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This study focuses on a formulation within the theory of porus media for continuum multicomponent modeling of bacterial driven methane oxidation in a porous landfill cover layer which consists of a porous solid matrix (soil and bacteria) saturated by a liquid (water) and gas phase. The solid, liquid, and gas phases are considered as immiscible constituents occupying spatially their individual volume fraction. However, the gas phase is composed of three components, namely methane (CH4), oxygen (O2), and carbon dioxide (CO2). A thermodynamically consistent constitutive framework is derived by evaluating the entropy inequality on the basis of Coleman and Noll [8], which results in constitutive relations for the constituent stress and pressure states, interaction forces, and mass exchanges. For the final set of process variables of the derived finite element calculation concept we consider the displacement of the solid matrix, the partial hydrostatic gas pressure and osmotic concentration pressures. For simplicity, we assume a constant water pressure and isothermal conditions. The theoretical formulations are implemented in the finite element code FEAP by Taylor [29]. A new set of experimental batch tests has been created that considers the model parameter dependencies on the process variables; these tests are used to evaluate the nonlinear model parameter set. After presenting the framework developed for the finite element calculation concept, including the representation of the governing weak formulations, we examine representative numerical examples.

A continuum triphase model (i.e., a solid filled with fluid containing nutrients) based on the theory of porous media (TPM) is proposed for the phenomenological description of growth and remodeling phenomena in isotropic and transversely isotropic biological tissues. In this study, particular attention is paid on the description of the mass exchange during the stress-strain-and/or nutrient-driven phase transition of the nutrient phase to the solid phase. In order to define thermodynamically consistent constitutive relations, the entropy inequality of the mixture is evaluated in analogy to Coleman and Noll (Arch Ration Mech Anal 13: [167][168][169][170][171][172][173][174][175][176][177][178] 1963). Thereby, the choose of independent process variables is motivated by the fact that the resulting phenomenological description derives both a physical interpretability and a comprehensive description of the coupled processes. Based on the developed thermodynamical restrictions constitutive relations for stress, mass supply and permeability are proposed. The resulting system of equation is implemented into a mixed finite element scheme. Thus, we obtain a coupled calculation concept to determine the solid motion, inner pressure as well as the solid, fluid and nutrient volume fractions.

In this paper the volume fraction concept in the porous media theory is considered. First, the historical development of this important concept is traced. Thereupon, the concept of volume fractions is formulated and the transport theorems, which govern the transformations of volume elements from one placement to another, are derived. An essential feature is the incorporation of the effect of incompressibility into the transport theorems. This is only possible by the introduction of a new multiplicative decomposition of the deformation gradients of the constituents into parts describing the deformations of the realistic (true) materials and parts including the changes of the pores.

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