Network Models Of Porous Media Research Articles

Local Hydraulic Resistance in Heterogeneous Porous Media

AbstractWe examine the validity of the commonly used Hagen‐Poiseuille model of local resistance of porous media using direct numerical simulations. We provide theoretical arguments that highlight possible limitations of this model and formulate a new constitutive model that is based on the circularity of iso‐pressure surfaces. We compare the performance of both models on three different three‐dimensional artificial porous media. We show that the new model improves the root‐mean‐squared‐relative error from , and for the HP model to , and for the three porous media respectively. We anticipate that our approach may find broad application in network models of porous media that are typically build from 3D images with intricate pore geometries.

Network model of porous media – Review of old ideas with new methods

The paper takes up the old ideas of describing porous media with several tube and network models. The well-known models received from literature gave a good concept of dewatering equilibria resulting in capillary pressure curves and pore size distributions (PoSD). However, numerical methods and measurement techniques were not sophisticated allowing to evaluate the models appropriately.In this work, a numerical method based on statistics is introduced to validate the network model of FATT from 1956: The porous filter cake structure is implemented as a matrix, which elements represent the pore size correlating with the capillary entrance pressure for each pore. The input for the calculations can be any mathematical approximation of a PoSD, which can be derived from capillary pressure tests or micro computer tomography (µCT) analysis of the filter cake.A procedure based on the concept of FATT is presented to generate dewatering equilibria for different applied pressures. Therefore, the elements of the matrix are checked to be ‘dewatered’ regarding to their size, position, the applied pressure level and the progress of dewatering. The network model known from literature is improved by implementing additional conditions for the description of physical phenomena, such as the formation of residual bridge liquid or hydrodynamic isolated areas.X-ray microscopy, mercury intrusion tests and laboratory desaturation experiments by using semipermeable membranes for capillary pressure tests are used to validate the pore size distribution. The different results are integrated into the matrix model as starting parameters. For the laboratory experiments, the PoSD is calculated from the measured capillary pressure curve, using the distributed tube model and the YOUNG-LAPLACE-equation on an equal basis to the established mercury intrusion analysis. However, with the tomography measurements, it is possible to determine PoSD using different defined geometry elements fitting inside the pore space. The force balance is evaluated at the pore entrance by using the wetting line of the pore throat. The direct measurement of the void geometry allows the calculation of the pressure distribution without the LAPLACIAN assumptions.In this way, the difference between experimental, measured and modelled PoSDs is emphasised to validate the old (and improved) ideas of network models describing porous media.

Generation of Spatially Correlated Network Models of Porous Media

A Monte Carlo (MC) algorithm to generate pore networks with spatial correlation of pore sizes is developed. The algorithm is illustrated for model porous media described as a two-dimensional square lattices of nodes and bonds which represent pore bodies and pore throats, respectively. We seek a spatial rearrangement of pore throats that minimizes the difference between the semivariogram of the network and a given model semivariogram. The spatial correlation of the initially uncorrelated network increases by means of MC moves where the diameters of a pair of randomly chosen pores are interchanged. The algorithm preserves the initial pore size distribution. Finally, the correlated pore throat networks are used to derive correlated pore body networks once a given body-to-throat size ratio is set. A simple parallel implementation of the algorithm is presented.

A novel architecture for pore network modelling with applications to permeability of porous media

Network models of porous media are beneficial for predicting the evolution of macroscopic mass transport properties. This work proposes a novel bi-regular network model based on truncated octahedral support. With the model, pore systems with different pore coordination spectra for a given average coordination number can be constructed to match experimental data. This feature and the maximum allowed pore coordination of 14 make the proposed model more realistic and flexible than the existing models based on cubic supports. Limestone is used as a study material to illustrate the model performance. Experimental pore space data for this material obtained by X-ray tomography (CT) are used for constructing microstructure-informed model realisations. It is demonstrated that with these realisations the experimentally measured permeability of the material can be predicted. The model is further used to assess the effects of pore connectivity and porosity on the permeability. It is shown that the effect of pore connectivity is substantially stronger than the effect of porosity. A strategy for calculating the evolution of permeability with damage-related pore space changes is also described. The results reveal the effects of the mechanical properties of the medium on the permeability evolution as a function of damage evolution. Developments of this strategy are suggested for deriving mechanism-based constitutive laws for engineering applications.

