Capillary Bundle Research Articles

Whole Organ Volumetric Sensing Ultrasound Localization Microscopy for Characterization of Kidney Structure.

Glomeruli are the filtration units of the kidney and their function relies heavily on their microcirculation. Despite its obvious diagnostic importance, an accurate estimation of blood flow in the capillary bundle within glomeruli defies the resolution of conventional imaging modalities. Ultrasound Localization Microscopy (ULM) has demonstrated its ability to image in-vivo deep organs in the body. Recently, the concept of sensing ULM or sULM was introduced to classify individual microbubble behavior based on the expected physiological conditions at the micrometric scale. In the kidney of both rats and humans, it revealed glomerular structures in 2D but was severely limited by planar projection. In this work, we aim to extend sULM in 3D to image the whole organ and in order to perform an accurate characterization of the entire kidney structure. The extension of sULM into the 3D domain allows better localization and more robust tracking. The 3D metrics of velocity and pathway angular shift made glomerular mask possible. This approach facilitated the quantification of glomerular physiological parameter such as an interior traveled distance of approximately 7.5 ±0.6 microns within the glomerulus. This study introduces a technique that characterize the kidney physiology which can serve as a method to facilite pathology assessment. Furthermore, its potential for clinical relevance could serve as a bridge between research and practical application, leading to innovative diagnostics and improved patient care.

An investigation of the behaviour of polymer fluids in the American Petroleum Institute filter press

The American Petroleum Institute (API) filter press test has been used for decades in the construction industry as part of the quality control regime for bentonite-based excavation support fluids. The industry has carried over the use of this test to polymer fluids despite the lack of published evidence of its suitability for these fluids and the very different mechanisms by which polymer fluids and bentonite slurries achieve excavation support. This paper presents the first systematic investigation of this issue through a combination of laboratory testing and theoretical analysis. The investigation demonstrates the very different behaviours of bentonite slurries and polymer fluids. In contrast to the results for bentonite slurries, API filter press results for polymers are shown to be highly sensitive to the filter paper used. In particular, repeatability testing revealed a substantial variation in the polymer fluid loss rates attributable to three primary factors: (a) the filter paper pore size; (b) filter paper damage resulting from the applied test pressure; (c) apparent ‘clogging’ of the filter paper pore space. Furthermore, the study demonstrates the poor repeatability of the API filter press test for polymer fluids even when filter papers of the same type are used. Interestingly, analysis of polymer flow with respect to filter paper pore size and the applied pressure showed that the filter papers were behaving as porous media rather than a simple bundle of capillaries; their behaviour could not be modelled using a simple capillary bundle model. Importantly, this finding shows that the filter press may provide a rapid method of assessing the apparent viscosity of polymer fluids in porous media at high shear rates – data that cannot be obtained by rotational viscometry, and would otherwise require resort to permeameter testing of coarse soils. The investigation demonstrates that the filter press test is not useful for the on-site quality control of polymer fluids but, given the theory presented in the paper, it can be a useful laboratory tool that provides valuable insight into polymer fluid flow behaviour in soils of high hydraulic conductivity, the most challenging soils for polymer fluid support.

The immiscible to miscible transition and its consequences for 3-phase displacements in porous media of arbitrary wettability: Comparison of 3D pore network results with basic theory

