Renormalization group methods in subsurface hydrology: overview and applications in hydraulic conductivity upscaling

The renormalization group (RG) approach is a powerful theoretical framework, more suitable for upscaling strong heterogeneity than low-order perturbation expansions. Applications of RG methods in subsurface hydrology include the calculation of (1) macroscopic transport parameters such as effective and equivalent hydraulic conductivity and dispersion coefficients, and (2) anomalous exponents characterizing the dispersion of contaminants due to long-range conductivity correlations or broad (heavy-tailed) distributions of the groundwater velocity. First, we review the main ideas of RG methods and their hydrological applications. Then, we focus on the hydraulic conductivity in saturated porous media with isotropic lognormal heterogeneity, and we present an RG calculation based on the replica method. The RG analysis gives rigorous support to the exponential conjecture for the effective hydraulic conductivity [Water Resour. Res. 19 (1) (1983) 161]. Using numerical simulations in two dimensions with a bimodal conductivity distribution, we demonstrate that the exponential expression is not suitable for all types of heterogeneity. We also introduce an RG coarse-grained conductivity and investigate its applications in estimating the conductivity of blocks or flow domains with finite size. Finally, we define the fractional effective dimension, and we show that it justifies fractal exponents in the range 1−2/dα<1 (where d is the actual medium dimension) in the geostatistical power average.     

Another look at the conceptual fundamentals of porous media upscaling

We suggest a critical look at the epistemic foundations of the porous media upscaling problem that focuses on conceptual processes at work and not merely on form manipulations. We explore the way in which critical aspects of scientific methodology make their appearance in the upscaling context, thus generating useful effective parameters in practice. The fons et origo of our approach is a conceptual blending of knowledge states that requires the revision of the traditional method of scientific argument underlying most upscaling techniques. By contrast to previous techniques, the scientific reasoning of the proposed upscaling approach is based on a stochastic model that involves teleologic solutions and stochastic logic integration principles. The syllogistic form of the approach has important advantages over the traditional reasoning scheme of porous media upscaling, such as: it allows the rigorous derivation of the joint probability distributions of hydraulic gradients and conductivities across space; it imposes no restriction on the functional form of the effective parameters or the shape of the probability laws governing the random media (non-Gaussian distributions, multiple-point statistics and non-linear models are automatically incorporated); it relies on sound methodological principles rather than being ad hoc; and it offers the rational means for integrating the multifarious core knowledge bases and uncertain site-specific information sources about the subsurface system. Previous upscaling results are derived as special cases of the proposed upscaling approach under limited conditions of porous media flow, a fact that further demonstrates the generalization power of the approach. Our hope is that looking at the upscaling problem in this novel way will direct further attention to the methodological exploration of the problem at the length and the detail that it deserves.I would like to thank Drs. A. Kolovos and D.T. Hristopulos for their valuable comments. The work was supported by grants from the Army Research Office (Grant no. DAAG55–98–1-0289) and the National Institute of Environmental Health Sciences (P42-ES05948 & P30-ES10126).     

渗流方程的三维粗化算法

本文提出了地下流体渗流问题的三维解粗化算法,在粗网格内流体压强分布用直接解法求解三维渗流方程,用这些解计算粗网格的等效渗透率,在流体流速大的区域仍采用精细网格的计算方法.用所得等效渗透率计算了粗化网格的渗流场的压强分布,结果表明渗流方程的三维粗化解非常逼近采用精细网格的解,但计算的速度比采用精细网格提高了100多倍.     

