Showing papers in "Transport in Porous Media in 2019"

Flow in Fractured Porous Media: A Review of Conceptual Models and Discretization Approaches

Abstract: The last decade has seen a strong increase of research into flows in fractured porous media, mainly related to subsurface processes but also in materials science and biological applications, as connected fractures can totally dominate flow patterns. Due to the fractures’ characteristics as approximately planar discontinuities with an extreme size-to-width ratio, they challenge standard macroscale mathematical and numerical modeling of flow based on averaging. Thus, over the last decades, various, and also fundamentally different, modeling approaches have been developed. This paper reviews common conceptual models and discretization approaches for flow in fractured porous media, with an emphasis on the dominating effects the fractures have on flow processes. In this context, the paper discusses the tight connection between physical and mathematical modeling and simulation approaches. Extensions and research challenges related to transport, multi-phase flow and fluid-solid interaction are also commented on.

Porous Media Characterization Using Minkowski Functionals: Theories, Applications and Future Directions

Abstract: An elementary question in porous media research is in regard to the relationship between structure and function. In most fields, the porosity and permeability of porous media are properties of key interest. There is, however, no universal relationship between porosity and permeability since not only does the fraction of void space matter for permeability but also the connectivity of the void fraction. With the evolution of modern day X-ray microcomputed tomography (micro-CT) and advanced computing, it is now possible to visualize porous media at an unprecedented level of detail. Approaches in analyzing micro-CT data of porous structures vary in the literature from phenomenological characterization to network analysis to geometrical and/or topological measurements. This leads to a question about how to consistently characterize porous media in a way that facilitates theoretical developments. In this effort, the Minkowski functionals (MF) emerge from the field of statistical physics where it is evident that many physical processes depend on the geometry and topology of bodies or multiple bodies in 3D space. Herein we review the theoretical basis of the MF, mathematical theorems and methods necessary for porous media characterization, common measurement errors when using micro-CT data and recent findings relating the MF to macroscale porous media properties. This paper is written to provide the basics necessary for porous media characterization and theoretical developments. With the wealth of information generated from 3D imaging of porous media, it is necessary to develop an understanding of the limitations and opportunities in this exciting area of research.

Mixing-Limited Reactions in Porous Media

Abstract: Mixing-driven reactions in porous media are ubiquitous and span natural and engineered environments, yet predicting where and how quickly reactions occur is immensely challenging due to the complex and nonuniform nature of porous media flows. In particular, in many instances, there is an enormous range of spatial and temporal scales over which reactants can mix. This paper aims to review factors that affect mixing-limited reactions in porous media, and approaches used to predict such processes across scales. We focus primarily on the challenges of mixing-driven reactions in porous media at pore scales to provide a concise, but comprehensive picture. We balance our discussion between state-of-the-art experiments, theory and numerical methods, introducing the reader to factors that affect mixing, focusing on the bracketing cases of transverse and longitudinal mixing. We introduce the governing equations for mixing-limited reactions and then summarize several upscaling methods that aim to account for complex pore-scale flow fields. We conclude with perspectives on where the field is going, along with other insights gleaned from this review.

Beyond Kozeny–Carman: Predicting the Permeability in Porous Media

Abstract: Various processes such as heterogeneous reactions or biofilm growth alter a porous medium’s underlying geometric structure. This significantly affects its hydrodynamic parameters, in particular the medium’s effective permeability. An accurate, quantitative description of the permeability is, however, essential for predictive flow and transport modeling. Well-established relations such as the Kozeny–Carman equation or power law approaches including fitting parameters relate the porous medium’s porosity to a scalar permeability coefficient. Opposed to this, upscaling methods directly enable calculating the full, potentially anisotropic, permeability tensor. As input, only the geometric information in terms of a representative elementary volume is needed. To compute the porosity–permeability relations, supplementary cell problems must be solved numerically on this volume and their solutions must be integrated. We apply this approach to provide easy-to-use quantitative porosity–permeability relations that are based on representative single grain, platy, blocky, prismatic soil structures, porous networks, and real geometries obtained from CT-data. As a discretization method, we use discontinuous Galerkin method on structured grids. To make the relations explicit, interpolation of the obtained data is used. We compare the outcome with the well-established relations and investigate the ranges of the validity. From our investigations, we conclude whether Kozeny–Carman-type or power law-type porosity–permeability relations are more reasonable for various prototypic representative elementary volumes. Finally, we investigate the impact of a microporous solid matrix onto the permeability.

