Finite element implementation of stress-jump and stress-continuity conditions at porous-medium, clear-fluid interface

The boundary conditions at the interface between clear-fluid and porous-medium domains are very important for solving flow through an open domain adjoining a porous medium. In this Galerkin finite-element (FE) based simulation of such interface flows employing Stokes and Brinkman equations, the traditional interfacial condition based on the continuity of stress in fluid and porous media is compared with the stress-jump condition proposed by Ochoa-Tapia and Whitaker using the rigorous volume averaging method. A novel FE formulation employing a second-order adjustable tensor is proposed to implement this new stress-jump condition for full three-dimensional flows. The paper also clarifies the hitherto obscure relationship between flow variables in the fluid and porous media for the conventional stress-continuity condition. In the first validation study involving numerical predictions of flow parallel to the interface, our FE implementation of the new stress-jump condition agree very well with the analytical solution for flow parallel to the interface, thereby proving the soundness of our adjustable tensor approach. Similar excellent results were obtained for FE implementation of the stress-continuity condition as well. A good match with analytical solution for a constant cross-flow superimposed on the parallel flow was also achieved while differences in velocity profiles near the interfaces were studied for the two conditions. Lastly a complex 3D flow simulation involving a fluid and porous media interface within the unit-cell of a non-crimp stitched fiber mat, used in liquid composite molding process during the manufacture of composite materials, is undertaken. The permeability of this dual-scale fibrous porous medium, estimated using the newly implemented stress-jump condition, agrees well with the experimental result thereby pointing to the accuracy of the FE implementation of the condition. Our simulations reveal that the stress-jump condition leads to a much smaller boundary layer within porous medium near the interface as compared to the stress-continuity condition, and hence to a lower, more accurate net flow-rate through the unit cell. However the two interfacial conditions yield similar results with a decrease in the porosity.     

A multiscale Eulerian–Lagrangian localized adjoint method for transient advection–diffusion equations with oscillatory coefficients

We develop a multiscale Eulerian–Lagrangian localized adjoint method for transient linear advection– diffusion equations with oscillatory coefficients, which arise in mathematical models for describing flow and transport through heterogeneous porous media, composite material design, and other applications.     

Microfluidic systems for the analysis of viscoelastic fluid flow phenomena in porous media

In this study, two microfluidic devices are proposed as simplified 1-D microfluidic analogues of a porous medium. The objectives are twofold: firstly to assess the usefulness of the microchannels to mimic the porous medium in a controlled and simplified manner, and secondly to obtain a better insight about the flow characteristics of viscoelastic fluids flowing through a packed bed. For these purposes, flow visualizations and pressure drop measurements are conducted with Newtonian and viscoelastic fluids. The 1-D microfluidic analogues of porous medium consisted of microchannels with a sequence of contractions/expansions disposed in symmetric and asymmetric arrangements. The real porous medium is in reality, a complex combination of the two arrangements of particles simulated with the microchannels, which can be considered as limiting ideal configurations. The results show that both configurations are able to mimic well the pressure drop variation with flow rate for Newtonian fluids. However, due to the intrinsic differences in the deformation rate profiles associated with each microgeometry, the symmetric configuration is more suitable for studying the flow of viscoelastic fluids at low De values, while the asymmetric configuration provides better results at high De values. In this way, both microgeometries seem to be complementary and could be interesting tools to obtain a better insight about the flow of viscoelastic fluids through a porous medium. Such model systems could be very interesting to use in polymer-flood processes for enhanced oil recovery, for instance, as a tool for selecting the most suitable viscoelastic fluid to be used in a specific formation. The selection of the fluid properties of a detergent for cleaning oil contaminated soil, sand, and in general, any porous material, is another possible application.     

A high-performance lattice Boltzmann implementation to model flow in porous media

We examine the problem of simulating single and multiphase flow in porous medium systems at the pore scale using the lattice Boltzmann (LB) method. The LB method is a powerful approach, but one which is also computationally demanding; the resolution needed to resolve fundamental phenomena at the pore scale leads to very large lattice sizes, and hence substantial computational and memory requirements that necessitate the use of massively parallel computing approaches. Common LB implementations for simulating flow in porous media store the full lattice, making parallelization straightforward but wasteful. We investigate a two-stage implementation consisting of a sparse domain decomposition stage and a simulation stage that avoids the need to store and operate on lattice points located within a solid phase. A set of five domain decomposition approaches are investigated for single and multiphase flow through both homogeneous and heterogeneous porous medium systems on different parallel computing platforms. An orthogonal recursive bisection method yields the best performance of the methods investigated, showing near linear scaling and substantially less storage and computational time than the traditional approach.     

