The dual porosity model

Abstract

In the past years, various researchers have devoted their attention to modelling groundwater flow in dual-porosity of aquifers. A system of partial differential equations has been suggested and applied in many scenarios with some limitations. In the past, researchers used fractional and classical derivatives as differential operators to model the problems. However, these models cannot depict the crossover effect and the randomness of the geological formations with great precision. In recent years, researchers suggested piecewise differential and integral operators which these operators are defined in a given interval. The passage of a process from one to another is called crossover. The concept of stochastic differential equations was then presented to include the effect of randomness into mathematical formulation. Partial equations depicting groundwater flow within a dual-porosity aquifer were modified. To achieve this, we considered fractional differential and integral operators with different kernels including power law, exponential decay and the generalized Mittag-Leffler functions. Furthermore, randomness was incorporated in the modified model to obtain a piecewise fractional stochastic system of partial differential equations. The numerical method based on the Newton polynomial interpolation was applied to solve the obtained systems of partial differential equations numerically. Matlab was then used towards the generation of simulations. For a fractional order starting from 0.95 to 0.7, the results displayed high flow velocity within fracture, this behaviour is known as long range dependency, also within the matrix soil, we also observed high flow meaning in this case the fractional derivative is expressing the flow within a matrix rock with high transmissivity. For fractional order ranging from 0.7 to 0.5, the results normal flow behaviours; water flows with average velocity within the matrix rock and moderate- high velocity within the fractures. And, from 0.5 to 0.2, we observed slow flow behaviour, the mathematical equations are exhibiting flow with low velocities, expressing the flow within a shale with limited or no transmissivity.