Session Q35: Porous Media Flows: Upscaling & Modeling

Upscaling the Navier-Stokes Equation for Turbulent Flows in Porous Media Using a Volume Averaging Method

Turbulent flows through porous media are encountered in a number of natural and engineered systems. Many attempts to close the Navier-Stokes equation for such type of flow have been made, for example using RANS models and double averaging. On the other hand, Whitaker (1996) applied volume averaging theorem to close the macroscopic N-S equation for low Re flow. In this work, the volume averaging theory is extended into the turbulent flow regime to posit a relationship between the macroscale velocities and the spatial velocity statistics in terms of the spatial averaged velocity only. Rather than developing a Reynolds stress model, we propose a simple algebraic closure, consistent with generalized effective viscosity models (Pope 1975), to represent the spatial fluctuating velocity and pressure respectively. The coefficients (one 1st order, two 2nd order and one 3rd order tensor) of the linear functions depend on averaged velocity and gradient. With the data set from DNS, performed with inertial and turbulent flows (pore Re of 300, 500 and 1000) through a periodic face centered cubic (FCC) unit cell, all the unknown coefficients can be computed and the closure is complete. The macroscopic quantity calculated from the averaging is then compared with DNS data to verify the upscaling.   

Energy-based upscaling of immiscible two-phase flow in porous media: flow regimes and applicability conditions

Empirical or theoretical extensions of Darcy’s law for immiscible two-phase flow have shown significant limitations in properly modelling the flow at the continuum-scale. We tackle this problem by proposing a set of upscaled equations based on pore-scale flow regimes, i.e., the topology of flowing phases. The incompressible Navier-Stokes equation is upscaled by means of multi-scale expansions and its closures derived from the mechanical energy balance for different flow regimes at the pore-scale. We also derive the applicability conditions of the upscaled equations based on the order of magnitude of relevant dimensionless numbers, i.e., Eotvos, Reynolds, Capillary, Froude numbers, and the viscosity and density ratio of the system. We demonstrate that the classical two-phase Darcy law is recovered for a limited range of operative conditions and it is compatible only with the connected-pathway flow regime, while additional terms accounting for interfacial and wall interactions should be incorporated to model accurately ganglia or drop traffic flow.   

A metamodel for the apparent permeability tensor of three-dimensional porous media in the inertial regime

In the description of the homogenized flow through a porous medium saturated by a fluid, the apparent permeability tensor is one of the most important parameters to evaluate. In this work we compute numerically the apparent permeability tensor for a 3D porous medium constituted by rigid cylinder using the VANS (Volume-Averaged Navier-Stokes) theory. Such a tensor varies with the Reynolds number, the mean pressure gradient orientation and the porosity. A database is created exploring the space of the above parameters. Including the two Euler angles that define the mean pressure gradient is extremely important to capture well possible 3D effects. Based on the database, a kriging interpolation metamodel is used to obtain an estimate of all the tensor components for any input parameters. Preliminary results of the flow in a porous channel based on the metamodel and the VANS closure are shown; the use of such a reduced order model together with a numerical code based on the equations at the macroscopic scale permit to maintain the computational times to within reasonable levels.   

Effective viscosity in Brinkman equation and stress condition at the interface between a porous medium and a pure fluid

We examine the flow parallel to the interface between a porous medium and a pure fluid. When Darcy’s law is used to describe the momentum transport in the porous layer, the classic Beavers-Joseph condition relates the shear rate and the slip velocity at the interface with a slip parameter that depends on the structure of the porous surface. When the Brinkman equation is used, the averaged velocity is continuous at the interface, however the fluid shear stress across the interface commonly experiences a jump. This shear stress jump can be expressed in terms of the slip velocity at the interface divided by a length characterized by the square root of the permeability, a dimensionless stress jump coefficient, and the effective viscosity introduced in the Brinkman equation. In this work, we explore methods to compute numerically the values of effective viscosity for given porous structures, and study the momentum transfer from the clear fluid onto the solid structure at the interface.   

Modeling filtration and fouling with a microstructured membrane filter

Membrane filters find widespread use in diverse applications such as A/C systems and water purification. While the details of the filtration process may vary significantly, the broad challenge of efficient filtration is the same: to achieve finely-controlled separation at low power consumption. The obvious resolution to the challenge would appear simple: use the largest pore size consistent with the separation requirement. However, the membrane characteristics (and hence the filter performance) are far from constant over its lifetime: the particles removed from the feed are deposited within and on the membrane filter, fouling it and degrading the performance over time. The processes by which this occurs are complex, and depend on several factors, including: the internal structure of the membrane and the type of particles in the feed. We present a model for fouling of a simple microstructured membrane, and investigate how the details of the microstructure affect the filtration efficiency. Our idealized membrane consists of bifurcating pores, arranged in a layered structure, so that the number (and size) of pores changes in the depth of the membrane. In particular, we address how the details of the membrane microstructure affect the filter lifetime, and the total throughput.   

Stochastic approach to model fouling in membrane filters with complex pore morphology

Membrane filters are widely used in industrial applications to remove contaminants and undesired impurities (particles) from a solvent. During the filtration process the membrane internal void area becomes fouled with impurities and as a consequence the filter performance deteriorates, a process that depends on filter internal structure, particle concentration and flow dynamics. The complexity of membrane internal morphology and the random nature of the particle dynamics in the flow make the filtration and fouling challenging to predict; nonetheless, mathematical modeling can play a key role in investigating filter fouling, and in suggesting design modifications for more efficient filtration. To date, many models have been proposed to describe the effects of complexity of membrane structure, and the stochasticity of particle dynamics individually but very few studies focus on both together. In this work, we present an idealized mathematical model, in which a membrane consists of a series of bifurcating pores. Pores decrease in size as the membrane is traversed and particles are removed from the feed by adsorption within pores (which shrinks them) and stochastic sieving (pore blocking by large particles).   

