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.     

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.     

Dynamic capillary effects in heterogeneous porous media

In standard multi-phase flow models on porous media, a capillary pressure saturation relationship developed under static conditions is assumed. Recent experiments have shown that this static relationship cannot explain dynamic effects as seen for example in outflow experiments. In this paper, we use a static capillary pressure model and a dynamic capillary pressure model based on the concept of Hassanizadeh and Gray and examine the behavior with respect to material interfaces. We introduce a new numerical scheme for the one-dimensional case using a Lagrange multiplier approach and develop a suitable interface condition. The behavior at the interface is discussed and verified by various numerical simulations.     

Solving the Buckley–Leverett equation with gravity in a heterogeneous porous medium

Immiscible two‐phase flow in porous media can be described by the fractional flow model. If capillary forces are neglected, then the saturation equation is a non‐linear hyperbolic conservation law, known as the Buckley–Leverett equation. This equation can be numerically solved by the method of Godunov, in which the saturation is computed from the solution of Riemann problems at cell interfaces. At a discontinuity of permeability this solution has to be constructed from two flux functions. In order to determine a unique solution an entropy inequality is needed. In this article an entropy inequality is derived from a regularisation procedure, where the physical capillary pressure term is added to the Buckley‐Leverett equation. This entropy inequality determines unique solutions of Riemann problems for all initial conditions. It leads to a simple recipe for the computation of interface fluxes for the method of Godunov. This revised version was published online in July 2006 with corrections to the Cover Date.     

Numerical modeling of two-phase hysteresis combined with an interface condition for heterogeneous porous media

This paper presents a numerical implementation of two-phase capillary hysteresis and its combination with a capillary interface condition for the treatment of heterogeneities. The hysteresis concepts chosen in this work are first implemented in a node-centered FV discretization scheme and subsequently combined with the interface condition that predicts sharp saturation discontinuities at material interfaces, based on a pressure equilibrium concept. This approach allows for the approximation of history-dependent, and at the same time discontinuous, saturations at material interfaces. The resulting model provides a well-defined evolution of the hysteretic capillary pressure–saturation relationships at material interfaces that is independent of the grid spacing. As demonstrated with a simple 1-D example, this concept therefore offers the advantage that the solution of a two-phase flow problem involving hysteresis does not relate to the grid resolution at the material interfaces.     

Hydro‐mechanical modeling of cohesive crack propagation in multiphase porous media using the extended finite element method

In this paper, a numerical model is developed for the fully coupled hydro‐mechanical analysis of deformable, progressively fracturing porous media interacting with the flow of two immiscible, compressible wetting and non‐wetting pore fluids, in which the coupling between various processes is taken into account. The governing equations involving the coupled solid skeleton deformation and two‐phase fluid flow in partially saturated porous media including cohesive cracks are derived within the framework of the generalized Biot theory. The fluid flow within the crack is simulated using the Darcy law in which the permeability variation with porosity because of the cracking of the solid skeleton is accounted. The cohesive crack model is integrated into the numerical modeling by means of which the nonlinear fracture processes occurring along the fracture process zone are simulated. The solid phase displacement, the wetting phase pressure and the capillary pressure are taken as the primary variables of the three‐phase formulation. The other variables are incorporated into the model via the experimentally determined functions, which specify the relationship between the hydraulic properties of the fracturing porous medium, that is saturation, permeability and capillary pressure. The spatial discretization is implemented by employing the extended finite element method, and the time domain discretization is performed using the generalized Newmark scheme to derive the final system of fully coupled nonlinear equations of the hydro‐mechanical problem. It is illustrated that by allowing for the interaction between various processes, that is the solid skeleton deformation, the wetting and the non‐wetting pore fluid flow and the cohesive crack propagation, the effect of the presence of the geomechanical discontinuity can be completely captured. Copyright © 2012 John Wiley & Sons, Ltd.     