A pseudo two-phase model of colloid transport in porous media

In this article we suggest a new phenomenological mathematical model for the groundwater transport of colloid particles through porous media which is able to describe some significant effects experimentally observed but not captured within the framework of the classic approach. Our basic idea is to consider both the pure water and the colloid suspension as two thermodynamic phases. Using the network models of porous media, we simulated numerically the transport process at the pore-scale. By averaging the result derived, we have obtained the relative permeabilities for both phases, the percolation threshold for suspension flow, and the effective suspension viscosity. Due to specific laws of colloid particles repartition between various classes of pores, the relative permeability of suspension happens to be a highly nonlinear function of saturation, very far from the diagonal straight line. This determines a difference between the macroscale phase velocities. The suspension velocity is shown to be higher than that of water in major cases, only if the colloid particles are not too large. The suggested model predicts and describes in a closed form the effect of colloid transport facilitation observed experimentally.

Effects of three different network models on the filter coefficient of brownian particles

The dynamic behavior of granular filtration is investigated in the present paper by using the Brownian dynamic simulation method to track the individual particles moving through the 2-D network model of porous media in the filter bed. The scope includes three different network structures, modified square, triangular and honeycomb, all with the same initial permeability, and two types of DLVO interaction energy curves. The specific filtration functions of F 1( α′, σ) and F 2( β′, σ) are determined from the breakthrough curve of effluent concentration and the pressure drop. When comparing those three different network models with the same interaction energy curve, it is found that the honeycomb network can filtrate particles the easiest, because of its lowest coordination number, so its corresponded specific deposit σ is the highest. When comparing with the available experimental data under the unfavorable deposition conditions, it is found that the present triangular network gives smaller discrepancy than that of the convective diffusion model.

Pore-scale network modelling of flow propagators derived from pulsed magnetic field gradient spin echo NMR measurements in porous media

Pulsed magnetic field gradient spin echo (PGSE) NMR experiments carried out on porous media where fluid flow is occurring, may be analysed to give the propagator, P( ξ, t), This quantity is the ensemble density distribution of particle (nuclei) displacements in a chosen direction, ξ, in a given time interval, t. These displacements arise as a result of both the convection and diffusion of molecules in the flowing fluid. The propagator can be derived for various displacement times, t, and hence these gradually probe a wider domain of the pore-scale velocity field within the porous medium. Such measurements can be performed separately on the oil and water phases in a two-phase flowing system. The interpretation and modelling of these single- and two-phase propagators in terms of the pore-scale flow field within the porous medium presents a difficult and interesting scientific challenge. In this paper, we model the main qualitative features of the experimentally measured propagators for both single- and two-phase flow using connected 3D pore network models of porous media. The calculated flow field within such models shows some non-trivial and qualitatively correct predictions about real flow fields in porous media. The propagator is modelled directly by incorporating transport due to both convection and diffusion for large numbers of marker particles (the nuclei) for both single- and two-phase flow. In the latter case, the transport within each of the two immiscible phases (oil and water) has been modelled in their separate pore occupancy networks. The network model captures most of the qualitative features for both single- and two-phase propagators thus giving us the capability to clearly interpret the respective flowing and non-flowing fractions of the oil (non-wetting) and water (wetting) phases in two-phase experiments. Therefore, our findings offer a powerful approach — PGSE NMR experiments and associated pore-scale modelling — for understanding and characterising two-phase flow through porous media.

Determination of relative permeabilities using the network models of porous media

An analytical method for obtaining the relative permeabilities of multiphase flows in porous media is presented. Porous media is modelled by a network of capillaries. The percolation and effective-medium models are used to calculate the network conductivity. Newtonian and plastic flow cases are considered. An application of the method to predict the efficiency of the polymer flooding is shown. The computed results are verified by means of experimental data.

Estimation of particle size distribution from a network model of porous media : G.P.Mathews et al. (University of Plymouth, Devon, UK.) Powder Tech., vol 104, no 2, 1999, 169–179

Estimation of particle size distribution from a network model of porous media : G.P.Mathews et al. (University of Plymouth, Devon, UK.) Powder Tech., vol 104, no 2, 1999, 169–179

Principles of the analytic description of fluid flow in network models of porous media

A method of calculating the conductivity of capillary networks which combines percolation concepts of conductivity in the neighborhood of the flow threshold with the results obtained using the effective medium model is proposed. The conductivity calculated theoretically is compared with experimental data. Water and oil phase permeabilities for combined flow through a capillary network are determined as an example of the use of the relations obtained.

Estimating particle size distributions from a network model of porous media

A new method is described for estimating the particle size distribution of consolidated porous materials using a network model of void structure based on mercury porosimetry. It is compared to the only other comparable method, a modification of that developed by Mayer and Stowe [R.P. Mayer, R.A. Stowe, J. Coll. Sci. 20 (1965) 893] for obtaining the radii of uniform spheres in regular packings. The methods are tested on unconsolidated materials, namely four very different sands, a sample of glass beads of narrow size range and a binary mixture of the coarsest and finest of the sands. The new method compares favourably with the modified Mayer and Stowe technique when estimating the particle sizes of the unmixed sands, and is an improvement for the glass beads and the sand mixture. The network model used to estimate particle size distributions also successfully models the experimental liquid permeabilities of the samples to within one order of magnitude.

