Upscaling the effect of heterogeneities in porous media is crucial for macroscopic flow predictions, with widespread applications in energy and environmental settings. In this study, we derive expressions for the upscaled flow properties of a porous medium with a vertical heterogeneity, using a combination of asymptotic analysis and numerical simulations. Then, we use these upscaled expressions to describe the dynamic flooding of an aquifer, where the classic Buckley-Leverett formulation is modified to account for heterogeneities. In particular, we show that heterogeneities can modify flooding speeds significantly, and we discuss the implications of these results in the case of carbon dioxide sequestration.
We present a model for upscaling the time-dependent effective retardation factor in hierarchical porous media with multimodal reactive mineral facies. The model extends the approach by Deng et al. (2013) in which they expanded a Lagrangian-based stochastic theory presented by Rajaram (1997) in order to describe the scaling effect of retardation factor. They used a first-order linear approximation in deriving their model to make the derivation tractable. Importantly, the linear approximation is known to be valid only to variances of 0.2. In this article we show that the model can be derived wit…
The typical temporal resolution used in modern simulations significantly exceeds characteristic time scales at which the system is driven. This is especially so when systems are simulated over time-scales that are much longer than the typical temporal scales of forcing factors. We investigate the impact of space-time upscaling on reactive transport in porous media driven by time-dependent boundary conditions whose characteristic time scale is much smaller than that at which transport is studied or observed at the macroscopic level. The focus is on transport of a reactive solute undergoing diff…
The macroscopic spreading and mixing of solute plumes in saturated porous media is ultimately controlled by processes operating at the pore scale. Whilst the conventional picture of pore-scale mechanical dispersion and molecular diffusion leading to persistent hydrodynamic dispersion is well accepted, this paradigm is inherently two-dimensional (2D) in nature and neglects important three-dimensional (3D) phenomena. We discuss how the kinematics of steady 3D flow at the porescale generate chaotic advection, involving exponential stretching and folding of fluid elements,the mechanisms by which i…
Detailed understanding of the coupling between fluid flow and solid deformation in porous media is crucial for the development biomedical devices and novel energy technologies relating to a wide range of geological and biological processes. Well established models based on poroelasticity theory exist for describing coupled fluid-solid mechanics. However, these models are not adapted to describe systems with multiple fluid phases or 'hybrid-scale' systems containing both solid-free regions and porous matrices. To address this problem, we present a novel computational fluid dynamics approach bas…
Mineral dissolution in porous media coupled with single- or multi-phase flows is pervasive in natural and engineering systems including carbon capture and sequestration (CCS) and acid stimulation in reservoir engineering. Dissolution of minerals occurs as chemicals in the solid phase is transformed into ions in the aqueous phase, effectively modifying the physical, hydrological and geochemical properties of the solid matrix, and thus leading to the strong coupling between local dissolution rate and pore-scale flow. However, our fundamental understanding of this coupling effect at the pore leve…
An open-sourced multiphase Darcy-Brinkman approach is proposed to simulate two-phase flow in hybrid systems containing both solid-free regions and porous matrices. This micro-continuum model is rooted in elementary physics and volume averaging principles, where a unique set of partial differential equations is used to represent flow in both regions and scales. The crux of the proposed model is that it tends asymptotically towards the Navier-Stokes volume-of-fluid approach in solid-free regions and towards the multiphase Darcy equations in porous regions. Unlike existing multiscale multiphase s…
Reactive transport in saturated/unsaturated porous media is numerically upscaled to the space-time scale of a hypothetical measurement through coarse-grained space-time (CGST) averages. The reactive transport is modeled at the fine-grained Darcy scale by the actual number of molecules involved in reactions which undergo advective and diffusive movements described by global random walk (GRW) simulations. The CGST averages verify identities similar to a local balance equation which allow us to derive expressions for the flow velocity and the intrinsic diffusion coefficient in terms of averaged m…
