Well-Induced Stream Depletion and Groundwater Return Flow: Estimating Impact Schedules with a Finite-Difference Spreadsheet

Robinson, Samuel Collin

doi:10.1061/(ASCE)WR.1943-5452.0001230

Open access

Technical Notes

Apr 30, 2020

Well-Induced Stream Depletion and Groundwater Return Flow: Estimating Impact Schedules with a Finite-Difference Spreadsheet

Author: Samuel Collin Robinson, M.ASCE [email protected]Author Affiliations

Publication: Journal of Water Resources Planning and Management

Volume 146, Issue 7

https://doi.org/10.1061/(ASCE)WR.1943-5452.0001230

PDF

Abstract

Allocation of water from stream-aquifer systems must account for stream depletion from well withdrawal and for accretion from groundwater return flow. Both are subject to attenuation and delay. Transient impact assessment tools include: simplified analytical solutions; special finite-difference groundwater modeling software, such as MODFLOW; and the Delayed Impact Calculator (Calculator), a spreadsheet programmed by the author for streamlined finite-difference estimation of dynamic schedules. Well-induced stream depletion results from the Calculator are compared to results from established tools for a previously documented scenario. Similar comparisons are described for modified cases. Calculator results match MODFLOW (version 1.12.00) results (

y = 1.0000 x

,

r^{2} = 1.0000

, 3 sets, 90 pairs each). The finite-difference tools estimate somewhat quicker stream response to groundwater impulse than do analytical solutions derived for ideal semi-infinite aquifers. A previously documented schedule estimate for a simple groundwater return flow scenario is confirmed by the Calculator. An example of Calculator utility is given, treating return flow to a reach above an active surface diversion.

Introduction

Integrating administration of surface water and groundwater enables maximization of overall benefit from finite resources. Effective allocation of water accounts for stream depletion from well withdrawals and for accretion from groundwater return flow, including attenuation and delay (Colorado General Assembly 2018). These phenomena are not typically apt for direct measurement.

Aquifer test results from well pumping give information useful for estimation of transient impact timing (Pattle Delamore Partners Limited and Environment Canterbury 2000). Available tools for schedule estimation include analytical solutions, developed by Theis, Glover, Hunt and others (Hunt 2014), as well as special modeling software, developed by the United States Geological Survey (Harbaugh 2005). Where stream-aquifer system characteristics do not conform to oversimple idealization, analytical solutions are not appropriate. Often, field acquisition of input data adequate to avail complex modeling is prohibited by cost. These circumstances present opportunity to deploy a streamlined finite-difference tool. Spreadsheets have given platform to this class of instrument for decades (Olsthoorn 1985; Hancock and Heaney 1987).

The Delayed Impact Calculator (Calculator) is a Microsoft Excel 2010 workbook, programmed by the author to estimate stream depletion and accretion schedules using a finite-difference approach within a modest scope of complexity, inspired by the work of Bittinger (1967). The Calculator can be obtained for free online from the HydroShare repository (Robinson 2020).

The remainder of the present document gives: description of methods employed by the Calculator, presentation of well withdrawal and groundwater return flow examples, and comparison of results.

Delayed Impact Calculator Methods

The Calculator quantifies stream gain or loss and provides response functions that can be superimposed on general aquifer dynamics (Bittinger 1967). Where cumulative effects are so significant or aquifer behavior varies so greatly, relative to baseline conditions, that superposition is not appropriate, a regional groundwater model should be employed instead (Pattle Delamore Partners Limited and Environment Canterbury 2000).

The Calculator treats scenarios defined within these conditions: stream and aquifer share a direct hydraulic connection; perennial streamflow exists; stream piezometric head remains constant; groundwater flow transmission occurs by differential piezometric head pushing water horizontally through a single principal aquifer layer; aquifer hydraulic conductivity is isotropic; and the system is bounded on sides and bottom by impermeable material that passes no-flow. Intermittent pumping, irregular aquifer dimensions, user-configured impermeable barriers, meandering stream alignments, and spatially-varied geological characteristics are permitted.

