Massachusetts-Rhode Island Water Science Center

U.S. Geological Survey, Open-File Report 98-415A

By Paul M. Barlow and Allen F. Moench

A product of the Ground-Water Resources Program

This report is available for viewing and printing as five Portable Document Format files. The outside cover can also be viewed and printed. Select the files you are interested in and click on each link to open.

1. Titles and Contacts (2 pages)2. Preface, Contents, and Conversions (10 pages)

3. Report body (54 pages)

4. Attachment 1 - Derivation of Analytical Solutions (22 pages)

5. Attachment 2 - Documentation of Computer Programs STLK1 and STWT1 (10 pages)

Outside front cover (1 page)

Analytical solutions to the ground-waterflow equation are derived for ten cases of hydraulic interaction between a stream and a confined, leaky, or water-table aquifer. The ten aquifer types for which analytical solutions are derived are: a semi-infinite or finite-width confined aquifer; a semi-infinite or finite-width leaky aquifer with constant head overlying the aquitard; a semi-infinite or finite-width leaky aquifer with an impermeable layer overlying the aquitard; a semi-infinite or finite-width leaky aquifer overlain by a water-table aquitard; and a semi-infinite or finite-width water-table aquifer. All aquifer types allow for the presence or absence of a uniform semipervious streambank. Of primary interest are newly derived solutions for water-table aquifers and for leaky aquifers overlain by water-table aquitards.

Two computer programs are described that evaluate the analytical solutions for time-varying stream-stage or recharge stresses that are specified by the user. The programs can simulate the effects of stream-stage fluctuations for all aquifer types. However, simulation of basin-wide recharge or evapotranspiration at the water table is permitted only for water-table aquifers and leaky aquifers overlain by a water-table aquitard. For these aquifer types, effects of recharge or evapotranspiration can be simulated alone or in combination with stream-stage fluctuations. The computer programs use the convolution relation to calculate changes in ground-water levels at an observation well or observation piezometer, seepage rates at the stream-aquifer boundary, and bank storage. The program designated STLK1 was developed for application to confined and leaky aquifers, and the program designated STWT1 was developed for application to water-table aquifers. The programs can be applied to the analysis of a passing flood wave, determination of ground-water discharge rates in response to recharge, determination of aquifer hydraulic properties, design of streamaquifer data-collection networks, and testing of numerical-model computer codes. Instructions are provided for constructing the necessary data-input files for the programs, and three sample problems are described to provide examples of the uses of the programs.

Abstract

Introduction

Purpose and Scope

Description of Stream-Aquifer Systems

Previous Studies

General Theoretical Background

Governing Differential Equation and Initial and Boundary Conditions

Convolution Relations

Presentation of Analytical Solutions

Confined and Leaky Aquifers

Assumptions

Boundary-Value Problems

Laplace Transform Analytical Solutions

Water-Table Aquifers

Assumptions

Boundary-Value Problems

Laplace Transform Analytical Solutions

Evaluation of Analytical Solutions for Step Input

Confined and Leaky Aquifers

Water-Table Aquifers

Computer Programs STLK1 and STWT1—Implementation of Analytical Solutions for Time-Varying Inputs

Discretization of Convolution Relations

Instructions for Preparing Data-Input Files

Program STLK1

Program STWT18

Result and Plot Files

Sample Problems

Sample Problem 1—Response of a Confined Aquifer to a Sinusoidal Flood Wave

Sample Problem 2—Response of a Water-Table Aquifer to a Sinusoidal Flood Wave

Sample Problem 3—Response of a Water-Table Aquifer to Recharge

Summary

References Cited

Attachment 1. Derivation of Analytical Solutions

Step Response for Flow from a Semi-Infinite Confined or Leaky Aquifer

Head Distribution Due to a Step Change in Stream Stage

Dimensionless Seepage at Streambank Due to Step Change in Stream Stage

Step Response for Flow from a Finite-Width Confined or Leaky Aquifer with a

Semipervious Streambank

Head Distribution Due to a Step Change in Stream Stage

Dimensionless Seepage at Streambank Due to Step Change in Stream Stage

Step Response for Flow from a Semi-Infinite Water-Table Aquifer

Head Distribution Due to a Step Change in Stream Stage

Dimensionless Seepage at Streambank Due to Step Change in Stream Stage

Step Response for Flow from a Finite-Width Water-Table Aquifer with a Semipervious Streambank

Head Distribution Due to a Step Change in Stream Stage

Dimensionless Seepage at Streambank Due to Step Change in Stream Stage

Attachment 2. Documentation of Programs STLK1 and STWT177

Program STLK1

Narrative

List of Program Variables

Program STWT1

Narrative

List of Program Variables

1-4. Schematic diagrams showing:

