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Quantitative Elements of Physical Hydrology

The Hydrologic Cycle:Mass Balance and Flux in the

Water Cycle

© John F. HermanceJanuary 25, 2007

Contact information:Jack HermanceEnvironmental Geophysics/HydrologyDepartment of Geological SciencesBrown University, Providence, RI 02912-1846Tel: 401-863-3830e-mail: John_Hermance@Brown.Edu

Discussion Summary

Objective: To understand the inter-relationships of the principal hydrological processes in a watershed.

These are:PrecipitationEvaporation

& transpirationDepression storageInfiltrationOverland flow

Hortonian flowSaturated flow

InterflowThroughflowGroundwater flowStreamflow generation

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Begin by defining the conventions usedin our visualization graphics.

Consider a genericwatershed . . .

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A subsection of the stream . . .

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We need to understand the interaction of these

elements . . .

Elements of the Hydrologic Cycle

Factors affecting the behavior of water and its movement through the watershed:

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Quantitative Elements of Physical Hydrology

© John F. HermanceJanuary 25, 2007

An Introduction to Mass Transport:Total flux across a boundary

Quantitative Elements of Physical Hydrology

© John F. HermanceJanuary 25, 2007

Flow across a surface elementFlow across a surface element

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How many flux lines pass through aunit area?

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How many flux lines pass through aunit area?

How many flux lines pass through aunit area?

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In this case,qnormal = qactual cos θ

The number of flux lines passing through a unit area depends on the geometrical relationship between the direction of the flux vector and the orientation of the area.

This, in turn, is abbreviated.qn = Area . qactual cos θbecomes:

qn = A . qactual

(Note that vectors are shownin bold, "areas" have direction,and this vector product is calleda "scalar", "dot" or "inner" product)

(Projection of A on to B, orB on to A.)

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Quantitative Elements of Physical Hydrology

© John F. HermanceJanuary 25, 2007

Application of these concepts tostreamflow discharge.

Application of these concepts tostreamflow discharge.

Consider a stream in plan (2D) view.Consider a stream in plan (2D) view.

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What is the total stream discharge Q?What is the total stream discharge Q?

First, . . . we need to ask how is stream discharge Q measured?

First, . . . we need to ask how is stream discharge Q measured?

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Standard (USGS) proceduresemploy rotating cups.

(Lance Ramsbey - USGS)

Standard (USGS) proceduresemploy rotating cups.

(Lance Ramsbey - USGS)

What is the total stream discharge Q?What is the total stream discharge Q?

A rotating cup (pin-wheel) flow-meterA rotating cup (pin-wheel) flow-meter

(Number of turns/minute is proportional to flow velocity)

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This is a scalar flow measurement.

This is a scalar flow measurement.

A practical application: Suppose the hydrologist has limitedaccess to the stream. Only this profile, because of depth, etc..

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But, ... the hydrologist wants to know the total discharge Q at C-D;and, for estimating erosion effects, the mean velocity of theflow at C-D.

Suppose we know the cross-sectional area A of the stream, and the average velocity of the stream V, only along the profile A-B.

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Determine Flux:Q (stream discharge) = V A cos θ A = Cross-sectional area

A Question (?):Assume that V, the average velocity of the stream, is 1 ft/s.What would be the magnitude of that component of V normal (i.e. perpendicular) to the line A-B?

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A Question (?):Assume that along A-B, the cross-sectional area of the stream is 500 ft2, and the average velocity of the stream is 1 ft/s.What is the total discharge Q across the line A-B?

A practical application: Suppose the hydrologist has limitedaccess to the stream.

Determine Total Flux:Q (stream discharge) = V A cos θ A = Cross-sectional area

V = 1.0 ft/sA = 500 sq ftcos (45 deg) = 0.7Q = 350 cfs

This is the total stream discharge.

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Next Question (?):What is the total discharge Q (cfs) across the line (through thearea?) C-D? (A = 400 ft2)

Final Question (?):What is the average velocity V?

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Question (?):What is the total discharge Q (cfs) across the line (through the area?) C-D? (A = 400 ft2)

The total stream discharge isFor A-B:V = 1.0 ft/s (measured)A(true) = 350 sq ftQ = 350 cfs

For C-D, Q is the same.