A discontinuous Galerkin method for two-phase flow in a porous medium enforcing H(div) velocityand continuous capillary pressure

We consider the slightly compressible two-phase flow problem in a porous medium with capillary pressure. The problem is solved using the implicit pressure, explicit saturation (IMPES) method, and the convergence is accelerated with iterative coupling of the equations. We use discontinuous Galerkin to discretize both the pressure and saturation equations. We apply two improvements, which are projecting the flux to the mass conservative H(div)-space and penalizing the jump in capillary pressure in the saturation equation. We also discuss the need and use of slope limiters and the choice of primary variables in discretization. The methods are verified with two- and three-dimensional numerical examples. The results show that the modifications stabilize the method and improve the solution.     

多孔介质两相系统毛细压力与饱和度关系试验研究

两相系统毛细压力-饱和度(h~S)关系曲线的确定是多孔介质多相流动研究的基础。采用简易试验装置对理想和实际介质中水-气和油-水两相系统中的h~S关系曲线进行了测定。试验结果表明,对于相同两相系统,多孔介质孔隙度愈小,同一毛细压力对应的饱和度相应愈大;对于不同两相系统,理想介质的关系曲线在一定毛细压力以下平缓,较大毛细压力时陡直,实际介质关系曲线走势相对较陡。分析结果表明,水-气和油-水两相系统的实测数据符合Parker等提出的基于van Genuchten(1980)关系式的折算理论;应用折算理论,可以在同一多孔介质某一两相系统h~S关系已知的情况下较好地估计同一孔隙度条件下其它两相系统的h~S关系曲线。     

Streamline method for resolving sharp fronts for complex two-phase flow in porous media

In this paper, we present a fast streamline-based numerical method for the two-phase flow equations in high-rate flooding scenarios for incompressible fluids in heterogeneous and anisotropic porous media. A fractional flow formulation is adopted and a discontinuous Galerkin method (DG) is employed to solve the pressure equation. Capillary effects can be neglected in high-rate flooding scenarios. This allows us to present an improved streamline approach in combination with the one-dimensional front tracking method to solve the transport equation. To handle the high computational costs of the DG approximation, domain decomposition is applied combined with an algebraic multigrid preconditioner to solve the linear system. Special care at the interior interfaces is required and the streamline tracer has to include a dynamic communication strategy. The method is validated in various two- and three-dimensional tests, where comparisons of the solutions in terms of approximation of flow front propagation with standard fully implicit finite-volume methods are provided.     

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.     

Micromechanical approach to unsaturated water flow in structured geomaterials by two-scale computations

This article presents a micromechanical approach to the problem of unsaturated water flow in heterogeneous porous media in transient conditions. The numerical formulation is based on the two-scale model obtained previously by periodic homogenization. It allows for a coupled solution of the non-linear flow equations at macroscopic and microscopic scales and takes into account the macroscopic anisotropy of the medium and the local non-equilibrium of the capillary pressure. The model was applied to simulate two-dimensional water infiltration at constant flux into an initially dry medium containing inclusions of square and rectangular shapes. The numerical results showed the influence of the inclusion–matrix conductivity ratio and the local geometry on the macroscopic behavior. The influence of the conductivity ratio manifested itself by the acceleration or retardation of the onset of the macroscopic water flux at the outlet, while the local geometry (anisotropy) significantly affected the macroscopic spatial distribution of the water flux. Such type of approach can be extended to simulate coupled phenomena (for example hydro-mechanical problems) with evolving local geometry.     

Numerical simulation of ultrasonic waves in reservoir rocks with patchy saturation and fractal petrophysical properties

We simulate the propagation of ultrasonic waves in heterogeneous poroviscoelastic media saturated by immiscible fluids. Our model takes into account capillary forces and viscous and mass coupling effects between the fluid phases under variable saturation and pore fluid pressure. The numerical problem is solved in the space–frequency domain using a finite element procedure and the time–domain solution is obtained by a numerical Fourier transform. Heterogeneities due to fluid distribution and rock porosity–permeability are modeled as stochastic fractals, whose spectral densities reproduce saturation an petrophysical variations similar to those observed in reservoir rocks. The numerical experiments are performed at a central frequency of 500 kHz, and show clearly the effects of the different heterogeneities on the amplitudes of shear and compressional waves and the importance of wave mode conversions at the different interfaces.     

