ce 374 k – hydrology - university of texas at austin dt de = − • conservation of mass – mass...
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CE 374 K – Hydrology
Systems and Continuity
Daene C. McKinney
DecisionSupportSystem
Data Measurement
Precipitation, Temperature, Humidity, StreamflowWater Quality, Groundwater, Snow pack,
Evapotranspiration
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Data Processing &
Archiving
Data baseData modelData display
Analysis
Rainfall/runoff, Flooding, Hydraulics, Water Allocation,
Water Pollution, Environmental Flows
Decision Making
MCDMOperating rulesExpert systemOptimization, WarningsRisk management, Dispute Resolution
Decision Implementation
Infrastructure control, Institutional policies & incentivesWarnings, Alarms
River Basin Management
Water Resources Planning and Management
• Identification, formulation and analysis of projects and designs
• Based on scientific, legal, ethical, economic, …, concepts• Problems Considered
– Municipal and industrial supply– Irrigation– Flood control– Hydroelectric power– Navigation– Water quality– Recreation– Fisheries– Drainage & sediment control– Preservation and enhancement of natural water areas,
ecological diversity, archeology, etc
Water Resource Systems Analysis
• Water resources problems are– Complex, interconnected, and overlapping– Involving water allocations, economic development, and
environmental preservation
• Systems analysis– Break complex system down into components and analyze
the interactions between the components– Central method used in water resources planning
System
Inputs, I Outputs, Q
Parameters, β
Policies or controls, α
Transformation function
Q(t) = Ω(α, β) * I(t)
Some Systems:WatershedAquiferDevelopment AreaDetention Basin
System Transformation Function
• Mathematical model• Typically a set of algebraic equations • Derived from differential equations of
– Conservation of Mass (e.g., continuity)– Conservation of Momentum (e.g., Manning)– Conservation of Energy (e.g., friction loss)
Q(t) = Ω(α, β) * I(t)
System Characteristics
• Linear vs nonlinear– Linear - superposition is valid
• If I1 Q1 and I2 Q2
• Then I1 +I2 Q1 + Q2
• Lumped vs distributed parameter (spatially varying)
• Steady-state vs transient (time dependent)• Deterministic vs stochastic (random)
Atmospheric Moisture
Interception
Snowpack
Surface
Soil Moisture
Groundwater
Streams and Lakes
Runoff
RainSnow
Evaporation
Evapotranspiration
Evaporation
Throughfall and Stem Flow
Snowmelt
Infiltration
Overland Flow
Percolation
Groundwater Flow
Channel Flow
Pervious ImperviousWatershedBoundary
Energy
Hydrologic Processes(Precipitation, Evaporation, Infiltration, Runoff)
• Transform the distribution of water in the hydrologic cycle
• Governed by fundamental conservation principles
• Reynold’s Transport Theorem allows us to derive these fundamental principles
Atmospheric Moisture
Interception
Snowpack
Surface
Soil Moisture
Groundwater
Streams and Lakes
Runoff
RainSnow
Evaporation
Evapotranspiration
Evaporation
Throughfall and Stem Flow
Snowmelt
Infiltration
Overland Flow
Percolation
Groundwater Flow
Channel Flow
Pervious ImperviousWatershedBoundary
Energy
Atmospheric Moisture
Interception
Snowpack
Surface
Soil Moisture
Groundwater
Streams and Lakes
Runoff
RainSnow
Evaporation
Evapotranspiration
Evaporation
Throughfall and Stem Flow
Snowmelt
Infiltration
Overland Flow
Percolation
Groundwater Flow
Channel Flow
Pervious ImperviousWatershedBoundary
Energy
Systems• Laws of Mechanics
– Written for systems– System = arbitrary quantity
of mass of fixed identity– Fixed quantity of mass, m
0=dtdm
dtmd )( VFr
r= dt
dWdtdQ
dtdE
−=
• Conservation of Mass
– Mass is conserved and does not change
• Momentum– If surroundings
exert force on system, mass will accelerate
• Energy– If heat is added
to system or work is done by system, energy will change
Control Volumes
• Solid Mechanics– Follow the system,
determine what happens to it
• Fluid Mechanics– Consider the behavior in a
specific region or Control Volume
• Convert System approach to CV approach– Look at specific regions,
rather than specific masses• Reynolds Transport
Theorem– Relates time derivative of
system properties to rate of change of property in CV
)(extensiveenergymomentum,mass,=
∫ ∀=∫=CVCV
ddmB βρβ
)(intensivemassunitperofamount BdmdB
=
=β
Reynolds Transport Theorem
( ) ( )
( ) ( ) ( ) ( )t
BBt
BBt
BBBBdtdB
tIttIIItIIttIIt
tIIIttIIIIIt
Δ−
+Δ−
=
Δ+−+
=
Δ+Δ+→Δ
Δ+→Δ
0
0
lim
lim
I II III
∫∫ ⋅+∫∫∫ ∀=CSCV
dddtd
dtdB AV
rrβρβρ
Continuity Equation(Conservation of Mass)
∫∫ ⋅+∫∫∫ ∀=CSCV
dddtd
dtdB AV
rrβρβρ
1system; theof mass === dmdMMB β
∫∫ ⋅+∫∫∫ ∀=CSCV
dddtd AV
rrρρ0
constantif =ρ
0=∫∫ ⋅+∫∫∫ ∀CSCV
dddtd AV
rr
)()( tQtIdtdS
−=
0)()( =−+ tItQdtdS
Inflow – Outflow = Change in Storage
Discrete Time Continuity
)()( tQtIdtdS
−=
dttQdttIdS )()( −=
∫−∫=∫Δ
Δ−
Δ
Δ−−
tj
tj
tj
tj
S
SdttQdttIdS
j
j )1()1()()(
1
jjjj QISS −=− −1
jjjj QISS −+= −1Volume of water in storage at the end of the next time period Δt, Sj, equals the volume in storage at the beginning of that period, Sj-1, plus the volume of inflow, Ij-1, minus the volume of outflow, Qj-1
),(termeachofUnits333
sft
sm
TL
=
),(termeachofUnits 333 ftmL=
Shoal Creek Flood Memorial Day 1981
• Normal flow = 90 gpm• Storm peak = 6.5 million gpm• 13 lives lost
Shoal Creek Flood Memorial Day 1981
• 6.31 in. of rain fell uniformly over 7.03 sq. mi.• What was the equivalent volume of water?