Reactive transport upscaling at the Darcy scale: A new flow rate based approach raises the unsolved issue of porosity upscaling

This article demonstrates that permeability upscaling, which can require complex techniques, is not necessary to significantly decrease the CPU time in reactive transport modeling. CPU time depends more on the geochemistry than the flow calculation. Flow rate upscaling is proposed as an alternate method to permeability upscaling, which is more suited to time-consuming flow resolution. To apply this method, a finite volume approach is most convenient.Considering the equality of flow as the equivalence criterion, when the coarse grid overlays the fine grid, flow rate upscaling leads, by construction, to the exact results, whereas the accuracy of permeability upscaling methods often depends on specific conditions. Some focus is put on the limitations of a common permeability upscaling technique, the simplified renormalization. In stationary flow, the gain in CPU time is the same for both flow rate upscaling and permeability upscaling. In transient flow, flow rate upscaling is slightly less time-efficient but the ratio between both CPU times decreases when the geochemistry is more complex.Working with an accurate flow rate field in the upscaled case reveals that porosity upscaling is a surprisingly tricky issue. Solution mixing is induced and residence times can be significantly affected. These changes have potentially important consequences on reactive transport modeling. They are not specific to the flow rate upscaling method; they are a general issue. Some simplified cases, assuming a homogeneous mineralogy, are examined. At this stage, a simple heuristic method is proposed, which yields reliable results under particular conditions (high heterogeneity). Porosity upscaling remains an open research field.     

Time-of-flight (TOF)-based two-phase upscaling for subsurface flow and transport

Subsurface formations are characterized by heterogeneity over multiple length scales, which can have a strong impact on flow and transport. In this paper, we present a new upscaling approach, based on time-of-flight (TOF), to generate upscaled two-phase flow functions. The method focuses on more accurate representations of local saturation boundary conditions, which are found to have a dominant impact (in comparison to the pressure boundary conditions) on the upscaled two-phase flow models. The TOF-based upscaling approach effectively incorporates single-phase flow and transport information into local upscaling calculations, accounting for the global flow effects on saturation, as well as the local variations due to subgrid heterogeneity. The method can be categorized into quasi-global upscaling techniques, as the global single-phase flow and transport information is incorporated in the local boundary conditions. The TOF-based two-phase upscaling can be readily integrated into any existing local two-phase upscaling framework, thus more flexible than local–global two-phase upscaling approaches developed recently. The method was applied to permeability fields with different correlation lengths and various fluid-mobility ratios. It was shown that the new method consistently outperforms existing local two-phase upscaling techniques, including recently developed methods with improved local boundary conditions (such as effective flux boundary conditions), and provides accurate coarse-scale models for both flow and transport.     

A new upscaling method for fractured porous media

We present a method to determine equivalent permeability of fractured porous media. Inspired by the previous flow-based upscaling methods, we use a multi-boundary integration approach to compute flow rates within fractures. We apply a recently developed multi-point flux approximation Finite Volume method for discrete fracture model simulation. The method is verified by upscaling an arbitrarily oriented fracture which is crossing a Cartesian grid. We demonstrate the method by applying it to a long fracture, a fracture network and the fracture network with different matrix permeabilities. The equivalent permeability tensors of a long fracture crossing Cartesian grids are symmetric, and have identical values. The application to the fracture network case with increasing matrix permeabilities shows that the matrix permeability influences more the diagonal terms of the equivalent permeability tensor than the off-diagonal terms, but the off-diagonal terms remain important to correctly assess the flow field.     

Use of the Fourier-Laplace transform and of diagrammatical methods to interpret pumping tests in heterogeneous reservoirs

Advances in computer power and in reservoir characterization allow simulation of pressure transients in complex reservoirs generated stochastically. Generally, interpretation of these transient gives useful information about the reservoir hydraulic properties: a major goal is to interpret these transients in the stochastic context. First we ensemble average the pressure over all the random permeability field realizations to derive an equation which drives the ensemble averaged pressure. We use the Fourier transform in space and the Laplace transform in time, in conjuction with a perturbation series expansion in successive powers of the permeability fluctuations to obtain an explicit solution. The Nth order term of this series involves the hydrodynamic interaction between N permeability heterogeneities and after averaging we obtain an expansion containing correlation functions of permeability fluctuations of increasing order.Next, Feynman graphs are introduced allowing a more attractive graphical interpretation of the perturbation series. Then series summation techniques are employed to reduce the graph number to be summed at each order of the fluctuation expansion. This in turn gives useful physical insights on the homogenization processes involved. In particular, it is shown that the sum of the so-called ‘one-particle irreducible graphs’ gives the kernel of a linear integro-differential equation obeyed by the ensemble average pressure. All the information about the heterogeneity structure is contained in this renormalized kernel, which is a limited range function.This equation on its own is the starting point of useful asymptotic results and approximations. In particular it is shown that interpretation of pumping tests yields the steady-state equivalent permeability after a sufficiently long time for an infinite reservoir, as expected.     

