R help - Apr 2024 - Question regarding reservoir volume and water level

So, you know how to get volume for given water level. 

For the reverse problem, you get in trouble because of the nonlinearity inherent
in the dependence of surface area on the level.

I don't think there is a simple solution to this, save for mapping out the
volume as a function of water level and solving equations for the water level
using (say) uniroot(). Which may actually suffice for practical purposes.

For small changes, finding the derivative of the relation is easy: d(volume) =
Area * d(level) and this can be used as an approximate relation as long as the
Area remains nearly constant.

However generic questions like doubling the volume are impossible to answer
without knowledge of the reservoir shape. E.g. in a cylindrical reservoir
halving the water level also halves the volume, but in a conical reservoir,
halving the level leaves only 1/8 of the volume.

-pd


> On 8 Apr 2024, at 05:55 , javad bayat <j.bayat194 at gmail.com>
wrote:
> 
> Dear all;
> Many thanks for your replies. This was not homework. I apologize.
> Let me explain more.
> There is a dam constructed in a valley with the highest elevation of 1255
> m. The area of its reservoir can be calculated by drawing a polygon around
> the water and it is known.
> I have the Digital Elevation Model (DEM) of the region (reservoir and its
> surrounding area). I have calculated the volume of the current reservoir
> (7e6 m3) using the following codes.
> library(raster)
> library(terra)
> library(exactextractr)
> library(dplyr)
> library(sf)
> # Calculate volume for polygon
> # Read the DEM raster file
> r <- rast("E:/...DEM.tif")
> # Read the polygon shapefile
> p <- st_read("E:/...Dam.shp")
> 
> r <- crop(r, extent(p))
> r <- mask(r, p)
> 
> # Extract the cells in each polygon and calculate the area of each cell
> x <- exact_extract(r, p, coverage_area = TRUE)
> # Extract polygon values as a dataframe
> x1 = as.data.frame(x[1])
> head(x1)
> x1 = na.omit(x1)
> # Calculate the height above the minimum elevation in the polygon
> x1$Height = max(x1[,1]) - x1[,1]
> # Calculate the volume of each cell
> x1$Vol = x1[,2] * x1[,3]
> sum(x1$Vol)
> x2 = x1[,c(1,2,4)]
> x2 = sort(x2,'value')
> head(x2)
> x3 <- aggregate(Vol ~ value, data = x2, FUN = sum)
> x4 <- aggregate(coverage_area ~ value, data = x2, FUN = sum)
> x5 = cbind(x3, Area = x4[,2])
> library(dplyr)
> x6 <- x5 %>%
>  mutate(V_sum = cumsum(Vol)) %>%
>  mutate(A_sum = cumsum(Area))
> plot(x6$value~x6$V_sum)
> 
> And I thought that it is possible to get the elevation for a specific
> volume by linear model between elevation and volume, as follow:
> 
> # Get a linear model between elevation and the volume
> lm1 <- lm(value ~ V_sum, data = x6)
> d <- data.frame(V_sum = 14e6)  #
> predict(lm1, newdata = d)
> 
> But it is not possible through the LM.
> Now I want to know what would be the water level in the reservoir if the
> reservoir volume doubled or we adding a known volume to it?
> Also what would be the volume if the water level increases to 1250 m?
> 
> I would be more than happy if you help me to do this.
> Sincerely
> 
> On Mon, Apr 8, 2024 at 12:23?AM <avi.e.gross at gmail.com> wrote:
> 
>> John,
>> 
>> Your reaction was what my original reaction was until I realized I had
to
>> find out what a DEM file was and that contains enough of the kind of
>> depth-dimension data you describe albeit what may be a very irregular
cross
>> section to calculate for areas and thence volumes.
>> 
>> If I read it correctly, this can be a very real-world problem worthy of
a
>> solution, such as in places like California where they had a tad more
rain
>> than usual and some reservoirs may overflow. Someone else provided what
>> sounds like a mathematical algorithm but my guess is what is needed
here is
>> perhaps less analytic since there may be no trivial way to create
formulas
>> and take integrals and so on, but simply an approximate way to
calculate
>> incremental volumes for each horizontal "slice" and keep
adding or
>> subtracting them till you reach a target and then read off another
variable
>> at that point such as depth.
>> 
>> Some care must be taken as water level has to be relative to something
and
>> many natural reservoirs have no unique bottom level. Some water may
also be
>> stored underground and to the side and pour in if the level lowers or
can
>> be
