Anna Russian

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.

Dual control of flow field heterogeneity and immobile porosity on non-Fickian transport in Berea sandstone

Both flow field heterogeneity and mass transfer between mobile and immobile domains have been studied separately for explaining observed anomalous transport. Here we investigate non-Fickian transport using high-resolution 3-D X-ray microtomographic images of Berea sandstone containing microporous cement with pore size below the setup resolution. Transport is computed for a set of representative elementary volumes and results from advection and diffusion in the resolved macroporosity (mobile domain) and diffusion in the microporous phase (immobile domain) where the effective diffusion coefficient is calculated from the measured local porosity using a phenomenological model that includes a porosity threshold ( ϕθ) below which diffusion is null and the exponent n that characterizes tortuosity-porosity power-law relationship. We show that both flow field heterogeneity and microporosity trigger anomalous transport. Breakthrough curve (BTC) tailing is positively correlated to microporosity volume and mobile-immobile interface area. The sensitivity analysis showed that the BTC tailing increases with the value of ϕθ, due to the increase of the diffusion path tortuosity until the volume of the microporosity becomes negligible. Furthermore, increasing the value of n leads to an increase in the standard deviation of the distribution of effective diffusion coefficients, which in turn results in an increase of the BTC tailing. Finally, we propose a continuous time random walk upscaled model where the transition time is the sum of independently distributed random variables characterized by specific distributions. It allows modeling a 1-D equivalent macroscopic transport honoring both the control of the flow field heterogeneity and the multirate mass transfer between mobile and immobile domains.

Time domain random walks for hydrodynamic transport in heterogeneous media

We derive a general formulation of the time domain random walk (TDRW) approach to model the hydrodynamic transport of inert solutes in complex geometries and heterogeneous media. We demonstrate its formal equivalence with the discretized advection-dispersion equation and show that the TDRW is equivalent to a continuous time random walk (CTRW) characterized by space-dependent transition times and transition probabilities. The transition times are exponentially distributed. We discuss the implementation of different concentration boundary conditions and initial conditions as well as the occurrence of numerical dispersion. Furthermore, we propose an extension of the TDRW scheme to account for mobile-immobile multirate mass transfer. Finally, the proposed TDRW scheme is validated by comparison to analytical solutions for spatially homogeneous and heterogeneous transport scenarios.

Self-averaging and ergodicity of subdiffusion in quenched random media

We study the self-averaging properties and ergodicity of the mean square displacement m(t) of particles diffusing in d dimensional quenched random environments which give rise to subdiffusive average motion. These properties are investigated in terms of the sample to sample fluctuations as measured by the variance of m(t). We find that m(t) is not self-averaging for d<2 due to the inefficient disorder sampling by random motion in a single realization. For d≥2 in contrast, the efficient sampling of heterogeneity by the space random walk renders m(t) self-averaging and thus ergodic. This is remarkable because the average particle motion in d>2 obeys a CTRW, which by itself displays weak ergodicity breaking. This paradox is resolved by the observation that the CTRW as an average model does not reflect the disorder sampling by random motion in a single medium realization.

Temporal scaling of groundwater discharge in dual and multicontinuum catchment models

This paper presents a multicontinuum approach to model fractal temporal scaling of catchment response in hydrological systems. The temporal scaling of discharge is quantified in frequency domain by the transfer function HðxÞ, which is defined as the ratio between the spectra of catchment response and recharge time series. The transfer function may scale with frequency x as HðxÞ x2b. While the classical linear and Dupuit models predict exponents of b52 and b51, observations indicate scalings with noninteger exponents b. Such behaviors have been described by multifractal models, which, however, often lack a relation to the medium characteristics. We revisit and extend the classical linear Dupuit aquifer models and discuss their physical meanings in the light of the resulting aquifer dynamics. On the basis of these classical models, we derive a multicontinuum approach that provides physical recharge models which are able to explain fractal behaviors with exponents 1=2 < b < 2. Furthermore, this approach allows to link the fractal dynamics of the discharge process to the physical aquifer characteristics as reflected in the distribution of storage time scales. We systematically analyze the catchment responses in the proposed multicontinuum models, and identify and quantify the time scales which characterize the dynamics of the catchment response to recharge.

Scale dependence of the hydraulic properties of a fractured aquifer estimated using transfer functions

This is the peer reviewed version of the following article: [Pedretti, D., A. Russian, X. Sanchez-Vila, and M. Dentz (2016), Scale dependence of the hydraulic properties of a fractured aquifer estimated using transfer functions, Water Resour. Res., 52, 5008–5024, doi:10.1002/2016WR018660. ], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/2016WR018660/abstract. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.

Multi-continuum Approach to Modelling Shale Gas Extraction

Production rates in horizontal shale gas wells display declines that are controlled by the low permeability and the intrinsic heterogeneity of the shale matrix. We present an original multi-continuum approach that yields a physical model able to reproduce the complexity of the decreasing gas rates. The model describes the dynamics of gas rate as function of the physical reservoir parameters and geometry, while the shale matrix heterogeneity is accounted for by a stochastic description of transmissivity field. From the 3D (Dimensional) problem setting, including the heterogeneous shale matrix, the fractures generated by the hydrofracking operations, as well as the production well characteristics, we establish an effective upscaled 1D model for the gas pressures in fracture and matrix as well as the volumetric flux. We analyse the decline curves behaviour, and we identify the time scales that characterize the dynamics of the gas rate decline using explicit analytical Laplace space solutions of the upscaled process model. Asymptotically, the flux curves decrease exponentially, while in an intermediate regime we find a power-law behaviour, in which the flux scales with a power law in time as

Erratum: Self-averaging and ergodicity of subdiffusion in quenched random media [Phys. Rev. E93, 010101(R) (2016)]

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