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Modern pore-scale modeling has the ability to simulate flow and particle (i.e., colloids, nanoparticles) transport at the pore- and particle-scale in realistic porous media. The pore-structure of a material can be obtained through x-ray micro tomography (XCT) or a similar three-dimensional (3D) imaging technique. From these data sets, fluid and particle transport can be simulated by direct numerical simulation of the fundamental equations of motion. In this paper, we employ the finite element method to simulate Stokes flow and, after the flow field is resolved, Lagrangian particle tracking (LPT) to track the fate and transport of nanoparticles (NPs). The presented methodology for direct numerical modeling allowed the simulation of NP transport in real materials (i.e., natural or engineered samples that experiments can be performed on rather than a computer-generated porous medium) in larger domains or with more particles than previous works. XCT images of a micromodel and a Berea sandstone were used as the computational domains to analyze the effect of particle diameter, attractive and repulsive surface forces, flow rate, surface capacity, and XCT image-based mineralogy on particle transport. In the micromodel simulations, in the presence of attractive surface forces, NP effluent recovery increased as particle diameter increased and as flow rate increased – findings qualitatively consistent with published experimental data. The use of XCT image-based mineralogy to spatially distribute attachment sites (i.e., clay) in the Berea showed no difference in particle retention when compared to a random distribution of attachment sites. These results are being used to help understand ongoing micromodel experiments and to design continuum-scale models of NP transport.

We present a review of pore-scale simulations of immiscible fluid transport with focus on two of the most popular approaches: lattice Boltzmann modeling for direct simulations on digital models of the pore space and simulations on network models extracted from the pore space. This review focuses on covering basic theory and implementation strategies and gives the readers input and motivation to start their own pore-scale simulations and relate them to realistic porous media. We present a review of recent and relevant applications and how a digital workflow that combines advanced pore-scale imaging and simulations can give very useful input to different fields of science and industry, including reservoir characterization. Given the large span in methods and applications, this review does not aim to cover all methods or applications. However, it covers popular methods and describes to some extent their applicability to different types of transport problems.

Offshore drilling and production operations can result in spills or leaks of hydrocarbons into seabed sediments, which can potentially contaminate these sediments with oil. If this oil trapped later migrates to the water surface it has the potential for negative environmental impacts. For proper contingency planning and to avoid larger consequences in the environment, it is essential to understand mechanisms and rates for hydrocarbon migration from oil containing sediments to the water surface as well as how much will remain trapped in the sediments. It is believed that the amount of oil transported out of the sediment can be affected by tidal pumping, a common form of Subterranean Ground Water Discharge (SGD). However, we could find no study investigating the phenomenon of fluid flow in subsea sediments saturated with oil and the effects of tidal pumping. This study presents an experimental investigation of tidal pumping to determine if it is a possible mechanism to describe the appearance of an oil on the ocean surface above a sediment bed containing oil. An experimental apparatus was constructed of clear PVC pipe allowing for oil migration to be monitored as it flowed out of a sand pack containing oil, while tidal pressure oscillations were applied in three different manners. The effect of tidal pumping was simulated via compression of air above the water (which simulated the increasing static head from tidal exchange). Experimental results show that sustained oil release occurred from all tests, and tests with oscillating pressure produced for longer periods of time. Furthermore, the experimental results showed that the oil migration rate was affected by grain size, oil saturation, and oscillation wave type. In the static experiments, a linear relationship between grain size and permeability was observed, as is well-known in fluid flow in porous medium. However, the oil recovery does not show a linear relationship with viscosity, as the oil recovery only changed by 50% for a nearly 400% variation in viscosity. In all oscillating experiments the rate and ultimate recovery was less than the comparable static experiments. This leads to the preliminary conclusion that with an oscillating pressure on top of a sand pack, movement of a non-replenishing source of oil is suppressed by pressure oscillation.

This paper describes recent work on image-based network modeling of multiphase flow. The algorithm expands the range of flow scenarios and boundary conditions that can be implemented using dynamic network modeling, the most significant advance being the ability to model simultaneous injection of immiscible fluids under either transient or steady-state conditions using non-periodic domains. Pore-scale saturation distributions are solved rigorously from two-phase mass conservation equations simultaneously within each pore. Results show that simulations using a periodic network fail to track saturation history because periodic domains limit how the bulk saturation can evolve over time. In contrast, simulations using a non-periodic network with fractional flow as the boundary condition can account for behavior associated with both hysteresis and saturation history, and can capture phenomena such as the long pressure and saturation tails that are observed during dynamic drainage processes. Results include a sensitivity analysis of relative permeability to different model variables, which may provide insight into mechanisms for a variety of transient, viscous dominated flow processes.