Diffusion in Porous Media: Phenomena and Mechanisms

Abstract: Two distinct but interconnected approaches can be used to model diffusion in fluids; the first focuses on dynamics of an individual particle, while the second deals with collective (effective) motion of (infinitely many) particles. We review both modeling strategies, starting with Langevin’s approach to a mechanistic description of the Brownian motion in free fluid of a point-size inert particle and establishing its relation to Fick’s diffusion equation. Next, we discuss its generalizations which account for a finite number of finite-size particles, particle’s electric charge, and chemical interactions between diffusing particles. That is followed by introduction of models of molecular diffusion in the presence of geometric constraints (e.g., the Knudsen and Fick–Jacobs diffusion); when these constraints are imposed by the solid matrix of a porous medium, the resulting equations provide a pore-scale representation of diffusion. Next, we discuss phenomenological Darcy-scale descriptors of pore-scale diffusion and provide a few examples of other processes whose Darcy-scale models take the form of linear or nonlinear diffusion equations. Our review is concluded with a discussion of field-scale models of non-Fickian diffusion.

Prediction of Porosity and Permeability Alteration Based on Machine Learning Algorithms

Abstract: The objective of this work is to study the applicability of various machine learning algorithms for the prediction of some rock properties which geoscientists usually define due to special laboratory analysis. We demonstrate that these special properties can be predicted only basing on routine core analysis (RCA) data. To validate the approach, core samples from the reservoir with soluble rock matrix components (salts) were tested within 100 + laboratory experiments. The challenge of the experiments was to characterize the rate of salts in cores and alteration of porosity and permeability after reservoir desalination due to drilling mud or water injection. For these three measured characteristics, we developed the relevant predictive models, which were based on the results of RCA and data on coring depth and top and bottom depths of productive horizons. To select the most accurate machine learning algorithm, a comparative analysis has been performed. It was shown that different algorithms work better in different models. However, two-hidden-layer neural network has demonstrated the best predictive ability and generalizability for all three rock characteristics jointly. The other algorithms, such as support vector machine and linear regression, also worked well on the dataset, but in particular cases. Overall, the applied approach allows predicting the alteration of porosity and permeability during desalination in porous rocks and also evaluating salt concentration without direct measurements in a laboratory. This work also shows that developed approaches could be applied for the prediction of other rock properties (residual brine and oil saturations, relative permeability, capillary pressure, and others), of which laboratory measurements are time-consuming and expensive.

Modeling Oil Recovery in Mixed-Wet Rocks: Pore-Scale Comparison Between Experiment and Simulation

Abstract: To examine the need to incorporate in situ wettability measurements in direct numerical simulations, we compare waterflooding experiments in a mixed-wet carbonate from a producing reservoir and results of direct multiphase numerical simulations using the color-gradient lattice Boltzmann method. We study the experiments of Alhammadi et al. (Sci Rep 7(1):10753, 2017. https://doi.org/10.1038/s41598-017-10992-w ) where the pore-scale distribution of remaining oil was imaged using micro-CT scanning. In the experiment, in situ contact angles were measured using an automated algorithm (AlRatrout et al. in Adv Water Resour 109:158–169, 2017. https://doi.org/10.1016/j.advwatres.2017.07.018 ), which indicated a mixed-wet state with spatially non-uniform angles. In our simulations, the pore structure was obtained from segmented images of the sample used in the experiment. Furthermore, in situ measured angles were also incorporated into our simulations using our previously developed wetting boundary condition (Akai et al. in Adv Water Resour 116(March):56–66, 2018. https://doi.org/10.1016/j.advwatres.2018.03.014 ). We designed six simulations with different contact angle assignments based on experimentally measured values. Both a constant contact angle based on the average value of the measured values and non-uniform contact angles informed by the measured values gave a good agreement for fluid pore occupancy between the simulation and the experiment. However, the constant contact angle assignment predicted 54% higher water effective permeability after waterflooding than that estimated for the experimental result, whereas the non-uniform contact angle assignment gave less than 1% relative error. This means that to correctly predict fluid conductivity in mixed-wet rocks, a spatially heterogeneous wettability state needs to be taken into account. The novelty of this work is to provide a direct pore-scale comparison between experiments and simulations employing experimentally measured contact angles, and to demonstrate how to use measured contact angle data to improve the predictability of direct numerical simulation, highlighting the difference between the contact angle required for the simulation of dynamic displacement process and the contact angle measured at equilibrium after waterflooding.

