Proper modeling of diffusion in fractured reservoirs

Diffusion may play a key role in a number of oil recovery processes such as heavy oil and naturally fractured reservoirs. In fractured media, several laboratory experiments and numerical studies showed that CO2 injection can improve recovery. Molecular diffusion, gravity drainage, and oil swelling are the main contributing mechanisms. Proper modeling of diffusion of hydrocarbon mixtures at the reservoir PVT and geological conditions is not a trivial task. The challenge is in computing the diffusion coefficients for the non-ideal multicomponent mixtures in oil and gas phases, and in physically accurate modeling of the diffusion driving force. One common approach in most simulators is to use the classical Fick's law which simplifies the multicomponent diffusion fluxes by only considering the main-diffusion (diagonal) terms and neglecting the cross-diffusion (off-diagonal) terms. The diffusion fluxes are assumed independent and the diffusion driving force of each component is proportional to the component self concentration gradient. In this work, we demonstrate analytically and numerically that this simplified approach may have a major inconsistency related the flux balance constraint and, in some applications, it may fail to capture the right direction of diffusion as a result of neglecting the dragging effect. We propose an alternative model based on the generalized Fick's law in which diffusion coefficients are calculated as a function of temperature, pressure, and composition. The proposed approach can be seen equivalent to the Maxwell-Stephan model in which the diffusion driving force is the chemical potential instead of he composition gradient. We also tackle another problem that may occur in fractured media when fractures get fully saturated with gas in an under-saturated oil surrounding. Intra-phase gas and oil diffusions will not be initiated due to the discontinuity of phases between the fracture and the rock matrix. The proposed approach in the literature that allows for direct gas-to-oil diffusion may not have a sound bases for issues related to the driving force and the estimation of the mass transfer coefficients. We provide a solution for the cross-phase diffusion flux based on the assumption of having chemical equilibrium at the gas-oil contact. Several numerical examples are provided.