Statistical characteristics of flow as indicators of channeling in heterogeneous porous and fractured media

Abstract

We introduce two new channeling indicators D_ic and D_cc based on the Lagrangian distribution of flow rates. On the basis of the participation ratio, these indicators characterize the extremes of both the flow-tube width distribution and the flow rate variation along flow lines. The participation ratio is an indicator biased toward the larger values of a distribution and is equal to the normalized ratio of the square of the first-order moment to the second-order moment. Compared with other existing indicators, they advantageously provide additional information on the flow channel geometry, are consistently applicable to both porous and fractured media, and are generally less variable for media generated using the same parameters than other indicators. Based on their computation for a broad range of porous and fracture permeability fields, we show that they consistently characterize two different geometric properties of channels. D_ic gives a characteristic scale of low-flow zones in porous media and a characteristic distance between effectively flowing structures in fractured cases. D_cc gives a characteristic scale of the extension of high-flow zones in porous media and a characteristic channel length in fractured media. D_ic is mostly determined by channel density and permeability variability. D_cc is, however, more affected by the nature of the correlation structure like the presence of permeability channels or fractures in porous media and the length distribution in fracture networks.

Introduction

Spatial heterogeneity in hydraulic conductivity affects fluid flow and solute transport in complex natural media like fractured media [38], alluvial systems [11] and strongly heterogeneous porous media [28] and has been a subject of research for decades ([8] and references therein). It is a function of contrasts between high permeability and low permeability values. As flow tends to avoid low-k zones for high-k zones, heterogeneity induces the development of preferential flow paths [20], [23] also called “paths of least resistance” [39], along which flow is focused. Their effects on upscaled/effective hydrologic properties have been observed in laboratory and numerical studies. Fogg [10] performed a numerical study on the hydraulic conductivity distribution in the Wilcox aquifer and suggests that flow is mainly controlled by the continuity and connectivity of sand deposits rather than by local hydraulic conductivity values. Hanor [16] drew similar conclusions for the Livingston site. Silliman [34] illustrated the formation of preferential flow paths with laboratory experiments. Knudby et al. [22] and, Ronayne and Gorelick [30] showed how the estimate of aquifer properties, like the effective permeability of a system, should take channeling into account. Ronayne et al. [31] used statistical channeling properties to estimate aquifer parameters in a system affected by channeling. Similarly, Kerrou et al. [19] showed that not accounting explicitly for channeling in a sequential self-calibration approach resulted in flow underestimation and strong deviations in capture zone estimates. Trinchero et al. [37] showed that for moderate heterogeneities, both the connectivity of high-k values and apparent porosity are key in predicting transport times efficiently. Although channeling is important for flow and transport properties, its quantification remains a matter of debate. Two types of indicators have been proposed: indicators derived from the comparison of upscaled hydraulic properties with their small-scale counterparts, and statistical indicators calculated from the permeability and flow fields. The first category of estimators is based on hydraulic properties that are sensitive to channeling. The simplest estimator is the effective permeability, K_eff, known to be sensitive to flow organization [14]. In 2D multi-log-Gaussian isotropic weakly-correlated fields, the equivalent permeability is equal to the geometric mean $K_{\mathrm{eff}}=K_{\mathrm{g}}$ [27]. If the connectivity of the higher-K zones is greater than that of the lower-K zones, K_eff is larger than $K_{\mathrm{g}}$ [32] within the limit that $K_{\mathrm{eff}}⩽K_{\mathrm{a}}$ (where $K_{\mathrm{a}}$ is the arithmetic mean) [40]. The type of average measured by the power averaging exponent CF₁ [9], [18] has thus been considered as a measure of channeling [20]:

$${\mathit{CF}}_1:K_{\mathrm{eff}}={\left(\frac{1}{V}{\int }_{V}K(x{)}^{{\mathit{CF}}_1}\mathit{dV}\right)}^{\frac{1}{{\mathit{CF}}_1}}$$

CF₁ varies between −1 and 1 for the harmonic and arithmetic means, respectively, and is equal to zero for the geometric mean corresponding to isotropic weakly-correlated multi-Gaussian fields. As transport is also strongly affected by channeling, breakthrough curve properties have been proposed as estimators of the channeling degree [41]. Knudby and Carrera [20] used the ratio CT₁ of the average arrival time $\overline{t}$ to the time at which 5% of the solute have broken through the domain boundary t₅:

$${\mathit{CT}}_1=\overline{t}/t_5$$

When preferential flow paths exist, $t_5$ becomes much smaller than $\overline{t}$, ${\mathit{CT}}_1$ increases and the field should be considered as increasingly connected. The apparent hydraulic diffusivity has been proposed as an intermediary characteristics between flow and transport connectivities [21]. Park et al. [29] suggested that the normalized travel time and distance be used to investigate preferential flow.

The second category of estimators uses statistical characteristics of the permeability field or of the flow field. N-point spatial connectivity statistics are dedicated to the measurement of connectivity and were applied to permeability fields to estimate the presence of high-k connected patterns [17], [24]. Western et al. [42] used a directional multi-point geostatistical indicator and showed that it could capture the difference between random and channeled fields with similar k-distributions, unlike non-directional indicators. Frippiat et al. [13] suggested that the presence of preferential flow paths or flow barriers could be identified using head and flow variances, since head variance is negatively correlated to connectivity while flow variance is positively correlated to the effective permeability increase. Bruderer-Weng et al. [3] used the multifractal spectrum of the flow field to quantify channeling in heterogeneous pipe networks. The distribution of flow has also been used for quantifying channeling in fractured networks [6].

The multiplicity of the proposed indicators shows that channeling cannot be restricted to a single simple characteristic. The concept of channeling also strongly depends on the application targeted. The relevant use of channeling indicators probably differ between flow and transport applications [33]. In this study, we focused first on the geometrical characterization of channels, i.e. on the channels themselves rather than on their consequences in terms of flow or transport. In this respect, the first category of indicators based on equivalent medium properties are limited by the fact that they measure the consequences of channeling rather than channeling itself. The limitation of the indicators based on permeability statistics arises from the measurement of a single cause of channeling cause (the connectivity of high-k zones) where channeling is also induced by the variability of permeability [26]. The advantage of those indicators based on the statistical properties of the flow field is the measurement of channeling itself. As opposed to the multifractal dimensions and the variance of head or flow, we look for indicators based on the geometrical properties of the channels that additionally identify channeling consistently in both porous and fractured media.

Even though channeling occurs under many different circumstances, it has two recurrent characteristics. First, flow is localized within a few structures. Second, channeling locally maintains high flow rates over long distances. On the basis of these two characteristics, we aimed at defining quantitative channeling indicators that meet the three following constraints. First, they must be globally consistent with the visually intuitive classification of channeling. Second, they must provide a quantification of channeling. Third, they must be applicable simultaneously to porous and fractured media.

We define two new indicators in Section 2. We compute their value for the broad range of synthetic fields introduced in Section 3. In Section 4, we analyze first their consistency with the expected ranking of channeling and then their dependency on the permeability correlation structures. Finally, we compare them to other existing indicators in Section 5.