Development of Open-Source Porous Media Simulators: Principles and Experiences

Abstract: By its very nature, research into multi-physical processes occurring in porous and fractured media requires a collaborative approach. An interdisciplinary approach has led to the adoption of collaborative software development paradigms in this field relying on software for scientific computing as research infrastructures. The development of open-source software has become a cornerstone of computational approaches in academia and has even spawned successful business models in the commercial world. This article is geared toward readers who want to learn more about potential benefits of open-source software in porous media research and who want to familiarize themselves with typical workflows required to become an active contributor to or user of open-source solutions for porous media simulation. The article puts general principles, motivations and concepts into the specific context of experiences and lessons learned from the authors developing the open-source software projects OpenGeoSys and DuMu $$^{\text {x}}$$ .

Theory and Applications of Macroscale Models in Porous Media

Abstract: Systems dominated by heterogeneity over a multiplicity of scales, like porous media, still challenge our modeling efforts. The presence of disparate length- and time-scales that control dynamical processes in porous media hinders not only models predictive capabilities, but also their computational efficiency. Macrosopic models, i.e., averaged representations of pore-scale processes, are computationally efficient alternatives to microscale models in the study of transport phenomena in porous media at the system, field or device scale (i.e., at a scale much larger than a characteristic pore size). We present an overview of common upscaling methods used to formally derive macroscale equations from pore-scale (mass, momentum and energy) conservation laws. This review includes the volume averaging method, mixture theory, thermodynamically constrained averaging, homogenization, and renormalization group techniques. We apply these methods to a number of specific problems ranging from food processing to human bronchial system, and from diffusion to multiphase flow, to demonstrate the methods generality and flexibility in handling different applications. The primary intent of such an overview is not to provide a thorough review of all currently available upscaling techniques, nor a complete mathematical treatment of the ones presented, but rather a primer on some of the tools available for upscaling, the basic principles they are based upon, and their specific advantages and drawbacks, so to guide the reader in the choice of the most appropriate method for particular applications and of the most relevant technical literature.

Reduced-Order Modeling of Subsurface Multi-phase Flow Models Using Deep Residual Recurrent Neural Networks

Abstract: We present a reduced-order modeling technique for subsurface multi-phase flow problems building on the recently introduced deep residual recurrent neural network (DR-RNN) (Nagoor Kani et al. in DR-RNN: a deep residual recurrent neural network for model reduction. ArXiv e-prints, 2017). DR-RNN is a physics-aware recurrent neural network for modeling the evolution of dynamical systems. The DR-RNN architecture is inspired by iterative update techniques of line search methods where a fixed number of layers are stacked together to minimize the residual (or reduced residual) of the physical model under consideration. In this manuscript, we combine DR-RNN with proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM) to reduce the computational complexity associated with high-fidelity numerical simulations. In the presented formulation, POD is used to construct an optimal set of reduced basis functions and DEIM is employed to evaluate the nonlinear terms independent of the full-order model size. We demonstrate the proposed reduced model on two uncertainty quantification test cases using Monte Carlo simulation of subsurface flow with random permeability field. The obtained results demonstrate that DR-RNN combined with POD–DEIM provides an accurate and stable reduced model with a fixed computational budget that is much less than the computational cost of standard POD–Galerkin reduced model combined with DEIM for nonlinear dynamical systems.

Multi-layered Porous Foam Effects on Heat Transfer and Entropy Generation of Nanofluid Mixed Convection Inside a Two-Sided Lid-Driven Enclosure with Internal Heating

Abstract: Mixed convection of Cu-water nanofluid inside a two-sided lid-driven enclosure with an internal heater, filled with multi-layered porous foams is studied numerically and its heat transfer and entropy generation number are evaluated. Use of multi-layered porous media instead of homogeneous ones is capable of heat transfer enhancement, by weakening flow where does not impose a pivotal role on heat transfer and amplifying the flow in regions where have more effects on the heat transfer. Eight different arrangements of porous layers are considered and the two-phase mixture model is implemented to simulate nanofluid mixed convection inside the cavity. Results are presented in terms of stream functions, isotherms, Nusselt and entropy generation number for the eight cases considering various Richardson numbers (Ri = 10−4 to 103) and nanofluid concentrations (φ = 0 to 0.04). Results indicate that using the multi-layered porous material can confine flow vortices in the vicinity of the moving walls and could enhance the heat transfer up to 17 percent (with respect to the case using homogeneous porous material with the highest permeability), such that this enhancement is more in lower Ri values (stronger convective effects). Entropy generation number also increases by nanofluid volume fraction increment and Ri decrement. Cases with a higher heat transfer rate also have the higher entropy generation number. In addition, an increase of volume fraction decreases the relative entropy generation number (S*) for low Ri number, while contrary fact observed for high Ri values.

