Boltzmann transformation of radial two-phase black oil model for tight oil reservoirs

Abstract

Tight oil accumulates in impermeable reservoir rocks, often shale or tight sandstones. The flow behaviour of tight oil in unconventional reservoirs is described by peculiar complexities such as the typical low permeability to viscosity ratio and the dissolution of some gases in the oil phase. Reservoir simulations that consider these complexities negligible stand the potential of poorly characterizing the reservoir flow dynamics. The adoption of similarity transformation effectively reduces the complexities associated with the flow equations through spatial variable (r) and temporal variable (t). The Boltzmann variable left(xi =dfrac{r}{sqrt{t}}right) is introduced to facilitate the reformulation of transient two-phase flow phenomenon in a radial geometry. The technique converts the two-phase Black oil model (thus highly nonlinear partial differential equations (PDEs)) to ordinary differential equations (ODEs). The resulting ODEs present a reduced form on the flow model which is solved by finite difference approximations (the Implicit-Pressure-Explicit-Saturation (IMPES)) scheme. No loss of vital flow characteristics was observed between the Black oil model and the similarity transform flow model. Furthermore, the similarity approach facilitated the determination of pressure and saturation equations as unique functions of the Boltzmann variable. This derivation is applied to an infinitely acting reservoir where the Boltzmann variable tends to infinity (xi rightarrow infty). Finally, this case study’s analytical solution formulated critical relations for fluid flow rate and cumulative production, which are useful for single-phase flow analysis.