Transport in Porous Media

Jafarizadeh, Azadeh; Ahmadzadeh, MohammadAli; Mahmoudzadeh, Sajad; Panjepour, Masoud

doi: 10.1007/s11242-022-01895-0pmid: N/A

The present study investigates the effects of the geometric properties of porous media on fluid flows and uses a combination of computational fluid dynamics (CFD) and machine learning (ML) methods to develop new models for predicting the coefficients in the Forchheimer equation (ΔP/L = αv + βv2). For this purpose, CFD simulations are performed to assess the effects of foam structural properties on fluid flows for each of the 217 foam types tested. In this research, the Voronoi tessellation method is used to investigate such foam physical properties as porosity, pore diameter, strut diameter, and specific surface area. In all the simulations, air is used as the fluid entering the metallic foams at different superficial velocities but at a constant temperature of 300 K. Pressure gradient as well as Darcy (α) and non-Darcy (β) flow coefficients are then calculated for each foam using the equation and the simulation results. It is shown that the values thus obtained strongly depend on the geometric properties of the porous medium. A second aspect of the study involves resolving the problems of the computation cost due to the complex geometries of foams that result in too many computational grids in the proposed method. This was addressed via machine learning (ML) regression models to develop a continuous model based on foam intrinsic properties. In this approach, models are proposed for estimating α and β coefficients to be used in calculating pressure drop in different metallic foams. The results show that the ridge regularization regression and ordinary least squares are robust models for predicting the coefficients based on foam geometric properties. Moreover, the model is capable of calculating both the values for Reynolds number and friction factor in the continuous range, whereby the flow type can also be determined. Finally, the results obtained from the models indicate the efficacy of the proposed approach for the study of fluid flows in large-scale porous media with minimum errors and satisfactory accuracy.

Jin, Zhihe; Yuan, Fan

doi: 10.1007/s11242-023-01904-wpmid: N/A

This work presents a double-permeability poroelasticity model for fluid flows in both the microvascular and interstitial networks in a biological tissue. In the newly developed model, both networks are modeled as porous structures with distinct permeabilities and porosities. The microvascular and the interstitial fluid pressures are hydraulically as well as mechanically coupled together. The numerical results for the steady-state flow in a one-dimensional capillary bed using some preliminary material parameters show that the vascular pressure decreases almost linearly from the arteriole-side to the venule-side. The interstitial fluid pressure (IFP) is elevated by an increase in the venule-side vascular pressure as well as by a decrease in the lymphatic drainage capability. Under a transient flow condition induced by a sudden drop in the venule-side vascular pressure, the IFP may pop up during a very short period of time before decreasing to the reduced steady-state value at long times due to the mechanical coupling between the vascular pressure and IFP which acts much faster than the hydraulic coupling between the two pressures through the vascular walls. Oscillatory mechanical load may produce comparable IFP and promote fluid exchange between the microvessels and interstitium. Finally, a perturbation analysis reveals that a boundary layer for the IFP develops near the tissue boundary. For the first-order approximation, the vascular pressure is decoupled from the IFP and the IFP may be obtained with the first-order vascular pressure as a source.

Magyari, Eugen

doi: 10.1007/s11242-022-01897-ypmid: N/A

Recently a new empirical model for the capillary pressure in porous media has been proposed (see Foroughi et al. in Transp Porous Media 145:683–696, 2022). In the present Note some of the mathematical features of this model are discussed in more detail. The focus is the lowest slope dP¯c/dSemin\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\left| {{\text{d}}\,\overline{P}_{{\text{c}}} /{\text{d}}\,S_{{\text{e}}} } \right|_{{{\text{min}}}}$$\end{document} of the pressure–saturation curves P¯c=P¯cSe\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\overline{P}_{{\text{c}}} = \overline{P}_{{\text{c}}} \left( {S_{{\text{e}}} } \right)$$\end{document}. In this respect an absolute minimum, as well as dual saturation-exponents have been detected.

Knyazeva, A. G.; Maslov, A. L.

doi: 10.1007/s11242-023-01901-zpmid: N/A

Shale is sedimentary rock formed by the accumulation of pelagic sediments, minerals and their further transformation. The pyrolysis of oil shale can yield liquid hydrocarbons and a combustible gas that can be used as energy sources. There are various methods of oil shale thermal processing. However, methods that do not require the extraction of rock to the surface (in situ methods) are of special interest. A mathematical model of in situ oil shale heating and decomposition is presented. In comparison with known papers, new model describes processes under “non-equilibrium” conditions, when gas pressure in pores is determined by temperature and composition change. Model also allows describing the flow of the reacting mixture forming product in the presence of a production well and can be used to determine the technological parameters of the well and the heating process that provide maximum valuable product extraction rate.

Bansal, A.; Suthar, Om P.

doi: 10.1007/s11242-022-01898-xpmid: N/A

Thermal convection in a Newtonian fluid-saturated horizontal porous medium is studied using the linear stability analysis in the present study. The porous medium is uniformly rotating about a vertical axis, and the fluid and porous matrix are out of thermal equilibrium. The horizontal boundaries are assumed to be subjected to time-periodic temperatures with heating from below. The extended Darcy law, which includes the Coriolis force and time derivative terms, is used to model the linear momentum conservation equation. A deviation in the critical Darcy-Rayleigh number is calculated as a function of governing parameters, and the impact of those is illustrated graphically to understand the effect of modulation on the onset of convection, mainly when the porous matrix and fluid are not in local thermal equilibrium. It is noted that, at low-frequency symmetric modulation, the instability can be enhanced by rotation. In contrast, in the case of asymmetric modulation, the stability can be enhanced by rotation.