Convective mixing in formations with horizontal barriers

It has been shown that convective mixing in porous media flow is important for applications such as saltwater intrusion and geological storage of carbon dioxide. In the latter case, dissolution from the injected phase to the resident brine is assisted by convective mixing, which leads to enhanced storage security through reduced buoyancy. Here, we focus on the effect of horizontal barriers on the efficiency of convective mixing. Previous investigations of the effect of heterogeneity on mixing efficiency have focused on random permeability fields or barriers of small extent compared to the intrinsic finger wavelength. The effect of horizontal barriers of larger extent, such as mudstone inclusions or thin shale deposits, has not been given sufficient attention. We perform detailed numerical investigations to represent the continuous solution of this problem in semi-infinite domains with barriers arranged in a periodic manner. The results show that mass flux into the domain, which is a measure of the efficiency of redistribution of the solute, is inversely proportional to the barrier length and proportional to the horizontal and vertical aperture between the barriers, for the cases studied. The flow structure is complex, and it depends not only on the total area of barriers but also largely on the distribution of barriers. Therefore, neither simple analytical models nor simple upscaling methods that lack information about the flow paths, can be used to predict the behavior. However, we compute the effective vertical permeability by flow-based upscaling and show that it can be used to directly obtain a first-order approximation to the mass flux into the domain.     

On the predictivity of pore-scale simulations: Estimating uncertainties with multilevel Monte Carlo

A fast method with tunable accuracy is proposed to estimate errors and uncertainties in pore-scale and Digital Rock Physics (DRP) problems. The overall predictivity of these studies can be, in fact, hindered by many factors including sample heterogeneity, computational and imaging limitations, model inadequacy and not perfectly known physical parameters. The typical objective of pore-scale studies is the estimation of macroscopic effective parameters such as permeability, effective diffusivity and hydrodynamic dispersion. However, these are often non-deterministic quantities (i.e., results obtained for specific pore-scale sample and setup are not totally reproducible by another “equivalent” sample and setup). The stochastic nature can arise due to the multi-scale heterogeneity, the computational and experimental limitations in considering large samples, and the complexity of the physical models. These approximations, in fact, introduce an error that, being dependent on a large number of complex factors, can be modeled as random. We propose a general simulation tool, based on multilevel Monte Carlo, that can reduce drastically the computational cost needed for computing accurate statistics of effective parameters and other quantities of interest, under any of these random errors. This is, to our knowledge, the first attempt to include Uncertainty Quantification (UQ) in pore-scale physics and simulation. The method can also provide estimates of the discretization error and it is tested on three-dimensional transport problems in heterogeneous materials, where the sampling procedure is done by generation algorithms able to reproduce realistic consolidated and unconsolidated random sphere and ellipsoid packings and arrangements. A totally automatic workflow is developed in an open-source code [1], that include rigid body physics and random packing algorithms, unstructured mesh discretization, finite volume solvers, extrapolation and post-processing techniques. The proposed method can be efficiently used in many porous media applications for problems such as stochastic homogenization/upscaling, propagation of uncertainty from microscopic fluid and rock properties to macro-scale parameters, robust estimation of Representative Elementary Volume size for arbitrary physics.     