>> used to escape if the level rises.
>> 
>> 
>> -----Original Message-----
>> From: R-help <r-help-bounces at r-project.org> On Behalf Of
Sorkin, John
>> Sent: Sunday, April 7, 2024 3:08 PM
>> To: Rui Barradas <ruipbarradas at sapo.pt>; javad bayat
<j.bayat194 at gmail.com
>>> ;
>> R-help <R-help at r-project.org>
>> Subject: Re: [R] Question regarding reservoir volume and water level
>> 
>> Aside from the fact that the original question might well be a class
>> exercise (or homework), the question is unanswerable given the data
given
>> by
>> the original poster. One needs to know the dimensions of the reservoir,
>> above and below the current waterline. Are the sides, above and below
the
>> waterline smooth? Is the region currently above the waterline that can
>> store
>> water a mirror image of the region below the waterline? Is the region
above
>> the reservoir include a flood plane? Will the additional water go into
the
>> flood plane?
>> 
>> The lack of required detail in the question posed by the original
poster
>> suggests that there are strong assumptions, assumptions that typically
>> would
>> be made in a class-room example or exercise.
>> 
>> John
>> 
>> John David Sorkin M.D., Ph.D.
>> Professor of Medicine, University of Maryland School of Medicine;
>> Associate Director for Biostatistics and Informatics, Baltimore VA
Medical
>> Center Geriatrics Research, Education, and Clinical Center;
>> PI Biostatistics and Informatics Core, University of Maryland School of
>> Medicine Claude D. Pepper Older Americans Independence Center;
>> Senior Statistician University of Maryland Center for Vascular
Research;
>> 
>> Division of Gerontology and Paliative Care,
>> 10 North Greene Street
>> GRECC (BT/18/GR)
>> Baltimore, MD 21201-1524
>> Cell phone 443-418-5382
>> 
>> 
>> 
>> 
>> ________________________________________
>> From: R-help <r-help-bounces at r-project.org> on behalf of Rui
Barradas
>> <ruipbarradas at sapo.pt>
>> Sent: Sunday, April 7, 2024 10:53 AM
>> To: javad bayat; R-help
>> Subject: Re: [R] Question regarding reservoir volume and water level
>> 
>> ?s 13:27 de 07/04/2024, javad bayat escreveu:
>>> Dear all;
>>> I have a question about the water level of a reservoir, when the
volume
>>> changed or doubled.
>>> There is a DEM file with the highest elevation 1267 m. The lowest
>> elevation
>>> is 1230 m. The current volume of the reservoir is 7,000,000 m3 at
1240 m.
>>> Now I want to know what would be the water level if the volume
rises to
>>> 1250 m? or what would be the water level if the volume doubled
>> (14,000,000
>>> m3)?
>>> 
>>> Is there any way to write codes to do this in R?
>>> I would be more than happy if anyone could help me.
>>> Sincerely
>>> 
>>> 
>>> 
>>> 
>>> 
>>> 
>>> 
>>> 
>> Hello,
>> 
>> This is a simple rule of three.
>> If you know the level l the argument doesn't need to be named but
if you
>> know the volume v then it must be named.
>> 
>> 
>> water_level <- function(l, v, level = 1240, volume = 7e6) {
>>   if(missing(v)) {
>>     volume * l / level
>>   } else level * v / volume
>> }
>> 
>> lev <- 1250
>> vol <- 14e6
>> 
>> water_level(l = lev)
>> #> [1] 7056452
>> water_level(v = vol)
>> #> [1] 2480
>> 
>> 
>> Hope this helps,
>> 
>> Rui Barradas
>> 
>> 
>> --
>> Este e-mail foi analisado pelo software antiv?rus AVG para verificar a
>> presen?a de v?rus.
>> http://www.avg.com/
>> 
>> ______________________________________________
>> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see
>> https://stat.ethz.ch/mailman/listinfo/r-help
>> PLEASE do read the posting guide
>> http://www.r-project.org/posting-guide.html
>> and provide commented, minimal, self-contained, reproducible code.
>> 
>> ______________________________________________
>> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see
>> https://stat.ethz.ch/mailman/listinfo/r-help
>> PLEASE do read the posting guide
>> http://www.R-project.org/posting-guide.html
>> and provide commented, minimal, self-contained, reproducible code.
>> 
>> 
> 
> -- 
> Best Regards
> Javad Bayat
> M.Sc. Environment Engineering
> Alternative Mail: bayat194 at yahoo.com
> 
> 	[[alternative HTML version deleted]]
> 
> ______________________________________________
> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
-- 
Peter Dalgaard, Professor,
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Office: A 4.23
Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com

