtained using an algorithm based on an nonuniform grid. Results for some values of the governing parameters when Rayleigh number is equal to 10^4 are provided."

A numerical approach to a 2D porous-medium mathematical model: Application to an atherosclerosis problem

This work deals with the study and implementation of a Local Space-Time DG ADER approach, in the finite volume framework with Weighted Essentially Non-Oscillatory (WENO) reconstruction in space, in the context of 2D porous media problems. The particular application performed concerns a biomedical problem dealing with the first stages of atherosclerosis disease, where the artery is considered as a porous medium, although the methodology described can be applied to other situations. The mathematical model on which this research is based is given by a system of two-dimensional nonlinear reaction-diffusion equations with a nonlinear source term in one of the equations, which is a variant of the model proposed originally in N. El Khatib, S. Genieys & V. Volpert (2007) Math. Model. Nat. Phenom., vol 2(2), pp. 126–141 [1]. The 2D model considered in this work has two main features: (i) the artery is taken as a porous medium and (ii) a nonlinear non-homogeneous Neumann boundary condition is incorporated, aimed to account for the recruitment of immune cells through the upper boundary, as a response to the production of cytokines. Certain theoretical properties of the stationary solutions and the evolution solution are stated and proved. Some of them are also verified with the numerical simulation results.

Generalized finite difference method based meshless analysis for coupled two-phase porous flow and geomechanics

This paper applies GFDM to analyze coupled two-phase flow and geomechanics for the first time, to our knowledge, is also the first meshless solver for two-phase fluid-solid coupling problems in porous media. Firstly, the implicit GFDM-based discrete schemes of the hyperbolic two-phase flow equation and elliptic stress equilibrium equation coupled by Biot's poroelastic theory are derived. Then the nonlinear solver based on Newton's method is used to solve the fluid pressure, water saturation, and displacement at each node simultaneously. Two technical details are found critical to the implementation of this method. The one is using different radii of the node influence domain to discretize the flow equation and the stress equilibrium equation respectively, and the other one is the selection criterion of the two radii of the node influence domain, which improves the implementation flexibility of the multi-physics coupling calculation, reduces the dissipation error of the calculation results of the convection-dominated water saturation distribution, and ensure a low computational cost. Several numerical test cases are implemented to analyze the computational efficiency and accuracy, and illustrate that GFDM can achieve good computational performance in case of regular domains, irregular domains, and different point clouds.Overall, this work may provide an important reference for building a GFDM-based general-purpose numerical simulator for multi-physics flow problems in porous media.

Homogenized lattice Boltzmann model for simulating multi-phase flows in heterogeneous porous media

A homogenization approach for the simulation of multi-phase flows in heterogeneous porous media is presented. It is based on the lattice Boltzmann method and combines the grayscale with the multi-component Shan–Chen method. Thus, it mimics fluid–fluid and solid–fluid interactions also within pores that are smaller than the numerical discretization. The model is successfully tested for a broad variety of single- and two-phase flow problems. Additionally, its application to multi-scale and multi-phase flow problems in porous media is demonstrated using the electrolyte filling process of realistic 3D lithium-ion battery electrode microstructures as an example. The approach presented here shows advantages over comparable methods from literature. The interfacial tension and wetting conditions are independent and not affected by the homogenization. Moreover, all physical properties studied here are continuous even across interfaces of porous media. The method is consistent with the original multi-component Shan–Chen method (MCSC). It is as stable as the MCSC, easy to implement, and can be applied to many research fields, especially where multi-phase fluid flow occurs in heterogeneous and multi-scale porous media.

Well-posedness and numerical approximation of steady convection-diffusion-reaction problems in porous media

We study a class of steady nonlinear convection-diffusion-reaction problems in porous media. The governing equations consist of coupling the Darcy equations for the pressure and velocity fields to two equations for the heat and mass transfer. The viscosity and diffusion coefficients are assumed to be nonlinear depending on the temperature and concentration of the medium. Well-posedness of the coupled problem is analyzed and existence along with uniqueness of the weak solution is investigated based on a fixed-point method. An iterative scheme for solving the associated fixed-point problem is proposed and its convergence is studied. Numerical experiments are presented for two examples of coupled convection-diffusion-reaction problems. Applications to radiative heat transfer and propagation of thermal fronts in porous media are also included in this study. The obtained results show good numerical convergence and validate the established theoretical estimates.

