Permeability-porosity Relation Research Articles

Compaction and sintering effects on scaling law of permeability-porosity relation of powder materials

Analysis and understanding of the permeability-porosity relationship of powder materials by cold compaction and solid-state hot sintering is very important for material design and optimization. This work presents a pore-scale numerical framework, including a discrete element method for powder cold compaction, a cellular automata algorithm with curvature-driving redistribution for solid-state hot sintering and a lattice Boltzmann method for fluid flow simulation, to study the permeability-porosity relationship of materials by cold compaction and solid-state hot sintering processes. The results show that the cubic scaling law always holds for the porous materials by cold compaction, while the hot sintering process decreases the permeability dramatically at low porosity. An exponential decay function has been proposed as a correction factor to the permeability-porosity relationship of materials by compaction & sintering processes, which predicts the permeability asymptotically approaching to the cubic scaling law at high porosity while agreeing well with both numerical and experimental data at low porosity. The microstructure evolution analysis shows that the small pores that connect large pores may vanish by sintering, which causes the remarkable permeability decrease at low porosity. The excellent agreements between the numerical predictions and the experimental measurements suggest that the proposed numerical framework provides a powerful tool for analysis and optimization of material designs.

Permeability-porosity relation during erosion-induced water inrush: Experimental and theoretical investigations

• Large-scale seepge-erosion tests were conducted to explore the permeability-porosity evolution characteristics during erosion-induced water inrush process. • Two types of permeability-porosity relations were observed, in relation to the common erosion process (CEP) and mutation erosion process (MEP). • Evaluations of four popular permeability-porosity relations indicated their inapplicabilities for both of CEP and MEP. • A new permeability-porosity model considering the process-dependent behavior is proposed. During the construction of tunnelling in soil ground, erosion-induced water inrush is one of the most serious geohazards, during which the migration of soil particles can substantially alter soil porosity and permeability. Understanding the permeability-porosity ( k-φ ) relation is crucial for predicting the rate of water inflow and the possibility of water inrush. The applicability of existing k-φ models in this problem has not been evaluated. In this study, the k-φ relation was investigated by a series of laboratory seepage-erosion tests on soil specimens with different Talbot indexes and initial porosities. Based on the experimental results, an evaluation was made on the performance of four popular k-φ models, including the Kozeny-Carman (K-C), Kruger, Ives-Pienvichitr (I-P), and Verma and Pruess (V-P) models. Two types of seepage-erosion processes were observed, including the common erosion and mutation erosion processes (CEP and MEP). CEP and MEP mean the processes with and without the occurrence of water inrush, respectively. During the CEP associated with low Talbot index and porosity, the permeability of soil was significantly overestimated by the V-P model and slightly underestimated by the other three models. During the MEP associated with high Talbot index and porosity, the erosion led to water inrush through three stages: slow and rapid evolution as well as stable stages. Significant underestimates of soil permeability were caused by all the models, thereby leading to high risk. None of these models was able to well capture the k-φ relation during different erosion-flow processes for both CEP and MEP. To address this problem, a power function with one adjustable exponent for the k-φ relation was proposed and verified.

Simulation of flow characteristics and development of permeability model in fractured-vuggy carbonate reservoir

Fractured-vuggy carbonate reservoirs have complex pore structure, significant heterogeneity, and irregular permeability-porosity relation, which make existing evaluation methods unable to accurately calculate the permeability. To investigate the flow characteristics of carbonate rocks with triple-porosity, 3D digital rock models were constructed from X-ray computed tomography (CT) scanning images of carbonate cores collected in the field, and the physical properties of rocks were obtained by combining digital rock analysis (DRA) and nuclear magnetic resonance (NMR) experiment. The effects of fractures and vugs on the permeability of carbonate rocks was investigated using the Stokes-Brinkman method, and then a new fractured-vuggy carbonate permeability model was developed by artificially adding vugs and fractures. The results indicate that the carbonate pores can be divided into matrix pores, vugs, and fractures according to their sizes by combining digital rock analysis and T2 spectrums. The transport resistance is effectively reduced by fractures and vugs, which leads to the change of the flow path. For tight carbonate rocks, the dominant flow path formed through fractures is a crucial factor affecting permeability. On the contrary, unconnected vugs have less effect on permeability. Finally, the new permeability model was applied to the logging interpretation of field data. The new model is in better agreement with the permeability analysis of field cores, which greatly improves the permeability calculation accuracy of the triple-porosity reservoir in the study area.

