Numerical Solution of Plane-Radial Boundary Value Inverse Problem for the Equation of Non-Stationary Relaxation Filtration of Fluid in a Porous Medium

Abstract

The paper addresses a numerical solution of a plane–radial boundary inverse problem for the relaxation filtration equation describing fluid flow in an elastically deformable porous medium. The study is motivated by the widespread occurrence of relaxation filtration in hydrogeology, oil and gas production, and subsurface hydromechanics, where reliable identification of medium parameters and reconstruction of unknown boundary actions are essential. Such boundary inverse problems are ill-posed: small perturbations in the input data may cause large deviations in the recovered solution, which makes stable and accurate numerical techniques crucial. The inverse problem is solved using De Souza’s marching method; however, computational experiments show that its accuracy strongly depends on the distance between the measurement point providing the “initial data” and the target boundary. As this distance increases, the error grows due to error accumulation and the intrinsic instability of boundary reconstruction. To improve stability and reduce errors, smoothing splines are employed. The spline approximation effectively suppresses high-frequency noise and stabilizes the recovery of boundary values. As a result, more stable numerical solutions with acceptable accuracy are obtained even under substantial data errors, demonstrating the promise of combining marching methods with smoothing procedures for ill-posed boundary inverse problems of relaxation filtration.