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A spectral method is developed to study the steady and unsteady flow of fluid into a line sink from a horizontally confined aquifer, and the results are compared to solutions obtained implementing the finite element package COMSOLTM. The aquifer or drain is considered to be confined below so that the solutions are fundamentally unsteady. Comparison is made between the two methods in determining the drawdown of the surface. [ABSTRACT FROM AUTHOR]

In this paper we present and analyze a fully-mixed formulation for the coupled problem arising in the interaction between a free fluid and a poroelastic medium. The flows in the free fluid and poroelastic regions are governed by the Stokes and Biot equations, respectively, and the transmission conditions are given by mass conservation, balance of stresses, and the Beavers-Joseph-Saffman law. We apply dual-mixed formulations in both domains, where the symmetry of the Stokes and poroelastic stress tensors is imposed by setting the vorticity and structure rotation tensors as auxiliary unknowns. In turn, since the transmission conditions become essential, they are imposed weakly by introducing the traces of the fluid velocity, structure velocity, and the poroelastic media pressure on the interface as the associated Lagrange multipliers. The existence and uniqueness of a solution are established for the continuous weak formulation, as well as a semidiscrete continuous-in-time formulation with non-matching grids, together with the corresponding stability bounds. In addition, we develop a new multipoint stress-flux mixed finite element method by involving the vertex quadrature rule, which allows for local elimination of the stresses, rotations, and Darcy fluxes. Well-posedness and error analysis with corresponding rates of convergence for the fully-discrete scheme are complemented by several numerical experiments. [ABSTRACT FROM AUTHOR]

In this work, we develop and analyze an explicit finite volume scheme for a one-dimensional nonlinear, degenerate, convection–diffusion equation having application in petroleum reservoir. The main difficulty is that the solution typically lacks regularity due to the degenerate nonlinear diffusion term. We analyze a numerical scheme corresponding to explicit discretization of the diffusion term and a Godunov scheme for the advection term. L∞{L}^{\infty } stability under appropriate CFL conditions and BV{\rm{BV}} estimates are obtained. It is shown that the scheme satisfies a discrete maximum principle. Then we prove convergence of the approximate solution to the weak solution of the problem, and we mount convergence results to a weak solution of the problem in L1{L}^{1}. Results of numerical experiments are presented to validate the theoretical analysis.

Analytic element method plays a crucial role to solve fluid flow problems in porous media having isotropic hydraulic conductivity. Flow problems in anisotropic domains can be modeled by means of an equivalent isotropic domains by appropriate scaling of the coordinate transformation. Once analytic solutions are obtained for the equivalent isotropic domain, they can be mapped back to the original anisotropic domain. In this article we use a suitable coordinate transformation to solve the problem of potential flow in anisotropic domain. If the standard analytic solution for steady radial flow towards a well is considered with coordinate transformation, the resulting solutions correctly models the far field but they do not meet the constant head boundary condition at the boundary of a well. This is a significant drawback if one is interested in the flow field close to the well or wish to estimate the head at the pumping well. In this paper, using conformal mapping, we develop a technique to overcome this difficulty on the boundary of the well. [ABSTRACT FROM AUTHOR]

Membrane filtration of feed containing multiple species of particles is a common process in the industrial setting. In this work, we propose a model for filtration of a suspension containing multiple particle species (concrete examples of our model are shown in two and three species), each with different affinities for the material of the porous filter membrane. Using the pore shape within the membrane as a design objective, we formulate a number of optimization problems pertaining to effective separation of desired and undesired particles in the special case of two-particle species and we present results showing how properties such as feed composition affect the optimal filter design. In addition, we propose a novel multi-stage filtration strategy, which provides a significant mass yield improvement for the desired particles, and, surprisingly, higher purity of the product as well. [ABSTRACT FROM AUTHOR]

