Éditeur(s) : HAL CCSD American Physical Society (APS)
Résumé : International audience
Diffusion in natural and engineered media is quantified in terms of stochastic models for the heterogeneity-induced fluctuations of particle motion. However, fundamental properties such as ergodicity and self-averaging and their dependence on the disorder distribution are often not known. Here, we investigate these questions for diffusion in quenched disordered media characterized by spatially varying retardation properties, which account for particle retention due to physical or chemical interactions with the medium. We link self-averaging and ergodicity to the disorder sampling efficiency Rn, which quantifies the number of disorder realizations a noise ensemble may sample in a single disorder realization. Diffusion for disorder scenarios characterized by a finite mean transition time is ergodic and self-averaging for any dimension. The strength of the sample to sample fluctuations decreases with increasing spatial dimension. For an infinite mean transition time, particle motion is weakly ergodicity breaking in any dimension because single particles cannot sample the heterogeneity spectrum in finite time. However, even though the noise ensemble is not representative of the single-particle time statistics, subdiffusive motion in q≥2 dimensions is self-averaging, which means that the noise ensemble in a single realization samples a representative part of the heterogeneity spectrum.
ISSN: 2470-0053

hal-01622668
https://hal.archives-ouvertes.fr/hal-01622668
https://hal.archives-ouvertes.fr/hal-01622668v2/document
https://hal.archives-ouvertes.fr/hal-01622668/file/russian2017.pdf
DOI : 10.1103/PhysRevE.96.022156

Éditeur(s) : HAL CCSD Springer Verlag (Germany) Springer Verlag
Résumé : 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 (t^{-beta }), where (beta ) reflects the medium heterogeneity. We use this solution to fit a set of real data displaying distinctly different decline trends and study the sensitivity of the model to the reservoir parameters in order to identify their respective controls at the different stages of the decline curve dynamics. Results indicate that the initial value of the gas rate is determined by the transmissivity of the fractures and the initial pressure of the gas in the shale matrix. The latter causes mainly a shift in the entire decline curve. The early time of decline curve shape is primarily controlled by the fracture properties (compressibility and transmissivity). During the main part of the economically valuable production times, i.e. before the production rate drops exponentially, the decline curve is strongly controlled by the properties of the shale rocks including their heterogeneity, which is modelled by two parameters describing the non-Fickian pressure diffusion effects in a stochastic framework.
ISSN: 0169-3913

hal-01217247
https://hal.archives-ouvertes.fr/hal-01217247
DOI : 10.1007/s11242-015-0504-y

Éditeur(s) : HAL CCSD American Geophysical Union
Résumé : International audience
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.
ISSN: 0043-1397

insu-00952537
https://hal-insu.archives-ouvertes.fr/insu-00952537
https://hal-insu.archives-ouvertes.fr/insu-00952537/document
https://hal-insu.archives-ouvertes.fr/insu-00952537/file/russian.pdf
DOI : 10.1002/2013WR014255

Éditeur(s) : HAL CCSD American Geophysical Union
Résumé : International audience
Mobile-immobile mass transfer is widely used to model non-Fickian dispersion in porous media. Nevertheless, the memory function, implemented in the sink/source term of the transport equation to characterize diffusion in the matrix (i.e., the immobile domain), is rarely measured directly. Therefore, the question can be posed as to whether the memory function is just a practical way of increasing the degrees of freedom for fitting tracer test breakthrough curves or whether it actually models the physics of tracer transport. In this paper we first present a technique to measure the memory function of aquifer samples and then compare the results with the memory function fitted from a set of field-scale tracer tests performed in the same aquifer. The memory function is computed by solving the matrix diffusion equation using a random walk approach. The properties that control diffusion (i.e., mobile-immobile interface and immobile domain cluster shapes, porosity, and tortuosity) are investigated by X-ray microtomography. Once the geometry of the matrix clusters is measured, the shape of the memory function is controlled by the value of the porosity at the percolation threshold and of the tortuosity of the diffusion path. These parameters can be evaluated from microtomographic images. The computed memory function compares well with the memory function deduced from the field-scale tracer tests. We conclude that for the reservoir rock studied here, the atypical non-Fickian dispersion measured from the tracer test is well explained by microscale diffusion processes in the immobile domain. A diffusion-controlled mobileimmobile mass transfer model therefore appears to be valid for this specific case.
ISSN: 0043-1397

insu-00373705
https://hal-insu.archives-ouvertes.fr/insu-00373705
https://hal-insu.archives-ouvertes.fr/insu-00373705/document
https://hal-insu.archives-ouvertes.fr/insu-00373705/file/Gouze_et_al-2008-Water_Resources_Research.pdf
DOI : 10.1029/2007WR006690

