1.1. Analytical Methods for Evaluation of Stream Depletion
 Stream depletion (SD) is one of the most widely used hydrogeological concepts developed in the twentieth century for water resources management. Recent droughts and the proliferation of large-capacity wells for irrigation have renewed interest in the SD concept. In the United States, tens of thousands of wells capable of pumping over 1000 m3/d are located in alluvial valleys. Vast water withdrawals have dramatically changed local and regional water budgets of aquifers and streams. For example, maps comparing perennial streams in Kansas in the 1960s with those of the 1990s show a marked decrease in the length of streamflow [Sophocleous, 1997].
 Wells upset the dynamic equilibrium of the water budget that existed in predevelopment conditions. A decrease of groundwater drainage into a stream or increase of stream water losses into the aquifer are examples of changes in natural discharge or discharge [Theis, 1940, 1941; Sophocleous, 1997; Bredehoeft, 1997, 2002]. The sum of these two terms required for a transition to a new dynamic equilibrium under groundwater pumping is sometimes referred as “capture.” Hantush  introduced the term “stream depletion” as synonymous with “capture” to characterize changes in natural groundwater discharge to the streams. This term is sometimes applied to direct water losses (fluxes) from streams that are hydraulically connected with the pumped aquifers [e.g., Wilson, 1993].
 Direct measurements of stream depletion rate (SDR) using stream discharge data are difficult and rare [Sophocleous et al., 1988; Hunt et al., 2001; Nyholm et al., 2002, 2003; Kollet and Zlotnik, 2003]. Their results are fraught with uncertainties due to the runoff variability and available accuracy of discharge measurements. Therefore mathematical modeling is commonly used for SDR evaluation. In cases where appropriate information is available, numerical modeling allows one to determine the SDR and other water budget items using the drawdown and runoff characteristics for calibration [e.g., Nyholm et al., 2002, 2003]. However, this information is often limited, so more simple analytical models are generally used for SDR evaluation.
 There are four analytical solutions available for estimating SDR that differ in their descriptions of streambed properties and degree of penetration. The dimensionless SDR function D can be defined as a fraction of pumping rate Q,
where qS is stream depletion rate, which depends on time t, well characteristics, including distance from stream bank d, and aquifer parameters (hydraulic conductivity K, thickness b, storativity S, etc.). Jenkins  introduced the characteristic timescale ta, which sometimes is called the “stream depletion factor,”
where T = Kb is transmissivity. Only a short summary of the equations is presented below, since hydrogeological conditions for applications of these solutions have been summarized elsewhere [e.g., Barlow and Moench, 1998; Zlotnik and Huang, 1999; Hunt, 1999; Zlotnik et al., 1999; Butler et al., 2001].
 1. A stream fully penetrates a uniform aquifer with an impermeable horizontal base. Theis  and Glover and Balmer  derived the SDR function DTGB for a well with a pumping rate Q at the distance d from the stream as follows:
After t > 100 ta, more than 94% of groundwater withdrawal Q is supplied by the SDR; eventually this fraction reaches 100%. A short review of this solution is given by Wallace et al. .
 2. A stream fully penetrates a uniform aquifer; the aquifer and streambed have contrasting hydraulic conductivity. Hantush  accounted for a partial stream penetration and properties of streambed sediments by introducing a fictitious thin incompressible “vertical” layer of reduced hydraulic conductivity KS (KS < K) and thickness mS. The SDR function (DH) utilizes retardation coefficient BS, which accounts for streambed properties:
Compared with the Theis-Glover-Balmer solution, the pace of the SDR increase over time is slower, and the term “retardation” properly describes the later onset of the 100%. However, to highlight the hydrogeological context of this coefficient, the term “streambed leakage coefficient” for BS is more appropriate. A short review of this solution is given by Darama .
 3. A stream with a streambed of finite thickness negligibly penetrates the aquifer; aquifer and streambed have contrasting hydraulic conductivities. Zlotnik et al. , Hunt , and Butler et al.  obtained the SDR for this realistic streambed geometry. In the particular case of a very shallow stream of finite width W, the SDR function can be presented in a closed form [Zlotnik et al., 1999, equations (11) and (12)]:
Note that equation (6) is identical to the Hantush solution DH in equation (5), but a different form of streambed leakage coefficient BS more realistically represents the streambed and the water fluxes. In the case of a small stream width (W ≪ 2B) the expression for BS simplifies to
and the problem reduces to the Hunt [1999, equation (20)] solution.
