Crossover from Nonlinearity Controlled to Heterogeneity Controlled Mixing in Two-Phase Porous Media Flows

We use high resolution Monte Carlo simulations to study the dispersive mixing in two-phase, immiscible, porous media flow that results from the interaction of the nonlinearities in the flow equations with geologic heterogeneity. Our numerical experiments show that distinct dispersive regimes occur depending on the relative strength of nonlinearity and heterogeneity. In particular, for a given degree of multiscale heterogeneity, controlled by the Hurst exponent which characterizes the underlying stochastic model for the heterogeneity, linear and nonlinear flows are essentially identical in their degree of dispersion, if the heterogeneity is strong enough. As the heterogeneity weakens, the dispersion rates cross over from those of linear heterogeneous flows to those typical of nonlinear homogeneous flows.     

Modelling Two-Phase Incompressible Flow in Porous Media Using Mixed Hybrid and Discontinuous Finite Elements

In this paper, we present a numerical model for simulating two-phase (oil–water and air–water) incompressible and immiscible flow in porous media. The mathematical model which is based on a fractional flow formulation is formed of two nonlinear partial differential equations: a mean pressure equation and a water saturation equation. These two equations can be solved in a sequential manner. Two numerical methods are used to discretize the equations of the two-phase flow model: mixed hybrid finite elements are used to treat the pressure equation, h-based Richards' equation and the diffusion term in the saturation equation, the advection term in the saturation equation is treated with the discontinuous finite elements. We propose a better way to calculate the nonlinear coefficients contained in our equations on each element of the discretized domain. In heterogeneous porous media, the saturation becomes discontinuous at the interface between two porous media. We show in this paper how to use the capillary pressure–saturation relationship in order to handle the saturation jump in the mixed hybrid finite element method. The two-phase flow simulator is verified against analytical solutions for some flow problems treated by other authors.     

Numerical Homogenization of Two-Phase Flow in Porous Media

Homogenization has proved its effectiveness as a method of upscaling for linear problems, as they occur in single-phase porous media flow for arbitrary heterogeneous rocks. Here we extend the classical homogenization approach to nonlinear problems by considering incompressible, immiscible two-phase porous media flow. The extensions have been based on the principle of preservation of form, stating that the mathematical form of the fine-scale equations should be preserved as much as possible on the coarse scale. This principle leads to the required extensions, while making the physics underlying homogenization transparent. The method is process-independent in a way that coarse-scale results obtained for a particular reservoir can be used in any simulation, irrespective of the scenario that is simulated. Homogenization is based on steady-state flow equations with periodic boundary conditions for the capillary pressure. The resulting equations are solved numerically by two complementary finite element methods. This makes it possible to assess a posteriori error bounds.     

低渗透岩芯水驱油试验相似理论

以相似理论为基础,考虑低渗透多孔介质非达西渗流特性,研究了油水两相渗流模拟试验理论。应用方程分析法,推导出油水两相渗流控制方程的无量纲形式,获得了低渗透岩芯水驱油模拟试验的相似准则。采用隐式求解压力、显式求解饱和度的方法（IMPES）对无量纲控制方程进行求解,得到包含相似准数的数值模拟器。通过敏感性分析,得到了每个相似准数对试验结果的影响程度,并通过实际低渗透岩芯的水驱油试验结果进行了验证。结果表明,敏感因子大的相似准数对试验结果影响较大,反之影响则较小。因此,在设计物理模拟试验时,当无法满足模型和原型所有相似准数相等时,应该优先满足敏感因子较大的相似准数相等。     

Modeling compositional compressible two-phase flow in porous media by the concept of the global pressure

We derive a new formulation for the compositional compressible two-phase flow in porous media. We consider a liquid–gas system with two components: water and hydrogen. The formulation considers gravity, capillary effects, and diffusivity of each component. The main feature of this formulation is the introduction of the global pressure variable that partially decouples the system equations. To formulate the final system, and in order to avoid primary unknowns changing between one-phase and two-phase zones, a second persistent variable is introduced: the total hydrogen mass density. The derived system is written in terms of the global pressure and the total hydrogen mass density. The system is capable of modeling the flows in both one and two-phase zones with no changes of the primary unknowns. The mathematical structure is well defined: the system consists of two nonlinear parabolic equations, the global pressure equation, and the total hydrogen mass density equation. The derived formulation is fully equivalent to the original one. Numerical simulations show ability of this new formulation to model efficiently the phase appearance and disappearance.     