The theory of the immiscible to miscible transition and its consequences for 3-phase displacements in a system of arbitrary wettability distribution has been studied in a previous paper using a “process based” model through the simplest case possible; a capillary bundle (CB) model. As expected, the CB model was not able to capture many realistic phenomena such as phase trapping, film and layer-flow, topological effects etc. However, it was recognized that it should be able to indicate certain expected qualitative trends in 3-phase displacements in real 3D connected networks.The current paper extends that earlier work by showing the consequences of changing the model of the porous medium to a 3D connected lattice pore network model (PNM) to capture phase trapping and rock connectivity. Initially a simple 3D regular cubic bond lattice is used for this initial work to make a clear comparison with the CB model. Two important characteristics of 3D network refer to (i) displacement phase accessibility, and (ii) defending phase trapping/escape. Three trapping/escape cases based on the probability of invading phase accessibility and defending phase escape have been defined in the 3D lattice network. Networks with different coordination numbers (z = 6 and 4) were studied in order to mimic different rock connectivity and study how and in which way the applied rules result in specific changes in 3-phase saturation paths, 3-phase pore occupancies and phase trapping. Again the focus is on the changes which are predicted to occur in all these quantities as the system moves from an immiscible (IFT σgo ∼ “large”) to a miscible (σgo → 0) case.In this paper, a weakly oil-wet (WOW) wettability distribution is considered and various phase invasions (gas, water, and oil) in the corresponding 2-phase systems are investigated. Both saturation paths and pore occupancy sequences of different phase invasions under different miscibility conditions are presented for both the CB and PNM models. It is found that having a well-connected network and considering a relaxed rule on defending and displacing phases, the results are quite similar to that of the CB model. Also, applying restricted trapping/escaping rules and less connected network mean deviation from the CB model, but the underlying physics of the immiscible to miscible transitions and the central physical parameter of 3-phase displacements in all cases are the same as those observed for the CB model.

Estimation of gas diffusion coefficient for gas/oil-saturated porous media systems by use of early-time pressure-decay data: An experimental/numerical approach

This paper presented a novel numerical method for estimating the gas diffusion coefficient based on the early-time pressure-decay data. Experimentally, “flooding–soaking” procedures were developed to perform the gas diffusion in an oil-saturated tight core under different gas phase volume conditions. After flooding, the capillary bundle model was used to calculate the oil–gas contact area. The early-time pressure-decay data of the gas phase were monitored and recorded during the soaking process. Theoretically, a non-equilibrium inner boundary condition coupled with the characteristics of experimental early-time pressure had been incorporated to develop a diffusion model for a gas/oil-saturated tight core system. Based on gas-phase mass balance equations and gas equation of state, the diffusion coefficients were optimized once the discrepancy between experimental data and numerical solutions was minimized. According to the estimated results in this study, the CH4 diffusion coefficients were 3.74 × 10−11 and 3.86 × 10−11 m2/s in tight core saturated with crude oil, respectively. Moreover, the oil–gas contact area significantly impacts the diffusion flux in oil-saturated porous media. Specifically, an additional 10% contact area results in a 75% increase in CH4 diffusion mass. In addition, with the application of our proposed model to CH4/bitumen and CO2/bitumen systems, the diffusion coefficients were in close agreement with the results reported in previous literature, indicating that the proposed model was applicable to both gas/liquid and gas/liquid-saturated porous media systems.

Response to the comments on “Cellular aggregation dictates universal spreading behaviour of a whole-blood drop on a paper strip”

In 2023, we published a research article in the Journal of Colloidal and Interface Science, based on our experimental findings and substantiating scaling arguments leading to a simple theoretical insight on the effect of red blood cell (RBC) aggregation on the wicking behaviour of a finite volume of blood as it navigates through the porous passages of a paper matrix (Laha et al., 2023). Of late, we received comments from Li (2024), which offered certain suggestions regarding the possible improvement of the capillary bundle model as considered in our article for analyzing the transport of blood through the paper pores. Herein, we provide a detailed discussion on each of the points raised by Li (2024) and rationalize our views in further details in addition to the contents already provided in our concerned article.

Comment on “Cellular aggregation dictates universal spreading behaviour of a whole-blood drop on a paper strip”

Laha et al. studied the diffusive behavior of a whole-blood drop on filter paper using the generalized capillary bundle model. However, some model parameters should be further refined to accurately reflect the physics involved in this diffusion process. Moreover, citations are missing for some key equations. Addressing these aspects will improve the model applicability to this application and benefit readers in accessing more accurate and detailed information.

