# A Concept of Fuzzy Dual Permeability of Fractured Porous Media

## Abstract

**:**

^{2}) ponded tests at the Hell’s Half Acre site, mesoscale (56 m

^{2}) ponded tests at the Box Canyon site, and a large-scale infiltration test (31,416 m

^{2}) at the Radioactive Waste Management Complex at INL. Methods of fuzzy clustering and fuzzy regression were applied to describe the time-depth waterfront penetration and to characterize the phenomena of rapid flow through a predominantly fractured component and slow flow through a predominantly porous matrix component. The concept of fuzzy dual permeability is presented using a series of fuzzy membership functions of the waterfront propagation with depth and time. To describe the time variation of the flux, a fuzzy Horton’s model is presented. The developed concept can be used for the uncertainty quantification in flow and transport in geologic media.

## 1. Introduction

^{2}) ponded tests at the Hell’s Half Acre site [15]; mesoscale ponded tests at the Box Canyon site [14]; and a large-scale test at the Radioactive Waste Management Complex at Idaho National Laboratory [7,25].

## 2. Summary of Field Ponded Infiltration Tests

#### 2.1. HHA Infiltration Tests

#### 2.2. Box Canyon Infiltration Tests

^{2}. The infiltration tests included one 2-week test, three 2-day tests, and one 4-day test. Overall, all tests showed a decrease in the infiltration rate with time, presumably because of mechanical or microbiological clogging of fractures and the vesicular basalt in the near-surface zone, as well as the effect of entrapped air.

_{f}∗ t + (1/β) (i

_{o}− i

_{f}) [1 − exp (−β ∗ t)]

_{f}is final (constant) flux, i

_{o}is the initial flux, and β is the decay parameter called Horton’s coefficient. The model parameters used for calculations are given in Table 1. This model describes experimental data very well, as the correlation coefficients determined by fitting Horton’s cumulative flux to data collected in the field for all tests were 0.99 [14]. (The fuzzy analysis of the parameters of the Horton equation is presented in Section 4.2.)

#### 2.3. Large-Scale Infiltration Test at the Radioactive Waste Management Complex at INL

^{2}in area) with the application of various tracers [7,25]. Figure 3 illustrates the breakthrough curves (BTCs) of selenium, which was used as a tracer, at the different monitoring wells. These BTCs demonstrate significant variations in the water travel time at different locations. Observations showed that the tracer travel time to the outer Ring C of monitoring wells, located more than 100 ft from the inner Ring B, is in the same range as that to inner Ring B. The tracer breakthrough curves were used for the evaluation of the water arrival times to different depths, and the data were analyzed in Section 4.1 below along with the data from the HHA and Box Canyon tests.

## 3. Methods of Fuzzy Data Analysis

#### 3.1. General Idea of Fuzzy Data Analysis and a Fuzzy Membership Function

_{A}(x))|x $\in $ X}, where µ

_{A}(x) is called the membership function for the fuzzy set A, and X is referred to as the universe of discourse. The membership function A is used to associate each element x $\in $ X with a value within the interval [0, 1]. A fuzzy membership function expresses an imprecise, rather than exact, quantity, as is the case with “ordinary” (single-valued) numbers. In other words, an FMF is a function in a domain that is a specified set of real numbers. Each numerical value in the FMF domain is assigned a specific grade of membership within the range from 0, representing the smallest possible grade, to 1, representing the largest possible grade. A fuzzy membership function generally bounds a probability distribution function (PDF). In Figure 4, a fuzzy number shown in red is said to “enclose” a probability distribution shown in the blue color. A fuzzy number also expresses the idea that what is probable must preliminarily be possible.

#### 3.2. Fuzzy Clustering and Fuzzy Regression

## 4. Results of Fuzzy Data Analysis

#### 4.1. Fuzzy Clustering and Fuzzy Regression of the Time-Depth Waterfront Penetration

#### 4.2. Fuzzy Analysis of the Infiltration Rate and Parameters of Horton’s Infiltration Model

_{f}, i

_{o}, and β parameters (as described in Section 3.1), and the results are shown in Figure 7. Figure 7a,b are plotted using the same scale of the x-axis to visually compare the ranges of the infiltration rate at the small-scale HHA site and the mesoscale Box Canyon site. These figures demonstrate the larger ranges of the area-averaged infiltration ranges of i

_{o}and i

_{f}at the HHA site, which can be explained by the fast infiltration through the near-surface weathered fractured basalt, and which corresponds to the fast waterfront penetration cluster shown in Figure 4.

