Geir Terje Eigestad

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.

Symmetry and M-Matrix Issues for the O-Method on an Unstructured Grid

More sophisticated discretization methods than the traditional control-volume finite-difference methods, have been proposed by Aavatsmark et al. in recent papers for solving the mass balance equations for porous media flow. These methods are based on a local representation of fluxes across cell-edges of control volumes (CVs). This paper will focus on mathematical properties of the discrete operator that arises when an elliptic term of the form −∇⋅(K∇p) is discretized based on these discretization principles.

Numerical Modelling of Capillary Transition Zones

For a large number of reservoirs, a vertical transition zone between water and oil exists. In this zone, both water saturation and capillary pressure vary with height. Traditionally one assumes that there is a relation between capillary pressure and water saturation, given by a Pc-S primary drainage curve prior to any production in the reservoir. The vertical fluid distribution is found by assuming equilibrium between capillary forces and gravity. This paper will focus on numerical modelling of the fluid distribution as production is started in a reservoir. As wells may be open and shut during the (field) lifetime of a reservoir, both imbibition and drainage may occur in different parts of a reservoir. The numerical model will take into account the irreversibility of imbibition and drainage, commonly known as hysteresis, which applies for both capillary pressure and relative permeability. Our test examples will deal with three different production rate regimes; capillary-dominated, capillary-viscous and viscous. We investigate the fluid distribution in the transition zones as the wells are shut down and equilibrium again is reached for the different cases. The tests will show that the fluid distribution differs for different injection- and production rates. For the case where the production in the reservoir is very close to equilibrium, we also show how the fluid distribution can be found analytically.

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.

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.

Streamline methods on fault adapted grids for risk assessment of storage of CO2 in geological formations

Streamline methods have shown to be effective for reservoir characterization and simulation. In this work we will develop methodology which allows for tracing of streamlines in fractured or faulted media including anthropogenic faults such as abandoned wells. The basis for a streamline method is a sequential splitting of the coupled pressure and saturation equations. A mass-conservative discretization, which handles general faulted grids in a consistent manner, will be used for the pressure equation. The fact that the saturation equation is solved along the streamlines, makes accurate tracing of streamlines essential. In earlier work we have developed streamline tracing on structured and unstructured matching grids. Here we present an extension to grids adapting to faulted media. For CO2 injection, probabilistic analysis within a risk assessment framework requires multiple realizations. Therefore, fast numerical methods, such as streamline simulation, are needed for screening. The work is motivated in part by the need to assess potential of geological storage of CO2 and is also highly relevant for reservoir simulation.