Transport in Porous Media

Cilia are hair-like structures that move in unison with the purpose of propelling fluid. They are found, for example, in the human bronchiole respiratory system and molluscs. Here, we validate a novel model of fluid flow due to the movement of cilia in a fixed computational domain. We consider two domains, a porous medium and a free-fluid domain and numerically solve the Stokes–Brinkman system of equations where the cilia geometry and velocity are input and the velocity of fluid due to the movement of cilia is determined. The cilia velocities and geometry are approximated using human lung cilia experimental data available in the literature. We use a mixed finite element method of Taylor-Hood type to calculate the fluid velocities in a three-dimensional domain. The results are validated in a simple case by comparison with an exact solution with good agreement. This problem can be used as a benchmark for the movement of fluid phases due to the self-propelled movement of a solid phase in a porous medium.

Cilia, hair-like, organelles that are found in the respiratory tract (nasal cavity, pharynx, trachea, and bronchi) rhythmically beat to clear mucus from the airways. Here, we formulate a novel model of fluid flow due to the movement of cilia, and in the companion paper, Part II, the model is numerically solved under simplifying assumptions using physical data from lung bronchi. The model is based on a porous media model, modified so that instead of fluid moving through a solid porous structure, the solid moves the fluid. Two macroscale regions are considered: a porous medium and a free-fluid domain. We use hybrid mixture theory to derive the governing equations so that we have a broader understanding of the assumptions used to obtain the model. The resulting model is the classical Brinkman Stokes equations generalized to account for the movement of the cilia. The model can be used as a prototype to determine the movement of fluid due to the given movement of a solid component of a porous material.

In this manuscript, we consider a transport system with a dechlorination reaction network, in which tetrachloroethylene (PCE) reacts to produce trichloroethylene (TCE), TCE reacts to form three daughter products, cis-1,2-dichloroethylene (cis-1,2-DCE), trans-1,2-dichloroethylene (trans-1,2-DCE), and 1,1-dichloroethylene (1,1-DCE), three DCEs further react to produce vinyl chloride (VC), finally VC reacts to produce ethylene (ETH). Because the partial differential equation describing the reactive transport of VC, is coupled by three reactant concentrations, currently the problem must be solved numerically. Following Lu et al. (2003), we extend the analytical solution from five species to the entire PCE reaction network. Using the singular value decomposition (SVD) the system of transport equations with convergent reactions is decoupled into seven orthogonal subsystems. Previously published analytical solutions of single species transport become the basic solutions in the transformed domain for each independent subsystem. The solutions in real concentration domain are obtained using the inverse transform. The solution derived in this study can then be used instead of Sun et al. (1999) in BIOCHLOR for simulating more realistic systems of biodegradation and reactive transport.

In a porous material, both the pressure drop across a bubble and its speed are nonlinear functions of the fluid velocity. Nonlinear dynamics of bubbles in turn affect the macroscopic hydraulic conductivity, and thus the fluid velocity. We treat a porous medium as a network of tubes and combine critical path analysis with pore-scale results to predict the effects of bubble dynamics on the macroscopic hydraulic conductivity and bubble density. Critical path analysis uses percolation theory to find the dominant (approximately) one-dimensional flow paths. We find that in steady state, along percolating pathways, bubble density decreases with increasing fluid velocity, and bubble density is thus smallest in the smallest (critical) tubes. We find that the hydraulic conductivity increases monotonically with increasing capillary number up to Ca ∼ 10−2, but may decrease for larger capillary numbers due to the relative decrease of bubble density in the critical pores. We also identify processes that can provide a positive feedback between bubble density and fluid flow along the critical paths. The feedback amplifies statistical fluctuations in the density of bubbles, producing fluctuations in the hydraulic conductivity.

Rock fractures transmit underground gas effectively once they have sufficient widths and interconnection. However, the fracture geometries needed for gas transport are strongly influenced by surrounding pressure conditions. In order to inspect and quantify the influence of surrounding pressure, we design and manufacture a set of gas flow apparatus that can be connected to the MTS815 material testing system, which provides loads and exhibits external pressures in the experiment. With the apparatus and MTS815, we test the fractured samples of sandstone and coal and obtain their relationship between permeability and external pressure. In particular, our permeability calculation based on collection of gas flux and pressure difference has involved the influence of non-Darcian flow. In addition, our study also includes a numerical simulation in the RFPA software platform to display the internal field changes of cracks. The results show that fracture permeability strongly depends on confining pressure, and a critical pressure probably occurs, about 1.5–2 MPa in our experiments, to split each of the permeability curves into two stages, a slow climb and an exponential rush. As a complement, the numerical simulation also demonstrates one more stage for the permeability curve, the post-rush steady fluctuation.

Sudden changes in isotopic tracer concentration in pore waters have been interpreted as indicating barriers to vertical advective flow through porous rocks in the subsurface, e.g. step changes in $$^{87}\hbox {Sr}/^{86}$$ Sr ratio are often used in the oil and gas industry as a signature of reservoir compartmentalisation. This study shows that this is not necessarily the case. It can take millions of years for such step changes to equilibrate by diffusion if there is no flow resulting from pressure or density gradients even in high permeability, high porosity rocks, particularly if the water saturation is low. Changes in tracer concentration gradients can be good indicators of changes in porosity (or water saturation) between layers. In contrast changes in sorption without a change in porosity are almost impossible to identify. The time taken for concentration gradients to equilibrate is affected by the layer properties but can be quickly estimated from the harmonic average of the effective diffusion coefficient for each layer and a simple analytical expression for a homogeneous system. This was achieved by performing a sensitivity analysis on different layer properties (porosity contrast, saturation contrast, sorption contrast, thickness ratio) using existing analytical solutions for diffusion in layered systems.