Transport in Porous Media

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Accounting for poro-mechanical effects in full-field reservoir simulation studies and uncertainty quantification workflows using complex reservoir models is challenging, mainly because of the high computational cost. We hence introduce an alternative approach that couples hydrodynamics through existing flow diagnostics simulations with poro-mechanics to screen the impact of coupled poro-mechanical processes on reservoir performance without significantly increasing computational overheads. In flow diagnostics, time-of-flight distributions and influence regions can be used to characterise the flow field in the reservoir, which depends on the distribution of petrophysical properties that are altered due to production-induced changes in pore pressure and effective stress. These extended flow diagnostics calculations hence enable us to quickly screen how the dynamics in the reservoirs (e.g. reservoir connectivity, displacement efficiency, and well allocation factors) are affected by the complex interactions between poro-mechanics and hydrodynamics. Our poro-mechanically informed flow diagnostics account for steady-state and single-phase flow conditions based on the poro-elastic theory and assume that the reservoir does not contain fractures. Fluid flow and rock deformation calculations are coupled sequentially. The equations are discretised using a finite-volume method with two-point flux-approximation and the virtual element method, respectively. The solution of the coupled system considers stress-dependent permeabilities. Due to the steady-state nature of the calculations and the effective proposed coupling strategy, these calculations remain computationally efficient while providing first-order approximations of the interplay between poro-mechanics and hydrodynamics, as we demonstrate through a series of case studies. The extended flow diagnostic approach hence provides an efficient complement to traditional reservoir simulation and uncertainty quantification workflows and enable us to assess a broader range of reservoir uncertainties.
Ostwald ripening of gas bubbles is a thermodynamic process for mass transfer, which is important for both foam enhanced oil recovery and geological CO2 storage. We present a methodology for simulating Ostwald ripening of gas ganglia surrounded by liquid in arbitrary pore geometries. The method couples a conservative level set model for capillary-controlled displacement and a ghost-bubble technique that calculates mass transfer based on difference in chemical potentials. The methodology is implemented in a software framework for parallel computations. As a validation of the model, we show that simulations of bubble ripening in a pore throat connecting two pore bodies are consistent with previously reported trends in similar geometries. Then we investigate the impact of gas type, compressibility factor, and local capillary pressure on gas-bubble ripening in various water-wet pore geometries. The results confirm that gas solubility and compressibility factor are proportional to the rate of mass transfer. Our simulations suggest that Ostwald ripening has largest impact in heterogeneous or fractured porous structures where differences in gas-bubble potentials are high. However, if the liquid separating the gas bubbles is also a disconnected phase, which can happen in intermediate-wet porous media, the resulting local capillary pressure can limit the coarsening and stabilise smaller bubbles. Finally, we simulated Ostwald ripening on a 3-D pore-space image of sandstone containing a residual gas/water configuration after imbibition. Characterization of gas-bubble morphology during the coarsening shows that large ganglia get more ramified at the expense of small spherical ganglia that cease to exist.
Reactive transport numerical modelling is used to explore chemical interactions between fluids and solids associated with a broad range of subsurface processes. It is a powerful tool extensively used in geoscientific applications. However, simulations often require large computational times. This is due to the fact that they typically involve slow and iterative calculations of geochemical reactions, even when these are based on a very similar set of input values. In this work, we present a generic pipeline to implement, train and validate Machine Learning algorithms to be used as surrogate models of the geochemical reactions. The resulting surrogate models are subsequently used in a novel reactive transport modelling framework. As a proof of concept, a reactive transport model with Machine Learning-based geochemistry is used to simulate the three-dimensional hydrothermal dolomitization of a fractured carbonate reservoir. The model encompasses fluid flow, heat transfer, solute transport and chemical reactions. The accuracy and efficiency of the proposed approach are evaluated by comparison with a conventional reactive transport modelling tool. It is shown that the reactive transport model based on artificial neural networks provides a substantial reduction of the computational burden, with a speedup of one order of magnitude, while providing considerable accuracy. It is concluded that the proposed framework is particularly appealing for large-scale simulations of thermo-hydro-geochemical systems. The limitations of the proposed methodology are discussed, and potential mitigation strategies are proposed for future work.
Motivated by some discrepancies in the comparison between the rheometric viscosity function of shear-thinning fluids and the apparent Darcy viscosity function obtained in porous media simulations/experiments, we propose a model for the flow in porous media with a non-uniform pore distribution in the pore scale, i.e., inside the Representative Elementary Volume (REV). Since different pore sizes inside the REV would lead to different characteristic shear rates, the heterogeneity of the pores in the elementary scale can be responsible for the mentioned discrepancies. Indeed, simulations in a plug of a porous medium have shown that the apparent viscosity function decreases as we increase the standard deviation distribution of the diameter of idealized cylindrical pores. The ratio of the mobility of the heterogeneous pore distribution to the mobility of the homogeneous one is plotted as a function of the power-law index of the non-Newtonian viscosity function. The dimensionless analysis has shown to be a powerful tool in the investigation, collapsing the information in several cases. In addition, our results reveal that a non-uniform pore size distribution can be an important source of the discrepancy between the viscosity function obtained in a rheometric device and the one obtained from the Darcy equation, as reported in the literature.
Modeling of fluid flow in porous media is a pillar in geoscience applications. Previous studies have revealed that heterogeneity and fracture distribution have considerable influence on fluid flow. In this work, a numerical investigation of two-phase flow in heterogeneous fractured reservoir is presented. First, the discrete fracture model is implemented based on a hybrid-dimensional modeling approach, and an equivalent continuum approach is integrated in the model to reduce computational cost. A multilevel adaptive strategy is devised to improve the numerical robustness and efficiency. It allows up to 4-levels adaption, where the adaptive factors can be modified flexibly. Then, numerical tests are conducted to verify the the proposed method and to evaluate its performance. Different adaptive strategies with 3-levels, 4-levels and fixed time schemes are analyzed to evaluate the computational cost and convergence history. These evaluations demonstrate the merits of this method compared to the classical method. Later, the heterogeneity in permeability field, as well as initial saturation, is modeled in a layer model, where the effect of layer angle and permeability on fluid flow is investigated. A porous medium containing multiple length fractures with different distributions is simulated. The fine-scale fractures are upscaled based on the equivalent approach, while the large-scale fractures are retained. The conductivity of the rock matrix is enhanced by the upscaled fine-scale fractures. The difference of hydraulic property between homogeneous and heterogeneous situations is analyzed. It reveals that the heterogeneity may influence fluid flow and production, while these impacts are also related to fracture distribution and permeability.
of potential applications of quantum computational algorithms to problems in geoscience
Characterization and modeling of geomedia, computing their effective flow, transport, elastic, and other properties, and simulating various phenomena that occur there constitute some of the most intensive calculations in science and engineering. Over the past twenty five years, however, development of quantum computers has made great progress, and powerful quantum algorithms have been developed for simulating many important and computationally difficult problems, with the potential for enormous speed-up over the most efficient classical algorithms. This perspective describes such algorithms and discusses their potential applications to problems in geoscience, ranging from reconstruction and modeling of geomedia, to simulating fluid flow by the Stokes and Navier-Stokes equations, or lattice Boltzmann and lattice gas methods, numerical solution of the advection-diffusion equation, pattern recognition in and analysis of big data, machine learning methods, and image processing. Although several hurdles remain that must be overcome before practical computations with quantum computers, and in particular those associated with geoscience, become possible, such as designing fault-tolerant quantum computers, which is still far into the future, noisy intermediate-scale computers are already available whose capabilities have been demonstrated for various problems.
