Modeling Ground Water Interactions with Surface Water

1
Modeling Ground Water Interactions with Surface
Water
Western States Water Council
Water Information Management Systems
MeetingAlbuquerque, New Mexico September 13, 2006
David E. PrudicU.S. Geological SurveyCarson
City, Nevada
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When the wells dry, we know the worth of water
Benjamin Franklin
Add streams and this statement embodies the
history of development in the Western United
States during the past two centuries
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Ground-Water Interactions with Surface Water is
Ever Changing
Modeling ground-water interactions with surface
water is difficult because of variability in
climate, and changing practices in the
development of surface-water and ground-water
supplies
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History of Development
Knowing the history of an area is key to
understanding and modeling ground water
interactions with surface water
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Development is Divided into Three Periods

• Local diversion of water from streams for
  mining and irrigation of areas in adjacent flood
  plains
• Development of large reservoirs and complex
  irrigation systems for areas distant from the
  flood plains
• Development of deep turbine pumps and pivot
  irrigation systems for ground-water irrigation

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First Period

• Excess diversion of streams led to the
  development of laws and the Doctrine of Prior
  Appropriation
• Decreased annual stream flow
• Raised ground-water levels and increased
  ground-water discharge to streams during periods
  of low flow

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Second Period

• Development of a Concentrated Hydraulic Society
  to utilize and maintain infrastructure
• Decreased annual flows of streams
• Increased ground water in storage around
  reservoirs
• Increased ground-water levels (storage) in areas
  distant from rivers
• Localized water logging, creation of wetlands,
  and construction of drains
• Increased diffuse ground-water discharge to
  evapotranspiration near areas of irrigation

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Third Period

• Development of a Distributed Hydraulic Society
  from the drilling and pumping of thousands of
  individual wells with minimum regulation
• Decreased ground-water levels and storage
• Decrease in streamflows slow to develop because
  of large ground-water storage capacity

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Modeling Ground-Water and Surface Water
Interaction

• Modeling approach traditionally has depended on
  perspectiveone either asked
• How does surface water influence ground-water
  flow?
• or
• How does ground water influence the surface-water
  flow?

S. A. Leake
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Surface-Water Hydraulics Simulation Model
Black Box
Groundwater
S. A. Leake
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Ground-Water Flow Model
Surface-water processes
S. A. Leake
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Numerical Simulation of Surface Water and Ground
Water Interaction

• Many questions regarding effects of ground-water
  withdrawals on surface water are difficult to
  answer because
• Aquifers that interact with surface water are
  often heterogeneous and
• Both surface-water and ground-water flows can
  change in time and space

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Ground-Water Flow Equation
The equation is simply an expression of mass
balance
S. A. Leake
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Ground-Water Flow Equation, W term
The W term is flow rate per unit volume of
aquifer added to or taken from ground-water
system.
Q3
Q1
Q2
Most interaction between ground water and surface
water is lumped into the W term
S. A. Leake
15
Common types of surface-water boundaries in
ground-water models

• Constant or specified flow
• Constant or specified head
• Head-dependent flow Q f(h)May be reasonable
  for large lakes and rivers not affected by
  changes in ground-water flow because quantity of
  surface water is not part of calculation

S. A. Leake
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Constant or specified flow boundaries

• Flow rate is specified in model grid cell as
  known value of recharge or discharge
• Model computes head at boundary location

Example Divide total streamflow loss or gain
into parts belonging in each cell
NOTE Make sure model-computed head is reasonable
at these locations!
S. A. Leake
17
Constant or specified head boundaries

• Head in model grid cell corresponding to
  surface-water location is specified as elevation
  of water surface
• Flow rate to or from boundary is computed from
  adjacent model grid cells
• Does not require a W term in flow equation

Example Determine average stream stage in each
cell traversed by stream. Assign stage values as
specified head in cells.
NOTE Make sure model-computed flow is reasonable
at these locations!
S. A. Leake
18
Head-dependent flow boundaries

• A functional relationship between head in aquifer
  and flow to or from boundary is derived, usually
  by Darcys law
• Function is made part of W term
• Most model programs, such as MODFLOW have several
  such functions built in
• The ground-water head now occurs in the W term,
  possibly adding difficulty to the solution

