Estimating Data Frequent Situation: Data observed at discrete points - - PowerPoint PPT Presentation

Estimating Data

Frequent Situation: Data observed at discrete points pi. Estimate to be made for another point q.

r1 r2 r3 r4 r5 r6 p1 p2 p3 p4 p5 p6 q

Two simple options. Constant and Linear interpolation.

r1 r2 r3 r4 r5 r6 p1 p2 p3 p4 p5 p6 Linear Interpolation Constant Approximation q

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MyWatershed-estimating total rainfall

Rain−gauges MyWatershed

Shown here is my watershed with the locations of rain-gauges. Estimate the total rainfall over my watershed (in cubic-meters . Question: What should I assume as the rainfall at point p? Heuristic: Assign to each point p, the rainfall at the closest gauge.

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The Voronoi

a b d c bisector perpendicular

region(c): All points for which c is the closest. Note that it depends on the presence of other points.

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The Domain decomposition

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MyWatershed-the overlay

MyWatershed g(i) A(i)

Draw your watershed on a graph-paper. Let g(i) be a gauge and let the reading at g(i) be r(i). We want to find all points p for which the closest point is g(i).

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MyWatershed-the overlay

MyWatershed g(i) A(i)

Draw your watershed on a graph-paper. Let g(i) be a gauge and let the reading at g(i) be r(i). We want to find all points p for which the closest point is g(i). Compute the polygon P(i) by the method of bisectors. Let A(i) be the fraction of the area lying inside my waterhsed.

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MyWatershed-the construction

MyWatershed g(i) A(i) Ignore

Measure A(i) using the graph

• paper. Ignore area outside the

watershed. The sum

i A(i) = A the total

area of the watershed. Average rainfall r = A(i)r(i) A(i)

Finally...

Total Volumne= A.r

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Domain Decomposition

Division of the domain into non-overlapping triangles

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Internal Section Formula

x 1−x y 1−y a b c p q r

f (p) = (1 − x) · f (a) + x · (y · f (c) + (1 − y) · f (b))

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Delaunay-Voronai Dual Decomposition

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Other Options

MyWatershed g(i) A(i) a,f(a) b,f(b) c,f(c) x,f(x) x=u1.a+u2.b+u3.c f(x)=u1.f(a)+u2.f(b)+u3.f(c) u1+u2+u3=1

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Measuring Stream-flows

V-notch weir. Suitable for small streams. A V-notch is inserted in the stream so that there is sufficient head behind the V-notch. Measurements are taken

• n the height of the

stream-level on the V-notch. Flow: cu.m./s is given by an empirical relationship. For a 90-degree V -notch:Q = 2.5H5/2 where Q in cu.ft/s, and H is ht. of head above crest. Example: If H = 0.25ft then Q = 0.078 cu.ft/s.

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Measuring Stream-flows

For larger streams Use a stick-mounted flow-meter. Select a stream cross-section. Follow a schedule

• f measurements

at various depths and points on the cross-section. Use formula to compute flow.

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Measuring Stream-flows

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Flow in Open-Channel

Mannings Eqn. V = (1.49R2/3S1/2)/n where V is average velocity in ft/s R is surface-area/wet-perimeter in ft. S is the slope of the water and n is as below: Mountain streams 0.04 winding stream 0.035 natural streams 0.025 unlined canals 0.02 smooth concrete 0.012 Example (Fetter) : An aquaduct is with a slope of 5ft/mile and with a rectangular cross-section of 50ft and water depth of 8ft. What is the average velocity? R = (50×8)/66 = 6.06. S = 5/(1760 × 3) = 0.000947. n = 0.02. V = 3.048ft/s

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Another Problem

Mumbai needs 3000 mega-liters/day which come from lakes about 100 km away and about 500 ft above Mumbai in

• elevation. Estimate the the number of pipes needed to transfer

this water, if the diameter of these pipes is 2m. Estimate the total flow and the height of water in the nalla

• pposite Hostel 5 on a rainy day.

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Diversion Based Irrigation

Small bandaharas within stream and small canals to take away water. Few gates and flood based canal delivery. extraction through wells

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Diversion Based Irrigation-A Section

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Measuring other flows

Infiltration: Standard models. Also Infiltrometer which measures infiltration and conductivity, a hydrogeological term.