## 5. Conclusions

^{2}scale), mesoscale infiltration tests conducted at Box Canyon (56 m

^{2}), and the LSIT at the Radioactive Waste Management Complex (200 m radius). The application of the fuzzy systems analysis, specifically, the fuzzy clustering and regression, to the characterization of the time-depth waterfront penetration and the infiltration rate time series helped characterize the phenomena of fuzzy dual permeability, which are caused by rapid flows through predominantly fractured components and slow flows through predominantly matrix components. Incorporation of the fuzzy systems approach into hydrogeological analysis and modeling is expected to provide a significant payoff for solving problems that are characterized by imprecise, vague, and ambiguous information, and it can be used for the management of water resources, decision-making, and risk analysis to characterize the uncertainty phenomena of flow and transport in geological formations. Further investigations can be conducted to put the concept of fuzzy permeability in association with neural net models and evolutionary algorithms.

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) A schematic of the small-scale (0.5 m

^{2}) Hell’s Half Acre ponded infiltration test [15], and (

**b**) graphs of the cumulative water flux plotted using data collected in individual pans during the ponded infiltration tests at the HHA site (solid and dashed lines correspond to three clusters of the individual pans; explanation will be given in Section 4.2).

**Figure 2.**Time variations of the cumulative total volume of water infiltrated during the Box Canyon tests, which were calculated from the Horton equation with the parameters given in Table 1.

**Figure 4.**Illustration of a concept of the fuzzy membership function (a possibility function, which is in red) that is defined using the four points (a, b, c, and d), in comparison with the probability distribution function (shown in blue).

**Figure 5.**Fuzzy clustering of water travel time vs. depth from three field ponded infiltration tests in fractured basalt: (

**a**) the scatter plot of the depth vs. time of the waterfront penetration, based on the data from the three ponded infiltration tests, (

**b**) fuzzy membership functions of the depths corresponding to the clusters shown in (

**a**), and (

**c**) fuzzy membership functions of the time corresponding to the clusters shown in (

**a**,

**b**). The colors of FMFs in (

**c**) correspond to the clusters identified in (

**b**). The vertical and horizontal dashed blue lines represent an example of mapping between the depth and time for the clusters “Near-surface weathered basalt”.

**Figure 6.**(

**a**) Results of the application of fuzzy regression of the waterfront penetration along the fast flow paths (blue color) and slow flow paths (red color). Solid blue and red lines are fuzzy regression lines and dashed blue and red lines are 95% CIs. (

**b**) FMFs of time of the clusters, and a dashed black line is the trapezoidal FMF that corresponds to the water arrival at the depth of 30 m. The trapezoidal FMF with support = [3.098,19.2837] and core = [6.5,14.2]. The colors of FMFs in (

**b**) are the same as in Figure 5.

**Figure 7.**Comparison of the fuzzy membership functions of the area-averaged Horton’s model parameters i

_{f}and i

_{o}for the HHA and Box Canyon infiltration tests. Dashed lines are calculated PDFs, and solid lines are fuzzy membership functions. In (

**a**,

**b**), the red color is the maximum infiltration rate, and the blue is the minimum infiltration rate. In (

**c**) the red is Horton’s β for the HHA tests, and the blue is for the Box Canyon tests.

**Table 1.**Parameters of the Horton equation used to describe the cumulative water flux during the Box Canyon tests [14].

Test | i_{o} | i_{f} | b |
---|---|---|---|

96-1 | 11.01 | 0 | 0.17 |

97-1 | 9.96 | 2.16 | 1.32 |

97-2 | 17.7 | 2.7 | 4.02 |

97-3 | 6.74 | 1.63 | 1.17 |

97-4 | 4.16 | 0.63 | 0.21 |

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**MDPI and ACS Style**

Faybishenko, B.
A Concept of Fuzzy Dual Permeability of Fractured Porous Media. *Water* **2023**, *15*, 3752.
https://doi.org/10.3390/w15213752

**AMA Style**

Faybishenko B.
A Concept of Fuzzy Dual Permeability of Fractured Porous Media. *Water*. 2023; 15(21):3752.
https://doi.org/10.3390/w15213752

**Chicago/Turabian Style**

Faybishenko, Boris.
2023. "A Concept of Fuzzy Dual Permeability of Fractured Porous Media" *Water* 15, no. 21: 3752.
https://doi.org/10.3390/w15213752