Porous Medium Domain Research Articles

A discontinuous Galerkin method for the Brinkman–Darcy-transport problem

In this paper, a weighted discontinuous Galerkin finite element method and an upwind format are presented to solve the coupled Brinkman–Darcy flow and transport model. The flow in a highly permeable region of the model is governed by the Brinkman equations, and the percolation in the porous media is controlled by the Darcy equations, which are coupled by three interface conditions. The permeability coefficients in this model are strongly nonhomogeneous, anisotropic, and discontinuous; the transport equation is a convection-dominated problem; and the velocity field in the porous media domain does not satisfy the divergence-free condition. A weighted discontinuous Galerkin finite element method is used to solve the complex permeability coefficient problem and an upwind scheme is used to solve the convection dominated problem. The interface conditions can be naturally incorporated into the discrete formulation without introducing additional variables. Optimal error estimates are obtained for the semi-discretization and the full discretization with the backward Euler scheme in suitable energy norm. A series of numerical experiments are provided to illustrate the proposed method, including testing the convergence and accuracy of various types of meshes; simulating fluid flow in complex porous media such as obstacles, layered media, and curved interfaces; and studying the coupled flow behavior of surface-subsurface flows with contaminant transport.

TURBULENT FLOW SIMULATION OF THE NONNEWTONIAN FLUIDS THROUGH A POROUS MEDIUM USING LES TURBULENCE MODEL

In this study, fully turbulent flow of non-Newtonian fluids is studied in a square periodic array as a porous medium, with porosity ranging from 0.5 to 0.84. Large eddy simulation (LES) is used as an effective turbulence model at a Reynolds number of 40,000. The governing equations are solved by Ansys Fluent commercial software for various porosities. The power-law and Carreau approaches are employed to model the dilatant and pseudo-plastic non- Newtonian fluids. In this study, the power-law and Carreau models are also compared. The results showed that an increase in porosity has a significant effect on flow parameters. Specifically, as porosity increases, vorticities, pressure, and velocity values increase within the computational domain. Moreover, it was observed that the Carreau model has a greater ability to obtain more realistic results in porous media domain and complex geometries.

Magnetophoresis of paramagnetic metal ions in porous media.

We report a numerical investigation of the magnetophoresis of solutions containing paramagnetic metal ions. Using a simulated magnetic field of a superconducting magnet and the convection-diffusion model, we study the transport of transition metal salts through a porous medium domain. In particular, through a detailed comparison of the numerical results of magnetophoretic velocity and ion concentration profiles with prior published experiments, we validate the model. Subsequent to model validation, we perform a systematic analysis of the model parameters on the magnetophoresis of metal ions. Magnetophoresis is quantified with a magnetic Péclet number Pem. Under a non-uniform magnetic field, Pem initially rises, exhibiting a local maximum, and subsequently declines towards a quasi-steady value. Our results show that both the initial and maximum Pem values increase with increasing magnetic susceptibility, initial concentration of metal solutes, and ion cluster size. Conversely, Pem decreases as the porosity of the medium increases. Finally, the adsorption of metal salts onto the porous media surface is modeled through a dimensionless Damkohler number Daad. Our results suggest that the adsorption significantly slows the magnetophoresis and self-diffusion of the paramagnetic metal salts, with a net magnetophoresis velocity dependent on the kinetics and equilibrium adsorption properties of the metal salts. The latter result underscores the crucial role of adsorption in future magnetophoresis research.

Wave Characteristics over a Dual Porous Submerged Breakwater Using a Fully Nonlinear Numerical Wave Tank with a Porous Domain

This study developed a two-dimensional fully nonlinear numerical wave tank (FN-NWT) to examine the nonlinear interaction between waves and dual submerged porous structures. Using the FN-NWT, not only reflection and transmission coefficients, but also wave deformation/force depending on porosity were investigated. The FN-NWT was developed using the boundary element method (BEM), and consisted of a fluid domain and a porous medium domain. Darcy’s law or the non-Darcy (Forchheimer) flow equation were applied to the flow passing through the porous domain. The wave reflection coefficient of the porous submerged structures agreed well with the given experimental data when using Forchheimer flow boundary conditions. Excessive attenuation of the transmitted wave occurred when Darcy’s condition was employed. The difference in each coefficient due to the spacing of the submerged structure was reduced in the porous structure compared with the non-porous structure. The difference according to the incident wave height was clearly revealed in the transmission coefficient. The developed dual-domain FN-NWT can be applied to investigate the nonlinear interaction between waves and porous structures as a first-cut design tool.

