Investigating a Flood with EXCEL

Feb 14, 2014APEC Lesson Study Conference

Hee-chan Lew & Chan-hee AnKorea National University of Education

Introduction

• Teaching material for 6 hours for 10th graders and for ad-vanced secondary students for investigating a flood with EXCEL based on the following principles:

• Investigating a flood with Excel, students must use mathe-matics concepts or skills they have learned in schools.

• Students need to use computer technology (Excel) in order to use complex data as realistic as possible.

• Students must appreciate that mathematics is really valu-able in handling natural disasters like issuing a flood fore-cast.

• Students must understand that real situation is much com-plicated to consider more variables than in our materials.

• Students must handle Excel by themselves by putting for-mulas on proper cells after understanding some mathemat-ical concepts.

• This material are supposed to be used in the textbooks based on the next national curriculum to be revised in 2016.

Basic information (I hour): Characteristics of a Flood in Korea

• During Summer time of July to September, Korea has 2-4 typhoons carrying a heavy rain: Current average rainfall in summer time is 723.2mm and it is almost 55 % of the annual rainfall.

• Two thirds of Korean Peninsular is a moun-tain area, consisting of granite and gneiss mainly to make soils difficult to keep a wa-ter.

Current Damage by Big Typhoons.

Typhoon PeriodRainfall per

one day(mm)

Damage

Persons

Property

(Million

US$)

Lusa 2002.8.30~9.1870.5

(Gang Reung)246 5,148

‘Mae-Mi’ 2003.9.12~9.13453.0

(Nam Hae)131 4,223

Process of a flood forecast

• Korea Water Resources Corporation(KERC) • Functions of KERC:• It gets rainfall, an inflow and an outflow of

main dams and rivers; measures water levels of main rivers; controls a storage of main dams or reservoirs in advance; issues a flood forecast

• Here, students understand that some data for some key concepts are important to control a flood.

flood investigation

• Three kinds of flood investigation• reservoir : for draining waters from reser-

voirs/dams for a flood control in advance (2 hour material)

• channel or river: for issuing a flood fore-cast in the river area(I hour material)

• watershed : for forecasting a maximum outflow of the watershed(2 hour material)

• Storage Equation to represent I, O, S

Reservoir Investigation

• Inflow (I): Amount of water flowing in per second ( ㎥ /sec)

• Outflow (O): Amount of water flowing out per second ( ㎥ /sec)

• Storage (S): Amount of water kept in the reservoir ( ㎥ )

Relation among I, O, S

• Students can discuss with the follow-ing concepts:

• If I-O > 0, the ratio of change for S is +

• If I-O < 0, the ratio of change for S is -

•

Finding Storage using Excel

• −𝐼 𝑂=∆ /∆ is the Storage Equation I 𝑆 𝑡to represent the ratio of change for Storage

• Problem 1: When I and O at given times are provided, find Storage for the time interval and accumulated Storage using EXCEL.

• ..\..\..\Problem 1.xlsx

Storage Equation II

• From Problem 1, students understand that in the discrete situation, we need to use the Storage Equation II rather than the storage Equation I:

∆ −∆𝑡 𝑡=𝑆2−𝑆1

S of a Dam according to Water lev-els

• Average section method• If we know areas of sections at the two

given water levels whose difference is small, we can find Storage between the two levels by multiplying the average sec-tion of the two sections and the level dif-ference.

Example of finding Storage of a dam according to Water levels

• Problem 2: Find Storage of the trun-cated conic shaped dam of which the bottom area at 30 m high is 2000m2 and the gradient of the side is 1:6. by increasing water levels by 0.2m each.

O of a dam according to Water lev-els

• There are formulas for the outflow according to the water levels

• water-gate:

• no water-gate:

Here, is the water amount coefficient, is the width of the

waterway (m), is the distance from the water level to the

structures (m), is an acceleration of gravity(m/sec2)

Example: Finding Outflow of a dam according to Water levels

• Problem 3: If the dam has no water-gate, the height of the dam is 31m and the width of the water gate is 10m, find the overflow according to water levels. Assume that the coeffi-cient C of water amount is 1.6.