Real gas flow through heterogeneous porous media: theoretical aspects of upscaling

We consider the problem of upscaling transient real gas flow through heterogeneous bounded reservoirs. One of the commonly used methods for deriving effective permeabilities is based on stochastic averaging of nonlinear flow equations. Such an approach, however, would require rather restrictive assumptions about pressure-dependent coefficients. Instead, we use Kirchhoff transformation to linearize the governing stochastic equations prior to their averaging. The linearized problem is similar to that used in stochastic analysis of groundwater flow. We discuss the effects of temporal localization of the nonlocal averaged Darcy's law, as well as boundary effects, on the upscaled gas permeability. Extension of the results obtained by means of small perturbation analysis to highly heterogeneous porous formations is also discussed.     

Estimates of effective permeability for non-Newtonian fluid flow in randomly heterogeneous porous media

An understanding of the interplay between non-Newtonian effects in porous media flow and field-scale domain heterogeneity is of great importance in several engineering and geological applications. Here we present a simplified approach to the derivation of an effective permeability for flow of a purely viscous power–law fluid with flow behavior index n in a randomly heterogeneous porous domain subject to a uniform pressure gradient. A standard form of the flow law generalizing the Darcy’s law to non-Newtonian fluids is adopted, with the permeability coefficient being the only source of randomness. The natural logarithm of the permeability is considered a spatially homogeneous and correlated Gaussian random field. Under the ergodic hypothesis, an effective permeability is first derived for two limit 1-D flow geometries: flow parallel to permeability variation (serial-type layers), and flow transverse to permeability variation (parallel-type layers). The effective permeability of a 2-D or 3-D isotropic domain is conjectured to be a power average of 1-D results, generalizing results valid for Newtonian fluids under the validity of Darcy’s law; the conjecture is validated comparing our results with previous literature findings. The conjecture is then extended, allowing the exponents of the power averaging to be functions of the flow behavior index. For Newtonian flow, novel expressions for the effective permeability reduce to those derived in the past. The effective permeability is shown to be a function of flow dimensionality, domain heterogeneity, and flow behavior index. The impact of heterogeneity is significant, especially for shear-thinning fluids with a low flow behavior index, which tend to exhibit channeling behavior.     

Upscaling of Forchheimer flows

In this work we propose upscaling method for nonlinear Forchheimer flow in heterogeneous porous media. The generalized Forchheimer law is considered for incompressible and slightly-compressible single-phase flows. We use recently developed analytical results (Aulisa et al., 2009) [1] and formulate the resulting system in terms of a degenerate nonlinear flow equation for the pressure with the nonlinearity depending on the pressure gradient. The coarse scale parameters for the steady state problem are determined so that the volumetric average of velocity of the flow in the domain on fine scale and on coarse scale are close. A flow-based coarsening approach is used, where the equivalent permeability tensor is first evaluated following streamline methods for linear cases, and modified in order to take into account the nonlinear effects. Compared to previous works (Garibotti and Peszynska, 2009) [2], (Durlofsky and Karimi-Fard) [3], this approach can be combined with rigorous mathematical upscaling theory for monotone operators, (Efendiev et al., 2004) [4], using our recent theoretical results (Aulisa et al., 2009) [1]. The developed upscaling algorithm for nonlinear steady state problems is effectively used for variety of heterogeneities in the domain of computation. Direct numerical computations for average velocity and productivity index justify the usage of the coarse scale parameters obtained for the special steady state case in the fully transient problem. For nonlinear case analytical upscaling formulas in stratified domain are obtained. Numerical results were compared to these analytical formulas and proved to be highly accurate.     

Binary upscaling—the role of connectivity and a new formula

Although recognized as important, measures of connectivity (i.e. the existence of high-conductivity paths that increase flow and allow for early solute arrival) have not yet been incorporated into methods for upscaling hydraulic conductivities of porous media. We present and evaluate a binary upscaling formula that utilizes connectivity information. The upscaled hydraulic conductivity (K) of binary media is determined as a function of the proportions and conductivities of the two materials, the geometry of the inclusions, and the mean distance between them. The use of a phase interchange theorem renders the formula equally applicable to two-dimensional media with inclusions of low K and high K as compared with the matrix. The new upscaling formula is tested on two-dimensional binary random fields spanning a broad range of spatial correlation structures and conductivity contrasts. The computed effective conductivities are compared to what is obtained using self-consistent effective medium theory, the coated ellipsoids approximation, and to a streamline approach. It is shown that, although simple, the proposed formula performs better than available methods for binary upscaling. The use of connectivity information leads to significantly improved behavior close to the percolation threshold. The proposed upscaling formula depends exclusively on parameters that are obtainable from field investigations.     

