Fractal scaling analysis of groundwater dynamics in confined aquifers

Abstract

Abstract. Groundwater closely interacts with surface water and even climate systems in most hydroclimatic settings. Fractal scaling analysis of groundwater dynamics is of significance in modeling hydrological processes by considering potential temporal long-range dependence and scaling crossovers in the groundwater level fluctuations. In this study, it is demonstrated that the groundwater level fluctuations in confined aquifer wells with long observations exhibit site-specific fractal scaling behavior. Detrended fluctuation analysis (DFA) was utilized to quantify the monofractality, and multifractal detrended fluctuation analysis (MF-DFA) and multiscale multifractal analysis (MMA) were employed to examine the multifractal behavior. The DFA results indicated that fractals exist in groundwater level time series, and it was shown that the estimated Hurst exponent is closely dependent on the length and specific time interval of the time series. The MF-DFA and MMA analyses showed that different levels of multifractality exist, which may be partially due to a broad probability density distribution with infinite moments. Furthermore, it is demonstrated that the underlying distribution of groundwater level fluctuations exhibits either non-Gaussian characteristics, which may be fitted by the Lévy stable distribution, or Gaussian characteristics depending on the site characteristics. However, fractional Brownian motion (fBm), which has been identified as an appropriate model to characterize groundwater level fluctuation, is Gaussian with finite moments. Therefore, fBm may be inadequate for the description of physical processes with infinite moments, such as the groundwater level fluctuations in this study. It is concluded that there is a need for generalized governing equations of groundwater flow processes that can model both the long-memory behavior and the Brownian finite-memory behavior.

Time–space fractional governing equations of transient groundwater flow in confined aquifers: Numerical investigation

“…The numerical results demonstrated that the proposed time–space fractional governing equation for groundwater flow in confined aquifers might provide a new perspective on modelling groundwater flow and on interpreting the dynamics of groundwater level fluctuations. For example, these findings, especially the observed heavy‐tailed groundwater heads under fractional powers of time and/or space derivatives, may explain the non‐Gaussian observations of the groundwater level fluctuations (Joelson et al, ; Tu et al, ). Additionally, the numerical results may imply that the newly derived fractional groundwater governing equation may help explain the observed heavy‐tailed solute transport behaviour in groundwater flow by incorporating nonlocal or long‐range dependence of the underlying groundwater flow field (Kavvas, Ercan, et al, ; Kavvas, Tu, et al, ).…”

“…As stated in Ercan and Kavvas (), the fractional orders of a given practical problem closely relate to the boundary conditions and physical conditions of the problem. Moreover, the fractional time and space derivative orders may be partially explained by the physical mechanisms behind the spatial/temporal fractal scaling or long‐range dependence behaviour of groundwater flow, as fractal structures within time series of groundwater level fluctuations have been found in the literature (Joelson et al, ; Tu et al, ; Yu et al, ).…”

“…Moreover, although the Brownian models can describe the behaviour of groundwater level fluctuations in many cases, groundwater level fluctuations also show the presence of long‐range dependence and heavy‐tailed behaviour, which may not be explained by the Brownian models (Joelson, Golder, Beltrame, Néel, & Di Pietro, ; Z. Li & Zhang, ; Yu, Ghasemizadeh, Padilla, Kaeli, & Alshawabkeh, ). Hence, generalized governing equations of the groundwater flow process, which can be flexibly applied to cases both within the Brownian context and those outside, are needed (Kavvas, Ercan, et al, ; Kavvas, Tu, et al, ; Tu, Ercan, & Kavvas, ). Time–space fractional governing equations for groundwater flow would be a good candidate to model both the Brownian finite memory behaviour and the long‐range dependence phenomena of groundwater flow processes at various scales, due to the capabilities of fractional differential equations.…”

Fractional governing equations of transient groundwater flow in unconfined aquifers with multi-fractional dimensions in fractional time

“…The β-order Caputo fractional derivative D kβ a f (x) of a function f (x) may be defined as (Odibat and Shawagfeh, 2007;Podlubny, 1998;Usero, 2008, andLi et al, 2009)…”

“…The fractional time and space derivatives are estimated in the same manner as seen in Tu et al (2018), where the Caputo fractional space and time derivatives in the fractional governing equation are estimated by the numerical algorithm in Odibat (2009) and the algorithm reported by Murio (2008), respectively. The Caputo fractional space deriva-…”