A fully-coupled discontinuous Galerkin method for two-phase flow in porous media with discontinuous capillary pressure

In this paper, we formulate and test numerically a fully-coupled discontinuous Galerkin (DG) method for incompressible two-phase flow with discontinuous capillary pressure. The spatial discretization uses the symmetric interior penalty DG formulation with weighted averages and is based on a wetting-phase potential/capillary potential formulation of the two-phase flow system. After discretizing in time with diagonally implicit Runge-Kutta schemes, the resulting systems of nonlinear algebraic equations are solved with Newton’s method and the arising systems of linear equations are solved efficiently and in parallel with an algebraic multigrid method. The new scheme is investigated for various test problems from the literature and is also compared to a cell-centered finite volume scheme in terms of accuracy and time to solution. We find that the method is accurate, robust, and efficient. In particular, no postprocessing of the DG velocity field is necessary in contrast to results reported by several authors for decoupled schemes. Moreover, the solver scales well in parallel and three-dimensional problems with up to nearly 100 million degrees of freedom per time step have been computed on 1,000 processors.     

Vertex-centred discretization of multiphase compositional Darcy flows on general meshes

This paper concerns the discretisation on general 3D meshes of multiphase compositional Darcy flows in heterogeneous anisotropic porous media. Extending Coats’ formulation [15] to an arbitrary number of phases, the model accounts for the coupling of the mass balance of each component with the pore volume conservation and the thermodynamical equilibrium and dynamically manages phase appearance and disappearance. The spatial discretisation of the multiphase compositional Darcy flows is based on a generalisation of the Vertex Approximate Gradient scheme, already introduced for single-phase diffusive problems in [24]. It leads to an unconditionally coercive scheme for arbitrary meshes and permeability tensors. The stencil of this vertex-centred scheme typically comprises 27 points on topologically Cartesian meshes, and the number of unknowns on tetrahedral meshes is considerably reduced, compared with the usual cell-centred approaches. The efficiency of our approach is exhibited on several examples, including the nearwell injection of miscible CO₂ in a saline aquifer taking into account the vaporisation of H₂O in the gas phase as well as the precipitation of salt.     

Finite volume approximation for an immiscible two-phase flow in porous media with discontinuous capillary pressure

We consider an immiscible incompressible two-phase flow in a porous medium composed of two different rocks so that the capillary pressure field is discontinuous at the interface between the rocks. This leads us to apply a concept of multivalued phase pressures and a notion of weak solution for the flow which have been introduced in Cancès and Pierre (SIAM J Math Anal 44(2):966–992, 2012). We discretize the problem by means of a numerical algorithm which reduces to a standard finite volume scheme in each rock and prove the convergence of the approximate solution to a weak solution of the two-phase flow problem. The numerical experiments show in particular that this scheme permits to reproduce the oil-trapping phenomenon.     

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 discontinuous Galerkin method for a diffuse-interface model of immiscible two-phase flows with soluble surfactant

Computational Geosciences - A numerical method using discontinuous polynomial approximations is formulated for solving a phase-field model of two immiscible fluids with a soluble surfactant. The...     

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。     

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.     

Finite element analysis of land subsidence above depleted reservoirs with pore pressure gradient and total stress formulations

The solution of the poroelastic equations for predicting land subsidence above productive gas/oil fields may be addressed by the principle of virtual works using either the effective intergranular stress, with the pore pressure gradient regarded as a distributed body force, or the total stress incorporating the pore pressure. In the finite element (FE) method both approaches prove equivalent at the global assembled level. However, at the element level apparently the equivalence does not hold, and the strength source related to the pore pressure seems to generate different local forces on the element nodes. The two formulations are briefly reviewed and discussed for triangular and tetrahedral finite elements. They are shown to yield different results at the global level as well in a three‐dimensional axisymmetric porous medium if the FE integration is performed using the average element‐wise radius. A modification to both formulations is suggested which allows to correctly solve the problem of a finite reservoir with an infinite pressure gradient, i.e. with a pore pressure discontinuity on its boundary. Copyright © 2001 John Wiley & Sons, Ltd.     

Modeling physical properties of sandstone reservoirs by blending 2D image analysis data with 3D capillary pressure data

We show that the connectivity e.g., the association between 3D pores and throat sizes, of a set of samples from a North Sea (Alwyn) sandstone reservoir cannot be modeled by taking into account only morphological criteria. This result is obtained by combining 2D image analysis data, Pore Types, and 3D capillary pressure curves using an approach developed by Ehrlich et al. (1991a–b) and McCreesh et al. (1991). Our result suggests that the permeability, in this particular reservoir, does not arise solely from the morphology of the samples.