In this paper, we design a Co-flow microchannel capillary device to numerically simulate the generation of double-emulsion droplets by a level set method. The three velocity vortices generated during the generation of double-emulsion droplets and the pressure variation on the z-axis were systematically investigated. The effects of the dimensionless numbers Cain, Caout, and Wein on the volumes of the inner droplets and outer droplets are discussed. The results show that three velocity vortices can be generated during the generation of double-emulsion droplets due to the inhomogeneous force on the droplets. The internal pressure of the inner droplet is much larger than the internal pressure of the outer droplet, and the larger the volume of the droplet the smaller the internal pressure of the droplet. With the increase in the Cain, the volume of the outer droplet increases and the volume of the inner droplet increases and then decreases. As the Caout increases, the volume of the outer and inner droplets decreases significantly. With the increase in the Wein, the volume of the inner droplet increases significantly, and the volume of the outer droplet increases and then decreases. In view of the above phenomenon that the volume of droplet increases and then decreases, we can conclude that the two factors, velocity and pressure, affect the volume of droplet at the same time.
Viscous fingering in porous media occurs when the ( miscible or immiscible ) displacing fluid has a lower viscosity than the displaced fluid. For example, immiscible fingering is observed in experiments where water displaces a much more viscous oil. Modelling the observed fingering patterns in immiscible viscous fingering has proven to be very challenging, which has often been identified as being due to numerical issues. However, in a recent paper (Sorbie et al. in Transp. Porous Media 133:331–359, 2020) suggested that the modelling issues are more closely related to the physics and formulation of the problem. They proposed an approach based on the fractional flow curve, $${f}_{w}^{*}$$ f w ∗ , as the principal input, and then derived relative permeabilities which give the maximum total mobility. Sorbie et al . were then able to produce complex, well-resolves immiscible finger patterns using elementary numerical methods. In this paper, this new approach to modelling immiscible viscous fingering is tested by performing direct numerical simulations of previously published experimental water/oil displacements in 2D sandstone porous media. Experiments were modelled at adverse viscosity ratios ( $${\mu }_{o}/{\mu }_{w}$$ μ o / μ w ), with oil viscosities ranging from μ o = 412 to 7000 cP, i.e. for a viscosity ratio range, ( $${\mu }_{o}/{\mu }_{w}$$ μ o / μ w ) $$\sim$$ ∼ 400–7000. These experiments have extensive production data as well as in situ 2D immiscible fingering images, measured by X-ray scanning. In all cases, very good quantitative agreement between experiment and modelling results is found, providing a strong validation of the new modelling approach. The underlying parameters used in the modelling of these unstable immiscible floods, the $${f}_{w}^{*}$$ f w ∗ functions, show very consistent and understandable variation with the viscosity ratio, ( $${\mu }_{o}/{\mu }_{w}$$ μ o / μ w ).
This paper deals with how to solve chemical dissolution-front instability problems, which are nonlinearly coupled by subsurface pore-fluid flow, reactive mass transport and porosity evolution processes in fluid-saturated porous media, through using two different computational schemes. In the first computational scheme, porosity, pressure of the pore-fluid and concentration of the solute are used as fundamental unknown variables to describe a chemical dissolution system, so that it can be named as the PPC scheme. In the second computational scheme, porosity, velocity of the pore-fluid and concentration of the solute are used as fundamental unknown variables to describe a chemical dissolution system, so that it can be named as the PVC scheme. Since the finite element equations of a chemical dissolution-front instability problem on the basis of using the PPC scheme is available, the main focus of this study is on deriving the finite element equations of a chemical dissolution-front instability problem on the basis of using the PVC scheme. In particular, analytical solutions for the property matrices of a four-node rectangular element have been derived and used in both the PVC scheme and the PPC scheme. Through comparing the computational simulation results obtained from using both the PPC scheme and the PVC scheme, it has demonstrated that: (1) if the chemical dissolution system is in a stable state, then the PPC scheme is superior to the PVC scheme because the PPC scheme uses less computational efforts than the PVC scheme; (2) if the chemical dissolution system is in an unstable state, then the PVC scheme is superior to the PPC scheme because the PVC scheme yields more accurate computational simulation results than the PPC scheme.
A continuous-time random walk (CTRW) reactive transport model is used to study the impact of physical heterogeneity on the effective reaction rates in porous media in a sample of length 15 cm over timescales up to 108 s (3 years). The model has previously been validated using nuclear magnetic resonance (NMR) measurements during dissolution of a limestone. The model assumes first-order reaction. We construct three domains with increasing physical heterogeneity and study dissolution at four Péclet numbers, Pe = 0.0542, 0.542, 5.42 and 54.2. We characterize signatures of physical heterogeneity in the three porous media using velocity distributions and show how these imprint on the signatures of particle displacement, namely particle propagator distributions. In addition, we demonstrate the ability of our CTRW model to capture the impact of physical heterogeneity on the longitudinal dispersion coefficient over several orders of magnitude in space and time. Reactive transport simulations show that the effective reaction rates depend on (i) initial physical heterogeneity and (ii) transport conditions. For all heterogeneities and Pe, the late-time reaction rate exhibits a time dependence t-a with a≠0.5 that indicates the persistence of incomplete mixing. We show that the higher the initial heterogeneity, the lower the late-time reaction rate. A decrease in Pe promotes mixing by diffusion over advection, resulting in higher reaction rates. The post-dissolution propagators indicate an increase in the degree of non-Fickian transport. Overall, we establish a framework to demonstrate and quantify the impact of physical heterogeneity on transport and effective reaction rates in porous media.
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Influence of percolativity on effective medium predictions for the normalized electrical resistivity (formation factor) F of anisotropic shaly sands as a function of porosity ϕ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\phi }}$$\end{document}. The experimental data are from Table 1 (squares) and Table 3 (circles) in Waxman and Smits (1968) (see pages 110-112 in this reference for a detailed description of the samples). The two curves bounding the data points are solutions of a generalized effective medium approximation based on anisotropic percolativity with d=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=1$$\end{document} and e=0.45\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e=0.45$$\end{document}. The short-dashed upper bound is Fx=Fy\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_x=F_y$$\end{document} and the short-dash-dotted lower bound is Fz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_z$$\end{document}. The three curves on the right are the Bruggeman effective medium theory and its anisotropic generalization to aligned oblate ellipsoids. The isotropic Bruggeman (1935) theory with Fx=Fy=Fz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_x=F_y=F_z$$\end{document} is shown as the solid line. The generalized anisotropic effective medium approximation from Schwartz [1994, eq.(8)] for aligned oblate ellipsoids with d=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=1$$\end{document} and e=0.45\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e=0.45$$\end{document} is shown as the long-dashed line for Fx=Fy\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_x=F_y$$\end{document} and the long-dash-dotted line for Fz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_z$$\end{document}
Connectivity and connectedness are nonadditive geometric functionals on the set of pore scale structures. They determine transport of mass, volume or momentum in porous media, because without connectivity there cannot be transport. Percolativity of porous media is introduced here as a geometric descriptor of connectivity, that can be computed from the pore scale and persists to the macroscale through a suitable upscaling limit. It is a measure that combines local percolation probabilities with a probability density of ratios of eigenvalues of the tensor of local percolating directions. Percolativity enters directly into generalized effective medium approximations. Predictions from these generalized effective medium approximations are found to be compatible with apparently anisotropic Archie correlations observed in experiment.
