borehole: Borehole Function

In gek: Gradient-Enhanced Kriging

Borehole Function

Description

The borehole function is defined by

f_{\rm borehole}(x) = \frac{2 \pi T_u (H_u - H_l)}{\log(r/r_w)\left(1 + \frac{2 L T_u}{\log(r / r_w) r_w^2 K_w} + \frac{T_u}{T_l}\right)}

with x = (r_w, r, T_u, H_u, T_l, H_l, L, K_w).

Usage

borehole(x)
boreholeGrad(x)

Arguments

x	a numeric vector of length 8 or a numeric matrix with n rows and 8 columns.

Details

The borehole function calculates the water flow rate \rm [m^3/yr] through a borehole.

Input	Domain	Distribution	Description
r_w	[0.05, 0.15]	\mathcal{N}(0.1, 0.0161812)	radius of borehole in \rm m
r	[100, 50\,000]	\mathcal{LN}(7.71, 1.0056)	radius of influence in \rm m
T_u	[63070, 115600]	\mathcal{U}(63070, 115600)	transmissivity of upper aquifer in \rm m^2/yr
H_u	[990, 1100]	\mathcal{U}(990, 1110)	potentiometric head of upper aquifer in \rm m
T_l	[63.1, 116]	\mathcal{U}(63.1, 116)	transmissivity of lower aquifer in \rm m^2/yr
H_l	[700, 820]	\mathcal{U}(700, 820)	potentiometric head of lower aquifer in \rm m
L	[1120, 1680]	\mathcal{U}(1120, 1680)	length of borehole in \rm m
K_w	[9855, 12045]	\mathcal{U}(9855, 12045)	hydraulic conductivity of borehole in \rm m/yr

Note, \mathcal{N}(\mu, \sigma) represents the normal distribution with expected value \mu and standard deviation \sigma and \mathcal{LN}(\mu, \sigma) is the log-normal distribution with mean \mu and standard deviation \sigma of the logarithm. Further, \mathcal{U}(a,b) denotes the continuous uniform distribution over the interval [a,b].

Value

borehole returns the function value of borehole function at x.

boreholeGrad returns the gradient of borehole function at x.

Author(s)

Carmen van Meegen

References

Harper, W. V. and Gupta, S. K. (1983). Sensitivity/Uncertainty Analysis of a Borehole Scenario Comparing Latin Hypercube Sampling and Deterministic Sensitivity Approaches. BMI/ONWI-516, Office of Nuclear Waste Isolation, Battelle Memorial Institute, Columbus, OH.

Morris, M., Mitchell, T., and Ylvisaker, D. (1993). Bayesian Design and Analysis of Computer Experiments: Use of Derivatives in Surface Prediction. Technometrics, 35(3):243–255. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/00401706.1993.10485320")}.

See Also

gekm for another example.

Examples

# List of inputs with their distributions and their respective ranges
inputs <- list("r_w" = list(dist = "norm", mean =  0.1, sd = 0.0161812, min = 0.05, max = 0.15),
	"r" = list(dist = "lnorm", meanlog = 7.71, sdlog = 1.0056, min = 100, max = 50000),
	"T_u" = list(dist = "unif", min = 63070, max = 115600),
	"H_u" = list(dist = "unif", min = 990, max = 1110),
	"T_l" = list(dist = "unif", min = 63.1, max = 116),
	"H_l" = list(dist = "unif", min = 700, max = 820),
	"L" = list(dist = "unif", min = 1120, max = 1680),
	# for a more nonlinear, nonadditive function, see Morris et al. (1993)
	"K_w" = list(dist = "unif", min = 1500, max = 15000))

# Function for Monte Carlo simulation
samples <- function(x, N = 10^5){
	switch(x$dist,
		"norm" = rnorm(N, x$mean, x$sd),
		"lnorm" = rlnorm(N, x$meanlog, x$sdlog),
		"unif" = runif(N, x$min, x$max))
}

# Uncertainty distribution of the water flow rate
set.seed(1)
X <- sapply(inputs, samples)
y <- borehole(X)
hist(y, breaks = 50, xlab = expression(paste("Water flow rate ", group("[", m^3/yr, "]"))), 
	main = "", freq = FALSE)

gek documentation built on April 4, 2025, 12:35 a.m.