Permeability-Porosity Relation During Erosion-Induced Water Inrush Experimental and Theoretical Investigations

Permeability-Porosity Relation During Erosion-Induced Water Inrush Experimental and Theoretical Investigations

Scaling Law of Permeability and Porosity for Fluid Transport Phenomena in Porous PCM Media

The present paper reports the numerical results of fluid flow in porous phase change materials (PCM) media. This is an important topic in potential scientific, technological and engineering field’s especially latent heat storage. In this paper, we are only interested in the correlation between permeability and porosity of the porous media and not in latent storage. Fluid flow is characterized by many parameters mainly permeability and porosity. Many models have been proposed for the study of this phenomenon over the years. However, it can be modeled using the complex model that studies the characteristics of pore microstructure and fluid flow in porous media. This model is more accurate and realistic compared to previous models. It predicts permeability and porosity with a good agreement with experimental data. In this paper, the complex model is used to determine the impact of the tortuosity and the density of capillary distribution on the relation between permeability and porosity and check their scaling laws with universal exponents independently of other parameters. The results show that the permeability-porosity relation is proportional to the standard deviation of capillary distribution and its density. The tortuosity affects porosity proportionally, and permeability inversely. The relation between porosity and permeability follows a power law with universal exponents β = 4.06 ± 0.12 for different values of the expectation of distribution, the density of capillaries and the tortuosity; and β = 1.69 ± 0.01 for different values of the standard deviation, density of capillaries and tortuosity. The universality of these exponents further validates the complex model with various previous experimental and numerical studies.

Clot Permeability, Agonist Transport, and Platelet Binding Kinetics in Arterial Thrombosis

The formation of wall-adherent platelet aggregates is a critical process in arterial thrombosis. A growing aggregate experiences frictional drag forces exerted on it by fluid moving over or through the aggregate. The magnitude of these forces is strongly influenced by the permeability of the developing aggregate; the permeability depends on the aggregate's porosity. Aggregation is mediated by formation of ensembles of molecular bonds; each bond involves a plasma protein bridging the gap between specific receptors on the surfaces of two different platelets. The ability of the bonds existing at any time to sustain the drag forces on the aggregate determines whether it remains intact or sheds individual platelets or larger fragments (emboli). We investigate platelet aggregation in coronary-sized arteries using both computational simulations and invitro experiments. The computational model tracks the formation and breaking of bonds between platelets and treats the thrombus as an evolving porous, viscoelastic material, which moves differently from the background fluid. This relative motion generates drag forces which the fluid and thrombus exert on one another. These forces are computed from a permeability-porosity relation parameterized from experimental measurements. Basing this relation on measurements from occlusive thrombi formed in our flow chamber experiments, along with other physiological parameter values, the model produced stable dense thrombi on a similar timescale to the experiments. When we parameterized the permeability-porosity relation using lower permeabilities reported by others, bond formation was insufficient to balance drag forces on an early thrombus and keep it intact. Under high shear flow, soluble agonist released by platelets was limited to the thrombus and a boundary layer downstream, thus restricting thrombus growth into the vessel lumen. Adding to the model binding and activation of unactivated platelets through von Willebrand-factor-mediated processes allowed greater growth and made agonist-induced activation more effective.