In the upscaling from pore- to continuum (Darcy) scale, reaction and deposition phenomena at the solid-liquid interface of a porous medium have to be represented by macroscopic reaction source terms. The effective rates can be computed, in the case of periodic media, from three-dimensional microscopic simulations of the periodic cell. Several computational and semi-analytical models have been studied in the field of colloid filtration to describe this problem. They often rely on effective deposition rates defined by simple linear reaction ODEs, neglecting the advection-diffusion interplay, and…
Over the last two decades, lattice Boltzmann methods have become an increasingly popular tool to compute the flow in complex geometries such as porous media. In addition to single phase simulations allowing, for example, a precise quantification of the permeability of a porous sample, a number of extensions to the lattice Boltzmann method are available which allow to study multiphase and multicomponent flows on a pore scale level. In this article we give an extensive overview on a number of these diffuse interface models and discuss their advantages and disadvantages. Furthermore, we shortly r…
In disordered porous media, two-phase flow of immiscible fluids (biphasic flow) is organized in patterns that sometimes exhibit fractal geometries over a range of length scales, depending on the capillary, gravitational and viscous forces at play. These forces, as well as the boundary conditions, also determine whether the flow leads to the appearance of fingering pathways, i.e., unstable flow, or not. We present here a short review of these aspects, focusing on drainage and summarizing when these flows are expected to be stable or not, what fractal dimensions can be expected, and in which ran…
Pore-network modeling is a widely used predictive tool for pore-scale studies in various applications that deal with multiphase flow in porous media. Despite recent improvements to enable pore-network modeling on simplified pore geometry extracted from rock images and its computational efficiency compared to direct numerical simulation methods, there are still limitations to modeling a large representative pore-network for heterogeneous cores. These are due to the technical limits on sample size to discern void space during imaging and computational limits on pore-network extraction algorithms…
Biofilm growth in porous media: experiments, computational modeling at the porescale, and upscaling
Oregon State University（俄勒冈州立大学）；University of Warsaw（华沙大学）；Brookhaven National Laboratory（布鲁克海文国家实验室）；Helmholtz Centre for Environmental Research（赫尔霍兹环境研究所）
Transport in multiphase flow through porous media plays a central role in many biological, geological, and engineered systems. Here, we use numerical simulations of transport in immiscible two-phase flow to investigate dispersion in multiphase porous media flow. While dispersion in the main flow direction is similar to that of a single-phase flow and is governed by the porous media structure, we find that transverse dispersion exhibits fundamentally different dynamics. The repeated activation and deactivation of different flow pathways under the effect of capillary forces lead to intermittent …
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；Xiangtan University（湘潭大学）；Academy of Mathematics and System Sciences, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；Shenzhen International Center for Industrial and Applied Mathematics（深圳国际工业与应用数学中心）；Shenzhen Research Institute of Big Data（深圳大数据研究院）
The present article discusses the impact of liquid coolant subcooling on multiphase fluid flow and boiling heat transfer in porous media with internal heat generation. Although extremely relevant and important, only limited studies are available in the open literature on the effects of coolant subcooling in heat-generating porous media and hence, this requires a detailed analysis. The analysis is carried out using a developed computational model of multiphase fluid flow through clear fluid and porous media considering the relevant heat transfer and mass transfer phenomena. Results suggest that…
We report an approach to fully visualize the flow of two immiscible fluids through a model three-dimensional (3D) porous medium at pore-scale resolution. Using confocal microscopy, we directly image the drainage of the medium by the non-wetting oil and subsequent imbibition by the wetting fluid. During imbibition, the wetting fluid pinches off threads of oil in the narrow crevices of the medium, forming disconnected oil ganglia. Some of these ganglia remain trapped within the medium. By resolving the full 3D structure of the trapped ganglia, we show that the typical ganglion size, and the tota…
Dynamic multiphase flow triggers chaotic mixing in porous media
University of Oslo（奥斯陆大学）；Norwegian University of Science and Technology（挪威科技大学）；University of Copenhagen（哥本哈根大学）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；Univ. Savoie Mont Blanc（萨瓦蒙布朗大学）；Univ. Gustave Eiffel（古斯塔夫·埃菲尔大学）；Université de Rennes（雷恩大学）