Effects of Image Resolution on Sandstone Porosity and Permeability as Obtained from X-Ray Microscopy

Abstract: Estimating porosity and permeability for porous rock is a vital component of reservoir engineering, and imaging techniques have to date focused on methodologies to match image-derived flow parameters with experimentally identified values. Less emphasis has been placed on the trade-off between imaging complexity, computational time, and error in identifying porosity and permeability. Here, the effect of image resolution on the permeability derived from micro-X-ray microscopy (micro-XRM) is discussed. A minicore plug of Bentheimer sandstone is imaged at a resolution of $$1024 \times 1024 \times 1024$$ voxels, with a voxel size of 1.53 $${\upmu }\hbox {m}$$ , and progressively rebinned to as low as 32 voxels per side (voxel size 48.96 $${\upmu }\hbox {m}$$ ). Pore-scale flow is modeled using the finite volume method in the open-source program OpenFOAM®. A sharp drop in permeability between images with a voxel size of 24 and 12 $${\upmu }\hbox {m}$$ suggests that an optimal speed/resolution trade-off may be found. The primary source of error is due to reassignment of voxels along the pore–solid interface and the subsequent change in pore connectivity. We observe the error in permeability and porosity due to both image resolution and thresholding values in order to find a method that balances an acceptable error range with reasonable computation time.

Reliability of Algorithms Interpreting Topological and Geometric Properties of Porous Media for Pore Network Modelling

Abstract: Pore network models (PNMs) offer a computationally efficient way to analyse transport in porous media. Their effectiveness depends on how well they represent the topology and geometry of real pore systems, for example as imaged by X-ray CT. The performance of two popular algorithms, maximum ball and watershed, is evaluated for three porous systems: an idealised medium with known pore throat properties and two rocks with different morphogenesis—carbonate and sandstone. It is demonstrated that while the extracted PNM simulates simple flow (permeability) with acceptable accuracy, their topological and geometric properties are significantly different. This suggests that such PNM may not serve more complex studies, such as reactive/convective transport of contaminants or bacteria, and further research is necessary to improve the interpretation of real pore spaces with networks. Linear topology–geometry relations are derived and presented to stimulate development of more realistic PNM.

Microfluidics for Porous Systems: Fabrication, Microscopy and Applications

Abstract: No matter how sophisticated the structures are and on what length scale the pore sizes are, fluid displacement in porous media can be visualized, captured, mimicked and optimized using microfluidics. Visualizing transport processes is fundamental to our understanding of complex hydrogeological systems, petroleum production, medical science applications and other engineering applications. Microfluidics is an ideal tool for visual observation of flow at high temporal and spatial resolution. Experiments are typically fast, as sample volume is substantially low with the use of miniaturized devices. This review first discusses the fabrication techniques for generating microfluidics devices, experimental setups and new advances in microfluidic fabrication using three-dimensional printing, geomaterials and biomaterials. We then address multiphase transport in subsurface porous media, with an emphasis on hydrology and petroleum engineering applications in the past few decades. We also cover the application of microfluidics to study membrane systems in biomedical science and particle sorting. Lastly, we explore how synergies across different disciplines can lead to innovations in this field. A number of problems that have been resolved, topics that are under investigation and cutting-edge applications that are emerging are highlighted.

Micro-continuum Framework for Pore-Scale Multiphase Fluid Transport in Shale Formations

Abstract: A micro-continuum simulation framework is proposed to study the complex pore-scale dynamics associated with hydrocarbon recovery from shale gas. The model accounts for the presence of immiscible fluid phases and for transport mechanisms in the nanoporous structures including slip flow, adsorption, surface and Knudsen diffusion. We employ the concept of sub-grid models to simulate the transport phenomena in shale gas. Specifically, we use high-resolution FIB–SEM images that provide information on the spatial distribution of the minerals, resolved pore space, and sub-resolution porous regions. The model is used to investigate several production scenarios at the pore-scale. In one setting, the organic matter is in direct contact with a micro-crack; in the other setting, clay regions are sandwiched between the organic matter and the “open” crack. The simulations show that it is important to account for the presence of multiple immiscible fluid phases because they can play a critical role in hydrocarbon production from shale-gas formations both in terms of production rate and in terms of residual mass of hydrocarbon. Moreover, we show that, because of wettability conditions, the rate of hydrocarbon recovery, as well as the ultimate recovery, depends strongly on the spatial distribution of the kerogen and clay in the vicinity of the micro-cracks.