Finite-Difference Formulation

Given the listed conditions, application of Darcy’s law and conservation of volume yields the following version of the transient groundwater flow equation, still a nonlinear differential formula (Bittinger 1967)

\frac{\partial}{\partial X} (K h \frac{\partial H}{\partial X}) + \frac{\partial}{\partial Y} (K h \frac{\partial H}{\partial Y}) = S \frac{\partial H}{\partial t} + \frac{Q}{(\partial X) (\partial Y)}

(1)

Finite-difference expression allows numerical approximation of the analytically intractable form by representing the continuous function with discrete values corresponding to specific times and locations. The Calculator divides time into periods. Those treated here are 1 day or longer, specified in either Days or nominal Months of 30.4375 days. Periods are subdivided into computation steps. Lowercase n is the time step index. Spatial dimensions are divided into horizontal increments in the

X

and

Y

directions. Head, here denoting vertical distance from datum to piezometric surface, is represented at nodes in the horizontal center of each cell. The distance between adjacent nodes is half of the sum of the pair of grid increments. Lower case i and j are node indices. For unconfined aquifers, saturated thickness updates with each time step, either at the same time by recursion, or after step end, per specification. The Calculator offers several types of average for use with varied thickness and conductivity, for example, the distance-weighted arithmetic mean

{(\bar{K h})}_{i + 1} = \frac{{(K h \frac{Δ X}{2})}_{i} + {(K h \frac{Δ X}{2})}_{i + 1}}{\frac{Δ X_{i}}{2} + \frac{Δ X_{i + 1}}{2}}

(2)

Flow through a streambed is determined by the difference between stream channel piezometric head and aquifer head, as well as by permeability of the bed material. The relationship at play is Darcy’s law, with parameters described by Harbaugh (2005), on pages 6-6 and 6-8, from which this equation was adapted

Q_{R} = \frac{K^{'} L W (H - H_{R})}{M}

(3)

Implicit Representation

Discretization enables several different finite-difference representations of Eq. (1). In consideration of mathematical stability, the Calculator employs a backward-difference approach that approximates flow during a time step based on head condition at step end. This requires implicit expression, meaning relationship of multiple unknowns. Equations for adjacent locations go together as a system that can be solved simultaneously.

The backward-difference form presented by Bittinger (1967) is for uniform spatial increments and permissive streambeds. Reconfiguration extends utility. Distances between nodes are substituted for increment lengths, distance-weighted averaging is enabled, and streambed characteristics are included. The latter is accomplished by replacing

Q

with

Q_{I}

, for groundwater impulse, or

Q_{R}

, for flow through the streambed, as appropriate. After rearrangement, Eq. (4) results. Unspecified indices are understood to be

i

,

j

, and

n

A H_{n + 1, j - 1} + B H_{n + 1, i - 1} + C H_{n + 1, j + 1} + D H_{n + 1, i + 1} - E H_{n + 1} = F

(4)

A = \frac{{(\bar{K h})}_{j - 1} Δ X}{(\frac{Δ Y_{j - 1} + Δ Y}{2})}

B = \frac{{(\bar{K h})}_{i - 1} Δ Y}{(\frac{Δ X_{i - 1} + Δ X}{2})}

C = \frac{{(\bar{K h})}_{j + 1} Δ X}{(\frac{Δ Y_{j + 1} + Δ Y}{2})}

D = \frac{{(\bar{K h})}_{i + 1} Δ Y}{(\frac{Δ X_{i + 1} + Δ X}{2})}

E = A + B + C + D + \frac{(Δ X) (Δ Y) S}{Δ t} + \frac{K^{'} L W}{M}

F = Q_{I} - \frac{(Δ X) (Δ Y) S H}{Δ t} - \frac{K^{'} L W H_{R}}{M}

Head differential across the streambed, (

H_{n + 1} - H_{R}

), appears in Eq. (4): although split into parts, with one put to each side of the equal sign for solution. Future head values are determined together and then used to compute stream depletion or accretion.

Matrix Composition

Eq. (4) is written for each location in the grid representing the aquifer, with reference to head and parameters at that location and each adjacent location. A factor of

- 1

is applied to both sides of each equation. The resulting coefficient values are used to populate a matrix consisting of a row and column for each grid location, augmented by a constants column. The format for a single time step of a 3 by 3 grid is shown as Fig. 1. Coefficient subscripts are omitted to aid legibility. A value of each coefficient is calculated specific to each row.

Fig. 1. Augmented matrix for a simple $3 \times 3 (\times 1)$ grid.

Solution Algorithm

The system of equations can be solved by any of several techniques. In the interest of efficiency, the Modified Incomplete Cholesky Preconditioned Conjugate Gradient (MICCG) method is employed. MICCG was described by Hill (1990). The Calculator implements the procedure with a user-specified closure threshold.

