A Porous Media Model for the Description of Adaptive Bone Remodelling

Bones are strong and lightweight structures, which mainly consist of extracellular bone matrix. The bone remodelling is a process of resorption followed by replacement of the bone matrix with small changes in shape, which allow the bones to adapt according to the local loading situation. In the context of the Theory of Porous Media (TPM), a consistent model of bone tissue is introduced, which is able to describe the local accretion and reduction of the extracellular bone matrix. To this end, the bone is treated as an aggregate of two immiscible constituents. In this biphasic macroscopic model, the aggregate consists of the extracellular bone matrix and cells summarised to a solid phase and an interstitial fluid phase comprising nutrients, metabolites and bone precursors. The addition and removal of bone matrix is described by a mass exchange between the constituents, which depends upon the local strain of the material. Additionally, the growth energy is introduced as a non-mechanical quantity, which measures the average amount of chemical energy available for cell metabolism [1, 2], and thus, controls the growth process. The presented numerical example illustrates the fundamental effects of bone remodelling under varying boundary conditions. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

多孔介质热弥散系数的分形模型北大核心CSCD

热弥散系数是与流体的物性和多孔介质结构有关的,表征多孔介质传热传质强弱的重要参数.该文建立了分形多孔介质的孔喉结构模型,研究了在孔喉结构处流体由湍流状态变为层流状态的局部水头损失和速度弥散效应,在考虑微观孔喉结构和速度弥散效应的影响下,推导了热弥散系数关系式.研究表明,热弥散系数与孔喉比、孔喉结构个数和迂曲分形维数成正比,与孔隙率和面积分形维数成反比.进一步研究发现,孔喉比在1~150范围内对速度弥散效应有显著影响,流体在孔喉结构处存在局部水头损失,导致速度弥散效应增强,热弥散系数增大.     

Large-Time Behaviour of Solutions of the Exterior-Domain Cauchy–Dirichlet Problem for the Porous Media Equation with Homogeneous Boundary Data

The title of this paper states precisely what the subject is. The first part of the paper concerns the radially-symmetric problem in the exterior of the unit ball. It is shown that in time the solution of the problem converges to one of two specific self-similar solutions of the porous media equation, dependent upon the dimensionality of the problem. Moreover, the free boundary of the solution converges to that of the self-similar solution. The critical space dimension is two, for which there is no distinction between the self-similar solutions, and the form of the convergence is exceptional. The technique used is a comparison principle involving a variable that is a weighted integral of the solution. The second part of the paper is devoted to the problem in an arbitrary spatial domain with no conditions of symmetry. A special invariance principle and the results obtained for the radially-symmetric case are used to determine the large-time behaviour of solutions and their free boundaries. This behaviour is decidedly different from when the boundary data are fixed and not homogeneous.     

Large-Time Behaviour of Solutions of the Exterior-Domain Cauchy–Dirichlet Problem for the Porous Media Equation with Homogeneous Boundary Data

The title of this paper states precisely what the subject is. The first part of the paper concerns the radially-symmetric problem in the exterior of the unit ball. It is shown that in time the solution of the problem converges to one of two specific self-similar solutions of the porous media equation, dependent upon the dimensionality of the problem. Moreover, the free boundary of the solution converges to that of the self-similar solution. The critical space dimension is two, for which there is no distinction between the self-similar solutions, and the form of the convergence is exceptional. The technique used is a comparison principle involving a variable that is a weighted integral of the solution. The second part of the paper is devoted to the problem in an arbitrary spatial domain with no conditions of symmetry. A special invariance principle and the results obtained for the radially-symmetric case are used to determine the large-time behaviour of solutions and their free boundaries. This behaviour is decidedly different from when the boundary data are fixed and not homogeneous.     

Periodic Solutions to Porous Media Equations of Parabolic-elliptic Type

This paper is concerned with a equation, which is a model of filtration in partially saturated porous media, with mixed boundary condition of Dirichlet-Neumann type {∂_tb(u) - ∇ • a [∇u + k(b(u))] = f \qquad in \quad (0, ∞) × Ω u = h(t, x) \qquad on \quad (0, ∞) × Γ_0 v • a [∇u + k(b(u))] = g(t, x) \qquad on \quad (0, ∞) × Γ_1 We have proved that there exists one and only one periodic solution of the problem under the data f, g and h with same period. Moreover, we have proved that the unique periodic solution ω is asymptotically statble in the sense that for any solution u of the problem b(u(t)) - b(ω(t)) → 0\qquad in L²(Ω) as t → ∞.     

