Two-phase Flow In Heterogeneous Media Research Articles

A vertex scheme for two-phase flow in heterogeneous media

This paper presents the numerical solution of immiscible two-phase flows in porous media, obtained by a first-order finite element method equipped with mass-lumping and flux upwinding. The unknowns are the physical phase pressure and phase saturation. Our numerical experiments confirm that the method converges optimally for manufactured solutions. For both structured and unstructured meshes, we observe the high-accuracy wetting saturation profile that ensures minimal numerical diffusion at the front. Performing several examples of quarter-five spot problems in two and three dimensions, we show that the method can easily handle heterogeneities in the permeability field. Two distinct features that make the method appealing to reservoir simulators are: (i) maximum principle is satisfied, and (ii) local mass error is small and remains below nonlinear solver tolerance.

Adaptive mixed-hybrid and penalty discontinuous Galerkin method for two-phase flow in heterogeneous media

In this paper, we present a hybrid method, which consists of a mixed-hybrid finite element method and a penalty discontinuous Galerkin method, for the approximation of a fractional flow formulation of a two-phase flow problem in heterogeneous media with discontinuous capillary pressure. The fractional flow formulation is comprised of a wetting phase pressure equation and a wetting phase saturation equation which are coupled through a total velocity and the saturation affected coefficients. For the wetting phase pressure equation, the continuous mixed-hybrid finite element method space can be utilized due to a fundamental property that the wetting phase pressure is continuous. While it can reduce the computational cost by using less degrees of freedom and avoiding the post-processing of velocity reconstruction, this method can also keep several good properties of the discontinuous Galerkin method, which are important to the fractional flow formulation, such as the local mass balance, continuous normal flux and capability of handling the discontinuous capillary pressure. For the wetting phase saturation equation, the penalty discontinuous Galerkin method is utilized due to its capability of handling the discontinuous jump of the wetting phase saturation. Furthermore, an adaptive algorithm for the hybrid method together with the centroidal Voronoi–Delaunay triangulation technique is proposed. Five numerical examples are presented to illustrate the features of proposed numerical method, such as the optimal convergence order, the accurate and efficient velocity approximation, and the applicability to the simulation of water flooding in oil field and the oil-trapping or barrier effect phenomena.

Convergence of Discontinuous Galerkin Methods for Incompressible Two-Phase Flow in Heterogeneous Media

A class of discontinuous Galerkin methods with interior penalties is presented for incompressible two-phase flow in heterogeneous porous media with capillary pressures. The semidiscrete approximate schemes for fully coupled system of two-phase flow are formulated. In highly heterogeneous permeable media, the saturation is discontinuous due to different capillary pressures, and therefore, the proposed methods incorporate the capillary pressures in the pressure equation instead of saturation equation. By introducing a coupling approach for stability and error estimates instead of the conventional separate analysis for pressure and saturation, the stability of the schemes in space and time and a priori hp error estimates are presented in the $L^2(H^1)$ for pressure and in the $L^\infty(L^2)$ and $L^2(H^1)$ for saturation. Two time discretization schemes are introduced for effectively computing the discrete solutions.

Are upwind techniques in multi-phase flow models necessary?

Two alternatives of primary variables are compared for two-phase flow in heterogeneous media by solving fully established benchmarks. The first combination utilizes pressure of the wetting fluid and saturation of the non-wetting fluid as primary variables, while the second employs capillary pressure of the wetting fluid and pressure of the non-wetting fluid. While the standard Galerkin finite element method (SGFEM) is known to fail in the physical reproduction of two-phase flow in heterogeneous media (unless employing a fully upwind correction), the second scheme with capillary pressure as a primary variable without applying an upwind technique produces correct physical fluid behaviour in heterogeneous media, as observed from experiments.