Using the author’s desktop personal computer, computation of each example presented below was completed by the Calculator in 5 s or less. Complex scenarios, with larger grids and more time steps, take longer. Unexpected computation time is avoided by starting with modest numbers of increments and periods, then working up.

Response Output

Although heads are calculated first, the principal output is an impact schedule table that gives computed volumes of stream accretion or depletion by period. Both total and zone response are presented, allowing segregation by stream reach. Secondary output type is user-selected. Options include: Period Ratio, or stream response volume each period divided by groundwater impulse during the same period; Response Ratio, stream response volume each period divided by impulse over the entire span; and Cumulative Ratio, response divided by impulse as totals through the end of each period. For visualization, volume output is plotted versus input on an impact schedule column chart.

Comparison Standards

Impact schedule calculation tool performance can be investigated by comparison of results. Computations made for previously documented scenarios, modified scenarios, and a fresh scenario offer insight. Results documented in official publications provide useful standards, as do results from established calculation tools.

Documented Estimates

Results for a well-induced stream depletion scenario, fit for comparison, are presented by Pattle Delamore Partners Limited and Environment Canterbury (2000). Groundwater return flow results suitable for comparison are given by Bittinger (1967). These documented estimates provide touchstone support to further investigation.

Analytical Tool

Several established tools implement analytical solutions. The Alluvial Water Accounting System is such a tool. It employs the Glover-Balmer solution, and is maintained by the Integrated Decision Support Group at Colorado State University. For the present effort, another tool was obtained for comparison, a spreadsheet programmed with several analytical solutions for stream depletion, called Streamdepletionv3.xls (Environment Canterbury 2019).

Special Software

MODFLOW is a specialized groundwater analysis computer program accepted as a standard. It applies finite-difference techniques capable of handling highly complex scenarios of large scope (Harbaugh 2005). The United States Geological Survey offers the program with its ModelMuse interface (Winston 2019).

Well-Induced Stream Depletion Analyses

Well-induced stream depletion occurs where net removal of water from an aquifer hydraulically connected to a surface stream causes diminution of streamflow (Barlow and Leake 2012). A depletion scenario documented by Pattle Delamore Partners Limited and Environment Canterbury (2000, p. 43) has a single well, completed in an unconfined aquifer. The average rate of net withdrawal of groundwater is given as

20 L / s

. The well is located 200 m from a stream that is 5 m wide. The aquifer hydraulic conductivity is

80 m / day

, thickness is 10 m, and 0.1 is the storage coefficient. These parameters are common to the following comparisons.

Restrictive Streambed

Streambed characteristics are given as thickness of 1 m and vertical hydraulic conductivity of

0.8 m / day

. Depletion rate ratios are reported at 0.83, 5, 30, 60, and 90 days after pump start. Those at or after 1 day are of interest. With 100% Irrigation Efficiency to designate all pumped water as withdrawn from the aquifer, the Hunt (1999) worksheet of Streamdepletionv3.xls gives results identical to those reported.

For comparison of like quantities, depletion ratios reflecting instantaneous rates are converted to ratios reflecting approximate volumes, since finite-difference calculations give results for periods instead of instants. Conversion is accomplished by averaging output from period beginning and period end. As example, for the 5th day, that is from time = 4 to time = 5, the Streamdepletionv3.xls output ratios of 0.1176 and 0.1465, are averaged to yield 0.1321, a slightly different number than the 0.15 reported by rounding 0.1465. The converted Period Ratios are presented by the Hunt (1999) column of Table 1.

Table 1. Depletion estimates, restrictive streambed,

K = 80 m / day

Day	Hunt (1999)	MODFLOW	Calculator
5	0.1321	0.1350	0.1350
30	0.4496	0.4643	0.4643
60	0.5731	0.5916	0.5916
90	0.6379	0.6562	0.6562

Note: These are ratios of stream depletion to well pumping by volume each day.