Water engineer here. The standard approach is to 1) get the storage vs. 
elevation data from the designers of the reservoir or, barring that, 2) 
get the bathymetry data from USBR or state DWR, or, if available, get 
the DEM data from USGS if the survey was done before the reservoir was 
built or 3) get a boat+sonar with GPS? +lots of time and survey the 
bottom elevation yourself. Put the xyz data into ArcGIS and have it 
create the bottom surface, then, with several elevations, integrate the 
xyz data from Z to the bottom to find the storage. Plot the storage at 
each water surface to get an idea of the shape and then use 
lm(Elevation~f(Storage) where f(Storage) may be a cubic or quartic 
polynomial. Then double the Storage and calculate Elevation. This type 
of thing is done everyday by hydrologists.

Good luck

David K Stevens, PhD, PE, Professor
Civil and Environmental Engineering
Utah Water Research Laboratory
Utah State University
8200 Old Main Hill
Logan, UT 84322-8200
david.stevens at usu.edu
(435) 797-3229 (office)

On 4/9/2024 8:01 AM, peter dalgaard wrote:
> So, you know how to get volume for given water level.
>
> For the reverse problem, you get in trouble because of the nonlinearity
inherent in the dependence of surface area on the level.
>
> I don't think there is a simple solution to this, save for mapping out
the volume as a function of water level and solving equations for the water
level using (say) uniroot(). Which may actually suffice for practical purposes.
>
> For small changes, finding the derivative of the relation is easy:
d(volume) = Area * d(level) and this can be used as an approximate relation as
long as the Area remains nearly constant.
>
> However generic questions like doubling the volume are impossible to answer
without knowledge of the reservoir shape. E.g. in a cylindrical reservoir
halving the water level also halves the volume, but in a conical reservoir,
halving the level leaves only 1/8 of the volume.
>
> -pd
>
>
>
>> On 8 Apr 2024, at 05:55 , javad bayat <j.bayat194 at gmail.com>
wrote:
>>
>> Dear all;
>> Many thanks for your replies. This was not homework. I apologize.
>> Let me explain more.
>> There is a dam constructed in a valley with the highest elevation of
1255
>> m. The area of its reservoir can be calculated by drawing a polygon
around
>> the water and it is known.
>> I have the Digital Elevation Model (DEM) of the region (reservoir and
its
>> surrounding area). I have calculated the volume of the current
reservoir
>> (7e6 m3) using the following codes.
>> library(raster)
>> library(terra)
>> library(exactextractr)
>> library(dplyr)
>> library(sf)
>> # Calculate volume for polygon
>> # Read the DEM raster file
>> r <- rast("E:/...DEM.tif")
>> # Read the polygon shapefile
>> p <- st_read("E:/...Dam.shp")
>>
>> r <- crop(r, extent(p))
>> r <- mask(r, p)
>>
>> # Extract the cells in each polygon and calculate the area of each cell
>> x <- exact_extract(r, p, coverage_area = TRUE)
>> # Extract polygon values as a dataframe
>> x1 = as.data.frame(x[1])
>> head(x1)
>> x1 = na.omit(x1)
>> # Calculate the height above the minimum elevation in the polygon
>> x1$Height = max(x1[,1]) - x1[,1]
>> # Calculate the volume of each cell
>> x1$Vol = x1[,2] * x1[,3]
>> sum(x1$Vol)
>> x2 = x1[,c(1,2,4)]
>> x2 = sort(x2,'value')
>> head(x2)
>> x3 <- aggregate(Vol ~ value, data = x2, FUN = sum)