Existence and convergence of a discontinuous Galerkin method for the incompressible three-phase flow problem in porous media

Abstract This paper presents and analyzes a discontinuous Galerkin method for the incompressible three-phase flow problem in porous media. We use a first-order time extrapolation, which allows us to solve the equations implicitly and sequentially. We show that the discrete problem is well posed, and obtain a priori error estimates. Our numerical results validate the theoretical results, i.e., the algorithm converges with first order.

Two Different Computational Schemes for Solving Chemical Dissolution-Front Instability Problems in Fluid-Saturated Porous Media

Two Different Computational Schemes for Solving Chemical Dissolution-Front Instability Problems in Fluid-Saturated Porous Media

An upwind generalized finite difference method (GFDM) for meshless analysis of heat and mass transfer in porous media

In this paper, an upwind GFDM is developed for coupled heat and mass transfer problems in porous media. GFDM is a meshless method that can obtain the difference schemes of spatial derivatives by using Taylor expansion in local node influence domains and the weighted least squares method. The first-order single-point upstream scheme in the FDM/FVM-based reservoir simulator is introduced to GFDM to form the upwind GFDM, based on which a sequential coupled discrete scheme of the pressure diffusion equation and the heat convection-conduction equation is solved to obtain pressure and temperature profiles. This paper demonstrates that this method can be used to obtain the meshless solution of the convection–diffusion equation with a stable upwind effect. For porous flow problems, the upwind GFDM is more practical and stable than the method of manually adjusting the influence domain based on the prior information of the flow field to achieve the upwind effect. Two types of calculation errors are analyzed, and three numerical examples are implemented to illustrate the good calculation accuracy and convergence of the upwind GFDM for heat and mass transfer problems in porous media and indicate the increase in the radius of the node influence domain will increase the calculation error of temperature profiles. Overall, the upwind GFDM discretizes the computational domain using only a point cloud that is generated with much less topological constraints than the generated mesh, but achieves good computational performance as the mesh-based approaches, and therefore has great potential to be developed as a general-purpose numerical simulator for various porous flow problems in domains with complex geometry.

Parallel fully coupled methods for bound-preserving solution of subsurface flow and transport in porous media

As more powerful supercomputer systems with lots of memory and a large number of computing cores become available, the family of fully coupled algorithms is drawing more attention in scientific and engineering applications, due to its impressive robustness and scalability in extreme-scale simulations. In this work, we introduce and study some parallel domain decomposition preconditioned generalized Newton algorithms for solving the fully coupled and bound-preserving formulation of the subsurface flow and transport problem in porous media. In the approach, we present the active–set reduced–space (ASRS) method to guarantee the nonlinear consistency of the fully coupled system in a monolithic way, and meanwhile ensure the boundedness requirement of the solution. Furthermore, we focus on the application of the overlapping additive Schwarz preconditioning technique to accelerate the linear convergence of Newton iterations and be beneficial to the scalability of the inner linear solver. We present some numerical experiments to demonstrate the parallel scalability of the proposed algorithm on the Shaheen-II supercomputer with thousands of processors. In particular, the numerical results also show that the fully coupled framework has the nature of bound preservation and is efficient for the proposed reservoir flow problems.

A computationally efficient SPH framework for unsaturated soils and its application to predicting the entire rainfall-induced slope failure process

The first and fully validated smoothed particle hydrodynamics (SPH) model is presented to tackle coupled flow–deformation problems in unsaturated porous media that undergo large deformation and post-failure behaviour. Unlike the commonly adopted double-layer SPH framework for saturated soils, this paper presents a three-phase single-layer SPH model capable of predicting anisotropic seepage flows through porous media and their complete time-dependent transition from unsaturated to saturated states, as well as their influence on the mechanical behaviour of the porous media and vice versa. The mathematical framework is developed based on Biot's mixture theory and discretised using the authors’ recently developed novel SPH approximation scheme for the second derivatives of a field quantity. The soil is modelled using a suction-dependent elastoplastic constitutive model, expressed in terms of effective stress and suction. In addition, an adaptive two-timescale scheme is proposed for the first time to address existing challenges in solving coupled-flow large-deformation problems that involve a significant difference in the timescale required for the solid and fluid phases. The capability of the proposed SPH model was demonstrated through fundamental consolidation tests and a large-scale rainfall-induced slope failure experiment. Very good agreements with theoretical solutions and experimental results are achieved, suggesting that the proposed SPH model can be readily extended to solve a wide range of large-scale geotechnical applications involving coupled unsaturated seepage–deformation problems.