Fluid blobs in an immiscible Newtonian fluid flowing in a capillary tube with varying radius show highly nonlinear behavior. We consider here a generalization of previously obtained results to blobs of non-Newtonian fluids. We compute here the yield pressure drop and the mean flow rate in two cases: (i) When a single blob is injected, (ii) When many blobs are randomly injected into the tube. We find that the capillary effects emerge from the non-uniformity of the tube radius and contribute to the threshold pressure for flow to occur. Furthermore, in the presence of many blobs the threshold value depends on the number of blobs and their relative distances which are randomly distributed. For a capillary fiber bundle of identical parallel tubes, we calculate the probability distribution of the threshold pressure and the mean flow rate. We consider two geometries: tubes of sinusoidal shape, for which we derive explicit expressions, and triangular-shaped tubes, for which we find that essential singularities are developed. We perform numerical simulations confirming our analytical results.
Because of their extreme heterogeneity at multiple scales, carbonate rocks present a great challenge for studying and managing oil reservoirs. Depositional processes and diagenetic alterations of carbonates may have produced very complex pore structures and, consequently, variable fluid storage and flow properties of hydrocarbon reservoirs. To understand the impact of mineralogy on the pore system, we analyzed four carbonate rock samples (coquinas) from the Morro do Chaves Formation in Brazil. For this study, we used thin sections and XRD for their mineralogical characterization, together with routine core analysis, NMR, MICP and microCT for the petrophysical characterizations. The samples revealed very similar porosity values but considerably different permeabilities. Samples with a relatively high quartz content (terrigenous material) generally had lower permeabilities, mostly caused by more mineral fragmentation. Samples with little or no quartz in turn exhibited high permeabilities due to less fragmentation and more diagenetic actions (e.g., dissolution of shells). Results confirm that carbonate minerals are very susceptible to diagenesis, leading to modifications in their pore body and pore throat sizes, and creating pores classified as moldic and vug pores, or even clogging them. For one of the samples, we acquired detailed pore skeleton information based on microCT images to obtain a more complete understanding of its structural characteristics.
The evolution of pore structure caused by particle retention is a function of heterogeneity and non-linear coupling of particle transport and fluid flow. Pore-scale modelling enable us to elucidate the role of various mechanisms controlling particle transport and deposition. This study incorporated the Eulerian-Lagrangian approach to investigate the spatial and temporal deposition of particles using a benchmark data set for an artificial column made of glass beads for validation. The velocity field and trajectory of particles were determined by solving the Navier-Stokes and momentum balance equations. When the mean diameter of particles is smaller than the image voxel size, several particles are required to occupy a pore voxel. Particles with low velocity that cannot escape from the adhesion forces of surfaces are considered as deposited. The solid volume fraction of pore voxels adjacent to solid voxels changes dynamically through particle deposition. The role of surface deposition and clogging mechanisms during various experimental simulation scenarios was analyzed using an image-based technique. Mean injection velocity, particle size, surface adhesion forces, and surface roughness are considered as sensitivity parameters. The results show that the clogging mechanism was responsible for the structure permeability impairment rather than the surface deposition, when particle size and surface adhesion forces increased. However, the clogging mechanism did not affect permeability when surface roughness increased. Particle retention shows a maximum value around a critical velocity where the spatial particle retention switched from filter cake development to homogenous retention.
In a deep geological repository for the long-term containment of radioactive waste, the engineered barriers and host clay rock inhibit the migration of gases, due to their low permeability and high gas entry pressure. Some experiments in the literature have focused on the measurement of gas entry pressure \((P_{\text {g,e}}\), but there is a lack of 2-phase flow (water–gas) modeling studies that include entry pressure effects in such porous media. In the present work, the modified Van Genuchten–Mualem model (Vogel et al. 2000) is extended to two-phase flow, incorporating the capillary entry pressure parameter \((P_{\text {c,e}})\), and a new data analysis approach is developed in order to characterize the water–gas constitutive relations (saturation curve, water permeability curve, gas permeability curve). This constitutive model is then implemented in the iTOUGH2 code (Wainwright and Finsterle 2016 in Global sensitivity and data-worth analyses in iTOUGH2: User’s guide) with a change of primary variables to be described below (capillary pressure is set as primary state variable instead of gas saturation). After regression tests for verifying the change of primary variables in iTOUGH2, two problems were modeled: first, numerical flow experiments were performed in a clay soil (code-to-code benchmark tests, and comparisons focused on entry pressure effects); secondly, water–gas migration was modeled based on an in situ gas injection experiment (PGZ1) performed in the French URL (Underground Research Laboratory) of Bure. Sensitivity analyses show that gas entry pressure is an important controlling factor which should not be neglected in simulations of gas migration in clayey materials.
The dynamic characteristic of bone is its ability to remodel itself through mechanobiological responses. Bone regeneration is triggered by mechanical cues from physiological activities that generate structural strain and cause bone marrow movement. This phenomenon is crucial for bone scaffold when implanted in the cancellous bone as host tissue. Often, the fluid movement of bone scaffold and cancellous bone is studied separately, which does not represent the actual environment once implanted. In the present study, the fluid flow analysis properties of bone scaffold integrated into the cancellous bone at different skeletal sites are investigated. Three types of porous bone scaffolds categorized based on pore size configurations: 1 mm, 0.8 mm and hybrid (0.8 mm interlaced with 0.5 mm) were used. Three different skeletal sites of femoral bone were selected: neck, lateral condyle and medial condyle. Computational fluid dynamics was utilized to analyze the fluid flow properties of bone scaffold integrated cancellous bone. The results of this study reveal that the localization and maximum value of shear stress in an independent bone scaffold are significantly different compared to the bone scaffold integrated with cancellous bone by about 160% to 448% percentage difference. Low shear stress and high permeability were found across models that have higher Tb.Sp (trabecular separation). Specimen C and femoral lateral condyle showed the highest permeability in their respective category.
X-ray micro-computed tomography (micro-CT) has been widely leveraged to characterise the pore-scale geometry of subsurface porous rocks. Recent developments in super-resolution (SR) methods using deep learning allow for the digital enhancement of low-resolution (LR) images over large spatial scales, creating SR images comparable to high-resolution (HR) ground truth images. This circumvents the common trade-off between resolution and field-of-view. An outstanding issue is the use of paired LR and HR data, which is often required in the training step of such methods but is difficult to obtain. In this work, we rigorously compare two state-of-the-art SR deep learning techniques, using both paired and unpaired data, with like-for-like ground truth data. The first approach requires paired images to train a convolutional neural network (CNN), while the second approach uses unpaired images to train a generative adversarial network (GAN). The two approaches are compared using a micro-CT carbonate rock sample with complicated micro-porous textures. We implemented various image-based and numerical verifications and experimental validation to quantitatively evaluate the physical accuracy and sensitivities of the two methods. Our quantitative results show that the unpaired GAN approach can reconstruct super-resolution images as precise as the paired CNN method, with comparable training times and dataset requirements. This unlocks new applications for micro-CT image enhancement using unpaired deep learning methods; image registration is no longer needed during the data processing stage. Decoupled images from data storage platforms can be exploited to train networks for SR digital rock applications. This opens up a new pathway for various applications related to multi-scale flow simulations in heterogeneous porous media.