Next, the scenario is analyzed by MODFLOW with a single-layer grid, Fig. 2. During the first few days, analysis is sensitive to calculation time step. As the number of steps increase, the degree of sensitivity decreases. The difference between the day 5 Period Ratio found with 1 step per day and the ratio found with 4 steps per day is 4.4% of the 4-step result, while the difference from 4 to 8 steps is just 0.7% of the smaller 8-step result. 8 steps per day are specified to provide refined resolution, without making step-by-step calculation review impractical. The Well Package is invoked, with direct specification of a

- 1,728 m^{3} / day

pumping rate from 0 to 90 days. The River package is given constant stage, as well as direct conductance values specific to each stream cell. Computation of the 720 time steps takes about 7 s. River leakage output is multiplied by time step and aggregated by day, then divided by pumping, to yield Period Ratios, given by the MODFLOW column of Table 1.

Fig. 2. Finite-difference grid for well pumping example.

Comparison of the Hunt (1999) and MODFLOW results reveals that the two approaches give similar estimates; however, MODFLOW estimates quicker depletion than does the analytical tool. The day 90 ratio is 2.9% greater. This shows that a grid gives only approximate representation of continuous features, and that assumption of semi-infinite extent is similarly fraught. Early MODFLOW results are sensitive to increment specification, particularly dimensions of the cell holding the well. Depletion estimates during the 90-day period are only minimally altered by quadrupling grid domain area. Depending on conductivity, bed type, and pumping rate, later results are sensitive to domain size. Before day 365, the limited extent of the Fig. 2 grid would occasion compounding drawdown that would dominate system behavior. For long simulation of intensive pumping, domain area should match physical aquifer extent.

The Calculator is similarly configured, with same-step thickness update and distance-weighted harmonic mean characteristic averaging selected. Stream head and bed level are both set equivalent to initial aquifer head. Incision adjustment is “None.” Solution threshold order is 3. Period Ratio secondary output is selected.

1,728 m^{3}

of well pumping per day is input as positive by convention. Computation takes 5 s. Output is presented for the reported days by the Calculator column of Table 1. Inspection reveals that Calculator results match MODFLOW results. Results for each of 90 days are correlated statistically. The relationship between the estimates is

y = m x

, where

m

represents proportionality, here 1.0000. The coefficient of determination or square of the correlation coefficient,

r^{2}

, is also 1.0000, so all variation of output from one tool can be predicted to a high degree of precision by output from the other (Lomax 1998). Calculator results and MODFLOW results are the same.

Permissive Streambed

Information on streambed characteristics is often lacking. Also, in some cases, streambed permeability does not restrict flow to or from groundwater. Hence, the equivalent Theis-Jenkins and Glover and Balmer solutions are still in wide use (Hunt 2014). Therefore, calculations are repeated for a permissive bed. This condition is inherent to the Theis-Jenkins worksheet of Streamdepletionv3.xls. With streambed vertical conductivity adjusted to 1,000 times aquifer conductivity, the Hunt (1999) worksheet gives results equivalent to the Theis-Jenkins worksheet. Application of the same adjustment to MODFLOW and the Calculator gives the results presented by Table 2. Here, there is less difference between analytical and finite-difference estimates, just 0.7% at day 90. Again, Calculator results correlate 1:1 with MODFLOW results. The Calculator also offers a direct permissive bed option.

Table 2. Depletion estimates, permissive streambed,

K = 80 m / day

Day	Theis-Jenkins	MODFLOW	Calculator
5	0.4543	0.4525	0.4525
30	0.7709	0.7788	0.7788
60	0.8376	0.8441	0.8441
90	0.8673	0.8733	0.8733

Note: These are ratios of stream depletion to well pumping by volume each day.

Higher Conductivity

To provide perspective, the permissive streambed scenario is modified by changing the saturated horizontal hydraulic conductivity of the aquifer to

160 m / day

. While conductivity varies over a wider range, more dramatic change would necessitate further modification to keep the scenario at once meaningful and reasonable. The doubling gives a substantial difference without occasioning cascading complexity. Table 3 presents results for the modified scenario. For this case, MODFLOW estimates the day 90 depletion ratio as 2.4% greater than Theis-Jenkins. As shown previously, Calculator results statistically match MODFLOW results to a high degree of precision shown by Fig. 3 (

y = 1.0000 x

,

r^{2} = 1.0000

, 90 pairs).

Table 3. Depletion estimates, permissive streambed,

K = 160 m / day

Day	Theis-Jenkins	MODFLOW	Calculator
5	0.5966	0.6024	0.6023
30	0.8369	0.8459	0.8459
60	0.8848	0.8964	0.8964
90	0.9059	0.9279	0.9279

Note: These are ratios of stream depletion to well pumping by volume each day.