An Adaptive Grid Method for a Non-Equilibrium PDE Model from Porous Media

An adaptive grid method is applied to a PDE model from geo-hydrology. Due to the higher mixed-order derivative, non-monotone waves can appear which could represent similar structures as observed in laboratory experiments [5, 16, 18]. The effectiveness of the adaptive grid, which is based on a smoothed equidistribution principle, is shown compared to uniform grid simulations. On a uniform grid (numerical) oscillating non-monotone waves may appear which are not present in the adaptive grid.     

两相流体非线性渗流模型及其应用

基于三参数非线性渗流运动定律、质量守恒定律及椭圆渗流的概念,建立了低渗透介质中两相流体椭圆非线性渗流数学模型,运用有限差分法与外推法求得了其解,导出了两相流体椭圆非线性渗流条件下油井见水前后开发指标的计算公式,进行了实例分析。结果表明:非线性渗流对含水饱和度分布影响较大;非线性渗流使得水驱油推进速度比线性渗流的快,使油井见水时间提前,使得石油开发指标变差;非线性渗流使得同一时刻的压差比线性渗流的大,使石油开发难度加大。这为低渗油藏垂直裂缝井开发工程提供了科学依据。     

The Rain on Underground Porous Media Part I: Analysis of a Richards Model

The Richards equation models the water flow in a partially saturated underground porous medium under the surface. When it rains on the surface, boundary conditions of Signorini type must be considered on this part of the boundary. The authors first study this problem which results into a variational inequality and then propose a discretization by an implicit Euler’s scheme in time and finite elements in space. The convergence of this discretization leads to the well-posedness of the problem.     

具吸收项的多孔隙介质方程解的渐近性

本文研究了具吸收项的多孔隙介质方程解的渐近性质,部分地解答了P.L.Pelelier等人关于此问题的一个猜测。     

Structural Stability for the Brinkman Equations of Porous Media

Explicit a priori continuous dependence estimates are derived for the Brinkman equations for non-isothermal flow in porous media. Continuous dependence on the cooling coefficient is shown when a boundary condition of Newton cooling type is employed. Continuous dependence on the model itself is proved when the Boussinesq model is allowed to change to one appropriate to penetrative convection. The final result derives an a priori continuous dependence estimate for the heat supply and body force.     

多孔介质中颗粒性态分析

分析了流经多孔介质的、充满颗粒材料的、混合液流动中的颗粒相性态.试图建立扩散和耗散过程模型,量化了在本征体积平均时出现的偏差项,从而导出流动方程.     

微极性多组分多孔介质材料的混合物理论

将描述多组分系统的复合混合物理论与微极性连续介质力学理论相结合,建立了描述微极性多组分多孔介质材料的混合物理论.假定系统由多组分的微极性弹性固体和多组分微极性粘性流体组成.给出由混合物理论建立的系统的平衡方程.依据热力学第二定律以及本构假设建立了系统的本构方程,并使场方程闭合.为考虑固相的压缩性,在液相自由能函数中引入液相体积分数作为内变量,得到动力相容条件,用以限制固、液两相界面压力差的变化.最后,基于线性化理论得到线性化的本构方程和场方程,建立了考虑介质微极性的热-水力-力学组分输运模型.此理论框架可以运用到可变形多孔介质中污染物、药物以及农药输运等问题中,所得到的微极性多组分多孔介质系统的闭合场方程经退化后,可变为固、流相都为单一组分的多孔介质系统场方程,它与Eringen得到的结果一致.     

A Method for Numerical Simulation of Haline Convective Flows in Porous Media as Applied to Geology

Computational Mathematics and Mathematical Physics - A numerical code for simulating haline convective flows in porous media based on the finite difference method on a staggered nonuniform grid is...     

A Quasi-Projection Analysis for Elastic Wave Propagation in Fluid-Saturated Porous Media

This paper deals with the superconvergence phenomena for Galerkin approximations of solutions of Biot's dynamic equations describing elastic wave propagation in fluid-saturated porous media. An asymptotic expansion to high order of Galerkin solutions is used to derive these results.