In jkennel/hydrorecipes: Hydrogeology steps
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>"
)
library(hydrorecipes)
Generalized radial flow model (Barker, 1988)
time <- 1:2000
flow_rate <- c(rep(0.001, 500),
rep(0.002, 500),
rep(0.0, 1000))
dat <- data.frame(time, flow_rate)
# radial (flow_dimension = 2 Theis)
grf <- recipe(time~flow_rate, dat) |>
step_aquifer_grf(time = time,
flow_rate = flow_rate,
thickness = 1.0,
radius = 20,
specific_storage = 1e-5,
hydraulic_conductivity = 1e-3,
flow_dimension = 2) |>
plate()
plot(aquifer_grf~time, grf,
type = "l",
ylab = "drawdown",
xlab = "")
Anisotropic Theis flow model (Papadopulos, 1965)
time <- 1:2000
flow_rate <- c(rep(0.001, 500),
rep(0.002, 500),
rep(0.0, 1000))
dat <- data.frame(time, flow_rate)
# along major direction
theis_aniso_x <- recipe(time~flow_rate, dat) |>
step_aquifer_theis_aniso(time = time,
flow_rate = flow_rate,
thickness = 1.0,
distance_x = 20,
distance_y = 0,
specific_storage = 1e-5,
hydraulic_conductivity_major = 1.0e-3,
hydraulic_conductivity_minor = 1.0e-4,
major_axis_angle = 0.0) |>
plate()
# along minor direction
theis_aniso_y <- recipe(time~flow_rate, dat) |>
step_aquifer_theis_aniso(time = time,
flow_rate = flow_rate,
thickness = 1.0,
distance_x = 0,
distance_y = 20,
specific_storage = 1e-5,
hydraulic_conductivity_major = 1.0e-3,
hydraulic_conductivity_minor = 1.0e-4,
major_axis_angle = 0.0) |>
plate()
# midway between minor and major
theis_aniso_xy <- recipe(time~flow_rate, dat) |>
step_aquifer_theis_aniso(time = time,
flow_rate = flow_rate,
thickness = 1.0,
distance_x = 20 / (sqrt(2)),
distance_y = 20 / (sqrt(2)),
specific_storage = 1e-5,
hydraulic_conductivity_major = 1.0e-3,
hydraulic_conductivity_minor = 1.0e-4,
major_axis_angle = 0.0) |>
plate()
# isotropic higher K
theis_aniso_iso_high <- recipe(time~flow_rate, dat) |>
step_aquifer_theis_aniso(time = time,
flow_rate = flow_rate,
thickness = 1.0,
distance_x = 20,
distance_y = 0,
specific_storage = 1e-5,
hydraulic_conductivity_major = 1.0e-3,
hydraulic_conductivity_minor = 1.0e-3,
major_axis_angle = 0.0) |>
plate()
# isotropic lower K
theis_aniso_iso_low <- recipe(time~flow_rate, dat) |>
step_aquifer_theis_aniso(time = time,
flow_rate = flow_rate,
thickness = 1.0,
distance_x = 20,
distance_y = 0,
specific_storage = 1e-5,
hydraulic_conductivity_major = 1.0e-4,
hydraulic_conductivity_minor = 1.0e-4,
major_axis_angle = 0.0) |>
plate()
plot(aquifer_theis_aniso~time, theis_aniso_iso_low,
type = "l",
ylab = "drawdown",
xlab = "",
lty = 2,
col = "red")
points(aquifer_theis_aniso~time, theis_aniso_iso_high,
type = "l",
ylab = "drawdown",
xlab = "",
lty = 2)
points(aquifer_theis_aniso~time, theis_aniso_x,
type = "l",
ylab = "drawdown",
xlab = "")
points(aquifer_theis_aniso~time, theis_aniso_y,
type = "l",
ylab = "drawdown",
xlab = "",
col = "red")
points(aquifer_theis_aniso~time, theis_aniso_xy,
type = "l",
ylab = "drawdown",
xlab = "",
col = "blue")
# Early time
plot(aquifer_theis_aniso~time, theis_aniso_iso_low,
type = "l",
ylab = "drawdown",
xlab = "",
lty = 2,
xlim = c(0, 50),
ylim = c(0, 1),
col = "red")
points(aquifer_theis_aniso~time, theis_aniso_iso_high,
type = "l",
ylab = "drawdown",
xlab = "",
lty = 2)
points(aquifer_theis_aniso~time, theis_aniso_x,
type = "l",
ylab = "drawdown",
xlab = "")
points(aquifer_theis_aniso~time, theis_aniso_y,
type = "l",
ylab = "drawdown",
xlab = "",
col = "red")
points(aquifer_theis_aniso~time, theis_aniso_xy,
type = "l",
ylab = "drawdown",
xlab = "",
col = "blue")
Hantush-Jacob leaky aquifer (Hantush-Jacob, 1955)
time <- 1:2000
flow_rate <- c(rep(0.001, 500),
rep(0.002, 500),
rep(0.0, 1000))
# high
dat <- data.frame(time = 1:2000, flow_rate = flow_rate)
hj <- recipe(flow_rate~time, dat) |>
step_aquifer_leaky(time,
flow_rate,
leakage = 100,
radius = 100,
storativity = 1e-6,
transmissivity = 1e-4) |>
plate()