Thermo-hydro-mechanical modeling of fracturing porous media with two-phase fluid flow using X-FEM technique

In this paper, a fully coupled thermo-hydro-mechanical model is presented for two-phase fluid flow and heat transfer in fractured/fracturing porous media using the extended finite element method. In the fractured porous medium, the traction, heat, and mass transfer between the fracture space and the surrounding media are coupled. The wetting and nonwetting fluid phases are water and gas, which are assumed to be immiscible, and no phase-change is considered. The system of coupled equations consists of the linear momentum balance of solid phase, wetting and nonwetting fluid continuities, and thermal energy conservation. The main variables used to solve the system of equations are solid phase displacement, wetting fluid pressure, capillary pressure, and temperature. The fracture is assumed to impose the strong discontinuity in the displacement field and weak discontinuities in the fluid pressure, capillary pressure, and temperature fields. The mode I fracture propagation is employed using a cohesive fracture model. Finally, several numerical examples are solved to illustrate the capability of the proposed computational algorithm. It is shown that the effect of thermal expansion on the effective stress can influence the rate of fracture propagation and the injection pressure in hydraulic fracturing process. Moreover, the effect of thermal loading is investigated properly on fracture opening and fluids flow in unsaturated porous media, and the convective heat transfer within the fracture is captured successfully. It is shown how the proposed computational model is capable of modeling the fully coupled thermal fracture propagation in unsaturated porous media.     

Numerical modeling of multiphase fluid flow in deforming porous media: A comparison between two- and three-phase models for seismic analysis of earth and rockfill dams

In this paper, a fully coupled numerical model is presented for the finite element analysis of the deforming porous medium interacting with the flow of two immiscible compressible wetting and non-wetting pore fluids. The governing equations involving coupled fluid flow and deformation processes in unsaturated soils are derived within the framework of the generalized Biot theory. The displacements of the solid phase, the pressure of the wetting phase and the capillary pressure are taken as the primary unknowns of the present formulation. The other variables are incorporated into the model using the experimentally determined functions that define the relationship between the hydraulic properties of the porous medium, i.e. saturation, relative permeability and capillary pressure. It is worth mentioning that the imposition of various boundary conditions is feasible notwithstanding the choice of the primary variables. The modified Pastor–Zienkiewicz generalized constitutive model is introduced into the mathematical formulation to simulate the mechanical behavior of the unsaturated soil. The accuracy of the proposed mathematical model for analyzing coupled fluid flows in porous media is verified by the resolution of several numerical examples for which previous solutions are known. Finally, the performance of the computational algorithm in modeling of large-scale porous media problems including the large elasto-plastic deformations is demonstrated through the fully coupled analysis of the failure of two earth and rockfill dams. Furthermore, the three-phase model is compared to its simplified one which simulates the unsaturated porous medium as a two-phase one with static air phase. The paper illustrates the shortcomings of the commonly used simplified approach in the context of seismic analysis of two earth and rockfill dams. It is shown that accounting the pore air as an independent phase significantly influences the unsaturated soil behavior.     

Towards a thermodynamics of immiscible two-phase steady-state flow in porous media

We propose that steady-state two-phase flow in porous media may be described through a formalism closely resembling equilibrium thermodynamics. This leads to a Monte Carlo method that will be highly efficient in studying two-phase flow under steady-state conditions numerically. This work was partially supported by the Norwegian Research Council through grants nos. 154535/432 and 180296/S30.     

Multidimensional upstream weighting for multiphase transport in porous media

Truly multidimensional methods for hyperbolic equations use flow-based information to determine the computational stencil, as opposed to applying one-dimensional methods dimension by dimension. By doing this, the numerical errors are less correlated with the underlying computational grid. This can be important for reducing bias in flow problems that are inherently unstable at simulation scale, such as in certain porous media problems. In this work, a monotone, multi-D framework for multiphase flow and transport in porous media is developed. A local coupling of the fluxes is introduced through the use of interaction regions, resulting in a compact stencil. A relaxed volume formulation of the coupled hyperbolic–elliptic system is used that allows for nonzero residuals in the pressure equation to be handled robustly. This formulation ensures nonnegative masses and saturations (volume fractions) that sum to one (Acs et al., SPE J 25(4):543–553, 1985). Though the focus of the paper is on immiscible flow, an extension of the methods to a class of more general scalar hyperbolic equations is also presented. Several test problems demonstrate that the truly multi-D schemes reduce biasing due to the computational grid.     