Flow Mechanism and Simulation Approaches for Shale Gas Reservoirs: A Review

Abstract: The past two decades have borne remarkable progress in our understanding of flow mechanisms and numerical simulation approaches of shale gas reservoir, with much larger number of publications in recent 5 years compared to that before year 2012. In this paper, a review is constructed with three parts: flow mechanism, reservoir models and numerical approaches. In mechanism, it is found that gas adsorption process can be concluded into different isotherm models for various reservoir basins. Multi-component adsorption mechanisms are taken into account in recent years. Flow mechanism and equations vary with different Knudsen numbers, which could be figured out in two ways: molecular dynamics (MD) and lattice Boltzmann method (LBM). MD has been successfully applied in the study of adsorption, diffusion, displacement and other mechanisms. LBM has been introduced in the study of slippage, Knudsen diffusion and apparent permeability correction. The apparent permeability corrections are introduced to improve classic Darcy’s model in matrix with low velocities and fractures with high velocities. At reservoir-scale simulation, gas flow models are presented with multiple porosity classified into organic matrix with nanopores, organic matrix with micropores, inorganic matrix and natural fractures. A popular trend is to incorporate geomechanism with flow model in order to better understand the shale gas production. Finally, to solve the new models based on enhanced flow mechanisms, improved macroscopic numerical approaches, including the finite difference method and finite element method, are commonly used in this area. Other approaches like finite volume method and fast matching method are also developed in recent years.

Upscaling of Anomalous Pore-Scale Dispersion

Abstract: We study the upscaling of advective pore-scale dispersion in terms of the Eulerian velocity distribution and advective tortuosity, both flow attributes, and of the average pore length, a medium attribute. The stochastic particle motion is modeled as a time-domain random walk, in which particles move along streamlines in equidistant spatial steps with random velocities and thus random transition times. Particle velocities describe stationary spatial Markov processes, which evolve along streamlines on the mean pore length. The streamwise motion is projected onto the mean flow direction using tortuosity. This upscaled stochastic particle model predicts accurately the (non-Fickian) transport dynamics obtained from direct numerical simulations of particle transport in a three-dimensional digitized Berea sandstone sample. It captures all aspects of transport and sheds light on the dependence of the upscaled transport behavior on the flow heterogeneity and the initial particle distribution, which are critical for the accurate modeling of dispersion from the pre-asymptotic to asymptotic regimes.

Experimental Investigation of the Mechanical Behavior and Permeability of 3D Printed Sandstone Analogues Under Triaxial Conditions

Abstract: Consolidated drained triaxial tests arise as one of the most exhaustive methods to quantify the strength, volumetric behavior and failure process of rocks. Understanding the compressibility of a rock matrix and the permeability evolution induced by the effects of confining stress is essential to achieve a better understanding of the productive behavior and performance of enhanced hydrocarbon recovery methods in natural reservoirs. This study investigates the suitability of using of reservoir sandstone analogues, 3D printed with silica sand, to analyze the behavior of natural rocks. A comprehensive mechanical and hydraulic characterization of 3D printed silica sand is addressed by subjecting the sandstone analogues to various levels of confining stress and measuring their cumulative volumetric deformation and permeability evolution at each compression stage. Experimental results demonstrate that 3D printing technology can reproduce porous media that resembles the mechanical behavior of natural reservoir rocks. Nonetheless, some divergences are encountered with the properties of 3D printed analogues when compared to natural reservoir rocks. The 3D printed sandstone analogues were found to be more compressible and permeable than widely studied reservoirs such as Berea Sandstone. Efforts are made to optimize the 3D printing process of the rock analogues to overcome the differences encountered in the mechanical and hydraulic behavior.

Validating the Generalized Pore Network Model Using Micro-CT Images of Two-Phase Flow

Abstract: A reliable prediction of two-phase flow through porous media requires the development and validation of models for flow across multiple length scales. The generalized network model is a step towards efficient and accurate upscaling of flow from the pore to the core scale. This paper presents a validation of the generalized network model using micro-CT images of two-phase flow experiments on a pore-by-pore basis. Three experimental secondary imbibition datasets are studied for both sandstone and carbonate rock samples. We first present a quantification of uncertainties in the experimental measurements. Then, we show that the model can reproduce the experimental fluid occupancies and saturations with a good accuracy, which in some cases is comparable with the similarity between repeat experiments. However, high-resolution images need to be acquired to characterize the pore geometry for modelling, while the results are sensitive to the initial condition at the end of primary drainage. The results provide a methodology for improving our physical models using large experimental datasets which, at the pore scale, can be generated using micro-CT imaging of multiphase flow.

