## 1. Introduction

[2] Groundwater flow in an aquifer is a dynamic system which is usually recharged by precipitation and discharges through evapotranspiration (ET) and as base flow to rivers and creeks. Fluctuations of the hydraulic head (water level) are dynamic responses of an aquifer to the changes in recharge and discharge, and thus contain significant amounts of information about the nature of recharge and discharge processes. Valuable insights about groundwater recharge and discharge may be obtained by studying fluctuations of water levels in monitoring wells. Compared with groundwater recharge, ET, and base flow, water levels in monitoring wells are relatively easy to measure. Extensive data of groundwater levels sampled at various time periods of many years or decades are often available. In the United States, for example, groundwater levels in over 25,000 observation wells have been monitored by the U.S. Geological Survey sometime since the last century.

[3] Natural fluctuations in groundwater levels have been investigated traditionally with deterministic approaches [e.g., *Jacob*, 1943; *Dagan*, 1964; *Pinder et al.*, 1969] and lately with stochastic methods. One of the important stochastic methods, the spectral method, has been applied to study temporal and spatial variations of groundwater quantity and quality [e.g., *Gelhar*, 1974; *Bakr et al.*, 1978; *Gelhar and Axness*, 1983; *Duffy et al.*, 1984; *Duffy and Gelhar*, 1985, 1986; *Jin and Duffy*, 1994]. A critical assumption in many studies with stochastic approaches, including stationary spectral analyses, is that the spatial and temporal variations of the hydraulic head and contaminant concentration are wide-sense stationary [e.g., *Dagan*, 1989; *Gelhar*, 1993]. Nonstationary effects, e.g., flow in a bounded domain and transport of a finite contaminant plume, need to be considered in many applications. One of the effective ways to deal with nonstationarity is so-called “evolutionary spectra,” which have been used since 1950s in time series analysis [*Cramer*, 1951; *Priestley*, 1981] and applied to groundwater problems recently [*Li and McLaughlin*, 1991, 1995]. This nonstationary spectral method is adopted in this paper to study temporal scaling of the hydraulic head fluctuations due to natural groundwater recharge and discharge.

[4] Despite its importance, little attention has been given to scaling of temporal variations of groundwater levels, although scaling for spatial variations of the hydraulic conductivity and dispersivity has been investigated extensively [e.g., *Gelhar*, 1986; *Arya et al.*, 1988; *Wheatcraft and Tyler*, 1988; *Cushman and Ginn*, 1993; *Kemblowski and Wen*, 1993; *Dagan*, 1994; *Sahimi*, 1993; *di Federico and Neuman*, 1995; *Neuman*, 1990, 1994, 1995; *Molz and Boman*, 1995; *Rajaram and Gelhar*, 1995; *Zhan and Wheatcraft*, 1996; *Liu and Molz*, 1996; *Zhang et al.*, 1996; *Zhang and Lin*, 1998; *di Federico and Zhang*, 1999; *Mohanty*, 1999; *Boufadel et al.*, 2000; *Zhang and di Federico*, 2000; *Tennekoon et al.*, 2003]. Most recently, *Zhang and Schilling* [2004] discovered that temporal scaling in the time series of water level fluctuations may exist by carrying out spectral analyses of the hourly hydraulic head data observed over a 4–year period at seven monitoring wells in the Walnut Creek watershed in Iowa. They found that the hydraulic head in an aquifer may fluctuate as a fractal in time in response to either a white noise or a fractal recharge process, depending on how quickly the aquifer responds to recharge events and on the physical parameters of the aquifer (i.e., transmissivity and specific yield). They found that the recharge process at the Walnut Creek watershed may be a white noise process and that the base flow in Walnut Creek and four other watersheds has temporal scaling and the base flow spectrum has two distinct slopes with a breakpoint.

[5] Their findings are further investigated in this study by theoretical analyses with a nonstationary spectral method and by numerical simulations of transient groundwater flow under a hydrogeological condition similar to that at the Walnut Creek site, Iowa, using a one-dimensional transient groundwater flow model with a white noise recharge process. The hydraulic heads and discharge (base flow) to a creek in both homogeneous and heterogeneous aquifers are simulated. The theoretical and simulation results support the finding of *Zhang and Schilling* [2004] that the hydraulic head fluctuations due to natural recharge and discharge may be a temporal fractal and an aquifer may act as a fractal filter which takes stationary recharge inputs and produces fractal head fluctuations and base flow variations. In the following we will present first the measured data and their analysis, then theoretical derivations and numerical simulations, and finally some conclusions.