Numerical study of transpiration cooling at different outlet angles and hole pattern

The effects of transpiration cooling depends on the distribution of micropores in porous materials. The work simplifies the porous medium to a micro-scale pore plate structure that is densely organized, based on the idea of the capillary bundle model. The effects of hole pattern and hole outlet angle on transpiration cooling are investigated using numerical simulation. It is discovered that the hole outlet angle mostly affects the homogeneity of temperature distribution and has minimal effect on the surface cooling efficiency. The temperature uniformity index dropped by 35.9% yet the surface cooling efficiency only declined by 1.1% when the outlet angle was lowered from 45° to −45°. Furthermore, the temperature uniformity and cooling efficiency are directly affected by the hole pattern. When the long axis of the elliptical hole is parallel to the mainstream, it can achieve the best temperature uniformity; however, when the long axis is perpendicular to the mainstream direction, it can achieve higher cooling efficiency. Third, the material’s permeability will be decreased to varying degrees depending on the hole pattern and hole outlet angle. The results have important reference significance for the design of porous materials used for transpiration cooling.

Optimization‐based pore network modeling approach for determination of hydraulic conductivity function of granular soils

AbstractA wide range of applications of unsaturated hydraulic conductivity is well known in geotechnical, hydrological, and agricultural engineering fields. The standard prediction models for hydraulic conductivity function overlook the complexity of soil pore structure and employ a simplistic approach based on the bundle of capillary tubes. This study proposes an alternative approach employing pore network models calibrated to match soil water retention data to predict the hysteretic hydraulic conductivity function of granular soils. A novel approach to constructing a multidirectional pore network built on an irregular lattice with variable coordination numbers is presented for the realistic representation of soil voids. The geometric and topological parameters of the pore network model are optimized using the genetic algorithm, and adequate pore‐scale processes (piston‐like advance and corner flow during drainage and piston‐like advance, pore body filling, and snap‐off during imbibition) are modeled to get reasonable predictions of hysteretic hydraulic conductivity functions over the entire suction range of granular soils. Comparisons between the pore network model results, standard physically based models, and measured data for a variety of granular soils show that the proposed pore network has a superior performance over other models and compares favorably to the experimental data.

Quantifying the Effect of Pore‐Size Dependent Wettability on Relative Permeability Using Capillary Bundle Model

AbstractRelative permeability is a key parameter for characterizing the multiphase flow dynamics in porous media at macroscopic scale while it can be significantly impacted by wettability. Recently, it has been reported in microfluidic experiments that wettability is dependent on the pore size (Van Rooijen et al., 2022). To investigate the effect of pore‐size‐dependent wettability on relative permeability, we propose a theoretical framework informed by digital core samples to quantify the deviation of relative permeability curves due to wettability change. We find that the significance of impact is highly dependent on two factors: (i) the function between contact angle and pore size (ii) overall pore size distribution. Under linear function, this impact can be significant for tight porous media with a maximum deviation of 1,000%.

Productivity Model Study of Water-Bearing Tight Gas Reservoirs Considering Micro- to Nano-Scale Effects

Tight sandstone is rich in micron- and nano-scale pores, making the two-phase flow of gas and water complex. Establishing reliable relative permeability and productivity models is an urgent issue. In this study, we first used a slip model to correct the gas phase’s no-slip Hagen–Poiseuille equation for nano- and micropores. Then, combined with the fractal theory of porous media and the tortuous capillary bundle model, we established two-phase relative permeability models for nanopores and micropores. These relative permeability models comprehensively consider the gas slippage effect, the initiation pressure gradient, the pores’ fractal characteristics, and water film mechanisms. Based on these models, we developed a three-region coupling productivity model for water-bearing tight gas reservoirs with multi-stage fractured horizontal wells. This productivity model considered the micro- and nano-scale effects and the heterogeneity of fracture networks. Then, the model was solved and validated with a field case. The results indicated that the three-region composite unsteady productivity model for water-bearing tight gas reservoirs, which incorporated micro- and nano-scale effects (with consideration of micro-scale and nano-scale phenomena in the fluid flow), could accurately predict a gas well’s productivity. An analysis of the factors influencing productivity showed that ignoring the micro- and nano-scale effects in water-bearing tight gas reservoirs will underestimate the reservoir’s productivity. The initial water saturation, the two-phase flow’s initiation pressure gradient, and capillary force are all negatively correlated with the productivity of gas wells, while the conductivity of the fractures is positively correlated with gas well productivity.