Microstructure Effects on Effective Gas Diffusion Coefficient of Nanoporous Materials

Abstract: In this work, we develop a numerical framework for gas diffusion in nanoporous materials including a random generation-growth algorithm for microstructure reconstruction and a multiple-relaxation-time lattice Boltzmann method for solution of diffusion equation with Knudsen effects carefully considered. The Knudsen diffusion is accurately captured by a local diffusion coefficient computed based on a corrected Bosanquet-type formula with the local pore size determined by the largest sphere method. A robust validation of the new framework is demonstrated by predicting the effective gas diffusion coefficient of microporous layer and catalyst layer in fuel cell, which shows good agreement with several recent experimental measurements. Then, a detailed investigation is made of the influence on effective gas Knudsen diffusivity by many important microstructure factors including morphology category, size effect, structure anisotropy, and layering structure effect. A widely applicable Bosanquet-type empirical relation at the Darcy scale is found between the normalized effective gas diffusion coefficient and the average Knudsen number. The present work will promote the understanding and modeling of gas diffusion in nanoporous materials and also provide an efficient platform for the optimization design of nanoporous systems.

Multi-scale Model of Reactive Transport in Fractured Media: Diffusion Limitations on Rates

Abstract: Reactive transport in fractured media is conceptualized as a multi-scale problem that couples a pore-scale component, which comprises Navier–Stokes flow, multi-component transport and aqueous equilibrium in the fracture, and a Darcy-scale component, which comprises multi-component diffusive transport, aqueous equilibrium and mineral reactions in the porous matrix. The model that implements this multi-scale approach builds on an existing pore-scale model and is able to capture complex fracture geometries with the embedded-boundary method. The embedded boundary acts as the interface between pore- and Darcy-scale domains. Adaptive mesh refinement is used to match resolutions at the interface while using coarser resolution away from the interface when not needed in the Darcy-scale domain. The new model is validated and then compared to results from a pore-scale model. Multi-scale model results are shown to be equivalent to pore-scale results under diffusion-controlled reactions in the pore scale and very fast dissolution in the Darcy scale. The multi-scale model provides a more accurate solution for a given resolution as it effectively sets the equilibrium concentrations as boundary conditions. The multi-scale model is capable to capture flow channelization observed in an experimental fractured core and, at the same time, limitations in the dissolution of calcite by diffusive transport through an altered porous layer. Discrepancies in effluent calcium concentrations between the multi-scale results and results from a reduced-dimension Darcy-scale model for this fractured core experiment are attributed to the solution of the flow field and the gradients that develop inside the fracture. Discrepancies in effluent magnesium concentrations exemplify the limitations of the approach because the multi-scale model requires calibration of reactive surface areas as Darcy-scale continuum models.

Simulation of Gas Production from Multilayered Hydrate-Bearing Media with Fully Coupled Flow, Thermal, Chemical and Geomechanical Processes Using TOUGH + Millstone. Part 1: Numerical Modeling of Hydrates

Abstract: TOUGH + Millstone has been developed for the analysis of coupled flow, thermal and geomechanical processes associated with the formation and/or dissociation of CH4-hydrates in geological media. It is composed of two constituent codes: (a) a significantly enhanced version of the TOUGH + HYDRATE simulator, V2.0, that accounts for all known flow, physical, thermodynamic and chemical processes associated with the behavior of hydrate-bearing systems undergoing changes and includes the most recent advances in the description of the system properties, coupled seamlessly with (b) Millstone V1.0, a new code that addresses the conceptual, computational and mathematical shortcomings of earlier codes used to describe the geomechanical response of these systems. The capabilities of TOUGH + Millstone are demonstrated in the simulation and analysis of the system flow, thermal and geomechanical behavior during gas production from a realistic complex offshore hydrate deposit. In the first paper of this series, we discuss the physics underlying the T + H hydrate simulator, the constitutive relationships describing the physical, chemical (equilibrium and kinetic) and thermal processes, the states of the $${\hbox {CH}}_4 + {\hbox {H}}_2 \hbox {O}$$ system and the sources of critically important data, as well as the mathematical approaches used for the development of the of mass and energy balance equations and their solution. Additionally, we provide verification examples of the hydrate code against numerical results from the simulation of laboratory and field experiments.

