Network Modeling of Non-Darcy Flow Through Porous Media

Darcy's law is inadequate for describing high-velocity gas flow in porous media, which occurs in the near well-bore region of high capacity gas and condensate reservoirs. This study is directed at understanding the non-Darcy flow behavior. A pore-level network model has been developed to describe high velocity flow. The inputs to the model are pore size distributions and network coordination numbers. The outputs are permeability, non-Darcy coefficient, tortuousity and porosity. The additional pressure gradient term is found to be proportional to the square of the velocity in accordance with the Forchheimer's equation. The correlation between the non-Darcy coefficient and other flow properties (the permeability, the porosity and the tortuousity) is found to depend on the morphological parameters being changed. General correlations are derived between these flow properties.     

Does ITZ Influence Moisture Transport in Concrete?

Simulations of nuclear magnetic resonance (NMR) signal from fluids contained in porous media (such as rock cores) need to account for both enhanced surface relaxation and the presence of internal magnetic field gradients due to magnetic susceptibility contrast between the rock matrix and the contained fluid phase. Such simulations are typically focussed on the extraction of the NMR T₂ relaxation distribution which can be related to pore size and indirectly to system permeability. Discrepancies between such NMR signal simulations on digital rock cores and associated experimental measurements are however frequently reported; these are generally attributed to spatial variations in rock matric composition resulting in heterogeneously distributed NMR surface relaxivities (ρ) and internal magnetic field gradients. To this end, a range of synthetic sediments composed of variable mixtures of quartz and garnet sands were studied. These two constituents were selected for the following reasons: they have different densities allowing for ready phase differentiation in 3D μCT images of samples to use as simulation lattices and they have distinctly different ρ and magnetic susceptibility values which allow for a rigorous test of NMR simulations. Here these 3D simulations were used to calculate the distribution of internal magnetic field gradients in the range of samples, these data were then compared against corresponding NMR experimental measurements. Agreement was reasonably good with the largest discrepancy being the simulation predicting weak internal gradients (in the vicinity of the quartz sand for mixed samples) which were not detected experimentally. The suite of 3D μCT images and associated experimental NMR measurements are all publicly available for the development and validation of NMR simulation efforts.

     

Time-Dependent Methane Diffusion Behavior in Coal: Measurement and Modeling

In-depth understanding of how methane diffuses in porous media like coal is critical for enhanced coalbed methane recovery and coal seam methane drainage. However, the classic unipore gas diffusion model can only describe methane diffusion behavior in coal at the early time diffusion stage. Describing the whole methane diffusion behavior in coal has not been reasonably addressed in the coalbed methane field. Considering the large surface-to-volume ratio of coal, the authors proposed a time-dependent gas diffusion model to describe the whole diffusion process. This work first proposed a power–law relationship between diffusion coefficient and time according to previous studies of the time-dependent gas diffusion process in porous media. Then, both the analytical solution and its approximation solution for the time-dependent gas diffusion model were derived. Six methane diffusion tests in crushed coal were conducted to validate the proposed model under different pressures (0.5, 1.5, 2.5 MPa) and temperatures (20 and \(30\,{^{\circ }}\hbox {C}\)). Modeling results show that the time-dependent gas diffusion model is superior to the classic unipore gas diffusion model for describing the whole methane diffusion process in coal. Increasing temperature always accelerates methane diffusion in coal; the higher the temperature the larger the diffusion coefficient.     