Analysis of A New Model for Unsaturated Flow in Porous Media Including Hysteresis and Dynamic Effects

A new model for unsaturated flow in porous media, including capillary hysteresis and dynamic capillary effects, is analyzed. Existence and uniqueness of solutions are established and qualitative and quantitative properties of (particular) solutions are analyzed. Some results of numerical computations are given. The model under consideration incorporates simple ‘play’-type hysteresis and a dynamic term (time-derivative with respect to water content) in the capillary relation. Given an initial water content distribution, the model determines which parts of the flow domain are in drainage and which parts are in imbibition. The governing equations can be recast into an elliptic problem for fluid pressure and an evolution equation for water content. Standard methods are used to obtain numerical results. A comparison is given between J.R. Philip's semi-explicit similarity solution for horizontal redistribution in an infinite one-dimensional domain and solutions of the new model.     

Sequential implicit vertex approximate gradient discretization of incompressible two-phase Darcy flows with discontinuous capillary pressure

This work deals with sequential implicit schemes for incompressible and immiscible two-phase Darcy flows which are commonly used and well understood in the case of spatially homogeneous capillary pressure functions. To our knowledge, the stability of this type of splitting schemes solving sequentially a pressure equation followed by the saturation equation has not been investigated so far in the case of discontinuous capillary pressure curves at different rock type interfaces. It will be shown here to raise severe stability issues for which stabilization strategies are investigated in this work. To fix ideas, the spatial discretization is based on the Vertex Approximate Gradient (VAG) scheme accounting for unstructured polyhedral meshes combined with an Hybrid Upwinding (HU) of the transport term and an upwind positive approximation of the capillary and gravity fluxes. The sequential implicit schemes are built from the total velocity formulation of the two-phase flow model and only differ in the way the conservative VAG total velocity fluxes are approximated. The stability, accuracy and computational cost of the sequential implicit schemes studied in this work are tested on oil migration test cases in 1D, 2D and 3D basins with a large range of capillary pressure parameters for the drain and barrier rock types. It will be shown that usual splitting strategies fail to capture the right solutions for highly contrasted rock types and that it can be fixed by maintaining locally the pressure saturation coupling at different rock type interfaces in the definition of the conservative total velocity fluxes. The numerical investigation of the sequential schemes is also extended to the widely used finite volume Two-Point Flux Approximation spatial discretization.

     

岩石裂隙毛管压力-饱和度关系曲线的试验研究

介绍了三峡花岗岩体裂隙毛管压力-饱和度试验。试验采用互不溶混驱替法。试验结果表明,在渗流基本特征方面,裂隙非饱和渗流毛管压力-饱和度关系曲线与空隙介质水分特征曲线具有相似性,如毛管压力-饱和度关系曲线的滞后现象;湿润流体(水)的排泄曲线具有进气压和束缚水饱和度;非湿润流体的吸湿曲线具有残余饱和度。这种相似性表明,孔隙介质非饱和渗流的研究成果可用于裂隙非饱和渗流,孔隙介质水分特征曲线的解析模型,可用于研究裂隙毛管压力-饱和度关系曲线和拟合毛管压力-饱和度排泄曲线的试验数据。     

Numerical modeling of two-phase binary fluid mixing using mixed finite elements

Diffusion coefficients of dense gases in liquids can be measured by considering two-phase binary nonequilibrium fluid mixing in a closed cell with a fixed volume. This process is based on convection and diffusion in each phase. Numerical simulation of the mixing often requires accurate algorithms. In this paper, we design two efficient numerical methods for simulating the mixing of two-phase binary fluids in one-dimensional, highly permeable media. Mathematical model for isothermal compositional two-phase flow in porous media is established based on Darcy’s law, material balance, local thermodynamic equilibrium for the phases, and diffusion across the phases. The time-lag and operator-splitting techniques are used to decompose each convection–diffusion equation into two steps: diffusion step and convection step. The Mixed finite element (MFE) method is used for diffusion equation because it can achieve a high-order and stable approximation of both the scalar variable and the diffusive fluxes across grid–cell interfaces. We employ the characteristic finite element method with moving mesh to track the liquid–gas interface. Based on the above schemes, we propose two methods: single-domain and two-domain methods. The main difference between two methods is that the two-domain method utilizes the assumption of sharp interface between two fluid phases, while the single-domain method allows fractional saturation level. Two-domain method treats the gas domain and the liquid domain separately. Because liquid–gas interface moves with time, the two-domain method needs work with a moving mesh. On the other hand, the single-domain method allows the use of a fixed mesh. We derive the formulas to compute the diffusive flux for MFE in both methods. The single-domain method is extended to multiple dimensions. Numerical results indicate that both methods can accurately describe the evolution of the pressure and liquid level.     

