Techniques and Methods 6–A40

## Documentation of the Surface-Water Routing (SWR1) Process for Modeling Surface-Water Flow with the U.S. Geological Survey Modular Groundwater Model (MODFLOW–2005)

### ABSTRACT

A flexible Surface-Water Routing (SWR1) Process that solves the continuity equation for one-dimensional and two-dimensional surface-water flow routing has been developed for the U.S. Geological Survey three-dimensional groundwater model, MODFLOW–2005. Simple level- and tilted-pool reservoir routing and a diffusive-wave approximation of the Saint-Venant equations have been implemented. Both methods can be implemented in the same model and the solution method can be simplified to represent constant-stage elements that are functionally equivalent to the standard MODFLOW River or Drain Package boundary conditions.

A generic approach has been used to represent surface-water features (reaches) and allows implementation of a variety of geometric forms. One-dimensional geometric forms include rectangular, trapezoidal, and irregular cross section reaches to simulate one-dimensional surface-water features, such as canals and streams. Two-dimensional geometric forms include reaches defined using specified stage-volume-area-perimeter (SVAP) tables and reaches covering entire finite-difference grid cells to simulate two-dimensional surface-water features, such as wetlands and lakes. Specified SVAP tables can be used to represent reaches that are smaller than the finite-difference grid cell (for example, isolated lakes), or reaches that cannot be represented accurately using the defined top of the model.

Specified lateral flows (which can represent point and distributed flows) and stage-dependent rainfall and evaporation can be applied to each reach. The SWR1 Process can be used with the MODFLOW Unsaturated Zone Flow (UZF1) Package to permit dynamic simulation of runoff from the land surface to specified reaches. Surface-water/groundwater interactions in the SWR1 Process are mathematically defined to be a function of the difference between simulated stages and groundwater levels, and the specific form of the reach conductance equation used in each reach. Conductance can be specified directly or calculated as a function of the simulated wetted perimeter and defined reach bed hydraulic properties, or as a weighted combination of both reach bed hydraulic properties and horizontal hydraulic conductivity. Each reach can be explicitly coupled to a single specific groundwater-model layer or coupled to multiple groundwater-model layers based on the reach geometry and groundwater-model layer elevations in the row and column containing the reach.

Surface-water flow between reservoirs is simulated using control structures. Surface-water flow between reaches, simulated by the diffusive-wave approximation, can also be simulated using control structures. A variety of control structures have been included in the SWR1 Process and include (1) excess-volume structures, (2) uncontrolled-discharge structures, (3) pumps, (4) defined stage-discharge relations, (5) culverts, (6) fixed- or movable-crest weirs, and (7) fixed or operable gated spillways. Multiple control structures can be implemented in individual reaches and are treated as composite flow structures.

Solution of the continuity equation at the reach-group scale (a single reach or a user-defined collection of individual reaches) is achieved using exact Newton methods with direct solution methods or exact and inexact Newton methods with Krylov sub-space methods. Newton methods have been used in the SWR1 Process because of their ability to solve nonlinear problems. Multiple SWR1 time steps can be simulated for each MODFLOW time step, and a simple adaptive time-step algorithm, based on user-specified rainfall, stage, flow, or convergence constraints, has been implemented to better resolve surface-water response. A simple linear- or sigmoid-depth scaling approach also has been implemented to account for increased bed roughness at small surface-water depths and to increase numerical stability. A line-search algorithm also has been included to improve the quality of the Newton-step upgrade vector, if possible.

The SWR1 Process has been benchmarked against one- and two-dimensional numerical solutions from existing one- and two-dimensional numerical codes that solve the dynamic-wave approximation of the Saint-Venant equations. Two-dimensional solutions test the ability of the SWR1 Process to simulate the response of a surface-water system to (1) steady flow conditions for an inclined surface (solution of Manning’s equation), and (2) transient inflow and rainfall for an inclined surface. The one-dimensional solution tests the ability of the SWR1 Process to simulate a looped network with multiple upstream inflows and several control structures. The SWR1 Process also has been compared to a level-pool reservoir solution. A synthetic test problem was developed to evaluate a number of different SWR1 solution options and simulate surface-water/groundwater interaction.

The solution approach used in the SWR1 Process may not be applicable for all surface-water/groundwater problems. The SWR1 Process is best suited for modeling long-term changes (days to years) in surface-water and groundwater flow. Use of the SWR1 Process is not recommended for modeling the transient exchange of water between streams and aquifers when local and convective acceleration and other secondary effects (for example, wind and Coriolis forces) are substantial. Dam break evaluations and two-dimensional evaluations of spatially extensive domains are examples where acceleration terms and secondary effects would be significant, respectively.

First posted August 17, 2012

Joseph D. Hughes
USGS Florida Water Science Center
10500 University Center Drive, Suite 215
Tampa, FL 33612

http://fl.water.usgs.gov/

### Suggested citation:

Hughes, J.D., Langevin, C.D., Chartier, K.L., and White, J.T., 2012, Documentation of the Surface-Water Routing (SWR1) Process for modeling surface-water flow with the U.S. Geological Survey Modular Ground-Water Model (MODFLOW–2005): U.S. Geological Survey Techniques and Methods, book 6, chap. A40 (Version 1.0), 113 p.

### Contents

Preface

Acknowledgments

Abstract

Introduction

Purpose and Scope

Process Overview and Highlights

Mathematical Representation of Surface-Water Routing

Diffusive-Wave Approximation

Reservoir-Routing Approximation

SWR1 Finite-Difference Approximation

SWR1 Process Concepts and Implementation Details

Spatial and Temporal Discretization

Flow Terms for Reach Groups

Procedure for Specifying a Tilted Pool

Numerical Controls

Aquifer Evapotranspiration Beneath Two-Dimensional Reaches

Procedure for Coupling the SWR1 Process with MODFLOW

Surface-Water Budgets

Output Files

Routing Option Assumptions and Limitations

Aquifer-Reach Exchange Assumptions and Limitations

Tips for Designing SWR1 Models

Test Simulations

Summary

References Cited

Appendix 1. Notation Used in this Report

Appendix 2. Data Input Instructions for the SWR1 Process

Appendix 3. Numerical Solution of Surface-Water Routing Applied in the SWR1 Process

Appendix 4. SWR1 Process Preprocessor for Reach Segmentation and Connectivity

Appendix 5. List of Selected Input Datasets for Test Simulation 5

U.S. Department of the Interior | U.S. Geological Survey
URL: http://pubsdata.usgs.gov/pubs/tm/6a40/index.html
Page Contact Information: GS Pubs Web Contact