Éditeur(s) : HAL CCSD American Geophysical Union
Résumé : International audience
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 ( inline image) 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 inline image, 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.
ISSN: 0043-1397

hal-01278285
https://hal.archives-ouvertes.fr/hal-01278285
DOI : 10.1002/2015WR017645

Éditeur(s) : HAL CCSD Elsevier
Résumé : The correct characterization of the effective reactive transport dynamics is an important issue for modeling reactive transport on the Darcy scale, specifically in situations in which reactions are localized, that is when different reactions occur in different portions of the porous medium. Under such conditions the conventional approach of homogenizing only the porous medium chemistry is not appropriate. We consider here reactive transport in a porous medium that is characterized by mass transfer between a mobile and a distribution of immobile regions. Chemical and physical heterogeneities are reflected by distributions of kinetic reaction rate constants and residence times in the immobile zones. We derive an effective reactive transport equation for the mobile solute that is characterized by non-local physical mass transfer and reaction terms. Specifically, chemical heterogeneity is upscaled in terms of a reactive memory function that integrates both chemical and physical heterogeneity. Mass transfer limitations due to physical heterogeneity yield effective kinetic rate coefficients that can be much smaller than the volumetric average of the local scale coefficients. These results help to explain and quantify the often reported discrepancy between observed field reaction rate constants and the ones obtained under well mixed laboratory conditions. Furthermore, these results indicate that transport under physical and chemical heterogeneity cannot be upscaled separately.
ISSN: 0169-7722

hal-00617674
https://hal.archives-ouvertes.fr/hal-00617674
DOI : 10.1016/j.jconhyd.2010.06.002

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
Random walk (RW) methods are recurring Monte Carlo methods used to modelconvective and diffusive transport in complex heterogeneous media. Many applications canbe found, including fluid mechanic, hydrology and chemical reactors modeling. These methodsare easy to implement, very versatile and flexible enough to become appealing for manyapplications because they generally overlook or deeply simplify the building of explicit complexmeshes required by deterministic methods. RWprovides a good physical understandingof the interactions between the space scales of heterogeneities and the transport phenomenaunder consideration. In addition, they can result in efficient upscaling methods, especiallyin the context of flow and transport in fractured media. In the present study, we review theapplications of RW to several situations that cope with diverse spatial scales and differentinsights into upscaling problems. The advantages and downsides of RW are also discussed,thus providing a few avenues for further works and applications.
ISSN: 0169-3913

insu-01308060
https://hal-insu.archives-ouvertes.fr/insu-01308060
DOI : 10.1007/s11242-016-0693-z

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
We study diffusion in a heterogeneous medium that is characterized by spatially varying diffusion properties from a random walk point of view. We show that an inhomogeneous continuous time random walk (CTRW) with a spatially variable exponential transition time distribution solves the spatially discretized heterogeneous diffusion equation. This demonstrates the equivalence of the widely used time-domain random walk (TDRW) scheme and spatially inhomogeneous CTRW and at the same time provides a demonstration of the formal equivalence of the TDRW particle formulation and the heterogeneous diffusion equation. Based on this equivalence, we develop a TDRW method for heterogeneous diffusion under spatially variable multirate mass transfer properties. We discuss the implementation of these schemes and study the diffusion behavior in the presence of traps that are characterized by a truncated power-law trapping time distribution.
ISSN: 0309-1708

hal-00767566
https://hal.archives-ouvertes.fr/hal-00767566
DOI : 10.1016/j.advwatres.2012.07.015

Éditeur(s) : HAL CCSD American Geophysical Union
Résumé : International audience
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.
ISSN: 0043-1397

hal-01356319
https://hal.archives-ouvertes.fr/hal-01356319
DOI : 10.1002/2015WR018511