Global sensitivity analysis using multi-resolution polynomial chaos expansion for coupled Stokes–Darcy flow problems

Determination of relevant model parameters is crucial for accurate mathematical modelling and efficient numerical simulation of a wide spectrum of applications in geosciences. The conventional method of choice is the global sensitivity analysis (GSA). Unfortunately, at least the classical Monte-Carlo based GSA requires a high number of model runs. Response surfaces based techniques, e.g. arbitrary Polynomial Chaos (aPC) expansion, can reduce computational effort, however, they suffer from the Gibbs phenomena and high hardware requirements for higher accuracy. We introduce GSA for arbitrary Multi-Resolution Polynomial Chaos (aMR-PC) which is a localized aPC based data-driven polynomial discretization. The aMR-PC allows to reduce the Gibbs phenomena by construction and to achieve higher accuracy by means of localization also for lower polynomial degrees. We apply these techniques to perform the sensitivity analysis for the Stokes–Darcy problem which describes fluid flow in coupled free-flow and porous-medium systems. We consider the Stokes equations in the free-flow region, Darcy’s law in the porous-medium domain and the classical interface conditions across the fluid–porous interface including the conservation of mass, the balance of normal forces and the Beavers–Joseph condition for the tangential velocity. This coupled problem formulation contains four uncertain parameters: the exact location of the interface, the permeability, the Beavers–Joseph slip coefficient and the uncertainty in the boundary conditions. We carry out the sensitivity analysis of the coupled model with respect to these parameters using the Sobol indices on the aMR-PC expansion and conduct the corresponding numerical simulations.

A pressure-robust mixed finite element method for the coupled Stokes–Darcy problem

A pressure-robust numerical method for the coupled Stokes–Darcy problem is proposed. We use the continuous piecewise linear polynomial space combined with the lowest order Raviart–Thomas space to discretize the Stokes equation in the fluid region and the lowest order Raviart–Thomas elements to discretize the Darcy equation in the porous media domain. We also perform a divergence-free reconstruction of the test function for the right-hand term. Our method has good properties of convergence, pressure-robustness and local mass conservation in the sense of projection. A discrete inf-sup condition and error estimates for the numerical scheme are demonstrated, the error of velocity is independent of viscosity and pressure. Finally, the theoretical analysis is validated with several examples, demonstrating the superiority of the pressure-robust scheme.

Magnetohydrodynamic mixed convection of TiO2–Cu/water between the double lid-driven cavity and a central heat source surrounding by a wavy tilted domain of porous medium under local thermal non-equilibrium

The magnetohydrodynamic (MHD) mixed convection of heat and mass transfer is carried out using finite difference method applied inside a tilted porous cavity saturated with a hybrid nanofluid due to the presence of the double-moving lid and the heat sources. In contrast to the earlier research, various effects which are recognized by heat generation in the local thermal non-equilibrium case at the extended Brinkman Darcy model subjected to inclined magnetic field are thoroughly examined numerically. For instance, unusual observations of the cold mass surrounding the heat source emphasize that the maximum fluid temperature highly depends on the forced convection. Additionally, solid-phase temperature acts in accordance to the heat source location while fluid temperature is agitated by the moveable sides which points up the disparity at the thermal energy transportation. However, the transfer of heat and mass at the model requires a specific conduct due to the existence of damping factors. The magnetic field, for example, suppresses the fluid flow. Moreover, the thermal non-equilibrium condition deteriorates the global heat generation.