A Hierarchical Sampling for Capturing Permeability Trend in Rock Physics

Among all properties of reservoir rocks, permeability is an important parameter which is typically measured from core samples in the laboratory. Due to limitations of core drilling all over a reservoir, simulation of rock porous media is demanded to explore more scenarios not seen in the available data. One of the most accurate methods is cross correlation based simulation (CCSIM) which recently has broadly applied in geoscience and porous media. The purpose of this study is producing realizations with the same permeability trend to a real sample. Berea sandstone sample is selected for this aim. Permeability results, extracted from smaller sub-samples of the original sample, showed that classic Kozeny–Carman permeability trend is not suitable for this sample. One reason can be due to lack of including geometrical and fractal properties of pore-space distribution in this equation. Thus, a general trend based on fractal dimensions of pore-space and tortuosity of the Berea sample is applied in this paper. Results show that direct 3D stochastic modeling of porous media preserves porous structure and fractal behavior of rock. On the other hand, using only 2D images for constructing the 3D pore structures does not reproduce the measured experimental permeability. For this aim, a hierarchical sampling is implemented in two and three steps using both 2D and 3D stochastic modeling. Results showed that two-step sampling is not suitable enough, while the utilized three-step sampling occurs to be show excellent performance by which different models of porous media with the same permeability trend as the Berea sandstone sample can be generated.     

Permeability of Porous Media from Simulated NMR Response

Nuclear Magnetic Resonance (NMR) is an increasingly popular well-logging tool in petroleum industry because it is the only tool that attempts to estimate formation permeability. In this paper, spatially correlated porous media are generated. Permeabilities of these media are computed by the lattice Boltzmann method. NMR relaxation responses are simulated by a random walk technique and formation factors are computed by solving a Laplacian equation. The testing of commonly used NMR permeability correlations shows that three conditions should be met for the validity of these correlations. The surface relaxivity should not vary significantly. The formation factor should depend only on porosity. And the characteristic pore body radius should be proportional to the characteristic throat radius. The correlations are improved by including surface relaxivity and formation factor.     

Evaporation heat transfer in thin biporous media

 This paper deals with the evaporation heat transfer mechanism in thin biporous media that have two characteristic capillary pore radii. The character of the two levels of pore sizes allows the liquid phase to easily occupy the void space of the small pores and vapor phase to occupy the void space of the big pores. Compared with mono-porous media, biporous media increase the number of small evaporating menisci with high heat transfer performance. Evaporation heat transfer in pores of porous media is analyzed in detail. The results indicate that the average heat transfer coefficient increases with the capillary pore size reduction. Under the assumption of the uniform structure of biporous media, a calculation method to predict heat transfer performance for the evaporation in thin biporous media is given. The preliminary results reflect the behavior of observed vaporization heat transfer in thin biporous media well. Received on 22 February 2000     

An Algorithm for 3D Pore Space Reconstruction from a 2D Image Using Sequential Simulation and Gradual Deformation with the Probability Perturbation Sampler

The construction of a faithful 3D pore space model of a porous medium that could reproduce the macroscopic behavior of that medium is of great interest in various fields including medicine, material science, hydrology and petroleum engineering. A computationally efficient algorithm is developed that uses the probability perturbation method and sequential multiple-point statistics simulations to generate 3D stochastic and equiprobable representations of random porous media when only a 2D thin section image is available. By employing the probability perturbation method as a gradual deformation technique, the pore patterns of a single 2D image are deformed to generate a series of 2D stochastically simulated images. The 3D pore structure is then generated by simply stacking the 2D-simulated images. The quality of the 3D reconstruction is critically dependent on the rate of deformation and a simple general procedure for choosing this parameter is presented. Various criteria such as porosity, two-point auto-correlation function, multiple-point connectivity function, local percolation probability, absolute permeability obtained by lattice-Boltzmann method (LBM), formation factor and two-phase relative permeability calculations are used to validate the results. The method is tested on two random porous solids; Berea Sandstone and synthetic Silica, for which directly measured 3D micro-CT images are available. The stochastically reconstructed 3D pore space preserves the low- and high-order spatial statistics, the macroscopic flow properties and the microstructure of the 3D micro-CT images.     