Shale gas reservoir is a complex multi-scale system containing micro-nanopores and micro-fractures. Understanding shale gas transport mechanism in fractured porous media is important to predict shale gas production performance accurately. This paper established a shale gas production prediction model, considering gas rarefaction effects, adsorption, diffusion, and stress sensitivity. The variation in production and drainage patterns with production time by multi-stage and multi-cluster fracturing considering fracture hit was studied by using this model. In addition, the influences of connecting hydraulic fractures, natural fracture conductivity, and stress sensitivity on shale gas production are discussed. When the spacing between the connecting fractures exceeds 4 stages (176 m), the production of the child well and parent well tends to be stable as the spacing between connecting hydraulic fractures increases. The child well production decreases, and the parent well production increases by considering fracture hits. The cumulative production of both parent and child wells increases with the increase in natural fracture conductivity. The results show that the production of parent and child wells considering stress sensitivity is, respectively, 14.31% and 18.73% lower than that without considering stress sensitivity. The key findings of this study can be expected to provide theoretical supports for the shale gas transport mechanisms in fractured porous media.
Schematic for a two-dimensional linear stability problem. Phase I (the invading phase) displaces phase II (defending phase) from the left, with an interface between the two phases
Solution space of Eq. 18 for incompressible flow, absent of gravity and capillary forces
Dimensionless decay rate as a function of dimensionless wavenumber for compressible flow of three different compressibility ratios. a Solution for two mobility ratios for compressibility ratio of 0.1, b solution for two mobility ratios for compressibility ratio of 1, c solution for two mobility ratios for compressibility ratio of 10
We use linear stability theory to investigate the effect of fluid compressibility on interface stability during a dissipative displacement (Darcy flow). We find that compressibility changes the perturbation growth rate as a function of perturbation wavenumber. Our results indicate that both favorable (less than unity) and unfavorable (greater than unity) mobility ratios will always lead to positive maximum growth rate, which traditionally is recognized as the criterion for instability. We conclude, however, that in the case of favorable mobility ratio, the maximum perturbation growth rate is always smaller than the unperturbed growth rate naturally existing in compressible displacements. The interface will still be stable because the perturbation will never exceed the background flow. Therefore, compressibility does not change the stability of displacements, which is ultimately determined by mobility ratio.
The present work is dedicated to determining the effective permeability of doubly porous materials made of a solid phase comprising a network of interconnected pores at the nanoscale and a network of non-interconnected pores at the microscopic scale. The fluid flow at microscopic scale through the solid phase containing nanopores is described by the Darcy law, while fluid flow in the nano- and microscopic pores is governed by the Stokes equations. A two-scale homogenization approach is proposed to estimate the effective permeability of doubly porous materials in question. In the nanoscopic-to-microscopic upscaling, a micromechanical model based on the generalized self-consistent scheme (GSCS) is elaborated to estimate the microscopic permeability. In the microscopic-to-macroscopic upscaling, the equivalent inclusion method combined with the dilute, Mori–Tanaka, differential schemes is used to obtain different estimates of the macroscopic permeability. In the two-scale homogenization approach elaborated, the pore size and shape effects as well as the solid/fluid interface influence are taken into account. The results given by the proposed two-scale homogenization approach are discussed and compared with the relevant numerical results provided by the finite element method.
Darcy’s law for porous media transport is given a new local thermodynamic basis in terms of the grand potential of confined fluids. The local effective pressure gradient is determined using non-equilibrium molecular dynamics, and the hydraulic conductivity and permeability are investigated. The transport coefficients are determined for single-phase flow in face-centered cubic lattices of solid spheres. The porosity changed from that in the closest packing of spheres to near unity in a pure fluid, while the fluid mass density varied from that of a dilute gas to a dense liquid. The permeability varied between 5.7×10-20m2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$5.7 \times {10^{-20}} \hbox {m}^2$$\end{document} and 5.5×10-17m2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$5.5 \times {10^{-17}} \hbox {m}^2$$\end{document}, showing a porosity-dependent Klinkenberg effect. Both transport coefficients depended on the average fluid mass density and porosity but in different ways. These results set the stage for a non-equilibrium thermodynamic investigation of coupled transport of multi-phase fluids in complex media.
Scheme of particle transport processes through continuum heterogeneous domain. The left image is a computed velocity field. The right-top image displays the advection path of two particles. The right-bottom image shows the diffusion possibilities of a single particle
Scheme of stratified models using three velocity classes: a Random Walk Particle b Isochronal Water method using Water Mixing Approach formulation and c Water Parcels method using Multi-Advective Water Mixing Approach formulation
a Scalar dissipation rate and b apparent dispersion for continuum injection. Dashed black line and yellow line display the analytical solution of the apparent dispersion (Haber and Mauris 1988) and the asymptotic dispersion (Aris 1956), respectively
Comparison of Particle Random Walk model and Markov model in distance for the return probability. The Markov model is defined for spatial increment of x = 0.02
Modeling solute transport in heterogeneous porous media faces two challenges: scale dependence of dispersion and reproducing mixing separately from spreading. Both are crucial since real applications may require km scales whereas reactions, often controlled by mixing, may occur at the pore scale. Methods have been developed in response to these challenges, but none has satisfactorily characterized both processes. In this paper, we propose a formulation based on the Water Mixing Approach extended to account for velocity variability. Velocity is taken as an independent variable, so that concentration depends on time, space and velocity. Therefore, we term the formulation the Multi-Advective Water Mixing Approach. A new mixing term between velocity classes emerges in this formulation. We test it on Poiseuille’s stratified flow using the Water Parcel method. Results show high accuracy of the formulation in both dispersion and mixing. Moreover, the mixing process exhibits Markovianity in space even though it is modeled in time.
In order to assess the capability of a filter skeleton to retain flowing particles transported from eroded soil, different models based on continuous or discrete medium are available. The porous medium is often described by the grain size distribution, whereas constriction size distribution (CSD) is the key parameter governing the filtration process. This study describes the filter constrictions analysis and its application to the reduction after filtration. This investigation involves combined hole erosion-filtration experiments describing internal erosion of a base soil and particles filtration by a granular medium. The combination of experimental data of measured porosity and analytical results of the CSD was used to evaluate constrictions size reduction subsequently to filtration mechanism. The filtration depth was estimated according to retained particle mass and porosity reduction, which is evaluated from the evolution of hydraulic conductivity. The analysis of obtained results showed the occurrence of a non-uniform constriction reduction, suggesting an effective filtration depth.