Enhancing energy efficiency of PCM wall in winter with novel PCM-DLCB coupling design

In order to enhance the thermal performance of the phase change material (PCM) wall during winter, a novel PCM and double-layer capillary bundles (PCM-DLCB) coupling wall is proposed. The PCM-DLCB wall utilizes the capillary bundle near the outer surface to capture solar energy during sunny winter days. The heat is then transferred to the capillary bundle near the inner surface through a circulating pump, promoting the melting of the PCM. Optimization and long-term operation comparison research were conducted using numerical simulation to investigate the potential of the PCM-DLCB wall to improve energy efficiency during winter. Five structural parameters, including the type of PCM, the position of the capillary bundle, the position and thickness of the PCM layer, the spacing of the capillary bundles, and the flow velocity in the capillary bundle, were selected as variables for structural and operation optimization. The optimization target was the accumulated heat gain from the inner surface of the wall. The research results indicate that: (1) The most suitable PCM is RT-18HC; (2) Placing the PCM layer between the double-layer capillary bundle yields better results compared to other arrangements; (3) Optimal energy saving is achieved when the outer capillary bundle is positioned 6 mm away from the outer surface and the inner capillary bundle is positioned 8 mm away from the inner surface; (4) The optimal thickness of the PCM layer is 10 mm; (5) The optimal flow velocity in the capillary bundle is approximately 0.04 m/s. Furthermore, long-term operation results based on a typical hot summer and cold winter climate zone reveal that the optimized PCM-DLCB wall provides a 7.80 % increase in indoor heat gain compared to an ordinary PCM wall during winter, and an 18.4 % increase compared to a wall without PCM.

The immiscible to miscible transition and its consequences for 3-phase displacements in porous media of arbitrary wettability: Basic theory

In this paper, a “process-based” 3-phase model is applied to explore how multiphase fluid flow behaviour in two non-uniform wettability systems, named weakly oil-wet (WOW) and mixed-wet large (MWL), change when gas and oil move towards miscibility (σgo → 0). In the course of the theoretical development, we indicate how parts of the theory have been validated and confirmed satisfactorily by summarising experimental data from the literature. To demonstrate how the complexities of the three-phase physics interact with the system wettability, we use a capillary bundle (CB) model for the porous medium. This is sufficient to show the main processes which occur in the transition from immiscible to miscible behaviour. Even although the CB model is the simplest possible case for a porous medium, the results demonstrate a high level of unpredictable complexity which cover key parameters of the physics of 3-phase flow within porous media. In addition, for each case, the results point to the specific physical parameter that controls the phase separation lines. In some cases, gas injection or oil injection in a WOW system, the phase displacement changes quite radically as the system goes towards miscibility since there is a transition in wetting order from O/W/G to O/G/W. The differences and similarities between saturation paths in WOW and MWL are described in detail. For a given pore size distribution (PSD), the 3-phase displacements are controlled by the parameters of the rock wettability structure (see text) and the 3 interfacial tensions. Moving towards miscibility, 3-phase fluid displacement for gas injection and oil invasion are controlled by IFTs, while for water flooding, wettability structure plays the key role at all miscibility conditions. Although the simple CB model cannot describe all phenomena, such as phase trapping, the phase displacement paths, still the pore occupancies are qualitatively correct. Results of various phase injection in WOW are broadly consistent with pore occupancy sequences observations from pore scale X-Ray image analysis in literature despite simplicity of the theory which is validated in terms of its underling physics.

