Global random walk solvers for fully coupled flow and transport in saturated/unsaturated porous media

In this article, we present new random walk methods to solve flow and transport problems in saturated/unsaturated porous media, including coupled flow and transport processes in soils, heterogeneous systems modeled through random hydraulic conductivity and recharge fields, processes at the field and regional scales. The numerical schemes are based on global random walk algorithms (GRW) which approximate the solution by moving large numbers of computational particles on regular lattices according to specific random walk rules. To cope with the nonlinearity and the degeneracy of the Richards equation and of the coupled system, we implemented the GRW algorithms by employing linearization techniques similar to the -scheme developed in finite element/volume approaches. The resulting GRW -schemes converge with the number of iterations and provide numerical solutions that are first-order accurate in time and second-order in space. A remarkable property of the flow and transport GRW solutions is that they are practically free of numerical diffusion. The GRW solvers are validated by comparisons with mixed finite element and finite volume solvers in one- and two-dimensional benchmark problems. They include Richards’ equation fully coupled with the advection-diffusion-reaction equation and capture the transition from unsaturated to saturated flow regimes.publishedVersio

Water Quality Simulation with Particle Tracking Method

In the numerical simulation of fluid flow and solute transport in porous media, finite element method (FEM) has long been utilized and has been proven to be efficient. In this work, an alternative approach called random walk particle tracking (RWPT) method is proposed. In this method, a finite number of particles represent the distribution of a solute mass. Each particle carries a certain fraction of the total mass and moves in the porous media according to the velocity field. The proposed RWPT model is established on a scientific software platform OpenGeoSys (OGS), which is an open source initiative for numerical simulation of thermo-hydro-mechanical-chemical (THMC) processes in porous media. The flow equation is solved using finite element method in OGS. The obtained hydraulic heads are numerically differentiated to obtain the velocity field. The particle tracking method does not solve the transport equation directly but deals with it in a physically stochastic manner by using the velocity field. Parallel computing concept is included in the model implementation to promote computational efficiency. Several benchmarks are developed for the particle tracking method in OGS to simulate solute transport in porous media and pore space. The simulation results are compared to analytical solutions and other numerical methods to test the presented method. The particle tracking method can accommodate Darcy flow as it is the main consideration in groundwater flow. Furthermore, other flow processes such as Forchheimer flow or Richards flow can be combined with as well. Two applications indicate the capability of the method to handle theoretical real-world problems. This method can be applied as a tool to elicit and discern the detailed structure of evolving contaminant plumes.Bei der numerischen Simulation von Strömung und Stofftransport in porösen Medien hat die Nutzung der Finite-Elemente-Methode (FEM) eine lange Tradition und wird sich als effizient erweisen. In dieser Arbeit wird ein alternativer Ansatz, die random walk particle tracking (RWPT) Methode vorgeschlagen. Bei diesem Verfahren stellt eine endliche Anzahl von Partikeln die Verteilung eines gelösten Stoffes dar. Jedes Teilchen trägt einen bestimmten Bruchteil der Gesamtmasse und bewegt sich in den porösen Medien gemäß des Geschwindigkeitsfeldes. Das vorgeschlagene RWPT Modell basiert auf der wissenschaftlichen Softwareplattform OpenGeoSys (OGS), die eine Open-Source-Initiative für die numerische Simulation thermo-hydro-mechanisch-chemischen (THMC) in porösen Medien darstellt. Die Strömungsgleichung wird in OGS mit der Finite-Elemente-Methode gelöst. Der Grundwasserstand wird numerisch berechnet, um das Geschwindigkeitsfeld zu erhalten. Die Partikel-Tracking-Methode löst die Transportgleichung nicht direkt, sondern befasst sich mit ihr in einer physikalisch stochastische Weise unter Nutzung des Geschwindigkeitsfeldes. Zur Berücksichtigung der Recheneffizienz ist ein Parallel Computing-Konzept in der Modell-Implementierung enthalten. Zur Simulation des Stofftransports in porösen Medien und im Porenraum wurden mehrere Benchmarks für die Partikel-Tracking-Methode in OGS entwickelt. Die Simulationsergebnisse werden mit analytischen Lösungen und andere numerische Methoden verglichen, um die Aussagefähigkeit des vorgestellten Verfahrens zu bestätigen. Mit der Partikel-Tracking-Methode kann die Darcy-Strömung gelöst werden, die das wichtigste Kriterium in der Grundwasserströmung ist. Außerdem bewältigt die Methode auch andere Strömungsprozesse, wie die Forchheimer-Strömung und die Richards-Strömung. Zwei Anwendungen zeigen die Leistungsfähigkeit der Methode bei der prinzipiellen Handhabung von Problemen der realen Welt. Die Methode kann als ein Instrument zur Aufdeckung Erkennung der detaillierte Struktur von sich entwickelnden Schadstofffahnenangewendet werden

