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The direct proportionality between the flow rate and the pressure gradient of creeping flows was experimentally discovered by H. Darcy in the 19th century and theoretically justified a couple of decades ago using upscaling methods such as volume averaging or homogenization. X-ray computed micro-tomography (CMT) and pore-scale numerical simulations...
Citations
... The velocity and pressure variables only need to be solved within the fluid region using the Navier-Stokes equations. Specifically, for low Reynolds numbers and constant temperature conditions, the fluid is treated as incompressible [18,25,30]. However, for high Reynolds number, incidentally, high Mach number, or when temperatures vary, the fluid is considered compressible. ...
With the rising cost of energy and the advancement of corporate social responsibility, there is a growing interest
in addressing the challenge of recovering and storing high-temperature waste heat. Sensible heat storage in
packed beds stands out as a cost-effective and seemingly straightforward solution for high-temperature Thermal
Energy Storage (TES). Engineering models developed to design low-temperature TES systems were tentatively
used to design this new generation of high-temperature systems. Delving into the physics of coupled heat
and mass transfer reveals a lack of validation of this approach. This study seeks to establish a comprehensive
bottom-up methodology - from the particle scale up to the system level - to provide informed and validated
engineering models for the design of high-temperature TES systems. To achieve this goal, we developed
a multi-scale numerical model to explore the physics of heat and momentum transfer in packed-bed TES
systems. At the microscopic scale (pore/particle), we consider the flow of a compressible high-temperature gas
between the particles, coupled to transient heat conduction within the particles, with particular attention given
to incorporating accurate temperature-dependent viscosity for the gas phase and thermal conductivity and
density for both solid and gas phases. At the macroscopic scale (engineering), we propose a high-temperature
extension of state-of-the-art two-equation TES models. The governing equations considered are the volume-
average conservation laws for gas-mass, gas-momentum and energy of both phases.
... Secondly, the lack of data on macroscopic properties, namely, the permeability and Forchheimer tensors is particularly challenging for the relevance of the models. In previous work, [20], micro-scale simulations were performed to solve the Navier-Stokes equations under the Darcy flow assumption, and the permeability tensor was predicted based on the results. The permeability was then compared with values obtained by Borner et al. [21] using direct simulation Monte Carlo methods, revealing an error of nearly 42%. ...
... Other parameters are the gas viscosity and density . Two main approaches can be utilized to estimate the permeability and Forchheimer correction tensors numerically: one based on direct micro-scale simulations using classical Navier-Stokes equations, and the other on upscaling theories and solving of the associated closure problems [14,17,20,24,25]. In cases involving periodic media or when dealing with representative elementary volumes, both methods provide the same results. ...
... However, defining a numerical approach for nonperiodic anisotropic porous media still remains necessary. In the first method, the permeability tensor, as derived from Darcy's law, is determined by solving the Navier-Stokes equations to obtain pressure and velocity terms which are then suitably averaged [20,25]. In the second method, the permeability tensor and Forchheimer correction tensor can be numerically estimated by solving a closure problem on a periodic unit cell representative of the structure, such as arrays of spheres [14,17] or a digital structure based on tomographic images of porous media, such as porous rocks [24]. ...
... In order to make the domain periodic, A thin layer of fluid mesh around the domain is added. This method, which is evaluated and discussed in [36], allows solving the closure problem on non-periodic domains with good accuracy. The geometry is always approximated using a Cartesian mesh and this creates mesh slots when the geometry is approached. ...
... The fluid-solid thermal conductivity ratio is still used as being equal to 100. The characteristic distance used for the Peclet number is the average distance between two pores given by [36] and equal to 135 µm. The results obtained are presented in Fig. 13. ...
... In a porous media, the average gas velocity is obtained from the Darcy's law [36,42,43] = − 1 1 · ∇ (27) where is the gas pressure, the void volume fraction, the gas viscosity and is the second-order tensor permeability. Assuming a perfect gas mixture, the following equation is obtained for pressure and considering creeping flows in the continuum regime in the pores, the equation is rewritten as follows ...
... Information content evaluation is needed not only for stochastic reconstructions. Physical properties of any material depend on its structure, the boundary conditions [17,[59][60][61] and representativeness. The latter quality is tricky, but in many cases crucial to evaluate [62][63][64]. ...
