Jean-Raynald De Dreuzy

Hydraulic properties of two‐dimensional random fracture networks following a power law length distribution: 1. Effective connectivity

Natural fracture networks involve a very broad range of fractures of variable lengths and apertures, modeled, in general, by a power law length distribution and a lognormal aperture distribution. The objective of this two-part paper is to characterize the permeability variations as well as the relevant flow structure of two-dimensional isotropic models of fracture networks as determined by the fracture length and aperture distributions and by the other parameters of the model (such as density and scale). In this paper we study the sole influence of the fracture length distribution on permeability by assigning the same aperture to all fractures. In the following paper [de Dreuzy et al., this issue] we study the more general case of networks in which fractures have both length and aperture distributions. Theoretical and numerical studies show that the hydraulic properties of power law length fracture networks can be classified into three types of simplified model. If a power law length distribution n (l) ∼ l−a is used in the network design, the classical percolation model based on a population of small fractures is applicable for a power law exponent a higher than 3. For a lower than 2, on the contrary, the applicable model is the one made up of the largest fractures of the network. Between these two limits, i.e., for a in the range 2–3, neither of the previous simplified models can be applied so that a simplified two-scale structure is proposed. For this latter model the crossover scale is the classical correlation length, defined in the percolation theory, above which networks can be homogenized and below which networks have a multipath, multisegment structure. Moreover, the determination of the effective fracture length range, within which fractures significantly contribute to flow, corroborates the relevance of the previous models and clarifies their geometrical characteristics. Finally, whatever the exponent a, the sole significant scale effect is a decrease of the equivalent permeability for networks below or at percolation threshold.

Hydraulic properties of two-dimensional random fracture networks following a power law length distribution: 2. Permeability of networks based on lognormal distribution of apertures

The broad length and aperture distributions are two characteristics of the heterogeneity of fractured media that make difficult, and even theoretically irrelevant, the application of homogenization techniques. We propose a numerical and theoretical study of the consequences of these two properties on the permeability of bidimensional synthetic fracture networks. We use a power law for the model of length distribution and a lognormal model for aperture distribution. We have especially studied the two endmost models for which length and aperture are (1) independent and (2) perfectly positively correlated. For the model without correlation between length and aperture we show that the permeability can be adequately characterized by a power-averaging function whose parameters are detailed in the text. In contrast, for the model with correlation we show that the prevailing parameter is the correlation when the power law length exponent a is lower than 3, whereas the random structure of the network is a second-order parameter. We also determine the permeability scaling and the scale dependence of the flow pattern structure. Three types of scale effects are found, depending exclusively on the geometrical properties of the network, i.e., on the length distribution parameter a. For a larger than 3, permeability decreases for scales below a definite correlation length and becomes constant above. We show in this case that a correlation between length and aperture does not fundamentally change the permeability model. In all other cases the correlation entails much larger-scale effects. For a in the range 1-3 in the case of an absence of correlation and for a in the range 2-3 in the case of correlation, permeability increases and tends to a limit, whereas the flow structure is channeled when permeability increases and tends to homogenize when permeability tends to its limit. We note that this permeability model is consistent with natural observations of permeability scaling. For a in the range 1-2, in the case of correlation, permeability increases with scale with no apparent limit. We characterize the channeled flow pattern, and we show that permeability may increase even when flow is distributed in several independent structures.

Asymptotic dispersion in 2D heterogeneous porous media determined by parallel numerical simulations

We determine the asymptotic dispersion coefficients in 2D exponentially correlated lognormally distributed permeability fields by using parallel computing. Fluid flow is computed by solving the flow equation discretized on a regular grid and transport triggered by advection and diffusion is simulated by a particle tracker. To obtain a well-defined asymptotic regime under ergodic conditions (initial plume size much larger than the correlation length of the permeability field), the characteristic dimension of the simulated computational domains was of the order of 103 correlation lengths with a resolution of ten cells by correlation length. We determine numerically the asymptotic effective longitudinal and transverse dispersion coefficients over 100 simulations for a broad range of heterogeneities s 2 [0, 9], where s 2 is the lognormal permeability variance. For purely advective transport, the asymptotic longitudinal dispersion coefficient depends linearly on s 2 for s 2 1 and the asymptotic transverse dispersion coefficient is zero. Addition of homogeneous isotropic diffusion induces an increase of transverse dispersion and a decrease of longitudinal dispersion.

Flow Simulation in Three-Dimensional Discrete Fracture Networks

In fractured rocks, fluid flows mostly within a complex arrangement of fractures. Both the fracture network structure and its hydraulic properties are determined at first order by the broad range of fracture lengths and densities. To handle the observed wide variety of fracture properties and the lack of direct fracture visualization, we develop a general and efficient stochastic numerical model for discrete fracture networks (DFNs) in a three-dimensional (3D) computational domain. We present an original conforming mesh generation method addressing the penalizing configurations stemming from close fractures and acute angles between fracture intersections. Flows are subsequently computed by using a mixed hybrid finite element (MHFE) method. The lack of direct fracture knowledge is treated by Monte-Carlo simulations requiring simulations with a large number of networks with various characteristics. We analyze the complexity in size and in time for the computation of flow in 3D DFNs meshed with our method and compare with the complexities for 2D rectangular domains meshed with a regular grid. We find out that complexity in size is similar whereas complexity in time is slightly larger for DFNs than for 2D regular domains.