Complex hydraulic conductance of a pore between nodes j and m, with g being the real conductivity and iCω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$iC\omega $$\end{document} representing the capacitance effect
Experiments, numerical simulations, and analytical models for simple models of porous media, such as a single pore and spatially-periodic models, have provided evidence that the dynamic, frequency-dependent permeability of porous media, when rescaled by its static value, may follow a universal function of the suitably-rescaled frequency, independent of the morphology of the pore space. No approach has, however, been developed to prove or refute the universality for a general model of a heterogeneous porous medium. We propose two approaches to analyze the problem. One is based on a dynamic effective-medium approximation (EMA) for d-dimensional networks of interconnected pores as the model of porous media, characterized by a pore-size or pore-conductance distribution. The EMA is accurate when the heterogeneity of the pore space is not very strong. The second approach is based on the critical-path analyzis that provides accurate estimates of the permeability when the pore space is highly heterogeneous. We show that both approaches predict that the rescaled frequency-dependent permeability is a universal function of the rescaled frequency. Thus, the two approaches together strongly support the universality of the rescaled dynamic permeability in any porous medium. The implications for the frequency-dependent electrical conductivity, the formation factor, and the diffusion and dispersion coefficients of porous media are also discussed.
In this paper, a numerical model is developed for the assessment of carbon dioxide transport through naturally fractured cap-rocks during CO2 sequestration in underground aquifers. The cap-rock contains two types of fracture with different length scales: micro-cracks (fissures) and macro-cracks (faults). The effect of micro-cracks is incorporated implicitly by modifying the intrinsic permeability tensor of porous matrix, while the macro-cracks are modeled explicitly using the extended finite element method (X-FEM). The fractured porous medium is decomposed into the porous matrix and fracture domain, which are occupied with two immiscible fluid phases, water and CO2. The flow inside the matrix domain is governed by the Darcy law, while the flow within the fracture is modeled using the Poiseuille flow. The mass conservation law is fulfilled for each fluid phase at both porous matrix and fracture domain; moreover, the mass exchange between the matrix and fracture is guaranteed through the integral formulation of mass conservation law. Applying the X-FEM technique, the explicit representation of macro-cracks is modeled by enriching the standard finite element approximation space with an enrichment function. Finally, several numerical examples of CO2 injection into a brine aquifer below a naturally fractured cap-rock are modeled in order to investigate the effects of cracks’ aperture and orientation as well as the temperature of aquifer and the depth of injection on the leakage pattern of the carbon dioxide through the cap-rock.
Supercritical CO2 (SC-CO2), with many particular properties, has been considered a fracturing fluid for exploiting unconventional hydrocarbon recently, which can also contribute to achieving partly geological sequestration of carbon. However, understanding the changes in mechanical attributes and microstructure of formation rock during CO2 injection is essential for the processes of fracture propagation, carbon storage, and hydrocarbon flow in porous media. In this study, a series of soaking experiments between tight sandstone and dry/water/brine-SC-CO2 are conducted. Several quantification techniques, such as XRD, SEM, three-dimensional (3D) laser scanning confocal microscope, etc., are adopted to investigate the variation of mineral components, mechanical properties, and rock microstructure. The XRD results indicate that after the treatment of SC-CO2, the relative contents of carbonate and feldspar minerals in the tight sandstone decrease, while the quartz and clay minerals increase. Through the microstructure analysis, three main mechanisms in the interaction between rocks and brine/SC-CO2 are obtained: strong dissolution of rock structure, new mineral precipitation generation, and CO2 adsorption causing expansion. Moreover, the quantitative topographic results show that the maximum height, root mean square, and fractal dimension of rock surface are decreased to different degrees after SC-CO2 interactions. The SC-CO2 has a deterioration effect on the mechanical properties of rock, and it will further intensify the rock damage with the introduction of water and brine. The porosity of rocks increased slightly by 2.15% under the interaction of dry/SC-CO2, while it decreased, respectively, by 5.4% and 11.64% under the interaction of water/SC-CO2 and brine/SC-CO2. The permeability increases gradually after the interaction of dry/water/brine-SC-CO2. The reduction of mechanical strength induced by SC-CO2 would be conductive to lowering the initiation pressure of rock during the fracturing operation, but it will affect the geological stability of reservoir rock.
We investigate transport of an inert solute in multidimensional porous media characterized by spatially variable hydraulic conductivity. Through the use of a GPU-accelerated solute transport simulator based on the Random Walk Particle Tracking technique, we show how different factors such as the degree of heterogeneity, flow dimensionality and source zone configurations impact mixing. Solute mixing is quantified in terms of the temporal evolution of the plume’s statistics (mean, variance and probability density function) and the dilution index. Our analysis show that mixing is strongly affected by the above mentioned factors. We also compare the probability distributions obtained from the numerical simulations with the beta distribution. Despite the discrepancies at very low concentrations, our results show that the fitting with the beta distribution is improved for transport in three-dimensional settings originating from a vertical planar source. To improve the fit at low concentrations, we utilize two one-to-one variable transformation, namely the logarithm and power law transformations. Results demonstrate that the Pareto-type IV and the uniform distributions are capable to capture the lower tail of the cumulative distribution function. Numerical results are validated against existing analytical solution for both homogeneous and heterogeneous media.
Production simulation from fractured shale reservoirs is often performed by simplifying the hydraulic fractures as rectangular planes with homogeneous aperture. This study investigates the effects of heterogeneous fracture aperture and proppant distribution in realistic, non-rectangular fractures on the multi-phase production from shales. The heterogeneous hydraulic fractures are generated with the GEOS multiphysics simulator under realistic 3D stress field. These fractures are embedded into the TOUGH+ multi-phase flow simulator for production simulation. The results emphasize the importance of flow barriers within the hydraulic fractures, due both to low-aperture regions caused by the stress-shadow effect and the settling of proppant. The production rate is particularly sensitive to aperture heterogeneity if the flow barriers are close to the wellbore such that a great portion of fracture volume is isolated from the well. A stage-to-stage comparison reveals that production from different stages could vary significantly because the local stress field leads to different fracture area and aperture. The use of proppant prevents fracture closure, but if the propped regions are far from the well, they do not enhance production because flow barriers between these regions and the well act as bottlenecks. The present study highlights the importance of incorporating aperture heterogeneity into production simulation, provides insights on the relationship between flow barriers, proppant concentration, and well production, and proposes a practical method to mitigate numerical difficulties when modeling heterogeneous fractures.
This work presents a new model to describe the initiation of groundwater flash vaporization during mining within an active geothermal system. To account for the phase transition from single-phase liquid water to two-phase water–steam mixture involved in flash vaporization, this model applies pressure and enthalpy to define the thermodynamic state of water. Pressure and enthalpy based equations of state formulations relating to the properties of water in different states are developed. A robust numerical solver is also proposed for simulating hydrodynamic response of groundwater to flash vaporization. Results illustrate steam expansion effect in flash vaporization initiation process, which has not been identified before. By comparing the results with the multiphase flow model in which phase transition is not considered, the authors demonstrate the necessity of incorporating the phase transition in flash vaporization simulation for future geothermal hazard controls during mining within an active geothermal system.
