Coarse graining equations for flow in porous media: a HaarWavelets and renormalization approach

Coarse graining of equations for flow in porous media is an important aspect in modelling permeable subsurface geological systems. In the study of hydrocarbon reservoirs as well as in hydrology, there is a need for reducing the size of the numerical models to make them computationally efficient, while preserving all the relevant information which is given at different scales. In the first part, a new renormalization method for upscaling permeability in Darcy’s equation based on Haar wavelets is presented, which differs from other wavelet based methods. The pressure field is expressed as a set of averages and differences, using a one level Haar wavelet transform matrix. Applying this transform to the finite difference discretized form of Darcy’s law, one can deduce which permeability values on the coarse scale would give rise to the average pressure field. Numerical simulations were performed to test this technique on homogeneous and heterogeneous systems. A generalization of the above method was developed designing a hierarchical transform matrix inspired by a full Haar wavelet transform, which allows us to describe pressure as an average and a set of progressively smaller scale differences. Using this transform the pressure solution can be performed at the required level of detail, allowing for different resolutions to be kept in different parts of the system. A natural extension of the methods is the application to two-phase flow. Upscaling mobility allows the saturation profile to be calculated on the fine or coarse scale while based on coarse pressure values. To conclude, an alternative approach to upscaling in multi-phase flow is to upscale the saturation equation itself. Taking its Laplace transform, this equation can be reduced to a simple eigenvalue problem. The wavelet upscaling method can now be applied to calculate the upscaled saturation profile, starting with fine scale velocity data

Geostatistical and stochastic study of flow and tracer transport in the unsaturated zone at Yucca Mountain

Yucca Mountain has been proposed by the U.S. Department of Energy as the nation’s long-term, permanent geologic repository for spent nuclear fuel or high-level radioactive waste. The potential repository would be located in Yucca Mountain’s unsaturated zone (UZ), which acts as a critical natural barrier delaying arrival of radionuclides to the water table. Since radionuclide transport in groundwater can pose serious threats to human health and the environment, it is important to understand how much and how fast water and radionuclides travel through the UZ to groundwater. The UZ system consists of multiple hydrogeologic units whose hydraulic and geochemical properties exhibit systematic and random spatial variation, or heterogeneity, at multiple scales. Predictions of radionuclide transport under such complicated conditions are uncertain, and the uncertainty complicates decision making and risk analysis. This project aims at using geostatistical and stochastic methods to assess uncertainty of unsaturated flow and radionuclide transport in the UZ at Yucca Mountain. Focus of this study is parameter uncertainty of hydraulic and transport properties of the UZ. The parametric uncertainty arises since limited parameter measurements are unable to deterministically describe spatial variability of the parameters. In this project, matrix porosity, permeability and sorption coefficient of the reactive tracer (neptunium) of the UZ are treated as random variables. Corresponding propagation of parametric uncertainty is quantitatively measured using mean, variance, 5th and 95th percentiles of simulated state variables (e.g., saturation, capillary pressure, percolation flux, and travel time). These statistics are evaluated using a Monte Carlo method, in which a three-dimensional flow and transport model implemented using the TOUGH2 code is executed with multiple parameter realizations of the random model parameters

Anomalous transport in the crowded world of biological cells

A ubiquitous observation in cell biology is that diffusion of macromolecules and organelles is anomalous, and a description simply based on the conventional diffusion equation with diffusion constants measured in dilute solution fails. This is commonly attributed to macromolecular crowding in the interior of cells and in cellular membranes, summarising their densely packed and heterogeneous structures. The most familiar phenomenon is a power-law increase of the MSD, but there are other manifestations like strongly reduced and time-dependent diffusion coefficients, persistent correlations, non-gaussian distributions of the displacements, heterogeneous diffusion, and immobile particles. After a general introduction to the statistical description of slow, anomalous transport, we summarise some widely used theoretical models: gaussian models like FBM and Langevin equations for visco-elastic media, the CTRW model, and the Lorentz model describing obstructed transport in a heterogeneous environment. Emphasis is put on the spatio-temporal properties of the transport in terms of 2-point correlation functions, dynamic scaling behaviour, and how the models are distinguished by their propagators even for identical MSDs. Then, we review the theory underlying common experimental techniques in the presence of anomalous transport: single-particle tracking, FCS, and FRAP. We report on the large body of recent experimental evidence for anomalous transport in crowded biological media: in cyto- and nucleoplasm as well as in cellular membranes, complemented by in vitro experiments where model systems mimic physiological crowding conditions. Finally, computer simulations play an important role in testing the theoretical models and corroborating the experimental findings. The review is completed by a synthesis of the theoretical and experimental progress identifying open questions for future investigation.Comment: review article, to appear in Rep. Prog. Phy