A topology Nash game for tumoral antiangiogenesis

Tumoral angiogenesis plays a critical role in the process of growth of cancerous tumors. We consider activators and inhibitors of angiogenesis as players of a Nash game. From the activator’s point of view, the medium formed by existing blood vessels, surrounding tissue, and the tumor outer surface is seen as a porous medium. In contrast, the inhibitor sees the same medium as an elastic structural environment. We then define a competition between activator and inhibitor distributions. The aim of the activator is to minimize the pressure drop, while the inhibitor’s aim is to minimize the elastic compliance of the surrounding host tissue or, in a duel version, minimize the drainage of the tumoral neovascularization. Numerical results illustrate how (theoretical) tumors develop multiple channels as an optimal response to the optimal distribution of inhibitors.     

A Lagrangian Scheme for the Solution of Nonlinear Diffusion Equations Using Moving Simplex Meshes

A Lagrangian numerical scheme for solving nonlinear degenerate Fokker–Planck equations in space dimensions \(d\ge 2\) is presented. It applies to a large class of nonlinear diffusion equations, whose dynamics are driven by internal energies and given external potentials, e.g. the porous medium equation and the fast diffusion equation. The key ingredient in our approach is the gradient flow structure of the dynamics. For discretization of the Lagrangian map, we use a finite subspace of linear maps in space and a variational form of the implicit Euler method in time. Thanks to that time discretisation, the fully discrete solution inherits energy estimates from the original gradient flow, and these lead to weak compactness of the trajectories in the continuous limit. Consistency is analyzed in the planar situation, \(d=2\). A variety of numerical experiments for the porous medium equation indicates that the scheme is well-adapted to track the growth of the solution’s support.     

MHD flow in porous medium induced by torsionally oscillating disk

The unsteady laminar flow of an incompressible, viscous, electrically conducting fluid in porous medium fully saturated with the liquid and bounded by torsionally oscillating disk in the presence of a transverse magnetic field has been computed. It is assumed that the flow between the disk and the porous medium is governed by Navier-Stokes equation and that in the porous medium by Brinkman equation. Flows in the two regions are matched at the interface by assuming that the velocity and stress components are continuous at it. Approximate solutions of the flow characteristics are obtained. Numerical results are presented graphically and discussed.     

Numerical analysis of a new model for the processes of the one-dimensional case of metal crystallization

The numerical analysis of a new model of metal crystallization is performed. The novelty of the model, which has recently appeared in publications, consists in modeling that is simultaneously carried out at several scale levels, from micro-to macroscales. Although experimental studies of metal crystallization revealed many details of metal crystallization, there is no comprehensive theoretical view on this phenomenon. In the model employed in this work, the space of the crystallizing alloy is assumed to be a porous medium, in which the disturbance propagation is described by Biot’s type of equations. The formation of crystal nuclei is described by a modified Cahn-Hilliard equation. A numerical scheme is constructed and its convergence is demonstrated. It is shown that various crystallization modes can be set by varying the parameters. We intend to investigate a multidimensional variant in our subsequent research, using multiprocessor computer complexes.     

Numerical analysis of a finite volume scheme for two incompressible phase flow with dynamic capillary pressure

In this paper, we propose and analyze the convergence of a TPFA (Two Points Flux Approximation) finite volume scheme to approximate the two incompressible phase flow with dynamic capillary pressure in porous media. The fully implicit scheme is based on nonstandard approximation on mobilities and capillary pressure on the dual mesh. We derive a discrete variational formulation and we present a new result of convergence in a two or three dimensional porous medium. In comparison with static capillary pressure, the non-equilibrium capillary model requires more powerful techniques; especially the discrete energy estimates are not standard.     

冻结站中盐水流量和温度的模糊解耦控制

针对现有冻结站盐水流量和温度控制方法存在控制精度低的问题,提出了一种盐水流量和温度的模糊解耦控制方法。首先根据盐水流量和温度控制精度的不同,采用模糊控制法对流量和温度进行模糊控制;然后分析和研究盐水流量和温度间的耦合关系,采用解耦控制法对模糊控制输出的流量和温度进行解耦控制。仿真结果表明,该控制方法无需对盐水流量和温度建立精确数学模型,可实现盐水流量和温度的解耦控制,控制精度高。     

Visualization of connectivity in octree based models

An interesting problem in the oil and gas industry is the visualization of the movement of oil and gas in porous media. An example of such a medium is a rock sample with some distribution of holes (pores) connected by channels (pore throats), the solid parts of the rock are called grains. In our work we have simulated the porous medium using a pointer-based octree, representing these pores and grains. This data structure allows us to model the connectivity of the pores and thus visualize fluid penetration within the medium. Whereas earlier models represent a serious simplifications or two dimensional homogeneous layers, our model provides us with a statistically accurate distribution in three dimensions and a more accurate representation of the connectivity. In this paper we present our data structure and the techniques which were used to create models of porous media and their porous networks. Next, we present algorithms for connectivity in octrees and we show how to apply them to modelling and visualization of fluid penetration in porous media.     

