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We consider two-phase flow and transport in heterogeneous porous media, where the permeability is modeled as a random field. To assess the resulting uncertainty of flow and transport, multilevel Monte Carlo (MLMC) is used. MLMC involves two types of errors: a sampling error and a discretization error, which, under reasonable assumptions, can be conveniently estimated during execution. In this work, we address the question of how to balance the two errors. Furthermore, a simple MLMC parallelization strategy is presented. A numerical comparison of MLMC with standard Monte Carlo (MC) reveals significant gains in computational efficiency at equivalent accuracy. [ABSTRACT FROM AUTHOR]

Summary The multiscale nature of geological formations is reflected in the flow and transport behaviors of the pore fluids. For example, multiple pathways between different locations in the porous medium are usually present. The topology, length, and strength of these flow paths can vary significantly, and the total flow at a given location can be the result of contributions from a wide range of pathways between the points of interest. We use a high-resolution pore network of a natural porous formation as an example of the multiscale connectivity of the pore space. A single continuum model can capture the contributions from all the flow paths properly only if the control volume (computational cell) is much larger than the longest pathway. However, depending on the densities and lengths of these long pathways, choosing the appropriate size of the control volume that allows for a single continuum description of the properties, such as conductivity and transmissibility, may conflict with the desire to resolve the flow field properly. To capture the effects of the multiscale pathways on the flow, a non-local continuum model is described. The model can represent non-local effects, for which Darcy’s law is not valid. In the limit where the longest connections are much smaller than the size of the control volume, the model is consistent with Darcy’s law. The non-local model is used to describe the flow in complex pore networks. The pressure distributions obtained from the non-local model are compared with pore-network flow simulations, and the results are in excellent agreement. Importantly, such multiscale flow behaviors cannot be represented using the local Darcy law. [ABSTRACT FROM AUTHOR]

Abstract: We consider two phase flow and transport in heterogeneous porous media with uncertain permeability distribution. The resulting transport uncertainty is assessed by means of multilevel Monte Carlo (MLMC). In contrast to the Monte Carlo (MC) method, which operates on one specific numerical grid with one numerical solver, MLMC samples from a hierarchy of grids or numerical solvers. In this work, the MLMC performance resulting from a hierarchy consisting of a finite volume transport solver and a streamline-based solver is compared to a purely grid-based hierarchy. Unlike the established grid-based MLMC method, our solver-based MLMC method operates on the same numerical grid and therefore avoids difficulties related to the upscaling of permeability fields or boundary conditions on coarser grids. For a two dimensional test case with log-normal permeability distribution, both MLMC approaches are compared to a MC reference run. At equivalent accuracy, significant speedups of MLMC with respect to MC are achieved. [Copyright &y& Elsevier]

The probability density function (PDF) of the local concentration of a contaminant, or tracer, is an important component of risk assessment in applications that involve flow in heterogeneous subsurface formations. In this paper, a novel joint velocity-concentration PDF method for tracer flow in highly heterogeneous porous media is introduced. The PDF formalism accounts for advective transport, pore-scale dispersion (PSD), and molecular diffusion. Low-order approximations (LOAs), which are usually obtained using a perturbation expansion, typically lead to Gaussian one-point velocity PDFs. Moreover, LOAs provide reasonable approximations for small log conductivity variances (i.e., σ Y2 < 1). For large σ Y2, however, the one-point velocity PDFs deviate significantly from the Gaussian distribution as demonstrated convincingly by several Monte Carlo (MC) simulation studies. Furthermore, the Lagrangian velocity statistics exhibit complex correlations that span a wide range of scales, including long-range correlations due to the formation of preferential flow paths. Both non-Gaussian PDFs and complex long-range correlations are accurately represented using Markovian velocity processes (MVPs) in the proposed joint PDF method. LOA methods can be generalized to some extent by presuming a certain shape for the concentration PDF (e.g., a β PDF fully characterized by the concentration mean and variance). The joint velocity-concentration PDF method proposed here does not require any closure assumptions on the shape of the marginal concentration PDF. The Eulerian joint PDF transport equation is solved numerically using a computationally efficient particle-based approach. The PDF method is validated with high-resolution MC reference data from Caroni and Fiorotto (2005) for saturated transport in velocity fields, which are stationary in space and time, for domains with σ Y2 = 0.05, 1, and 2 and Péclet numbers ranging from 100 to 10,000. PSD is modeled using constant anisotropic dispersion coefficients in both the reference MC computations and our PDF method. [ABSTRACT FROM AUTHOR]