The nonlinear shallow water equations (SWEs) are widely used to model the unsteady water flows in rivers and coastal areas. In this work, we present a novel class of locally conservative, entropy stable and well-balanced discontinuous Galerkin (DG) methods for the nonlinear shallow water equation with a non-flat bottom topography. The major novelty of our work is the use of velocity field as an independent solution unknown in the DG scheme, which is closely related to the entropy variable approach to entropy stable schemes for system of conservation laws proposed by Tadmor (in: Tezduyar, Hughes T (eds) Proceedings of the winter annual meeting of the American Society of Mechanical Engineering 1986) back in 1986, where recall that velocity is part of the entropy variable for the shallow water equations. Due to the use of velocity as an independent solution unknown, no specific numerical quadrature rules are needed to achieve entropy stability of our scheme on general unstructured meshes in two dimensions. The proposed DG semi-discretization is then carefully combined with the classical explicit strong stability preserving Runge–Kutta (SSP–RK) time integrators (Gottlieb et al. in SIAM Rev. 43, 89–112, 2001) to yield a locally conservative, well-balanced, and positivity preserving fully discrete scheme. Here the positivity preservation property is enforced with the help of a simple scaling limiter. In the fully discrete scheme, we re-introduce discharge as an auxiliary unknown variable. In doing so, standard slope limiting procedures can be applied on the conservative variables (water height and discharge) without violating the local conservation property. Here we apply a characteristic-wise TVB limiter (Cockburn and Shu in J Comput Phys 141:199–224, 1998) on the conservative variables using the Fu-Shu troubled cell indicator (Fu and Shu in J Comput Phys 347:305–327, 2017) in each inner stage of the Runge–Kutta time stepping to suppress numerical oscillations. This fully discrete can be readily applied to various SWEs simulations without dry areas where the water height is close to zero. The case with dry areas need further special attention, where the velocity approximation can be unphysically large near cells with a small water height, which may eventually crashes the simulation if no special treatment is used near these cells. Here we propose a simple wetting/drying treatment for the velocity update without violating the local conservation property to enhance the robustness of the overall scheme. One- and two-dimensional numerical experiments are presented to demonstrate the performance of the proposed methods. [ABSTRACT FROM AUTHOR]

We consider a model for the flow of two immiscible and incompressible fluid phases in a porous medium. A surfactant is dissolved in one of the fluid phases, and its concentration at the interface separating the two fluids can change the surface tension. At the scale of pores, we assume that the flow is governed by the Navier-Stokes equations, while for the phase separation, a Cahn-Hilliard phase-field model is adopted. Using formal homogenization, we derive a two-scale model describing the averaged behaviour of the system at the larger Darcy scale, where effective quantities are found through local (cell) problems at the smaller pore scale. For this two-scale model, we formulate a numerical scheme and present numerical results highlighting the influence of the solute-dependent surface tension. [ABSTRACT FROM AUTHOR]

In this paper we propose the use of a continuous data assimilation algorithm for miscible flow models in a porous medium. In the absence of initial conditions for the model, observed sparse measurements are used to generate an approximation to the true solution. Under certain assumption of the sparse measurements and their incorporation into the algorithm it can be shown that the resulting approximate solution converges to the true solution at an exponential rate as time progresses. Various numerical examples are considered in order to validate the suitability of the algorithm. [ABSTRACT FROM AUTHOR]

The present study investigates the influence of uniform magnetic field on the flow of a Newtonian fluid sandwiched between two micropolar fluid layers through a rectangular (horizontal) porous channel. Fluid flow in the every region is steady, incompressible and the fluids are immiscible. Uniform magnetic field is applied in a direction perpendicular to the direction of fluid motion. The governing equations of micropolar fluid are expressed in Eringen's approach and further modified by Nowacki's approach. For respective porous channels, expressions for linear velocity, microrotations, stresses (shear and couple) are obtained analytically. Continuity of velocities, continuity of microrotations and continuity of stresses are employed at the porous interfaces; conditions of no slip and no spin are applied at the impervious boundaries of the composite channel. Numerical values of flow rate, wall shear stresses and couple stresses at the porous interfaces are evaluated by MATHEMATICA and listed in tables. Graphs of flow rate and fluid velocity are plotted and their behaviors discussed. [ABSTRACT FROM AUTHOR]

