Non-local and localized analyses of non-reactive solute transport in bounded randomly heterogeneous porous media: Theoretical framework

Morales-Casique, Eric; Neuman, Shlomo P.; Guadagnini, Alberto

doi:10.1016/j.advwatres.2005.10.002

Cited by 98 publications

(69 citation statements)

References 62 publications

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“…It is known that perturbative solutions of advective-dispersion equations can sometimes result in bimodal behavior of averaged concentration fields [Jarman and Russell, 2003;Morales-Casique et al, 2006;Jarman and Tartakovsky, 2008]. However, herein we do not encounter such concerns.…”

Section: Average Flow Equationmentioning

confidence: 54%

Effective two‐phase flow in heterogeneous media under temporal pressure fluctuations

Bolster

Dentz

Carrera

2009

Water Resources Research

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[1] We study the combined effect of spatial heterogeneity and temporal pressure fluctuations on the dispersion of the saturation of a displacing fluid during injection into an immiscible one. In a stochastic modeling framework, we define two different dispersion quantities, one which measures the front uncertainty due to temporal pressure fluctuations and another one which quantifies the actual spreading and dispersion of the saturation front in a typical medium realization. We derive an effective large-scale flow equation for the saturation of the displacing fluid that is characterized by a nonlocal dispersive flux term. From the latter we derive measures for the spread of the saturation front due to spatiotemporal fluctuations. Our analysis demonstrates that temporal fluctuations enhance the front uncertainty but not the average spreading of the saturation front. The analytical developments are complemented by numerical simulations of the full two-phase flow problem. Correct assessment of the spread of the saturation front is of importance to several applications, including the assessment of the sequestration potential of a carbon dioxide storage site, for example. Spreading enhances the surface area between the fluids, which in turn enhances the dissolution and entrapment efficiency. The latter facilitates reactions and thus the sequestration of CO 2 in stable forms. Our study provides a theoretical basis for the design of injection strategies to optimize dispersion and to minimize uncertainty of the saturation front.

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Section: Average Flow Equationmentioning

confidence: 54%

Effective two‐phase flow in heterogeneous media under temporal pressure fluctuations

Bolster

Dentz

Carrera

2009

Water Resources Research

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“…Many papers have documented the improved fits that may be generated (see the extensive review by Berkowitz et al [2006]). These studies validate the fact that nonlocality is the by-product of upscaling [Neuman, 1993;Cushman and Ginn, 1993;Morales-Casique et al, 2006;Dentz and Bolster, 2010]. It is a simple notion that upscaling replaces detailed velocity information with memory kernels in spatial and temporal convolutions.…”

Section: Introductionsupporting

confidence: 50%

Comparison of Fickian and temporally nonlocal transport theories over many scales in an exhaustively sampled sandstone slab

Major

Benson

Revielle

et al. 2011

Water Resources Research

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[1] It is not a compelling argument, solely on the basis of a better fit to solute breakthrough curve (BTC) data, that a temporally nonlocal model is necessary to simulate transport in an advection-dominated system. One may counter that the classical advection-dispersion equation (ADE) is a valid model at some small scale and that the detailed hydraulic conductivity (K) data must be well-represented : then the nonlocality is only a result of upscaling and loss of information. But is the nonlocal model demonstrably necessary at all scales? We examine the experiment conducted by Klise et al. (2008) in which a 30.5 Â 30.5 cm slab of relatively homogeneous, cross-bedded sandstone was exhaustively sampled for K. The slab was sealed, saturated with potassium iodide, and X-rayed 10 times while being flushed with fresh water. The 8,649 air-permeameter measurements were down-and upscaled to make finer and coarser grids on which the velocity field was solved and the ADE applied. The optimized parameters in the ADE were found to scale predictably, most notably the longitudinal dispersivity ( L ), which grew linearly with upscaling. But at all levels of up-and downscaling, including the original K measurement scale of 0.33 cm, the ADE did not adequately represent the late-time tails. The temporally nonlocal, timefractional ADE (t-FADE) was applied and the optimized parameters ( L and the immobile capacity ) did not depend on scale. The better fit provided by the t-FADE in the late BTC tails did not bring about a sacrificed fit elsewhere in the BTC. Furthermore, the optimized ADE and t-FADE solutions do not converge at the smallest scale, directly implying that the temporal nonlocality is a necessary model component. We conclude that the logical inference ''if the ADE is valid in heterogeneous material, then there is tailing in the BTC'' is not a proof that the reverse is true. We provide a clear counterexample. A corollary is that a mismatch between data and a discretized solution to the ADE does not imply that more data will improve fits or predictive ability.

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“…Alternative transport models are proposed to capture the effects of spatial heterogeneities in fractured rocks and the effects of flow channeling or velocity dispersion [14,15]. The models have extended to the mass transfer models with time-or space-dependent dispersion coefficients (e.g., [16]), the multi-rate mass transfer models [17], the continuous time random walk approach [18], the time-domain random walk approach, the fractional advection-dispersion equation approach [19], and the stochastic approach [20].…”

Section: Introductionmentioning

confidence: 99%

Mathematical Modeling of Non-Fickian Diffusional Mass Exchange of Radioactive Contaminants in Geological Disposal Formations

Suzuki

Fomin

Чугунов

et al. 2018

Water

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Abstract:Deep geological repositories for nuclear wastes consist of both engineered and natural geologic barriers to isolate the radioactive material from the human environment. Inappropriate repositories of nuclear waste would cause severe contamination to nearby aquifers. In this complex environment, mass transport of radioactive contaminants displays anomalous behaviors and often produces power-law tails in breakthrough curves due to spatial heterogeneities in fractured rocks, velocity dispersion, adsorption, and decay of contaminants, which requires more sophisticated models beyond the typical advection-dispersion equation. In this paper, accounting for the mass exchange between a fracture and a porous matrix of complex geometry, the universal equation of mass transport within a fracture is derived. This equation represents the generalization of the previously used models and accounts for anomalous mass exchange between a fracture and porous blocks through the introduction of the integral term of convolution type and fractional derivatives. This equation can be applied for the variety of processes taking place in the complex fractured porous medium, including the transport of radioactive elements. The Laplace transform method was used to obtain the solution of the fractional diffusion equation with a time-dependent source of radioactive contaminant.

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Non-local and localized analyses of non-reactive solute transport in bounded randomly heterogeneous porous media: Theoretical framework

Cited by 98 publications

References 62 publications

Effective two‐phase flow in heterogeneous media under temporal pressure fluctuations

Effective two‐phase flow in heterogeneous media under temporal pressure fluctuations

Comparison of Fickian and temporally nonlocal transport theories over many scales in an exhaustively sampled sandstone slab

Mathematical Modeling of Non-Fickian Diffusional Mass Exchange of Radioactive Contaminants in Geological Disposal Formations

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