An Analysis of the Error Associated to Single and Double Exponential Approximations of Theoretical Poroelastic Models

Abstract

The temporal evolutions of displacements, strains, stresses, and fluid pressure in a poroelastic material under sustained compression are theoretically expressed as sums of an infinite number of exponentials. However, estimation of an infinite number of time constants is impractical in experimental settings. In the past, empirical models containing a finite number of exponentials have been proposed and used to approximate the theoretical poroelastic models. At the present time, however, the degree of error associated with such approximations is unclear. In this paper, we present an analysis of the error encountered when approximating a poroelastic model containing an infinite number of exponentials with a single or double exponential model. As a testing platform, the presented error analysis is applied to the estimation of effective Poisson’s ratio (EPR) and fluid pressure in a uniform cylindrical poroelastic sample and a cylindrical poroelastic sample containing an inclusion under stress relaxation. Our results show that, when the infinite number of exponentials in the theoretical models are approximated with finite number of exponentials, significant error is invoked only in the first few time samples of the EPR and fluid pressure, while the error is negligible for the remaining time samples. We also show that, when estimating clinically relevant mechanical parameters such as Poisson’s ratio or the product of aggregate modulus and interstitial permeability, such approximation invokes small error (<3%) in comparison to the general model with infinite number of exponentials. Therefore, such approximation may be acceptable in techniques aiming at reconstructing mechanical parameters from poroelastographic data.