A practical middle ground between a single-well analytical solution and a full basin-scale numerical model. ADFs combine an analytical depletion curve with a geometric rule for splitting depletion among many stream reaches, scaling efficiently to thousands of wells.
SGMA Depletion Series Page 04 of 07
The problem ADFs solve
The classic analytical solutions (page 02) work for one well and one straight, infinite stream. A real SGMA basin has hundreds of wells and a branching stream network. A full numerical model can answer the question, but takes months of effort and is awkward for "what if we move this well" analyses.
ADFs let you keep the analytical machinery, but route each well's depletion onto a real stream network using one of a handful of geometric apportionment rules.
Three-step process
Compute depletion curve for each well using an analytical solution (Hunt, Glover, etc.) — gives Qs(t)/Q.
Apportion that depletion to nearby stream reaches using one of several geometric methods (see below).
Sum across wells for each reach to get the total depletion as a function of time.
Implemented in the open‑source R package streamDepletr (Zipper, 2020).
Interactive: how does each method apportion a well?
Drag the well around the stream network. The three methods below show different ways of splitting the well's stream depletion among the reaches it can influence. None of them are wrong — they encode different geometric assumptions about how groundwater finds its way to streams.
Drag the orange well in the map. Reach colors show apportioned fraction (darker = higher). The bar chart on the right gives the numerical split.
Figure 1. Left: apportioned fraction of well's total depletion to each reach at the current time. Right: total depletion fraction Qs/Q over time, split by reach. Reaches outside the search radius receive zero apportionment. After Zipper et al. (2019, 2021).
The three apportionment methods
Inverse Distance (ID)
Each reach gets a weight proportional to 1/di, then weights are normalized. Distance is measured to the closest point on each reach.
fi = (1/di) / Σ (1/dj)
Tends to spread depletion widely. Use when the well is in a network of multiple, similar-size reaches.
Inverse Distance Squared (IDS²)
Same idea, but distance squared. Closer reaches dominate more strongly.
fi = (1/di2) / Σ (1/dj2)
Concentrates depletion in the nearest reach. Default in streamDepletr; matches numerical-model benchmark best in Zipper et al. (2019) tests.
Thiessen (Web)
Build Thiessen polygons around vertices/reaches; weight each reach by the area of its polygon that lies within the search radius around the well.
fi ∝ Areai ∩ Search
Topology-aware — accounts for which reaches are "shielded" by intervening reaches. Better in stream networks with parallel reaches.
Strengths, weaknesses, and benchmarks
Where ADFs shine
Many wells, many streams — runs in seconds where MODFLOW takes hours.
Scenario testing — easy to add/remove wells, change pumping, rerun.
Open-source & reproducible — code, parameters, and assumptions are auditable.
Adequate for first-order GSP screening — published benchmarks against MODFLOW show median R² > 0.8 across diverse settings (Zipper et al. 2019).
Bridge between Theis and MODFLOW — useful for prioritizing where a fuller numerical model is needed.
Where ADFs struggle
Strongly heterogeneous aquifers (paleochannels, faults) — apportionment rule isn't aware of K variability.
Boundary effects near basin edges or impermeable contacts.
Multilayer / leaky systems where vertical anisotropy matters.
Disconnected streams (page 01) — assumption of constant connection breaks.
Tightly clustered wells where superposition adequacy needs checking.
When the answer needs to defend a specific compliance threshold — usually requires numerical confirmation.
A practical workflow for an SGMA basin
Map ISW reaches — identify gaining/losing-connected segments from gaging, temperature, isotopes, and head data.
Inventory wells — locations, pumping rates and schedules (often the hardest data step).
Estimate aquifer parameters — T, Sy/S, and streambed conductance λ; can be reach-specific.
Run an ADF with all three apportionment methods to bracket uncertainty.
Identify "hot" reaches and wells — those contributing the most to depletion.
Confirm with a numerical model for high-stakes management decisions (e.g., curtailing specific wells).
Key references in the project library
Zipper, S.C., Dallemagne, T., Gleeson, T., Boerman, T.C. & Hartmann, A. (2019). Rapid and accurate estimates of streamflow depletion caused by groundwater pumping using analytical depletion functions. WRR 55, 5807–5829.
Zipper, S.C. (2020). Introduction to streamDepletr (R package vignette).
Zipper, S.C., Farmer, W.H., Brookfield, A. et al. (2021). Streamflow depletion estimation for conjunctive water management in a heavily-stressed aquifer using analytical depletion functions.
Li, Q., Zipper, S.C., Hyndman, D.W. et al. (2021). Too many streams and not enough time or money: analytical depletion functions for streamflow depletion estimates. Groundwater.
Huggins, X., Zipper, S.C., Hyndman, D.W. et al. (2018). Streamflow depletion modeling: methods for an adaptable and conjunctive water management decision support tool. JAWRA.
Reeves, H.W., Hamilton, D.A., Seelbach, P.W. & Asher, A.J. (2009). Ground-Water-Withdrawal Component of the Michigan Water-Withdrawal Screening Tool. USGS SIR 2009‑5003.
Tolley, D. et al. (2020). Streamflow depletion analysis for an intermittent stream, Scott Valley, CA — California ADF application.
Zipper, S.C., Carah, J.K., Dillis, C. et al. (2019). Cannabis and residential groundwater pumping impacts on streamflow and ecosystems in Northern California. Environ. Res. Commun.