02 — Stream Depletion: Concept & Analytical Solutions

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Figure 1. Plan view of a pumping well at distance D from a stream. The drawdown cone (orange) expands with time; the analytical solutions calculate the resulting reduction in streamflow by superposing a mirror "image" injection well across the stream. Adapted from Theis (1941), Glover & Balmer (1954).

The core insight

A well doesn't pump "from" the stream the instant it turns on. It pumps from storage in the aquifer. But the resulting drawdown cone eventually intercepts the stream, replacing storage release with captured streamflow.

The fraction of pumping that comes from the stream — the depletion fraction Q_s/Q — grows from 0 toward 1 over a time scale set by the stream depletion factor SDF = S·D²/T.

SDF = S · D² / T
S = storativity (–) · D = distance to stream (L) · T = transmissivity (L²/T)

Jenkins (1968) showed that at t = SDF, about 28% of pumping has been captured; at t = 10·SDF, ~75%. The factor compresses three parameters into a single time constant for first-order screening.

Interactive analytical solution explorer

Inputs

Pumping rate Q 500 gpm

Distance to stream D 500 ft

Transmissivity T 5,000 ft²/d

Storativity S 0.10

Streambed conductance λ 100 ft/d

Solution

Diagnostics

Stream depletion factor SDF	— days
Time to 28% depletion (≈ SDF)	— days
Time to 50% depletion	— days
Time to 90% depletion	— days
Depletion rate at 1 yr	— gpm (—% of Q)
Depletion volume at 1 yr	— AF
Depletion volume at 5 yr	— AF

Units: gpm = gallons per minute; AF = acre-feet; conversions assume 1 gpm = 0.1923 AF/yr.

Figure 2. Left: depletion rate as a fraction of pumping rate vs. time. Right: cumulative depletion volume as a fraction of cumulative pumping. Both use the same parameter set. The Glover curve is the upper bound; Hunt with finite streambed conductance reduces and delays depletion. Solutions assume horizontal, homogeneous, isotropic aquifer with a fully penetrating linear stream.

The solution family — what to use when

Theis (1941)

First demonstration that a stream and a pumping well form a coupled system; introduced the image‑well analogy. Foundational concept paper; not a stand-alone depletion formula.

Use: conceptual reference.

Glover & Balmer (1954)

Closed-form solution for the image‑well configuration: fully penetrating stream, no streambed resistance, homogeneous confined or unconfined aquifer. Gives the depletion upper bound.

Q_s/Q = erfc(√(SD²/4Tt))

Use: screening; first‑order estimates.

Jenkins SDF (1968)

Tabulated dimensionless rate and volume functions of t/SDF; turned Glover into a desktop calculation. Still the foundation of many state screening tools (e.g., Colorado, Nebraska).

Use: rapid screening; regulatory permitting.

Hantush (1965)

Adds a semipervious streambed — a clogging layer that resists exchange. The stream is no longer a perfect "constant head" boundary; depletion is delayed and reduced.

Use: alluvial reaches with measurable streambed clogging.

Hunt (1999, 2003)

Unified analytical solution with a streambed conductance λ. Hunt (1999) for unconfined aquifers; Hunt (2003) extends to semiconfined. The most commonly used solution in modern depletion screening tools.

λ = K_bed·W / b_bed

Use: default for analytical screening.

Specialized extensions

Zlotnik (2004): maximum depletion rate in leaky systems. Butler & Tsou (2003): finite-width streams, shallow penetration. Chen & Yin (2004): partial penetration. Yeh et al. (2008): wedge-shaped aquifers. Hunt (2008): springs.

Use: when the assumptions of Hunt are clearly violated.

Assumptions to test before you trust the answer

The classic solutions assume:

Aquifer is horizontal, infinite, homogeneous, isotropic.
Stream is straight, infinite, fully penetrating (or with a known streambed conductance for Hunt/Hantush).
Pumping is at a constant rate from a single well.
Stream stage is not affected by depletion (constant boundary head).
Aquifer remains connected to the stream throughout (no disconnection).
No recharge, no other boundaries.

When to upgrade beyond analytical

Multiple wells with overlapping cones → use superposition or go numerical.
Pumping varies seasonally → Theis with variable Q (segment + superpose) or numerical.
Stream goes dry / disconnects → analytical breaks down (see Brunner 2009 / 2011).
Layered aquifers, dipping beds, bedrock pinchouts → numerical (MODFLOW, IWFM, ParFlow).
Stream meanders sharply, partial penetration matters → see Chen & Yin (2004), Huang et al. (2015).
Multiple streams in the capture area → Analytical Depletion Functions (see page 04).

Key references in the project library

Theis, C.V. (1941). The effect of a well on the flow of a nearby stream. Transactions, AGU 22(3): 734–738.
Glover, R.E. & Balmer, G.G. (1954). River depletion resulting from pumping a well near a river. Transactions, AGU 35(3): 468–470.
Hantush, M.S. (1965). Wells near streams with semipervious beds. JGR 70(12): 2829–2838.
Jenkins, C.T. (1968). Computation of rate and volume of stream depletion by wells. USGS TWRI 4‑D1.
Hunt, B. (1999). Unsteady stream depletion from groundwater pumping. Groundwater 37(1): 98–102.
Hunt, B. (2003). Unsteady stream depletion when pumping from a semiconfined aquifer. J. Hydrologic Eng. 8(1): 12–19.
Zlotnik, V. (2004). A concept of maximum stream depletion rate for leaky aquifers in alluvial valleys. WRR 40, W06507.
Butler, J.J., Zlotnik, V.A. & Tsou, M.-S. (2001, 2003). Finite-width streams of shallow penetration. KGS reports.
Chen, X. & Yin, Y. (2004). Semianalytical solutions for stream depletion in partially penetrating streams. Groundwater 42(1): 92–96.
Hunt, B. (2008). Improved spring depletion solution and analysis.
Huang, C.-S., Yang, S.-Y. & Yeh, H.-D. (2018). Review of analytical models for stream depletion: guide to model selection. J. Hydrology.
Barlow, P.M. & Leake, S.A. (2012). Streamflow depletion by wells. USGS Circular 1376 (Appendix B compiles the analytical solutions in consistent notation).