This paper continues the study of the use of spline functions for problems involving heat and fluid flow in porous media. Thus, we investigate the numerical solution of the problem of free convection in a square cavity filled with a fluid-saturated porous medium having constant temperatures on the side walls, using a spline collocation method. We present the mathematical model and obtain the governing dimensionless equations together with boundary conditions. Then the problem is solved using cubic spline functions approximations. We analyse the streamlines, isotherms and the numerical results of the local and average Nusselt numbers, for different values of the governing parameters. The results obtained are successfully validated when compared with previously reported results from the open literature. This numerical method has several advantages, such as its speed of convergence and the fact that it is easier to use when the boundary conditions are given for the derivatives of the unknown functions, no extra approximations being necessary. Also, when using cubic spline functions, the discretization process leads to a sparse linear system, which is solvable much more efficiently, thus improving the accuracy of the numerical results and reducing the cost of implementation. The present numerical results can help other researchers in finding approximate solutions of problems in heat transfer arising in many applications in industry. [ABSTRACT FROM AUTHOR]

Soil salinization is a major cause of soil degradation and hampers plant growth. For soils saturated with saline water, the evaporation of water induces accumulation of salt near the top of the soil. The remaining liquid gets an increasingly larger density due to the accumulation of salt, giving a gravitationally unstable situation, where instabilities in the form of fingers can form. These fingers can, hence, lead to a net downward transport of salt. We here investigate the appearance of these fingers through a linear stability analysis and through numerical simulations. The linear stability analysis gives criteria for onset of instabilities for a large range of parameters. Simulations using a set of parameters give information also about the development of the fingers after onset. With this knowledge, we can predict whether and when the instabilities occur, and their effect on the salt concentration development near the top boundary. [ABSTRACT FROM AUTHOR]

Multidomain mixed pairs of coupled stationary constrained boundary value problems, are formulated variationally in frameworks of reflexive Banach spaces as macro-hybrid systems, and analyzed. Interior synchronizing and interface coupling transmission constraints are modeled in terms of Lagrange multipliers as solutions of dual variational subpotential maximal monotone inclusions. Existence and uniqueness results are established via resolvent fixed-point characterizations of the corresponding primal and dual macro-hybridized problems. Multimedia mixed mechanical subsurface-surface Darcy/Stokes incompressible flow coupled pairs with intrinsic distributed control constraints exemplify the general macro-hybrid mixed systems. Coupling dual-primal and primal-dual interface macro-hybrid mixed variational conditions are lastly developed and applied to the Darcy/Stokes multimedia pair model. [ABSTRACT FROM AUTHOR]

We study the homogenization of a double porosity diffusion problem in a thin highly heterogeneous composite medium formed by two materials separated by an imperfect interface, where the solution and its flux exhibit jumps. By applying homogenization techniques specific to the thin periodic domain under study, one derives the limit problem. [ABSTRACT FROM AUTHOR]

We consider a model for Cosserat–Bingham fluids. In contrast to the classical Bingham fluid, the Cosserat–Bingham fluids include symmetric and antisymmetric stresses and two types of plug zones. In the article we demonstrate the global-in-time solvability of the three dimensional initial boundary value problem for the Cosserat–Bingham model. [ABSTRACT FROM AUTHOR]

A well known approximation for the displacement of two immiscible fluids in a porous medium is the Hele-Shaw model. In experiments it was observed that a liquid with variable viscosity, introduced between the two initial fluids, can minimize the Saffman-Taylor instability. In some works an attempt was made to replace the variable viscosity liquid with a sequence of several immiscible liquids with constant viscosities. We prove that the linear stability analysis of this multi-layer Hele-Shaw model leads us to an ill-posed problem. [ABSTRACT FROM AUTHOR]

