Ivar Aavatsmark

Monotonicity of control volume methods

Robustness of numerical methods for multiphase flow problems in porous media is important for development of methods to be used in a wide range of applications. Here, we discuss monotonicity for a simplified problem of single-phase flow, but where the simulation grids and media are allowed to be general, posing challenges to control-volume methods. We discuss discrete formulations of the maximum principle and derive sufficient criteria for discrete monotonicity for arbitrary nine-point control-volume discretizations for conforming quadrilateral grids in 2D. These criteria are less restrictive than the M-matrix property. It is shown that it is impossible to construct nine-point methods which unconditionally satisfy the monotonicity criteria when the discretization satisfies local conservation and exact reproduction of linear potential fields. Numerical examples are presented which show the validity of the criteria for monotonicity. Further, the impact of nonmonotonicity is studied. Different behavior for different discretization methods is illuminated, and simple ideas are presented for improvement in terms of monotonicity.

Numerical Convergence of the MPFA O-Method for General Quadrilateral Grids in Two and Three Dimensions

This paper presents the MPFA O-method for quadrilateral grids, and gives convergence rates for the potential and the normal velocities. The convergence rates are estimated from numerical experiments. If the potential is in H 1+α , α>0, the found L 2 convergence order on rough grids in physical space is min{2, 2α} for the potential and min{1, α} for the normal velocities. For smooth grids the convergence order for the normal velocities increases to min{2,α}. The O-method is exact for uniform flow on rough grids. This also holds in three dimensions, where the cells may have nonplanar surfaces.

Sufficient criteria are necessary for monotone control volume methods

Control volume methods are prevailing for solving the potential equation arising in porous media flow. The continuous form of this equation is known to satisfy a maximum principle, and it is desirable that the numerical approximation shares this quality. Recently, sufficient criteria were derived guaranteeing a discrete maximum principle for a class of control volume methods. We show that most of these criteria are also necessary. An implication of our work is that no linear nine-point control volume method can be constructed for quadrilateral grids in 2D that is exact for linear solutions while remaining monotone for general problems.

Efficient multiphysics modelling with adaptive grid refinement using a MPFA method

A sequential solution procedure is used to simulate compositional two-phase flow in porous media. We employ a multiphysics concept that adapts the numerical complexity locally according to the underlying processes to increase efficiency. The framework is supplemented by a local refinement of the simulation grid. To calculate the fluxes on such grids, we employ a combination of the standard two-point flux approximation and a multipoint flux approximation where the grid is refined. This is then used to simulate a large-scale example related to underground CO2 storage.

Errors in the upstream mobility scheme for countercurrent two-phase flow in heterogeneous porous media

In reservoir simulation, the upstream mobility scheme is widely used for calculating fluid flow in porous media and has been shown feasible for flow when the porous medium is homogeneous. In the case of flow in heterogeneous porous media, the scheme has earlier been shown to give erroneous solutions in approximating pure gravity segregation. Here, we show that the scheme may exhibit larger errors when approximating flow in heterogeneous media for flux functions involving both advection and gravity segregation components. Errors have only been found in the case of countercurrent flow. The physically correct solution is approximated by an extension of the Godunov and Engquist–Osher flux. We also present a new finite volume scheme based on the local Lax–Friedrichs flux and test the performance of this scheme in the numerical experiments.

A Compact MPFA Method with Improved Robustness

MPFA methods were introduced to solve control- volume formulations on general grids. While these methods are general in the sense that they may be applied to any grid, their convergence properties vary. An important property for multiphase flow is the monotonicity of the numerical elliptic operator. In a recent paper, conditions for monotonicity on quadrilateral grids have been developed. These conditions indicate that MPFA formulations which lead to smaller flux stencils, are desirable for grids with high aspect ratio or severe skewness and for media with strong anisotropy or strong heterogeneity. We introduce a new MPFA method for quadrilateral grids termed the L-method. The methodology is valid for general media. For homogeneous media and uniform grids, this method has four-point flux stencils and seven-point cell stencils in two dimensions. The reduced stencil appears as a consequence of adapting the method to the closest neighboring cells. We have tested the convergence and monotonicity properties for this method, and compared it with the O-method. For moderate grids the convergence rates are the same, but for rough grids with large aspect ratios, the convergence of the O-methods is lost, while the L-method converges with a reduced convergence rate. The L-method has a somewhat larger monotonicity range than the O-methods, but the dominant difference is that when monotonicity is lost, the O-methods may give large oscillations, while the oscillations with the L-method are small or absent.

A generalized cubic equation of state with application to pure CO 2 injection in aquifers

A generalized cubic equation of state is given. The Peng-Robinson and the Soave-Redlich-Kwong equations are special cases of this equation. The generalized equation of state is precisely as simple and computationally efficient as these classical equations. Through comparison with the Span-Wagner equation for CO 2, we obtain an improved density accuracy in predefined temperature-pressure domains. The generalized equation is then verified through two relevant examples of CO 2 injection and migration. Comparisons are made with other standard cubic EOS in order to show the range of solutions obtained with less accurate EOS.

Interpretation of well-cell pressures on hexagonal grids in numerical reservoir simulation

Peaceman’s equivalent well-cell radius for 2D square grids has been generalized to 2D grids consisting of regular hexagons. The development consists of the following steps. Firstly, the analytical solution for the pressure drop between injector and producer for wells in a seven-spot pattern is determined. Secondly, this solution is compared with the numerical solution on hexagonal grids for a sixth of a seven-spot pattern. Finally, the equivalent well-cell radius is calculated, and its asymptotic behavior for infinitely fine grids is derived. The results are valid for both steady-state and unsteady-state conditions.

Interpretation of well-cell pressures on stretched hexagonal grids in numerical reservoir simulation

Peaceman’s equivalent well-cell radius for 2D Cartesian grids has been generalized to 2D uniform Voronoi grids consisting of stretched hexagons in an isotropic medium. An analytical expression for the equivalent well-cell radius for infinitely fine grids is derived. The derivation is performed by comparison of analytical and numerical solution for boundary value problems with one or two wells. Since the well-cell radius varies slowly with the grid fineness, the found formula can be considered representative for all grid sizes.

Adaptive streamline tracing for streamline simulation on irregular grids

Streamline methods have shown to be effective for reservoir simulation. Streamline simulation relies on an efficient an accurate calculation of streamlines and time-of-flight coordinates (TOF). Streamlines are commonly computed on a cell-by-cell basis using a flux interpolation in the semi-analytical Pollocks method. An alternative method is a grid-cell corner-point velocity interpolation, which is able to reproduce uniform flow in three spatial dimensions. For this method numerical integration of streamlines is required. The shape of streamlines are affected by a change in velocity direction over a cell, whereas TOF is sensitive to variation in the absolute value of the velocity. Hence, an adaptive method should be used to control the error in the numerical integration of the velocity. In this work, we propose a method for adaptive step size selection for numerical integration of streamlines, and compare it with existing methods.