Monte Carlo (MC) studies of flow in heterogeneous porous formations, in which the log-conductivity field is multi-Gaussian, have shown that as the log-conductivity variance σ Y2 increases beyond about 0.5, the one-point velocity probability density functions (PDFs) deviate significantly from Gaussian behavior. The velocity statistics become more complex due to the formation of preferential flow paths, or channels, as σ Y2 increases. Methods that employ low-order approximations (e.g., truncated perturbation expansions) are limited to small σ Y2 and are unable to represent the complex velocity statistics associated with σ Y2 > 1. Here a stochastic transport model for highly heterogeneous domains (i.e., σ Y2 > 1) is proposed. In the model, the Lagrangian velocity components of tracer particles are represented using continuous Markovian stochastic processes in time. The Markovian velocity process (MVP) model is described using a set of first-order ordinary and stochastic differential equations, which are easy to solve using a particle-based method. Once the MVP model is calibrated based on velocity statistics from MC simulations of the flow (hydraulic head and velocity), the MVP-based model can be used to describe the evolution of the tracer concentration field accurately and efficiently. Specifically, the MVP model is validated using MC simulations of longitudinal and transverse tracer spreading due to a point-like injection in the domain. We demonstrate that for σ Y2 > 1, the ensemble-averaged tracer cloud remains markedly non-Gaussian for relatively large travel distances from the point source. The MVP transport model captures this behavior and reproduces the velocity statistics quite accurately. [ABSTRACT FROM AUTHOR]

• Produce large random flow or pore networks from a given template or base network. • Spatial heterogeneity of nodes or pores are preserved from the base network. • Connectivity and throat-/pore-statistics are preserved as well. • Bounded and unbounded, that is periodic networks can be generated. • Tests involving homogeneous and highly heterogeneous networks are documented. Over the past years, tomographic scanning techniques like micro-CT have become popular for the acquisition of high-fidelity void-space geometries of natural porous media (e.g., Bultreys et al., 2016; Raeini et al., 2017). Limitations both in computing time and memory prohibit, however, direct numerical simulations of flow and transport in large resp. detailed sample geometries. Flow or pore networks derived from scans alleviate this limitation, but still necessitate a methodology to extrapolate to larger samples. In this work, we present a network generation algorithm that is particularly suited for heterogeneous irregular networks. While emulating from an existing base network new networks of equal or larger sizes, the outlined algorithm scales approximately linearly with the network node or pore count and maintains (1) node connectivity resp. pore coordination-number statistics, (2) geometrical pore/throat properties, as well as (3) the potentially inhomogeneous spatial clustering of pores. While existing methods address the first two properties, the third point is crucial especially in heterogeneous media to match flow/transport properties like the permeability that have a strong dependence on the spatial distance between connected pores. Moreover, the cubical networks generated by our algorithm are periodic in all spatial directions, thus eliminating topological boundary effects, which are not present in natural media. Bounded networks of arbitrary sizes can then be recovered by cutting the generated networks and thus flow/transport processes at larger scales can be studied while incorporating physically-based descriptions of pore-scale processes. [ABSTRACT FROM AUTHOR]

• Non-local Darcy theory with conductivity distribution is extended for finite domains. • A robust method for the extraction of the conductivity distribution is presented. • Validation of extended theory against data from networks with different connectivity. This work is concerned with the intricate interplay between node or pore pressures and connection or throat conductivities in flow or pore networks. A setting similar to pore networks is given by fracture networks. Recently, a non-local generalization of Darcy's law for flow and transport in porous media was presented in the context of unbounded or periodic pore networks. In this work, we first outline a robust method for the extraction of the hydraulic conductivity distribution, which is at the heart of the non-local Darcy formulation. Second, a theory for mean pressure and flow in bounded networks is outlined. Predictions of that theory are validated against numerical network results and it is demonstrated that the theory works well for networks with high connectivity involving pores with high coordination numbers. For other networks, improvements to the outlined theory are proposed and their accuracy is assessed. [ABSTRACT FROM AUTHOR]

We devise an efficient methodology to provide a universal statistical description of advection-dominated dispersion (Péclet→∞) in natural porous media including carbonates. First, we investigate the dispersion of tracer particles by direct numerical simulation (DNS). The transverse dispersion is found to be essentially determined by the tortuosity and it approaches a Fickian limit within a dozen characteristic scales. Longitudinal dispersion was found to be Fickian in the limit for bead packs and superdiffusive for all other natural media inspected. We demonstrate that the Lagrangian velocity correlation length is a quantity that characterizes the spatial variability for transport. Finally, a statistical transport model is presented that sheds light on the connection between pore-scale characteristics and the resulting macroscopic transport behavior. Our computationally efficient model accurately reproduces the transport behavior in longitudinal direction and approaches the Fickian limit in transverse direction. [ABSTRACT FROM AUTHOR]