A Review of Fracturing and Enhanced Recovery Integration Working Fluids in Tight Reservoirs

Tight reservoirs, characterized by low porosity, low permeability, and difficulty in fluid flow, rely on horizontal wells and large-scale hydraulic fracturing for development. During fracturing, a significant volume of fracturing fluid is injected into the reservoir at a rate far exceeding its absorption capacity. This not only serves to create fractures but also impacts the recovery efficiency of tight reservoirs. Therefore, achieving the integration of fracturing and enhanced recovery functions within the working fluid (fracturing-enhanced recovery integration) becomes particularly crucial. This study describes the concept and characteristics of fracturing-enhanced recovery integration and analyzes the types and features of working fluids. We also discuss the challenges and prospects faced by these fluids. Working fluids for fracturing-enhanced recovery integration need to consider the synergistic effects of fracturing and recovery; meet the performance requirements during fracturing stages such as fracture creation, proppant suspension, and flowback; and also address the demand for increased recovery. The main mechanisms include (1) enlarging the effective pore radius, (2) super-hydrophobic effects, and (3) anti-swelling properties. Fracturing fluids are pumped into fractures through pipelines, where they undergo complex flow in multi-scale fractures, ultimately seeping through capillary bundles. Flow resistance is influenced by the external environment, and the sources of flow resistance in fractures of different scales vary. Surfactants with polymerization capabilities, biodegradable and environmentally friendly bio-based surfactants, crosslinking agents, and amino acid-based green surfactants with outstanding properties will unleash their application potential, providing crucial support for the effectiveness of fracturing-enhanced recovery integration working fluids. This article provides important references for the green, efficient, and sustainable development of tight oil reservoirs.

Saturation Equation: An Analytical Expression for Partial Saturation during Wicking Flow in Paper Microfluidic Channels.

The design and fabrication of paper-based microfluidic devices is critically dependent on modeling fluid flow through porous paper membranes. A commonly observed phenomenon is partial saturation, i.e., regions of the paper membrane not being filled completely due to pores of different sizes. The most comprehensive model to date of partial saturation during wicking flow in paper is the Richards equation. However, the solution to the Richards equation requires numerical solvers like COMSOL, which makes it largely inaccessible to the paper microfluidics and lateral flow assay community. There is therefore a need for a simple and appropriate model of partial saturation in paper membranes, easily usable by the wider research community. In the current work, we present an approach to model paper membranes as a bundle of parallel capillaries whose radii follow a two-parameter log-normal distribution. Application of the Washburn equation to the bundle provides a distribution of fluid fronts, which can be used to calculate saturation. Using this approach, we developed the "saturation equation"─an explicit analytical expression to calculate saturation as a function of space and time in 1D wicking flow. Experimentally obtained data for spatiotemporal saturation for four different paper materials were fit to this analytical model to obtain parameters for each material. Results obtained from this analytical model match well with both experimental data and numerical results obtained from the Richards equation. The availability of an explicit analytical expression for partial saturation will enable incorporation of the critical phenomenon of partial saturation in the design of paper microfluidic devices.

The PDI model system for parameterizing soil hydraulic properties

AbstractThe Peters–Durner–Iden (PDI) model system for describing soil hydraulic properties (SHP) has been developed over a decade. Inspired by Rien van Genuchten's seminal work, the PDI system focuses on an efficient and simple parameterization of water retention curves and hydraulic conductivity curves (HCC) across the entire soil moisture spectrum. By combining capillary and non‐capillary components for water retention and conductivity, it aims to reconcile mathematical simplicity and insights on water adsorption and film flow in soils. Recent developments have reduced the number of free parameters of the conductivity model to zero, enhancing the model's applicability in cases of limited data availability. The first reduction was achieved by a prediction of absolute non‐capillary conductivity based on the consideration of film and corner flow on the pore scale, and the second by a prediction of absolute capillary conductivity by a capillary bundle model. This allows a complete characterization of SHP over the entire moisture range with only four retention curve parameters. The inclusion of a maximum pore size in the capillary conductivity model prevents an unrealistic drop of the HCC near saturation. This paper provides a comprehensive overview of the PDI model system, emphasizing its conceptual features and mathematical details. An Excel sheet and a Python code stored in a repository are provided for accessibility.