◮ slope, soil properties, vegetation.

Transpiration: Standard data from experimental plots. Also FAO and agriculture department.

◮ Typically depends on wind velocity, air temperature, humidity

and also plant properties.

◮ Typically about 100 to 200 times of wieght gained by plant. For

crops, about 3mm per day.

• Evaporation. From soil as well as water bodies. 1mm-5mm per
• day. Depends on air temperature, humidity and velocity.

Seepage, Groundwater flows: Depends on conductivity and hydraulic heads. Darcy’s law.

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Thanks

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Water and Development

Part 2c: Sub-surface and Groundwater

Milind Sohoni

www.cse.iitb.ac.in/∼sohoni email: sohoni@cse.iitb.ac.in

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Objectives

Sub-surface and Groundwater: Stocks and Flows. How does GW and SSW function as stocks? Sub-surface water (a.k.a. Soil Moisture) in the top few meters. Groundwater: deeper, saturated. Complex interaction between SSW and GW. What are the basic mechanisms (laws and models) by which they work? What are the key parameters to describe these and how are these measured?

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Groundwater

• Deep. Accessed through wells and

bore-wells. Water-Table: important concept. How much water is available through-out the year? Specific Yield Does it depend on the nature of soil/rock underneath? Aquifer How do different wells interact? Conductivity

• Ground

WaterTable Well

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Porosity: Soil as a container

Porosity: The fraction of empty space with a soil. em Depends on configuration. Porosity depends on the regularity of particle size. The more sorted the particles, the higher the porosity. May change across different areas and different depths.

High Porosity Low

Sand 0.1mm-1mm Silt 0.005mm-0.1mm Clay < 0.005mm

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Moisture

Moisture: The volume fraction of wet soil which is water. Water exists in within the voids and is either (i) loosely held, or (ii) tightly held by soil particles. Soil moisture n increases with depth and reaches its theoretical maximum of proposity p. This is called saturation. At this point, soil moisture equals porosity.

p moisture depth saturation

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Saturation

The region below is called the saturated region. The region above is the unsaturated region. This depth is called the depth of the water-table. At this depth, water appears spontaneously in a dug-well. Saturated water can be extracted easily. Unsaturated region: important for plants and microbes. Groundwater also flows just as ordinary water, albeit at different rates. Groundwater flows eventually go to streams, rivers and oceans.

• Ground

WaterTable Well

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Moisture when it rains:

When the rain falls (a) Before Rains: surface moisture less than porosity. (b) Start of Rain: surface mosture starts increasing: Infiltration phase. (c) Saturation: Surface saturates: Run-Off phase. (d) Rain Stops: Moisture descends and joins water-table by gravity.

(a) (b) (c) (d) Depth Water−Table

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Porosity and Soil Moisture

Key Quantites Soil Moisture: Fraction of soil-volume filled with water. Porosity of a soil: Maximum possible value of soil moisture. Take a fixed volume V sample of soil.

◮ Use a standard gouge, scoop, screw or core.

Let Ws be its weight. Let Wd be the weight of the sample after oven-drying. Let Ww be the weight of the sample after immersing it in water till it gets saturated. Let ρ be the density of water. Porosity p = Ww − Wd ρV Moisture n = Ws − Wd ρV

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Porosity and Specific Yield

Porosity: The volume fraction of void to solid in dried sample. Saturation: When these voids are fully filled with water. Specific Yield Sy: the ration of the colume of water that drains from a rock owing to gravity, to the total rock volumne.

• Q

h2 h1

h1, h2 resp., are the heights of the saturated layer. Q is the volume of the water discharged to reach h2 from h1. Sy =

Q (h1−h2)A

Caution: rock above hi is wet, but unsaturated.

• Lab. setup: Takes a lot of time for

water to drip.

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Specific Yield

Importance: This is actually the fraction which is accessible. Note 1: In accessible voids are NOT counted in porosity. Note 2: To access full n-fraction, oven heating was required. Clearly Sy ≤ n, the porosity and Sr = n − Sy Sr is called the Specific Retentivity. Sr is largely due to the adhesion of water molecules to the rock layer. Specific Yield of a well : to be done later. Some Specific Yields Clay 2 Sandy Clay 7 Silt 18 Fine Sand 21 Medium Sand 26 Fine Gravel 25

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