Physically representative network models have been used for years for investigating pore-scale behavior in porous materials. However, the technology has remained largely in the research domain, with limited application to industrial processes. In this work, we introduce two algorithms that are derived from well-tested network modeling techniques, but provide unique capabilities for rapid assessment of important porous material properties from high-resolution 3D images. The first is a fast, fully automated algorithm for determining the characteristic length scale for three continuum parameters: porosity, permeability, and electrical resistivity (or formation factor). This information is important for assessing the size of the computational domain needed for image-based modeling. It operates by computing relevant continuum properties from consecutively smaller subdomains extracted from a larger pore network model. Resulting data trends are plotted so that the relevant characteristic lengths can be inferred....

[1] The concept of coupling pore-scale and continuum-scale models for subsurface flow has long been viewed as beneficial, but implementation has been slow. In this paper, we present an algorithm for direct coupling of a dynamic pore-network model for multiphase flow with a traditional continuum-scale simulator. The ability to run the two models concurrently (exchanging parameters and boundary conditions in real numerical time) is made possible by a new dynamic pore-network model that allows simultaneous injection of immiscible fluids under either transient-state or steady-state conditions. Allowing the pore-scale model to evolve to steady state during each time step provides a unique method for reconciling the dramatically different time and length scales across the coupled models. The model is implemented by embedding networks in selected gridblocks in the reservoir model. The network model predicts continuum-scale parameters such as relative permeability or average capillary pressure from first principles, which are used in the continuum model. In turn, the continuum reservoir simulator provides boundary conditions from the current time step back to the network model to complete the coupling process. The model is tested for variable-rate immiscible displacements under conditions in which relative permeability depends on flow rate, thus demonstrating a situation that cannot be modeled using a traditional approach. The paper discusses numerical challenges with this approach, including the fact that there is not a way to explicitly force pore-scale phase saturation to equal the continuum saturation in the host gridblock without an artificial constraint. Hurdles to implementing this type of modeling in practice are also discussed.

Image-based network modeling has become a powerful tool for modeling transport in real materials that have been imaged using X-ray computed micro-tomography (XCT) or other three-dimensional imaging techniques. Network generation is an essential part of image-based network modeling, but little quantitative work has been done to understand the influence of different network structures on modeling. We use XCT images of three different porous materials (disordered packings of spheres, sand, and cylinders) to create a series of four networks for each material. Despite originating from the same data, the networks can be made to vary over two orders of magnitude in pore density, which in turn affects network properties such as pore-size distribution and pore connectivity. Despite the orders-of-magnitude difference in pore density, single-phase permeability predictions remain remarkably consistent for a given material, even for the simplest throat conductance formulas. Detailed explanations for this beneficial attribute are given in the article; in general, it is a consequence of using physically representative network models. The capillary pressure curve generated from quasi-static drainage is more sensitive to network structure than permeability. However, using the capillary pressure curve to extract pore-size distributions gives reasonably consistent results even though the networks vary significantly. These results provide encouraging evidence that robust network modeling algorithms are not overly sensitive to the specific structure of the underlying physically representative network, which is important given the variety image-based network-generation strategies that have been developed in recent years.

Microtomography is an emerging technique for particle and particulate-materials characterization. To use this technology effectively, robust and accurate computational algorithms are needed to compute relevant particle properties, including particle surface area and particle-particle contact area. However, the most accurate algorithms that have been developed for computing the exposed (void/solid) surface area in a microtomography image cannot be used directly for computing surface areas or particle-particle contact areas for individual particles in a dense packing. This paper presents an algorithm for extracting particle contact areas from a digitized, segmented image of a packed granular material, which in turn can be used to find individual particle surface areas (even if the complete surfaces are not exposed because of contacts in the packing). Results show that small errors in the binary surface-area computations are magnified in the course of determining particle contact areas; the total error in the computation depends mainly on the size of the contact area in voxel units.

