Streamline simulators are gaining increasing acceptance because of their computational advantage, intuitive appeal and ability to visualize the flow patterns in 3D. Much of the current industry experience has been limited to black oil simulation, specifically two phase water-oil displacements under incompressible or slightly compressible conditions. However, given the favorable computational scaling properties of streamline models, the potential advantage for compositional simulation can be even more compelling. Although several papers have discussed compositional streamline formulation, they all suffer from a major limitation, particularly for compressible flow. The decoupling of the compositional pressure and transport equations, which form the basis for streamline simulation, has been accomplished under severely restrictive assumptions.
In this paper, we examine the implications of these assumptions on the accuracy of compositional streamline simulation using a novel and rigorous treatment of compressibility. Our compositional streamline formulation builds on the recent work of Cheng et al.1 and uses the concept of effective density to redefine the bi-streamfunctions and decouple the 3D conservation equations to 1D transport equations for overall compositions along streamlines. The effective density accounts for volume changes with pressure and can be conveniently traced along streamlines. The streamline equations are solved using a third order TVD scheme to minimize numerical dispersion. The phase compositions and saturations are obtained via thermodynamic flash calculations.
We propose a novel ‘optimal’ gridding strategy for efficient solution of the 1D compositional equations along streamlines. The approach is based on a bias-variance trade-off of ‘slowness’ along streamlines and adaptively coarsens the grid for the 1D solution without compromising accuracy. This significantly reduces the number of flash calculations during the 1D solutions. We compare the results from our proposed formulation with finite difference simulation and also, existing compositional streamline formulations in the literature to highlight the importance of the rigorous treatment of compressibility effects. Finally, we examine the scaling behavior of the streamline and finite difference models for compositional simulation to identify potential computational advantages for large-scale field applications.
In many applications, a black oil representation of the reservoir fluids is inadequate. These include depletion of gas condensate and volatile oil reservoirs and also enhanced oil recovery processes such as enriched miscible gas injection, carbon dioxide flooding and chemical flooding. Specifically, when the fluid properties are dependent on both phase composition and pressure, we have to resort to compositional simulation. Such simulations involve the solution of the mass conservation equation in conjunction with phase equilibrium calculations to determine phase compositions, phase pressures and saturations.2–7 The additional capabilities of compositional simulation also make it more expensive in terms of computation time and memory. This makes the potential benefit of streamline based compositional simulation even more compelling than for black oil or for two phase waterflood.
We already know that streamline models in many situations can outperform conventional finite difference simulation in terms of computational speed. However, most applications have been limited to incompressible or slightly compressible flow and under convection dominated flow conditions.5–14 The underlying incompressibility assumption allows us to easily decouple the pressure and conservation equations by introducing a time of flight coordinate.10 By applying the time of flight as a spatial coordinate, the multi-dimensional conservation equations are reduced to a series of 1D equations along streamlines and can be solved using a relatively large time step compared to the original grid block based equations. The streamline-based solution can also reduce numerical dispersion and grid orientation effects leading to improved accuracy of the solution.