Fig. 3. Correlation of estimated ratios of stream depletion, permissive bed, $K = 160 m / d a y$ .

Groundwater Return Flow Analyses

Groundwater return flow occurs where deep percolation or well injection let residual water accrue to an aquifer hydraulically connected to a surface stream, causing streamflow to be greater than it would be otherwise. In many cases the water was taken from the stream prior to use, hence the term return flow; however, the label has come to be applied regardless of water source. Groundwater return flow from deep percolation of leftover irrigation water is different than surface return flow from field runoff, or tailwater, principally in the delay and attenuation undergone prior to impacting the stream (e.g., Jones and Cech 2009, p. 110).

Documented Case

Model 153 lends its simplicity to concise description (Bittinger 1967, Appendixes C and D). Return flow from irrigation of one side of an alluvial valley accrues to a stream with permissive bed. The model is made up of 1 row for the stream, with width of 30.48 m, and 16 rows for the adjacent irrigated area, with widths of 201.17 m. Tight bedrock gives bottom to the alluvium, at a depth of 15.24 m below the initial water table. Specific yield is 0.20, and hydraulic conductivity is given as “Permeability” of

0.003048 m / s

, or

263.35 m / day

. Time periods of 10 days are computed with a single step each. About 0.03048 m of water enters the aquifer by deep percolation over the first period and no water enters the aquifer thereafter. These data were input to the Calculator with a row length of 201.17 m and equivalent deep percolation volume of

19,736 m^{3}

during just the first period (0 thereafter). Next-step aquifer property update timing and Response Ratio secondary output were specified. The ratios output by the Calculator are the same as the documented results.

Return Flow by Reach

Delayed impact analysis can include complicating factors, such as surface water administration requiring differentiation of impact by stream reach, heterogeneous streambed permeability, and impermeable barriers. The Calculator empowers operators with elementary knowledge of Excel to analyze custom configurations including these features.

An instructive example has the site configuration and dimensions presented by Fig. 4. Uniform ground surface elevation is at the datum, initial aquifer head is 10 m below, and aquifer bottom is 53 m below. Specific yield is 0.20, and saturated horizontal hydraulic conductivity is

60 m / day

. Separate response zones are designated, Zone 1 being the reach of interest, upstream of a surface water diversion structure. The River has a restrictive bed, 2 m thick, with vertical hydraulic conductivity of

0.03 m / day

. The creek bed is 1 m thick, with hydraulic conductivity of

5 m / day

. Bed width is cell width. Stream head and bed elevation are both made equivalent to initial aquifer head. A variable series of 84 single-month percolation periods divided into 10 steps each is specified. Deep percolation arrives steadily throughout each period of the irrigation season and accrues to the aquifer over 4 seasons at the various impulse volumes given in cubic meters each month, from April through October, by Fig. 5. During the next 3 years there is no irrigation. How much does deep percolation contribute to river flow that can be taken for beneficial use by the surface water diversion structure after irrigation has stopped?

Fig. 4. Finite-difference grid for return flow by reach example.

Fig. 5. Impact schedule chart for return flow by reach example.

Distance-weighted harmonic mean characteristic averages and same-step update timing are chosen, as is a solution threshold order of 3. Incision adjustment is “None.” Computation completes during 4 s. Fig. 5 shows deep percolation input alongside stream accretion output. Inspection reveals that returns are markedly lagged, to different degrees by reach, as manifest by each time series having its own seasonality. Although the percolation schedule peaks during July, in year 4, total stream accretion peaks during October, and Reach 1 accretion peaks during December.

For a sample answer to the question about groundwater return flow contribution to surface diversion supply, preceding deep percolation is found to cause

230 m^{3}

of stream accretion in Zone 1 during period 51, or March of the 5th year, despite the passage of several months without percolation.

Conclusion

Transient well-induced stream depletion and groundwater return flow phenomena often defy direct measurement. Instead, water use plans are informed by engineering estimates of lagged impact schedules. Several types of groundwater flow analysis tools are available for schedule calculation. Where need overcomes economic limitation, analysis of aquifers with significant vertical flow or several distinct layers is accomplished by MODFLOW or other elaborate tools. Some simple cases lend themselves to treatment by tools based on idealized analytical solutions. For sites that are less ideal, the Delayed Impact Calculator allows treatment of a multitude of single-layer physical configurations. Calculator format provides streamlined data input and output processing features. Output from the Calculator exhibits a 1:1 correlation to output from MODFLOW. Given adequate domain, finite-difference estimates of stream response are close to, although somewhat greater than, estimates made by analytical solutions. Elementary knowledge of Excel is adequate to use the Calculator to efficiently estimate custom impact delay schedules for sound water allocation. Suggestions for improvement of the Calculator are welcome.