Efficient numerical modeling of liquid infusion into a porous medium partitioned by impermeable perforated interlayers

AbstractNumerical modeling of flow through porous media and the simulation of liquid flow through orifices, channels and perforated walls, membranes, interlayers find applications in various fields. However, the mesh refinement needed to describe the detail at the scale of orifices within a domain multiple orders of magnitude larger raises numerical challenges. The present work proposes a pragmatic solution to model perforated layers partitioning a large porous media domain using 1D elements to model the holes and connect the 3D elements which represent the porous media. As an illustration, the approach is applied in liquid composite molding processes, and to the processing of large thick panels toughened with perforated interlayers. However, this work could be adopted in numerous fields. The combination of 3D and 1D elements to manage components with different dimensions has been used before, however no proper analysis of the loss of accuracy introduced has been conducted to our knowledge. A systematic parametric study is conducted to quantify the impact of the length of the domain, the number of interlayers, the diameter of the holes and the viscosity of the fluid on the loss of accuracy. Meshing rules and directions are provided to improve the accuracy of the simulations.

Derivation and analysis of a 1D porous medium flow solver embedded in a two-domain model for 2D and 3D transpiration cooling

• A 1D porous medium flow solver is derived and analyzed. • The 1D solver is embedded in a two-domain model for 2D/3D transpiration cooling. • Differences between results of the new model and a full 2D/3D model are negligible. • Transpiration cooling problems are solved significantly faster with the new model. For the numerical investigation of transpiration cooling problems, often two-domain models are used. Different systems of governing equations are solved in a hot gas and a porous medium domain, and both domains are coupled with each other by imposing boundary conditions on the interface. This work presents a novel approach which splits the porous medium domain into 1D problems for Darcy-Forchheimer flow under local thermal nonequilibrium. The fluid and solid temperature solutions are computed analytically, and only one ordinary differential equation has to be solved to determine fluid density and Darcy velocity. Unique solutions to the equations of the 1D model are verified to exist under physically reasonable assumptions. The solutions of the 1D problems are assembled to one 2D or 3D solution. In a postprocessing, slip and adiabatic side walls are modeled despite being nonexistent in 1D. Numerical 2D and 3D test cases together with experimental data verify the suitability of the new assembled-1D approach for the simulation of transpiration-cooled hot gas flows with orthogonal cooling gas injection. Furthermore, the test cases demonstrate the significant computational speed-up related to a full-2D/3D approach.

Towards hybrid two‐phase modelling using linear domain decomposition

AbstractThe viscous flow of two immiscible fluids in a porous medium on the Darcy scale is governed by a system of nonlinear parabolic equations. If infinite mobility of one phase can be assumed (e.g., in soil layers in contact with the atmosphere) the system can be substituted by the scalar Richards model. Thus, the porous medium domain may be partitioned into disjoint subdomains where either the full two‐phase or the simplified Richards model dynamics are valid. Extending the previously considered one‐model situations we suggest coupling conditions for this hybrid model approach. Based on an Euler implicit discretization, a linear iterative (L‐type) domain decomposition scheme is proposed, and proved to be convergent. The theoretical findings are verified by a comparative numerical study that in particular confirms the efficiency of the hybrid ansatz as compared to full two‐phase model computations.

Dimensionless Characterization to Estimate Horizontal Groundwater Velocity from Temperature–Depth Profiles in Aquifers

The outcome of a dimensionless characterization study in a two-dimensional porous media domain in which groundwater flows at a constant horizontal velocity is presented in this report. Using spatial discrimination, the dimensionless groups that govern the solution patterns are determined from dimensionless governing equations. As a boundary condition on the surface, the case of constant temperature is studied. From the mathematical deduction of the groups, a characteristic horizontal length emerges. This length determines the region in which temperature–depth profiles are affected by flow. Existing analytical solutions have been shown to be invalid due to the severe assumption that the horizontal thermal gradient has a constant value. Therefore, universal solutions based on pi theorem have been obtained for the characteristic horizontal length, temperature field, temperature–depth profiles and horizontal temperature profiles. Dependencies between dimensionless groups have been depicted by universal curves, abacuses and surfaces. These graphical solutions are used in an easy way to estimate groundwater velocity from experimental temperature measurements in the form of an inverse problem. In addition, an easy and fast protocol for estimating fluid flow velocity and groundwater inlet temperature from temperature profile measurements is proposed. This protocol is applied in a scenario of groundwater discharge from a quaternary aquifer to a salty lagoon located in the southeast of Spain.