Efficient numerical solution of micro–macro models for multicomponent transport and reaction problems in porous media

ABSTRACT Transport and reaction of dissolved species in hierarchically structured complex media, e.g. porous media, can be described by micro–macro models (models with distributed micro-structure, models). These models offer advantages compared to pure micro- or pure macroscale models but exhibit, compared to pure macroscale models, a strongly increased numerical complexity. We show how parallel computing together with Schur complement techniques can make those models numerically feasible.

Enthalpy-based immersed boundary-lattice Boltzmann model for solid-liquid phase change in porous media under local thermal non-equilibrium condition

Heat transfer enhancement structures are adopted to improve the performance of latent heat thermal energy storage (LHTES) systems, such as metallic porous matrix, multi-tube, and fin. The metallic porous matrix and complex geometries of the structures bring difficulties to the numerical studies of the solid-liquid phase transition in the enhanced LHTES system. In this work, an enthalpy-based immersed boundary (IB)-lattice Boltzmann (LB) model is proposed for solid-liquid phase change problems in porous media under the local thermal non-equilibrium (LTNE) condition. In this model, to reduce the numerical diffusion across the solid-liquid interface, a two-relaxation-time (TRT)-LB model is developed to obtain the temperature field of phase change material (PCM). Two single-relaxation-time (SRT)-LB equations (LBEs) are employed for the velocity field of the liquid PCM and the temperature field of the porous matrix, respectively. The partially saturated method is used for the non-slip boundary condition on the moving solid-liquid interface. Different discrete source term schemes are employed for source terms induced by surface heat transfer and IB. Based on the source term treatment, the multi-direct-forcing IB-LB method is employed for the implementation of velocity, enthalpy and temperature boundary conditions on the complex boundary. The proposed model is verified by four cases: one-dimensional conduction melting under the LTNE condition, natural convection in a metallic porous matrix-filled cavity, conduction melting in an annulus filled with metallic porous matrix, and constrained melting in an isothermal circular cylinder without porous media. As an example of application, a metallic porous matrix-enhanced LHTES system with different multi-tube arrangements at different Rayleigh numbers (Ra) and porosities is investigated.

Fully implicit, stabilised, three-field material point method for dynamic coupled problems

This study presents the formulation and implementation of a fully implicit stabilised Material Point Method (MPM) for dynamic problems in two-phase porous media. In particular, the proposed method is built on a three-field formulation of the governing conservation laws, which uses solid displacement, pore pressure and fluid displacement as primary variables (u–p–U formulation). Stress oscillations associated with grid-crossing and pore pressure instabilities near the undrained/incompressible limit are mitigated by implementing enhanced shape functions according to the Generalised Interpolation Material Point (GIMP) method, as well as a patch recovery of pore pressures – from background nodes to material points – based on the same Moving Least Square Approximation (MLSA) approach investigated by Zheng et al. [1]. The accuracy and computational convenience of the proposed method are discussed with reference to several poroelastic verification examples, spanning different regimes of material deformation (small versus large) and dynamic motion (slow versus fast). The computational performance of the proposed method in combination with the PARDISO solver for the discrete linear system is also compared to explicit MPM modelling [1] in terms of accuracy, convergence rate, and computation time.

The two-phase Stefan problem with anomalous diffusion

The non-local in space two-phase Stefan problem (a prototype in phase change problems) can be formulated via a singular nonlinear parabolic integro-differential equation which admits a unique weak solution. This formulation makes Stefan problem to be part of the General Filtration Problems; a class which includes the Porous Medium Equation. In this work, we prove that the weak solutions to both Stefan and Porous Media problems are continuous.