 Butler et al.  extended this approach to include the effects of large-scale heterogeneity and finite alluvial valley width on estimates of SDR and drawdown. It was shown numerically that this semianalytical method and an accompanying code are accurate for many cases having anisotropic conditions and varying degrees of aquifer penetration.
 4. A stream fully penetrates a uniform unconfined aquifer and partially penetrates an aquitard beneath; water is pumped from a well in an adjacent semiconfined aquifer with an impermeable horizontal base. Hunt  developed a two-term solution:
where ta is calculated using the parameters of the semiconfined aquifer (instead of the unconfined aquifer) and Δ incorporates the aquitard parameters.
 All of these approaches share one essential trait: They predict that the stream will supply 100% of the groundwater withdrawal after a sufficiently long pumping period. Application of these solutions for water resources management without considering hydrogeological conditions may lead to overestimation of stream depletion.
1.2. Leaky Aquifers and the Concept of Maximum Stream Depletion Rate (MSDR)
 In some cases, groundwater withdrawals in alluvial aquifers can be partially supplied by leakage from adjacent aquifers. Indeed, the base of many alluvial aquifers consists of low permeability bedrock that can be considered an aquitard (Figure 1). In these instances, the absence of water budget data for alluvial aquifers evolves into the assumption of negligible flow from the aquifer base for practical purposes. However, there are many situations where alluvial aquifers are in hydraulic connection with adjacent aquifers. “This occurs in rivers of the Gulf Coast and in the High Plains. In this case, there may be a significant contribution from the underlying sediments to the baseflow of the stream” [Larkin and Sharp, 1992, p. 1609]. Sharp [1988, p. 278] noticed that recharge from bedrock aquifers may be an important factor in the water budget of alluvial aquifers, “…but because alluvium is usually more permeable, the effects are less pronounced. The major evidence for this type of recharge shows in water chemistry.” Zones of anomalous water chemistry were found in the alluvium of the Missouri River, the Ohio River, and the Arkansas River, but in smaller alluvial systems recharge from the bedrock was found to be proportionately more important, especially in carbonate bedrock terrain. Low aquitard permeability causes difficulties in quantifying this recharge [e.g., Neuzil, 1994].
 Such conditions are common in alluvial valleys in the midwestern United States. For example, in the Platte River (Missouri River basin), fine sediments of eolian origin separate large areas of the alluvial aquifer from the sand and gravel High Plains Aquifer [e.g., McGuire and Kilpatrick, 1998]. Similar hydrostratigraphic conditions exist in various geologic environments (e.g., Florida [Motz, 1998], Netherlands [Heij, 1989], Hong Kong [Jiao and Tang, 1999]).
 To investigate the SDR in such hydrogeological conditions, I consider a stream in an alluvial valley, which is separated from the lower aquifer by a unit of thickness mA and low hydraulic conductivity KA (Figure 2). Sometimes, the hydraulic conductivity of the underlying aquifer can be comparable to the hydraulic conductivity of the alluvial aquifer K, and the underlying aquifer serves as a source bed. This concept of the source bed will be discussed in light of the groundwater budget later.
 These hydrogeological conditions introduce a new paradigm in analytical SDR assessment. The emphasis shifts from consideration of the streambed properties and stream partial penetration to the effect of induced recharge of the alluvial aquifer from a source bed (leakage). Therefore the major purpose of this paper is to investigate the effects of induced recharge on stream depletion. I will demonstrate that the maximum SDR may vary in the range from 0 to 100% of the pumping rate of an individual well. I define maximum stream depletion rate (MSDR) as a maximum fraction of the pumping rate supplied by stream depletion.
 The objectives of this paper are as follows: (1) to derive an estimate of the transient SDR induced by an individual well for streams in leaky aquifers; (2) to estimate the maximum SDR (MSDR), which is reached in steady state conditions; (3) to develop estimates of the induced recharge (the leakage from the lower source-bed aquifer); and (4) to assess the effects of the alluvial valley width on the MSDR.