( )
hours 8in gallonsmillion 770gal921,908,770
gal/ft48052.7*ft525,055,103
ft/mi5280*mi03.7*in12ft1*in31.6
33
22
==
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
Time (hr)
Runo
ff (c
fs)
0
1
2
3
Prec
ip (i
n)
Example – Shoal Creek• Given:
– Incremental precipitation over the watershed, pulse– Streamflow measured at the outlet, continuous
• Find: Storage as function of time
• Convert streamflow to pulse data
• Average streamflow over time interval
• Equivalent depth over the watershed
• Continuity Eq. ( )∑ −+==
j
iiij QISS
10
hour21
=Δt
( ) tQQ ii Δ+ +121
( ) tQQA ii Δ+ +12
11
111200010 ;;0 QISSQISSS −+=−+==
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
Time (hr)
Run
off (
cfs)
0
1
2
3
Prec
ip (i
n)
Shoal Creek Flood
Time Time Incremental Instantaneous Incremental Incremental Cumulative Interval Precip Streamflow Streamflow Storage Storage
j t Ij Q(t) Qj ΔSj Sj
hr in cfs in in in0.0 0 203 0.00
1 0.5 0.15 246 0.02 0.13 0.132 1.0 0.26 283 0.03 0.23 0.363 1.5 1.33 828 0.06 1.27 1.624 2.0 2.20 2323 0.17 2.03 3.655 2.5 2.08 5697 0.44 1.64 5.296 3.0 0.20 9531 0.84 -0.64 4.657 3.5 0.09 11025 1.13 -1.04 3.618 4.0 0.00 8234 1.06 -1.06 2.559 4.5 0.00 4321 0.69 -0.69 1.85
10 5.0 0.00 2246 0.36 -0.36 1.4911 5.5 0.00 1802 0.22 -0.22 1.2712 6.0 0.00 1230 0.17 -0.17 1.1013 6.5 0.00 713 0.11 -0.11 1.0014 7.0 0.00 394 0.06 -0.06 0.9315 7.5 0.00 354 0.04 -0.04 0.8916 8.0 0.00 303 0.04 -0.04 0.86
6.31
01 SSS −=Δ
1I 1Q
1112 QISS −+=
( )∑ −+==
j
iiij QISS
10
0.0
0.5
1.0
1.5
2.0
2.5
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
Streamflow (in)Precip (in)
Shoal Creek Flood
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
Change in Storage (in)
0.0
1.0
2.0
3.0
4.0
5.0
6.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
Storage (in)
Shoal Creek at Northwest Park, Austin, Texas, May 24-25, 1981Area= 7.03 mi2 195985152
Time Time Incremental Instantaneous Incremental Incremental Cumulative Interval Precip Streamflow Streamflow Storage Storage
j t Ij Q(t) Qj ΔSj Sj
hr in cfs in in in0.0 0 203 0.00
1 0.5 0.15 246 0.02 0.13 0.132 1.0 0.26 283 0.03 0.23 0.363 1.5 1.33 828 0.06 1.27 1.624 2.0 2.20 2323 0.17 2.03 3.655 2.5 2.08 5697 0.44 1.64 5.296 3.0 0.20 9531 0.84 -0.64 4.657 3.5 0.09 11025 1.13 -1.04 3.618 4.0 0.00 8234 1.06 -1.06 2.559 4.5 0.00 4321 0.69 -0.69 1.85
10 5.0 0.00 2246 0.36 -0.36 1.4911 5.5 0.00 1802 0.22 -0.22 1.2712 6.0 0.00 1230 0.17 -0.17 1.1013 6.5 0.00 713 0.11 -0.11 1.0014 7.0 0.00 394 0.06 -0.06 0.9315 7.5 0.00 354 0.04 -0.04 0.8916 8.0 0.00 303 0.04 -0.04 0.86
6.31