Sources and SinksSources and Sinks

Based on Slides Prepared By

Eileen Poeter, Colorado School of Mines

Boundary ConditionsBoundary Conditions

Types of Boundary ConditionsTypes of Boundary Conditions1) 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 Three basic types of Boundary Conditionsof 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.

DIRICHLETDIRICHLETConstant Head & Constant Head & Specified Head Boundaries 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

NEUMANNNEUMANNNo Flow and No Flow and Specified Flow BoundariesSpecified 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.

CAUCHYCAUCHYHead Dependent FlowHead 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 SurfaceFree 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 SurfaceSeepage 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 Natural and Artificial BoundariesBoundaries

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 Natural and Artificial BoundariesBoundaries

BOUNDARY TYPE NATURAL EXAMPLES

ARTIFICIAL USES

CONSTANT Fully Penetrating Surface Water Features

Distant Boundary (Line of unchanging hydraulic head contour) or

SPECIFIED HEAD SPECIFIED FLOW Precipitation/Recharge Flowline

Pumping/Injection Wells Divide Impermeable material Subsurface Inflow

HEAD DEPENDENT FLOW

Rivers Distant Boundary (Line of unchanging hydraulic head contour)

Springs (drains)   Evapotranspiration  Leakage From a Reservoir or Adjacent Aquifer

 

Natural Hydrologic FeaturesNatural Hydrologic Features as Boundaries 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

Natural No-Flow BoundaryNatural 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 Note ground-water divide shifts after development—may or may after development—may or may

not be a good no-flow BCnot be a good no-flow BC

T.E. Reilly, 2000

Practical ConsiderationsPractical Considerations

• Boundary conditions must be assigned to every point on the boundary surface

• Modeled boundary conditions are usually greatly simplified compared to actual conditions

Sources and SinksSources and Sinks

•Injection and Pumping Wells•Flux across the Water Table•Leakage

Sources and SinksSources and Sinks

•Same model options used to represent boundary conditions are used to represent internal sources and sinks

•Specified flux for wells•Specified head for lakes, rivers, drains

Wells—an internal boundary Wells—an internal boundary condition at a point—thought of condition at a point—thought of

as a stress to the systemas 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– Node may represent entire thickness of a cell

(which often represents an aquifer)• MODFLOW’s MNW package remedies some issues

Flux across the Water Table Flux across the Water Table (free surface boundary)(free surface boundary)

• Treated as an internal source or sink (Dupuit assumptions)– May vary with time and space– Array(s) of flux values

• Variety of methods for estimating water table fluxes

• Directly to/from surface or through unsaturated zone

Estimating water table fluxesEstimating water table fluxes• Recharge

– Infiltrated water that crosses the water table and becomes part of the ground water system

• Discharge– Ground water that moves upward across the

water table and discharges to the surface or to the unsaturated zone

RechargeRecharge• No one universally applicable method

• Traditionally based on percentage of precipitation

• Significant spatial and temporal variations

• Accommodated by Zones

• Rates adjusted during calibration

• Often largest unknown

Recharge MethodsRecharge Methods• Water Balance Models

– With MF either pre-process or code modifications

– PRMS and GS-FLOW

• Unsaturated Zone Models

DischargeDischarge• Evapotranspiration occurs when

– Plants remove water from the water table– Direct evaporation from the water table

• May be head-dependent (if water-table too far below land surface ET is unlikely)

• Varies in space and time

LeakageLeakage• Movement of water through a layer of material that has a

lower hydraulic conductivity– Enters or leaves depending on relative head

– Magnitude and direction may vary in time

– Source: aquifer, river, or lake

– Sink: aquifer, river, lake, drain, springs, or seeps

• Type of head-dependent boundary– General head boundaries

– Drains (only act as sinks)

Ground-water / Surface-water Ground-water / Surface-water InteractionInteraction

• 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 How a stream could interact with could interact with the ground-water the ground-water

systemsystem

T.E. Reilly, 2000