Flow and entrapment of dense nonaqueous phase liquids in physically and chemically heterogeneous aquifer formations

The migration and entrapment of dense nonaqueous phase liquids (DNAPLs) in aquifer formations is typically believed to be controlled by physical heterogeneities. This belief is based upon the assumption that permeability and capillary properties are determined by the soil texture. Capillarity and relative permeability, however, will also depend on porous medium wettability characteristics. This wettability may vary spatially in a formation due to variations in aqueous phase chemistry, contaminant aging, and/or variations in mineralogy and organic matter distributions. In this work, a two-dimensional multiphase flow simulator is modified to simulate coupled physical and chemical formation heterogeneity. To model physical heterogeneity, a spatially correlated permeability field is generated, and then related to the capillary pressure-saturation function according to Leverett scaling. Spatial variability of porous medium wettability is assumed to be correlated with the natural logarithm of the intrinsic permeability. The influence of wettability on the hysteretic hydraulic property relations is also modeled. The simulator is then employed to investigate the potential influence of coupled physical and chemical heterogeneity on DNAPL flow and entrapment. For reasonable ranges of wettability characteristics, simulations demonstrate that spatial variations in wettability can have a dramatic impact on DNAPL distributions. Higher organic saturations, increased lateral spreading, and decreased depth of infiltration were predicted when the contact angle was varied spatially. When chemical heterogeneity was defined by spatial variation of organic-wet solid fractions (fractional wettability porous media), however, the resultant organic saturation distributions were more similar to those for perfectly water-wet media, due to saturation dependent wettability effects on the hydraulic property relations.     

Transformation of spatial and perturbation derivatives of travel time at a curved interface between two arbitrary media

We consider the partial derivatives of travel time with respect to both spatial coordinates and perturbation parameters. These derivatives are very important in studying wave propagation and have already found various applications in smooth media without interfaces. In order to extend the applications to media composed of layers and blocks, we derive the explicit equations for transforming these travel–time derivatives of arbitrary orders at a general smooth curved interface between two arbitrary media. The equations are applicable to both real–valued and complex–valued travel time. The equations are expressed in terms of a general Hamiltonian function and are applicable to the transformation of travel–time derivatives in both isotropic and anisotropic media. The interface is specified by an implicit equation. No local coordinates are needed for the transformation.     

Upscaling of flow in heterogeneous porous formations: Critical examination and issues of principle

We consider heterogeneous media whose properties vary in space and particularly aquifers whose hydraulic conductivity K may change by orders of magnitude in the same formation. Upscaling of conductivity in models of aquifer flow is needed in order to reduce the numerical burden, especially when modeling flow in heterogeneous aquifers of 3D random structure. Also, in many applications the interest is in average values of the dependent variables over scales larger or comparable to the conductivity length scales. Assigning values of the conductivity K_b to averaging domains, or computational blocks, is the topic of a large body of literature, the problem being of wide interest in various branches of physics and engineering. It is clear that upscaling causes loss of information and at best it can render a good approximation of the fine scale solution after averaging it over the blocks.The present article focuses on upscaling approaches dealing with random media. It is not meant to be a review paper, its main scope being to elucidate a few issues of principle and to briefly discuss open questions. We show that upscaling can be usually achieved only approximately, and the result may depend on the particular upscaling scheme adopted. The typically scarce information on the statistical structure of the fine-scale conductivity imposes a strong limitation to the upscaling problem. Also, local upscaling is not possible in nonuniform mean flows, for which the upscaled conductivity tensor is generally nonlocal and it depends on the domain geometry and the boundary conditions. These and other limitations are discussed, as well as other open topics deserving further investigation.     

