A Streamline Based Approach to Solution of Three-Phase Flow

Abstract

Two methods for calculation of three-phase compressible flow in a porous media using streamlines are presented. For simplicity, gravity and capillary effects are neglected. Introduction Various aspects of streamline computations have been reported in a number of recent papers. A review of the technique was given by King et.al.[7]. As many other authors [5][9][13], he emphasis on incompressible two-phase flow. Lately, however there have been a few contributions to the field addressing compressible flow [3][10]. In this paper, we consider three-phase compressible flow. The reason why compressibility has been neglected by most authors in the field, is that it represent a strong coupling between the pressure and the saturation equations. In a streamline method, the pressure is calculated first and defines the streamlines. The saturations are then propagated along those streamlines. To obtain a stable solution using an explicit finite difference method (FDM), there is a strong limitation on the time step length. Thus, it is crucial for the efficiency of the streamline method to establish a time stepping sequence where only a small number of pressure updates is needed compared to the number of time steps required by the saturation solver. While this has been done successfully for years for incompressible flow [4], it is still a challenge to obtain both accuracy and efficiency for compressible flow. We present two different approaches to handle the couplings between pressure and saturation. First, we describe a sequential IMPES type method, where we do additional steps for reducing the mass discrepancy error. Then an implicit method for both pressure and saturation along the streamlines is presented. The results are compared to the solutions obtained using an existing black-oil simulator. The Governing Equations Consider three-phase compressible flow in a porous media. For simplicity, the effects of gravity are neglected. Gravity can be accounted for using operator splitting [5] and introducing solvers for three-phase flow along gravity lines. This work is in progress, but will not be the issue of this paper. Also, capillary forces are neglected, since physical effects transverse to streamlines complicates the streamline approach. We will study a black-oil model, with three phases and three components. We will allow gas to dissolve in the oil phase while we assume the water and gas phase to consist of only water and gas respectively. The component conservation equations now read [11] ∂∂t(ϕSwbw)​∇⋅(bwfwv→t)=qw∂∂t(ϕSobo)​∇⋅(bofov→t)=qo∂∂t(ϕ(Sgbg​RsSobo))​∇⋅(bgfgv→t​Rsbofov→t)=qg(1) for water, oil and gas respectively. By summing up the component conservation equations, we obtain the pressure equation ct∂P∂t​∇⋅v→t=Q−b→⋅∇P(2) where v→t=λt∇P(3) is the Darcy velocity. Refer to the nomenclature for an explanation on notation.

Acceleration of Streamline Simulation Using Adaptive Mesh Refinement Along Streamlines

“…where G U, are given by (6) for the two-phase case and by (7) for the three-phase case, respectively. Upwinding in (8) is reflected by the fact that the fluxes are always evaluated at the left state (all characteristic speeds are non-negative).…”

“…A proven numerical method for simulation of large geological models is the streamline method [1][2][3][4][5][6][7][8][9][10]. The method employs the traditional pressure-saturation formulation of the flow in porous media [11].…”

Darcy-Stokes Streamline Simulation for the Tahe-Fractured Reservoir With Cavities

“…A streamline simulation method is then less sensitive to grid size and orientation. It has experienced a fast growth particularly in the recent decade (Pollock 1988;Bratvedt et al 1992;King et al 1993;Datta-Gupta and King 1995;Bratvedt et al 1996;Batycky et al 1997;Blunt et al 1996;Thiele et al 1996;Thiele et al 1997;Christie and Clifford 1997;King and Datta-Gupta 1998;Ingebrigtsen et al 1999 Donato and Blunt 2004;Cheng et al 2006). In particular, Batycky et al (1997) gave the history of streamline technical development; a streamline technical review was presented in the papers by Datta-Gupta (1998), Datta-Gupta (2000), and Thiele (2001); Di Donato and Blunt (2004) studied streamline simulation in a dual-porosity model; and Cheng et al (2006) provided streamline simulation in compressible fluids.…”

Streamline Simulation with Compressibility and Spatial and Dynamic Variation in Oil Composition

“…Despite several studies in this area (Thiele et al 1997;Cheng et al 2005;Kozlova et al 2006;Ingebrigtsen et al 1999;Osako and Datta-Gupta, 2007), most present practical applications are still based on the assumption of incompressibility or rate-controlled wells. To extend streamline simulation with oil composition variation to compressible cases, a formulation similar to the one proposed by Cheng et al (2006) and Osako and Datta-Gupta (2007) has been used.…”