Boundary Conditions

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Boundary Conditions
Based on Slides Prepared By Eileen Poeter,
Colorado School of Mines
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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)
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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.
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DIRICHLETConstant 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
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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,
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Example Constant Head
and an open pit mine is simulated by defining
heads in the pit area, to the elevation of the
pit bottom,
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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.
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Example Constant Head
Lesson Monitor the in-flow at constant head
boundaries and make calculations to assure
yourself the flow rates are reasonable.
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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,
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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,
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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.
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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.
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NEUMANNNo 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.
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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.
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Example SPECIFIED FLOW
Injection Well If the injection flow is too
large, calculated heads may be above the ground
surface in unconfined aquifer models.
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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.
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Example SPECIFIED FLOW
Lesson Monitor calculated heads at specified
flow boundaries to ensure that the heads are
physically reasonable.
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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.
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Example NO FLOW
Simplified model with no-flow boundary
representing the ground-water divide.
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Example NO FLOW
Use of a no-flow boundary in this manner may
cause problems When a ground water divide is
defined as a no-flow 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.
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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.
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CAUCHYHead 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.

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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.
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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.
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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.
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Natural and Artificial Boundaries
BOUNDARY TYPE NATURAL EXAMPLES ARTIFICIAL USES
CONSTANT Fully Penetrating Surface Water Features Distant Boundary (Line of unchanging hydraulic head contour)
or Fully Penetrating Surface Water Features Distant Boundary (Line of unchanging hydraulic head contour)
SPECIFIED HEAD Fully Penetrating Surface Water Features Distant Boundary (Line of unchanging hydraulic head contour)
SPECIFIED FLOW Precipitation/Recharge Flowline
SPECIFIED FLOW Pumping/Injection Wells Divide
SPECIFIED FLOW Impermeable material Subsurface Inflow
HEAD DEPENDENT FLOW Rivers Distant Boundary (Line of unchanging hydraulic head contour)
HEAD DEPENDENT FLOW Springs (drains)  
HEAD DEPENDENT FLOW Evapotranspiration  
HEAD DEPENDENT FLOW Leakage From a Reservoir or Adjacent Aquifer  
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Boundary Condition Exercise
Theis (many assumptions ) and Theim (radial
flow to a well in confined conditions)
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Boundary Condition Exercise
Dupuit formula for radial flow under water table
conditions
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Boundary Condition Exercise
Example An oceanic island in a humid climate
permeable materials are underlain by relatively
impermeable bedrock
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Boundary Condition Exercise
Example An alluvial aquifer associated with a
medium-sized river in a humid climate the
aquiferis underlain and bounded laterally by
bedrock of low hydraulic conductivity
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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
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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
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Boundary Condition Exercise
Example A confined aquifer bounded above and
below by leaky confining beds
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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

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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

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How a stream could interact with the ground-water
system
T.E. Reilly, 2000
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South Carolina Well in the Piedmont
Nearby stream gage correlates with well.
(3)
Source, Bruce Campbell, USGS, SC, 2000
Drought
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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

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Note ground-water divide shifts after
developmentmay or may not be a good no-flow BC
T.E. Reilly, 2000
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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

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Wellsan internal boundary condition at a
pointthought 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

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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