Simulation of capillary pressure curves using bond correlated site percolation on a simple cubic network

A stochastic approach to network modelling has been used to simulate quasi-static immiscible displacement in porous media. Both number-based and volume-based network saturation results were obtained. Number-based results include: number-based saturation curves for primary drainage, secondary imbibition and secondary drainage, fluid distribution data, and cluster trapping history. Using pore structure data of porous media, it is possible to convert the number-based curves to capillary pressure — saturation relationships. Pore size distribution functions and pore shapes which are thought to closely represent Berea sandstone samples were used to predict the capillary curves. The physical basis of these calculations is a one-to-one correspondence between the cumulative node and bond index fractions in the network analysis, and the cumulative number-based distributions of pore body and pore throat diameters, respectively. The oil-water capillary pressure curve simulated for primary drainage closely resembles those measured experimentally. The agreement between the simulated and the measured secondary imbition and secondary drainage curves is less satisfactory.     

Calculating the Anisotropic Permeability of Porous Media Using the Lattice Boltzmann Method and X-ray Computed Tomography

A lattice Boltzmann (LB) method is developed in this article in a combination with X-ray computed tomography to simulate fluid flow at pore scale in order to calculate the anisotropic permeability of porous media. The binary 3D structures of porous materials were acquired by X-ray computed tomography at a resolution of a few microns, and the reconstructed 3D porous structures were then combined with the LB model to calculate their permeability tensor based on the simulated velocity field at pore scale. The flow is driven by pressure gradients imposed in different directions. Two porous media, one gas diffusion porous layer used in fuel cells industry and glass beads, were simulated. For both media, we investigated the relationship between their anisotropic permeability and porosity. The results indicate that the LB model is efficient to simulate pore-scale flow in porous media, and capable of giving a good estimate of the anisotropic permeability for both media. The calculated permeability is in good agreement with the measured date; the relationship between the permeability and porosity for the two media is well described by the Kozeny–Carman equation. For the gas diffusion layer, the simulated results showed that its permeability in one direction could be one order of magnitude higher than those in other two directions. The simulation was based on the single-relaxation time LB model, and we showed that by properly choosing the relaxation time, it could give similar results to those obtained using the multiple-relaxation time (MRT) LB method, but with only one third of the computational costs of MRTLB model.     

Stochastic Pore Network Generation from 3D Rock Images

Pore networks can be extracted from 3D rock images to accurately predict multi-phase flow properties of rocks by network flow simulation. However, the predicted flow properties may be sensitive to the extracted pore network if it is small, even though its underlying characteristics are representative. Therefore, it is a challenge to investigate the effects on flow properties of microscopic rock features individually and collectively based on small samples. In this article, a new approach is introduced to generate from an initial network a stochastic network of arbitrary size that has the same flow properties as the parent network. Firstly, we characterise the realistic parent network in terms of distributions of the geometrical pore properties and correlations between these properties, as well as the connectivity function describing the detailed network topology. Secondly, to create a stochastic network of arbitrary size, we generate the required number of nodes and bonds with the correlated properties of the original network. The nodes are randomly located in the given network domain and connected by bonds according to the strongest correlation between node and bond properties, while honouring the connectivity function. Thirdly, using a state-of-the-art two-phase flow network model, we demonstrate for two samples that the rock flow properties (capillary pressure, absolute and relative permeability) are preserved in the stochastic networks, in particular, if the latter are larger than the original, or the method reveals that the size of the original sample is not representative. We also show the information that is necessary to reproduce the realistic networks correctly, in particular the connectivity function. This approach forms the basis for the stochastic generation of networks from multiple rock images at different resolutions by combining the relevant statistics from the corresponding networks, which will be presented in a future publication.     

An Improved Method for Reconstructing the Digital Core Model of Heterogeneous Porous Media

The heterogeneous pore space of porous media strongly affects the storage and migration of oil and gas in the reservoir. In this paper, the cross-correlation-based simulation (CCSIM) is combined with the three-step sample method to reconstruct stochastically 3D models of the heterogeneous porous media. Moreover, the two-point and multiple-point connectivity probability functions are used as vertical constraint conditions to select the boundary points of pore and matrix, respectively. The heterogeneities of pore spaces of four rock samples are investigated, and then our methods are tested on the four samples. Quantitative comparison is made by computing various statistical and petrophysical properties for the original samples, as well as the reconstructed model. It was found that the results from CCSIM-TSS are obviously better than that from CCSIM. Finally, the analysis of the distance (ANODI) was used to measure of the variability between the realizations of the four rock samples. The results demonstrated that the results from CCSIM-TSSmp are better than that from CCSIM-TSStp as the complexity of connectivity and heterogeneities of pore spaces increase.     