Structures are abundant in both natural and human-made environments and usually studied in the form of images or scattering patterns. To characterize structures a huge variety of descriptors is available spanning from porosity to radial and correlation functions. In addition to morphological structural analysis, such descriptors are necessary for stochastic reconstructions, stationarity and representativity analysis. The most important characteristic of any such descriptor is its information content — or its ability to describe the structure at hand. For example, from crystallography it is well known that experimentally measurable S_2 correlation function lacks necessary information content to describe majority of structures. The information content of this function can be assessed using Monte-Carlo methods only for very small 2D images due to computational expenses. Some indirect quantitative approaches for this and other correlation function were also proposed. Yet, to date no methodology to obtain information content for arbitrary 2/3D image is available. In this work, we make a step towards developing a general framework to perform such computations analytically. We show, that one can assess the entropy of a perturbed random field and that stochastic perturbation of fields’ correlation function decreases its information content. In addition to analytical expression, we demonstrate that different regions of correlation function are in different extent informative and sensitive for perturbation. Proposed model bridges the gap between descriptor-based heterogeneous media reconstruction and information theory and opens way for computationally effective way to compute information content of any descriptor as applied to arbitrary structure.
... Image-based CFD simulations of Darcy's style experiments used to estimate permeability have been reported for the performances of carbon fibres and brain tissue in [34,14]. In [34], it is discussed that the creeping flow regime is valid for inlet velocities less than 0.1m/s and a Re < 0.5, within the study conducted by [14] an inlet velocity of 0.0025 m/s is used. ...
... Image-based CFD simulations of Darcy's style experiments used to estimate permeability have been reported for the performances of carbon fibres and brain tissue in [34,14]. In [34], it is discussed that the creeping flow regime is valid for inlet velocities less than 0.1m/s and a Re < 0.5, within the study conducted by [14] an inlet velocity of 0.0025 m/s is used. Darcy's law is used to estimate permeability; it has been found that the transition between Darcian and non-Darcian regimes occurs at an inlet velocity of 0.02m/s, the corresponding average Re in the domain is between 1-2, corresponding to a pressure drop through the sample of 0.0021MPa (0.02bar). ...
The meniscal tissue is a layered material with varying properties influenced by collagen content and arrangement. Understanding the relationship between structure and properties is crucial for disease management, treatment development, and biomaterial design. The internal layer of the meniscus is softer and more deformable than the outer layers, thanks to interconnected collagen channels that guide fluid flow. To investigate these relationships, we propose a novel approach that combines Computational Fluid Dynamics (CFD) with Image Analysis (CFD-IA). We analyze fluid flow in the internal architecture of the human meniscus across a range of inlet velocities (0.1mm/s to 1.6m/s) using high-resolution 3D micro-computed tomography scans. Statistical correlations are observed between architectural parameters (tortuosity, connectivity, porosity, pore size) and fluid flow parameters (Re number distribution, permeability). Some channels exhibit Re values of 1400 at an inlet velocity of 1.6m/s, and a transition from Darcy's regime to a non-Darcian regime occurs around an inlet velocity of 0.02m/s. Location-dependent permeability ranges from 20-32 Darcy. Regression modelling reveals a strong correlation between fluid velocity and tortuosity at high inlet velocities, as well as with channel diameter at low inlet velocities. At higher inlet velocities, flow paths deviate more from the preferential direction, resulting in a decrease in the concentration parameter by an average of 0.4. This research provides valuable insights into the fluid flow behaviour within the meniscus and its structural influences.
... In Eq.1, the permeability can be determined by experimental measurements or numerical simulations [31]. The effective conductivity, , is known to be bounded by the arithmetic ( = ( + ) ) and the harmonic ( = ( ∕ + ∕ ) −1 ) averages of the solid and gas conductivities. ...
... The sample dimension and the architectural properties of Calcarb are listed in Table 2. Due to the manufacturing process, there are clusters of fibers made of five to ten fibers. The mean diameter of the fiber clusters has been shown to be the most relevant characteristic length to compute the Reynolds number as it triggers the formations of eddies in the pores [31]. It will be used in the modeling section. ...
... Considering the gas velocity and density ranges provided in the previous section, is found to vary from 0.95 to 4.2. In a previous study, it was proved that Darcy's law remains valid with an error under 1% for < 5 [31]; hence, in this work, we will use Darcy's law. The coupling between the different regions is done by considering the conservation of mass and the continuity of temperatures and heat fluxes at the interfaces. ...
