A new formulation of immiscible compressible two-phase flow in porous media

A new formulation is proposed to describe immiscible compressible two-phase flow in porous media. The main feature of this formulation is the introduction of a global pressure. The resulting equations are written in a fractional flow formulation and lead to a coupled system which consists of a nonlinear parabolic (the global pressure equation) and a nonlinear diffusion–convection one (the saturation equation) which can be efficiently solved numerically. To cite this article: B. Amaziane, M. Jurak, C. R. Mecanique 336 (2008).     

From Single-Phase to Compositional Flow: Applicability of Mixed Finite Elements

In this paper we discuss the formulation of the governing equations that describe flow of fluids in porous media. Various types of fluid flow, ranging from single-phase flow to compositional flow, are considered. It is shown that all the differential equations governing these types of flow can be effectively rewritten in a fractional flow formulation; i.e., in terms of a global pressure and saturation (or saturations), and that mixed finite element methods can be accurately exploited to solve the pressure equation. Numerical results are presented to see the performance of the mixed methods for the flow equations in three space dimensions.     

Analysis of one-dimensional horizontal two-phase flow in geothermal reservoirs

In the absence of capillarity the single-component two-phase porous medium equations have the structure of a nonlinear parabolic pressure (equivalently, temperature) diffusion equation, with derivative coupling to a nonlinear hyperbolic saturation wave equation. The mixed parabolic-hyperbolic system is capable of substaining saturation shock waves. The Rankine-Hugoniot equations show that the volume flux is continuous across such a shock. In this paper we focus on the horizontal one-dimensional flow of water and steam through a block of porous material within a geothermal reservoir. Starting from a state of steady flow we study the reaction of the system to simple changes in boundary conditions. Exact results are obtainable only numerically, but in some cases analytic approximations can be derived. When pressure diffusion occurs much faster than saturation convection, the numerical results can be described satisfactorily in terms of either saturation expansion fans, or isolated saturation shocks. At early times, pressure and saturation profiles are functionally related. At intermediate times, boundary effects become apparent. At late times, saturation convection dominates and eventually a steady-state is established. When both pressure diffusion and saturation convection occur on the same timescale, initial simple shock profiles evolve into multiple shocks, for which no theory is currently available. Finally, a parameter-free system of equations is obtained which satisfactorily represents a particular case of the exact equations.     

Stable Propagation of Saturation Overshoots for Two-Phase Flow in Porous Media

Propagation of saturation overshoots for two-phase flow of immiscible and incompressible fluids in porous media is analyzed using different computational methods. In particular, it is investigated under which conditions a given saturation overshoot remains stable while moving through a porous medium. Two standard formulations are employed in this investigation, a fractional flow formulation and a pressure–saturation formulation. Neumann boundary conditions for pressure are shown to emulate flux boundary conditions in homogeneous media. Gravity driven flows with Dirichlet boundary conditions for pressure that model infiltration into heterogeneous media with position-dependent permeability are found to exhibit pronounced saturation overshoots very similar to those seen in experiment.     

Nonlinear waves in fluid flow through a viscoelastic tube

A system of nonlinear equations for describing the perturbations of the pressure and radius in fluid flow through a viscoelastic tube is derived. A differential relation between the pressure and the radius of a viscoelastic tube through which fluid flows is obtained. Nonlinear evolutionary equations for describing perturbations of the pressure and radius in fluid flow are derived. It is shown that the Burgers equation, the Korteweg-de Vries equation, and the nonlinear fourth-order evolutionary equation can be used for describing the pressure pulses on various scales. Exact solutions of the equations obtained are discussed. The numerical solutions described by the Burgers equation and the nonlinear fourth-order evolutionary equation are compared.     

非牛顿流体两相渗流问题的摄动解

本文用奇异摄动法结合正则摄动法求解了考虑毛管力因素时多孔介质中弱非牛顿流体的两相驱替问题,得到了分流函数和湿相饱和度的渐近解析解。所得结果同数值解和经典的牛顿流体两相渗流结果进行了比较,并着重讨论了非牛顿因素的影响。     

Streamline computations for porous media flow including gravity

In this paper we consider porous media flow without capillary effects. We present a streamline method which includes gravity effects by operator splitting. The flow equations are treated by an IMPES method, where the pressure equation is solved by a (standard) finite element method. The saturation equation is solved by utilizing a front tracking method along streamlines of the pressure field. The effects of gravity are accounted for in a separate correction step. This is the first time streamlines are combined with gravity for three-dimensional (3D) simulations, and the method proves favourable compared to standard splitting methods based on fractional steps. By our splitting we can take advantage of very accurate and efficient 1D methods. The ideas have been implemented and tested in a full field simulator. In that context, both accuracy and CPU efficiency have tested favourably.     

