A solitary wave in a porous medium

The one-phase Darcy continuity equation, including the quadratic gradient term, is considered. The exact linearization of the equation is found by a functional transformation for an arbitrary spatial dimension in the limit case where the constant fluid compressibility is much more dominant than the constant compressibilities of the reservoir parameters.The equation permits a solution representing a localized wave travelling through a one-dimensional reservoir without changing its form. This is the actual long-time limit of the transient solution for a constant sandface-rate injection of a compressible fluid with a constant compressibility if the fluid is much more compressible than the matrix. A solitary wave solution is not possible for production.A fully developed solitary wave would appear only for very high pressure increases, but the first signs of the emerging solitary wave are detectable at the sandface for moderate pressure increases which can appear under physical reservoir conditions.Latin symbols a Dimensionless wave propagation velocity - A _N Sandface area (N = 0, 1, 2) - c ₁, c ₂ Sums of compressibilities - c _x Generic (generalized) compressibility - c Fluid compressibility - c _h Reservoir height (i.e. bulk volume) compressibility (N = 0, 1) - c _k , c , c Generalized compressibilities - D Spatial reservoir dimensionality (D = 1, 2, 3) - f Fractional change of p _n1 due to nonlinear effects - h Reservoir height (proportional to bulk volume for N = 0, 1) - Horizontal reservoir width (N = 0) - k Reservoir permeability - K _N Constant with dimension of pressure (N = 0, 1, 2) - n Sum index - N Integer variable (N = D – 1) - p Reservoir pressure - p^* Overburden pressure - p _D Dimensionless (scaled) version of p - p ₀ Initial pressure - q Volumetric flow rate referred to sandface - r Radial (or linear) spatial distance from center of well - r _w Well radius - r _e External reservoir radius (or length) from center of well - t Time variable - t _f Injection/production time corresponding to fraction f - T Cole-Hopf-transformed version of dimensionless pressure y - u Rescaled (dimensionless) version of v _D - v Darcy velocity - v _d Dimensionless (scaled) version of v - x Generic symbol in compressibility expression (also used for auxiliary function and for auxiliary variable) - y Rescaled (dimensionless) version of p _D - z Dimensionless (scaled) version of r Greek symbols Coefficient of inertial resistance - Variable in wave solution for y - p _n1 Absolute change in physical sandface pressure due to production or injection - _p Pressure change over (dimensionless) distance behind and far away from front - _r Physical distance at constant time corresponding to - Characteristic (dimensionless) width of solitary wave - Formation porosity - ₁, ₂ Integration constants - Dimensionless (scaled) length of finite reservoir - Fluid viscosity - Fluid density - Dimensionless (scaled) version of t - Wave solution for dimensionless pressure y - Integer variable (±1) distinguishing between production and injection     

Asymptotic Behavior of the Incompressible Navier-Stokes Fluid with Degree of Freedom in Porous Medium

The authors study the asymptotic behavior of the incompressible Navier-Stokes fluid with degree of freedom in the porous medium in R~n with n = 2 or 3. They derive the Darcy law as ε, the character size of the hole, tends to zero. Moreover, the authors obtain the expression of the degree of freedom from the homogenized model.     

Generalized Darcy–Oseen resolvent problem

In this paper, we study the well‐posedness of a coupled Darcy–Oseen resolvent problem, describing the fluid flow between free‐fluid domains and porous media separated by a semipermeable membrane. The influence of osmotic effects, induced by the presence of a semipermeable membrane, on the flow velocity is reflected in the transmission conditions on the surface between the free‐fluid domain and the porous medium. To prove the existence of a weak solution of the generalized Darcy–Oseen resolvent system, we consider two auxiliary problems: a mixed Navier–Dirichlet problem for the generalized Oseen resolvent system and Robin problem for an elliptic equation related to the general Darcy equations. © 2016 The Authors Mathematical Methods in the Applied Sciences Published by John Wiley & Sons Ltd.     