X-ray computed tomography (CT) is increasingly used to characterize the morphology of water distribution in gas diffusion layers (GDLs) for polymer electrolyte fuel cell (PEFC) applications. The resulting images can provide access to critical performance data for GDLs, including internal water contact angle distributions, water saturation, water cluster size, and pore-size distributions. Given the propensity for unimodal grayscale pixel distributions in X-ray CT images, basic image processing techniques like thresholding, erosion, and dilation are often insufficient. To address this issue, we used machine learning algorithms to segment X-ray CT image stacks of GDLs, comparing the performance of basic image processing with decision tree learning (via Trainable WEKA Segmentation) and convolutional neural networks (CNNs) (via U-Net and MSDNet). The training methods and classification features for each algorithm were varied and evaluated against a GDL sample with a semi-bimodal pixel distribution (SGL 10BA) and a more difficult, unimodal sample (EP40T). The optimal combinations for each algorithm were then applied to segment a GDL sample with a microporous layer (MPL), an SGL 10BC, as MPL-containing GDLs are generally preferred in PEFCs. We found that decision tree learning, aside from being the easiest to use, exhibited the best performance for each of the four phases—pores, water, GDL, and MPL—based on F1 scores. Based on the wide collection of literature, properly trained CNNs should produce significantly better results. However, obtaining such results may require substantially more investment to determine the optimal algorithm for a particular scenario.
Pressure reduction following uplift may lead to dissociation of gas hydrates. The dynamics of hydrate dissociation in such settings, however, are poorly understood. We used TOUGH+HYDRATE to investigate the response of gas hydrates to an uplift of 0.009 myr $$^{-1}$$ - 1 over the last 8 kyrs, the approximate end of the postglacial sea-level rise. Geological parameters for the simulations are based on hydrate deposits from the Nankai Trough subduction zone. Our results suggest stabilisation from endothermic cooling, elevated pore pressure, and pore water freshening significantly slows hydrate dissociation such that the hydrate remains in place at its pre-uplift level. A shallower hydrate layer forms from upward-migrating gas when assuming moderate to high permeability (10 $$^{-15}$$ - 15 and 10 $$^{-13}$$ - 13 m $$^{2}$$ 2 ), while gas remains trapped for low permeability (10 $$^{-17}$$ - 17 m $$^{2}$$ 2 ). In the latter case, we predict elevated pore pressure with potential implications for seafloor stability. Our findings suggest that following uplift, hydrates may exist outside the predicted regional gas hydrate stability field for thousands of years.
Screening tools such as BioScreen, BioChlor, ATRANS, AT123D-AT, ArcNLET, and Hydroscape are routinely employed to simulate the three-dimensional transport of reactive contaminants in groundwater. These tools estimate contaminant plume concentrations either using exact semi-analytical solutions or the approximate closed-form Domenico analytical solution. Semi-analytical solutions involve numerical integration procedures that can be mathematically challenging and computationally demanding. To overcome this, screening tools often use the approximate closed-form Domenico solution. However, the approximate Domenico solution introduces significant errors under realistic values of longitudinal dispersion, especially at plume locations beyond the advective front. Recently, an improved closed-form approximation to the three-dimensional reactive transport problem was developed using the concept of characteristic residence time. However, this solution was only applicable for a rectangular area source subject to a Dirichlet boundary condition. This severely restricts the use and applicability of the closed-form approximate solution to solve practically relevant simplified groundwater contaminant transport problems. Here, we present a library of six exact semi-analytical solutions for point, line, and area sources (three source geometries) under Dirichlet and Cauchy boundary conditions (two boundary conditions). Additionally, we develop approximate closed-form analytical solutions for all six solutions using the characteristic residence time concept. Our approximate solutions match well with the exact solutions under a wide range of parameter and domain conditions. We extend our analytical solutions to include the effects of linear equilibrium sorption, source decay, and pulse source input. Our analytical solution library facilitates the application of screening tools for a wide range of practically relevant simplified groundwater reactive contaminant transport problems.
For improved operating conditions of a polymer electrolyte membrane (PEM) fuel cell, a sophisticated water management is crucial. Therefore, it is necessary to understand the transport mechanisms of water throughout the cell constituents especially on the cathode side, where the excess water has to be removed. Pore-scale modeling of diffusion layers and gas distributor has been established as a favorable technique to investigate the ongoing processes. Investigating the interface between the cathode layers, a particular challenge is the combination and interaction of the multi-phase flow in the porous material of the gas diffusion layer (GDL) with the free flow in the gas distributor channels. The formation, growth and detachment of water droplets on the hydrophobic, porous surface of the GDL have a major influence on the mass, momentum and energy exchange between the layers. A dynamic pore-network model is used to describe the flow through the porous GDL on the pore-scale. To capture the droplet occurrence and its influence on the flow, this dynamic two-phase pore-network model is extended to capture droplet formation and growth at the surface of the GDL as well as droplet detachment due to the gas flow in the gas distributor channels. In this article, the developed model is applied to single- and multi-tube systems to investigate the general drop behavior. These rather simple test-cases are compared to experimental and numerical data available in the literature. Finally, the model is applied to a GDL unit cell to analyze the interaction between two-phase flow through the GDL and drop formation at the interface between GDL and gas distributor channel.
Capillary number, understood as the ratio of viscous force to capillary force, is one of the most important parameters in enhanced oil recovery (EOR). It continues to attract the interest of scientists and engineers, because the nature and quantification of macroscopic capillary forces remain controversial. At least 41 different capillary numbers have been collected here from the literature. The ratio of viscous and capillary force enters crucially into capillary desaturation experiments. Although the ratio is length scale dependent, not all definitions of capillary number depend on length scale, indicating potential inconsistencies between various applications and publications. Recently, new numbers have appeared and the subject continues to be actively discussed. Therefore, a short review seems appropriate and pertinent.
Capillary trapping is considered as one of the safest geologic CO2 storage mechanisms due to its hydrodynamic stability. However, the thermodynamic stability of capillary trapping was questioned by our recent work (Xu in Geophys Res Lett 46(23):13804–13813, 2019). Gravity induces the top bubbles to grow at expense of bottom bubbles through the diffusion of dissolved gas components, that finally may form a gas cap and posing a risk of leakage, even in absence of convection. Here, we improve the gravity-induced ripening model introduced earlier and conduct theoretical and numerical analysis. Four regimes of bubble ripening are identified according to the modified Bond number and initial gas saturation, resulting in different scaling between the equilibrium time and the length scale. Vertical heterogeneity is also shown to have a great impact on the ripening process. When the permeability gradient is downward, the capillary pressure gradient competes with the gravitational gradient and results in a complex gas redistribution behavior. Capillarity dominates in a short time, while gravitational potential determines the global saturation profile in long term. This work provides a new physical perspective in evaluating CO2 sequestration security and has a potential application in other porous systems that gas generated and evolve under strong external fields.