Network modeling is a useful tool for investigating pore-scale behavior and in some cases for determining macroscopic information such as permeability, relative permeability, and capillary pressure. Physically representative network models are particularly useful because quantitative and predictive results can be obtained. In the past, network models have been used as stand-alone tools for predicting flow behavior at the pore scale. In these cases, simple boundary conditions such as a pressure gradient in one direction are generally imposed on the network. However, with the increasing emphasis on multiscale modeling techniques, the real potential of network models is as a bridge from the pore to the continuum scale. In this context, continuum-scale and pore-scale models are used jointly; pore-scale behavior is upscaled and substituted into a continuum-scale simulator. Methods for integrating these techniques are being developed, and one important question is how to match boundary conditions for the two scales. In this work, physically representative network models created from computer-generated sphere packings are coupled to adjacent continuum-scale models. By coupling the two regions, realistic boundary conditions are enforced, which reflect the heterogeneity of the packed bed as well as the resistance of the adjacent medium. Results of the direct coupling show that both pore-scale phenomena and macroscopic behavior (such as flowrate) are significantly different than when these same parameters are obtained by implementing simple (decoupled) boundary conditions.

Fixed beds of cylindrical particles are important in chemical engineering applications, but their packing structures are not as well understood or as well characterized as sphere packings. In this work, X-ray microtomography is used to obtain 3D images of 1.8 mm diameter equilateral cylinders in a 23 mm cylindrical container over a range of bulk porosities. A novel algorithm is used to computationally reconstruct the packings, resulting in data sets that give the location and orientation of each cylinder in the imaged packings. Extensive analysis has been performed, including bulk and local porosities, radial distribution functions, and parameters describing local and global ordering. The major factors affecting packing structure are the overall packing density and the proximity to the wall. At the highest overall packing densities, near-wall porosity becomes nearly equal to interior porosity, and significant global ordering occurs near the wall. For a vertical container, global ordering is characterized by the alignment of the particles with an orthogonal coordinate system that has one axis coincident with r (as defined by the container) and the other two axes in the z – θ plane, but rotated 45 ∘ with the horizontal. The observed structures are relevant in the context of flow maldistribution and heat transfer in fixed beds.

Using a new computational model, we have studied the dynamics and coalescence of a pair of two-dimensional droplets in pressure-driven flow through a constricted capillary tube, which is a prototype problem for the analysis of the interaction of emulsion droplets in porous media. We present simulations that quantify the effects of various system parameters on the droplet stability. These include the capillary number, the interfacial tension, the suspended-to-suspending-phase viscosity ratio, the valence and concentration of added electrolytes, the droplet-to-pore-size ratio, the pore-body-to-throat-size ratio, and the type of pore geometry. Our simulations show that the capillary number Ca plays an important role in determining whether the drops coalesce. At low Ca , drops deform only slightly and coalescence occurs at the entrance of the pore throat, whereas significant deformation enables the drops move through the pore without coalescence at high Ca . Coalescence is favored at intermediate values of the viscosity ratio. The destabilizing effect of added electrolytes is found to be insignificant for 10-μm drops, but significant for micron-size drops. Among the geometric-related parameters, the drop-to-pore-size ratio is the most significant.

Traditional particle-characterization techniques require particles to be dispersed from their original form, which destroys important morphologic information in materials such as aggregates or porous materials. X-ray computed microtomography provides a powerful tool for non-destructive analysis. However, robust techniques for comprehensive material characterization of these images have not been developed. In this paper we present a new algorithm for the computer analysis of particulate materials from high-resolution tomography images. The key aspect of the algorithm is the assignment of every solid-phase voxel in the image to its associated particle in a physically representative manner, which is in essence a particle-scale computer reconstruction of the material. Once this digital reconstruction is obtained, a vast amount of morphologic information can be extracted, including parameters obtained by traditional particle-analysis techniques (e.g., particle-size distribution and porosity) as well as parameters not usually available (e.g., spatial correlations in particle size, particle aspect ratios, surface areas, and orientations, particle contacts, particle coordination numbers, and more). Additionally, the computer reconstruction allows for complex manipulations such as the comparison of a specific parameter for two different particle-size classes within the material. The paper includes validation of the algorithm using computer-generated packings, as well as an example using microtomography data from a real material.

