Transcript Conceptual Model Development

Boundary Conditions
Based on Slides Prepared By
Eileen Poeter, Colorado School of Mines
Types of Boundary Conditions
1) Specified Head: head is defined as a function of space and time
(could replace ABC, EFG)
Constant Head: a special case of specified head (ABC, EFG)
2) Specified Flow: could be recharge across (CD) or zero across (HI)
No Flow (Streamline): a special case of specified flow where the
flow is zero (HI)
3) Head Dependent Flow: could replace (ABC, EFG)
Free Surface: water-table, phreatic surface (CD)
Seepage Face: h = z; pressure = atmospheric at the ground surface (DE)
Three basic types
of Boundary Conditions
After:
Definition of Boundary and Initial Conditions in the Analysis of Saturated
Gournd-Water Flow Systems - An Introduction, O. Lehn Franke, Thomas E. Reilly,
and Gordon D. Bennett, USGS - TWRI Chapter B5, Book 3, 1987.
DIRICHLET
Constant Head &
Specified Head Boundaries
Specified Head:
Head (H) is defined as a function of time and
space.
Constant Head:
Head (H) is constant at a given location.
Implications:
Supply Inexhaustible, or Drainage Unfillable
Example: Constant Head
Example of Potential Problems Which May Result
From Misunderstanding / Misusing a Constant
Head Boundary
If heads are fixed at the ground surface to
represent a swampy area,
Example: Constant Head
and an open pit mine is simulated by defining
heads in the pit area, to the elevation of the pit
bottom,
Example: Constant Head
the use of constant heads to represent the swamp
will substantially overestimate in-flow to the pit.
This is because the heads are inappropriately
held high, while in the physical setting, the
swamp would dry up and heads would decline,
therefore actual in-flow would be lower. The
swampy area is caused by a high water table. It is
not an infinite source of water.
Example: Constant Head
Lesson: Monitor the in-flow at constant head
boundaries and make calculations to assure
yourself the flow rates are reasonable.
Example: Specified Head
Example of Potential Problems Which May Result
From Misunderstanding / Misusing a Specified
Head Boundary
When a well is placed near a stream, and the
stream is defined as a specified head,
Example: Specified Head
the drawdown may be underestimated, if the
pumping is large enough to affect the stream
stage. The specified flow boundary may supply
more water than the stream caries,
Example: Specified Head
and drawdowns should be greater, for the given
pumping rate. The stream stage, and flow rate,
should decrease to reflect the impact of the
pumping.
Example: Specified Head
Lesson: Monitor the in-flow at specified head
boundaries. Confirm that the flow is low enough
relative to the streamflow, such that stream
storage will not be affected.
NEUMANN
No Flow and
Specified Flow Boundaries
Specified Flow:
Discharge (Q) varies with space and time.
No Flow:
Discharge (Q) equals 0.0 across boundary.
Implications: H will be calculated as the value required to
produce a gradient to yield that flow, given a specified
hydraulic conductivity (K). The resulting head may be above
the ground surface in an unconfined aquifer, or below the base
of the aquifer where there is a pumping well; neither of these
cases are desirable.
Example: SPECIFIED FLOW
Example of Potential Problems Which May Result
From Misunderstanding / Misusing a Specified
Flow Boundary
In this example, the model represents a simple unconfined
aquifer with one well. Two cases are presented:
1) an injection well, and
2) a withdrawal (pumping) well.
Example: SPECIFIED FLOW
Injection Well: If the injection flow is too large,
calculated heads may be above the ground
surface in unconfined aquifer models.
Example: SPECIFIED FLOW
Withdrawal Well: If the withdrawal flow is too
large, calculated heads may fall below the bottom
of the aquifer, yet the model may still yield water.
Example: SPECIFIED FLOW
Lesson: Monitor calculated heads at specified
flow boundaries to ensure that the heads are
physically reasonable.
Example: NO FLOW
Example of Potential Problems Which May Result
From Misunderstanding / Misusing a No Flow
Boundary
When a no flow boundary is used to represent a ground
water divide, drawdown may be overestimated, and
although the model does not indicate it, there may be
impacts beyond the model boundaries.
Example: NO FLOW
Simplified model with no-flow boundary
representing the ground-water divide.
Example: NO FLOW
Use of a no-flow boundary in this manner may cause
problems: When a ground water divide is defined as a noflow boundary, the flow system on the other side of the
boundary cannot supply water to the well, therefore
predicted drawdowns will be greater than would be
experienced in the physical system. The no-flow boundary
prevents the ground water divide from shifting, implying
there drawdown is zero on the other side of the divide.
Example: NO FLOW
Lesson: Monitor head at no flow boundaries used
to represent flow lines or flow divides to ensure
the location is valid even after the stress is
applied.
CAUCHY
Head Dependent Flow
Head Dependent Flow:
H1 = Specified head in reservoir
H2 = Head calculated in model
Implications:
•If H2 is below AB, q is a constant and AB is the seepage face, but
model may continue to calculate increased flow.