Effects of pore structure and wettability on the electrical resistivity of partially saturated rocks—A network study

A network model of porous media is used to assess the effects of pore structure and matrix wettability on the resistivity of partially saturated rocks. Our focus is the magnitude of the saturation exponent n from Archie's law and the hysteresis in resistivity between drainage and imbibition cycles. Wettability is found to have the dominant effect on resistivity. The network model is used to investigate the role of a wetting film in water‐wet systems, and the behavior of oil‐wet systems. In the presence of a thin wetting film in water‐wet systems, the observed variation in n with saturation is reduced significantly resulting in lower n values and reduced hysteresis. This is attributed to the electrical continuity provided by the film at low‐water saturation between otherwise physically isolated portions of water. Oil‐wet systems, when compared with the water‐wet systems, are found to have higher n values. In addition, the oil‐wet systems exhibit a different form of hysteresis and more pronounced hysteresis. These differences in the resistivity response are attributed to differences in the pore scale distribution of water. The effects of pore structure are assessed by varying pore size distribution and standard deviation of the pore size distribution and considering networks with pore size correlation. The most significant parameter is found to be the pore size correlation. When the sizes of the neighboring pores of the network are correlated positively, the magnitude of n and hysteresis are reduced substantially in both the water‐wet and oil‐wet systems. This is attributed to higher pore accessibility in the correlated networks. The results of the present study emphasize the importance of conducting laboratory measurements on core samples with reservoir fluids and wettability that is representative of the reservoir. Hysteresis in resistivity can be present, particularly in oil‐wet systems, and should be considered in the interpretation of resistivity data.

The Extended Washburn Equation and Its Application to the Oil/Water Pore Doublet Problem

The equations of capillary rise or wetting-phase imbibition into a cylindrical capillary are frequently described by the Washburn equation. From this starting point, the familiar pore doublet model for imbibition into a wide/narrow pair of pores has been developed. This simple analytically soluble model can afford insights into the physics of immiscible water/oil displacements in porous media, which are of importance to waterflooding of oil reservoirs. Here, we reexamine the basis of both the Washburn equation and the formulation of the pore doublet model by incorporating a more rigorous treatment of the fluid mechanics. We have extended equations previously proposed by Szekely et al. to describe water/off displacement in a capillary and it transpires that certain additional inertial terms, which appear in the full formulation of the capillary displacement problem, may be particularly important at the pore size and aspect ratio commonly encountered within porous media. The main result of this additional physics is that the pore filling time and the relative filling order of large and small pores may change since these quantities depend on the pore aspect ratio in the extended formulation. When this fuller formulation is embedded within the extended pore doublet model, we find that the modified equations lead to different findings for certain cases, compared with previous well-known (analytical) solutions. The original pore doublet model is fully characterised by three quantities, the aspect ratio, r2/L, the ratio of capillary radii, r1/r2, and the fluid "supply," i.e., free supply or restricted supply, which is characterized by a velocity, V02. In the extended pore doublet model there is a fourth governing quantity, the pore scale Reynolds number, NRe , which describes the inertial terms in the extended Washburn equation. The resulting coupled differential equations for the extended pore doublet model do not admit an analytical solution and must be solved numerically. Under certain conditions of very restricted supply of wetting fluid (low V02), the extended model agrees very closely with the conventional pore doublet results. However, there are cases which are common at the pore scale (and pore aspect ratio) where significant differences are seen. The relevance of these findings to the modeling of imbibition processes within network models of porous media is discussed.

Network Modeling of Diffusion Coefficients for Porous Media: I. Theory and Model Development

AbstractGas diffusion often dominates constituent transport in porous media and is dependent on pore geometry, water content, and water distribution in a porous medium. Network models of porous media offer the ability to investigate the influence and interaction of pore‐scale porous media properties and fluid properties on macroscopic properties of the medium. This study was conducted to investigate the macroscopic relative gas diffusion coefficient vs. air‐filled porosity relationship (diffusion characteristic) of porous media using a network modeling approach. A cubic sphere‐and‐tube network model of porous media was adapted from petroleum engineering using Fick's law and the principle of conservation of mass to simulate one‐dimensional, steady‐state, isothermal, isobaric, molecular diffusion of a dilute binary gas in a nonadsorbing porous medium containing a single nonwetting fluid (air) and a single wetting fluid (water). The network model simulates hysteresis in air and water distributions in porous media for boundary drying and wetting curves of the soil water characteristic and demonstrates the effect of air‐filled porosity, Henry's law liquid‐gas partitioning coefficient, the ratio between gas‐ and liquid‐phase diffusion rates, and pore geometry on the diffusion characteristic.

