Numerical Investigation of Memory-Based Diffusivity Equation: The Integro-Differential Equation

The classical momentum balance equation discovered by Darcy in 1856 is expressed as the flux is proportional to the pressure gradient. However, the passage of the fluid through the porous matrix is very complex in general and hence may cause a local variation of the permeability. Thus, a one-dimensional model for an oil reservoir is introduced by considering the modification of conventional momentum balance equation. The modification is performed by introducing a derivative of fractional distributed orders as memory formalism. The fractional order is equivalent to a time-dependent diffusivity, and the distributed orders represent a variety of memory mechanisms to model the pressure response with a varied distribution of porosity and permeability. The time-domain and space-domain solutions are obtained by means of a numerical solution of the model equation. Results show that memory-based diffusivity equation has less pressure drops compared to Darcy model for a given distance and time. The differences in pressure drop between the two models become more significant when reservoir life becomes longer. The memory has an effect on the reservoir porosity and permeability which increases with time. If reservoir production continues, memory effect becomes more visible and contributesmore in pressure response, which may be considered as a memory-driven mechanism. The proposed model is validated usingMiddle East filed data. The findings of this research establish the contribution ofmemory in reservoir fluid flow through porous media.