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Finite difference discretizations of 1D poroelasticity equations with discontinuous coefficients are analyzed. A recently suggested FD discretization of poroelasticity equations with constant coefficients on staggered grid, [5], is used as a basis. A careful treatment of the interfaces leads to harmonic averaging of the discontinuous coefficients. Here, convergence for the pressure and for the displacement is proven in certain norms for the scheme with harmonic averaging (HA). Order of convergence 1.5 is proven for arbitrary located interface, and second order convergence is proven for the case when the interface coincides with a grid node. Furthermore, following the ideas from [3], modified HA discretization are suggested for particular cases. The velocity and the stress are approximated with second order on the interface in this case. It is shown that for wide class of problems, the modified discretization provides better accuracy. Second order convergence for modified scheme is proven for the case when the interface coincides with a displacement grid node. Numerical experiments are presented in order to illustrate our considerations.

Iterative solution of large scale systems arising after discretization and linearization of the unsteady non-Newtonian Navier–Stokes equations is studied. cross WLF model is used to account for the non-Newtonian behavior of the fluid. Finite volume method is used to discretize the governing system of PDEs. Viscosity is treated explicitely (e.g., it is taken from the previous time step), while other terms are treated implicitly. Different preconditioners (block–diagonal, block–triangular, relaxed incomplete LU factorization, etc.) are used in conjunction with advanced iterative methods, namely, BiCGStab, CGS, GMRES. The action of the preconditioner in fact requires inverting different blocks. For this purpose, in addition to preconditioned BiCGStab, CGS, GMRES, we use also algebraic multigrid method (AMG). The performance of the iterative solvers is studied with respect to the number of unknowns, characteristic velocity in the basic flow, time step, deviation from Newtonian behavior, etc. Results from numerical experiments are presented and discussed.

In this paper we consider numerical algorithms for solving a system of nonlinear PDEs arising in modeling of liquid polymer injection. We investigate the particular case when a porous preform is located within the mould, so that the liquid polymer flows through a porous medium during the filling stage. The nonlinearity of the governing system of PDEs is due to the non-Newtonian behavior of the polymer, as well as due to the moving free boundary. The latter is related to the penetration front and a Stefan type problem is formulated to account for it. A finite-volume method is used to approximate the given differential problem. Results of numerical experiments are presented. We also solve an inverse problem and present algorithms for the determination of the absolute preform permeability coefficient in the case when the velocity of the penetration front is known from measurements. In both cases (direct and inverse problems) we emphasize on the specifics related to the non-Newtonian behavior of the polymer. For completeness, we discuss also the Newtonian case. Results of some experimental measurements are presented and discussed.

In this work the problem of fluid flow in deformable porous media is studied. First, the stationary fluid-structure interaction (FSI) problem is formulated in terms of incompressible Newtonian fluid and a linearized elastic solid. The flow is assumed to be characterized by very low Reynolds number and is described by the Stokes equations. The strains in the solid are small allowing for the solid to be described by the Lame equations, but no restrictions are applied on the magnitude of the displacements leading to strongly coupled, nonlinear fluid-structure problem. The FSI problem is then solved numerically by an iterative procedure which solves sequentially fluid and solid subproblems. Each of the two subproblems is discretized by finite elements and the fluid-structure coupling is reduced to an interface boundary condition. Several numerical examples are presented and the results from the numerical computations are used to perform permeability computations for different geometries.

In soil mechanics assumption of only vertical subsidence is often invoked and this leads to the one-dimensional model of poroelasticity. The classical model of linear poroelasticity is obtained by Biot [1], detailed derivation can be found e.g., in [2]. This model is applicable also to modelling certain processes in geomechanics, hydrogeology, petroleum engineering (see, e.g., [3, 8], in biomechanics (e.g., [9, 10]), in filtration (e.g., filter cake formation, see [15, 16, 17]), in paper manufacturing (e.g., [11, 12]), in printing (e.g., [13]), etc. Finite element and finite difference methods were applied by many authors for numerical solution of the Biot system of PDEs, see e.g. [3, 4, 5] and references therein. However, as it is wellknown, the standard FEM and FDM methods are subject to numerical instabilities at the first time steps. To avoid this, discretization on staggered grid was suggested in [4, 5]. A single layer deformable porous medium was considered there. This paper can be viewed as extension of [4, 5] to the case of multilayered deformable porous media. A finite volume discretization to the interface problem for the classical one-dimensional Biot model of consolidation process is applied here. Following assumptions are supposed to be valid: each of the porous layers is composed of incompressible solid matrix, it is homogeneous and isotropic. Furthermore, one of two following assumptions is valid: porous medium is not completely saturated and ﬂuid is incompressible or porous medium is completely saturated and fluid is slightly compressible. The reminder of the paper is organised as follows. Next section presents the mathematical model. Third section is devoted to the dicsretization of the continuous problem. Fourth section contains the results from the numerical experiments.