On the Pressure Distribution in a Porous Media under a Spherical Loading Surface

The phenomenon of pressure generation and relaxation inside a porous media is widely observed in biological systems. Herein, we report a biomimetic study to examine the pressure distribution inside a soft porous layer when a spherical loaded surface suddenly impacts on it. A novel experimental setup was developed that includes a fully instrumented spherical piston and a soft fibrous porous layer underneath. Extensive experimental study was performed with different porous materials, different loadings and different sized loading surfaces. The pore pressure generation and the motion of the loading surface were recorded. A novel theoretical model was developed to characterize the pressure field during the process. Excellent agreement was observed between the experimental results and the theoretically predictions. It shows that the pressure generation is governed by the Brinkman parameter, $\alpha =$h/K$_{\mathrm{p}}^{\mathrm{0.5}}$, where h is the porous layer thickness, and K$_{\mathrm{p}}$ is the undeformed permeability. The study improves our understanding of the dynamic response of soft porous media under rapid compression. It has board impact on the study of transient load bearing in biological systems and industry applications.   

Viscous flow in and around a cavity surrounded by a concentric permeable patch

Steady viscous incompressible fluid flow in and around a spherical fluid cavity of radius $a$ surrounded by a permeable patch with thickness $b-a$ is investigated in the limit of low-Reynolds number. Our model uses the Stokes equations in the pure fluid regions and the Darcy law in the concentric permeable patch. Analytic solutions for the velocity and pressure fields are derived in singularity form involving the key parameters such as the Darcy permeability coefficient $k$ and the thickness of the permeable layer. The Faxen law for the hydrodynamical drag acting on the concentric spherical geometry due to an arbitrary incident flow is extracted from our singularity solutions. It is found that the thickness of the permeable layer and the permeability play a crucial role in controlling the drag. An expression for the mass of the fluid that enters the outer sphere is calculated by integrating the exterior radial velocity field. The hydrodynamic force on the concentric spherical shell due to the flow induced by a Stokeslet is also derived from our general expressions. Several special cases of interest are deduced from our exact analysis. The results are of some interest in the prediction of forces exerted on the walls in certain biological models with permeable layers.   

Optimal viscous damping of vibrating porous cylinders

We theoretically study small-amplitude oscillations of permeable cylinders immersed in an unbounded fluid. Specifically, we examine the effects of permeability and oscillation frequency on the damping coefficient, which is proportional to the power required to sustain the vibrations. Cylinders of both circular and non-circular cross-sections undergoing transverse and rotational vibrations are considered. Our calculations indicate that the damping coefficient often varies non-monotonically with the permeability. Depending on the oscillation period, the maximum damping of a permeable cylinder can be many times greater than that of an otherwise impermeable one. This might seem counter-intuitive at first since generally the power it takes to steadily drag a permeable object through the fluid is less than the power needed to drive the steady motion of the same but impermeable object. However, the driving power (or damping coefficient) for oscillating bodies is determined by not only the amplitude of the cyclic fluid force experienced by them but also by the phase shift between the force and their periodic motion. An increase in the latter is responsible for excess damping coefficient of vibrating porous cylinders.   

A robust control volume finite element method for high aspect ratio domains with dynamic mesh optimisation

It can be challenging to produce good quality meshes for models of heterogeneous porous media as domains can have very large aspect ratio and/or complex geometries. Here, a novel control volume finite element method (CVFEM) for simulating multi-phase flow in heterogeneous porous media with highly distorted meshes is presented. In this new formulation, velocity and saturation are discretised as in the classical CVFEM, whereas pressure is discretised using control volumes. The use of control volumes to discretise the pressure creates a pressure matrix that converges very efficiently even when large angle elements are present in the mesh. Heterogeneous geologic features are represented as volumes bounded by surfaces. Our approach conserves mass and the method converges efficiently using highly anisotropic meshes. Results are presented showing the robustness of the presented method for a set of highly heterogenenous and complex geometries, showing very high aspect ratio elements, and using the iterative solvers provided in PETSc. The novel control volume representation for pressure creates a pressure matrix that can be solved very efficiently independently of the elements shapes appearing in the domain, which is key when using dynamic mesh optimisation or high aspect ratio domains.   

Pre-Darcy flow in tight and shale formations

There are evidences that the fluid flow in tight and shale formations does not follow Darcy law, which is identified as pre-Darcy flow. Here, the unsteady linear flow of a slightly compressible fluid under the action of pre-Darcy flow is modeled and a generalized Boltzmann transformation technique is used to solve the corresponding highly nonlinear diffusivity equation analytically. The effect of pre-Darcy flow on the pressure diffusion in a homogenous formation is studied in terms of the nonlinear exponent, $m$, and the threshold pressure gradient, $G_{\mathrm{1}}$. In addition, the pressure gradient, flux, and cumulative production per unit area for different $m$ and $G_{\mathrm{1}}$ are compared with the classical solution of the diffusivity equation based on Darcy flow.