Professor Andro Mikelić is known for his seminal mathematical contributions to flow in porous media. In this presentation we summarize how his work has impacted the development of numerical models coupling flow and poromechanics arising in geosciences and biosciences applications such as subsidence events, carbon sequestration, groundwater remediation, hydrocarbon production, and hydraulic...
In this presentation we consider the equations of nonlinear poroelasticity derived from mixture theory. They describe the quasi-static mechanical behavior of a fluid saturated porous medium. The nonlinearity arises from the compressibility of the fluid and from the dependence of porosity and permeability on the divergence of the displacement. We point out some limitations of the model. In our...
We derive the new effective boundary condition for the fluid flow in domain with porous boundary. Starting from the Newtonian fluid flow through a domain with an array of small holes on the boundary, using the homogenization and the boundary layers, we find an effective law in the form of generalized Darcy law. If the pores geometry is isotropic, then the condition splits in Beavers-Joseph...
We study the behaviour of a system of equations that describes diffusion with chemical reactions caused by microorganisms in a double porosity medium. The objective is to find various classes of nontrivial limit solutions at large time (the patterns), especially simultaneous spatial-temporal patterns. In contrast to a classical reaction-diffusion system (RDS), our system contains four...
A three-scale model for flow in karst conduit networks in carbonate rocks is constructed based on a reiterated homogenization procedure. The first upscaling, performed from the high-fidelity flow model, is based on a topological model reduction considering a discrete network of conduits. The subsequent macroscopization procedure projects the reduced model into the cells of a coarse...
In this talk, we present an extension of the phase-field fracture propagation model to the immiscible two-phase flow fracture model, and with a transport problem. The flow model is derived by using the lubrication theory, and we provide the absolute and relative permeabilities with nonzero capillary pressure. The contribution in solid mechanics consists of displacements and a phase-field...
Building on the work of Andro Mikelic and Mary WHeeler, we propose a numerical scheme for an elliptic-parabolic system involving deformation and pressure in porous media. Existence and uniqueness of the solution have been proved by Mikelic et al; we will add some convergence results for ou numerical scheme.
This is joint work with Ludovic Goudenège and Danielle Hilhorst
We prove the existence of a weak solution to a fluid-poroelastic structure interaction problem in which the structure consists of two layers: a thin poroelastic plate layer in direct contact with Stokes flow, and a thick Biot layer sitting on top of the thin layer. In the (quasi-static) Biot layer the permeability is a nonlinear function of the fluid content. Existence of a weak solution is...
Exchange processes across a porous-medium free-flow interface occur in a wide range of environmental, technical and bio-mechanical systems. In the course of these processes, flow dynamics in the porous domain and in the free-flow domain exhibit strong coupling, often controlled by mechanisms at the common interfaces. Therefore, understanding the underlying processes is decisive. An example of...
In porous media and other complex media with different length scales, (periodic) homogenization has been successfully applied for several decades to arrive at macroscopic, upscaled models, which only keep the microscopic information by means of a decoupled computation of 'effective' parameters on a reference cell. The derivation of Darcy's law for flow in porous media is a prominent example....
The urgent need for a better quantitative understanding of physiological processes in cells, tissues and the whole body is particularly evident in the current pandemic, in which the SARS-CoV2 virus is upsetting vital processes in the infected individuals. The damages of the virus at the cellular level, initially mainly in the lung, are causing inflammation that can get disordered and lead to a...
Problems including reactive transport processes through thin layers with a heterogeneous structure play an important role in many applications, especially from biosciences, medical sciences, geosciences, and material sciences. In our contribution, we consider a nonlinear reaction--diffusion equation in a domain consisting of two bulk-domains, which are separated by a thin layer with a periodic...
In this presentation, we revisit our efforts to model fluid-filled fracture propagation in a porous medium. Several challenges and extensions in mathematical modelling as well as the design of numerical methods will be discussed. Along with these theoretical and algorithmic accomplishments, a computational framework IPACS has been developed, which is substantiated with some numerical simulations.
Catalytic membranes can degrade gaseous pollutants to clean gas via a catalytic reaction to achieve green emissions. A catalytic membrane is a three scale porous medium. Membranes used in catalytic filters usually have thickness centimeters or millimeters, and consist of active (washcoat) particles, inert material and microscale, micron size, pores. The washcoat particles are porous material...
This talk is devoted to a problem of model reduction for a class of reaction- diffusion-ODE systems. Such systems of equations arise, for example, in modeling of interactions between cellular processes and diffusing growth factors. Taking into account different time and space scales of the underlying processes leads to singularly perturbed problems. We focus on a shadow limit approximation for...
The talk will focus on rigorous homogenization results for a system of partial differential equations describing the transport of a N-component electrolyte in a dilute Newtonian solvent through a rigid random disperse porous medium. We will consider the nonlinear Poisson-Boltzmann equation in a random medium, describe the stochastic homogenization procedure and formulate the convergence...