Transport and intersection mixing in random fracture networks with power law length distributions

The importance of fracture intersection mixing rules, complete mixing and streamline routing, on simulated solute migration patterns in random fracture networks is assessed. For this purpose, and based on geological evidence, two-dimensional model networks having power law fracture length distributions and lognormal fracture permeability distributions are considered. Different fracture network structures are accounted for by the power law length distribution, ranging from networks composed of infinite length fractures to percolation networks with constant length fractures. Comparison of solute particle statistics shows that there is no significant difference between the bulk transport properties calculated with the two mixing rules. In fact, it is found that the choice of mixing assumptions has a significant influence in less than 5% of the total number of fracture intersections in most fracture networks. This result can be attributed to the small mean effective coordination number, defined as the number of branches connected to an intersection with nonzero flux. It is concluded that solute transport in fracture networks is strongly influenced by variability and uncertainty in parameters defining the geometrical structure of networks and that, by comparison, the choice of mixing rules at fracture intersections is of little importance.

Synthetic benchmark for modeling flow in 3D fractured media

Intensity and localization of flows in fractured media have promoted the development of a large range of different modeling approaches including Discrete Fracture Networks, pipe networks and equivalent continuous media. While benchmarked usually within site studies, we propose an alternative numerical benchmark based on highly-resolved Discrete Fracture Networks (DFNs) and on a stochastic approach. Test cases are built on fractures of different lengths, orientations, aspect ratios and hydraulic apertures, issuing the broad ranges of topological structures and hydraulic properties classically observed. We present 18 DFN cases, with 10 random simulations by case. These 180 DFN structures are provided and fully documented. They display a representative variety of the configurations that challenge the numerical methods at the different stages of discretization, mesh generation and system solving. Using a previously assessed mixed hybrid finite element method (Erhel et al., 2009a), we systematically provide reference flow and head solutions. Because CPU and memory requirements stem mainly from system solving, we study direct and iterative sparse linear solvers. We show that the most cpu-time efficient method is a direct multifrontal method for small systems, while conjugate gradient preconditioned by algebraic multrigrid is more relevant at larger sizes. Available results can be used further as references for building up alternative numerical and physical models in both directions of improving accuracy and efficiency.

Random Walk Methods for Modeling Hydrodynamic Transport in Porous and Fractured Media from Pore to Reservoir Scale

Random walk (RW) methods are recurring Monte Carlo methods used to model convective and diffusive transport in complex heterogeneous media. Many applications can be found, including fluid mechanic, hydrology and chemical reactors modeling. These methods are easy to implement, very versatile and flexible enough to become appealing for many applications because they generally overlook or deeply simplify the building of explicit complex meshes required by deterministic methods. RW provides a good physical understanding of the interactions between the space scales of heterogeneities and the transport phenomena under consideration. In addition, they can result in efficient upscaling methods, especially in the context of flow and transport in fractured media. In the present study, we review the applications of RW to several situations that cope with diverse spatial scales and different insights into upscaling problems. The advantages and downsides of RW are also discussed, thus providing a few avenues for further works and applications.

Conditioning of stochastic 3-D fracture networks to hydrological and geophysical data

The geometry and connectivity of fractures exert a strong influence on the flow and transport properties of fracture networks. We present a novel approach to stochastically generate three-dimensional discrete networks of connected fractures that are conditioned to hydrological and geophysical data. A hierarchical rejection sampling algorithm is used to draw realizations from the posterior probability density function at different conditioning levels. The method is applied to a well-studied granitic formation using data acquired within two boreholes located 6 m apart. The prior models include 27 fractures with their geometry (position and orientation) bounded by information derived from single-hole ground-penetrating radar (GPR) data acquired during saline tracer tests and optical televiewer logs. Eleven cross-hole hydraulic connections between fractures in neighboring boreholes and the order in which the tracer arrives at different fractures are used for conditioning. Furthermore, the networks are conditioned to the observed relative hydraulic importance of the different hydraulic connections by numerically simulating the flow response. Among the conditioning data considered, constraints on the relative flow contributions were the most effective in determining the variability among the network realizations. Nevertheless, we find that the posterior model space is strongly determined by the imposed prior bounds. Strong prior bounds were derived from GPR measurements and helped to make the approach computationally feasible. We analyze a set of 230 posterior realizations that reproduce all data given their uncertainties assuming the same uniform transmissivity in all fractures. The posterior models provide valuable statistics on length scales and density of connected fractures, as well as their connectivity. In an additional analysis, effective transmissivity estimates of the posterior realizations indicate a strong influence of the DFN structure, in that it induces large variations of equivalent transmissivities between realizations. The transmissivity estimates agree well with previous estimates at the site based on pumping, flowmeter and temperature data.

Particle-tracking simulations of anomalous transport in hierarchically fractured rocks

Complex topology of fracture networks and interactions between transport processes in a fracture and the ambient un-fractured rock (matrix) combine to render modeling solute transport in fractured media a challenge. Classical approaches rely on both strong assumptions of either limited or full diffusion of solutes in the matrix and simplified fracture configurations. We analyze fracture-matrix transport in two-dimensional Sierpinski lattice structures, which display a wide range of matrix block sizes. The analysis is conducted in several transport regimes that are limited by either diffusion or block sizes. Our simulation results can be used to validate the simplifying assumptions that underpin classical analytical solutions and to benchmark other numerical methods. They also demonstrate that both hydraulic and structural properties of fractured rocks control the residence time distribution.

Use of power averaging for quantifying the influence of structure organization on permeability upscaling in on-lattice networks under mean parallel flow

with the local permeability distribution variance s 2 is nonnegligible but remains small. It is equal to 0.09 for sparse networks and 0.14 for dense networks representing 4.5% and 7%, respectively, of the full possible range of w values. Power averaging is not strictly valid but gives an estimate of upscaling at a few percent. Here w depends slightly on the local permeability distribution shape beyond its variance but mostly on the morphological network structures. Most of the morphological control on upscaling for on‐lattice structures is local and topological and can be explained by the dependence on the average number of neighbor by points (effective coordination number) within the flowing structure (backbone).