Models for Ground Water Flow: A Numerical Comparison Between Richards' Model and the Fractional Flow Model

In this paper we compare two models for flow in porous media. The first is the well known Richards' equation, which is based on the assumption that the air in the unsaturated zone has infinite mobility. This means that it models a single phase. In the second and more general full two-phase approach, the air is considered as a separate phase. Here, we use the fractional flow equation.We study the difference between the two models numerically by varying the relative contribution of the different physical terms (the gravity and the total velocity) in the fractional flow equation. Richards' equation is considered as the limit of the fractional flow approach when the mobility of the air-phase tends to infinity. In particular, we are interested in determining the parameter intervals where the two models differ significantly, and we will quantify the asymptotic behavior.The equations are studied in the two-dimensional (2D) case. The study is based on a relative permeability depending quadratically on the saturation, and a capillary pressure expressed by a cubic function of the saturation.     

The Local Analysis of Changing Force Balances in Immiscible Incompressible Two-Phase Flow

The balance of viscous, capillary and gravity forces strongly affects two-phase flow through porous media and can therefore influence the choice of appropriate methods for numerical simulation and upscaling. A strict separation of the effects of these various forces is not possible due to the nature of the nonlinear coupling between the various terms in the transport equations. However, approximate prediction of this force balance is often made by calculation of dimensionless quantities such as capillary and gravity numbers. We present an improved method for the numerical analysis of simulations which recognises the changing balance of forces – in both space and time – in a given domain. The classical two-phase transport equations for immiscible incompressible flow are expressed in two forms: (i) the convection–diffusion-gravity (CDG) formulation where convection and diffusion represent viscous and capillary effects, respectively, (ii) the oil pressure formulation where the viscous effects are attributed to the product of mobility difference and the oil pressure gradient. Each formulation provides a different perspective on the balance of forces although the two forms are equivalent. By discretising the different formulations, the effect of each force on the rate of change of water saturation can be calculated for each cell, and this can be analysed visually using a ternary force diagram. The methods have been applied to several simple models, and the results are presented here. When model parameters are varied to determine sensitivity of the estimators for the balance of forces, the CDG formulation agrees qualitatively with what is expected from physical intuition. However, the oil pressure formulation is dominated by the steady-state solution and cannot be used accurately. In addition to providing a physical method of visualising the relative magnitudes of the viscous, gravity and capillary forces, the local force balance may be used to guide our choice of upscaling method.     

Approximate symmetry group classification for a nonlinear fractional filtration equation of diffusion-wave type

A one-dimensional nonlinear fractional filtration equation with the Riemann–Liouville time-fractional derivative is proposed for modeling fluid flow through a porous medium. This equation is derived under an assumption that the fluid has a fractional equation of state in which the fluid density depends on the time-fractional derivative of pressure. The obtained equation belongs to the diffusion-wave type of equations. A case when the order of fractional differentiation is close to an integer number is considered, and a small parameter is introduced into the fractional filtration equation under consideration. An expansion of the Riemann–Liouville time-fractional derivative into the series with respect to this small parameter is obtained. Using this expansion, a first-order approximation of the derived fractional filtration equation is performed. Next, the problem of approximate Lie point symmetry group classification for this approximate nonlinear filtration equation with a small parameter is studied. It is shown that approximate symmetry groups admitted by different realizations of the approximate filtration equation have much more dimensions than the corresponding exact Lie point symmetry groups admitted by unperturbed fractional diffusion-wave equations. Obtained classification results permit to construct approximate invariant solutions for the considered nonlinear time-fractional filtration equations.     

A fractional gradient representation of the Poincaré equations

A gradient representation and a fractional gradient representation of the Poincaré equations are studied. Firstly, the condition presented here for the Poincaré equation can be considered as a gradient system. Then, a condition under which the Poincaré equation can be considered as a fractional gradient system is obtained. Finally, two examples are given to illustrate applications of the result.     

