Identifying space-dependent coefficients and the order of fractionality in fractional advection diffusion equation

TL;DR

This paper presents an inverse method to identify space-dependent coefficients and the fractional order in a fractional advection diffusion equation, improving computational efficiency and demonstrating effectiveness on synthetic and real tracer data.

Contribution

It introduces an adjoint-based inverse approach for fractional PDEs that efficiently estimates space-dependent coefficients and fractional order from tracer data.

Findings

Method accurately recovers coefficients and fractional order on synthetic data.

Application to real tracer test data demonstrates practical utility.

Computational effort is significantly reduced using the adjoint 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.