Identifying Space-Dependent Coefficients and the Order of Fractionality in Fractional Advection–Diffusion Equation

Abstract

Tracer tests in natural porous media sometimes show abnormalities that suggest considering a fractional variant of the Advection Diffusion Equation supplemented by a time derivative of non-integer order. We are describing an inverse method for this equation: it finds the order of the fractional derivative and the coefficients that achieve minimum discrepancy between solution and tracer data. Using an adjoint equation divides the computational effort by an amount proportional to the number of freedom degrees, which becomes large when some coefficients depend on space. Method accuracy is checked on synthetical data, and applicability to actual tracer test is demonstrated.

FracFit: A robust parameter estimation tool for fractional calculus models

“…For both CTRW [ Cortis and Berkowitz , ] and MRMT [ Haggerty , ], publicly available computational toolboxes for parameter estimation exist. Alternative modeling approaches include spatial and temporal Markov models [ LeBorgne et al ., ; Meyer and Tchelepi , ] and the adjoint equation method [ Maryshev et al ., ]. The goal of this paper is to describe a new toolbox for fractional advection‐dispersion models [ Liu et al ., ; Schumer et al ., ; Meerschaert et al ., ].…”