Renormalization of miscible flow functions

Theory of Viscous Fingering in Two Phase, Three Component Flow

Abstract

When a mobile fluid, such as gas, displaces less mobile fluids, such as oil and water, a viscous instability develops and the injected fluid fingers through the system leading to early breakthrough and poor recovery. In a WAG process (Water Alternate Gas) water and a miscible solvent (gas) are injected into a reservoir containing water and oil. Compared with a miscible flood, when only solvent is injected, fingering can be supressed by the simultaneous injection of water, if it reduces the apparent mobility contrast between the injected and displaced fluids. In order to understand and control this instability, it is necessary to be able to predict the expected extent of fingering. The behaviour of a miscible flood, with only hydrocarbon flowing, can be modelled successfully using a Todd and Longstaff fractional flow. We modify the Todd and Longstaff model self-consistently to account for fingering in multi-component systems. The method is verified for WAG floods. The empirical flow equations are solved exactly in one dimension and the results are in excellent agreement with the averaged saturation and concentration profiles computed using two dimensional high resolution simulation, for a variety of injected water saturations, in both secondary and tertiary displacements.

Introduction

In miscible floods where only a single, hydrocarbon, phase is flowing, the degree of fingering can be successfully phase is flowing, the degree of fingering can be successfully predicted using empirical models for an averaged solvent predicted using empirical models for an averaged solvent fractional flow and phase mobility. The Todd and Longstaff approach, involves only one parameter, the effective mobility ratio, M eff (or equivalently, the mixing exponent). Using = 2/3, Todd and Longstaff were able to match the oil recovery from experiments on sand packs at several oil/solvent viscosity ratios. In secondary and tertiary WAG floods Christie and Newley have shown using direct simulation that the Todd and Longstaff parameter needs to be modified. In one special cue, matched velocity WAG, they were able to recalibrate successfully, but there was no unifying theory to account for fingering at any injection ratio. We assume that the Todd and Longstaff model can describe fingering in multicomponent flow and derive a prediction for the effective mobility by finding the total mobility contrast across the unstable front. The theoretical one dimensional saturation and concentration profiles for WAG floods are compared with the results of high resolution simulations in two dimensions. The simulations are performed on a very fine mesh, which is necessary to resolve accurately the viscous fingering. For two component flows, the simulator has been validated by comparison with experiments in bead packs and thin hetereogeneous rock slabs. The computed averaged profiles are in excellent agreement with the theoretical predictions for secondary and tertiary floods over a range of WAG ratios. The empirical models derived here can be used in full field simulators to account for sub-grid scale fingering. Alternatively, in a homogeneous system we can predict the averaged saturation and concentration profiles and production curves.