Consistent and Efficient Representation of Diffusive Mass Transfer in Fractured Reservoirs
Authors: Wu Cheng , Jessen Kristian 
Published: August 2020

Abstract
The performance of gas injection EOR in fractured reservoirs can be dictated by diffusive mass transfer between the fracture/matrix segments, particularly for high fracture-intensity and low matrix-permeability.
The classical Fick's law approach, with constant diffusion coefficients, is still a popular model for representing molecular diffusion in simulations, where the dragging effects between components are neglected. Another popular approach is based on the generalized Fick's law combined with the Maxwell-Stefan (MS) model, where component interactions are accounted for, and diffusion coefficients depend on composition, temperature and pressure.
Representation of the driving force for diffusive mass transfer is commonly done by one of the two approaches: Gradients in concentration or gradients in fugacity (or chemical potential). However, the accuracy and consistency of these are rarely studied. In addition, the gas-oil interphase diffusion, at the fracture-matrix interface, is normally simplified as inner-phase diffusion or modeled by assuming an average interface composition: Neither has a sound physical basis. Here, we provide a detailed analysis of the accuracy, consistency and efficiency of commonly used models to facilitate fit-for-purpose forecasting of EOR operations in fractured systems.
We consider a dual-porosity reservoir model in which the generalized Fick's law with diffusion coefficients from the MS approach is used to describe molecular diffusion. Gas-oil diffusion across the fracture-matrix interface is modeled based on two-film theory, where gas in fractures and oil in the matrix are assumed to equilibrate at the interface. The two formulations of driving forces for diffusion: concentration gradients (iterative approach) and chemical potential gradients (non-iterative approach), are compared in terms of CPU requirement as well as the impact of component sorting on the accuracy/consistency of the diffusive fluxes. In the application of film theory, proper calculation of the diffusion coefficients is discussed/demonstrated relative to the common application of bulk phase compositions or averaging across the gas/oil interface.
Single matrix block and field-scale calculation examples are provided to demonstrate that the generalized Fick's law and a comprehensive model for gas-oil interface diffusion, provide results that are significantly different from the conventional approaches. Furthermore, we demonstrate that diffusive fluxes, based on gradients in fugacity (non-iterative approach), are inconsistent with respect to the component sorting. We propose a strategy to mitigate this inconsistency, and demonstrate that adequate selection of compositions, used in evaluating diffusion coefficients, can resolve the inconsistency.
This work carefully analyzes the modeling errors and pitfalls related to the representation of molecular diffusion via the generalized Fick's law. Two driving-force models are compared in detail, and the inconsistency of the computationally efficient chemical potential approach is discussed. An approach to resolve the inconsistency is proposed/validated towards more accurate, efficient and consistent simulation of diffusive mass transfer in fractured reservoirs.