Minimize the Total Pumping Cost from an Aquifer to a Water Tank, via a Pipe Network
Supposing that a number of wells pumps water from an infinite confined aquifer, transferring the water through pipes to a water tank, using this package makes it possible to allocate the pumping flow rates so that the overall operational cost is minimized. The user can insert the locations of the wells, the pipe network, the aquifer characteristics and the water demand and get the allocation of the pumping flow rates, the hydraulic drawdowns and the head losses due to pipe frictions.
See Nagkoulis, N., Katsifarakis, K. Minimization of Total Pumping Cost from an Aquifer to a Water Tank, Via a Pipe Network. Water Resour Manage 34, 4147–4162 (2020) for more information
You can install the released version of wellsolve from CRAN with:
install.packages("wellsolve")And the development version from GitHub with:
# install.packages("devtools")
devtools::install_github("teldosas/wellsolve")This is a basic example which shows you how to solve a common problem:
library(wellsolve)
projected <- "+proj=utm +zone=34"
x0 = 650000
y0 = 4400000
# the first value is the tank and the rest are wells
xc <- c(0,100,180,100,700,800,900)
xc <- xc + x0 - 10000
yc <- c(0,0,0,80,0,0,900)
yc <- yc + y0 - 10000
# xstart is a wild guess about the result.
# This helps the algorithm search for the result in the correct area
xstart <- c(0.250,0.1,0.25,0.1,0.1,0.01)
nw=7 #number of wells + 1 (the tank)
# 6
# /
# /
# 3 /
# / /
# / /
# / /
# T---1----2------4----5
#
# T=Tank 1-6=Wells no. 1-6
links <- data.frame(matrix(ncol=2, nrow=(nw-1)))
links[1:(nw-1),1] <- 1:(nw-1)
links[1,2] <- 0
links[2,2] <- 1
links[3,2] <- 0
links[4,2] <- 2
links[5,2] <- 4
links[6,2] <- 2
Dlist <- list()
Dlist[1:6] <- 0.05
# number of extra wells
nwextra <- 2
xextra <- c(500,800)
xextra <- xextra + x0 - 10000
yextra <- c(500,300)
yextra <- yextra + y0 - 10000
qextra <- c()
qextra[1] <- 0
qextra[2] <- 0
result <- wellsolve(projected, T=0.0025 ,R=3000, ff=0.03, qtot<-0.5, r0=0.200,
Dlist, qextra, nw, xc, yc, links,
nwextra, xextra, yextra, xstart)## Algorithm parameters
## --------------------
## Method: Broyden Global strategy: double dogleg (initial trust region = -2)
## Maximum stepsize = 1.79769e+308
## Scaling: fixed
## ftol = 1e-08 xtol = 1e-08 btol = 0.01 cndtol = 1e-12
##
## Iteration report
## ----------------
## Iter Jac Lambda Eta Dlt0 Dltn Fnorm Largest |f|
## 0 5.462367e+11 7.410957e+05
## 1 N(6.8e-08) N 0.9898 0.1411 0.2822 3.263078e+10 1.769475e+05
## 2 B(8.8e-08) N 0.9893 0.0372 0.0372 8.693787e+09 7.796567e+04
## 3 B(5.3e-08) N 0.9960 0.0365 0.0730 1.061454e+09 3.210573e+04
## 4 B(3.8e-08) N 0.9321 0.0201 0.0201 3.871930e+08 1.946157e+04
## 5 B(2.3e-08) W 0.7108 0.9143 0.0201 0.0401 8.979170e+07 9.374749e+03
## 6 B(1.5e-08) N 0.9019 0.0318 0.0637 2.519811e+06 1.808672e+03
## 7 B(1.4e-08) N 0.9046 0.0030 0.0059 4.167393e+05 6.473717e+02
## 8 B(5.9e-09) N 0.9095 0.0049 0.0098 6.329708e+04 2.428495e+02
## 9 B(5.4e-09) N 0.9475 0.0004 0.0004 2.216752e+04 1.475947e+02
## 10 B(5.9e-09) N 0.8946 0.0001 0.0000 2.170522e+04 1.450197e+02
## 11 B(9.9e-10) W 0.0399 0.8497 0.0000 0.0000 2.108433e+04 1.411546e+02
## 12 B(5.6e-08) C 0.8873 0.0000 0.0000 2.049992e+04 1.395073e+02
## 13 B(7.1e-08) W 0.0021 0.2056 0.0000 0.0000 2.060405e+04 1.388003e+02
## 14 N(5.2e-07) N 0.2489 0.0017 0.0035 2.455097e+02 1.575401e+01
## 15 B(5.2e-07) N 0.2367 0.0002 0.0004 9.223712e+00 3.044597e+00
## 16 B(5.5e-07) N 0.6625 0.0001 0.0001 2.607577e-04 1.449910e-02
## 17 B(5.5e-07) N 0.3479 0.0000 0.0000 5.376011e-07 7.942852e-04
## 18 B(5.6e-07) N 0.3109 0.0000 0.0000 1.836902e-07 4.390565e-04
## 19 B(5.5e-07) N 0.2292 0.0000 0.0000 4.581097e-10 2.331423e-05
# Case A
Dlist[1:6] <- 2