NMR Measurements of Tortuosity in Partially Saturated Porous Media

The tortuosity (τ), defined in the present context as the ratio of the free diffusion coefficient to the restricted diffusion coefficient of a contained fluid, is an important but difficult to measure characteristic of a porous medium, particularly when it is partially saturated with water. We develop and apply methodology, based on nuclear magnetic resonance (NMR) pulsed field gradient techniques, to measure τ for various sandstone rock cores as a function of residual water fraction. The NMR methodology requires the use of bipolar pulsed field gradient stimulated echo pulse sequences to avoid systematic errors due to magnetic susceptibility differences and D₂O as a stationary immiscible water phase; this was selected as it provides no ¹H NMR signal. Tortuosity of the free pore space was successfully measured using liquid ethane as a probe fluid for three different sandstones over the full accessible range of residual water saturation. Generally, the tortuosity was observed to increase with residual water (D₂O) content; however, significant variations were observed between the different sandstones.     

Computing the Longtime Behaviour of NMR Propagators in Porous Media Using a Pore Network Random Walk Model

We present a pore network model combined with a random walk algorithm allowing the simulation of molecular displacement distributions in porous media as measured by NMR. A particular feature of this technique is the ability to probe the time evolution of these distributions. The objective is to predict the displacement behaviour for time intervals larger than the experimental observation time and explore the asymptotic dispersion regime at long times. Starting from 3D micro-CT images, we computed the variance of displacement distributions of water molecules in a Fontainebleau sand and found very good agreement of the time evolution of the variance with experimental data, without fitting parameter. The model confirms a weak superdispersion in the asymptotic regime. In addition, we conclude that, since pore network models do not take into account small scale features of the porous medium (e.g., surface roughness and grain shape), the origin of the observed superdispersion is mainly due to the topology and geometry of the porous medium.     

A stochastic model for filtration of particulate suspensions with incomplete pore plugging

A population balance model for particulate suspension transport with capture of particles by porous medium accounting for complete and incomplete plugging of pores by retained particles is derived. The model accounts for pore space accessibility, due to restriction on finite size particle movement through the overall pore space, and for particle flux reduction, due to transport of particles by the fraction of the overall flux. The novel feature of the model is the residual pore conductivity after the particle retention in the pore and the possibility of one pore to capture several particles. A closed system of governing stochastic equations determines the evolution of size distributions for suspended particles and pores. Its averaging results in the closed system of hydrodynamic equations accounting for permeability and porosity reduction due to plugging. The problem of deep bed filtration of a single particle size suspension through a single pore size medium where a pore can be completely plugged by two particles allows for an exact analytical solution. The phenomenological deep bed filtration model follows from the analytical solution.     

Determination of Nanoparticle Macrotransport Coefficients from Pore Scale Processes

Nanoparticle transport in porous media is modeled using a hierarchical set of differential equations corresponding to pore scale and macroscale. At the pore scale, movement and interaction of a single particle with a solid matrix is modeled using the advection–dispersion–sorption equation. A single nanoparticle entering the space encounters viscous, diffusion and surface forces. Surface forces (electrostatic and van der Waals forces) between nanoparticles and mineral grains appear as sorption propensity on solid matrix boundary condition. These local events are then transformed into a macroscale continuum by imposing periodic boundary conditions for contiguous unit cells representing porous media and using a scheme of moment analysis. At the macroscale, propagation and retention of particles are characterized by three position-independent coefficients: mean nanoparticle velocity vector \({\bar{\mathbf{U}}}^*\), macroscopic dispersion coefficient \({\bar{\mathbf{D}}}^*\), and mean nanoparticle retention rate constant \({\bar{K}}^*\). The modeling results are validated with a set of nanoparticle transport tests in porous microchips. We also present simulations of realistic porous media, where an actual image of sandstone samples is processed into binary tones. The representative unit cells are constructed from the resulting binary image by searching for areas within the sample with maximum similarities to the whole sample in terms of porosity and specific surface area, which are found to show strong correlations with the resulting \({\bar{\mathbf{U}}}^*\) and \({\bar{K}}^*\), respectively.     

