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This article is dedicated to the general finite-volume framework used to discretize and solve saddle-point problems of various physics. The framework applies the Ostrogradsky–Gauss theorem to transform a divergent part of the partial differential equation into a surface integral, approximated by the summation of vector fluxes over interfaces. The interface vector fluxes are reconstructed using the harmonic averaging point concept resulting in the unique vector flux even in a heterogeneous anisotropic medium. The vector flux is modified with the consideration of eigenvalues in matrix coefficients at vector unknowns to address both the hyperbolic and saddle-point problems, causing nonphysical oscillations and an inf-sup stability issue. We apply the framework to several problems of various physics, namely incompressible elasticity problem, incompressible Navier–Stokes, Brinkman–Hazen–Dupuit–Darcy, Biot, and Maxwell equations and explain several nuances of the application. Finally, we test the framework on simple analytical solutions. [ABSTRACT FROM AUTHOR]

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-valued outputs, motivated by ideas from the active subspace methods. The proposed approach does not require expensive gradient computations. We generate an efficient surrogate model by combining a truncated Karhunen-Loéve (KL) expansion of the output with polynomial chaos expansions, for the output KL modes, constructed in the reduced parameter space. We demonstrate the effectiveness of the proposed surrogate modeling approach with a comprehensive set of numerical experiments, where we consider a number of function-valued (temporally or spatially distributed) QoIs. [ABSTRACT FROM AUTHOR]

In this paper, we study the a posteriori error estimate corresponding to the Brinkman–Darcy–Forchheimer problem. We introduce the variational formulation discretized by using the finite element method. Then, we establish an a posteriori error estimation with two types of error indicators related to the discretization and to the linearization. Finally, numerical investigations are shown and discussed. [ABSTRACT FROM AUTHOR]

The Richards' equation is widely used to simulate the saturation distributions in porous medium. Due to the high nonlinear of hydraulic diffusivity, it is usually difficult to obtain the analytical solutions, especially for finite boundaries. In this paper, the Richards equation for horizontal infiltration problem with finite Dirichlet boundaries is solved. The whole infiltrating process is divided into three state: the free infiltrating state, the transition state and the steady one, in which the transition state is considered as the key problem for analytical solving. Based on Boltzmann transformation and series expansion technique, an intermediate variable is introduced and an approximate analytical solution is derived for transition state. In addition, the exact solutions for other states are also given in the appendix. The present solutions can be applied for arbitrary nonlinear hydraulic diffusivity in Richards' equation. Two examples for power law and exponential law diffusivities are shown to confirm the accuracy of present method. [ABSTRACT FROM AUTHOR]

The present study investigates the onset of penetrative convection in- duced by selective absorption of radiation in a magnetic nanofluid saturated porous medium. The influence of Brownian motion, thermophoresis, and magnetophoresis on magnetic nanofluid treatment is taken into consideration. The Darcy’s model is selected for the porous medium. We conduct a linear stability analysis to examine the onset of instability and evaluate the results for two different configurations, namely, when the layer is heated from below and when the layer is heated from above. The numerical investigations are carried out by applying the Chebyshev pseudospectral method. The effect of the porosity parameter E, parameter Y (represents the ratio of internal heating to boundary heating), Lewis number Le, concentration Rayleigh number Rn, Langevin parameter αL, width of nanofluid layer d, diffusivity ratio η, and modified diffusivity ratio NA is examined at the onset of convection. The results indicate that the convection commences easily with an increase in the value of Y, Le, and NA but opposite in the case with a decrease in the value of E, αL, η and d for both the two configurations. The parameter Rn advances the onset of convection when the layer is heated from below, while delays the onset of convection when the layer is heated from above.

Thermal convection in a horizontally isotropic bi-disperse porous medium (BDPM) uniformly heated from below is analysed. The combined effects of uniform vertical rotation and Brinkman law on the stability of the steady state of the momentum equations in a BDPM are investigated. Linear and nonlinear stability analysis of the conduction solution is performed, and the coincidence between linear instability and nonlinear stability thresholds in the L 2 -norm is obtained. [ABSTRACT FROM AUTHOR]