result <- wellsolve(projected, T=0.0025 ,R=3000, ff=0.03, qtot<-0.5, r0=0.200,
Dlist, qextra, nw, xc, yc, links,
nwextra, xextra, yextra, xstart)## Algorithm parameters
## --------------------
## Method: Broyden Global strategy: double dogleg (initial trust region = -2)
## Maximum stepsize = 1.79769e+308
## Scaling: fixed
## ftol = 1e-08 xtol = 1e-08 btol = 0.01 cndtol = 1e-12
##
## Iteration report
## ----------------
## Iter Jac Lambda Eta Dlt0 Dltn Fnorm Largest |f|
## 0 2.191820e+05 3.933785e+02
## 1 N(2.0e-04) N 0.9605 0.2779 0.5558 2.460913e-05 3.316578e-03
## 2 B(2.0e-04) N 0.9948 0.0000 0.0000 3.059305e-14 1.164155e-07
## 3 B(2.0e-04) N 0.9990 0.0000 0.0000 6.342875e-26 1.928310e-13
# Case LC1
result <- wellsolve(projected, T=0.0025 ,R=3000, ff=0, qtot<-0.5, r0=0.200,
Dlist, qextra, nw, xc, yc, links,
nwextra, xextra, yextra, xstart)## Algorithm parameters
## --------------------
## Method: Broyden Global strategy: double dogleg (initial trust region = -2)
## Maximum stepsize = 1.79769e+308
## Scaling: fixed
## ftol = 1e-08 xtol = 1e-08 btol = 0.01 cndtol = 1e-12
##
## Iteration report
## ----------------
## Iter Jac Lambda Eta Dlt0 Dltn Fnorm Largest |f|
## 0 2.191839e+05 3.933803e+02
## 1 N(2.0e-04) N 0.9605 0.2779 0.5558 5.736430e-13 8.167479e-07
## 2 B(2.0e-04) N 0.6520 0.0000 0.0000 9.087678e-28 2.842171e-14
# Case B
Dlist[1:6] <- 0.4
result <- wellsolve(projected, T=0.0025 ,R=3000, ff=0.03, qtot<-0.5, r0=0.200,
Dlist, qextra, nw, xc, yc, links,
nwextra, xextra, yextra, xstart)## Algorithm parameters
## --------------------
## Method: Broyden Global strategy: double dogleg (initial trust region = -2)
## Maximum stepsize = 1.79769e+308
## Scaling: fixed
## ftol = 1e-08 xtol = 1e-08 btol = 0.01 cndtol = 1e-12
##
## Iteration report
## ----------------
## Iter Jac Lambda Eta Dlt0 Dltn Fnorm Largest |f|
## 0 2.139104e+05 3.877135e+02
## 1 N(1.9e-04) N 0.9664 0.2662 0.5324 2.229651e+02 9.945352e+00
## 2 B(1.9e-04) N 0.9995 0.0068 0.0137 2.044975e+00 9.380462e-01
## 3 B(1.7e-04) N 0.9999 0.0006 0.0012 1.587382e-05 3.448635e-03
## 4 B(1.7e-04) N 0.9751 0.0000 0.0000 1.163745e-09 4.167468e-05
## 5 B(1.7e-04) N 0.6950 0.0000 0.0000 1.249868e-12 1.416031e-06
## 6 B(1.7e-04) N 0.6949 0.0000 0.0000 6.889993e-17 6.448097e-09
# Case C
qextra[1] <- 0.1
qextra[2] <- 0.4
result <- wellsolve(projected, T=0.0025 ,R=3000, ff=0.03, qtot<-0.5, r0=0.200,
Dlist, qextra, nw, xc, yc, links,
nwextra, xextra, yextra, xstart)## Algorithm parameters
## --------------------
## Method: Broyden Global strategy: double dogleg (initial trust region = -2)
## Maximum stepsize = 1.79769e+308
## Scaling: fixed
## ftol = 1e-08 xtol = 1e-08 btol = 0.01 cndtol = 1e-12
##
## Iteration report
## ----------------
## Iter Jac Lambda Eta Dlt0 Dltn Fnorm Largest |f|
## 0 2.154684e+05 3.812010e+02
## 1 N(1.9e-04) N 0.9739 0.2619 0.5239 2.378423e+02 1.049493e+01
## 2 B(1.9e-04) N 0.9989 0.0071 0.0141 2.182377e+00 9.653594e-01
## 3 B(1.7e-04) N 0.9996 0.0006 0.0012 2.172868e-05 4.698379e-03
## 4 B(1.7e-04) N 0.7206 0.0000 0.0000 3.098375e-08 1.748498e-04
## 5 B(1.7e-04) N 0.6279 0.0000 0.0000 4.350881e-12 2.631109e-06
## 6 B(1.7e-04) N 0.5538 0.0000 0.0000 5.567208e-16 2.117073e-08