S. A. Leake
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Real world
Ground-water model
Streambed sediments
Cross section
Head in stream
Head in cell
Plan view
Area of stream
Area encompassed by model cell
Model cell
S. A. Leake
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Ground-water model
Head in stream
Head in cell
Idealized prism of river bed sediments contained
in a model cell
Model cell
S. A. Leake
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Flow Through Streambed
An expression of flow through the streambed is
computed from Darcys Law as
Head here is river head, HRIV
Length, L
Head here is aquifer head, HAQ
Width, W
Thickness, b
Vertical hydraulic conductivity is, Kv
Q Csfr (HSTR-HAQ) where Csfr Kv (LW)/b
S. A. Leake
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Routing of Surface Water

• Surface routing programs are used to simulate
  changing location of flow and recharge to an
  aquifer without having to manually specify stage
  or flow in every reach
• Stage and streambed conductance term can vary in
  relation to channel dimensions and flow

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Typical Method of Routing Streamflows
Section 1
Flow direction
Section 2
Section 5
Section 3
Section 6
Section 4
SQin SQout DStorage
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Three Surface-Water Routing Programs Connected to
MODFLOW

• Streamflow-Routing Package (STR1) recently
  rewritten (SFR1 SFR2) to include different flow
  options, solute transport, and unsaturated
  flowMyself and others (1989 2004 and 2005)
• Diffusion Analogy Surface-Water Flow Model
  (DAFLOW) Harvey Jobson, USGS Open-file Report
  99-217
• Branch Surface-Water Flow Model (MODBRANCH) Eric
  Swain and E. J. Wexler, USGS TWRI Book 6, Chapter
  A6

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Streamflow-Routing Package (STR1, SFR1, and SFR2)

• Simple routing through network of streams
  assuming no change in storage and sum of inflow
  equals outflow
• Stream depth computed using different options
  assuming steady, uniform flow

Upstream flow
Precipitation
OverlandRunoff
ET
Downstream flow
Leakage
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DAFLOW
Continuity of Mass

• One-dimensional unsteady flow in network of open
  channels
• Solves equations for continuity of mass and
  momentum assuming no lateral inflow
• Approximates flow in a stream as reaches of
  steady uniform flow separated by transitions of
  unsteady flow

Continuity of Momentum
Q is discharge, U is velocity, A is cross-section
area, t is time L is channel Length, y is
depth, G is acceleration of gravity, Sf is
friction slope, and So is streambed slope
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DAFLOW
Transition of unsteady flow
Steady flow
Flow
Control volume
Q2, A2
t
t Dt
Steady flow
Friction slopeat time t
Q1, A1
DL
So
So
Flow into and out of control volume
,
and
where C is speed of moving wave
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ModBranch

• Combines one-dimensional unsteady flow model with
  MODFLOW
• Solves equations for continuity of mass and
  momentum and allows lateral inflow and outflow
  (known as St. Venant equations)
• Approximates changes in streamflow through
  finite-difference methods

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Model rows
Stream Network
Inflow
Segment 1
Diversion
Inflow
Segment 3
Canal
Segment 2
Point diversion
Model columns
Segment 5
Segment 4
Pipeline
Segment junction
Segment 6
Segment 7
Outflow
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Stream Reaches in a Cell

• Flow into and out of each reach calculated during
  every MODFLOW iteration
• Multiple stream reaches simulated for a model
  cell
• Does not simulate multiple model cells for one
  stream reach

GW node
Only one cell per reach
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Inflow to a Reach
Pipeline
Tributary stream
Flood wave
Surface runoff
Soil zone
Interflow
Water table
Precipitation
Ground-water discharge
Finite-difference cell
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Outflow from a Reach
Pipeline
Flood wave
Evaporation
Diversion
Streamflow out
Stream leakage
Stream leakage
Finite-difference cell
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Diverting Flow into a Segment

• Specified diversion
• Diversion rate reduce to available flow
• Diversion rate reset to zero when available flow
  less than specified diversion

• Specified flow fraction available in stream
• Flow diverted only when available flow exceeds a
  specified flow rate

Last two options originally program by Randy
Hanson, San Diego Office
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Stream Depth

• Stream depth is computed at midpoint of each
  stream reach
• Flow is computed at midpoint prior to computing
  stream depth except when a constant depth is
  specified

Cell node
Reachmidpoint
Q QIn 0.5Overland flow(PLw)
-0.5(ETLw)leakage to GW
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Stream Depth

