The Hydrologic Cycle: Mass Balance and Flux in the Water Cycle

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Quantitative Elements of Physical Hydrology The Hydrologic Cycle: Mass Balance and Flux in the Water Cycle Contact information: Jack Hermance Environmental Geophysics/Hydrology Department of

1 Quantitative Elements of Physical Hydrology The Hydrologic Cycle: Mass Balance and Flux in the Water Cycle Contact information: Jack Hermance Environmental Geophysics/Hydrology Department of Geological Sciences Brown University, Providence, RI Tel: John F. Hermance January 25, 2007 Discussion Summary Objective: To understand the inter-relationships of the principal hydrological processes in a watershed. These are: Precipitation Evaporation & transpiration Depression storage Infiltration Overland flow Hortonian flow Saturated flow Interflow Throughflow Groundwater flow Streamflow generation 1

2 Begin by defining the conventions used in our visualization graphics. Consider a generic watershed... 2

3 A subsection of the stream... 3

4 Elements of the Hydrologic Cycle We need to understand the interaction of these elements... Factors affecting the behavior of water and its movement through the watershed: 4

5 Quantitative Elements of Physical Hydrology An Introduction to Mass Transport: Total flux across a boundary John F. Hermance January 25, 2007 Quantitative Elements of Physical Hydrology Flow across a surface element John F. Hermance January 25,

6 How many flux lines pass through a unit area? 6

7 How many flux lines pass through a unit area? How many flux lines pass through a unit area? 7

8 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. In this case, q normal = q actual cos θ This, in turn, is abbreviated. q n = Area. q actual cos θ becomes: q n = A. q actual (Projection of A on to B, or B on to A.) (Note that vectors are shown in bold, "areas" have direction, and this vector product is called a"scalar", "dot" or "inner" product) 8

9 Quantitative Elements of Physical Hydrology Application of these concepts to streamflow discharge. John F. Hermance January 25, 2007 Consider a stream in plan (2D) view. 9

10 What is the total stream discharge Q? First,... we need to ask how is stream discharge Q measured? 10

11 Standard (USGS) procedures employ rotating cups. (Lance Ramsbey - USGS) What is the total stream discharge Q? A rotating cup (pin-wheel) flow-meter (Number of turns/minute is proportional to flow velocity) 11

12 This is a scalar flow measurement. A practical application: Suppose the hydrologist has limited access to the stream. Only this profile, because of depth, etc.. 12

13 But,... the hydrologist wants to know the total discharge Q at C-D; and, for estimating erosion effects, the mean velocity of the flow 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. 13

14 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? 14

15 A Question (?): Assume that along A-B, the cross-sectional area of the stream is 500 ft 2, and the average velocity of the stream is 1 ft/s. What is the total discharge Q across the line A-B? Determine Total Flux: Q (stream discharge) = V A cos θ A = Cross-sectional area V = 1.0 ft/s A = 500 sq ft cos (45 deg) = 0.7 Q = 350 cfs This is the total stream discharge. A practical application: Suppose the hydrologist has limited access to the stream. 15

16 Next Question (?): What is the total discharge Q (cfs) across the line (through the area?) C-D? (A = 400 ft 2 ) Final Question (?): What is the average velocity V? 16

17 The total stream discharge is For A-B: V = 1.0 ft/s (measured) A(true) = 350 sq ft Q = 350 cfs For C-D, Q is the same. Question (?): What is the total discharge Q (cfs) across the line (through the area?) C-D? (A = 400 ft 2 ) The total stream discharge is For A-B: V = 1.0 ft/s A(projected) = 350 sq ft (500 x cos θ) Q = 350 cfs For C-D, Q is the same. The average velocity at C-D is V(avg) = Q/A = 350/400 = ft/s Final Question (?): What is the average velocity V? 17

18 Quantitative Elements of Physical Hydrology Flow through a reference volume John F. Hermance January 25, 2007 Flux through a closed surface 18

19 Flux through a closed surface 19

20 Apply this concept to streamflow 20

21 Introduce a mathematical volume... Introduce a mathematical volume... 21

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23 (The convention is that the sense of a vector is positive (+), when it is directed out of a closed surface.) 23

24 Flux through a closed surface 24

25 Flux through a closed surface: Case 2 Flux through a closed surface: Case 2 How can you have more flux leaving a volume than entering? 25

26 Next consider the special case of storage within the volume. (Could be a lake, reservoir,stream reach, pond, groundwater reservoir, etc.) 26

27 Consider "storage" in a stream. 27

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29 Quantitative Elements of Physical Hydrology So what are the parameters of a watershed to which these concepts apply? John F. Hermance January 25,

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32 Application of the concept of residence time to a reservoir. 32

33 A Question (?): Assuming a steady-state total volume of 200,000,000 m 3, and an average inflow of streamflow, groundwater flow and precipitation of 20,000,000 m 3 / year, what is the average residence time of a drop of water in the reservoir? 33

34 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 (This partitioning is a question of discharge for later.) from the reservoir a) over the spillway, b) consumptive use, c) evaporation, d) groundwater seepage"? 34

35 We next use conservation of mass to quantify the water cycle. 35

36 Quantifying watershed processes. John F. Hermance January 25, 2007 John F. Hermance January 25, 2007 Flow elements of the local water cycle in a watershed 36

37 Mass Balance Relation #1 John F. Hermance January 25, 2007 P + Q swi + G in -(ET + G out + Q swo ) = 0 Mass Balance Relation #2 P + Q swi + G in -(ET + G out + Q swo ) = S/ t John F. Hermance January 25,

38 (Global view of the water mass balance.) John F. Hermance January 25, 2007 Note (for example): P = I + Q S John F. Hermance January 25,

39 Note (for example): P = I + Q S 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. Hermance January 25, 2007 Note (for example): or P = I + Q S ET = I - Q G John F. Hermance January 25,

40 Note (for example): P = I + Q S or ET = I - Q G Since (from the latter) I = ET + Q G... John F. Hermance January 25, 2007 Note (for example): P = I + Q S or ET = I - Q G Since (from the latter) I = ET + Q G Substitute for I in the 1st relation, to obtain John F. Hermance January 25, 2007 P = ET + Q S + Q G 40

Quantitative Elements of Physical Hydrology End of Presentation (The Hydrologic Cycle: Mass Balance and Flux in the Water Cycle) Contact information: Jack Hermance Environmental

41 Quantitative Elements of Physical Hydrology End of Presentation (The Hydrologic Cycle: Mass Balance and Flux in the Water Cycle) Contact information: Jack Hermance Environmental Geophysics/Hydrology Department of Geological Sciences Brown University, Providence, RI Tel: John F. Hermance January 25, 2007 (An alternative view.) 41

42 The water cycle on a watershed scale John F. Hermance January 25, 2007 The water cycle on a watershed scale John F. Hermance January 25,

Watershed Delineation

Foundations of Physical Hydrology Watershed Delineation Contact information: Jack Hermance Environmental Geophysics/Hydrology Department of Geological Sciences Brown University, Providence, RI 02912-1846