Session LK: Flows in Porous Media

Boundary Layer Thickness in Dual Potential Fluid Flow through Porous Medium

A transient, two-dimensional fluid flow through porous medium in a dual potential field has been studied analytically and experimentally. The field in a rectangular domain is created by placing two inlets: manifold along one of the edges at potential $\varphi _1 $, and the other inlet is a channel placed in the center of the domain perpendicular to the first inlet on the top surface at potential $\varphi _2 (x)$. For any location in the domain with potential $\varphi ({\rm {\bf x}}),{\rm {\bf x}}=(x,y)$, we define two potential differences $\Delta \varphi _1 =\varphi ({\rm {\bf x}})-\varphi _1 $ and $\Delta \varphi _2 =\varphi ({\rm {\bf x}})-\varphi _2 (x)$ with respect to the two inlets. Therefore, two distinct sub-regions of the porous medium exist, where $\Delta \varphi _1 <\Delta \varphi _2 $ and $\Delta \varphi _1 >\Delta \varphi _2 $. The interface between the regions satisfies $\Delta \varphi _1 =\Delta \varphi _2 $ which we define as a boundary layer of thickness $\delta (x)$. In the experiments, we varied: the channel cross-sectional area, medium width, and thickness. The same fluid of very high viscosity (to reduce mixing) was used at both inlets with one stream dyed; thus, visualizing the flow and the two distinct sub-regions. We have also developed an analytical model to predict the boundary layer thickness, $\delta (x)$. Both, implicit and explicit solutions are found, where the explicit solution is given in the form of the inverse Lambert function. The solution has only one physical constant which is a function of the pressure gradient and the directional permeabilities of the porous medium. A comparison between experimental and analytical results reveals an excellent agreement.   

Gas Hydrate Dissociation in the Ocean

Methane gas is known to exist in extremely large quantities below the sea floor in the sediment of the deep and cold oceanic and in permafrost regions. Due to the large hydrostatic pressure and cool temperatures the gas reacts with the surrounding water to form a crystalline substance known as a gas hydrate. The fate of these reserves is very important to climate on earth because methane is a much more efficient greenhouse gas then carbon dioxide. The dissociation process in general can occur by either a decrease in pressure or an increase in temperature. In this study we concentrate on the latter. Once the hydrate dissociates, water and free gas remain above the phase boundary, occupying a larger volume than the original solid, and are be transported through the sediment. We have modeled this physical mechanism using volume averaged equations in a porous medium with a coupled two-phase flow. The movement of the phase boundary, which is proportional to the rate of heat transfer to this interface, is modeled as a Stefan type melting problem. The resultant governing equations are solved numerically, using a front fixing method to fix the phase boundary, to determine the rate of gas flux through the sediment and the dissociation rate.   

Numerical Modeling of 2-D Diffusion-Advection-Reaction Flows in Marine Sediments

We develope and investigate a 2-D numerical simulation for diffusion-advection-reaction processes in marine sediments. The simulation program is written in open source Octave language using a semi-implicit numerical scheme for both unsteady and steady state cases. We analyze and compare the accuracy, performance and stability trade-offs of several numerical schemes. We also test the influence of temporal resolution and time variability. We test our numerical results against several known experimental data sets, including cases of total sulfate consumption and brine incursion. To validate the physical correctness of the obtained solution, we apply our program to simulations of marine pore water profiles [from Ocean Drilling Program Leg 182 in the Great Australian Bight] with focus on bacterial sulfate reduction. We compare the results of the 2-D simulation versus a 1-D and partial 2-D model. We demonstrate that full 2-D model accounting for the lateral flow intrusion into the sediment is required to correctly model the physics of the process.   

Capillary flow in weakly 3-dimensional conduits

A large literature exists for capillary driven flows along simple conduits of uniform cross section (i.e. right circular and regular polygonal cylinders). In this work flows along several conduits of increased geometric complexity are considered that also possess a taper or some other geometric change along the primary flow path. Experimental, numerical, and analytical results will be presented for select transient flows in such conduits where power law time dependencies transition between regimes depending on the time dependent boundary conditions. Such flows may be exploited for the passive manipulation of liquids in microfluidic systems on Earth and microgravity fluid systems aboard spacecraft.   

Reaction characteristics of miscible displacement in vertical capillary tubes

It is known that the spike is formed in the miscible displacement in capillary tubes under a certain condition. In the present study, miscible displacement with chemical reaction in vertical capillary tubes is investigated experimentally in the condition where the spike is formed. The effect of reactant concentration on the reaction characteristics is focused on. We have found that the initial reactant concentration ratio included in the more- and less-viscous liquids normalized by the stoichiometric ratio of the chemical reaction, \textit{$\phi $}, significantly influences the distribution of the products. When the reactant contained in the more-viscous liquid is much more concentrated than that in stoichiometry; \textit{$\phi $} $<<$ 1, the product is scarcely distributed inside the spike. In contrast, when the reactant contained in the less-viscous liquid is much more concentrated than that in stoichiometry; \textit{$\phi $} $>>$ 1, the product is obviously distributed inside the spike. We propose that the difference in the product distribution is caused by both two factors; the idiosyncratic shape (spike) of boundary region and the difference between the diffusion coefficient of interdiffusion of miscible two liquids and that of reactants in less viscous liquid.   