In many industrial applications, reactive filters, which remove toxic substances using chemical reactions, prove very efficient and versatile. One such example is the removal of sulphur dioxide from flue gas by converting it into liquid sulphuric acid. In this paper, we extend a model developed in Kiradjiev et al. (SIAM J Appl Math 82:591–619, 2021) for the operation of a reactive filter to consider the long-time behaviour and determine how the filter becomes clogged with liquid and becomes inefficient. We explore how the parameters that govern the operation of the filter affect its efficiency and discuss optimal filtration strategies. We also consider a distinguished asymptotic limit in terms of the key dimensionless parameters and obtain an explicit solution that compares well with the numerical solution to the full problem. Finally, we compare the results of our model with those from a simplified model for a reactive filter considered in Breward and Kiradjiev (J Eng Math, 129(14):1–28, 2021). [ABSTRACT FROM AUTHOR]

Under a sharp asymptotic growth condition at infinity, we prove a Liouville type theorem for the inhomogeneous porous medium equation, provided it stays universally close to the heat equation. Additionally, for the homogeneous equation, we show that for the conclusion to hold, it is enough to assume the sharp asymptotic growth at infinity only in the space variable. The results are optimal, meaning that the growth condition at infinity cannot be weakened. [ABSTRACT FROM AUTHOR]

In this study we examine the classical problem of fluid flow in an aquifer, obeying the transient Darcy–Forchheimer law. This problem is solved by using the symmetry properties of the governing equations (e.g. the mass balance equation and the transient Darcy–Forchheimer momentum equation) which enable us to transform the time and the space coordinates into one independent coordinate. According to our study the flow in the aquifer may be divided into two main components. One component is the steady part of the flow discharge and the other one is the transient part of the flow discharge. The obtained solution shows that the reduction in the above-mentioned transient part leads to the creation of three zones: (1) the “near zone” located near the inlet face to the aquifer and is characterized by a positive flow; (2) the “far zone” in the aquifer lying at an infinite distance from the inlet face where the flow is also positive and (3) the “intermediate backflow zone”, which is lying among the above-mentioned zones and is characterized by reverse flow. The results obtained in this study may be useful for understanding the transient flow process in the aquifer, stemming from the Darcy–Forchheimer flow, and for the prediction of the discharge and head distribution in the aquifer. [ABSTRACT FROM AUTHOR]

Vertical infiltration of water plays an important role in the recharged of contaminated water and enhanced moisture content in the unsaturated porous media. The mathematical model used for such type of phenomenon is Burger's equation. Unsaturated porous media are analyzed by solving Burger's equation using the variational iterative modeling and homotopy perturbation method. When considering all moisture contents, it appears that the cumulative coefficient is unchanged. It is also shown that the soil's moisture content decreases with depth (y) and time (t). The results indicate that this method is very efficient and can be useful to solve large-scale problems that arise in civil engineering, geology, material science, and fossil fuel problems. [ABSTRACT FROM AUTHOR]

The gradient discretisation method (GDM) is a generic framework designed recently, as a discretisation in spatial space, to partial differential equations. This paper aims to use the GDM to establish a first general error estimate for numerical approximations of parabolic obstacle problems. This gives the convergence rates of several well-known conforming and non-conforming numerical methods. Numerical experiments based on the hybrid finite volume method are provided to verify the theoretical results. [ABSTRACT FROM AUTHOR]

We consider a recent plate model obtained as a scaled limit of the three-dimensional Biot system of poro-elasticity. The result is a '2.5'-dimensional linear system that couples traditional Euler–Bernoulli plate dynamics to a pressure equation in three dimensions, where diffusion acts only transversely. We allow the permeability function to be time dependent, making the problem non-autonomous and disqualifying much of the standard abstract theory. Weak solutions are defined in the so-called quasi-static case, and the problem is framed abstractly as an implicit, degenerate evolution problem. Utilizing the theory for weak solutions for implicit evolution equations, we obtain existence of solutions. Uniqueness is obtained under additional hypotheses on the regularity of the permeability function. We address the inertial case in an appendix, by way of semigroup theory. The work here provides a baseline theory of weak solutions for the poro-elastic plate and exposits a variety of interesting related models and associated analytical investigations. [ABSTRACT FROM AUTHOR]