Fractal and Multifractal Characteristics of Pore Throats in the Bakken Shale

Abstract: To evaluate pore structures of the Bakken Shale, which is one of the most important factors that affect petrophysical properties, high-pressure mercury intrusion was employed in this study. Pore structures such as pore-throat size, pore-throat ratio, and fractal attributes are investigated in this major shale play. Pore-throat size from 3.6 to 200 um is widely distributed in these shale samples. Accordingly, pore-throat size distributions demonstrate the multimodal behavior within the samples. The whole pore-throat network can be divided into four clusters: one set of large pores, two transitional/intermediate pore groups, and one set of smaller pores. The fractal analysis revealed that fractal dimensions decrease as the pore-throat size decreases. The multifractal analysis demonstrated that as the maturity of the shale samples increases, pore-throat size distributions would become more uniform and pore structures tend to become more homogeneous. The results are compared to our previous results obtained from nitrogen gas adsorption for further verifications of fractal behavior. Finally, although fractal analysis of mercury intrusion and nitrogen gas adsorption were comparable, the results of multifractal analysis from these two methods were not identical.

A Brief Introduction to Convection in Porous Media

Abstract: This paper serves as a brief introduction to the longer introduction provided by the book by Nield and Bejan (NB). Attention is focussed on the modelling of the interaction of the heat transfer and the presence of a porous medium. Except for a brief mention, convection in unsaturated media is beyond the scope of this book and hence this review. Our coverage is mainly confined to single phase flow. The effects of radiation considered are confined to a contribution to volumetric heating produced by the absorption of radiation. Topics covered relate to both pore scale (microscale) and macroscale modelling and processes. Both theoretical and experiment studies are considered. Whilst NB contains a section on deformable media, the book is largely constrained to rigid porous media. This review is confined to matters concerning rigid porous media.

Pore-Scale Simulations of Single- and Two-Phase Flow in Porous Media: Approaches and Applications

Abstract: 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.

Comparison of Flow and Transport Experiments on 3D Printed Micromodels with Direct Numerical Simulations

Abstract: Understanding pore-scale flow and transport processes is important for understanding flow and transport within rocks on a larger scale. Flow experiments on small-scale micromodels can be used to experimentally investigate pore-scale flow. Current manufacturing methods of micromodels are costly and time consuming. 3D printing is an alternative method for the production of micromodels. We have been able to visualise small-scale, single-phase flow and transport processes within a 3D printed micromodel using a custom-built visualisation cell. Results have been compared with the same experiments run on a micromodel with the same geometry made from polymethyl methacrylate (PMMA, also known as Perspex). Numerical simulations of the experiments indicate that differences in experimental results between the 3D printed micromodel and the Perspex micromodel may be due to variability in print geometry and surface properties between the samples. 3D printing technology looks promising as a micromodel manufacturing method; however, further work is needed to improve the accuracy and quality of 3D printed models in terms of geometry and surface roughness.

3D Microscale Flow Simulation of Shear-Thinning Fluids in a Rough Fracture

Abstract: The shear-thinning fluid flow in rough fractures is of wide interest in subsurface engineering. Inertial effects due to flow regime, fracture aperture variations as well as fluid rheology affect the macroscopic flow parameters in an interrelated way. We present a 3D microscale flow simulation for both Newtonian and Cross power-law shear-thinning fluids through a rough fracture over a range of flow regimes, thus evaluating the critical Reynolds number above which the linear Darcy’s law is no longer applicable. The flow domain is extracted from a computed microtomography image of a fractured Berea sandstone. The fracture aperture is much more variable than any of the previous numerical or experimental work involving shear-thinning fluids, and simulations are 3D for the first time. We quantify the simulated velocity fields and propose a new correlation for shift factor (parameter relating in situ porous medium viscosity with bulk viscosity). The correlation incorporates tortuosity (parameter calculated either based only on fracture image or on detailed velocity field, if available) as well as a fluid-dependent parameter obtained from the analytical/semi-analytical solutions of the same shear-thinning fluids flow in a smooth slit. Our results show that the shift factor is dependent on both the fracture aperture distribution (not only the hydraulic/equivalent aperture) and fluid rheology properties. However, both the inertial coefficient and critical Reynolds number are functions of the fracture geometry only, which is consistent with a recent experimental study.