1. Ground-water discharge from a water-table aquifer to a partially penetrating, hydraulically connected stream: (A) laterally extensive (semi-infinite) aquifer; and (B) narrow aquifer of finite width

2. Types of aquifers for which analytical solutions are derived: (A) confined; (B) leaky, with a constant head overlying the aquitard; (C) leaky, with an impermeable layer overlying the aquitard; (D) leaky, overlain by a water-table aquitard; and (E) water table (unconfined)

3. Response of stream-aquifer system to flood wave: (A) rise of stream stage and seepage of streamflow into aquifer as bank storage; (B) stream-stage hydrograph; (C) seepage hydrograph; (D) ground-water-head hydrograph; and (E) bank-storage-volume hydrograph

4. Response of stream-aquifer system to a gradual recharge event: (A) rise of water table; (B) recharge hydrograph; (C) ground-water-head hydrograph; and (D) ground-water-discharge hydrograph

5-14.Schematic sections showing:

5. Semi-infinite, confined aquifer (A) without semipervious streambank material and (B) with semipervious streambank material

6. Finite-width, confined aquifer (A) without semipervious streambank material and (B) with semipervious streambank material

7. Semi-infinite, leaky aquifer with constant head overlying the aquitard (case 1) (A) without semipervious streambank material and (B) with semipervious streambank material

8. Finite-width, leaky aquifer with constant head overlying the aquitard (case 1) (A) without semipervious streambank material and (B) with semipervious streambank material

9. Semi-infinite, leaky aquifer with impermeable layer overlying the aquitard (case 2) (A) without semipervious streambank material and (B) with semipervious streambank material

10. Finite-width, leaky aquifer with impermeable layer overlying the aquitard (case 2) (A) without semipervious streambank material and (B) with semipervious streambank material

11. Semi-infinite, leaky aquifer overlain by a water-table aquitard (case 3) (A) without semipervious streambank material and (B) with semipervious streambank material

12. Finite-width, leaky aquifer overlain by a water-table aquitard (case 3) (A) without semipervious streambank material and (B) with semipervious streambank material

13. Semi-infinite, water-table aquifer (A) without semipervious streambank material and (B) with semipervious streambank material

14. Finite-width, water-table aquifer (A) without semipervious streambank material and (B) with semipervious streambank material

15-20. Graphs showing change in (A) ground-water head and (B) seepage rate to aquifer, for 1-foot increase in stream stage:

15. Semi-infinite confined aquifer with and without semipervious streambank material

16. Finite-width confined aquifer with and without semipervious streambank material

17. Finite-width and semi-infinite confined aquifers

18. Semi-infinite leaky aquifer with constant head overlying the aquitard

19. Semi-infinite leaky aquifers

20. Semi-infinite water-table aquifer without semipervious streambank material

21-22. Graphs showing:

21. Change in ground-water head for 1-foot increase in stream stage at several vertical positions in a semi-infinite water-table aquifer

22. (A) Change in ground-water head and (B) seepage rate to aquifer, for 1-foot increase in stream stage; semi-infinite water-table aquifer and leaky aquifer overlain by water-table aquitard

23. Schematic diagram showing continuous and discretized (A) stream-stage and (B) recharge hydrographs

24-27. Representations showing:

24. Example data-input file for program STLK1

25. Example data-input file for program STWT1

26. Example result file for program STLK1

27. Example plot file for program STLK

28-29. Graphs showing (A) ground-water head at observation well, (B) seepage rate between stream and aquifer, and (C) bank storage in aquifer, for a one-day sinusoidal flood wave:

28. Semi-infinite confined aquifer

29. Semi-infinite water-table aquifer

30. Graph showing ground-water discharge for a one-day recharge event, finite-width water-table aquifer

31. Representation showing example data-input file for program STWT1 for recharge event (sample problem 3)

1. Dimensionless variables and variable groupings for confined and leaky aquifers

2. Dimensionless variables and variable groupings for water-table aquifers

3. Parameters and dimensions of the hypothetical confined and leaky aquifers

4. Parameters and dimensions of the hypothetical water-table aquifer

5. Input data format for program STLK1

6. Input data format for program STWT1

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**Suggested Citation:**

Barlow, P.M., Moench, A.F., 1998, Analyitical Solutions and Computer Programs for Hydraulic Interaction of Stream-Aquifer Systems, United States Geological Survey, Open-File Report 98-415A, 85 p.

For more information about USGS activities in Massachusetts, visit the USGS Massachusetts-Rhode Island Water Science Center home page, or contact the science center director Wayne H. Sonntag.