In this manuscript we focus on the question: what is the correct notion of Stokes–Biot stability? Stokes–Biot stable discretizations have been introduced, independently by several authors, as a means of discretizing Biot's equations of poroelasticity; such schemes retain their stability and convergence properties, with respect to appropriately defined norms, in the context of a vanishing storage coefficient and a vanishing hydraulic conductivity. The basic premise of a Stokes–Biot stable discretization is: one part Stokes stability and one part mixed Darcy stability. In this manuscript we remark on the observation that the latter condition can be generalized to a wider class of discrete spaces. In particular: a parameter-uniform inf-sup condition for a mixed Darcy sub-problem is not strictly necessary to retain the practical advantages currently enjoyed by the class of Stokes–Biot stable Euler–Galerkin discretization schemes. [ABSTRACT FROM AUTHOR]

Temporal variability of boundary conditions is a common feature of certain fluid flows through a porous matrix, encountered, for instance, in landfill gas or natural gas collection, and sparging wells. Darcy's law subject to the weak compressibility of the fluid results in a non-linear partial differential equation for the pressure field. Slow variation admits asymptotic solutions for generic time dependence of the boundary forcing function, both as Dirichlet and Neumann conditions. Flow control strategies are suggested based on the asymptotic theory. A sealed outer domain boundary is identified as the configuration best amenable to full control of the pressure distribution via the induced well suction. Fast variation leads to a novel application of the classical compact support similarity solution in a domain with discontinuous parameters. Article Highlights: In most realistic cases the slow asymptotic regime governs the flow response to time dependent boundary conditions. With a sealed outer boundary the spatio-temporal variation is decoupled, enabling full control of the pressure field. Pressure time dependence obeys a similarity law throughout the domain with generic unsteady boundary conditions. [ABSTRACT FROM AUTHOR]

In the paper, the space second-order conservative characteristic finite difference method based on essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) interpolation for solving two-dimensional conservative convection–diffusion equations in divergence form is developed. Combining the splitting technique, characteristic difference and mass correction method, a two-dimensional convection–diffusion equations are changed into the two-dimensional parabolic equations, where the convection term and unsteady term are considered as one term. The solutions and fluxes on the staggered meshes are computed by the splitting implicit solution-flux coupled scheme. Numerical experiments are presented to illustrate mass conservation and convergence. [ABSTRACT FROM AUTHOR]

The onset of thermal convection in anisotropic rotating bidisperse porous media is investigated. The optimal result concerning the coincidence between linear instability and nonlinear stability thresholds with respect the L 2 -norm is obtained. [ABSTRACT FROM AUTHOR]

We consider an optimal control model of groundwater pollution due to agricultural activities, the objective of the optimal manager being the optimization of the trade-off between the fertilizer use and the cleaning costs. The size of a buffer zone defined around the water production sites for limiting pollution effects may also be chosen by the manager. The spread of the pollution from crops to captation wells is modeled using a convection–diffusion–reaction equation. The main hydrogeological features of the dynamics are taken into account: the process is convection-dominated, dispersion effects are included, generic nonlinear reaction terms are considered. The existence and the uniqueness of the optimal solution is proven. Using asymptotic analysis, we then rigorously prove that a one-dimensional static model can be substituted to the full dynamic 3D model for the long time study of the optimal solution. We prove that this new optimal control problem is also well-posed. All these theoretical results are used for characterizing the optimal buffer zone. [ABSTRACT FROM AUTHOR]

In this paper, based on two-grid discretizations, two kinds of local and parallel finite element methods are proposed and investigated for the coupled Stokes/Darcy model. Following the idea presented in Xu and Zhou (Math. Comput. 69, 881–909 1999). a classical local and parallel finite element method is proposed and investigated. To derive global continuous approximations, a new local and parallel finite element method is devised by combining the partition of unity. We theoretically analyze the resulting formulations and derive optimal error estimates. Numerical experiments are reported to assess the theoretical results. [ABSTRACT FROM AUTHOR]

Here, we examine the exciting forces for an arrangement of two coaxial vertical cylinders—a riding porous cylinder and a submerged bottom-mounted solid rigid cylinder. We take up two cases: first we consider a hollow porous cylinder at the top and secondly a solid porous cylinder at the top, and for both the cases, there is a solid rigid cylinder placed at the bottom. The present configuration may be observed as a wave energy device which can tap and transfer ocean wave energy to be used as non-conventional energy. A three-dimensional representation of the problem is developed, based on the familiar method of eigenfunction expansion under the assumption of linear water wave theory. The important porous boundary condition on the porous boundary can be defined by means of Darcy's law. The matching conditions across the linear interface between the adjacent fluid domains can be obtained through the continuity of pressure and velocity. Subsequently, after solving a system of linear equations, exciting forces and wave run-up for the upper and lower cylinders are calculated through the evaluated velocity potentials. Various numerical experiments show the effect of different parameters, such as porous coefficients, draft ratio, the ratio of radii of the upper and lower cylinders and the water depth on exciting force and wave run-up. It is also shown that higher porosity value of the upper cylinder results in higher energy loss conforming to the wave dissipation by the structure. The obtained results establish that appropriate values of different parameters may be considered in designing practical structures in ocean. Successful validation of the present model carried out with an available established one confirms the efficiency of the present model. The present system may be considered as a wave energy device in other problems where it can tap and transfer ocean wave energy to be used as non-conventional energy. [ABSTRACT FROM AUTHOR]