A review on measurement of the dynamic effect in capillary pressure

Studies have shown that multiphase flows in porous media are in a transient state with the dynamic effect in capillary pressure. The ignorance of the dynamic effect in capillary pressure might lead to some potential errors in engineering application. In this paper, recent advancements in the measurement of dynamic effect in capillary pressure are reviewed, focusing on the experimental setup, experimental method, and data interpretation method. The measuring technologies are developed both in accuracy and measurement range of porous media and fluids. State-of-the-art measurement technology can measure the dynamic effect in capillary pressure in tight rock and carbonate even with the existence of supercritical fluids. During the measurement process, it is essential to record the pressure difference between the wetting and the non-wetting phases and the in-situ fluid saturation. Besides, the end effect and the rapid change in the pressure at the flowing front need to be prevented. The fluid saturation can be measured through electrical resistivity under various injection pressures, which is applicable to determine the dynamic capillary pressure. The semipermeable elements are particularly the key material for testing the dynamic capillary pressure. Their installation site changes from the ends to the sides of the porous media domain. Additionally, the widely used mass balance-based method is not suitable to obtain the average fluid saturation for less permeable porous media, and calibration is required if the centroid-corrected phase average method and the intrinsic phase average method are used to compute the average dynamic capillary pressure. This paper suggests a reference for the evaluation of multiphase and unsaturated flow in porous media.

Replication Data for: Effect of varying atmospheric conditions on methane boundary-layer development in a free flow domain interfaced with a porous media domain

Replication Data for: Effect of varying atmospheric conditions on methane boundary-layer development in a free flow domain interfaced with a porous media domain

An assessment of turbulence transportation near regular and random permeable interfaces

Turbulent channel flow with a porous wall is investigated using direct numerical simulation, where the porous media domain consists of regular or random circular cylinder arrays. We compare the statistics and structure of the mean flow and turbulence in the channel flow with a bulk Reynolds number of 2500 and two porosities (φ=0.6 and 0.8) for the porous media. It is shown that the random interface significantly affects the dynamics of turbulence and the time-averaged flow. More intense mixing is observed near the random interface due to augmented form-induced shear stresses. Due to the strong dependence of induced flow direction on the interface geometry, we segmented the flow field into two types of areas based on the slope angle formed by the top-layer cylinders: the windward area and leeward area. The conditional average of turbulence kinematic energy budget over each type of area reveals their respective role in turbulence transportation more explicitly. In addition, we use finite-time Lyapunov exponents to inspect the Lagrangian coherent structures in the flow fields, which reveal the preferential fluid trajectories in the random porous medium geometry.

Conception the Fluid Flow Behavior within Oil Reservoir Rock by Using Computed Tomography (CT) Scan

The behavior of fluid flow has been studied during the different flow media over the past decades. In addition, the behavior of the flow of fluid through porous media has garnered much research interest. This paper sheds light on fissured rocks of oil reservoir media (as one of the porous media domain), and the effect of these fissured on fluid flow. In this article, the Finite Volume Method (FVM) has been used to visualize the behavior of single-phase fluid flow in an actual core according to the dualporosity dual permeability model. The study was conducted in two parts, the first was the image processing for one of the real oil reservoir fractured rock images, where the image was processed and simulated by ANSYSCFX software, and the results showed a complete visualizing of the fluid behavior during this domain. As for the other side, a simulation of a real reservoir rock belonging to the Al-Nour field in Iraq / Misan was made. The X-ray Computed Tomography (CT) scan has been used to convert the real fractured core to a dynamic domain. ANSYS-CFX program has been used and the results illustrated the pressure counter, the velocity counter, the velocity streamline, and the velocity vectors for the studied model in three dimensions. A comparison was made between the productivity index for fractured and non-fractured rock and the results explained that the presence of fracture can improve the productivity index to about 5.74%.

NITSCHE'S TYPE STABILIZATION FOR THE FULLY MIXED NAVIER-STOKES/DARCY PROBLEM

In this paper, we present and analyze a fully mixed finite element scheme for the Navier-Stokes/Darcy problem based on the Nitsche's type interface stabilizations, in the fluid region coupled with the porous media domain. The reasonable parameter $ \delta>0 $, which is independent of mesh size $ h $, will guarantee the stability and optimal convergence of our stabilized scheme. Moreover, we explicitly derive the dependence and requirement of the stabilization parameter $ \delta $ for the optimal error estimates, while the numerical tests support the stability and efficiency of this stabilized mixed method.