Interfacial area is an important factor during two-phase flow in porous media because mass-transfer mechanisms take place at the interfaces of immiscible phases. The objective of this work is to quantify how grain-size distribution affects the temporal development of interfacial area during two-phase flow through porous media. A two-phase lattice Boltzmann model (color gradient method) was used to simulate drainage (displacement of a wetting fluid by a non-wetting fluid) and imbibition (displacement of the non-wetting fluid by the wetting fluid) in an ensemble of two-dimensional porous media samples. Five groups of porous media, each comprising 20 realizations, were characterized by their median grain size (d50) and coefficient of uniformity (Cu). For all 100 realizations, simulations of drainage and imbibition were conducted until steady-state saturation was achieved, and interfacial area was monitored throughout the simulations. During both drainage and imbibition, the interfacial area initially increases with time until reaching a peak area, then decreases, and then plateaus at a steady-state value. Interfacial area is higher during imbibition than during drainage. The temporal evolution of interfacial area, as quantified by peak area and time to reach peak area, was similar in the three groups characterized by small grain size (d50 ≈ 7.7 lattice units) and relatively uniform grain-size distribution (Cu ≈ 1.21, 1.49, 1.85), for both drainage and imbibition. This suggests that, for the fluid conditions considered here, nonuniformity of grain size is not important below a certain threshold value of Cu. However, two groups with larger grain size (d50 ≈ 8.9 lattice units) and relatively nonuniform grain-size distribution (Cu ≈ 1.85, 2.29) exhibited differences from each other, suggesting that nonuniformity of grain size affects interfacial area when Cu is above a certain value. Furthermore, median grain size was observed to have important effects on temporal evolution of interfacial area.
Uncertainty is ubiquitous with multiphase flow in subsurface rocks due to their inherent heterogeneity and lack of in-situ measurements. To complete uncertainty analysis in a multi-scale manner, it is a prerequisite to provide sufficient rock samples. Even though the advent of digital rock technology offers opportunities to reproduce rocks, it still cannot be utilized to provide massive samples due to its high cost, thus leading to the development of diversified mathematical methods. Among them, two-point statistics (TPS) and multi-point statistics (MPS) are commonly utilized, which feature incorporating low-order and high-order statistical information, respectively. Recently, generative adversarial networks (GANs) are becoming increasingly popular since they can reproduce training images with excellent visual and consequent geologic realism. However, standard GANs can only incorporate information from data, while leaving no interface for user-defined properties, and thus may limit the representativeness of reconstructed samples. In this study, we propose conditional GANs for digital rock reconstruction, aiming to reproduce samples not only similar to the real training data, but also satisfying user-specified properties. In fact, the proposed framework can realize the targets of MPS and TPS simultaneously by incorporating high-order information directly from rock images with the GANs scheme, while preserving low-order counterparts through conditioning. We conduct three reconstruction experiments, and the results demonstrate that rock type, rock porosity, and correlation length can be successfully conditioned to affect the reconstructed rock images. The randomly reconstructed samples with specified rock type, porosity and correlation length will contribute to the subsequent research on pore-scale multiphase flow and uncertainty quantification.
The dynamic capillary effect, which is a common phenomenon of two-phase flow in porous media, has crucial importance in various fields. Several factors, such as permeability, grain size, temperature, fluid properties, and saturation conditions, have a significant impact on the observed dynamic capillary effect. This review systematically analyzes the influence of three categories of factors on the dynamic capillary effect, including properties of porous media and fluids, as well as certain external factors. The main conclusions of available researches are presented and analyzed for each influencing factor. The possible reasons for controversial results are discussed as well. This review shows that the pore throat size of porous media has the greatest impact on the dynamic capillary effect, which can lead to a variation of up to nearly five orders of magnitude in the value of the dynamic capillary coefficient. Meanwhile, the grain size, fracture aperture, and temperature cause the variation of the dynamic capillary coefficient by one or two orders of magnitude. The effect of other factors on the dynamic capillary effect is relatively weakened. Finally, the future perspectives, concluding remarks, and recommendations are presented.
I use a mechanical model of a soft body to study the dynamics of an individual fluid droplet in a random, non-wettable porous medium. The model of droplet relies on the spring-mass system with pressure. I run hundreds of independent simulations. I average droplets trajectories and calculate the averaged tortuosity of the porous domain. Results show that porous media tortuosity increases with decreasing porosity, similar to single-phase fluid study, but the form of this relationship is different. Supplementary information: The online version contains supplementary material available at 10.1007/s11242-021-01705-z.
Water-flooding aided by electromagnetic (EM) heating using microwaves (MW) has a great potential for heavy oil recovery. Earlier, we have shown experimentally and theoretically that EM radioactive energy is absorbed by water and converted into heat near the EM source. The heat is imparted into the oil phase while being transported deep into the porous medium (Paz et al. in Transp Porous Media 119(1):57–75, 2017). The lowering of water is primarily responsible for the improved oil recovery in this process. This paper develops a model describing electromagnetic heating-assisted water flooding (EMA WF) in a thin heavy oil reservoir. The model is solved numerically using a staggered algorithm joining Galerkin Least Square Finite Elements Method (GLS-FEM) and Kurganov–Tadmor Finite Volumes Method (KT-FVM). Numerical results were obtained for different types of oil. For the considered parameter values, the EM heating technique increments the oil production up to 56% after a 24-month injection. This increment is inversely proportional to the oils API gravity. The implementation was validated by comparing computational results with the simplified model’s analytical solution obtained using Conservation Laws theory and the Sturm–Liouville theory. Simple convergence analysis was also performed endorsing our numerical approach. Both analytical and numerical approaches were obtained for the two-dimensional geometry with two parallel horizontal wells. The temperature profile obtained through the simplified model’s analytical solution is close to the one obtained by simulations (less than 1.9% relative error).
Roughness of surfaces significantly influences how methane and water flow in shale nanopores. We perform molecular dynamics simulations to investigate the influence of surface roughness on pore-scale transport of pure methane as well as of two-phase methane–water systems with the water sliding as droplets over the pore surface. For single-phase methane flow, surface roughness shows a limited influence on bulk methane density, while it significantly reduces the methane flow capacity. In methane–water systems, the mobility of water is a strong function of surface roughness including a clear transition between immobile and mobile water droplets. For cases with mobile water, droplet sliding speeds were correlated with pressure gradient and surface roughness. Sliding water droplets hardly deform, i.e., there is little difference between their advancing and receding contact angle with structured roughness.
Due to the multi-scale pore size and complex gas-bound water distribution, it is challenging to accurately predict gas transport property in shale. Given the known heterogeneities, single-resolution pore-scale imaging is not reliable for representative pore structure characterization. In this study, the image-based shale multi-scale pore network model (MPNM) is proposed and the impacts of pore structure and relative humidity (RH) on gas transport are analyzed in detail. 3D binary images are constructed by the multiple-point statistics method from a section of low-resolution SEM image which covers the large-scale pore structure and fine-scale SEM images with the same physical size at high resolution. The maximal ball fitting method is applied to extract large-scale pore network model (LPNM) and fine-scale pore network models (FPNMs) from the 3D binary images, respectively. MPNM is obtained by merging the LPNM and FPNMs based on the proposed procedure. The confined gas-bound water distribution at different RH is calculated considering the disjoining pressure resulting from van der Waals force, electric double-layer interactions and structural force. Gas slippage in irregular pores is considered for gas transport. Pore structure parameters and gas permeabilities are calculated based on the MPNM, LPNM and FPNMs. Study results indicate that the gas permeability of MPNM is more close to the laboratory pressure pulse decay measured gas permeability of studied sample. Gas permeability decreases with the increasing RH and drops to zero at average pore radius less than 12 nm and RH larger than 0.7.