Identification of diffusion parameters in a nonlinear convection–diffusion equation using the augmented Lagrangian method

Numerical identification of diffusion parameters in a nonlinear convection–diffusion equation is studied. This partial differential equation arises as the saturation equation in the fractional flow formulation of the two-phase porous media flow equations. The forward problem is discretized with the finite difference method, and the identification problem is formulated as a constrained minimization problem. We utilize the augmented Lagrangian method and transform the minimization problem into a coupled system of nonlinear algebraic equations, which is solved efficiently with the nonlinear conjugate gradient method. Numerical experiments are presented and discussed. This work was partially supported by the Research Council of Norway (NFR), under grant 128224/431.     

Node-centered finite volume discretizations for the numerical simulation of multiphase flow in heterogeneous porous media

Three node-centered finite volume discretizations for multiphase porous media flow are presented and compared. By combination of these methods two additional discretization methods are generated. The ability of these schemes to describe flows at textural interfaces of different geologic formations is investigated. It was found that models with nonzero-entry pressures for the capillary pressure-saturation relationship in conjunction with the Box discretization may give rise to spurious oscillations for flows around low permeable lenses. Furthermore, the applicability and sensitivity of the discretization methods with regard to the used computational grids is discussed. The schemes are used for the numerical study of two-phase flow in porous media with zones of different material properties. This revised version was published online in July 2006 with corrections to the Cover Date.     

The representer method for two-phase flow in porous media

The representer method is applied to a one-dimensional two-phase flow model in porous media; capillary pressure and gravity are neglected. The Euler–Lagrange equations must be linearized, and one such linearization is presented here. The representer method is applied to the linear system iteratively until convergence, though a rigorous proof of convergence is out of reach. The linearization chosen is easy to calculate but does not converge for certain weights; however, a simple damping restores convergence at the cost of extra iterations. Numerical experiments are performed that illustrate the method, and quick comparison to the ensemble Kalman smoother is made. This research was supported by NSF grant EIA-0121523.     

Analysis and numerical approximation of Brinkman regularization of two-phase flows in porous media

We consider a system of nonlinear partial differential equations that arises in the modeling of two-phase flows in a porous medium. The phase velocities are modeled using a Brinkman regularization of the classical Darcy’s law. We propose a notion of weak solution for these equations and prove existence of these solutions. An efficient finite difference scheme is proposed and is shown to converge to the weak solutions of this system. The Darcy limit of the Brinkman regularization is studied numerically using the convergent finite difference scheme in two space dimensions as well as using both analytical and numerical tools in one space dimension. The results suggest that the Brinkman regularization may not approximate the accepted entropy solutions of the Darcy model and raise fundamental questions about the use of Brinkman type models in two-phase flows.     

Adjoint analysis of Buckley-Leverett and two-phase flow equations

This paper analyzes the adjoint equations and boundary conditions for porous media flow models, specifically the Buckley-Leverett equation, and the compressible two-phase flow equations in mass conservation form. An adjoint analysis of a general scalar hyperbolic conservation law whose primal solutions include a shock jump is initially presented, and the results are later specialized to the Buckley-Leverett equation. The non-convexity of the Buckley-Leverett flux function results in adjoint characteristics that are parallel to the shock front upstream of the shock and emerge from the shock front downstream of the shock. Thus, in contrast to the behavior of Burgers’ equation where the adjoint is continuous at a shock, the Buckley-Leverett adjoint, in general, contains a discontinuous jump across the shock. Discrete adjoint solutions from space-time discontinuous Galerkin finite element approximations of the Buckley-Leverett equation are shown to be consistent with the derived closed-form analytical solutions. Furthermore, a general result relating the adjoint equations for different (though equivalent) primal equations is used to relate the two-phase flow adjoints to the Buckley-Leverett adjoint. Adjoint solutions from space-time discontinuous Galerkin finite element approximations of the two-phase flow equations are observed to obey this relationship.     

A virtual element method for the two-phase flow of immiscible fluids in porous media

A primal C⁰-conforming virtual element discretization for the approximation of the bidimensional two-phase flow of immiscible fluids in porous media using general polygonal meshes is discussed. This work investigates the potentialities of the Virtual Element Method (VEM) in solving this specific problem of immiscible fluids in porous media involving a time-dependent coupled system of non-linear partial differential equations. The performance of the fully discrete scheme is thoroughly analysed testing it on general meshes considering both a regular problem and more realistic benchmark problems that are of interest for physical and engineering applications.