• Flow at midpoint is partly dependent on streambed
  leakage, which is dependent on stream depth that
  is dependent on flow
• Equation is repeatedly solved using Newtons
  method until
• Change in stream stage is less than a specified
  value usually 0.0001 or
• Average of last two stream stages after reaching
  a maximum of 50 iterations

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Options for Computing Stream DepthMannings
Equation for Wide Rectangular Channel
Q (C/n) A R2/3 S1/2
where A area of rectangular channel
(width depth)
R hydraulic radius Area divided by wetted
perimeter R depth when width (w) gtgtdepth)
C constant (1.0 for m3/s and 1.486 for ft3/s)
n Mannings roughness coefficient, dimensionless
Depth can be computed as
d (Q/n)/(CwS 1/2)3/5
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Eight Point Cross Section
The eight point cross section is divided into
three parts. Vertical walls are assumed at the
end of each cross section
Channel
Left bank
Right bank
Part 2
Width
Depth
Wetted perimeter
Depth, wetted perimeter, and width computed from
cross section given flow.
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Log-Log equation
The width (w), mean depth (d), and mean velocity
(v) each increase with respect to discharge (Q)
as power functions (Leopold, Wolman and Miller,
1964 and 1992, Fluvial Processes in
Geomorphology) WIDTH aQb, DEPTH cQf,
VELOCITY kQm where a, b, c, f, k, and m are
numerical coefficients because w d v Q
aQbcQf kQm Q and b f m 1 and a c
k 1
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Depth Computed from Table

• Table of depth and width versus flow is entered
  from data collected at a streamflow gage
• Program uses log interpolation between values
  except when computed flow is between 0 and first
  value in table

Width
Depth
Flow
10
0.5
5
15
1.3
50
20
1.8
100
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2.6
1,000
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Different Options can be used for Each Stream
Segment
Wide rectangular channel
Diversion canal - constant
Table
Eight-point cross section
Log-log equation
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Simulation of Unsaturated Flow Beneath Streams

• Details published in USGS Techniques and Methods
  6A-13 (Niswonger and Prudic, 2005)

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Simulating Vertical Unsaturated Flow Beneath
Streams (SFR2)

• Formulation is based on kinematic wave solution
  to Richards equation (Smith, 1983)
• - Avoid instabilities associated with
  conventional unsaturated flow models
• - Speeds up computational time for applications
  to large basin scale simulations

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Infiltration
Time
Water Content
Depth
Time 1
Time 2
Time 3
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Unsaturated Zone Divided into Compartments
Stream
High stage
6
Streambed
Depth, meters
Normal stage
0
Unsaturated zonecompartment
0
14
Width, meters
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Ground Water Interactions with Lakes are
Simulated Using the Lake Package (Merritt and
Konikow, 2000)
Pipeline
Tributary stream
Outlet stream
Surface runoff
Evaporation
Interflow
Precipitation
Lake cell

Ground-waterdischarge
Lakebed
Lake leakage
Aquifer cell with node
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Connection of Streams with a Lake
Model rows
Segment 1
1
1
2
Segment 2
2
3
4
3
6
5
4
5
6
7
7
Lake 1
Model columns
8
9
1
2
3
4
5
Segment 3
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Lake surface
Lake bottom
Lakebed thickness
Lakebed
Conductance terms
Distance from base of lakebed to point in aquifer
Aquifer
Cross-sectional area
Point in aquifer
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Lakes can Divide and Merge
High stageone lake, one water budget
Lake Surface
Saturated zone
Low stagetwo or more lakes with different water
budgets
Lake 2
Lake 1
Saturated zone
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Integration of PRMS with MODFLOWGSFLOW, Version
1.0
Streams and Lakes
Soil Zone in PRMS
Interflow
Soil-Moisture Dependent Flow
Surfacerunoff
Ground-water discharge
Ground-water discharge
Head-Dependent Flow
Soil-Moisture/Head-Dependent Flow
Gravity drainagethrough unsaturated zone
Leakage
Ground water
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Hydrologic Response Units and Finite-Difference
Grid, Sagehen Creek, Truckee, California
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Ground-Water Recharge
3-yr simulation period daily time steps
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Ground-Water Discharge Saturation Excess
3-yr simulation period daily time steps