A Feature-Based Stochastic Permeability of Shale: Part 1—Validation and Two-Phase Permeability in a Utica Shale Sample

Abstract: Estimate of permeability plays a crucial role in flow-based studies of fractured tight-rocks. It is well known that most of the flow through tight-rocks (e.g., shales) is controlled by permeable features (e.g., fractures, laminations, etc.), and there is negligible flow through the matrix. However, current approaches in the literature to model permeability of tight-rocks do not account for such features present within the rock ranging from micro-scale to field-scale. Current permeability modeling approach assumes a single continuum without considering the presence of permeable features within the matrix (e.g., micro-fractures) or outside the matrix (e.g., natural fractures). Although the laboratory-measured permeability implicitly captures discrete features present in that sample (e.g., fractures, laminations, micro-fractures), most of the permeability models proposed for shale do not account for these features. Fracture permeability in the literature is typically modeled using an ideal slit assumption; however, this highly overestimates its permeability because fractures in real medium are non-ideal in terms of their porosity and tortuosity, which affect their permeability. Additionally, the transition zone between fracture and matrix also affects the permeability of fracture. In this study, part of a two-part series, a new method to predict permeability of fractured shale by discretizing the medium into matrix (inorganic and organic) and fractures is presented. New analytical expressions of permeability are derived to account for non-ideal nature of porous medium and two-phase flow in fractures. Rock feature in each cell of the grid is identified as one of the three elements (organic matter, inorganic matter, or fracture), and permeability of that cell is estimated using a suitable analytical expression. This method allows estimating permeability at any scale of interest and more robustly than by a pure analytical approach. The proposed method is validated against local and global-scale measurements on three fractured samples from laboratory. Finally, the method is used to predict two-phase flow permeability of supercritical CO2 displacing water within a fracture in a Utica shale sample. The proposed two-phase flow permeability equations can be used as a quick analytical tool to predict relative permeability estimates of two-phase flow in fractured shale samples. In Part 2, the proposed method is used to estimate field-scale permeability through an optimization process that uses field-scale production and other readily available information.

Capillarity-Driven Oil Flow in Nanopores: Darcy Scale Analysis of Lucas–Washburn Imbibition Dynamics

Abstract: We present gravimetrical and optical imaging experiments on the capillarity-driven imbibition of silicone oils in monolithic silica glasses traversed by 3D networks of pores (mesoporous Vycor glass with 6.5 nm or 10 nm pore diameters). As evidenced by a robust square root of time Lucas–Washburn (L–W) filling kinetics, the capillary rise is governed by a balance of capillarity and viscous drag forces in the absence of inertia and gravitational effects over the entire experimental times studied, ranging from a few seconds up to 10 days. A video on the infiltration process corroborates a collective pore filling as well as pronounced imbibition front broadening resulting from the capillarity and permeability disorder, typical of Vycor glasses. The transport process is analyzed within a Darcy scale description, considering a generalized prefactor of the L–W law, termed Lucas–Washburn–Darcy imbibition ability. It assumes a Hagen–Poiseuille velocity profile in the pores and depends on the porosity, the mean pore diameter, the tortuosity and the velocity slip length and thus on the effective hydraulic pore diameter. For both matrices a reduced imbibition speed and thus reduced imbibition ability, compared to the one assuming the nominal pore diameter, bulk fluidity and bulk capillarity, can be quantitatively traced to an immobile, pore wall adsorbed boundary layer of 1.4 nm thickness. Presumably, it consists of a monolayer of water molecules adsorbed on the hydrophilic pore walls covered by a monolayer of flat-laying silicone oil molecules. Our study highlights the importance of immobile nanoscopic boundary layers on the flow in tight oil reservoirs as well as the validity of the Darcy scale description for transport in mesoporous media.

Effects of Grain Size on Deformation in Porous Media

Abstract: The solid system in deformable porous media undergoes deformation with the flow of fluid. In this paper, in order to study the micro-mechanism of the deformation, the solid system in the porous media is represented by a pack of spherical particles and simulated by discrete element method. The fluid system in the porous media is also simulated by computational fluid dynamics. To consider the fluid–particle interactions in the porous media, the above techniques are coupled and applied for simulating the solid deformation and fluid flow. Different models consisting of different particle sizes are studied in dry (without the presence of fluid) and wet states (with the flow of fluid). The results show that with the decrease in the particle size, the solid deformation declines, which imitates the actual deformation in the porous media. More importantly, the comparison between the dry and wet models indicates that the effect of the fluid on the particle system is diminishing with the smaller packed particles. The solid deformation tendency is quantified by the reduction in the values of some micro-mechanical properties, such as permeability (absolute and relative), porosity and pore-size distribution.