Nonlinear fluid flow laws for orthotropic porous media

Nonlinear fluid flow laws for orthotropic porous media are written in invariant tensor form. As usual in the theory of fluid flow through porous media [1, 2], the equations contain the flow velocity up to the second power. Expressions that determine the nonlinear resistances to fluid flow are presented and it is shown that, on going over from linear to nonlinear flow laws, the asymmetry effect may manifest itself, that is, the fluid flow characteristics may differ along the same straight line in the positive and negative directions. It is shown that, as compared with the linear fluid flow law for orthotropic media when for three symmetry groups a single flow law is sufficient, in nonlinear laws the anisotropy manifestations are much more variable and each symmetry group must be described by specific equations. A system of laboratory measurements for finding the nonlinear flow characteristics for orthotropic porous media is considered.     

A piezoelectric stress gauge for measuring dynamic stress in soil and rock media

A piezoelectric stress gauge is described in this paper. Its major performance data are: measuring range 10⁶–10⁸ Pa, response time less than 7μs; non-linearity within ±1% and total stress measurement error within ± 10%. It can be used for measuring dynamic stress in soil, rock and concrete media as well as dynamic force and dynamic pressure in fluids.     

General Anisotropic Effective Medium Theory for the Effective Permeability of Heterogeneous Reservoirs

One of the techniques to calculate the effective property of a heterogeneous medium is the effective medium theory. The present paper presents a general mathematical formulation for the effective medium approximation using a self-consistent choice of the effective permeability, to apply it to the case of a general anisotropic 2D medium and to the case of a 3D isotropic medium with randomly oriented ellipsoidal inclusions. The 2D results are compared with analytical results and with a homogenization technique with good result. The 3D correlations are used to derive percolation thresholds in two-phase systems with a large permeability contrast, which are compared to numerical results from the literature, also with good results.     

多孔介质有效应力原理研究

提出了基于多孔介质的一种新的有效应力原理,据此,当饱和土加载后,孔隙度的变化与孔孙水压力之间的关系是非线性的,它们之间的变化关系同样取决于初始孔隙度以及初始孔隙水压力。     

Effective Relative Permeabilities and Capillary Pressure for One-Dimensional Heterogeneous Media

The paper presents an analytical construction of effective two-phase parameters for one-dimensional heterogeneous porous media, and studies their properties. We base the computation of effective parameters on analytical solutions for steady-state saturation distributions. Special care has to be taken with respect to saturation and pressure discontinuities at the interface between different rocks. The ensuing effective relative permeabilities and effective capillary pressure will be functions of rate, flow direction, fluid viscosities, and spatial scale of the heterogeneities.The applicability of the effective parameters in dynamic displacement situations is studied by comparing fine-gridded simulations in heterogeneous media with simulations in their homogeneous (effective) counterparts. Performance is quite satisfactory, even with strong fronts present. Also, we report computations studying the applicability of capillary limit parameters outside the strict limit.     

裂缝性油藏流固耦合渗流

本文给出了考虑介质变形的双重孔隙介质流固耦合渗流模型,并考虑渗流参数随有效应力而变化的非线性双重孔隙介质流固耦合渗流,在此基础上,本文还推导了双重孔隙介质非线性系数非线性等流固耦合流流计算,并给出了算例。     

三维非均匀不稳定渗流方程的二维不均匀粗化算法

在无源汇条件下,根据流过某一个横截面的流体流量等于流过这一横截面内所有精细网格的流体流量之和这一特点提出了粗化网格等效渗透率的计算方法。在粗化区内,利用直接解法求解二维渗流方程,再用这些解合成粗化网格的三维合成解,并由合成解计算粗化网格的等效渗透率。根据精度的要求采用了不均匀网格粗化,在流体流速大的区域采用精细网格。利用所得等效渗透率计算了粗化网格的某三维非均匀不稳定渗流场的压降解,结果表明三维非均匀不稳定渗流方程的二维不均匀粗化解非常逼近采用精细网格的解,但计算的速度比采用精细网格提高了80倍。     

A semi‐discrete central scheme for scalar hyperbolic conservation laws with heterogeneous storage coefficient and its application to porous media flow

In this paper, we develop a new Godunov‐type semi‐discrete central scheme for a scalar conservation law on the basis of a generalization of the Kurganov and Tadmor scheme, which allows for spatial variability of the storage coefficient (e.g. porosity in multiphase flow in porous media) approximated by piecewise constant interpolation. We construct a generalized numerical flux at element edges on the basis of a nonstaggered inhomogeneous dual mesh, which reproduces the one postulated by Kurganov and Tadmor under the assumption of homogeneous storage coefficient. Numerical simulations of two‐phase flow in strongly heterogeneous porous media illustrate the performance of the proposed scheme and highlight the important rule of the permeability–porosity correlation on finger growth and breakthrough curves. Copyright © 2013 John Wiley & Sons, Ltd.     

The solid phase stress tensor in porous media mechanics and the Hill-Mandel condition

An assessment of the stress tensors used currently for the modeling of partially saturated porous media is made which includes concepts like total stress, solid phase stress, and solid pressure. Thermodynamically constrained averaging theory is used to derive the solid phase stress tensor. It is shown that in the upscaling procedure the Hill conditions are satisfied, which is not trivial. The stress tensor is then compared to traditional stress measures. The physical meaning of two forms of solid pressure and of the Biot coefficient is clarified. Finally, a Bishop-Skempton like form of the stress tensor is obtained and a form of the total stress tensor that does not make use of the effective stress concept.     