The onset of double-diffusive convection in horizontal porous layers filled by an electrically conducting fluid embedded in a magnetic field, is investigated on taking into account of Soret phenomenon. Sufficient conditions guaranteeing the onset of steady or oscillatory convection in a closed algebraic form are obtained. Conditions guaranteeing that Hopf convection in the absence of Soret phenomenon, turns into steady convection by the Soret parameter, have been found. [ABSTRACT FROM AUTHOR]

Hydrodynamic behaviour of slip flow and radially applied exponential time-dependent pressure gradient in a curvilinear concentric cylinder is examined. A two-step method of solution has been utilized in resolving the governing momentum equation. Accordingly, the exact solution of the time-dependent partial differential equation is derived in terms of the Laplace parameter. Afterwards, the Laplace domain solution is then inverted to time domain using a numerical-based inverting scheme known as Riemann-sum approximation. The effect of various dimensionless parameters involved in the problem on the Dean velocity, shear stresses and Dean vortices is discussed with the aid of graphs. It is found that maximum Dean velocity is due to an exponentially growing time-dependent pressure gradient and slip wall coefficient. Stability of the Dean vortices is achieved by suppressing time, wall slippage and inducing an exponentially decaying time-dependent pressure gradient. [ABSTRACT FROM AUTHOR]

The intrinsic permeability is a crucial parameter to characterise and quantify fluid flow through porous media. However, this parameter is typically uncertain, even if the geometry of the pore structure is available. In this paper, we perform a comparative study of experimental, semi-analytical and numerical methods to calculate the permeability of a regular porous structure. In particular, we use the Kozeny–Carman relation, different homogenisation approaches (3D, 2D, very thin porous media and pseudo 2D/3D), pore-scale simulations (lattice Boltzmann method, Smoothed Particle Hydrodynamics and finite-element method) and pore-scale experiments (microfluidics). A conceptual design of a periodic porous structure with regularly positioned solid cylinders is set up as a benchmark problem and treated with all considered methods. The results are discussed with regard to the individual strengths and limitations of the used methods. The applicable homogenisation approaches as well as all considered pore-scale models prove their ability to predict the permeability of the benchmark problem. The underestimation obtained by the microfluidic experiments is analysed in detail using the lattice Boltzmann method, which makes it possible to quantify the influence of experimental setup restrictions. [ABSTRACT FROM AUTHOR]

In this paper, we study an optimal control problem for nonlocal Cahn-Hilliard-Brinkman system, which models phase separation of binary fluids in porous media. The system evolves with regular potential in a two-dimensional bounded domain. We extend the existence of weak solution results for the system to prove the existence of strong solution under extra assumptions on the forcing term and initial datum. Further, using our regularity results, we study the tracking type optimal control problem. We prove the existence of optimal control and establish the first-order optimality condition. Lastly, we characterise optimal control in terms of the solution of the corresponding adjoint system. The existence of the solution for the adjoint system is also established. [ABSTRACT FROM AUTHOR]

The correct choice of interface conditions and effective parameters for coupled macroscale free-flow and porous-medium models is crucial for a complete mathematical description of the problem under consideration and for accurate numerical simulation of applications. We consider single-fluid-phase systems described by the Stokes–Darcy model. Different sets of coupling conditions for this model are available. However, the choice of these conditions and effective model parameters is often arbitrary. We use large-scale lattice Boltzmann simulations to validate coupling conditions by comparison of the macroscale simulations against pore-scale resolved models. We analyse three settings (lid-driven cavity over a porous bed, infiltration problem and general filtration problem) with different geometrical configurations (channelised and staggered distributions of solid grains) and different sets of interface conditions. Effective parameters for the macroscale models (permeability tensor, boundary layer constants) are computed numerically for each geometrical configuration. Numerical simulation results demonstrate the sensitivity of the coupled Stokes–Darcy problem to the location of the sharp fluid–porous interface, the effective model parameters and the interface conditions. [ABSTRACT FROM AUTHOR]