Simulation of the dynamic (frequency-dependent) permeability and electrical conductivity in porous materials based on the concept of an ensemble of pores

The class of models of porous media based on the concept of an ensemble of pores with a certain distribution of the main geometrical parameters (e.g., pore size) is studied. The case of the saturation of the pore space with a single-phase multicomponent fluid mixture is studied with and without taking into account the transfer of electric charges. Transfer laws are derived from the condition of decreasing free energy. The hydrodynamic connectivity of pores (and electrical conductivity) is described by two kernels: one kernel describes the connectivity of pores in space, and the other describes the connectivity of pores in the elementary macrovolume. The frequency dependences of the dynamic permeability determined in laboratory experiments and the electrical conductivity of the porous medium were determined using the concept of an ensemble of pores. The relationship between the models considered and relaxation filtration models is established.     

Digitally Reconstructed Porous Media: Transport and Sorption Properties

The basic aim of this work is to present a combination of techniques for the reconstruction of the porous structure and the study of transport properties in porous media. The disordered structure of porous systems like random sphere packing, Vycor glass and North Sea chalk, is represented by three-dimensional binary images. The random sphere pack is generated by a standard ballistic deposition procedure, while the chalk and the Vycor matrices by a stochastic reconstruction technique. The transport properties (Knudsen diffusivity, molecular diffusivity and permeability) of the resulting 3-dimensional binary domains are investigated through computer simulations. Furthermore, physically sound spatial distributions of two phases filling the pore space are determined by the use of a simulated annealing algorithm. The wetting and the non-wetting phases are initially randomly distributed in the pore space and trial-and-error swaps are performed in order to attain the global minimum of the total interfacial energy. The effective diffusivities of the resulting domains are then computed and a parametric study with respect to the pore volume fraction occupied by each phase is performed. Reasonable agreement with available data is obtained in the single- and multi-phase transport cases.     

The evaluation of the influence of various parameters on the velocity of elastic wave propagation in a rock medium

A full waveform recording in a borehole during acoustic logging makes it possible to determine the elastic parameters of a medium under in-situ conditions.The velocity of elastic wave propagation in rocks and elastic moduli are influenced by factors connected with its macrostructure and microstructure, as well as with rock overburden and porous pressure and temperature.The results of the calculations of the relationships between the elastic and reservoir parameters of sedimentary rocks are presented in this paper. The theoretical Kuster and Toksöz model has been applied.The influence of the porosity, the pore space coefficient, and the saturation of different media of porous rocks on elastic moduli and on compressional and shear wave propagation have been considered in this model. The complex composition of the skeleton and the influence of clay material in the porous rock are taken into account.     

A New Method for Generating Pore-Network Models of Porous Media

In this study, we have developed a new method to generate a multi-directional pore network for representing a porous medium. The method is based on a regular cubic lattice network, which has two elements: pore bodies located at the regular lattice points and pore throats connecting the pore bodies. One of the main features of our network is that pore throats can be oriented in 13 different directions, allowing a maximum coordination number of 26 that is possible in a regular lattice in 3D space. The coordination number of pore bodies ranges from 0 to 26, with a pre-specified average value for the whole network. We have applied this method to reconstruct real sandstone and granular sand samples through utilizing information on their coordination number distributions. Good agreement was found between simulation results and observation data on coordination number distribution and other network properties, such as number of pore bodies and pore throats and average coordination number. Our method can be especially useful in studying the effect of structure and coordination number distribution of pore networks on transport and multiphase flow in porous media systems.