Water Quality Simulation with Particle Tracking Method

In the numerical simulation of fluid flow and solute transport in porous media, finite element method (FEM) has long been utilized and has been proven to be efficient. In this work, an alternative approach called random walk particle tracking (RWPT) method is proposed. In this method, a finite number of particles represent the distribution of a solute mass. Each particle carries a certain fraction of the total mass and moves in the porous media according to the velocity field. The proposed RWPT model is established on a scientific software platform OpenGeoSys (OGS), which is an open source initiative for numerical simulation of thermo-hydro-mechanical-chemical (THMC) processes in porous media. The flow equation is solved using finite element method in OGS. The obtained hydraulic heads are numerically differentiated to obtain the velocity field. The particle tracking method does not solve the transport equation directly but deals with it in a physically stochastic manner by using the velocity field. Parallel computing concept is included in the model implementation to promote computational efficiency. Several benchmarks are developed for the particle tracking method in OGS to simulate solute transport in porous media and pore space. The simulation results are compared to analytical solutions and other numerical methods to test the presented method. The particle tracking method can accommodate Darcy flow as it is the main consideration in groundwater flow. Furthermore, other flow processes such as Forchheimer flow or Richards flow can be combined with as well. Two applications indicate the capability of the method to handle theoretical real-world problems. This method can be applied as a tool to elicit and discern the detailed structure of evolving contaminant plumes.Bei der numerischen Simulation von Strömung und Stofftransport in porösen Medien hat die Nutzung der Finite-Elemente-Methode (FEM) eine lange Tradition und wird sich als effizient erweisen. In dieser Arbeit wird ein alternativer Ansatz, die random walk particle tracking (RWPT) Methode vorgeschlagen. Bei diesem Verfahren stellt eine endliche Anzahl von Partikeln die Verteilung eines gelösten Stoffes dar. Jedes Teilchen trägt einen bestimmten Bruchteil der Gesamtmasse und bewegt sich in den porösen Medien gemäß des Geschwindigkeitsfeldes. Das vorgeschlagene RWPT Modell basiert auf der wissenschaftlichen Softwareplattform OpenGeoSys (OGS), die eine Open-Source-Initiative für die numerische Simulation thermo-hydro-mechanisch-chemischen (THMC) in porösen Medien darstellt. Die Strömungsgleichung wird in OGS mit der Finite-Elemente-Methode gelöst. Der Grundwasserstand wird numerisch berechnet, um das Geschwindigkeitsfeld zu erhalten. Die Partikel-Tracking-Methode löst die Transportgleichung nicht direkt, sondern befasst sich mit ihr in einer physikalisch stochastische Weise unter Nutzung des Geschwindigkeitsfeldes. Zur Berücksichtigung der Recheneffizienz ist ein Parallel Computing-Konzept in der Modell-Implementierung enthalten. Zur Simulation des Stofftransports in porösen Medien und im Porenraum wurden mehrere Benchmarks für die Partikel-Tracking-Methode in OGS entwickelt. Die Simulationsergebnisse werden mit analytischen Lösungen und andere numerische Methoden verglichen, um die Aussagefähigkeit des vorgestellten Verfahrens zu bestätigen. Mit der Partikel-Tracking-Methode kann die Darcy-Strömung gelöst werden, die das wichtigste Kriterium in der Grundwasserströmung ist. Außerdem bewältigt die Methode auch andere Strömungsprozesse, wie die Forchheimer-Strömung und die Richards-Strömung. Zwei Anwendungen zeigen die Leistungsfähigkeit der Methode bei der prinzipiellen Handhabung von Problemen der realen Welt. Die Methode kann als ein Instrument zur Aufdeckung Erkennung der detaillierte Struktur von sich entwickelnden Schadstofffahnenangewendet werden

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

Upscaling of a dual-permeability Monte Carlo simulation model for contaminant transport in fractured networks by genetic algorithm parameter identification

International audienceThe transport of radionuclides in fractured media plays a fundamental role in determining the level of risk offered by a radioactive waste repository in terms of expected doses. Discrete Fracture Networks (DFN) methods can provide detailed solutions to the problem of modeling the contaminant transport in fractured media. However, within the framework of the performance assessment (PA) of radioactive waste repositories, the computational efforts required are not compatible with the repeated calculations that need to be performed for the probabilistic uncertainty and sensitivity analyses of PA. In this paper, we present a novel upscaling approach, which consists in computing the detailed numerical fractured flow and transport solutions on a small scale and use the results to derive the equivalent continuum parameters of a lean, one-dimensional Dual-Permeability, Monte Carlo Simulation (DPMCS) model by means of a Genetic Algorithm search. The proposed upscaling procedure is illustrated with reference to a realistic case study of migration taken from literature