Network-inspired versus Kozeny\u2013Carman based permeability-porosity relations applied to Biot\u2019s poroelasticity model

Water injection in the aquifer induces deformations in the soil. These mechanical deformations give rise to a change in porosity and permeability, which results in non-linearity of the mathematical problem. Assuming that the deformations are very small, the model provided by Biot’s theory of linear poroelasticity is used to determine the local displacement of the skeleton of a porous medium, as well as the fluid flow through the pores. In this continuum scale model, the Kozeny–Carman equation is commonly used to determine the permeability of the porous medium from the porosity. The Kozeny–Carman relation states that flow through the pores is possible at a certain location as long as the porosity is larger than zero at this location in the aquifer. However, from network models it is known that percolation thresholds exist, indicating that the permeability will be equal to zero if the porosity becomes smaller than these thresholds. In this paper, the relationship between permeability and porosity is investigated. A new permeability-porosity relation, based on the percolation theory, is derived and compared with the Kozeny–Carman relation. The strongest feature of the new approach is related to its capability to give a good description of the permeability in case of low porosities. However, with this network-inspired approach small values of the permeability are more likely to occur. Since we show that the solution of Biot’s model converges to the solution of a saddle point problem for small time steps and low permeability, we need stabilisation in the finite element approximation.

Detection of flow units of Quseir Formation (Lower Campanian) as a potential reservoir using experimental correlations of capillary pressure derived parameters, Gebel el-Silsila, Egypt

Sandstone surface samples were collected from Quseir Fm., a member of Nubian Sandstone at Gebel el-Silsila and subjected to petrographical investigation using polarizing microscope. X-Ray diffraction was carried out to determine mineralogy. Scanning electron microscope also was utilized for the morphological characterization of the rock components. This sandstone petrofacies is mainly quartz arenite. The depositional environment of the studied sandstones is fluvial to fluvial-marine, and they suffered post-depositional alterations. Alterations include diagenesis (clay infiltration and original detrital grains disintegration), weathering, deterioration phenomena as erosion, voids existence due to dissolution and absence of bound materials. Mineralogical characterization indicated that quartz is the main constituent of sandstone in addition to kaolinite clay mineral as a cementing material. Petrophysically, the studied samples subjected to porosity and permeability measurements. The measurements showed that the samples are highly porous and permeable. Permeability–porosity relation displayed two groups (A and B). Although porosity values are close to great extent, permeability is quite different for the two groups due to the different distributions of pore throats, perhaps as a result of uneven clay distribution within the two groups of samples. Many capillary pressure-derived parameters were calculated using experimental correlations. The results confirmed the presence of two flow groups, macroport group (group A) and megaport group (group B) that is the best in terms of reservoir quality. The two groups reflect the excellent properties for the studied potential reservoir.

Study on flow parameters of fractal porous media in the high-velocity fluid flow regime

High-velocity fluid flow, which will result in the region of the wellbore or fracture, is generally in the turbulent flow regime and has drawn tremendous attention in petroleum engineering field. Turbulent factor is the key parameter, which is widely used to describe high-velocity flow in porous media. In this work, a theoretical model for turbulent factor in fractal porous media in the high-velocity fluid flow regime is developed. Moreover, a novel analytical expression for the permeability in porous media based on Wu's resistance model is also derived. Then, the analytical Kozeny-Carman constant with no empirical constant is obtained. The predictions of permeability-porosity relation by the current mathematical models have been validated by comparing with available experimental data. Furthermore, the effects of structural parameters of porous media on the curve of velocity and pressure drop are discussed in detail.

Fluid dynamics in porous media with Sailfish

In this work we show the application of Sailfish to the study of fluid dynamics in porous media. Sailfish is an open-source software based on the lattice-Boltzmann method. This application of computational fluid dynamics is of particular interest to the oil and gas industry and the subject could be a starting point for an undergraduate or graduate student in physics or engineering. We built artificial samples of porous media with different porosities and used Sailfish to simulate the fluid flow through them in order to calculate their permeability and tortuosity. We also present a simple way to obtain the specific superficial area of porous media using Python libraries. To contextualise these concepts, we analyse the applicability of the Kozeny–Carman equation, which is a well-known permeability–porosity relation, to our artificial samples.

Predicting water permeability in sedimentary rocks from capillary imbibition and pore structure

This study was financed by the Spanish Ministry of Science and Innovation CGL2011-25162. A pre-doctoral research fellowship was awarded to C. Pla for this project.