Le Beulze, Aurélie; Santos De Pera, Nathalie; Levaché, Bertrand; Questel, Mathias; Panizza, Pascal; Lequeux, François; Levant, Michael; Passade-Boupat, Nicolas

doi: 10.1007/s11242-023-01900-0pmid: N/A

We present a new experimental approach to further understand the injectivity impairment due to reinjection of produced water in an oilfield, containing residual oil and solids. A unique microfluidic setup with imposed flowrate is characterized by excellent reproducibility and allows one to determine the kinetics of external cake formation and the propagation of the damage inside the porous medium, similar to what happens at the injection wellbore. The growth rates of the external cake and that of the propagation of the internal damage exhibit discontinuities, likely related to a pressure buildup up to a threshold Laplace pressure above which the O/W Pickering droplets are pushed through, and which sets a limit to the cake growth. Finally, the external cake reaches a quasi-stationary thickness whose mechanisms are discussed below. Direct visualization readily achieved in microfluidic experiments, coupled with spatiotemporal image analysis, enables better spatial resolution than core flooding experiments and shows that the damage occurs in a small region close to the entry to the porous medium. These developments lead to the derivation of an analytical model of the damage formation. It appears that although very localized, this damage strongly decreases the global permeability of the whole porous medium. Finally, controlled temperature experiments permit to identify the variation of the viscosity of the oil droplets (or the viscosity ratio), as the primary mechanism by which temperature influences clogging. Clogging is slowed at high temperatures, but the final state is characterized by particle clogging and is thus irreversible.

Cheon, Hyejeong L.; Fyhn, Hursanay; Hansen, Alex; Wilhelmsen, Øivind; Sinha, Santanu

doi: 10.1007/s11242-022-01893-2pmid: N/A

We study immiscible two-phase flow of a compressible and an incompressible fluid inside a capillary tube of varying radius under steady-state conditions. The incompressible fluid is Newtonian and the compressible fluid is an inviscid ideal gas. The surface tension associated with the interfaces between the two fluids introduces capillary forces that vary along the tube due to the variation in the tube radius. The interplay between effects due to the capillary forces and the compressibility results in a set of properties that are different from incompressible two-phase flow. As the fluids move towards the outlet, the bubbles of the compressible fluid grows in volume due to the decrease in pressure. The volumetric growth of the compressible bubbles makes the volumetric flow rate at the outlet higher than at the inlet. The growth is not only a function of the pressure drop across the tube, but also of the ambient pressure. Furthermore, the capillary forces create an effective threshold below which there is no flow. Above the threshold, the system shows a weak nonlinearity between the flow rates and the effective pressure drop, where the nonlinearity also depends on the absolute pressures across the tube.

Knoch, Jonas; Gahn, Markus; Neuss-Radu, Maria; Neuß, Nicolas

doi: 10.1007/s11242-022-01896-zpmid: 36628266

In this paper, we derive an effective model for transport processes in periodically perforated elastic media, taking into account, e.g., cyclic elastic deformations as they occur in lung tissue due to respiratory movement. The underlying microscopic problem couples the deformation of the domain with a diffusion process within a mixed Lagrangian/Eulerian formulation. After a transformation of the diffusion problem onto the fixed domain, we use the formal method of two-scale asymptotic expansion to derive the upscaled model, which is nonlinearly coupled through effective coefficients. The effective model is implemented and validated using an application-inspired model problem. Numerical solutions for both, cell problems and macroscopic equations, are investigated and interpreted. We use simulations to qualitatively determine the effect of the deformation on the transport process.

Bez, Luiz F.; Leiderman, Ricardo; Souza, André; Azeredo, Rodrigo B. de V.; Pereira, André M. B.

doi: 10.1007/s11242-022-01894-1pmid: N/A

NMR techniques are key in the study of porous reservoir rock, both experimentally and numerically. The T2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$T_2$$\end{document} relaxation process, the most common application of NMR, measures the loss of coherence of transversal magnetization and strongly depends on the fluid/matrix interaction—thus providing useful insights into the pore size distribution of a rock sample. The pore space is often studied at the μm scale through the use of micro-CT images, which are formed by stacks of images with hundreds of thousands of pixels each, posing significant challenges to numerical simulations. In this paper, we present an image-based, fully explicit, and matrix-free finite element implementation for the simulation of the T2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$T_2$$\end{document} relaxation process that is capable of handling such large problems. The utilized mathematical model considers relaxation due to bulk effects and surface relaxivity, not taking into account the effects of magnetic field gradients. We propose the usage of stable time marching schemes that use hyperbolization as means of acquiring stability with large time-steps. We compare the numerical performance of different time-marching schemes, showing that the application of the Leap-Frog method in a hyperbolized form of the equation can give the best trade-off between memory use and numerical convergence. Additionally, we show that the use of a lumped mass matrix allows for a fully explicit and simpler implementation while adding negligible amounts of numerical error.Article HighlightsA novel image-based and matrix-free methodology combined with explicitstable methods permits the solution of large problems with low time andlow memory consumption.Performance of three explicit time-integration schemes is studied numericallywith different coefficients, grid sizes, and time-step sizes.The lumping of the mass matrix allows the implementation of a fully explicitscheme while adding negligible amounts of numerical error.