Impacts of Concentration Dependence of Diffusion Coefficient on VAPEX Drainage Rates

Abstract The Vapex analytical model is extended to cover situations when diffusion coefficients are dependent on concentration due to the extreme viscosity contrasts between the solutes and solvents. The new analytical model covers such situations along with the cases in which the diffusion coefficient and viscosity relate to each other under the Stokes-Einstein law. In the process, conceptual inconsistency in the past models is uncovered, and a new concept of the "average flow fraction of bitumen" in the flowing mixtures is introduced. The modelled result on overall functionality of the drainage rate of bitumen shows the square-root relationships to key reservoir parameters; it is unchanged from the past analytical models. These relationships are compared with the observed correlations of the scaled bitumen rates to these parameters in the numerous existing Vapex experimental data which cover a variety of conditions. Introduction Dunn et al1 Developed the theoretical model of the gravity drainage process for bitumen recovery known as Vapex based on the model of the steam-assisted gravity drainage (SAGD) process by Butler et al2. This model assumed that the diffusion coefficients of solvent-bitumen systems are constant similar to the case of thermal diffusivity; thus, the steady-state profiles of solvent concentration ahead of the solvent-bitumen interface is the smooth exponential decay towards an infinite distance. In reality, both the diffusion coefficients of both solvent and bitumen are strongly dependent upon compositions due to the extreme viscosity contrasts between the solutes and solvents. As a result, the observed concentration profiles in diffusion experiments exhibit the abrupt front-end profiles3. The theoretical endeavour here is to understand the impact of the non-exponential concentration profiles on the Vapex drainage and bitmen rates. Governing Mechanisms The most fundamental mechanism of the process is the gravity drainage caused by the density difference between the liquid-bitumen phase and the injected vapour phase. The drainage flow of the bitumen phase occurs only from viscosity reduction due to the impact of the injected solvent (or heat in the case of SAGD) of the otherwise semi-solid bitumen. Therefore, how the injected solvent penetrates into the bitumen phase in the reservoir is of the primary importance to the process. According to Fick's law, a material balance across a differential distance dx in this situation can be expressed as a continuity equation for the change of concentration (volume fraction is chosen), C, with time, t, by using the diffusion coefficient, D: Equation (1) (Available in full paper) If the draining interface with a fixed concentration is moving at a velocity U in a direction normal to the interface, at a steady-state condition, the concentration of invaded solvent, C, does not change with time at a given depth from the interface, ξ (a moving axis with the interface, ξ = x-Ut); then Equation (2) (Available in full paper)