Pore Space Microstructure Evolution of Regular Sphere Packings Undergoing Compaction and Cementation

We examine the pore space structure evolution of ordered uniform sphere packs: simple cubic (SC), body centered cubic (BCC), and face centered cubic (FCC), undergoing simple diagenetic processes that reduce their pore spaces. Focus is on the occurrence of pore space microstructure changes or transitions, which are followed through their characteristic or critical pore lengths (lc). For almost all the cubic packings undergoing either compaction or cementation there are no singularities in lc. This is a consequence of having a single pore shape controlling flow at all stages of the process. However, this is not so for the BCC packing under cementation, for which lc is non-monotonic exhibiting a kink at \({\phi \approx 0.1452}\) , the porosity at which the pore shape controlling flow switches to a different form and position. These results for uniform compaction/cementation complement our previous works on pore networks under random shrinkage. Kinks in lc as porosity decreases signal pore space microstructure transitions that anticipate sudden changes in the permeability–porosity relation as porosity decreases. The consequences are great; clearly lc is not a constant unless the diagenetic process is mild. A lc function of compaction/cementation advancement should be used above a transition and a different lc function below. For the sphere packs here, once the diagenetic process has reduced the pore space substantially, a lc function of compaction/cementation advancement is mandatory if we are to capture all significant flow features.

Laboratory measurements of porosity, permeability, resistivity, and velocity on Fontainebleau sandstones

The relations among the resistivity, elastic-wave velocity, porosity, and permeability in Fontainebleau sandstone samples from the Ile de France region, around Paris, France were experimentally revisited. These samples followed a permeability-porosity relation given by Kozeny-Carman’s equation. For the resistivity measurements, the samples were partially saturated with brine. Archie’s equation was used to estimate resistivity at 100% water saturation, assuming a saturation exponent, [Formula: see text]. Using self-consistent (SC) approximations modeling with grain aspect ratio 1, and pore aspect ratio between 0.02 and 0.10, the experimental data fall into this theoretical range. The SC curve with the pore aspect ratio 0.05 appears to be close to the values measured in the entire porosity range. The elastic-wave velocity was mea-sured on these dry samples for confining pressure between 0 and [Formula: see text]. A loading and unloading cycle was used and did not produce any significant hysteresis in the velocity-pressure behavior. For the velocity data, using the SC model with a grain aspect ratio 1 and pore aspect ratios 0.2, 0.1, and 0.05 fit the data at [Formula: see text]; pore aspect ratios ranging between 0.1, 0.05, and 0.02 were a better fit for the data at [Formula: see text]. Both velocity and resistivity in clean sandstones can be modeled using the SC approximation. In addition, a linear fit was found between the P-wave velocity and the decimal logarithm of the normalized resistivity, with deviations that correlate with differences in permeability. Combining the stiff sand model and Archie for cementation exponents between 1.6 and 2.1, resistivity was modeled as a function of P-wave velocity for these clean sandstones.

Three-Dimensional Characterization of Permeability at the Core Scale

In this article, we introduce an integrated method for characterizing permeability heterogeneity at the core scale. It combines the results of laboratory core flooding with already-developed field scale history matching techniques such as gradual deformation and pilot points. Prior to any experiment, X-ray computed tomography (CT) imaging techniques are used to obtain three-dimensional porosity distribution in cores. The samples are submitted to viscous, miscible displacement of water by water–glycerin mixture. The dynamic data collected during injection are the time variations in inlet–outlet pressure drop and three-dimensional CT-scan concentration maps of invading fluid collected at successive times. We develop an inversion or matching process which takes advantage of the available data to characterize the spatial distribution of permeability heterogeneities within core samples. Permeability is assumed to be related to porosity. This matching process involves two successive optimizations. First, an initial permeability guess derived from porosity is modified by varying deterministic parameters until the corresponding simulated pressure answer fits the measured pressure drop. Second, an extended optimization process with both deterministic and stochastic parameters is run to match pressure drop and concentration data. This methodology is applied to a synthetic example for which the permeability–porosity relation is known. It yields a three-dimensional permeability model reproducing the reference pressure and concentration maps. The methodology is also applied to experimental data. In this case, it provides three-dimensional permeability models leading to an improved, but perfectible data match. A major difficulty is the unknown relationship between permeability and porosity.

