Analytical Solution of Generalised Mandel’s Problem

¹ Mathematics Study Program, Bandung Institute of Technology, Bandung, Indonesia
² Actuarial Sciences Study Program, Bandung Institute of Technology, Bandung, Indonesia

^* e-mail: adilan.widyawa@itb.ac.id

Abstract

Mandel’s problem considers the time-dependent deformation of a fully saturated rectangular porous material with drainage at the lateral sides and steady load from the top and bottom. It plays a major role in the fields of geomechanics and hydrogeology. This paper presents analytical solutions to Mandel’s problem with compressible fluid and solid particles. The governing equations are developed based on the equation of equilibrium and the conservation of mass both for fluid and solid constituents. The analytical solution is obtained by applying the Laplace transform to the proposed governing equation with the pore pressure becoming the main variable. The result can serve as a benchmarking and calibration tool for both modelling and experimental data. To show the application of the proposed analytical solution, we perform simulations to study the influence of solid compressibility on the pore pressure distribution. We found that the pore pressure is significantly affected not only by the permeability but also the compressibility of the solid particles. Therefore, compared to the classical analytical solution of Mandel’s problem, in which the fluid and solid particles are incompressible, our analytical solution offers more general results and provides a deeper understanding of the mechanical behaviour of a porous material.