Transport in heterogeneous porous media

We present a new algorithm for modelling single phase transport of a tracer in porous media which demonstrates that structure on all scales affects macroscopic transport behaviour. We marry the robustness of the continuous time random walk (CTRW) framework with the simplicity of a Monte Carlo approach to reservoir simulation. We simulate transport as a series of particles transitioning between nodes with probability (t).dt that a particle will first arrive at a nearest neighbor in a time t to t + dt. To this end we first determine the mixing rules and transition probability ADE(t) for transport governed by the advection-dispersion equation (ADE) (Rhodes and Blunt, 2006). We validate our algorithm by simulating advective transport in bond percolation clusters at the critical point. We compute the histogram of flow speeds using the velocities from the bonds on the backbone and find the multifractal spectrum for two-dimensional lattices with linear dimension L _ 2000 and in three dimensions for L _ 250. We demonstrate that in the limit of large systems all the negative moments of the velocity distribution become ill-defined. However, to model transport, the velocity histogram should be weighted by the flux to obtain a well-defined mean travel time. Finally, we use CTRWtheory to demonstrate that anomalous transport is observed whose characteristics can be related to the multifractal properties of the system. We next demonstrate a pore-to-reservoir simulation methodology which is consistent across all scales of interest. At the micron scale, we fit a truncated power law (t) for the distribution of particle transition times from pore to pore simulations. To do this we use our transport algorithm on a geologically representative network model of Berea sandstone and compare the results to the explicit modelling of advection and molecular diffusion by Bijeljic and Blunt (2006). We find that the results are similar. We then demonstrate the effect of increasing pore scale heterogeneity on the power law exponent (_) by stretching the distribution of throat radii in our network model. We show that by increasing the spread of velocities within the network we decrease _ making the transport more anomalous - in keeping with the consensus currently in the literature. This (t) is then used to calculate transport on the mm to cm scale. We can then move up to the metre/grid block scale by using the transit time distribution from the mm-cm simulation to model transport in an explicit, geologically representative model of heterogeneity found within a grid block of the reservoir. From these numerical experiments we determine the (t) appropriate for transport on grid block scale systems characterized by Peclet (Pe) number and the type of heterogeneity within the system. This allows us to account for small scale uncertainty by interpreting (t) probabilistically and running simulations for different possible realizations of the reservoir heterogeneity. At the field scale, we represent the reservoir as an unstructured network of nodes connected by links. For each node-to-node transition, we use our upscaled (t) from a simulation of transport at the smaller scale. We account for small-scale uncertainty by parameterising (t) in terms of sub-scale heterogeneity and Peclet number. We demonstrate the methodology by finding a (t) for each scale of interest taking into consideration the relevant physics at that scale and using the appropriate function in a million-cell reservoir model. We show that the macroscopic behaviour can be very different from that predicted by assuming that the ADE operates at the small scale. Small-scale structure dramatically retards the advance of the plume with the particles becoming trapped in the slow moving pores/regions increasing breakthrough times by an order of magnitude compared to those predicted by the ADE

Particle Density Estimation with Grid-Projected Adaptive Kernels

The reconstruction of smooth density fields from scattered data points is a procedure that has multiple applications in a variety of disciplines, including Lagrangian (particle-based) models of solute transport in fluids. In random walk particle tracking (RWPT) simulations, particle density is directly linked to solute concentrations, which is normally the main variable of interest, not just for visualization and post-processing of the results, but also for the computation of non-linear processes, such as chemical reactions. Previous works have shown the superiority of kernel density estimation (KDE) over other methods such as binning, in terms of its ability to accurately estimate the "true" particle density relying on a limited amount of information. Here, we develop a grid-projected KDE methodology to determine particle densities by applying kernel smoothing on a pilot binning; this may be seen as a "hybrid" approach between binning and KDE. The kernel bandwidth is optimized locally. Through simple implementation examples, we elucidate several appealing aspects of the proposed approach, including its computational efficiency and the possibility to account for typical boundary conditions, which would otherwise be cumbersome in conventional KDE

Use of Global Sensitivity Analysis and Polynomial Chaos Expansion for Interpretation of Non-reactive Transport Experiments in Laboratory-Scale Porous Media

International audienceIn this work, we show how the use of global sensitivity analysis (GSA) in conjunction with the polynomial chaos expansion (PCE) methodology can provide relevant information for the interpretation of transport experiments in laboratory-scale heterogeneous porous media. We perform GSA by calculating the Sobol indices, which provide a variance-based importance measure of the effects of uncertain parameters on the output of a chosen interpretive transport model. The choice of PCE has the following two benefits: (1) it provides the global sensitivity indices in a straightforward manner, and (2) PCE can serve as a surrogate model for the calibration of parameters. The coefficients of the PCE are computed by probabilistic collocation. The methodology is applied to two nonreactive transport experiments available in the literature, while considering both transient and pseudo steady state transport regimes. This method allows a rigorous investigation of the relative effects and importance of different uncertain quantities, which include boundary conditions as well as porous medium hydraulic and dispersive parameters. The parameters that are most relevant to depicting the system's behavior can then be evaluated. In addition, one can assess the space-time distribution of measurement points, which is the most influential factor for the identifiability of parameters. Our work indicates that these methods can be valuable tools in the proper design of model-based transport experiments