•If H2 rises, H1 doesn't change in the model, but it may in the field.
•If H2 is less than H1, and H1 rises in the physical setting, then inflow is
underestimated.
•If H2 is greater than H1, and H1 rises in the physical setting, then
inflow is overestimated.
Free Surface
Free Surface:
h = Z, or H = f(Z)
e.g. the water table h = z
or a salt water interface
Note, the position of the boundary is not fixed!
Implications: Flow field geometry varies so transmissivity will
vary with head (i.e., this is a nonlinear condition). If the water table is
at the ground surface or higher, water should flow out of the model, as
a spring or river, but the model design may not allow that to occur.
Seepage Surface
Seepage Surface: The saturated zone intersects the
ground surface at atmospheric pressure and water
discharges as evaporation or as a downhill film of flow.
The location of the surface is fixed, but its length varies
(unknown a priori).
Implications: A seepage surface is neither a head or flowline.
Often seepage faces can be neglected in large scale models.
Natural and Artificial
Boundaries
It is most desirable to terminate your model at natural
geohydrologic boundaries. However, we often need to
limit the extent of the model in order to maintain the
desired level of detail and still have the model execute in a
reasonable amount of time.
Consequently models sometimes have artificial
boundaries.
For example, heads may be fixed at known water table
elevations at a county line, or a flowline or ground-water
divide may be set as a no-flow boundary.
Natural and Artificial
Boundaries
BOUNDARY TYPE
NATURAL
EXAMPLES
CONSTANT
Fully Penetrating
Distant Boundary (Line
Surface Water Features of unchanging hydraulic
head contour)
or
SPECIFIED HEAD
SPECIFIED FLOW
ARTIFICIAL USES
Precipitation/Recharge Flowline
Pumping/Injection Wells Divide
Impermeable material Subsurface Inflow
HEAD DEPENDENT
FLOW
Rivers
Springs (drains)
Evapotranspiration
Leakage From a
Reservoir or Adjacent
Aquifer
Distant Boundary (Line
of unchanging hydraulic
head contour)
Boundary Condition Exercise
Theis (many assumptions )
and Theim (radial flow to a well in confined conditions)
Boundary Condition Exercise
Dupuit formula for radial flow under water table conditions
Boundary Condition Exercise
Example: An oceanic island in a humid climate; permeable
materials are underlain by relatively impermeable bedrock
Boundary Condition Exercise
Example: An alluvial aquifer associated with a mediumsized river in a humid climate; the aquifer
is underlain and bounded laterally by bedrock of low
hydraulic conductivity
Boundary Condition Exercise
Example: An alluvial aquifer associated with an intermittent
stream in an arid climate; the aquifer is underlain and
bounded laterally by bedrock of low to intermediate
hydraulic conductivity
Boundary Condition Exercise
Example: A western valley with internal drainage in an arid
region; intermittent streams flow from surrounding
mountains towards a valley floor; a part of valley floor is
playa
Boundary Condition Exercise
Example: A confined aquifer bounded above and below by
leaky confining beds
Hydrologic Features as
Boundaries
• Boundary can be assigned to hydrologic
feature such as:
– Surface water body
• Lake, river, or swamp
– Water table
• Recharge and evapotranspiration or source/sink
specified head
– Impermeable surface
• Bedrock or permeable unit pinches out
Ground-water / Surface-water
Interaction
• Hydraulic head in aquifer can be equal to
elevation of surface-water feature or allowed to
leak to the surface-water feature
• Usually defined as a “Constant-Head” or
“Specified Head” Boundary or “Head-dependent
flow” boundary
• If elevation of SW changes, as with streams,
elevation of the boundary condition changes
How a stream
could interact with
the ground-water
system
T.E. Reilly, 2000
South Carolina Well in the Piedmont
Nearby stream gage correlates with well.
(3)
Source, Bruce Campbell, USGS, SC, 2000
Drought
No-Flow Boundary
• Hydraulic conductivity contrasts between
units
– Alluvium on top of tight bedrock
• Assume GW does not move across this
boundary
• Can use ground-water divide or flow line
Note ground-water divide shifts
after development—may or may
not be a good no-flow BC
T.E. Reilly, 2000
Water Table or “Flow” Boundary
• Intermittent areal recharge on water-table
– Moves through unsaturated zone
– Volume of water per unit area per unit of time entering
the GW system is specified
– May vary with time and space
• Evapotranspiration occurs when plants remove
water from the water-table
– May be head-dependent (if water-table too far below
land surface ET is unlikely
– Volume of water per unit area per unit of time leaving
the GW system as a function of depth to water is
specified
– May vary in space and time
Wells—an internal boundary
condition at a point—thought of
as a stress to the system
• A well is a specified flow rate at a point
– Can be pumping or injecting water
– Withdrawals or injection may vary in space
and time
Practical Considerations
• Boundary conditions must be assigned to
every point on the boundary surface
• Modeled boundary conditions are usually
greatly simplified compared to actual
conditions