Decoupled energy‐law preserving numerical schemes for the Cahn–Hilliard–Darcy system

We study two novel decoupled energy‐law preserving and mass‐conservative numerical schemes for solving the Cahn‐Hilliard‐Darcy system which models two‐phase flow in porous medium or in a Hele–Shaw cell. In the first scheme, the velocity in the Cahn–Hilliard equation is treated explicitly so that the Darcy equation is completely decoupled from the Cahn–Hilliard equation. In the second scheme, an intermediate velocity is used in the Cahn–Hilliard equation which allows for the decoupling. We show that the first scheme preserves a discrete energy law with a time‐step constraint, while the second scheme satisfies an energy law without any constraint and is unconditionally stable. Ample numerical experiments are performed to gauge the efficiency and robustness of our scheme. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 936–954, 2016     

Uncoupling evolutionary groundwater‐surface water flows using the Crank–Nicolson Leapfrog method

Consider an incompressible fluid in a region Ω_f flowing both ways across an interface into a porous media domain Ω_p saturated with the same fluid. The physical processes in each domain have been well studied and are described by the Stokes equations in the fluid region and the Darcy equations in the porous media region. Taking the interfacial conditions into account produces a system with an exactly skew symmetric coupling. Spatial discretization by finite element method and time discretization by Crank–Nicolson LeapFrog give a second‐order partitioned method requiring only one Stokes and one Darcy subphysics and subdomain solver per time step for the fully evolutionary Stokes‐Darcy problem. Analysis of this method leads to a time step condition sufficient for stability and convergence. Numerical tests verify predicted rates of convergence; however, stability tests reveal the problem of growth of numerical noise in unstable modes in some cases. In such instances, the addition of time filters adds stability. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Simulation of Coupled Viscous and Porous Flow Problems

Theoretical and numerical formulations are presented for the conjugate problem involving incompressible flow and flow in a saturated porous medium. The major focus of the work is the development of a generally applicable finite element method for the simulation of both fixed interface and evolving porous interface problems. The available alternatives for coupling Darcy and non-Darcy models to the Navier-Stokes equations have been studied and evaluated in a mixed finite element framework. Questions regarding convergence of the finite element method for porous flow models have been addressed. Numerical experiments on simple flow geometries have revealed the shortcomings of both the Darcy and Brinkman models. Application of the more realistic models to practical, multidimensional, flow studies has also been demonstrated.     

页岩气在超低渗介质中的渗流行为

基于页岩气藏复杂孔隙结构和页岩气在纳米孔隙表面的滑脱和吸附-解吸附等现象,通过引入表观渗透率,修正Darcy渗流模型,得到了页岩气渗流本构方程. 将计算结果与Darcy模型计算结果进行了比照,结果表明,在产量定常情形下,基于非Darcy渗流模型得到的井底压力高于Darcy流结果; Darcy流模型得到的压力衰减速度较快,改进后的渗流模型更能准确描述与表征页岩气渗流过程;研究结果可为页岩气藏的经营与管理提供基础参数.      

不同温度下丝网过滤器的渗透率试验研究

为了研究丝网型过滤器在不同温度下的阻力特性,对一种褶皱型不锈钢丝网过滤器进行了低温和常温下的渗透率试验研究.流动工质分别采用纯净的液氮和二氧化碳气体,测试两种温度下工质流经过滤器试验台的压降,从而利用总流伯努利方程以及层流下的达西公式计算两种工况下的渗透率.试验结果表明,85 K时的渗透率比283 K时低2个数量级,这主要是由于低温下不锈钢丝网发生冷缩,改变了流道结构并降低了孔隙率,从而导致渗透率降低且压降升高.     

确定多孔介质流动参数的格子Boltzmann方法

本文采用D2Q9不可压缩Lattice-Boltzmann模型模拟了多孔介质中的流场,该模型从介观层次上描述了多孔介质孔隙中的流动特性,验证了Darcy定律,给出了一种计算渗透率的简便方法,本文还运用图像处理方法来构造多孔介质,该方法构造的二维多孔介质能较好地体现实际多孔介质的几何特征。     

On the flow of fluids through inhomogeneous porous media due to high pressure gradients

Most porous solids are inhomogeneous and anisotropic, and the flows of fluids taking place through such porous solids may show features very different from that of flow through a porous medium with constant porosity and permeability. In this short paper we allow for the possibility that the medium is inhomogeneous and that the viscosity and drag are dependent on the pressure (there is considerable experimental evidence to support the fact that the viscosity of a fluid depends on the pressure). We then investigate the flow through a rectangular slab for two different permeability distributions, considering both the generalized Darcy and Brinkman models. We observe that the solutions using the Darcy and Brinkman models could be drastically different or practically identical, depending on the inhomogeneity, that is, the permeability and hence the Darcy number.