The flow of non-Newtonian fluids in packed beds and other porous media is important in several applications such as polymer processing, filtration, and enhanced oil recovery. Expressions for flowrate versus pressure gradient are desirable for a-priori prediction and for substitution into continuum models. In this work, physically representative network models are used to model the flow of shear-thinning fluids, including power-law and Ellis fluids. The networks are used to investigate the effects of fluid rheology and bed morphology on flow. A simple macroscopic model is developed for the flow of power-law and Ellis fluids in packed beds using results from the network model. The model has the same general functionality as those developed using the popular bundle-of-tubes approach. The constant β , which appears in these models, is often directly derived from the tortuosity and a simple representation of the porous media. It is shown here that this can lead to incorrect and ambiguous values of the constant. Furthermore, the constant is a weak function of the shear-thinning index, indicating that no single bundle-of-tubes could ever properly model flow for a wide variety of shear-thinning fluids. The macroscopic model is compared to experimental data for shear-thinning fluids available in the literature. The model fits the data well when β is treated as an experimental parameter. The best-fit values of β vary, which is expected because even the constant C in the Blake–Kozeny equation varies depending on the source consulted. Additionally, physical effects, such as adsorption and filtration, as well as rheological effects such as viscoelasticity may affect the value of β . We believe that in the absence of these effects, β equals approximately 1.46 for packed beds of uniform spheres at relatively moderate values of the shear-thinning index ( > 0.3 ) .

Network modeling has been performed to obtain quantitative and predictive results of the flow of yield-stress fluids in packed beds. Physically representative networks were used as the basis for the modeling, which have a one-to-one correspondence to computer-generated packed beds of spheres. The networks are able to account for the interconnectivity, heterogeneity, and converging/diverging geometry that are inherent in porous media. The approach can be used to model a wide range of non-Newtonian fluids, but the emphasis is for yield-stress fluids that can be represented using a Bingham model. For these fluids, a threshold pressure gradient is required to initiate flow, and flow at low pressure gradients is characterized by critical percolation behavior. Quantitative results of superficial velocity vs. pressure gradient are presented, and are compared to traditional bundle-of-tubes models, as well as limited experimental data available in the literature. Important differences are observed between the network model and the constitutive models. These are attributed mainly to heterogeneity and converging/diverging geometry, which are not accounted for in the semiempirical models. Comparison to experimental data is good for certain fluids. In other cases, the modeling suggests that effects other than fluid rheology may also have affected flow, such as adsorption or filtration. © 2004 American Institute of Chemical Engineers AIChE J, 50: 3034–3048, 2004

Many engineering problems require the estimation of mass transfer coefficients in porous materials. In heterogeneous materials or in cases where mass transfer sites are not spatially uniform, empirical equations for mass transfer coefficients vary widely, and the origin of these differences is not well understood. In this article, we use a stochastic algorithm to model mass transfer from single particles in a two-dimensional heterogeneous packed bed. The computed mass transfer coefficients are used to generate a distribution of local Peclet numbers in the bed. Detailed hydrodynamics are then used to interpret variations in the local Peclet number. The results show clear relationships between pore structure, streamline patterns, and mass transfer rates.

While network models of porous materials have traditionally been constructed using regular or disordered lattices, recent developments allow the direct modeling of more realistic structures such as sphere packings, microtomographic images, or computer-simulated materials. One of the obstacles in these newer approaches is the generation of network structures that are physically representative of the real systems. In this paper, we present and compare two different algorithms to extract pore network parameters from three-dimensional images of unconsolidated porous media systems. The first approach, which utilizes a pixelized image of the pore space, is an extension to unconsolidated systems of a medial-axis based approach (MA). The second approach uses a modified Delaunay tessellation (MDT) of the grain locations. The two algorithms are validated using theoretical packings with known properties and then the networks generated from random packing are compared. For the regular packings, both methods are able to provide the correct pore network structure, including the number, size, and location of inscribed pore bodies, the number, size, and location of inscribed pore throats, and the connectivity. Despite the good agreement for the regular packings, there were differences in both the spatial mapping and statistical distributions in network properties for the random packings. The discrepancies are attributed to the pixelization at low resolution, non-uniqueness of the inscribed pore-body locations, and differences in merging processes used in the algorithms, and serve to highlight the difficulty in creating a unique network from a complex, continuum pore space.