Displacement Mechanism and Flow Characteristics of Polymer Particle Dispersion System Based on Capillary Bundle Model

During the development of oil reservoirs, a rapid increase in water cut following reservoir flooding leads to inefficient or ineffective circulation of injected water, rendering a significant portion of the remaining oil in the reservoir inaccessible. The displacement method using polymer particle dispersion systems effectively solves the issue of rapid water breakthrough in oil reservoirs. Owing to the particle phase separation phenomenon, polymer particles can selectively penetrate into the larger pores where water circulation is inefficient, enhance their flow resistance, and thereby achieve equilibrium displacement along with an increased swept volume. This paper investigates the heterogeneous distribution of polymer particles within a porous medium, incorporates the red blood cell dendrite concentration distribution theory from biological fluid mechanics, and develops a mathematical model to delineate the viscosity characteristics of polymer particle dispersion systems, taking into account the phase separation phenomenon. Building on this foundation, it formulates a capillary bundle model for the polymer particle dispersion system specifically designed for oil displacement and proceeds to determine its relative permeability curve. Simulation outcomes reveal that at a water saturation level of 0.063, the concentration of polymer particles in fractured large pore capillaries is markedly elevated, yet capillaries with a pore size under 26 μm remain devoid of polymer particles. With the increase of water saturation, the concentration of polymer particles in large pore capillaries reduces, whereas it progressively augments in medium pore capillaries. Upon reaching a peak water saturation of 0.751, capillaries smaller than 18 μm are entirely free of polymer particles. These findings suggest that the heterogeneous distribution of polymer particles markedly inhibits the percolation capabilities of the dispersed system following a water phase breakthrough, facilitating the entry of more dispersion into oil-laden capillaries and thus enhancing the flow capacity of the oil phase.

Dynamic Pore Network Modeling of Imbibition in Real Porous Media with Corner Film Flow.

Wetting films can develop in the corners of pore structures during imbibition in a strongly wetting porous medium, which may significantly influence the two-phase flow dynamics. Due to the large difference in scales between main meniscus and corner film, accurate and efficient modeling of the dynamics of corner film remains elusive. In this work, we develop a novel two-pressure dynamic pore network model incorporating the interacting capillary bundle model to analyze the competition between the main meniscus and corner film flow in real porous media. A pore network with four-point star-shaped pore bodies and throat bonds is extracted from the real porous medium based on the pore shape factor and pore cross-sectional area, which is then decomposed into several layers of sub-pore networks, where the first layer of sub-pore network simulates the main meniscus flow while the upper layers characterize the corner film flow. The two-phase flow conductance of throat bonds for different layers of sub-pore networks are determined by high-resolution two-phase lattice Boltzmann modeling, thus inherently considering the viscous coupling effect. In addition, two artificial neural network models are developed to predict the two phases' flow conductance based on the shape of the throat cross section and the fluid properties. The accuracy of the developed model is validated with a lattice Boltzmann simulation of imbibition in a strongly wetting square tube. Then the model is used to simulate imbibition in a strongly wetting sandstone porous medium, and the competition between the main meniscus and the corner film flow is analyzed. The results show that with decreasing capillary number and viscosity ratio between wetting and nonwetting fluids, the development of the wetting corner film becomes more significant.