Final Question (?):What is the average velocity V?

The total stream discharge isFor A-B:V = 1.0 ft/sA(projected) = 350 sq ft (500 x cos θ)Q = 350 cfs

For C-D, Q is the same.

The average velocity at C-D isV(avg) = Q/A = 350/400 = 0.875 ft/s

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Quantitative Elements of Physical Hydrology

© John F. HermanceJanuary 25, 2007

Flow through a reference volumeFlow through a reference volume

Flux through a closed surface

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Flux through a closed surface

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Apply this concept to streamflow

Apply this concept to streamflow

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Introduce a “mathematical” volume . . .Introduce a “mathematical” volume . . .

Introduce a “mathematical” volume . . .Introduce a “mathematical” volume . . .

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(The convention is that the sense of a vector is positive (+),when it is directed out of a closed surface.)

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Flux through a closed surface

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Flux through a closed surface: Case 2

How can you have more flux leaving avolume than entering?

Flux through a closed surface: Case 2

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Next consider the special case of storagewithin the volume.

(Could be a lake, reservoir,stream reach, pond,groundwater reservoir, etc.)

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Consider "storage" in a stream.

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So what are the parameters ofa watershed to which

these concepts apply?

Quantitative Elements of Physical Hydrology

© John F. HermanceJanuary 25, 2007

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Application of the concept of residence timeto a reservoir.

Application of the concept of residence timeto a reservoir.

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A Question (?):Assuming a steady-state total volume of200,000,000 m3, and an average inflow ofstreamflow, groundwater flow and precipitationof 20,000,000 m3 / year, what is the averageresidence time of a drop of water in thereservoir?

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Another Question (?):What is the partitioning of discharge from the reservoir

a) over the spillway,b) consumptive use,c) evaporation,d) groundwater ”seepage"?

Another Question (?):What is the partitioning of discharge from the reservoir

a) over the spillway,b) consumptive use,c) evaporation,d) groundwater ”seepage"?

(This is a question for later.)(This is a question for later.)

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We next use conservation of mass to quantify the water cycle.

We next use conservation of mass to quantify the water cycle.

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Quantifying watershed processes.

© John F. HermanceJanuary 25, 2007

© John F. HermanceJanuary 25, 2007

Flow elements of the local water cyclein a watershed

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P + Qswi + Gin - (ET + Gout + Qswo) = 0© John F. Hermance

January 25, 2007

Mass Balance Relation #1

P + Qswi + Gin - (ET + Gout + Qswo) = ∆S/∆t© John F. Hermance

January 25, 2007

Mass Balance Relation #2

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(Global view of the water mass balance.)

© John F. HermanceJanuary 25, 2007

© John F. HermanceJanuary 25, 2007

Note (for example):P = I + QS

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© John F. HermanceJanuary 25, 2007

Note (for example):P = I + QS

Oftentimes, a water balance relation reflects the hydrologist’s point of view regarding a particular physical process.(What’s this “point of view”?)

© John F. HermanceJanuary 25, 2007

Note (for example):P = I + QS

orET = I - QG

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© John F. HermanceJanuary 25, 2007

Note (for example):P = I + QS

orET = I - QG

Since (from the latter)I = ET + QG . . .

© John F. HermanceJanuary 25, 2007

Note (for example):P = I + QS

orET = I - QG

Since (from the latter)I = ET + QG

Substitute for I in the 1st relation, to obtainP = ET + QS + QG

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Quantitative Elements of Physical Hydrology

© John F. HermanceJanuary 25, 2007

End of Presentation(The Hydrologic Cycle:

Mass Balance and Flux in the Water Cycle)

End of Presentation(The Hydrologic Cycle:

Mass Balance and Flux in the Water Cycle)

Contact information:Jack HermanceEnvironmental Geophysics/HydrologyDepartment of Geological SciencesBrown University, Providence, RI 02912-1846Tel: 401-863-3830e-mail: John_Hermance@Brown.Edu

(An alternative view.)

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© John F. HermanceJanuary 25, 2007

The water cycle ona watershed scale

© John F. HermanceJanuary 25, 2007

The water cycle ona watershed scale