Steady State Solutions of Immiscible Two-Phase Flow in Strongly Heterogeneous Media: Implications for Upscaling

Abstract

Abstract Steady state solutions of the classical equations for immiscible two phase flow (mass balance and Darcy's Law) have been used successfully in numerical upscaling in the limits of viscous or capillary dominated flow. At low flow rates in strongly water wet or mixed wet media, capillary pressure moves the fluids rapidly at the local scale. For viscous dominated flow, once the flood front has passed, local displacement is fast compared to the change in fractional flow. For upscaling, this fractional flow can be considered as a local boundary condition. The advantage of these steady state methods is that they are much faster than conventional dynamic methods such as those of Kyte and Berry (1975), Stone (1991), requiring what is effectively a single phase solution to the flow equations. More often, flow rates are between extremes and the above non-dynamic methods are no longer applicable. Steady state solutions can still be obtained though the numerical methods are more elaborate. Dynamic simulation can be performed until steady state is achieved with constant fractional flow boundary conditions. However, this is more time consuming than that required for conventional dynamic methods and thus a more direct method is required. We present a fully implicit method which will obtain steady state solutions directly and thus improve the speed of calculation. Simulations involving strong capillary forces depend heavily on internal boundary conditions defined by rapid change of either permeability or wettabillity. Continuity of capillary pressure must occur across these boundaries. The exception to this condition occurs for oil trapping (Van Duijn et al, 1995). For water wet rocks, this will happen for flow from high to low permeability rocks and the hydrocarbon will become trapped (Figure 1). In these conditions, the final steady state can be dependent on path and initial conditions. We demonstrate that capillary trapping depends not only on the form of the capillary pressure curves but also on the model boundary conditions as they vary in time. Results from the direct implicit steady state solver mentioned above are compared to those from dynamic simulation of waterflooding (Figure 2). Effective relative permeability curves from steady state upscaling techniques are compared to contrast the different degrees of capillary trapping. It is shown that fully implicit steady state solutions will provide a good approximation to the full fine scale model (Figure 3).