• ..\..\..\Problem 2&3.xlsx

Outflow according to the time

• Storage Equation III

• If we use Storage Equation III, when the amount of Inflow of each time, we can find the amount of Outflow according to the time.

Example of Finding Outflow accord-ing to the time

• Problem 4. There is a dam to keep water during flooding period. The bottom area of the dam is 4000 ㎡ and the dam has a vertical wall. When we know the data of I accord-ing to the given times, find O accord-ing to time. (We assume that the ini-tial dam water amount is 0)

• ..\..\..\Problem 4.xlsx

Channel Investigation

• Muskingum Method : This method was developed to forecast a flood on Muskingum area, Ohio, USA.

• Storage Equation IVThis method uses the storage equa-tion IV formed by the special relation with the weighting value between 𝑝Inflow and Outflow and the flowing index of the river : 𝑘

Equation of Outflow

• By using the equation IV and storage equation II, we can deduce the equa-tion of outflow of the river:

• ∆ −∆𝑡 𝑡=𝑆2−𝑆1

Example of Channel Investigation

• Problem 4. If we know the data of in-flow according to the given time, make a channel investigation by Muskingum method. We assume that

=0.25 and =1.5.𝑝 𝑘• ..\..\..\Problem 5.xlsx

Watershed investigation

• Watershed investigation is to forecast the final outflow of the watershed and its time after raining during some time interval.

• How to calculate the outflow of the area?

Linear n dams

• We have to assume that all amount of rainfall on the watershed flows through n hypothetical (non-exited) dams with linear arrangement in order to calculate outflow of the watershed for rainfall during some time interval.

Raining at an Instantaneous time

• We have to suppose that the watershed has the rainfall of 1 cm at a instanta-neous time to find the rainfall during some time interval. Why?

• For example, if this area has 1 cm rain-fall for 2 hours, we will interpret this rainfall as the average of two times 1 cm rainfall rained instantaneously at the beginning and the final of the 2 hour time interval.

Filling n dams

• We also assume that the instanta-neous rainfall fills n dams. It is possi-ble because the dams are hypotheti-cal and we have to decide the num-ber of n depending of rainfall.

• The rainfall rained at the instanta-neous time flows through n dams. The O of the (k-1)th dam is the I of the (k)th dam.

Relation between O and S

• Usually, O and S are in the linear rela-tion like the following equation:

• Here, is the storage coefficient and empirically known that=(delay time)/n.

• The delay time is the time difference between the time rain begins and the time of maximum outflow.

Equation for

• implies that • By storage equation I, • By the instant rainfall, because the

first dam was full, there is no inflow into the first dam, 1=0. So,

• Because the outflow of the first dam is the inflowof the second dam

Equation for

• Problem 7. Solve the above differen-tial equation to find

• Problem 8. Substituting the above equation for , find the equation for and repeat this process to find .

Example of finding maximum O

• Problem 9. Let’s the area of water-shed is 325 km2 and delaying time is 15 hours . Supposing that n=5 and n=6 respectively, compare their maximum outflows ( ㎥ /sec).

• ..\..\..\Problem 9.xlsx

maximum outflow and its time

date and timeRainfall

(mm)

Outflow

coefficientvalid rainfall (mm)

Aug 6

14:00~1600

16:00~18:00

18:00~20:00

27.3

56.7

31.4

0.55

0.60

0.7

27.3×0.55 ≒ 15.0

56.7×0.60 ≒ 34.0

31.4×0.7 ≒ 22.0

Problem10. Using the Excel data gotten in the case of n=5 in the problem 9, find the maximum outflow ( ㎥ /sec) of the watershed and its time under the following condition...\..\..\Problem10.xlsx

Epilog

• To use EXCEL spontaneously, students must know enough mathematical concepts and skills to represent a flood situation in math-ematical expression.

• Excel is a very useful tool for modeling situa-tions with mathematical concepts and skills.

• This semester, I will use these materials for high school students and revise it. Then I will report the result at my society or at the next APEC meetings.

Thank you for your atten-tion