Thermomechanical phase‐field fracture modeling of fluid‐saturated porous media

AbstractIn this work, the problem of brittle fracture in a fluid‐saturated porous material is extended by considering the non‐isothermal states of the sample. The temperature field will affect the problem in two aspects: 1) Temperature‐dependent material parameters, such as elasticity modulus (E) and critical energy release rate (Gc). 2) Thermal expansion due to thermo‐mechanical volume coupling. In hydraulic fracturing, we further study the effect of the temperature difference between the injected fluid and the surrounding porous media ambient on the crack behavior. The modeling of the porous media domain is based on the macroscopic theory of porous media (TPM), whereas the phase‐field method (PFM) is applied to approximate the sharp crack edges by diffusive ones. In the numerical implementation, the coupled system of partial differential equations will be solved using the FEM in order to simulate the heat transition in the crack and non‐crack regions.

Transport of Turbulence Across Permeable Interface in a Turbulent Channel Flow: Interface-Resolved Direct Numerical Simulation

Turbulence transportation across permeable interfaces is investigated using direct numerical simulation, and the connection between the turbulent surface flow and the pore flow is explored. The porous media domain is constructed with an in-line arranged circular cylinder array. The effects of Reynolds number and porosity are also investigated by comparing cases with two Reynolds numbers (Reapprox 3000,6000) and two porosities (varphi =0.5,0.8). It was found that the change of porosity leads to the variation of flow motions near the interface region, which further affect turbulence transportation below the interface. The turbulent kinetic energy (TKE) budget shows that turbulent diffusion and pressure transportation work as energy sink and source alternatively, which suggests a possible route for turbulence transferring into porous region. Further analysis on the spectral TKE budget reveals the role of modes of different wavelengths. A major finding is that mean convection not only affects the distribution of TKE in spatial space, but also in scale space. The permeability of the wall also have an major impact on the occurrence ratio between blow and suction events as well as their corresponding flow structures, which can be related to the change of the Kármán constant of the mean velocity profile.

Parametric Nonlinear Model Reduction Using K-Means Clustering for Miscible Flow Simulation

This work considers the model order reduction approach for parametrized viscous fingering in a horizontal flow through a 2D porous media domain. A technique for constructing an optimal low-dimensional basis for a multidimensional parameter domain is introduced by combining K-means clustering with proper orthogonal decomposition (POD). In particular, we first randomly generate parameter vectors in multidimensional parameter domain of interest. Next, we perform the K-means clustering algorithm on these parameter vectors to find the centroids. POD basis is then generated from the solutions of the parametrized systems corresponding to these parameter centroids. The resulting POD basis is then used with Galerkin projection to construct reduced-order systems for various parameter vectors in the given domain together with applying the discrete empirical interpolation method (DEIM) to further reduce the computational complexity in nonlinear terms of the miscible flow model. The numerical results with varying different parameters are demonstrated to be efficient in decreasing simulation time while maintaining accuracy compared to the full-order model for various parameter values.

Interpretation of multi-scale permeability data through an information theory perspective

Abstract. We employ elements of information theory to quantify (i) the information content related to data collected at given measurement scales within the same porous medium domain and (ii) the relationships among information contents of datasets associated with differing scales. We focus on gas permeability data collected over Berea Sandstone and Topopah Spring Tuff blocks, considering four measurement scales. We quantify the way information is shared across these scales through (i) the Shannon entropy of the data associated with each support scale, (ii) mutual information shared between data taken at increasing support scales, and (iii) multivariate mutual information shared within triplets of datasets, each associated with a given scale. We also assess the level of uniqueness, redundancy and synergy (rendering, i.e., information partitioning) of information content that the data associated with the intermediate and largest scales provide with respect to the information embedded in the data collected at the smallest support scale in a triplet. Highlights. Information theory allows characterization of the information content of permeability data related to differing measurement scales. An increase in the measurement scale is associated with quantifiable loss of information about permeability. Redundant, unique and synergetic contributions of information are evaluated for triplets of permeability datasets, each taken at a given scale.