Advection dominated transport processes in sub-surface formations are characterized by discontinuities in the fields of transported quantities, and realistic predictions are challenging for Eulerian transport schemes because they suffer from numerical diffusion. Henceforth, we have focused on developing a Lagrangian particle-tracking scheme for modeling advective solute transport in fractured media. To this end, we adopt an Embedded Discrete Fracture Model (EDFM) for fractured media with a permeable matrix. The flexibility to use non-conformal fracture–matrix discretizations makes EDFMs a compelling choice in field-scale flow problems. Unaffected by the numerical diffusion, Lagrangian transport schemes complement the potential of EDFMs by allowing the use of sufficiently large grid cell sizes for flow field computations. In an EDFM framework, the inter-continuum fluid mass exchange cannot be quantified by the particle trajectories/pathlines due to the unresolved fracture–matrix interfaces and different dimensionalities of the matrix and fracture discretizations. These constraints motivate the use of a stochastic particle-tracking scheme, and thus, we formulated a pathline-specific probability of inter-continuum particle transfer based on mass conservation of an elementary solute/fluid mass. The particle's transfer probability is calculated for the maximum residence time period in its associated fracture/matrix control volume, thus making the scheme time-adaptive. In addition, a conditional residence time distribution was derived, which dictates the timestamp of the particle transfer. First, we showcase that the tracking scheme preserves the initially homogeneous solute concentration field, suggesting that the probabilistic rules are consistent with the inter-continuum fluxes. Additionally, we illustrate the estimation of an evolving solute plume and compare the results with those of an Eulerian counterpart. The presented stochastic approach enables straightforward formulations of Lagrangian models for dynamic and sub-grid processes, e.g., solute interactions with the solid phase, the mapping of their effects onto large scale transport and additionally, be included in random walk models for dispersion. • Particle-tracking scheme suited for EDFMs, involving probabilistic particle transfers. • Formulation of particle-transfer probability and residence time distribution. • The probabilistic scheme is consistent with the fluxes and thus, conservative. [ABSTRACT FROM AUTHOR]

Abstract: Monte Carlo (MC) is a well known method for quantifying uncertainty arising for example in subsurface flow problems. Although robust and easy to implement, MC suffers from slow convergence. Extending MC by means of multigrid techniques yields the multilevel Monte Carlo (MLMC) method. MLMC has proven to greatly accelerate MC for several applications including stochastic ordinary differential equations in finance, elliptic stochastic partial differential equations and also hyperbolic problems. In this study, MLMC is combined with a streamline-based solver to assess uncertain two phase flow and Buckley–Leverett transport in random heterogeneous porous media. The performance of MLMC is compared to MC for a two dimensional reservoir with a multi-point Gaussian logarithmic permeability field. The influence of the variance and the correlation length of the logarithmic permeability on the MLMC performance is studied. [Copyright &y& Elsevier]

Solute transport observed in subsurface formations shows complex behavior, particularly in the presence of fractures. In this work, we focus on formations with fractures that are smaller compared to the domain of interest and which are distributed in a heterogeneous matrix at densities below the percolation threshold. We developed a simplified Lagrangian approach to characterize and predict advective transport in 2-D domains containing differently oriented fractures. To this end, we performed Monte Carlo simulation (MCS) studies using ensembles of random domain realizations and gathered/analyzed tracer particle displacement statistics. The series of displacement steps were defined by the locations of particles entering the matrix after traversing through one or several connected fractures. Then, we identify key correlation structures in the evolution of and between displacement step coordinates, namely, the step length, its orientation and the traverse time. Subsequently, a correlated random walk model was derived which is able to accurately reproduce longitudinal and transverse macrodispersion as recorded in the MCS. Finally, we explored the predictive capabilities of our stochastic model by simulating macrodispersion in stratified media composed of heterogeneous zones with different fracture orientations. • Statistical analysis of particle motion in 2-D heterogeneous continua with fractures. • Guided by the statistical analysis, development of a stochastic model for macrodispersion. • Longitudinal and transverse plume spreading are captured well by the model. [ABSTRACT FROM AUTHOR]

Abstract: The Karhunen–Loeve (KL) decomposition and the polynomial chaos (PC) expansion are elegant and efficient tools for uncertainty propagation in porous media. Over recent years, KL/PC-based frameworks have successfully been applied in several contributions for the flow problem in the subsurface context. It was also shown, however, that the accurate solution of the transport problem with KL/PC techniques is more challenging. We propose a framework that utilizes KL/PC in combination with sparse Smolyak quadrature for the flow problem only. In a subsequent step, a Lagrangian sampling technique is used for transport. The flow field samples are calculated based on a PC expansion derived from the solutions at relatively few quadrature points. To increase the computational efficiency of the PC-based flow field sampling, a new reduction method is applied. For advection dominated transport scenarios, where a Lagrangian approach is applicable, the proposed PC/Monte Carlo method (PCMCM) is very efficient and avoids accuracy problems that arise when applying KL/PC techniques to both flow and transport. The applicability of PCMCM is demonstrated for transport simulations in multivariate Gaussian log-conductivity fields that are unconditional and conditional on conductivity measurements. [Copyright &y& Elsevier]