In this work we consider a diffusion problem in a periodic composite having three phases: matrix, fibers and interphase. The heat conductivities of the medium vary periodically with a period of size εμ (ε > 0 and ε > 0) in the transverse directions of the fibers. In addition, we assume that the conductivity of the interphase material and the anisotropy contrast of the material in the fibers are of the same order ε2 (the so-called double-porosity type scaling) while the matrix material has a conductivity of order 1. By introducing a partial unfolding operator for anisotropic domains we identify the limit problem. In particular, we prove that the effect of the interphase properties on the homogenized models is captured only when the microstructural length scale is of order εβ with 0 < β ⩽ 1. [ABSTRACT FROM AUTHOR]

Resin infusion is a pressure-gradient-driven composite manufacturing process in which the liquid resin is driven to flow through and fill in the void space of a porous composite preform prior to the heat treatment for resin solidification. It usually is a great challenge to design both the infusion system and the infusion process meeting the manufacturing requirements, especially for large-scale components of aircraft and wind turbine blades. Aiming at addressing the key concerns about flow fronts and air bubble entrapment, the present study proposes a modelling framework of the multiphase flow of resin and air in a dual scale porous medium, i.e. a composite preform. A finite strain formulation is discussed for the fluid–solid interaction during an infusion process. The present study bridges the gap between the microscopic observation and the macroscopic modelling by using the averaging method and first principle method, which sheds new light on the high-fidelity finite element modelling. [ABSTRACT FROM AUTHOR]

We study a decoupling iterative algorithm based on domain decomposition for the time-dependent nonlinear Stokes–Darcy model, in which different time steps can be used in the flow region and in the porous medium. The coupled system is formulated as a space-time interface problem based on the interface condition for mass conservation. The nonlinear interface problem is then solved by a nested iteration approach which involves, at each Newton iteration, the solution of a linearized interface problem and, at each Krylov iteration, parallel solution of time-dependent linearized Stokes and Darcy problems. Consequently, local discretizations in time (and in space) can be used to efficiently handle multiphysics systems of coupled equations evolving at different temporal scales. Numerical results with nonconforming time grids are presented to illustrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, the counter – current imbibition phenomena in a heterogeneous porous media is studied with the consideration of two types of porous materials like volcanic and fine sand and Adomian decomposition method is applied to find the saturation of wetting phase and the recovery rate of the reservoir. A simulation result is developed here to study the effect of heterogeneity, capillarity and relative permeability on saturation rate and to obtain an optimum recovery rate of the reservoir with the choices of some interesting parametric value.

We study a regularised version of the magnetohydrodynamics (MHD) equations, the tamed MHD (TMHD) equations. They are a model for the flow of electrically conducting fluids through porous media. We prove existence and uniqueness of TMHD on the whole space R 3 , that smooth data give rise to smooth solutions and show that solutions to TMHD converge to a suitable weak solution of the MHD equations as the taming parameter N tends to infinity. Furthermore, we adapt a regularity result for the Navier–Stokes equations to the MHD case. [ABSTRACT FROM AUTHOR]

Robust discretization methods for (nearly-incompressible) linear elasticity are free of volume-locking and gradient-robust. While volume-locking is a well-known problem that can be dealt with in many different discretization approaches, the concept of gradient-robustness for linear elasticity is new: it assures that dominant gradient fields in the momentum balance do not lead to spurious displacements. We discuss both aspects and propose novel Hybrid Discontinuous Galerkin (HDG) methods for linear elasticity. The starting point for these methods is a divergence-conforming discretization. As a consequence of its well-behaved Stokes limit the method is gradient-robust and free of volume-locking. To improve computational efficiency, we additionally consider discretizations with relaxed divergence-conformity and a modification which re-enables gradient-robustness, yielding a robust and quasi-optimal discretization also in the sense of HDG superconvergence. [ABSTRACT FROM AUTHOR]

The paper discusses the numerical algorithm constructing a three-dimensional model for a flow of two-phase incompressible fluid caused by the mass force of gravity in a porous medium. The algorithm is based on a combination of a hybrid upwind method with an explicit scheme for determination of the saturation. The hybrid upwinding allows us to take into account flows of fluid of various nature (in this case, viscous and gravitational flows) separately, which is extremely important in the case of gravitational flow with opposite directions of phase flows. The explicit scheme being extremely simple in implementation provides a small dispersion of solutions on discontinuities. The proposed algorithm is illustrated by the results of numerical experiments demonstrating the monotonicity of the method considered in this paper. [ABSTRACT FROM AUTHOR]