This is the first of a series of two papers which studies the fractional porous medium equation on a Riemannian manifold with isolated conical singularities. In this article, we show R-sectoriality for the fractional powers of possibly non-invertible R-sectorial operators. Applications concern existence, uniqueness and maximal L q -regularity results for solutions of the fractional porous medium equation on manifolds with conical singularities. Space asymptotic behavior of the solutions close to the singularities is provided, and its relation to the local geometry is established. Our method extends the freezing-of-coefficients method to the case of non-local operators that are expressed as linear combinations of terms in the form of a product of a function and a fractional power of a local operator. [ABSTRACT FROM AUTHOR]

In this paper, a marker and cell (MAC) scheme is constructed for the free flow-porous media system with heat transport on non-uniform grids, where the Stokes-Darcy equation is employed to describe the free-flow and the porous regions. Error estimates for the velocity, pressure and temperature in different discrete norms are established rigorously and carefully by constructing discrete auxiliary functions. We obtain the second order superconvergence in the discrete L 2 norm for velocity, pressure and temperature and the second order superconvergence for some terms of the H 1 norm for the velocity on non-uniform grids. Several numerical examples verifying the theoretical results and illustrating the capabilities of the method are demonstrated. [ABSTRACT FROM AUTHOR]

Embedding geometries in structured grids allows a simple treatment of complex objects in fluid simulations. Various methods for embedding geometries are available. The commonly used Brinkman-volume-penalization models geometries as porous media, and approximates a solid object in the limit of vanishing porosity. In its simplest form, the momentum equations are augmented by a term penalizing the fluid velocity, yielding good results in many applications. However, it induces numerical stiffness, especially if high-pressure gradients need to be balanced. Here, we focus on the effect of the reduced effective volume (commonly called porosity) of the porous medium. An approach is derived, which allows reducing the flux through objects to practically zero with little increase of numerical stiffness. Also, non-slip boundary conditions and adiabatic boundary conditions are easily constructed. The porosity terms allow keeping the skew symmetry of the underlying numerical scheme, by which the numerical stability is improved. Furthermore, very good conservation of mass and energy in the non-penalized domain can be achieved, for which the boundary smoothing introduces a small ambiguity in its definition. The scheme is tested for acoustic scenarios, for near incompressible and strongly compressible flows. [ABSTRACT FROM AUTHOR]

A primal C0-conforming virtual element discretization for the approximation of the bidimensional two-phase flow of immiscible fluids in porous media using general polygonal meshes is discussed. This work investigates the potentialities of the Virtual Element Method (VEM) in solving this specific problem of immiscible fluids in porous media involving a time-dependent coupled system of non-linear partial differential equations. The performance of the fully discrete scheme is thoroughly analysed testing it on general meshes considering both a regular problem and more realistic benchmark problems that are of interest for physical and engineering applications. [ABSTRACT FROM AUTHOR]

A multigrid multilevel Monte Carlo (MGMLMC) method is developed for the stochastic Stokes–Darcy interface model with random hydraulic conductivity both in the porous media domain and on the interface. Three interface conditions with randomness are considered on the interface between Stokes and Darcy equations, especially the Beavers–Joesph interface condition with random hydraulic conductivity. Because the randomness through the interface affects the flow in the Stokes domain, we investigate the coupled stochastic Stokes–Darcy model to improve the fidelity. Under suitable assumptions on the random coefficient, we prove the existence and uniqueness of the weak solution of the variational form. To construct the numerical method, we first adopt the Monte Carlo (MC) method and finite element method, for the discretization in the probability space and physical space, respectively. In order to improve the efficiency of the classical single-level Monte Carlo (SLMC) method, we adopt the multilevel Monte Carlo (MLMC) method to dramatically reduce the computational cost in the probability space. A strategy is developed to calculate the number of samples needed in MLMC method for the stochastic Stokes–Darcy model. In order to accomplish the strategy for MLMC method, we also present a practical method to determine the variance convergence rate for the stochastic Stokes–Darcy model with Beavers–Joseph interface condition. Furthermore, MLMC method naturally provides the hierarchical grids and sufficient information on these grids for multigrid (MG) method, which can in turn improve the efficiency of MLMC method. In order to fully make use of the dynamical interaction between this two methods, we propose a multigrid multilevel Monte Carlo (MGMLMC) method with finite element discretization for more efficiently solving the stochastic model, while additional attention is paid to the interface and the random Beavers–Joesph interface condition. The computational cost of the proposed MGMLMC method is rigorously analyzed and compared with the SLMC method. Numerical examples are provided to verify and illustrate the proposed method and the theoretical conclusions. [ABSTRACT FROM AUTHOR]