桁架结构几何非线性问题的力密度法

桁架结构大都具有较强的几何非线;浊,受荷载后易出现较大的脯变形。提出一种基于力密度的针对桁架结构几何大变形问题的解法;引入杆件单元的力密度矩阵,推导出相应非线性方程的Jacobi矩阵;与有限单元法集成求解的思想相同,采用力密度矩阵建立结构变形后整体的精确非线性平衡方程。研究结果表明：应用Newton-Raphson迭代法求解,采用适当的迭代收敛精度可得到较精确的桁架结构位移解。     

A pressure‐based method with AUSM‐type fluxes for MHD flows at arbitrary Mach numbers

In this paper, we propose an extension of a PISO method, previously developed to solve the Euler equations, and which is here extended to the ideal magnetohydrodynamic (MHD) equations. By following a pressure‐based approach, we make use of the flexibility given by pressure equation for calculating flows at arbitrary Mach numbers. To handle MHD discontinuities, we have adapted the MHD‐Advection Upstream Splitting Method for our pressure‐based formulation. With the purpose of validation, four sets of test cases are presented and discussed. We start with the circularly polarized Alfvén waves that serves as a smooth flow validation. The second case is the 1‐D Riemann problem that is calculated using both 1‐D and 2‐D formulation of the MHD equations. The third and fourth problems are the Orszag–Tang vortex and the supersonic low‐ β cylinder allowing validation of the method in complex 2‐D MHD shock interaction. Copyright © 2013 John Wiley & Sons, Ltd.     

A numerical study of fluids with pressure‐dependent viscosity flowing through a rigid porous medium

Much of the work on flow through porous media, especially with regard to studies on the flow of oil, are based on ‘Darcy's law’ or modifications to it, such as Darcy–Forchheimer or Brinkman models. While many theoretical and numerical studies concerning flow through porous media have taken into account the inhomogeneity and anisotropy of the porous solid, they have not taken into account the fact that the viscosity of the fluid and drag coefficient could depend on the pressure in applications, such as enhanced oil recovery (EOR). Experiments clearly indicate that the viscosity varies exponentially with respect to the pressure and the viscosity can change, in some applications, by several orders of magnitude. The fact that the viscosity depends on pressure immediately implies that the ‘drag coefficient’ would also depend on the pressure. In this paper we consider modifications to Darcy's equation wherein the drag coefficient is a function of pressure, which is a realistic model for technological applications, such as EOR and geological carbon sequestration. We first outline the approximations behind Darcy's equation and the modifications that we propose to Darcy's equation, and derive the governing equations through a systematic approach using mixture theory. We then propose a stabilized mixed finite element formulation for the modified Darcy's equation. To solve the resulting nonlinear equations we present a solution procedure based on the consistent Newton–Raphson method. We solve representative test problems to illustrate the performance of the proposed stabilized formulation. One of the objectives of this paper is to show that the dependence of viscosity on the pressure can have a significant effect both on the qualitative and quantitative nature of the solution. Copyright © 2010 John Wiley & Sons, Ltd.     

Permanent Fronts in Two-Phase Flows in a Porous Medium

A family of exact solutions for a model of a one-dimensional horizontal flow of two immiscible, incompressible fluids in a porous medium, including the effects of capillary pressure, is obtained analytically by solving the governing singular parabolic nonlinear diffusion equation. Each solution has the form of a permanent front propagating with a constant velocity. It is shown that, for every propagation velocity, there exists a set of permanent fronts all of which are moving with this velocity in an inflowing wetting–outflowing non-wetting flow configuration. Global bifurcations of this set, with the front velocity as a bifurcation parameter, are investigated analytically and numerically in detail in the case when the permeabilities and the capillary pressure are linear functions of the wetting phase saturation. Main results for the nonlinear Brooks–Corey model are also presented. In both models three global bifurcations occur. By using a geometric dynamical system approach, the nonlinear stability of the permanent fronts is established analytically. Based on the permanent front solutions, an interpretation of the dynamics of an arbitrary front of finite extent in the model is given as follows. The instantaneous upstream (downstream) velocity of an arbitrary non-quasistationary front is equal to the velocity of a permanent front whose shape coincides up to two leading orders with the instantaneous shape of the non-quasistationary front at the upstream (respectively, downstream) location. The upstream and downstream locations of the front undergo instantaneous translations governed by modified nonsingular hyperbolic equations. The portion of the front in between these locations undergoes a diffusive redistribution governed by a nonsingular nonlinear parabolic diffusion equation. We have proposed a numerical approach based on a parabolic–hyperbolic domain decomposition for computing non-quasistationary fronts.     