Immiscible Two-Phase Flow in Porous Media: Effective Rheology in the Continuum Limit

AbstractWe consider steady-state immiscible and incompressible two-phase flow in porous media. It is becoming increasingly clear that there is a flow regime where the volumetric flow rate depends on the pressure gradient as a power law with an exponent larger than one. This occurs when the capillary forces and viscous forces compete. At higher flow rates, where the viscous forces dominate, the volumetric flow rate depends linearly on the pressure gradient. This means that there is a crossover pressure gradient that separates these two flow regimes. At small enough pressure gradient, the capillary forces dominate. If one or both of the immiscible fluids percolate, the volumetric flow rate will then depend linearly on the pressure gradient as the interfaces will not move. If none of the fluids percolate, there will be a minimum pressure gradient threshold to mobilize the interfaces and thereby get the fluids moving. We now imagine a core sample of a given size. The question we pose is what happens to the crossover pressure gradient that separates the power-law regime from the high-flow rate linear regime and the threshold pressure gradient that blocks the flow at low pressure gradients when the size of the core sample is increased. Based on analytical calculations using the capillary bundle model and on numerical simulations using a dynamical pore-network model, we find that the crossover pressure gradient and the threshold pressure gradient decrease with two distinct power laws in the size. This means that the power-law regime disappears in the continuum limit where the pores are infinitely small compared to the sample size.

Predicting the Electrical Conductivity of Partially Saturated Frozen Porous Media, a Fractal Model for Wide Ranges of Temperature and Salinity

AbstractThe quantitative determination of liquid water content and salinity in soils is crucial for the preservation of hydrological environments and engineering infrastructures, especially in frozen regions. Electrical conductivity, as a fundamental physical parameter in electrical and electromagnetic non‐destructive techniques, varies significantly with the physical and chemical properties, such as pore water conductivity, salinity, water saturation, and temperature. In this study, accounting for pore size and tortuous length following fractal distributions, we develop a new capillary bundle model for variation of electrical conductivity as a function of temperature in broad water saturation and salinity ranges. In this new model, we consider the contributions of bulk and surface conductivities to the total electrical conductivity. To test this model, a series of laboratory experiments were carried out for different initial water saturations and salinities using an electrical resistance apparatus and a nuclear magnetic resonance method. The experimental results show that unfrozen water saturation and ionic concentration affect the electrical conductivity of unsaturated frozen soils. Furthermore, the proposed model is capable of fitting the main trends of the experimental data from the literature and acquired in this study in unfrozen‐frozen conditions for different water contents. Relying on the proposed model, we also determine the expression of the apparent formation factor, which is significantly sensitive to porosity, water saturation, and temperature. The predicted values of the apparent formation factor also agree very well with the experimental data. This new capillary bundle model provides a new perspective in interpreting electrical monitoring to easily deduce changes in key variables in the cryosphere such as liquid water content and moisture gradients.

A new model for coal gas seepage based on fracture-pore fractal structure characteristics

Coal reservoir is a typical double pore medium, and the magnitude of its permeability is controlled by the parameters of the fracture pore structure. However, the relationship between fracture-pore parameters and permeability is not yet clear. Based on the fractal geometry theory, the fracture-pore morphology of coal body is characterized by tree bifurcation network and tortuous capillary bundle. This study also established a coal gas seepage flow model based on the fracture-pore fractal structure characteristics. The model includes the influence of coal structure parameters, different levels of original coal seam stress, and gas pressure on the gas seepage characteristics. Each parameter in the model has definite physical meaning and does not contain any empirical constants. Finally, the applicability of the model is evaluated by gas radial seepage experiment. It is shown that the gas permeability in the model is a function of fractal dimensions and structural parameters, and based on the sensitivity analysis of each parameter in the model, it is found that the tree diameter ratio β and length ratio γ of the bifurcation network have a significant impact on the permeability. When β increases from 0.1 to 0.9, the model permeability increases by 12 orders of magnitude. However, changes in the characteristic parameters of the pore structure have little effect on gas permeability. For dual porous media such as coal, the permeability of the fracture network dominates the overall permeability of the model, and small changes in the geometric structure of the fracture bifurcation network can lead to significant changes in permeability. The construction of this theoretical model provides a reliable basis for further improving coal-rock porous medium structure seepage models, as well as to better characterize the effect of gas drainage.