>> x4 <- aggregate(coverage_area ~ value, data = x2, FUN = sum)
>> x5 = cbind(x3, Area = x4[,2])
>> library(dplyr)
>> x6 <- x5 %>%
>>   mutate(V_sum = cumsum(Vol)) %>%
>>   mutate(A_sum = cumsum(Area))
>> plot(x6$value~x6$V_sum)
>>
>> And I thought that it is possible to get the elevation for a specific
>> volume by linear model between elevation and volume, as follow:
>>
>> # Get a linear model between elevation and the volume
>> lm1 <- lm(value ~ V_sum, data = x6)
>> d <- data.frame(V_sum = 14e6)  #
>> predict(lm1, newdata = d)
>>
>> But it is not possible through the LM.
>> Now I want to know what would be the water level in the reservoir if
the
>> reservoir volume doubled or we adding a known volume to it?
>> Also what would be the volume if the water level increases to 1250 m?
>>
>> I would be more than happy if you help me to do this.
>> Sincerely
>>
>> On Mon, Apr 8, 2024 at 12:23?AM <avi.e.gross at gmail.com> wrote:
>>
>>> John,
>>>
>>> Your reaction was what my original reaction was until I realized I
had to
>>> find out what a DEM file was and that contains enough of the kind
of
>>> depth-dimension data you describe albeit what may be a very
irregular cross
>>> section to calculate for areas and thence volumes.
>>>
>>> If I read it correctly, this can be a very real-world problem
worthy of a
>>> solution, such as in places like California where they had a tad
more rain
>>> than usual and some reservoirs may overflow. Someone else provided
what
>>> sounds like a mathematical algorithm but my guess is what is needed
here is
>>> perhaps less analytic since there may be no trivial way to create
formulas
>>> and take integrals and so on, but simply an approximate way to
calculate
>>> incremental volumes for each horizontal "slice" and keep
adding or
>>> subtracting them till you reach a target and then read off another
variable
>>> at that point such as depth.
>>>
>>> Some care must be taken as water level has to be relative to
something and
>>> many natural reservoirs have no unique bottom level. Some water may
also be
>>> stored underground and to the side and pour in if the level lowers
or can
>>> be
>>> used to escape if the level rises.
>>>
>>>
>>> -----Original Message-----
>>> From: R-help <r-help-bounces at r-project.org> On Behalf Of
Sorkin, John
>>> Sent: Sunday, April 7, 2024 3:08 PM
>>> To: Rui Barradas <ruipbarradas at sapo.pt>; javad bayat
<j.bayat194 at gmail.com
>>>> ;
>>> R-help <R-help at r-project.org>
>>> Subject: Re: [R] Question regarding reservoir volume and water
level
>>>
>>> Aside from the fact that the original question might well be a
class
>>> exercise (or homework), the question is unanswerable given the data
given
>>> by
>>> the original poster. One needs to know the dimensions of the
reservoir,
>>> above and below the current waterline. Are the sides, above and
below the
>>> waterline smooth? Is the region currently above the waterline that
can
>>> store
>>> water a mirror image of the region below the waterline? Is the
region above
>>> the reservoir include a flood plane? Will the additional water go
into the
>>> flood plane?
>>>
>>> The lack of required detail in the question posed by the original
poster
>>> suggests that there are strong assumptions, assumptions that
typically
>>> would
>>> be made in a class-room example or exercise.
>>>
>>> John
>>>
>>> John David Sorkin M.D., Ph.D.