Numerical solutions for flow in porous media

A numerical approach is proposed to model the flow in porous media using homogenization theory. The proposed concept involves the analyses of micro‐true flow at pore‐level and macro‐seepage flow at macro‐level. Macro‐seepage and microscopic characteristic flow equations are first derived from the Navier–Stokes equation at low Reynolds number through a two‐scale homogenization method. This homogenization method adopts an asymptotic expansion of velocity and pressure through the micro‐structures of porous media. A slightly compressible condition is introduced to express the characteristic flow through only characteristic velocity. This characteristic flow is then numerically solved using a penalty FEM scheme. Reduced integration technique is introduced for the volumetric term to avoid mesh locking. Finally, the numerical model is examined using two sets of permeability test data on clay and one set of permeability test data on sand. The numerical predictions agree well with the experimental data if constraint water film is considered for clay and two‐dimensional cross‐connection effect is included for sand. Copyright © 2003 John Wiley & Sons, Ltd.     

非离子表面活性剂对两相系统毛细压力和饱和度关系的影响

表面活性剂至水相的引入将对原先包含纯水两相系统的毛细压力和饱和度关系存在影响.对含有非离子表面活性剂Triton X-100的水-气和水-油两相系统中的毛细压力-饱和度关系曲线进行了试验测定.试验结果表明,同不含表面活性剂的纯水系列相比,在同一饱和度情况下,含有表面活性剂的Triton X-100(0.1%)系列对应的毛细压力水头值都有不同程度的减小,说明在Triton X-100存在的情况下,驱替出同样数量的湿润相体积所需的毛细压力值较小.以van Genuchten关系式为基础的拟合结果表明,在已知纯水系列毛细压力饱和度关系的情况下,对于Triton X-100-气系统的毛细压力饱和度关系,考虑界面张力降低作用引入折算系数得到的拟合值更接近于真实值;而对于Triton X-100-油系统拟合值接近真实值的程度则随多孔介质的不同而有所不同.     

Hybrid discretization of multi-phase flow in porous media in the presence of viscous,gravitational, and capillary forces

Multi-phase flow in porous media in the presence of viscous, gravitational, and capillary forces is described by advection diffusion equations with nonlinear parameters of relative permeability and capillary pressures. The conventional numerical method employs a fully implicit finite volume formulation. The phase-potential-based upwind direction is commonly used in computing the transport terms between two adjacent cells. The numerical method, however, often experiences non-convergence in a nonlinear iterative solution due to the discontinuity of transmissibilities, especially in transition between co-current and counter-current flows. Recently, Lee et al. (Adv. Wat. Res. 82, 27–38, 2015) proposed a hybrid upwinding method for the two-phase transport equation that comprises viscous and gravitational fluxes. The viscous part is a co-current flow with a one-point upwinding based on the total velocity and the buoyancy part is modeled by a counter-current flow with zero total velocity. The hybrid scheme yields C¹-continuous discretization for the transport equation and improves numerical convergence in the Newton nonlinear solver. Lee and Efendiev (Adv. Wat. Res. 96, 209–224, 2016) extended the hybrid upwind method to three-phase flow in the presence of gravity. In this paper, we present the hybrid-upwind formula in a generalized form that describes two- and three-phase flows with viscous, gravity, and capillary forces. In the derivation of the hybrid scheme for capillarity, we note that there is a strong similarity in mathematical formulation between gravity and capillarity. We thus greatly utilize the previous derivation of the hybrid upwind scheme for gravitational force in deriving that for capillary force. Furthermore, we also discuss some mathematical issues related to heterogeneous capillary domains and propose a simple discretization model by adapting multi-valued capillary pressures at the end points of capillary pressure curves. We demonstrate this new model always admits a consistent solution that is within the discretization error. This new generalized hybrid scheme yields a discretization method that improves numerical stability in reservoir simulation.