Coupled simulation of near-wellbore and reservoir models

It is well known that some near-wellbore physics such as formation damage, acid stimulation or near-well reservoir heterogeneity can have a big impact on well productivity. To evaluate this impact, near-wellbore numerical models have been developed to simulate near-wellbore physics using very fine gridblocks. However, near-well models are usually developed standalone and are decoupled from reservoir models. Using a standalone near-well model, which does not take into account production scenarios, does not allow us to correctly predict well injectivity or productivity. In this paper, we present a technique for the coupled simulation of a near-well flow model and a reservoir model in a simple and consistent way. This approach is applied to multiphase flow in 3D heterogeneous media. In this approach, data between these two models are updated through boundary conditions for the near-well model and numerical productivity indices (PI) for the reservoir model. Examples, including an example of two-phase flow in heterogeneous media, simulation of drilling induced formation damage in multi-layered reservoir and water shut-off treatment for a horizontal well, show that this coupled modelling gives quite satisfactory results.

Finite volume scheme for two-phase flows in heterogeneous porous media involving capillary pressure discontinuities

We study a one-dimensional model for two-phase flows in heterogeneous media, in which the capillary pressure functions can be discontinuous with respect to space. We first give a model, leading to a system of degenerated nonlinear parabolic equations spatially coupled by nonlinear transmission conditions. We approximate the solution of our problem thanks to a monotonous finite volume scheme. The convergence of the underlying discrete solution to a weak solution when the discretization step tends to 0 is then proven. We also show, under assumptions on the initial data, a uniform estimate on the flux, which is then used during the uniqueness proof. A density argument allows us to relax the assumptions on the initial data and to extend the existence-uniqueness frame to a family of solution obtained as limit of approximations. A numerical example is then given to illustrate the behavior of the model.

Stochastic Analysis of Immiscible Two-Phase Flow in Heterogeneous Media

Summary We present a stochastic analysis of immiscible two-phase flow (Buckley–Leverett displacement) in heterogeneous reservoirs. Since the detailed spatial variabilities of reservoir properties such as permeability and porosity cannot be described deterministically, both permeability and porosity, or either, are treated as random space functions. In turn, water saturation and oil production rate are also random variables. The main purpose of this study is to estimate the saturation field (by its expected value) and the associated uncertainty (by its standard deviation) for Buckley-Leverett displacement in such random media. The expected value and standard deviation may be used to construct confidence intervals for the saturation field in a reservoir and to evaluate the risk associated with a project caused by the incomplete knowledge of reservoir properties. Through transforming to coordinates attached to streamlines, the evaluation of saturation statistical moments is simplified to that of travel time and transverse displacement probability density functions. The latter do not vary with the two-phase composition but entirely depend on the total velocity under some conditions. Through some one- and two-dimensional examples, we found that the well-known discontinuities existing in saturation profiles and oil production curves for homogeneous media disappear in the case of heterogeneous media. This is due to heterogeneity fingering, or the so-called "heterogeneity induced dispersion." The effect of heterogeneity on oil production is the earlier water breakthrough and later oil arrival. The oil production is expected to take longer in heterogeneous media than in homogeneous media.

Monte Carlo Simulation of Linear Two-Phase Flow in Heterogeneous Media

In this paper an algorithmic formulation is given for the synthesis of linear two-phase systems with inherent variability. It is assumed that uncertainties in the flow system arise due to the heterogeneities of the porous medium. Methods of stochastic mechanics are adopted to offer a novel approach, whereby the absolute permeability is taken as a spatial stochastic process with a log-normal distribution at each point of the flow domain. A correlation structure is built into the model so as to achieve a realistic representation. In addition, a constraint on the invariance of the harmonic means of the random permeabilities is attained by way of topological scaling. A computer-synthesized waterflooding experiment is conducted to explore the effect of heterogeneity on output variables such as the oil pressure head and cumulative recovery. It is observed that output uncertainties can become significant even when the heterogeneities are small. By employing a least-squares procedure, the sensitivity of relative permeabilities to induced stochastic variations is also qualitatively investigated.