A VARIATIONAL INEQUALITY APPROACH FOR NON-STEADY SEEPAGE FLOW THROUGH THREE-DIMENSIONAL FRACTURE NETWORK

为了求解裂隙岩体有自由面非稳定渗流问题,将Darcy定律延拓至整个研究区域,使得潜在溢出边界条件满足Signorini型边界条件,建立了三维裂隙网络非稳定渗流问题的抛物型变分不等式（parabolic variational inequality,PVI）提法,并证明其与偏微分方程（partial differential equation,PDE）提法的等价性,从而将自由面上的流量条件以及潜在溢出边界上的互补条件转化成自然边界条件,降低该问题求解难度。同时给出了基于PVI提法的有限元数值求解方法,通过与交叉裂隙模型理论解的对比分析,证明了该方法的正确性。最后将该方法对含复杂三维裂隙网络的边坡进行非稳定渗流分析,计算结果表明该方法对于复杂裂隙网络求解具有较强的可靠性和适应性。     

A higher resolution edge‐based finite volume method for the simulation of the oil–water displacement in heterogeneous and anisotropic porous media using a modified IMPES method

In this article, we present a higher‐order finite volume method with a ‘Modified Implicit Pressure Explicit Saturation’ (MIMPES) formulation to model the 2D incompressible and immiscible two‐phase flow of oil and water in heterogeneous and anisotropic porous media. We used a median‐dual vertex‐centered finite volume method with an edge‐based data structure to discretize both, the elliptic pressure and the hyperbolic saturation equations. In the classical IMPES approach, first, the pressure equation is solved implicitly from an initial saturation distribution; then, the velocity field is computed explicitly from the pressure field, and finally, the saturation equation is solved explicitly. This saturation field is then used to re‐compute the pressure field, and the process follows until the end of the simulation is reached. Because of the explicit solution of the saturation equation, severe time restrictions are imposed on the simulation. In order to circumvent this problem, an edge‐based implementation of the MIMPES method of Hurtado and co‐workers was developed. In the MIMPES approach, the pressure equation is solved, and the velocity field is computed less frequently than the saturation field, using the fact that, usually, the velocity field varies slowly throughout the simulation. The solution of the pressure equation is performed using a modification of Crumpton's two‐step approach, which was designed to handle material discontinuity properly. The saturation equation is solved explicitly using an edge‐based implementation of a modified second‐order monotonic upstream scheme for conservation laws type method. Some examples are presented in order to validate the proposed formulation. Our results match quite well with others found in literature. Copyright © 2016 John Wiley & Sons, Ltd.     

Finite element formulation of slender structures with shear deformation based on the Cosserat theory

This paper addresses the derivation of finite element modelling for nonlinear dynamics of Cosserat rods with general deformation of flexure, extension, torsion, and shear. A deformed configuration of the Cosserat rod is described by the displacement vector of the deformed centroid curve and an orthogonal moving frame, rigidly attached to the cross-section of the rod. The position of the moving frame relative to the inertial frame is specified by the rotation matrix, parameterised by a rotational vector. The shape functions with up to third order nonlinear terms of generic nodal displacements are obtained by solving the nonlinear partial differential equations of motion in a quasi-static sense. Based on the Lagrangian constructed by the Cosserat kinetic energy and strain energy expressions, the principle of virtual work is employed to derive the ordinary differential equations of motion with third order nonlinear generic nodal displacements. A cantilever is presented as a simple example to illustrate the use of the formulation developed here to obtain the lower order nonlinear ordinary differential equations of motion of a given structure. The corresponding nonlinear dynamical responses of the structures are presented through numerical simulations using the MATLAB software. In addition, a MicroElectroMechanical System (MEMS) device is presented. The developed equations of motion have furthermore been implemented in a VHDL-AMS beam model. Together with available models of the other components, a netlist of the device is formed and simulated within an electrical circuit simulator. Simulation results are verified against Finite Element Analysis (FEA) results for this device.