Effect of Randomly Varying Capillary Pressure on Sharp Front Propagation and Instability

We analyze immiscible displacement in three-dimensional, randomly heterogeneous porous media. Previously, expressions were developed for the dynamics of such displacement using a sharp interface approach by treating capillary pressure at the front as a deterministic constant. In this work, we modify these expressions to account for randomness in the capillary pressure at the front. Assuming the capillary pressure to be correlated with permeability and porosity via the Leverett J-function, we can describe the spatial structure of the capillary pressure at the front from characteristics of the permeability and porosity spatial structures. Furthermore, we consider log permeability to be a homogeneous random field with a given mean, variance and covariance. The problem is cast in the form of integro-differential equations in which the parameters and domain of integration are random functions. These equations are then expanded in Taylor series about the mean position of the front, averaged and solved analytically for mean front position and its rate of propagation in one spatial dimension. Integro-differential equations are also developed for second ensemble moments and solved analytically in one spatial dimension for the variance of front position. Comparing results from the two modeling approaches reveals conditions under which variations in capillary pressure at the front become important. The results further show that the variations in capillary pressure at the front have a significant effect on front dynamics and instability at small propagation rates. As front propagation rate increases, this effect diminishes. Comparison with numerical Monte Carlo results indicates that our approach is accurate.