     

Immiscible two-phase Darcy flow model accounting for vanishing and discontinuous capillary pressures: application to the flow in fractured porous media

Fully implicit time-space discretizations applied to the two-phase Darcy flow problem leads to the systems of nonlinear equations, which are traditionally solved by some variant of Newton’s method. The efficiency of the resulting algorithms heavily depends on the choice of the primary unknowns since Newton’s method is not invariant with respect to a nonlinear change of variable. In this regard, the role of capillary pressure/saturation relation is paramount because the choice of primary unknowns is restricted by its shape. We propose an elegant mathematical framework for two-phase flow in heterogeneous porous media resulting in a family of formulations, which apply to general monotone capillary pressure/saturation relations and handle the saturation jumps at rocktype interfaces. The presented approach is applied to the hybrid dimensional model of two-phase water-gas Darcy flow in fractured porous media for which the fractures are modelled as interfaces of co-dimension one. The problem is discretized using an extension of vertex approximate gradient scheme. As for the phase pressure formulation, the discrete model requires only two unknowns by degree of freedom.     

A pore-scale numerical model for flow through porous media

A pore-scale numerical model based on Smoothed Particle Hydrodynamics (SPH) is described for modelling fluid flow phenomena in porous media. Originally developed for astrophysics applications, SPH is extended to model incompressible flows of low Reynolds number as encountered in groundwater flow systems. In this paper, an overview of SPH is provided and the required modifications for modelling flow through porous media are described, including treatment of viscosity, equation of state, and no-slip boundary conditions. The performance of the model is demonstrated for two-dimensional flow through idealized porous media composed of spatially periodic square and hexagonal arrays of cylinders. The results are in close agreement with solutions obtained using the finite element method and published solutions in the literature. Copyright © 1999 John Wiley & Sons, Ltd.     

多孔介质油水两相k~s~p关系数学模型的实验研究

数学模型是研究相对渗透率与饱和度关系曲线的重要方法。采用自行开发设计的人工平面多孔介质模型,测定了相对渗透率与饱和度的关系曲线。多孔介质选择粒径为0.5~1mm、1~2mm的标准砂,纯净的水为湿润相,用3号苏丹红染色的93#汽油为非湿润相,组成多孔介质油水两相流动系统。采用Van Genuchten and Mualeum(VGM)和Brooks-Corey-Burdine(BCB)两种数学模型计算相对渗透率与饱和度的关系曲线,通过比较两种数学模型计算结果之间和模型计算结果与实测结果的差异以及模型的应用、多相渗流系统自身特征,得出VGM、BCB两种数学模型计算结果符合实际情况,VGM模型应用过程更为简便,但VGM模型具有一定适用条件;在砂性多孔介质中,BCB模型计算相对渗透率与饱和度关系曲线更准确。     

模拟裂隙多孔介质中变饱和渗流的广义等效连续体方法

描述了一种计算裂隙多孔介质中变饱和渗流的广义等效连续体方法。这种方法忽略裂隙的毛细作用,设定一个与某孔隙饱和度相对应的综合饱和度极限值,并假定：（1）如果裂隙多孔介质的综合饱和度小于该极限值,水只在孔隙中存在并流动,而裂隙中则没有水的流动;（2）如果综合饱和度等于或大于该极限值,水将进入裂隙,并在裂隙内运动。分析比较了等效连续体模型的不同计算方法,并给出了一个模拟裂隙岩体中变饱和渗流与传热耦合问题的应用算例。结果表明,所述方法具有一般性,可以有效地模拟裂隙多孔介质中变饱和渗流的基本特征。     

利用时间分裂的错格伪谱法模拟地震波在基于改进BISQ模型的双相介质中的传播

改进的BISQ(Biot-Squirt)模型中各参数具有明确的物理意义和可实现性,在不引入特征喷流长度的情况下可将Biot流动和喷射流动两种力学机制有机地结合起来;而高精度的地震波场数值模拟技术是研究双相介质地震波传播规律的重要手段。本文从本构方程、动力学方程和动力学达西定律出发,推导了基于改进BISQ模型的双相各向同性介质的一阶速度--应力方程组;采用时间分裂错格伪谱法求该方程组的数值解,模拟半空间及层状双相介质中的地震波场。数值模拟结果表明:①与传统方法相比,时间分裂错格伪谱法波场数值模拟的精度更高,压制网格频散效果更好;②在非黏滞相界情况下,慢纵波呈传播性,而在黏滞相界情况下,慢纵波呈扩散性,以静态模式出现在震源位置;③双相介质分界面处,各类波型复杂的反射透射规律可由数值模拟结果清晰展现。