This work aims to provide a comprehensive and unified numerical analysis for a nonlinear system of parabolic variational inequalities (PVIs) subject to Dirichlet boundary condition. This analysis enables us to establish the existence of an exact solution to the considered model and to prove the convergence for the approximate solution and its approximate gradient. Our results are applicable for several conforming and nonconforming numerical schemes. [ABSTRACT FROM AUTHOR]

In this study we investigated unsteady mixed convection flow and heat transfer of radiating and reacting nanofluid with variable transport properties in a microchannel filed with a saturated porous medium by taking into account the convective boundary conditions. The Buongiorno's nanofluid flow model is used to study the effects of the Brownian motion and the thermophoresis. The governing highly nonlinear partial differential equations corresponding to the momentum, energy and concentration profiles have been formulated and solved numerically by utilizing the semi-discretization finite difference method. The effect of each governing thermophysical parameters on the microchannel hydrodynamic and thermal behaviors is discussed with the usage of graphs. The numerical results indicate that the velocity and temperature profiles show an increasing behavior with the variable viscosity parameter, Eckert number, thermal Grashof number, solutal Grashof number, Prandtl number and chemical reaction parameter, whereas the concentration profile increases with increasing values of variable thermal conductivity parameter, porous medium shape parameter, Forchheimer number, Brownian motion parameter, Schmidt number, Biot number and radiation parameter. Moreover, the result reveals that the skin friction coefficient increases with suction/injection Reynolds number, porous medium shape parameter, thermal Grashof number, Schmidt number and Brownian motion parameter but decreases with Eckert number, thermophoresis parameter, Biot number and radiation parameter. Both the heat transfer and the mass transfer rates at both sides of the microchannel walls are higher for large values of suction/injection Reynolds number, porous medium shape parameter and variable viscosity parameter, while both are lower for large values of Eckert number, variable thermal conductivity parameter and radiation parameter. Besides, Grashof number, Schmidt number and Biot number indicate an increasing effect on both the heat transfer and mass transfer rates at the cold wall of the microchannel. The numerical simulation also reveals that Brownian motion parameter and thermophoresis parameter show an opposite effect on both heat transfer and mass transfer rates at both sides of the microchannel walls. [ABSTRACT FROM AUTHOR]

In this paper, we consider a liquid–gas system with two components: water and hydrogen flow model in heterogeneous porous media with periodic microstructure taking into account kinetics in the mass transfer between the two phases. The particular feature in this model is that chemistry effects are taken into account. The microscopic model consists of the usual equations derived from the mass conservation laws of both fluids, along with the Darcy–Muskat and capillary pressure laws and the mass exchange is modeled as a source term in the equations. The problem is written in the terms of the phase formulation; i.e. the saturation of one phase, the pressure of the second phase, and the concentration of dissolved hydrogen in the liquid phase. The mathematical model consists in a system of partial differential equations: two degenerate nonlinear parabolic equations and one diffusion–convection equation. The major difficulties related to this new model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we obtain a nonlinear homogenized coupled system of three coupled partial differential equations with effective coefficients which are computed via solving cell problems. We give a rigorous mathematical derivation of the effective model by means of the two-scale convergence. [ABSTRACT FROM AUTHOR]

In this paper, based on two-grid discretizations and superposition principle, a new local and parallel finite element method is presented and studied for the coupled Stokes–Darcy model. Superposition principle is used to generate a series of local and independent subproblems to obtain a global continuous approximation. Optimal error estimates are derived. Numerical results are reported to support the theoretical findings. [ABSTRACT FROM AUTHOR]