Miscible viscous fingering of finite width samples with adsorption

Influence of adsorption on miscible viscous fingering of a finite width sample of higher viscous fluid displaced by a lower viscous fluid is studied in porous media. We assume that the viscosity of the sample depends on the mobile phase concentration $c_m$ of a given solute. This concentration $c_m$ is related to the stationary phase concentration $c_s$ of the solute adsorbed onto the porous matrix by a linear adsorption isotherm model. The model equations governed by Darcy's law coupled with the evolution equation for $c_m$ incorporating a retention factor $K$ are numerically integrated using a Fourier spectral method. In the presence of adsorption, fingering similar to that of the non-adsorptive case is observed but appears on a slower time scale and with a different wavelength. The fingering pattern with adsorption adopts a traveling wave solution with a speed which depends on the adsorption parameter $K$. Fingering contributes transiently to the broadening of the peak in time by increasing its variance in both adsorptive and non-adsorptive cases but the variance with adsorption is always smaller than without adsorption. Relevance of the results is discussed in relation with transport of viscous samples in chromatographic columns with retention.   

Effect of a punctual water jet underneath an unbounded porous media: an experimental and theoretical study

The response of a granular bed to the presence of a punctual water jet is studied experimentally and theoretically. The setup consists of a cylindrical tank of 24 cm diameter and 17 cm height which is partially filled with sand granules. The bottom of the tank features a central nozzle from where the water flows vertically into the tank. The setup is also designed in such a way that the size of the nozzle (jet) can be varied. The experiments show that three distinct regimes associated with the flow rate, Q, appear to outline the bed's behavior. For sufficiently small Q the bed remains motionless and acts as a porous media. It then becomes deformed if Q is sufficiently large. Finally it ``explodes'' and a locally fluidized bed limited to an area above the jet is observed if Q is increased further. The fluidized area (chimney) appears to have a roughly cylindrical shape. The results show that the onset for fluidization as well as the size of the chimney are almost independent of the size of the jet. On the basis of above observations, a theoretical model for the flow motion with and without fluidization is advanced.   

Direct Numerical Simulations of the onset of density-driven convection in anisotropic porous media

Among the many options available to mitigate carbon dioxide emissions in the short-term, geologic sequestration is being considered as one of the most promising ones. Carbon dioxide is injected into the earth in a supercritical phase where it's density is lower than that of the surrounding water. When carbon dioxide dissolves into the medium, the system has a gravitational instability leading to ``finger''-like convection patterns. Previous investigators have studied this problem in the framework of linear and global stability analyses. We present an improved linear stability analysis followed by detailed direct numerical simulations, using a pseudospectral method. We present results for the dependence of the critical time and critical wavenumber on the permeability anisotropy ratio. We also present an improved definition of the critical time which is more characteristic of the time scales observed for the onset of convection.   

Analysis of the effective viscosity in a lattice-Boltzmann model for flow in porous media

The lattice-Boltzmann method has been applied to study fluid flow in porous media. In the lattice-Boltzamnn modeling the macroscopic governing equations as well as the viscosity and the permeability are usually determined using a Chapman-Enskog expansion approach. However, this approach has been validated numerically only for a narrow range of the viscosity and the permeability. We have analytically derived the exact macroscopic governing equations of the lattice-Boltzmann model for the case of simple shear flows in porous media. We find that the effective viscosity in the governing equations is different from the one obtained from the Chapman-Enskog expansion. These findings have been validated numerically. We also find the surprising result that the effective viscosity is anisotropic. Possible improvements and applications of the model will be discussed.   

Polymer Electrolyte Membrane (PEM) Fuel Cells Modeling and Optimization

Performance of polymer electrolyte membrane (PEM) fuel cells is dependent on operating parameters and designing parameters. Operating parameters mainly include temperature, pressure, humidity and the flow rate of the inlet reactants. Designing parameters include reactants distributor patterns and dimensions, electrodes dimensions, and electrodes properties such as porosity, permeability and so on. This work aims to investigate the effects of various designing parameters on the performance of PEM fuel cells, and the optimum values will be determined under a given operating condition.A three-dimensional steady-state electrochemical mathematical model was established where the mass, fluid and thermal transport processes are considered as well as the electrochemical reaction. A Powell multivariable optimization algorithm will be applied to investigate the optimum values of designing parameters. The objective function is defined as the maximum potential of the electrolyte fluid phase at the membrane/cathode interface at a typical value of the cell voltage. The robustness of the optimum design of the fuel cell under different cell potentials will be investigated using a statistical sensitivity analysis. By comparing with the reference case, the results obtained here provide useful tools for a better design of fuel cells.   

A Two-Dimensional PEM Fuel Cell Model

Proton Exchange Membrane (PEM) fuel cell is a typical low temperature cell, where hydrogen and air are fed into the porous anodic electrode and cathodic electrode though the gas distributors on the bipolar plates, respectively. Activated by the catalyst on anode side, hydrogen will spilt into protons and electrons. Since only protons will be allowed to pass through the membrane, electrons must go through an external circuit. Electrons and protons meet air on cathode side to produce water and heat catalyzed by the catalyst on the cathode side. Numerical simulations are useful tools to describe the basic transport and electrochemical phenomena of PEM fuel cells. The goal of the present work is to develop 2-D computational models of PEM fuel cells, which take into account fluid flow, multi- species transport, current distribution and electrical potential. The velocity field in free channel described by Navier-Stokes equation and the velocity field in porous media described by Darcy’s Law are coupled along the channel-MEA interface. The governing differential equations are solved over a single computational domain, which consists of two gas channel layers, two gas diffusion layers, two catalyst layers as well as a membrane. The model is solved with commercial software COMSOL Multiphysics 3.2b. Parametric study will be conducted to analyze the effects of various parameters on the performance of PEM fuel cells. The results, including the mass concentration, the polarization curve and the velocity distribution, will be presented.