>>> Professor of Medicine, University of Maryland School of Medicine;
>>> Associate Director for Biostatistics and Informatics, Baltimore VA
Medical
>>> Center Geriatrics Research, Education, and Clinical Center;
>>> PI Biostatistics and Informatics Core, University of Maryland
School of
>>> Medicine Claude D. Pepper Older Americans Independence Center;
>>> Senior Statistician University of Maryland Center for Vascular
Research;
>>>
>>> Division of Gerontology and Paliative Care,
>>> 10 North Greene Street
>>> GRECC (BT/18/GR)
>>> Baltimore, MD 21201-1524
>>> Cell phone 443-418-5382
>>>
>>>
>>>
>>>
>>> ________________________________________
>>> From: R-help <r-help-bounces at r-project.org> on behalf of
Rui Barradas
>>> <ruipbarradas at sapo.pt>
>>> Sent: Sunday, April 7, 2024 10:53 AM
>>> To: javad bayat; R-help
>>> Subject: Re: [R] Question regarding reservoir volume and water
level
>>>
>>> ?s 13:27 de 07/04/2024, javad bayat escreveu:
>>>> Dear all;
>>>> I have a question about the water level of a reservoir, when
the volume
>>>> changed or doubled.
>>>> There is a DEM file with the highest elevation 1267 m. The
lowest
>>> elevation
>>>> is 1230 m. The current volume of the reservoir is 7,000,000 m3
at 1240 m.
>>>> Now I want to know what would be the water level if the volume
rises to
>>>> 1250 m? or what would be the water level if the volume doubled
>>> (14,000,000
>>>> m3)?
>>>>
>>>> Is there any way to write codes to do this in R?
>>>> I would be more than happy if anyone could help me.
>>>> Sincerely
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>> Hello,
>>>
>>> This is a simple rule of three.
>>> If you know the level l the argument doesn't need to be named
but if you
>>> know the volume v then it must be named.
>>>
>>>
>>> water_level <- function(l, v, level = 1240, volume = 7e6) {
>>>    if(missing(v)) {
>>>      volume * l / level
>>>    } else level * v / volume
>>> }
>>>
>>> lev <- 1250
>>> vol <- 14e6
>>>
>>> water_level(l = lev)
>>> #> [1] 7056452
>>> water_level(v = vol)
>>> #> [1] 2480
>>>
>>>
>>> Hope this helps,
>>>
>>> Rui Barradas
>>>
>>>
>>> --
>>> Este e-mail foi analisado pelo software antiv?rus AVG para
verificar a
>>> presen?a de v?rus.
>>> http://www.avg.com/
>>>
>>> ______________________________________________
>>> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more,
see
>>> https://stat.ethz.ch/mailman/listinfo/r-help
>>> PLEASE do read the posting guide
>>> http://www.r-project.org/posting-guide.html
>>> and provide commented, minimal, self-contained, reproducible code.
>>>
>>> ______________________________________________
>>> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more,
see
>>> https://stat.ethz.ch/mailman/listinfo/r-help
>>> PLEASE do read the posting guide
>>> http://www.R-project.org/posting-guide.html
>>> and provide commented, minimal, self-contained, reproducible code.
>>>
>>>
>> -- 
>> Best Regards
>> Javad Bayat
>> M.Sc. Environment Engineering
>> Alternative Mail: bayat194 at yahoo.com
>>
>> 	[[alternative HTML version deleted]]
>>
>> ______________________________________________
>> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see
>> https://stat.ethz.ch/mailman/listinfo/r-help
>> PLEASE do read the posting guide
http://www.R-project.org/posting-guide.html
>> and provide commented, minimal, self-contained, reproducible code.