Delta Cross Channel Gates

A “gate” formulation ———

Q = A3 √ 2gh

Matthai,H.F. 1967. Measurement of peak discharge at width contractions by indirect methods. Chapter A4 in Techniques of water-resources investigations of the United States Geological Survey.

Figure III.2.i-1. Delta Cross Channel and gates, circa 1950’s. Sacramento River is in the foreground.

Photo courtesy of Lloyd Peterson, USBR.

Figure III.2.i-3. USGS Monitoring locations in the vicinity of Delta Cross Channel, Sep 2003 to Nov 2004.

Sacramento River above DCC

Sacramento River below GS

Georgiana Slough

Delta Cross Channel

Sn

od

gra

ss S

lou

gh

Mo

kelu

mn

e R

iver

Nor

th F

ork

Sout

h Fo

rk

Dead

Horse

Cut

Figure III.2.i-4. Comparison of field measurements (blue rhombuses) in Delta Cross Channel and simulated flow (solid lines) based on gate equation for two gate coefficients. Flow estimates at maximum and minimum water depth are also shown as dashed lines.

USGS flow measurements are 15-minute averages. Stage measurements are instantaneous values. The stage difference in this plot is the average of two time-steps.

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

9,000

10,000

11,000

12,000

0.0 0.1 0.2 0.3 0.4

Stage difference (feet)

flo

w (

cfs)

w = 120'

= 0.65

= 1.00

Mass balance (continuity) and energy balance (Bernoulli equation) give

vd = [1 + f – (Ad / Au)2)]-½ [2 g (hu - hd)]½

where the friction loss term is assumed to take the form

hf = ½ f vd2

Binomial expansion about mean water level (and dropping h.o.t.s) give

Qd = Ad [2 g (hu - hd)]½ [1 + f – o2 (1 + d hd - u hu)]-½

where the ratio of cross-sections is assumed to take the form

= Ad / Au = o [1 + d hd - u hu + higher order terms (h.o.t.s)]

and

d ≈ wd / Ado and u ≈ wu / Auo

Semi-empirical coefficients to be calibrated: f, o2, u,d, Ado

Alternative Formulation

Figure III.2.i-5. Simulated flows in the Delta Cross Channel using open channel hydraulic formulation. Values of coefficients in Equation III.2.i-6 are given in Table III.2.i-2. Field data at 15-minute intervals span over 50 M2 tide cycles, from July 24 to August 18, 2004.

0

2,000

4,000

6,000

8,000

10,000

12,000

0 2,000 4,000 6,000 8,000 10,000 12,000

Measured (cfs)

Sim

ula

ted

flo

w (

cfs)

1 + f 2 au ad Ado

1.400 0.200 0.030 0.075 2732 sq.ft.

Figure III.2.i-6. Comparison of simulated flows in the Delta Cross Channel using gate-type formulation and open channel hydraulics formulation shown in Fig.III.2.i-4. Field data at 15-minute intervals span over 50 M2 tide cycles, from July 24 to August 18, 2004.

0

2,000

4,000

6,000

8,000

10,000

12,000

0 2,000 4,000 6,000 8,000 10,000 12,000

Measured (cfs)

Sim

ula

ted

flo

w (

cfs)

Gate coefficient = 1.00 Gate coefficient = 0.65

Open channel hydraulics

= 0.65

= 1.00

Figure III.2.i-7. Range of simulated flows in the Delta Cross Channel for a stage difference of ±¼” that estimated. Simulated flow are computed using the open channel hydraulics formulation (equation III.2.i-3). Only a small fraction of the data shown in Figs.III.2.i-4,5 are shown in this plot for better clarity.

(b) At high flow rates

0

2,000

4,000

6,000

8,000

10,000

12,000

4,000 5,000 6,000 7,000 8,000 9,000 10,000 11,000

Measured (cfs)

Sim

ula

ted

flo

w (

cfs)

Figure III.2.i-7. Range of simulated flows in the Delta Cross Channel for a stage difference of ±¼” that estimated. Simulated flow are computed using the open channel hydraulics formulation (equation III.2.i-3). Only a small fraction of the data shown in Figs.III.2.i-4,5 are shown in this plot for better clarity.

(a) At low flow rates

0

1,000

2,000

3,000

4,000

5,000

6,000

500 1,000 1,500 2,000 2,500 3,000 3,500 4,000

Measured (cfs)

Sim

ula

ted

flo

w (

cfs)

Observations – DCC gates formulation

• Current formulation is inappropriate

• An alternate formulation appears to simulate measured flow more closely

• Uncertainty in stage difference leads to large scatter at low flows