Regular Article

An application of a multiindex, time-fractional differential equation to evaluate heterogeneous, fractured rocks

¹ Raghavan Group, Inc., PO Box 52756, Tulsa, OK 74152, USA
² Kappa Engineering, 1500 City West Blvd, Suite 741, Houston, TX 77042, USA

^* Corresponding author: raghavan.raj@gmail.com

Received: 7 February 2022
Accepted: 6 December 2022

Abstract

A multiindex, distributed fractional differential equation is derived and solved in terms of the Laplace transformation. Potential applications of the proposed model include the study of fluid flow in heterogeneous rocks, the examination of bimodal fluid exchange between mobile-immobile regions in groundwater systems, the incorporation of the existence of liesegang bands in fractured rocks, and addressing the influences of faulted and other skin regions at interfaces, among others. Asymptotic solutions that reveal the structure of the resulting solutions are presented; in addition, they provide for ensuring the accuracy of the numerical computations. Fractional flux laws based on Continuous Time Random Walks (CTRW) serve as a linchpin to account for complex geological considerations that arise in the flow of fluids in heterogeneous rocks. Results are intended to be applied at the Theis scale when combined with geological/geophysical models and production statistics to all aspects of subsurface flow: production of geothermal and hydrocarbon fluids, injection of fluids into aquifers, geologic sequestration and hazardous waste disposal. Results may be extended to study the role of complex wellbores such as horizontal and fractured wells and more complex geological considerations such as faulted systems.