Dynamic analyses of multilayered poroelastic media using the generalized transfer matrix method

Since the generalized transfer matrix method overcomes the intrinsic instability of the Thomson–Haskell transfer matrix method for both high frequencies and/or thick layers, it can produce stable and accurate solutions for the dynamic analysis of viscoelastic media as well as anisotropic media. This paper extends the generalized transfer matrix method to the dynamic analysis of multilayered poroelastic media. Main improvements include the presentation of the concisely explicit general solutions for the dynamic analysis of multilayered poroelastic media and the derivation of an analytical inversion of 8×8 order layer matrix corresponding to the general solutions. The explicit solutions are valid for the dynamic analysis of one-dimensional, two-dimensional and three-dimensional poroelastic medium problems. In addition, an efficient recursive scheme is proposed for accurate determination of the equivalent interface sources for multilayered poroelastic media due to excitation by a source at an arbitrary depth. For the dynamic analysis of multilayered poroelastic media, the generalized transfer matrix method recursively transfers both the interface stiffness matrix and equivalent source starting from the bottom half space to the top layer, without resort to the numerical solution of the assembled global equations as the exact stiffness matrix method does. While keeping the simplicity of the Thomson–Haskell transfer matrix method, the generalized transfer matrix method is both accurate and stable. The related numerical examples have demonstrated that the generalized transfer matrix method is an alternative approach to conducting the dynamic analysis of multilayered poroelastic media.     

Coupling upscaling finite element method for consolidation analysis of heterogeneous saturated porous media

The coupling upscaling finite element method is developed for solving the coupling problems of deformation and consolidation of heterogeneous saturated porous media under external loading conditions. The method couples two kinds of fully developed methodologies together, i.e., the numerical techniques developed for calculating the apparent and effective physical properties of the heterogeneous media and the upscaling techniques developed for simulating the fluid flow and mass transport properties in heterogeneous porous media. Equivalent permeability tensors and equivalent elastic modulus tensors are calculated for every coarse grid block in the coarse-scale model of the heterogeneous saturated porous media. Moreover, an oversampling technique is introduced to improve the calculation accuracy of the equivalent elastic modulus tensors. A numerical integration process is performed over the fine mesh within every coarse grid element to capture the small scale information induced by non-uniform scalar field properties such as density, compressibility, etc. Numerical experiments are carried out to examine the accuracy of the developed method. It shows that the numerical results obtained by the coupling upscaling finite element method on the coarse-scale models fit fairly well with the reference solutions obtained by traditional finite element method on the fine-scale models. Moreover, this method gets more accurate coarse-scale results than the previously developed coupling multiscale finite element method for solving this kind of coupling problems though it cannot recover the fine-scale solutions. At the same time, the method developed reduces dramatically the computing effort in both CPU time and memory for solving the transient problems, and therefore more large and computational-demanding coupling problems can be solved by computers.     

Upscaling hydraulic conductivity based on the topology of the sub-scale structure

The hydraulic conductivity of heterogeneous porous media depends on the distribution function and the geometry of local conductivities at the smaller scale. There are various approaches to estimate the effective conductivity K_eff at the larger scale based on information about the small scale heterogeneity. A critical geometric property in this ‘upscaling’ procedure is the spatial connectivity of the small-scale conductivities. We present an approach based on the Euler-number to quantify the topological properties of heterogeneous conductivity fields, and we derive two key parameters which are used to estimate K_eff. The required coefficients for the upscaling formula are obtained by regression based on numerical simulations of various heterogeneous fields. They are found to be generally valid for various different isotropic structures. The effective unsaturated conductivity function K_eff (ψ_m) could be predicted satisfactorily. We compare our approach with an alternative based on percolation theory and critical path analysis which yield the same type of topological parameters. An advantage of using the Euler-number in comparison to percolation theory is the fact that it can be obtained from local measurements without the need to analyze the entire structure. We found that for the heterogeneous field used in this study both methods are equivalent.