... Information content evaluation is needed not only for stochastic reconstructions. Physical properties of any material depend on its structure, the boundary conditions [17,59,60,61] and representativeness. The latter quality is tricky, but in many cases crucial to evaluate [62,63,64]. ...
Structures are abundant in both natural and human-made environments and usually studied in the form of images or scattering patterns. To characterize structures a huge variety of descriptors is available spanning from porosity to radial and correlation functions. In addition to morphological structural analysis, such descriptors are necessary for stochastic reconstructions, stationarity and representativity analysis. The most important characteristic of any such descriptor is its information content - or its ability to describe the structure at hand. For example, from crystallography it is well known that experimentally measurable $S_2$ correlation function lacks necessary information content to describe majority of structures. The information content of this function can be assessed using Monte-Carlo methods only for very small 2D images due to computational expenses. Some indirect quantitative approaches for this and other correlation function were also proposed. Yet, to date no methodology to obtain information content for arbitrary 2D or 3D image is available. In this work, we make a step toward developing a general framework to perform such computations analytically. We show, that one can assess the entropy of a perturbed random field and that stochastic perturbation of fields correlation function decreases its information content. In addition to analytical expression, we demonstrate that different regions of correlation function are in different extent informative and sensitive for perturbation. Proposed model bridges the gap between descriptor-based heterogeneous media reconstruction and information theory and opens way for computationally effective way to compute information content of any descriptor as applied to arbitrary structure.
... Structure-property relationships (Sahimi, 2003;Torquato, 2002) are the basis of the pore-scale modeling technology that targets directly solving governing physical equations within the geometry of the materials under study or to simplify the relationships using various statistical approaches. In particular, flow and transport characteristics for porous media in the form of, for example, permeability tensors (Guibert et al., 2016;Gerke et al., 2019;Scandelli et al., 2022), relative permeabilities and capillary curves for multi-phase flow (Valvatne and Blunt, 2004;Patzek, 2001;Ryazanov et al., 2009), electrical properties (Khachkova et al., 2021) and dispersion tensors (Salles et al., 1993), are of utmost importance for applications in petroleum engineering (Blunt, 2017;Godinho et al., 2016), hydrology and soil physics (Barbosa et al., 2022a;Ferreira et al., 2022), food engineering (Derossi et al., 2019) and numerous other disciplines. The results of the pore-scale simulations can be used to optimize the parameters of the 2 sequestration, hydrocarbon recovery or soil irrigation, and to parameterize the continuum-scale models to describe the flow problems in the m-km scale. ...
The paper is focused on high efficiency Stokes solver that is applied to the incompressible flow in porous media. Computational domains are represented by binarized 3D computed tomography voxel models. The solution procedure is constructed around a fast algebraic multigrid solver that utilizes the power of graphics processing units (GPUs). In order to minimize memory footprint and accelerate the solver, a simple MAC-type staggered finite difference discretization is used and a coupled Stokes saddle-point type system is solved directly on a GPU. The MAC discretization is discussed, taking particular care of internal solid boundaries around pores. The method includes topological domain analysis on a GPU, which removes isolated volumes with no flow in the domain (based on the connected component labeling), depending on the boundary conditions. We consider various types of boundary conditions and efficient parallel strategies for the GPUs, including fast matrix assembly and residual regularization. Our method is extensively benchmarked both against analytical solutions and applied problems from digital rock physics in terms of computational wall time, precision, approximation order, and convergence. We demonstrate that it takes up to 5–23 s on a modern GPU card to obtain a solution with 1⋅10−6 error residuals for 3D geometries with 300–4503 voxels and a porosity range of 5–37%.
... Analyses and reviews of these methods can be found, for example, in [30][31][32][33][34][35][36]. Their applications to estimate tensorial permeabilities in numerical experiments or in the laboratory have been studied by many authors (including, for example, [23,25,[37][38][39][40][41][42][43][44]. Other works have focused on optimal estimation of heterogeneous reservoir properties such as permeability k(x, y). ...