A space-time generalized finite difference method for solving unsteady double-diffusive natural convection in fluid-saturated porous media

In this paper, the space-time generalized finite difference scheme is proposed to effectively solve the unsteady double-diffusive natural convection problem in the fluid-saturated porous media. In such a case, it is mathematically described by nonlinear time-dependent partial differential equations based on Darcy's law. In this work, the space-time approach is applied using a combination of the generalized finite difference, Newton-Raphson, and time-marching methods. In the space-time generalized finite difference scheme, the spatial and temporal derivatives can be performed using the technique for spatial discretization. Thus, the stability of the proposed numerical scheme is determined by the generalized finite difference method. Due to the property of this numerical method, which is based on the Taylor series expansion and the moving-least square method, the resultant matrix system is a sparse matrix. Then, the Newton-Raphson method is used to solve the nonlinear system efficiently. Furthermore, the time-marching method is utilized to proceed along the time axis after a numerical process in one space-time domain. By using this method, the proposed numerical scheme can efficiently simulate the problems which have an unpredictable end time. In this study, three benchmark examples are tested to verify the capability of the proposed meshless scheme.

Homogenized models of conduction-advection-radiation heat transfer in porous media

Upscaled models are derived from the periodic homogenization technique for conduction-convection-radiation problem in porous media for the case of an opaque solid and transparent fluid. Thermal radiation at the pore-scale level is first described through a differential view factor between elementary surfaces. Such heat transfer mode is coupled with the conduction-convection mechanisms by a radiative flux in the fluid phase and a jump of conductive fluxes at the solid/fluid interface. In the framework of the periodic homogenization procedure, a macroscopic model is built with upscaled effective coefficients. Both cases of closed and open pores are studied. In the second case the radiative transfer cannot be restricted to the central unit cell. The radiation at the macroscale only affects the effective conductivity. The validity of the approach is illustrated by some simple examples. The main objective of this work is to establish the macroscopic equation form of the conduction-convection-radiation problem and to understand how the radiation affects the macroscopic behavior.

Fully implicit local time-stepping methods for advection-diffusion problems in mixed formulations

This paper is concerned with numerical solution of transport problems in heterogeneous porous media. A semi-discrete continuous-in-time formulation of the linear advection-diffusion equation is obtained by using a mixed hybrid finite element method, in which the flux variable represents both the advective and diffusive flux, and the Lagrange multiplier arising from the hybridization is used for the discretization of the advective term. Based on global-in-time and nonoverlapping domain decomposition, we propose two implicit local time-stepping methods to solve the semi-discrete problem. The first method uses the time-dependent Steklov-Poincaré type operator and the second uses the optimized Schwarz waveform relaxation (OSWR) with Robin transmission conditions. For each method, we formulate a space-time interface problem which is solved iteratively. Each iteration involves solving the subdomain problems independently and globally in time; thus, different time steps can be used in the subdomains. The convergence of the fully discrete OSWR algorithm with nonmatching time grids is proved. Numerical results for problems with various Peclét numbers and discontinuous coefficients, including a prototype for the simulation of the underground storage of nuclear waste, are presented to illustrate the performance of the proposed local time-stepping methods.

Residual-based adaptivity for two-phase flow simulation in porous media using Physics-informed Neural Networks

This paper aims to provide a machine learning framework to simulate two-phase flow in porous media. The proposed algorithm is based on Physics-informed neural networks (PINN). A novel residual-based adaptive PINN is developed and compared with the residual-based adaptive refinement (RAR) method and with PINN with fixed collocation points. The proposed algorithm is expected to have great potential to be applied to different fields where adaptivity is needed. In this paper, we focus on the two-phase flow in porous media problem. We provide two numerical examples to show the effectiveness of the new algorithm. It is found that adaptivity is essential to capture moving flow fronts. We show how the results obtained through this approach are more accurate than using RAR method or PINN with fixed collocation points, while having a comparable computational cost.

A characteristic block-centered finite difference method for Darcy–Forchheimer compressible miscible displacement problem

In this paper, we construct, analyze, and numerically validate a characteristic block-centered finite difference method for Darcy–Forchheimer compressible miscible displacement problem in porous media. The block-centered finite difference method is used to discretize the miscible problem, where the pressure equation is described by the nonlinear Darcy–Forchheimer model, and the transport equation is addressed with the help of the characteristic method. A decoupled algorithm is designed for solving the resulting nonlinear system combined with Picard iteration. Our method is shown to be effective with the established theoretical analysis and several supporting numerical experiments.