Four aspects are important in the design of hydraulic lters. We distinguish between two cost factors and two performance factors. Regarding performance, filter eciencynd lter capacity are of interest. Regarding cost, there are production considerations such as spatial restrictions, material cost and the cost of manufacturing the lter. The second type of cost is the operation cost, namely the pressure drop. Albeit simulations should and will ultimately deal with all 4 aspects, for the moment our work is focused on cost. The PleatGeo Module generates three-dimensional computer models of a single pleat of a hydraulic lter interactively. PleatDict computes the pressure drop that will result for the particular design by direct numerical simulation. The evaluation of a new pleat design takes only a few hours on a standard PC compared to days or weeks used for manufacturing and testing a new prototype of a hydraulic lter. The design parameters are the shape of the pleat, the permeabilities of one or several layers of lter media and the geometry of a supporting netting structure that is used to keep the out ow area open. Besides the underlying structure generation and CFD technology, we present some trends regarding the dependence of pressure drop on design parameters that can serve as guide lines for the design of hydraulic lters. Compared to earlier two-dimensional models, the three-dimensional models can include a support structure.

This work presents the dynamic capillary pressure model (Hassanizadeh, Gray, 1990, 1993a) adapted for the needs of paper manufacturing process simulations. The dynamic capillary pressure-saturation relation is included in a one-dimensional simulation model for the pressing section of a paper machine. The one-dimensional model is derived from a two-dimensional model by averaging with respect to the vertical direction. Then, the model is discretized by the finite volume method and solved by Newton’s method. The numerical experiments are carried out for parameters typical for the paper layer. The dynamic capillary pressure-saturation relation shows significant influence on the distribution of water pressure. The behaviour of the solution agrees with laboratory experiments (Beck, 1983).

This paper concerns numerical simulation of flow through oil filters. Oil filters consist of filter housing (filter box), and a porous filtering medium, which completely separates the inlet from the outlet. We discuss mathematical models, describing coupled flows in the pure liquid subregions and in the porous filter media, as well as interface conditions between them. Further, we reformulate the problem in fictitious regions method manner, and discuss peculiarities of the numerical algorithm in solving the coupled system. Next, we show numerical results, validating the model and the algorithm. Finally, we present results from simulation of 3-D oil flow through a real car filter.

On a Multigrid Adaptive Refinement Solver for Saturated Non-Newtonian Flow in Porous Media A multigrid adaptive refinement algorithm for non-Newtonian flow in porous media is presented. The saturated flow of a non-Newtonian fluid is described by the continuity equation and the generalized Darcy law. The resulting second order nonlinear elliptic equation is discretized by a finite volume method on a cell-centered grid. A nonlinear full-multigrid, full-approximation-storage algorithm is implemented. As a smoother, a single grid solver based on Picard linearization and Gauss-Seidel relaxation is used. Further, a local refinement multigrid algorithm on a composite grid is developed. A residual based error indicator is used in the adaptive refinement criterion. A special implementation approach is used, which allows us to perform unstructured local refinement in conjunction with the finite volume discretization. Several results from numerical experiments are presented in order to examine the performance of the solver.

A non-linear multigrid solver for incompressible Navier-Stokes equations, exploiting finite volume discretization of the equations, is extended by adaptive local refinement. The multigrid is the outer iterative cycle, while the SIMPLE algorithm is used as a smoothing procedure. Error indicators are used to define the refinement subdomain. A special implementation approach is used, which allows to perform unstructured local refinement in conjunction with the finite volume discretization. The multigrid - adaptive local refinement algorithm is tested on 2D Poisson equation and further is applied to a lid-driven flows in a cavity (2D and 3D case), comparing the results with bench-mark data. The software design principles of the solver are also discussed.