A Mathematical Model for Short-Time Filtration in Poroelastic Media with Thermal Relaxation and Two Temperatures

Abstract

In this work, we derive a set of governing equations for a mathematical model of generalized thermoelasticity in poroelastic materials. This model predicts finite speeds of propagation of waves contrary to the model of coupled thermoelasticity where an infinite speed of propagation is inherent. Next, we prove the uniqueness of solution of these equations under suitable conditions. We also obtain a reciprocity theorem for these equations. A thermal shock problem for a half-space composed of a poroelastic material saturated with a liquid is then considered. The surface of the half-space is assumed to be traction free, permeable, and subjected to heating. The Laplace transform technique is used to solve the problem. Numerical results for the temperature in the elastic body and fluid, displacement of the elastic body, velocity of the fluid, and stresses for both components are obtained and represented graphically.

Fractional thermoelasticity applications for porous asphaltic materials

“…1, 2 , 3, 4, 5, 6. In these figures, the solid line represents the solution obtained in the frame of the dynamic coupled theory (t ¼ 0) (Biot 1955), and the dotted lines represent the solution obtained in the frame of the generalized thermoelasticity with thermal relaxation time t ¼ 1:0; s o ¼ ð 0:02Þ (Sherief and Hussein 2012), while the dashed lines represent the solution obtained in the frame of generalized thermoelasticity with fractional heat transfer 0 \ t \ 1; ð s o ¼ 0:02Þ (Ezzat 2012).…”

“…To eliminate these anomalies, Cattaneo (1948) and Vernotte (1958) proposed a damped version of Fourier's law by introducing a heat flux relaxation term, by taking Taylor's series to expand q is ðx i ; t þ s s Þ, q if ðx i ; t þ s f Þ and retaining terms up to the first order in s s and s f . The first well-known generalization of such a type (Sherief and Hussein 2012) …”

“…The heat Eqs. (16) and (17) for the two-phase system in the limiting case t ¼ 1 transforms to the work of Sherief and Hussein (2012) in the context generalized thermoelasticity with one relaxation time (Lord and Shulman 1967).…”

“…The problem of a fluidsaturated porous material has been studied for many years. A short list of papers pertinent to the present study includes Biot (1956a, b), Biot and Willis (1957), Deresiewicz and Skalak (1963), Nur and Byerlee (1971), Sherief and Hussein (2012), Youssef (2007).…”

Normal Mode Analysis to a Poroelastic Half-Space Problem under Generalized Thermoelasticity

“…Kumar and Devi (2008) investigated a porous, generalized thermoelastic medium subjected to thermomechanical boundary conditions permeated with various heat sources. Sherief and Hussein (2012) developed a set of governing equations that effectually create a mathematical model of generalized thermoelasticity in poroelastic materials, then, they used this model to solve a thermal shock problem regarding the use of half-space. Abbas and Youssef (2015) solved a two-dimensional problem of a porous material in the context of the fractional order generalized thermoelasticity theory with one relaxation time.…”