Once spilled into soils, dense nonaqueous phase liquids (DNAPLs) such as chlorinated solvents migrate deep into the subsurface because of their high density. Their downward migration typically continues until capillary forces balance gravitational forces or until essentially impermeable strata are reached. Efforts to mobilize the DNAPL for remediation purposes risks driving the contaminants deeper, which has spurred research for modifying buoyancy forces in situ. In this paper, a novel means of controlling the density of a DNAPL phase using polyaphrons is presented. Polyaphrons are a class of high internal phase ratio emulsions (HIPREs) that have unusual properties such as indefinite stability and flow properties through porous media. They provide a means of selectively delivering a light organic phase liquid to the vicinity of the DNAPL phase. Upon destabilization of the polyaphron by a polyvalent cation, the light internal phase mixes with the DNAPL to produce a nonaqueous phase of lower density than the original contaminant. The negative buoyancy of the DNAPL can thus be reversed. This approach holds great promise for manipulating DNAPL densities prior to or during remediation treatments.

This paper presents a new numerical model of reactive polymer flow in heterogeneous porous media. A moment representation of the log–normal polymer molecular weight distribution is used to model polymer as a multi-component species. Three leading moments are used to simulate the polymer transport and reaction processes in a two-dimensional porous medium. The 2D, multi-phase polymer flow model is based on a mass-transport equation for multi-component species and is coupled with kinetic models of the gelation process using an operator splitting scheme. The sensitivity of various parameters and constitutive equations is presented.

The modeling of fluid transport in fibrous materials is important for many applications. Most models operate at the continuum level, which requires an a priori knowledge of spatially averaged transport parameters. Alternatively, highly detailed models, in which the momentum equations are solved directly, require major simplifying assumptions. Thus, it is desirable to use intermediate-level techniques that model transport using first principles, but that are appropriate for real engineering processes. In this work, pore-scale network modeling is adapted for fibrous materials and tested for a large range of fibrous structures and solid volume fractions. A novel technique is used to generate prototype network structures from Voronoi diagrams. The Voronoi networks are coupled with two different multiphase flow algorithms, enabling the modeling of various displacement processes relevant to engineering. Permeability predictions agree well with known values. Effects of dynamics, wettability, and material structure on displacement were studied. This modeling technique not only allows for better quantification of how microscale properties affect macroscopic transport, but helps reduce the number of experiments required to predict continuum transport parameters for various materials and processes.

An algorithm is presented for constructing three-dimensional Delaunay tessellations in periodic domains. Applications include mesh generation for periodic transport problems and geometric decomposition for modelling particulate structures. The algorithm is a point insertion technique, and although the general framework is similar to point insertion in a convex hull, a number of new issues are introduced by periodicity. These issues are discussed in detail in the context of the computational algorithm. Examples are given for the tessellation of random points and random sphere packings. Performance data for the algorithm are also presented. These data show an empirical scaling of the computation time with size of O(N1.11) and tessellation rates of 7000–14000 tetrahedrons per second for the problems studied (up to 105 points). A breakdown of the performance is given, which shows the computational load is shared most heavily by two specific parts of the point-insertion procedure. Copyright © 2002 John Wiley & Sons, Ltd.

An algorithm is presented for solving the Stokes equation in large disordered two-dimensional porous domains. It is applied to random packings of discs, but the geometry can be essentially arbitrary. The approach includes the subdivision of the domain and a subsequent application of boundary integral equations to the subdomains. This gives a block diagonal matrix with sparse off-block components that arise from shared variables on internal subdomain boundaries. The global problem is solved using a biconjugate gradient routine with preconditioning. Results show that the effectiveness of the preconditioner is strongly affected by the subdomain structure, from which a methodology is proposed for the domain decomposition step. A minimum is observed in the solution time versus subdomain size, which is governed by the time required for preconditioning, the time for vector multiplications in the biconjugate gradient routine, the iterative convergence rate and issues related to memory allocation. The method is demonstrated on various domains including a random 1000-particle domain. The solution can be used for efficient recovery of point velocities, which is discussed in the context of stochastic modelling of solute transport

Network models are an effective means of incorporating pore-scale heterogeneity into flow models of porous materials. The drawback to these models used to be the inability to obtain quantitative macroscopic parameters representing larger (experimental-scale) media. However, recently developed modeling techniques, combined with more widely available computational resources, make the simulation of macroscopic parameters from a network approach viable. A network model for the slow flow of an incompressible fluid in disordered packed beds is presented. Fundamental fluid mechanics equations are solved at the pore scale and then translated to macroscopic behaviour using a network approach. The results reproduce experimental permeabilities and show excellent quantitative fits to residence time distributions for mechanical dispersion in real beds. Simulations of the RTD are of special interest, because they are definitive links between pore-scale flow behavior and macroscopic responses.