Viscous and gravitational fingering refer to flow instabilities in porous media that are triggered by adverse mobility or density ratios, respectively. These instabilities have been studied extensively in the past for 1) single-phase flow (e.g., contaminant transport in groundwater, first-contact-miscible displacement of oil by gas in hydrocarbon production), and 2) multi-phase immiscible and incompressible flow (e.g., water-alternating-gas (WAG) injection in oil reservoirs). Fingering in multiphase compositional and compressible flow has received much less attention, perhaps due to its high c…
Sub-resolution porosity (SRP) is an ubiquitous, yet often ignored, feature in Digital Rock Physics. It embodies the trade-off between image resolution and field-of-view, and it is a direct result of choosing an imaging resolution that is larger than the smallest pores in a heterogeneous rock sample. In this study, we investigate the impacts of SRP on multiphase flow in porous rocks. To do so, we use our newly developed Multiphase Micro-Continuum model to perform first-of-a-kind direct numerical simulations of two-phase flow in porous samples containing SRP. We show that SRP properties (porosit…
For both isothermal and thermal petroleum reservoir simulation, the Constrained Pressure Residual (CPR) method is the industry-standard preconditioner. This method is a two-stage process involving the solution of a restricted pressure system. While initially designed for the isothermal case, CPR is also the standard for thermal cases. However, its treatment of the energy conservation equation does not incorporate heat diffusion, which is often dominant in thermal cases. In this paper, we present an extension of CPR: the Constrained Pressure-Temperature Residual (CPTR) method, where a restricte…
A two-phase model and its application to wavefields numerical simulation are discussed in the context of modeling of compressible fluid flows in elastic porous media. The derivation of the model is based on a theory of thermodynamically compatible systems and on a model of nonlinear elastoplasticity combined with a two-phase compressible fluid flow model. The governing equations of the model include phase mass conservation laws, a total momentum conservation law, an equation for the relative velocities of the phases, an equation for mixture distortion, and a balance equation for porosity. They…
Derivation of macroscopic models for advection-diffusion processes in the presence of dominant heterogeneous (e.g., surface) reactions using homogenisation theory or volume averaging is often deemed unfeasible due to the strong coupling between scales that characterise such systems. In this work, we show how the upscaling can be carried out by applying and extending the methods presented in literature. The approach relies on the decomposition of the microscale concentration into a reactive component, given by the eigenfunction of the advection-diffusion operator, the associated eigenvalue whic…
Porous asphalt (PA) is an open-graded porous material with a porosity of 20%, allowing fast drainage of rain and improving driving and acoustic conditions. However, the high porosity leads to significant contact with water resulting in a shorter life expectancy. To improve the durability and performance of PA, the distribution of water and its residence time have to be understood which entails capturing diverse multiphase phenomena. For these reasons, a numerical study is performed to analyze in detail the fluid transport mechanisms at play in PA, towards estimating the liquid distribution ins…
It is well-known that wettability exerts fundamental control over multiphase flow in porous media, which has been extensively studied in uniform-wet porous media. In contrast, multiphase flow in porous media with heterogeneous wettability (i.e., mixed-wet) is less well-understood, despite its common occurrence. Here, we study the displacement of silicone oil by water in a mostly oil-wet porous media patterned with discrete water-wet clusters that have precisely controlled wettability. Surprisingly, the macroscopic displacement pattern varies dramatically depending on the details of wettability…
In this work, an efficient physics-constrained deep learning model is developed for solving multiphase flow in 3D heterogeneous porous media. The model fully leverages the spatial topology predictive capability of convolutional neural networks, and is coupled with an efficient continuity-based smoother to predict flow responses that need spatial continuity. Furthermore, the transient regions are penalized to steer the training process such that the model can accurately capture flow in these regions. The model takes inputs including properties of porous media, fluid properties and well controls…
Solute dispersion in magnetically influenced multiphase flow through a porous tube: axial transport and microrotational effects
Cooch Behar Panchanan Barma University（古奇比哈尔潘查南·巴尔马大学）；SRM Institute of Science and Technology（SRM科学技术研究所）；University of Galway（高威大学）；Indian Institute of Technology Hyderabad（印度理工学院海得拉巴分校）