Finite element modeling of free surface flow in variable porosity media

The aim of the present work is to present an overview of some numerical procedures for the simulation of free surface flows within a porous structure. A particular algorithm developed by the authors for solving this type of problems is presented. A modified form of the classical Navier–Stokes equations is proposed, with the principal aim of simulating in a unified way the seepage flow inside rockfill-like porous material and the free surface flow in the clear fluid region. The problem is solved using a semi-explicit stabilized fractional step algorithm where velocity is calculated using a 4th order Runge–Kutta scheme. The numerical formulation is developed in an Eulerian framework using a level set technique to track the evolution of the free surface. An edge-based data structure is employed to allow an easy OpenMP parallelization of the resulting finite element code. The numerical model is validated against laboratory experiments on small scale rockfill dams and is compared with other existing methods for solving similar problems.Peer ReviewedPostprint (author’s final draft

The Effects Of Diffusion, Capillary Heterogeneity, And Porous Media Morphology On CO2 Migration In Porous Media

The Modified Invasion Percolation (MIP) model was used to simulate the invasion of CO2 into a medium initially saturated with water and the style of capillary heterogeneity in combination with buoyancy and viscous forces were varied to yield 105 different simulation scenarios. Thereafter, the invasion of the CO2 gas is simulated using a finite difference model and the effective mass transfer coefficient is determined and linked to various styles of capillary heterogeneity. Results show that an increase in saturation does not result in an increase in mass transfer coefficient rather, the surface area, which in turn is controlled by the interplay of capillary heterogeneity effects and buoyancy forces, control the average mass transfer coefficient and the fractional mass left. We further correlated the mass transfer coefficient with the average surface area and saturation and we found that a very strong positive correlation coefficient exists between the surface area of the CO2 and mass transfer coefficient and a weak negative correlation coefficient exists between the average mass transfer coefficient and saturation exists, which improves with increasing stratification and increasing buoyancy forces. We also used a harmonic mean model to estimate the effective diffusivity in a Room Temperature Ionic Liquid (RTIL) membrane. A metallic porous membrane was thoroughly saturated with Emim(Tf2N) and placed in a diffusion chamber. CO2 gas was injected into the upper diffusion chamber and was alloto completely diffuse through the RTIL into the lower diffusion chamber. Lag-time technique was used to analyze the pressure-time data. Two different membrane configurations were used; 0.2-0.2&mgr; and 0.5-0.5&mgr;. Results show that the harmonic mean model reasonably predicts the effective diffusivity obtained from the experimental measurements

Application of upscaling methods for fluid flow and mass transport in multi-scale heterogeneous media : A critical review

Physical and biogeochemical heterogeneity dramatically impacts fluid flow and reactive solute transport behaviors in geological formations across scales. From micro pores to regional reservoirs, upscaling has been proven to be a valid approach to estimate large-scale parameters by using data measured at small scales. Upscaling has considerable practical importance in oil and gas production, energy storage, carbon geologic sequestration, contamination remediation, and nuclear waste disposal. This review covers, in a comprehensive manner, the upscaling approaches available in the literature and their applications on various processes, such as advection, dispersion, matrix diffusion, sorption, and chemical reactions. We enclose newly developed approaches and distinguish two main categories of upscaling methodologies, deterministic and stochastic. Volume averaging, one of the deterministic methods, has the advantage of upscaling different kinds of parameters and wide applications by requiring only a few assumptions with improved formulations. Stochastic analytical methods have been extensively developed but have limited impacts in practice due to their requirement for global statistical assumptions. With rapid improvements in computing power, numerical solutions have become more popular for upscaling. In order to tackle complex fluid flow and transport problems, the working principles and limitations of these methods are emphasized. Still, a large gap exists between the approach algorithms and real-world applications. To bridge the gap, an integrated upscaling framework is needed to incorporate in the current upscaling algorithms, uncertainty quantification techniques, data sciences, and artificial intelligence to acquire laboratory and field-scale measurements and validate the upscaled models and parameters with multi-scale observations in future geo-energy research.© 2021 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)This work was jointly supported by the National Key Research and Development Program of China (No. 2018YFC1800900 ), National Natural Science Foundation of China (No: 41972249 , 41772253 , 51774136 ), the Program for Jilin University (JLU) Science and Technology Innovative Research Team (No. 2019TD-35 ), Graduate Innovation Fund of Jilin University (No: 101832020CX240 ), Natural Science Foundation of Hebei Province of China ( D2017508099 ), and the Program of Education Department of Hebei Province ( QN219320 ). Additional funding was provided by the Engineering Research Center of Geothermal Resources Development Technology and Equipment , Ministry of Education, China.fi=vertaisarvioitu|en=peerReviewed

Cluster evolution in steady-state two-phase flow in porous media

We report numerical studies of the cluster development of two-phase flow in a steady-state environment of porous media. This is done by including biperiodic boundary conditions in a two-dimensional flow simulator. Initial transients of wetting and non-wetting phases that evolve before steady-state has occurred, undergo a cross-over where every initial patterns are broken up. For flow dominated by capillary effects with capillary numbers in order of $10^{-5}$, we find that around a critical saturation of non-wetting fluid the non-wetting clusters of size $s$ have a power-law distribution $n_s \sim s^{-\tau}$ with the exponent $\tau = 1.92 \pm 0.04$ for large clusters. This is a lower value than the result for ordinary percolation. We also present scaling relation and time evolution of the structure and global pressure.Comment: 12 pages, 11 figures. Minor corrections. Accepted for publication in Phys. Rev.