Effective hydraulic conductivities in unsaturated heterogeneous media by Monte Carlo simulation

When modeling flow and transport through unsaturated heterogeneous geological deposits, it may be neither computationally nor technically feasible to account for the actual heterogeneity in the simulations. One would fall short in terms of technical feasibility because there is simply no way that the entire spatial domain could be characterized (e.g., you cannot measure hydraulic conductivity at every location at a site). With respect to computational feasibility, the non-linear nature of the Richards equation (which is used to model the flow process) makes simulation of most sites extremely computationally intensive. The computational roadblock is being dismantled as computer hardware advances, but our inability to precisely characterize geological heterogeneity is expected to remain with us for a very long time. To address this problem, the analyst typically uses average or effective properties to model flow and transport behavior through heterogeneous media. In this paper, a variety of approaches for developing effective unsaturated flow properties are assessed. Computational results have been obtained which give the hydraulic conductivity ratios (K _parallel/K _nomal) for highly nonisotropic layered materials. These results are compared with analytical models. Good agreement was obtained for all soil saturation levels except for extremely dry conditions.This work was performed at Sandia National Laboratories, which is operated for the U.S. Department of Energy under contract number DE-AC04-76DP00789.     

Models of flow through fractured-porous anisotropic media

A mathematical model of the theory of flow through fractured-porous anisotropic media is considered. A new construction of the cross-flow function which depends on the direction of the action applied, i.e., on the difference between the pressure gradients in blocks and fractures, is given. This specification of the cross-flow function makes it possible to take into account the anisotropy of the seepage properties not only in Darcy’s law but also in the continuity equations and gives the mass-transfer law greater physical justification. Variants of the generalization of the Barenblatt-Zheltov-Kochina and Warren-Root models to include the case of arbitrary anisotropy are analyzed for the mathematical model proposed.     

The effective homogeneous behavior of heterogeneous porous media

The macroscopic equations that govern the processes of one- and two-phase flow through heterogeneous porous media are derived by using the method of multiple scales. The resulting equations are mathematically similar to the point equations, with the fundamental difference that the local permeabilities are replaced by effective parameters. The method allows the determination of these parameters from a knowledge of the geometrical structure of the medium and its heterogeneities. The technique is applied to determine the effective parameters for one- and two-phase flows through heterogeneous porous media made up of two homogeneous porous media.     

Thermal stretching in two-phase porous media: Physical basis for Maxwell model

An alternate yet general form of the classical effective thermal conductivity model (Maxwell model) for two-phase porous materials is presented, serving an explicit thermo-physical basis. It is demonstrated that the reduced effective thermal conductivity of the porous media due to non-conducting pore inclusions is caused by the mechanism of thermal stretching, which is a combination of reduced effective heat flow area and elongated heat transfer distance (thermal tortuosity).     

Accuracy of the Renormalization Method for Computing Effective Conductivities of Heterogeneous Media

The accuracy of the renormalization method for upscaling two-dimensional hydraulic conductivity fields is investigated, using two canonical 2 × 2 blocks: a checkerboard geometry and a geometry in which three of the cells have conductivity K ₁ and the other has conductivity K ₂. The predictions of the renormalization algorithm are compared to the arithmetic, harmonic and geometric means, as well as to theoretical predictions and finite element calculations. For the latter geometry renormalization works well over the entire range of the conductivity ratio K ₂/K ₁, but for the checkerboard geometry the error becomes unbounded as the conductivity ratio grows.     

含液多孔介质力学问题的边界元方法

提出了一种含液多孔介质力学问题的边界元求解方法.首先将问题分解为一系列含单孔流体夹杂的子问题,然后针对每个子问题建立了流体孔体积变化率与流体压力之问的函数关系,进一步采用边界元方法建立了以各流体孔压力为基本未知量的线性代数方程组,最后根据所求出的各流体孔的压力计算含液多孔介质内各点的位移、变形和应力.为了说明方法的有效...     

Wave propagation in fractured porous media

A theory of wave propagation in fractured porous media is presented based on the double-porosity concept. The macroscopic constitutive relations and mass and momentum balance equations are obtained by volume averaging the microscale balance and constitutive equations and assuming small deformations. In microscale, the grains are assumed to be linearly elastic and the fluids are Newtonian. Momentum transfer terms are expressed in terms of intrinsic and relative permeabilities assuming the validity of Darcy's law in fractured porous media. The macroscopic constitutive relations of elastic porous media saturated by one or two fluids and saturated fractured porous media can be obtained from the constitutive relations developed in the paper. In the simplest case, the final set of governing equations reduce to Biot's equations containing the same parameters as of Biot and Willis.Now at Izmir Institute of Technology, Anafartalar Cad. 904, Basmane 35230, Izmir, Turkey.     

Effective Parameter Interpretation and Extrapolation of Dispersion Simulations by Means of a Simple Two-Velocity Model

The investigation of dispersion by microscopic simulations yields a lot of detailed information. To identify characteristic behaviours, it is useful to condense this information into a few effective parameters, which describe the transport process in the model geometry on a larger scale. For this purpose, a very simple two-velocity model has been developed, which models the transition from reversible to irreversible spreading of a tracer volume. It is shown that this model is very similar to Taylor–Aris dispersion and that it is quite suitable to approximate the time dependence of dispersion. The model is applied to characterize the effect of dead end pores on dispersion with a single correlation parameter. Up to Péclet numbers of about 500, 'hold-up'-dispersion similar to Taylor–Aris-dispersion is found. The simulations have been performed by the lattice Bhatnagar–Gross–Krook (BGK) method, which is a particular type of cellular automata and therefore allows an easy implementation of complicated geometries. The fully irreversible asymptotic dispersion is reached in an exponential process, the parameters of which can be identified by the two-velocity model after the mixing has noticeably begun. These are used to extrapolate the process which reduces the computational effort by about one order of magnitude.