Analytical solution for anomalous diffusion in fractured nano-porous reservoirs

Abstract

This dissertation presents an analytical fluid flow model for multi-fractured horizontal wells in unconventional naturally fractured reservoirs. The solution builds on anomalous diffusion concept and fractional calculus to account for non-uniform velocity in highly disordered porous media. The mathematical model is derived for an isothermal, single phase and slightly compressible fluid. The model computes the transient bottomhole pressure solution for the well and is referred to as Tri-Linear Anomalous Diffusion and Dual-Porosity (TADDP) model. In this work, a time-dependent flux incorporating the fractional derivative of the process variable is utilized and combined with the classical mass balance equation. The temporal derivative power is used to describe the heterogeneity of the flow field. Fractional calculus allows accounting for non-localities of flow and the heterogeneity caused by the contrast between the properties of the rock matrix and natural fractures. The anomalous diffusion formulation is applied to the tri-linear model (TLM) by Ozkan et al. (2009) where the dual-porosity idealization is utilized to describe the naturally fractured reservoir between hydraulic fractures (the so-called stimulated reservoir volume, SRV) and fluid transfer takes place i) from tight rock matrix to natural fractures, ii) from natural fractures to the hydraulic fractures and iii) from hydraulic fractures to the wellbore. The option to use the conventional Darcy’s law for the flux is also available. Flow is treated independently in the rock matrix and natural fractures. In these domains, transport can be modeled by anomalous diffusion or normal diffusion. This approach provides flexibility in modeling natural fractures of discrete or sparse nature as well as modeling complex flow field in the matrix due to the presence of organic and inorganic content. Results show that the solution concurs with the established models and adheres to the general physics of fluid flow. The effects of anomalous diffusion at different reservoir regions can be interpreted from pressure data and delays in the flow can be addressed by adjusting the diffusion exponent. The model offers a range of answers compared to the conventional dual-porosity models and provides utility to model various flow heterogeneities and reservoir types.