Unsteady flow of a reactive variable viscosity non-Newtonian fluid through a porous saturated medium with asymmetric convective boundary conditions

This article examines the thermal effects in an unsteady flow of a pressure driven, reactive, variable viscosity, third-grade fluid through a porous saturated medium with asymmetrical convective boundary conditions. We assume that exothermic chemical reactions take place within the flow system and that the asymmetric convective heat exchange with the ambient at the surfaces follow Newton’s law of cooling. The coupled nonlinear partial differential equations governing the problem are derived and solved numerically using a semi-implicit finite difference scheme. Graphical results are presented and discussed qualitatively and quantitatively with respect to various parameters embedded in the problem.     

On numerical simulation of flow through oil filters

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.     

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

A nonlinear system with boundary-initial value conditions of convection–diffusion partial differential equations is presented to describe incompressible nuclear waste disposal contamination in porous media. The flow pressure is determined by an elliptic equation, the concentrations of brine and radionuclide are formulated by convection–diffusion equations, and the transport of temperature is defined by a heat equation. The pressure appears in convection–diffusion equations and heat equation in the form of Darcy velocity and controls the physical processes. The fluid pressure and velocity are solved by the conservative mixed volume element and the computation accuracy of Darcy velocity is improved one order. A combination method of the mixed volume element and the approximation of characteristics is applied to solve the brine and heat, where the diffusion is discretized by a mixed volume element method and the convection is treated by the method of characteristics. The characteristics can confirm strong computation stability at sharp fronts and it can avoid numerical dispersion and nonphysical oscillation. Larger time-steps along the characteristics are shown to result in smaller time-truncation errors than those resulting from standard methods. The mixed volume element method has the property of conservation on each element and it can obtain numerical solutions of the brine and adjoint vectors. The radionuclide is solved by a coupled method of characteristics and fractional step difference. The computational work is reduced greatly by decomposing a three-dimensional problem into three successive one-dimensional problems and using the algorithm of speedup. Using numerical analysis of priori estimates of differential equations, we demonstrate an optimal second order estimate in \(l^2\) norm. Numerical data are appropriate with the scheme and it is shown that the method is a powerful tool to solve the well-known problems in porous media.     

Combined Lattice Boltzmann and phase-field simulations for incompressible fluid flow in porous media

A Lattice-Boltzmann method for incompressible fluid flow is coupled with the dynamic equations of a phase-field model for multiple order parameters. The combined model approach is applied to computationally evaluate the permeability in porous media. At the boundaries between the solid and fluid phases of the porous microstructure, we employ a smooth formulation of a bounce-back condition related to the diffuse profile of the interfaces. We present simulations of fluid flow in both, static porous media with stationary non-moving interfaces and microstructures performing a dynamic evolution of the phase and grain boundaries. For the latter case, we demonstrate applications to dissolving grain structures with partial melt inclusions and computationally analyse the temporal evolution of the microporosity under wetting conditions at the melt-grain boundaries. In any development state of the material, the Darcy number and the hydraulic conductivity of the porous medium are evaluated for various types of fluid.     

Mathematical modeling and optimization of intratumor drug transport

A pseudohyperbolic problem of optimal control of intratumoral drug distribution is formulated. It takes into account the heterogeneity of tumor tissues and effects of convection diffusion in a fissured porous medium. A mathematical model constructed and the corresponding optimal control problem are shown to be correct. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 147–154, November–December 2007.     

Neuronal spatial learning

Neurons are electrically active structures determined by the evolution of ion-specific pumps and channels that allow the transfer of charges under the influence of electric fields and concentration gradients. Extensive studies of spike timing of neurons and the relationship to learning exist. However, the properties of spatial activations during action potential in the context of learning have to our knowledge not been consistently studied. We examined spatial propagation of electrical signal for many consecutive spikes using recorded information from tetrodes in freely behaving rats before and during rewarded T-maze learning tasks. Analyzing spatial spike propagation in expert medium spiny neurons with the charge movement model we show that electrical flow has directionality which becomes organized with behavioral learning. This implies that neurons within a network may behave as “weak learners” attending to preferred spatial directions in the probably approximately correct sense. Importantly, the organization of spatial electrical activity within the neuronal network could be interpreted as representing a change in spatial activation of neuronal ensemble termed “strong learning.” Together, the subtle yet critical modulations of electrical flow directivity during weak and strong learning represent the dynamics of what happens in the neuronal network during acquisition of a behavioral task.     

A multiscale preconditioner for stochastic mortar mixed finite elements

The aim of this paper is to introduce a new approach to efficiently solve sequences of problems that typically arise when modeling flow in stochastic porous media. The governing equations are based on Darcy’s law with a stochastic permeability field. Starting from a specified covariance relationship, the log permeability is decomposed using a truncated Karhunen–Loève expansion. Multiscale mortar mixed finite element approximations are used in the spatial domain and a nonintrusive sampling method is used in the stochastic dimensions. A multiscale mortar flux basis is computed for a single permeability, called a training permeability, that captures the main characteristics of the porous media, and is used as a preconditioner for each stochastic realization. We prove that the condition number of the preconditioned interface operator is independent of the subdomain mesh size and the mortar mesh size. Computational results confirm that our approach provides an efficient means to quantify the uncertainty for stochastic flow in porous media.