Notation

The following symbols are used in this paper:

$A - D$: flow transmission coefficients, each for a cell face, units of length-squared/time;
$E$: coefficient applicable to future head within a cell, units of length-squared/time;
$F$: expedited notation for determinate right side of flow equation, length-cubed/time;
$H$: potential energy as vertical piezometric aquifer head relative to datum, length;
$H_{R}$: piezometric head of free water surface in channel of river or stream, length;
$h$: water-transmitting vertical thickness of the aquifer, length;
$i$: node index along X axis;
$j$: node index along Y axis;
$K$: saturated horizontal hydraulic conductivity, length/time;
$K^{'}$: saturated vertical hydraulic conductivity of streambed material, length/time;
$\bar{K h}$: an average of conductivity and thickness products of adjacent cells, length-squared/time;
$L$: bed length, the longer of $Δ X$ and $Δ Y$ , length;
$M$: vertical thickness of bed material distinct from aquifer material, length;
$m$: coefficient of depletion volume estimate ratio proportionality, dimensionless;
$n$: time step index;
$Q$: rate of net flow out of aquifer, length-cubed/time;
$Q_{I}$: rate of groundwater impulse, positive out of aquifer, length-cubed/time;
$Q_{R}$: rate of flow through riverbed from aquifer, positive out of aquifer, length-cubed/time;
$r^{2}$: square of the Pearson Product-Moment Correlation Coefficient, dimensionless;
$S$: Storage Coefficient, dimensionless;
$t$: time;
$W$: bed width, the shorter of $Δ X$ and $Δ Y$ or a smaller user-specified value, length;
$X$: a horizontal distance, length;
$x$: depletion volume estimate ratio from a particular tool, dimensionless;
$Y$: the horizontal distance normal to $X$ , length;
$y$: depletion volume estimate ratio from another tool, dimensionless; and
$Δ$: the difference in value between locations one increment or step apart.

Data Availability Statement

All data used herein appear in the manuscript, figures, and tables. The computational tools can each be obtained for free online from the addresses given. The subject Calculator can be found in the HydroShare repository (Robinson 2020). It is a product of initiative without funding.

Mathematical processes employed by the Delayed Impact Calculator are described by the text. Visual Basic code that implements the calculations is protected to preserve integrity. All procedures of the code can be executed upon opening the workbook in a recent version of Excel with macros enabled.

Acknowledgments

The Delayed Impact Calculator Methods section includes material adapted from the instruction manual, another work by this author. Both include source references for citations of fact. Janet Alcon of Mesa County Libraries earned appreciation by facilitating access to literature. Anonymous reviewers and journal editors earned the gratitude of the author by making insightful suggestions for manuscript improvement.

References

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Information & Authors

Information

Published In

Journal of Water Resources Planning and Management

Volume 146 • Issue 7 • July 2020

Copyright

This work is made available under the terms of the Creative Commons Attribution 4.0 International license, https://creativecommons.org/licenses/by/4.0/.

History

Received: Feb 19, 2019

Accepted: Jan 13, 2020

Published online: Apr 30, 2020

Published in print: Jul 1, 2020

Discussion open until: Sep 30, 2020

Authors

Affiliations

Samuel Collin Robinson, M.ASCE [email protected]

P.E.

Professsional Engineer, Guide Water, P.O. Box 4033, Grand Junction, CO 81502. Email: [email protected]

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Abstract

Introduction

Delayed Impact Calculator Methods

Finite-Difference Formulation

Implicit Representation

Matrix Composition

Solution Algorithm

Response Output

Comparison Standards

Documented Estimates

Analytical Tool

Special Software

Well-Induced Stream Depletion Analyses

Restrictive Streambed

Permissive Streambed

Higher Conductivity

Groundwater Return Flow Analyses

Documented Case

Return Flow by Reach

Conclusion

Notation

Data Availability Statement

Acknowledgments

References

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History

Authors

Affiliations

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