We investigate the upscaling of diffusive transport parameters as function of pore scale material structure using a stochastic framework. We focus on sub-REV (representative elementary volume) scale where the complexity of pore space geometry leads to a significant scatter of transport observations. We study a large data set of sub-REV measurements on porosity and transport ability being a dimensionless parameter representing the ratio of diffusive flow through the porous volume and through an empty volume. We characterize transport ability as probability distribution functions (PDFs) of poros…
It is well known that the transient behavior during drainage or imbibition in multiphase flow in porous media strongly depends on the history and initial condition of the system. However, when the steady-state regime is reached and both drainage and imbibition take place at the pore level, the influence of the evolution history and initial preparation is an open question. Here, we present an extensive experimental and numerical work investigating the history dependence of simultaneous steady-state two-phase flow through porous media. Our experimental system consists of a Hele-Shaw cell filled …
We present modeling of an incompressible viscous flow through a fracture adjacent to a porous medium. We consider a fast stationary flow, predominantly tangential to the porous medium. Slow flow in such setting can be described by the Beavers-Joseph-Saffman slip. For fast flows, a nonlinear filtration law in the porous medium and a non- linear interface law are expected. In this paper we rigorously derive a quadratic effective slip interface law which holds for a range of Reynolds numbers and fracture widths. The porous medium flow is described by the Darcys law. The result shows that the inte…
Transport and particulate processes are ubiquitous in environmental, industrial and biological applications, often involving complex geometries and porous media. In this work we present a general population balance model for particle transport at the pore-scale, including aggregation, breakage and surface deposition. The various terms in the equations are analysed with a dimensional analysis, including a novel collision-induced breakage mechanism, and split into one- and two-particles processes. While the first are linear processes, they might both depend on local flow properties (e.g. shear).…
Multiphase flow in porous media occurs in several disciplines including petroleum reservoir engineering, petroleum systems' analysis, and CO$_2$ sequestration. While simulations often use a fully implicit discretization to increase the time step size, restrictions on the time step often exist due to non-convergence of the nonlinear solver (e.g. Newton's method). Here this problem is addressed for the Buckley-Leverett equations, which model incompressible, immiscible, two-phase flow with no capillary potential. The equations are recast as a gradient flow using the phase-field method, and a conv…
The multiphase flow mechanism in miscible displacement through porous media is an important topic in various applications, such as petroleum engineering, low Reynolds number suspension flows, dusty gas dynamics, and fluidized beds. To simulate such flows, volume averaging spatial operators are considered to incorporate pressure drag and skin friction experienced by a porous medium. In this work, a streamline-based Lagrangian methodology is extended for an efficient numerical approach to handle dispersion and diffusion of solvent saturation during a miscible flow. Overall pressure drag on the d…
GeoChemFoam is an open-source OpenFOAM-based numerical modelling toolbox that includes a range of custom packages to solve complex flow processes including multiphase transport with interface transfer, single-phase flow in multiscale porous media, and reactive transport with mineral dissolution. In this paper, we present GeoChemFoam's novel numerical model for simulation of conjugate heat transfer in micro-CT images of porous media. GeoChemFoam uses the micro-continuum approach to describe the fluid-solid interface using the volume fraction of fluid and solid in each computational cell. The ve…
We study modeling of two-phase flow in highly heterogeneous fractured and porous media. The flow behaviour is strongly influenced by mass transfer between a highly permeable (mobile) fracture domain and less permeable (immobile) matrix blocks. We quantify the effective two-phase flow behaviour using a multirate rate mass transfer (MRMT) approach. We discuss the range of applicability of the MRMT approach in terms of the pertinent viscous and capillary diffusion time scales. We scrutinize the linearization of capillary diffusion in the immobile regions, which allows for the formulation of MRMT …
Mathematical models based on probability density functions (PDF) have been extensively used in hydrology and subsurface flow problems, to describe the uncertainty in porous media properties (e.g., permeability modelled as random field). Recently, closer to the spirit of PDF models for turbulent flows, some approaches have used this statistical viewpoint also in pore-scale transport processes (fully resolved porous media models). When a concentration field is transported, by advection and diffusion, in a heterogeneous media, in fact, spatial PDFs can be defined to characterise local fluctuation…