Geothermal district heating development has been gaining momentum in Europe with numerous deep geothermal installations and projects currently under development. With the increasing density of geothermal wells, questions related to the optimal and sustainable reservoir exploitation become more and more important. A quantitative understanding of the complex thermo-hydraulic interaction between tightly deployed geothermal wells in heterogeneous temperature and permeability fields is key for a maximum sustainable use of geothermal resources. Motivated by the geological settings of the Upper Jurassic aquifer in the Greater Munich region, we develop a computational model based on finite element analysis and gradient-free optimization to simulate groundwater flow and heat transport in hot sedimentary aquifers, and numerically investigate the optimal positioning and spacing of multi-well systems. Based on our numerical simulations, net energy production from deep geothermal reservoirs in sedimentary basins by smart geothermal multi-well arrangements provides significant amounts of energy to meet heat demand in highly urbanized regions. Our results show that taking into account heterogeneous permeability structures and a variable reservoir temperature may drastically affect the results in the optimal configuration. We demonstrate that the proposed numerical framework is able to efficiently handle generic geometrical and geological configurations, and can be thus flexibly used in the context of multi-variable optimization problems. Hence, this numerical framework can be used to assess the extractable geothermal energy from heterogeneous deep geothermal reservoirs by the optimized deployment of smart multi-well systems. [ABSTRACT FROM AUTHOR]

Foam is used in enhanced oil recovery to improve the sweep efficiency by controlling the gas mobility. A common way to describe the foam displacement is by using population balance models, which consider the foam texture as part of the gas phase. Numerical simulation of such equations presents serious difficulties connected to the high non-linearity in the fractional flow. The linear kinetic model is studied mathematically for large initial reservoir water saturation and all possibilities of injection saturation. It was observed that the model contains some structural instabilities, i.e., a small variation in some parameters leads to qualitatively different solutions. One of these solutions presented localized decay in relative gas mobility, indicating that this behavior is due to the equations' mathematical properties. [ABSTRACT FROM AUTHOR]

Soil-based water filtering devices can be described by models of viscous flow in porous media coupled with an advection–diffusion–reaction system modelling the transport of distinct contaminant species within water, and being susceptible to adsorption in the medium that represents soil. Such models are analysed mathematically, and suitable numerical methods for their approximate solution are designed. The governing equations are the Navier–Stokes–Brinkman equations for the flow of the fluid through a porous medium coupled with a convection-diffusion equation for the transport of the contaminants plus a system of ordinary differential equations accounting for the degradation of the adsorption properties of each contaminant. These equations are written in meridional axisymmetric form and the corresponding weak formulation adopts a mixed-primal structure. A second-order, (axisymmetric) divergence-conforming discretisation of this problem is introduced and the solvability, stability, and spatio-temporal convergence of the numerical method are analysed. Some numerical examples illustrate the main features of the problem and the properties of the numerical scheme. [ABSTRACT FROM AUTHOR]

The incompressible miscible displacement of three-dimensional Darcy–Forchheimer flow is discussed in this paper, and the mathematical model is formulated by two partial differential equations, a Darcy–Forchheimer flow equation for the pressure and a convection-diffusion equation for the concentration. The pressure plays an important role and determines the Darcy–Forchheimer velocity. A conservative mixed finite element is used to approximate the pressure and the velocity, and the computational accuracy of Darcy–Forchheimer velocity is improved by one order. A second-order upwind fractional step difference scheme is adopted to obtain the concentration, and numerical oscillation and dispersion are eliminated. Computational work is reduced greatly by decomposing the whole computation into successive one-dimensional subcomputations, where the speedup solvers are used. Applying energy-norm method, operator decomposition, speedup algorithm and induction hypotheses, we derive an optimal second-order estimates in L 2 norm. Numerical experiments are given to show the theoretical accuracy and the applicability for solving actual problems. [ABSTRACT FROM AUTHOR]

We demonstrate some a priori estimates of a scheme using stabilization and hybrid interfaces applying to partial differential equations describing miscible displacement in porous media. This system is made of two coupled equations: an anisotropic diffusion equation on the pressure and a convection-diffusion-dispersion equation on the concentration of invading fluid. The anisotropic diffusion operators in both equations require special care while discretizing by a finite volume method SUSHI. Later, we present some numerical experiments. [ABSTRACT FROM AUTHOR]