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The river velocity used in fish migration calculations is related to river flow and pool geometry and varies with pool drawdown as a function of the volume. The pool is represented as an idealized channel having sloping sides and longitudinal sloping bottom. As a pool is drawn down, part of it may return to a free flowing stream that merges with a smaller pool at the downstream end of the reservoir. The submodel is illustrated in Fig. 19 and Fig. 20. Important parameters are as follows:
- Hu = full pool depth at the upstream end of the segment
- Hd = full pool depth at the downstream end of the segment
- L = pool length at full pool
- x = pool length at lowered pool
- E = pool elevation drop below full pool elevation
- W = pool width averaged over reach length at full pool
= average slope of the pool side - F = flow through the pool in kcfs
- Ufree = velocity of free flowing river.
Other parameters illustrated in Fig. 19 are used to develop the relationships between the parameters listed above and water velocity and pool volume. They are not named explicitly.
Pool Volume
Reservoir volume depends on elevation. Elevation is measured in terms of E, the elevation drop below the full pool level. The volume calculation is based on the assumptions that the width of the pool at the bottom and the pool side slopes are constant over pool length. As a consequence of these two assumptions, the pool width at the surface increases going downstream in proportion to the increasing depth of the pool downstream. When E >Hu, the drawn down elevation is below the level of the upstream end and the upper end of the segment becomes a free flowing river section that connects to a pool downstream in the segment. When E < Hu, the reservoir extends to the upper end of the segment and for mathematical convenience CRiSP.1 calculates a larger volume and subtracts off the excess. The volume relationship (as a function of elevation drop for E positive measured downward) is developed below.
The total volume is defined
(25)
First the equation for V1 is developed. Note that when E 
Hu the volume V1 divides into two parts
(26) where V' is a side volume and V" is the thalweg1 volume. They are defined
(27) where
(28)
(29)
. (30) Combining these terms, when E Hu it follows pool volume is
(31)
In terms of the fundamental variables in equations (26) to (31) this is
(32) for E Hu and x £ L.
When the pool elevation drop is less than the upper depth (so E < Hu and x = L) pool volume is V(E) where
(33)
The term V1(E) is the volume of the pool extended longitudinally above the dam, where the depth is Hu, so as to form the same triangular longitudinal cross-section as before. This is done so that the volume can still be expressed by eq (32). The term V2(E) is the excess volume of the portion of the pool above the dam and can be expressed
(34)
Summarizing, the volume relationship as a function of elevation drop, for E positive measured downward, is
(35)
where
(36)
The equation for full pool volume can be expressed
(37)
When the bottom width is zero the full pool volume becomes
(38)
Water Velocity
Water velocity through a reservoir is described in terms of the residence time T and the length of the segment L. The residence time in a segment depends on the amount of the reservoir that is pooled and free flowing (Fig. 20).
The equations for residence time are
(39)
where
- V(E) = pool volume (ft3) as a function of elevation drop E in feet
- F = flow in 1000 cubic feet per second or kcfs
- L = segment length in feet
- x = pool length defined by eq (28) and with units of feet
- Ufree = velocity of water in the free stream (kfs) (using the John Day River, the default value is 4.5 ft/s which is 4.5 x 10-3 kfs)
- T = residence time in this calculation is in kilo seconds or ks.
The velocity in the segment is
(40)
The velocity with the above units is in thousands of feet per second.
Combining equations eq (36), eq (39) and eq (40) the segment velocities are:
for E Hu
(41)
and for E < Hu
(42)
where
- U = average river velocity in ft/s
- Ufree = the velocity of a free flowing stream in ft/s
- F = flow in kcfs
- E = elevation drop (positive downward) in ft
- Hu = depth of the upper end of the segment in ft
- V1 and V2 = volume elements defined by eq (36)
Flow-Velocity Calibration
The calibration of the volume equation requires determining the average pool slope from the pool volume. The equation is the smaller angle of the two forms
(43)
where
- V(0) = pool volume at full pool
This scheme using eq(43) reflects the volume versus pool elevation relationship developed for each reservoir by the U.S. Army Corps of Engineers. Capacity versus elevation curves were obtained from several dams to check the accuracy of our volume model. The figures below show data points from these curves versus CRiSP's volume curve for two dams. Fig. 21 illustrates Lower Granite pool, with model coefficients of Hu = 40 ft., Hd = 118 ft, = 80.7o, L = 53 miles, W = 2000 ft, and Wanapum pool, with model coefficients Hu = 42ft., Hd = 116 ft, = 87.0o, L = 38 miles, W = 2996.1ft.
The water particle residence time in a segment is given in eq (39). The pool volume velocity/travel time equation was tested against particle travel calculations for Lower Granite Pool as reported by the Army Corps of Engineers in the Lower Granite Drawdown studies report (1993) (Fig. 22.)
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1A thalweg is the longitudinal profile of a canyon.
Columbia River Salmon Passage Model CRiSP.1.5 Theory, Calibration & Validation Manual
Copyright © 1996, Columbia Basin Research. All rights reserved.
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