In this paper, we study the dissolution of a porous formation made of soluble and insoluble materials with various types of Darcy-scale heterogeneities. Based on the assumption of scale separations, i.e., the convective and diffusive Damköhler numbers are smaller than certain limits which are documented in the paper, we apply large-scale upscaling to the Darcy-scale model to develop large-scale equations, which are used to describe the dissolution of porous formations with Darcy-scale heterogeneities. History-dependent closure problems are provided to get the effective parameters in the large-scale model. The large-scale model validity is tested by comparing numerical results for a 1D flow problem in a stratified system and a 2D flow problem in a nodular system to the Darcy-scale ones. The good agreement between results at Darcy and large scales shows the robustness of the large-scale model in representing the Darcy-scale results for the stratified system, even when the dissolution front is very sharp. Large-scale results for the nodular system represent satisfactorily the averaged Darcy-scale behaviors when the dissolution front is relatively thick, i.e., when model assumptions are satisfied, while there may be as expected some discrepancy generated between direct numerical simulations and large-scale results in the case of thin dissolution front. Overall, this study demonstrates the possibility of building a fully homogenized large-scale model incorporating dissolution history effects, and that the resulting large-scale model is capable to catch the main features of the Darcy-scale results within its applicability domain. Article highlights Large-scale model is developed for the dissolution of heterogeneous porous media, taking dissolution history effect into account. A sequential algorithm is proposed for the solution of effective mass exchange coefficient and effective permeability tensor. The large-scale model is validated for stratied and nodular systems.
Digital rock analysis provides us a powerful tool for predicting geophysical properties and studying fluid and interfacial transport mechanisms in rocks. However, people have to struggle and find a balance between scanning resolution and sample size due to current limitations of imaging technologies. With satisfaction of resolution requirement, the sample size has to be larger than the critical size of representative element volume (REV), so that the consequent pore-scale models are able to provide meaningful geophysical predictions for upscaling to Darcy-scale analysis. Following our previous work [Energies, 11: 1798, 2018] on REV size for single-phase flow, this work considers the critical size of REV for multiphase flow in porous media. A multiphase lattice Boltzmann model has been developed for simulation of two-phase immiscible flow. The relative permeability, which can be influenced by the capillary number and wettability, and the saturation of phases are calculated for upscaling. The critical size of REV for multiphase flow in porous media is therefore found and compared with that for single-phase flow. It is found that the REV size for the relative permeability–saturation curve of multiphase flow, which is influenced by the phase interaction and wettability, is beyond twice of that for the absolute permeability of single-phase flow in the present study. Article Highlights The critical size of REV for multiphase flow in porous media is determined by pore-scale modeling. The REV size of multiphase flow is beyond twice that of single-phase flow on the same porous structure. The REV size for the relative permeability–saturation curve is influenced by the phase interaction and wettability.
Polymer solution has extremely extensive applications in many natural and industrial processes, especially in oilfield development field, such as polymer flooding, fracturing and water shut-off. Thus, the study of flow behaviors of polymer in formation pores is significantly important. In this paper, the flow behaviors of the polymer droplets in the 3-D pore throat structure were systematically studied. Additionally, the influencing factors (polymer concentration, molecular weight and pore throat ratio, for instance) were investigated. As the increasing of polymer concentration and molecular weight, the polymer droplets were more difficult to break, which means the critical flow rate decreased and the average sizes of the first daughter droplets (FDD) were longer synchronously. Moreover, with the increase in pore throat ratio, the critical flow rate increased and the length of the FDD decreased. In addition, the prediction models of the length of the FDD with polymer concentration and pore throat ratio were established, respectively. The prediction model revealed that the length of the FDD satisfied an exponential relationship with the polymer concentration and a linear relationship with the pore throat ratio. Finally, the average size of droplets after macroscopic core flooding experiment was 8 μm and 17 μm when the polymer concentration was 0.01% and 0.1%, respectively. The results were consistent with the breakup behaviors of polymer droplets in the microscopic pore throat structure.
We investigate and classify possible analytical solutions for a simplified version of the foam bubble population model, by varying injection conditions and kinetic foam generation parameter. We prove that the behavior of the analytical solutions changes at the transition between two regions, similar to rarefaction-shock solutions for the Buckley-Leverett equation. In one region (region I), the solutions are in the form of traveling waves, in rather good agreement with CT scanned foam experiments and numerical simulations reported in Simjoo and Zitha (2015). In region II, however, corresponds to solutions as a sequence of waves: one spreading wave and one traveling wave. These corresponding flow profiles are different from those found so far in the experiments.
Uncertainty quantification and sensitivity analysis are crucial tools in the development and evaluation of mathematical models. In enhanced oil recovery, the co-injection of foam in porous media has been investigated through laboratory experiments and mathematical models as a promising technique for improving sweep efficiency. In this work, we study two mathematical models of foam flow in porous media. First, we present a foam quality-scan experiment using nitrogen and low concentration of an alpha-olefin sulfonate surfactant in brine using Indiana limestone carbonate core. Second, we evaluate the models based on their ability to represent the experimental data using inverse uncertainty quantification techniques. Third, the parameters’ estimated distributions are used to perform both forward uncertainty quantification and sensitivity analysis. We also present a detailed comparison of the models, and analyses on the experimental data, model discrepancy, and sources of uncertainties. The experimental results of foam apparent viscosity in carbonate rocks are consistent with other experiments in sandstones: The foam quality transition is present; the difference in apparent viscosity values is of the same magnitude as the difference in permeability. Propagation of uncertainties from the estimated parameter distributions through the models showed a good match between experimental data and model predictions. The sensitivity analysis showed that the model’s parameters play different roles and depend on the quantity of interest, the foam quality regime, and limiting water saturation. To summarize, this study provides essential information for possible improvements in the experiments and mathematical models of foam flow in EOR processes.
Miscible displacement of two-phase fluids in rough fractures is relevant to some industrial processes, including enhanced oil recovery and geological carbon sequestration. When a less viscous fluid displaces another more viscous fluid, finger instability occurs. Previous works focused on miscible displacement in porous media or Hele-Shaw, but the experimental study was rarely reported for rough fractures. Here, we perform visualization experiments of water displacing glycerol in a transparent fracture model to investigate the effects of flow rate and diffusion in miscible displacement. We quantify the displacement patterns using the sweep efficiency, the mixing length, and the relative contact area. We observe two distinct displacement regimes: dominant finger regime and multiple fingers regime. A critical Peclet number Pe is obtained to identify such two regimes. Below the critical Pe, the channel forms, and the displacement is the dominant finger regime, which results in low sweep efficiency and linearly growth of mixing length at late time. Above this critical Pe, intensive tip-splitting events result in the formation of dendritic displacement pattern, and the displacement is multiple fingers regime, slowing down the growth rate of mixing length at late time and contributes to the higher sweep efficiency. Our work shows a critical Pe that separates the two distinct regimes and improves our understanding of the evolution of the miscible displacement fronts in rough fractures.
Top-cited authors
Steffen Berg
  • Shell Global Solutions International B.V.
Ryan T. Armstrong
Hamdi Tchelepi
  • Stanford University
Martin J. Blunt
  • Imperial College London
Cyprien Soulaine
  • CNRS Orleans Campus