... k(x, y)/k 0 = 1 + 2a sin(x) cos(y) + a 2 cos 2 (y) (44) where k 0 is a reference value of permeability, obtained for instance at point x = 0, y = π/2 (and at all points located on horizontal lines y = π/2). In the remainder of this section, we take every where k 0 = 1, or equivalently, the permeability is rescaled by taking k(x, y) ← k(x, y)/k 0 . ...
... ∂ϕ/∂y = a cos(x) sin(y) / 1 + 2a sin(x) cos(y) + a 2 cos 2 (y) (47) Note that ∂ϕ/∂y represents the local head gradient ∇h(x, y) in our notations. Finally, note that, in fact, the permeability pattern k(x, y) of Equation (44) was inferred by Philip (1986) by solving exactly the following inverse problem: given the local flow pattern described by Equations (46) and (47), find the permeability field that satisfies this flow pattern: the result is the spatially distributed field k(x, y) given just above in Equation (44). Figure 4 displays the permeability pattern and the corresponding flow field of Philip (1986) for a = 1. These suggestions have not been pushed further for the present periodic pattern; the probabilistic or spatial integrals needed are cumbersome, and it seems that the procedures suggested just above have not been carried out yet in the literature for this quasi-analytical flow pattern. ...
When conducting numerical upscaling, either for a fractured or a porous medium, it is important to account for anisotropy because in general, the resulting upscaled conductivity is anisotropic. Measurements made at different scales also demonstrate the existence of anisotropy of hydraulic conductivity. At the “microscopic” scale, the anisotropy results from the preferential flatness of grains, presence of shale, or variation of grain size in successive laminations. At a larger scale, the anisotropy results from preferential orientation of highly conductive geological features (channels, fracture families) or alternations of high and low conductive features (stratification, bedding, crossbedding). Previous surveys of homogenization techniques demonstrate that a wide variety of approaches exists to define and calculate the equivalent conductivity tensor. Consequently, the resulting equivalent conductivities obtained by these different methods are not necessarily equal, and they do not have the same mathematical properties (some are symmetric, others are not, for example). We present an overview of different techniques allowing a quantitative evaluation of the anisotropic equivalent conductivity for heterogeneous porous media, via numerical simulations and, in some cases, analytical approaches. New approaches to equivalent permeability are proposed for heterogeneous media, as well as discontinuous (composite) media, and also some extensions to 2D fractured networks. One of the main focuses of the paper is to explore the relations between these various definitions and the resulting properties of the anisotropic equivalent conductivity, such as tensorial or non-tensorial behavior of the anisotropic conductivity; symmetry and positiveness of the conductivity tensor (or not); dual conductivity/resistivity tensors; continuity and robustness of equivalent conductivity with respect to domain geometry and boundary conditions. In this paper, we emphasize some of the implications of the different approaches for the resulting equivalent permeabilities.
This article explores the possibility to assess the flow and transport properties of loosely consolidated rock material—something that is very hard or impossible to achieve in the laboratory due to fragility of cores. We present two cases of weakly consolidated and unconsolidated rocks. We provide a solution based on pore-scale simulations and stochastic reconstructions using scanning electron (SEM) and grain optical microscopy images as input data. The hybrid reconstruction approach is based on 3D grain shape construction out of 2D optical images, packing of grains to match the target porosity measured from SEM imaging, and addition of clay and other cementing phases with the help of phase-recovery method. Note that standard digital rock protocol based on X-ray microtomography did not work for considered samples due to fine-grained particle size distribution (insufficient resolution of X-ray microtomography). After creation of 3D digital replicas of rock samples based on their SEM and optical microscopy images, we applied pore-scale modeling to obtain permeability and two-phase flow properties. Simulated permeability of 259 mD for the first sample was in surprisingly good agreement with laboratory measurements of 248 mD. For the second sample permeabilities deviated by an order of magnitude. After additional studies it was found that the mesh attached to the sample during measurements affected the results. After pore-scale simulations of the grain packing with the mesh we were able to achieve very good agreement with the experiment, confirming that the lab was basically exploring the properties of the mesh clogged with unconsolidated rock material. Thus, pore-scale hybrid rock structure reconstruction technique combined with pore-scale simulations was able to correct inaccurate laboratory assessment and obtain flow properties for unconsolidated rock sample. We believe the developed hybrid reconstruction technique to be robust enough to serve as a basis of the industrial technology for petrophysical studies of weakly and unconsolidated core material.
