Delphine Roubinet

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.

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.

Analytical models of heat conduction in fractured rocks

Discrete fracture network models routinely rely on analytical solutions to estimate heat transfer in fractured rocks. We develop analytical models for advective and conductive heat transfer in a fracture surrounded by an infinite matrix. These models account for longitudinal and transverse diffusion in the matrix, a two-way coupling between heat transfer in the fracture and matrix, and an arbitrary configuration of heat sources. This is in contrast to the existing analytical solutions that restrict matrix conduction to the direction perpendicular to the fracture. We demonstrate that longitudinal thermal diffusivity in the matrix is a critical parameter that determines the impact of local heat sources on fluid temperature in the fracture. By neglecting longitudinal conduction in the matrix, the classical models significantly overestimate both fracture temperature and time-to-equilibrium. We also identify the fracture-matrix Peclet number, defined as the ratio of advection timescale in the fracture to diffusion timescale in the matrix, as a key parameter that determines the efficiency of geothermal systems. Our analytical models provide an easy-to-use tool for parametric sensitivity analysis, benchmark studies, geothermal site evaluation, and parameter identification.

Discrete-dual-porosity model for electric current flow in fractured rock

The identification of fractures and the characterization of their properties are of critical importance in a wide variety of research fields and applications. To this end, geophysical methods are of significant interest as they can provide information regarding the spatial distribution of a number of subsurface physical properties in a rapid and noninvasive manner. Electrical resistivity surveying, in particular, has been shown in several previous investigations to exhibit sensitivity to the presence of fractures, suggesting that geoelectrical experiments may contain important information regarding how fractures are distributed and connected in the subsurface. However, a lack of suitable numerical modeling tools for electric current flow in fractured media has prevented a detailed and systematic exploration of this concept. To address this issue, we present a novel discrete-dual-porosity modeling approach that is specifically tailored to the electrical resistivity problem. With our approach, an analytical formulation for fracture-matrix current flow exchange at the fracture scale is integrated into a discrete-fracture-network model, which is then combined with a block-scale finite-volume representation of the rock matrix. Our methodology allows for low-cost and accurate simulation of electric current flow through both the fractures and matrix, and is readily applicable to complex fracture networks at relatively large scales. Although formulated here in two dimensions, this work represents an important first step toward investigating the effect of fracture-network characteristics on bulk electrical properties, as well as toward the simulation of geoelectrical survey data in realistic fractured-rock environments.

Streaming potential modeling in fractured rock: Insights into the identification of hydraulically active fractures

Numerous field experiments suggest that the self-potential (SP) geophysical method may allow for the detection of hydraulically active fractures and provide information about fracture properties. However, a lack of suitable numerical tools for modeling streaming potentials in fractured media prevents quantitative interpretation and limits our understanding of how the SP method can be used in this regard. To address this issue, we present a highly efficient two-dimensional discrete-dual-porosity approach for solving the fluid flow and associated self-potential problems in fractured rock. Our approach is specifically designed for complex fracture networks that cannot be investigated using standard numerical methods. We then simulate SP signals associated with pumping conditions for a number of examples to show that (i) accounting for matrix fluid flow is essential for accurate SP modeling and (ii) the sensitivity of SP to hydraulically active fractures is intimately linked with fracture-matrix fluid interactions. This implies that fractures associated with strong SP amplitudes are likely to be hydraulically conductive, attracting fluid flow from the surrounding matrix.

Particle Methods for Heat Transfer in Fractured Media

Quantitative understanding of heat transfer in fractured media is critical in a wide range of applications, including geothermal energy harvesting. Mathematical models of such systems must account for fluid flow and heat transfer in both complex fracture networks and the ambient rock matrix. Incorporation of individual fractures with millimeter-scale apertures into meter-scale computational domains on which continuum models are discretized would be computationally prohibitive even on modern supercomputers. By exploiting the similarities of the underlying mathematical structure of heat and mass transfer processes, we adopt a mesh-free time-domain particle-tracking method to model heat transfer in highly heterogeneous fractured media. The method is used to model heat extraction from geothermal reservoirs whose fracture networks exhibit fractal properties representative of faults and damage zones. We explore a range of fracture network properties and experimental conditions in order to study the impact of the fracture network topology and hydraulic regimes on heat transfer. Our results demonstrate anomalous behavior of heat transfer in fractured environments and a significant impact of the network topology on performance of geothermal reservoirs.