Network Modeling of Diffusion Coefficients for Porous Media: II. Simulations

AbstractGas diffusion often dominates constituent transport in porous media (PM) and is dependent on pore geometry, water content, and water distribution in PM. Network models of PM offer the ability to investigate the influence and interaction of pore‐scale PM properties and fluid properties on macroscopic properties of the system. This study was conducted to investigate the macroscopic relative gas diffusion coefficient vs. air‐filled porosity relationship or diffusion characteristic (DC) of PM using a network model. The network model was used to simulate DCs in wetting and drying PM containing air and water. A network size of nine by nine by nine spheres was used; increasing the network size to 19 by 19 by 19 produced essentially no change in the DC. The DC was independent of Henry's law gas‐liquid partition coefficient (H) for H values of 0.1, 1.0, and 5.0. The product HRw, where Rw is the ratio between the bulk gas‐ and liquid‐phase diffusion coefficients, strongly influences the DC when H values of 1000 to 10 000 are considered; this indicates that certain organic compounds have DCs independent of air‐filled porosity. Hysteresis in DCs was found in selected network cases, with the wetting DCs being greater than the drying DCs for most air‐filled porosities, in accord with some experimental results reported in the literature. Spatial correlation of network pore space was shown to bring the simulated DCs into better agreement with some experimental DCs.

Transport in capillary network models of porous media: theory and simulation

The recent correlated ramdom walk theory, for transport in random networks, is developed here also for cubic networks. A comparison of correlated ramdom walk theory and effective medium theory predictions for cubic networks with simulation results shows the latter to be more accurate in general. For random networks a hybrid alternative, combining the effective medium and correlated random walk theories, is formulated and found to compare better with simulation results. However, under normal conditions, in which percolation effects are not dominant, the correlated ramdom walk theory is found to be adequate. Under conditions involving positional variations in pore structure, or non-linear field-variable-dependent flux laws, the correlated random walk theory appears as the only analytical alternative. Based on this theory, an accurate semi-emperical expression for the conduction threshold is also developed, suggesting directions for its further development incorporating the considerations of disorder.

Fluid topology for invasion percolation in 3-D lattices

A computer algorithm for the displacement of one fluid by another in three-dimensional pore-throat network models of porous media allows the determination of fluid characteristics and distributions at any point in the displacement process. The flowing fluids are found to form two closely intertwined networks with very few regions of space containing only one fluid. Further, the fraction of dead-end invader, i.e. the ratio of the fraction of nonflowing to flowing pores is essentially constant throughout a process. The dendritic character of trapped, i.e. disconnected fluid is dependent on the degree of pore-throat correlation in the network. As a result, a given pore filled with one of the fluids is likely to have at least one neighbor filled with another fluid. Our observations imply that (1) fillable pores are very likely to be filled, and (2) clusters are not compact. The latter provides a ready explanation for the ease with which so-called oil ganglia are broken up rather than moved.

Statistical properties of immiscible fluid displacement in bond/site correlated network models of porous media : Maier, R; Laidlaw, W G Math GeolV23, N1, Jan 1991, P87–110

Statistical properties of immiscible fluid displacement in bond/site correlated network models of porous media : Maier, R; Laidlaw, W G Math GeolV23, N1, Jan 1991, P87–110

The statistical properties of immiscible fluid displacement in bond/site correlated network models of porous media

A statistical formulation is introduced which allows the determination of capillary pressure and relative permeability curves from lattice properties and, by extension, possibly core properties. Comparisons of statistical results to simulation results generally show good agreement for both drainage and imbibition. These processes turn out to be two different formulations of a single problem in one view presented here. Both agreement and disagreement between statistics and simulation results provide insight into the characteristics of the flow of immiscible fluids in porous media. A considerable amount of topological information on the fluids is also obtainable by these methods.

Depleted layer effects in polymer flow through porous media: II. Network calculations

This paper investigates the effects of surface exclusion in the rheology of polymers in network models of porous media. Here, the results developed for single capillary flow with depleted layers in Part I are incorporated into network calculations. Using input data based on the biopolymer, xanthan, a number of predictions are made on the macroscopic rheological behavior of solutions of this polymer in porous medium flow. It is found that the macroscopic “apparent slip” effect can be quite substantial even for fairly low depleted layer thicknesses compared with average pore size in the network.