Smoothed Particle Hydrodynamics Modeling of Transverse Flow in Randomly Aligned Fibrous Porous Media

The Lagrangian smoothed particle hydrodynamics (SPH) method is employed to obtain a meso-/micro-scopic pore-scale insight into the transverse flow across the randomly aligned fibrous porous media in a 2D domain. Fluid is driven by an external body force, and a square domain with periodic boundary conditions imposed at both the streamwise and transverse flow direction is assumed. The porous matrix is established by randomly embedding a certain number of fibers in the square domain. Fibers are represented by position-fixed SPH particles, which exert viscous forces upon, and contribute to the density variations of, the nearby fluid particles. An additional repulsive force, similar in form to the 12-6 Lennard-Jones potential between atoms, is introduced to consider the no-penetrating restraint prescribed by the solid pore structure. This force is initiated from the fixed solid material particle and may act on its neighboring moving fluid particles. Fluid flow is visualized by plotting the local velocity vector field; the meandering fluid flow around the porous microstructures always follow the paths of least resistance. The simulated steady-state flow field is further used to calculate the macroscopic permeability. The dimensionless permeability (normalized by the squared characteristic dimension of the fiber cross section) exhibits an exponential dependence on the porosity within the intermediate porosity range, and the derived dimensionless permeability—porosity relation is found to have only minor dependence on either the relative arrangement condition among fibers or the fiber cross section (shape or area).

Developing a new form of permeability and Kozeny–Carman constant for homogeneous porous media by means of fractal geometry

The semi-empirical Kozeny–Carman (KC) equation is the most famous permeability–porosity relation, which is widely used in the field of flow in porous media and is the starting point for many other permeability models. However, this relation has many limitations from its inception, and the KC constant is an empirical parameter which was proved to be not a constant. In this paper, we briefly reviewed the KC equation, its modifications and various models for the KC constant. We then derived an analytical expression for the permeability in homogeneous porous media based on the fractal characters of porous media and capillary model. The proposed model is expressed as a function of fractal dimensions, porosity and maximum pore size. The analytical KC constant with no empirical constant is obtained from the assumption of square geometrical model. Furthermore, a distinct linear scaling law between the dimensionless permeability and porosity is found. It is also shown that our analytical permeability is more closely related to the microstructures (fractal dimensions, porosity and maximum pore size), compared to those obtained from conventional methods and models.

Damage evolution and fluid flow in poroelastic rock

We present a formulation for mechanical modeling of the interaction between fracture and fluid flow. Our model combines the classic Biot poroelastic theory and a damage rheology model. The model provides an internally consistent framework for simulating coupled evolution of fractures and fluid flow together with gradual transition from brittle fracture to cataclastic flow in high-porosity rocks. The theoretical analysis, based on thermodynamic principles, leads to a system of coupled kinetic equations for the evolution of damage and porosity. A significant advantage of the model is the ability to reproduce the entire yield curve, including positive and negative slopes, in high-porosity rocks by a unified formulation. A transition from positive to negative values in the yield curve, referred to as a yield cap, is determined by the competition between the two thermodynamic forces associated with damage and porosity evolution. Numerical simulations of triaxial compression tests reproduce the gradual transition from localized brittle failure to distributed cataclastic flow with increasing pressure in high-porosity rocks and fit well experimentally measured yield stress for Berea sandstone samples. We modified a widely used permeability porosity relation by accounting for the effect of damage intensity on the connectivity. The new damage-permeability relation, together with the coupled kinetics of damage and porosity evolution, reproduces a wide range of realistic features of rock behavior. We constrain the model variables by comparisons of the theoretical predictions with laboratory results reporting porosity and permeability variation in rock samples during isotropic and anisotropic loading. The new damage-porosity-permeability relation enables simulation of coupled evolution of fractures and fluid flow and provides a possible explanation for permeability measurements in high-porosity rocks, referred to as the “apparent permeability paradox.”