plot(aquifer_leaky~time, hj, type = "l",
ylab = "drawdown",
xlab = "")
Barker-Herbert radially symmetric patch (Barker-Herbert, 1982)
Figure 2
formula <- as.formula(flow_rate~time)
q <- 0.05
t_1 <- 1.0
t_2 <- 1.0
s_1 <- 1e-5
s_2 <- 1e-7
r <- 10
R <- 100
time <- 10^seq(-4, 4, 0.01)
dat <- data.frame(time = time)
bh = recipe(formula = formula, data = dat) |>
step_aquifer_patch(time = time,
flow_rate = q,
thickness = 1.0,
radius = r,
radius_patch = R,
specific_storage_inner = s_1,
specific_storage_outer = s_2,
hydraulic_conductivity_inner = t_1,
hydraulic_conductivity_outer = t_2,
n_stehfest = 12L) |>
plate()
dim_time_scale = (4.0 * t_1) / (s_1 * r^2)
dim_s_scale = 4.0 * pi * t_1 / q
bh$dim_time <- bh$time * dim_time_scale
bh$dim_s <- bh$aquifer_patch * dim_s_scale
plot(dim_s~dim_time, bh, type = "l",
ylab = "drawdown (dimensionless)",
xlab = "time (dimensionless)",
log = "x",
ylim = c(0, 50))
bh = recipe(formula = formula, data = dat) |>
step_aquifer_patch(time = time,
flow_rate = q,
thickness = 1.0,
radius = r,
radius_patch = R,
specific_storage_inner = s_1,
specific_storage_outer = s_2,
hydraulic_conductivity_inner = t_1,
hydraulic_conductivity_outer = t_2 * 10,
n_stehfest = 12L) |>
plate()
dim_time_scale = (4.0 * t_1) / (s_1 * r^2)
dim_s_scale = 4.0 * pi * t_1 / q
bh$dim_time <- bh$time * dim_time_scale
bh$dim_s <- bh$aquifer_patch * dim_s_scale
points(dim_s~dim_time, bh, type = "l")
t_1 <- t_1 * 10.0
bh = recipe(formula = formula, data = dat) |>
step_aquifer_patch(time = time,
flow_rate = q,
thickness = 1.0,
radius = r,
radius_patch = R,
specific_storage_inner = s_1,
specific_storage_outer = s_2,
hydraulic_conductivity_inner = t_1,
hydraulic_conductivity_outer = t_2,
n_stehfest = 12L) |>
plate()
dim_time_scale = (4.0 * t_1) / (s_1 * r^2)
dim_s_scale = (4.0 * pi * t_1) / q
bh$dim_time <- bh$time * dim_time_scale
bh$dim_s <- bh$aquifer_patch * dim_s_scale
points(dim_s~dim_time, bh, type = "l")
Figure 3
formula <- as.formula(flow_rate~time)
t_1 <- 1.0
t_2 <- 1.0
s_1 <- 1e-5
s_2 <- 1e-5
bh = recipe(formula = formula, data = dat) |>
step_aquifer_patch(time = time,
flow_rate = q,
thickness = 1.0,
radius = r,
radius_patch = R,
specific_storage_inner = s_1,
specific_storage_outer = s_2,
hydraulic_conductivity_inner = t_1,
hydraulic_conductivity_outer = t_2,
n_stehfest = 12L) |>
plate()
dim_time_scale = (4.0 * t_1) / (s_1 * r^2)
dim_s_scale = 4.0 * pi * t_1 / q
bh$dim_time <- bh$time * dim_time_scale
bh$dim_s <- bh$aquifer_patch * dim_s_scale
plot(dim_s~dim_time, bh, type = "l",
ylab = "drawdown (dimensionless)",
xlab = "time (dimensionless)",
log = "x",
ylim = c(0, 50))
bh = recipe(formula = formula, data = dat) |>
step_aquifer_patch(time = time,
flow_rate = q,
thickness = 1.0,
radius = r,
radius_patch = R,
specific_storage_inner = s_1,
specific_storage_outer = s_2,
hydraulic_conductivity_inner = t_1,
hydraulic_conductivity_outer = t_2 * 10,
n_stehfest = 12L) |>
plate()
dim_time_scale = (4.0 * t_1) / (s_1 * r^2)
dim_s_scale = 4.0 * pi * t_1 / q
bh$dim_time <- bh$time * dim_time_scale
bh$dim_s <- bh$aquifer_patch * dim_s_scale
points(dim_s~dim_time, bh, type = "l")
t_1 <- t_1 * 10.0
bh = recipe(formula = formula, data = dat) |>
step_aquifer_patch(time = time,
flow_rate = q,
thickness = 1.0,
radius = r,
radius_patch = R,
specific_storage_inner = s_1,
specific_storage_outer = s_2,
hydraulic_conductivity_inner = t_1,
hydraulic_conductivity_outer = t_2,
n_stehfest = 12L) |>
plate()
dim_time_scale = (4.0 * t_1) / (s_1 * r^2)
dim_s_scale = (4.0 * pi * t_1) / q
bh$dim_time <- bh$time * dim_time_scale
bh$dim_s <- bh$aquifer_patch * dim_s_scale
points(dim_s~dim_time, bh, type = "l")
Figure 4
formula <- as.formula(flow_rate~time)
q <- 0.05
t_1 <- 1.0
t_2 <- 1.0
s_1 <- 1e-5
s_2 <- 1e-5
r <- 100
R <- 90
bh = recipe(formula = formula, data = dat) |>
step_aquifer_patch(time = time,
flow_rate = q,
thickness = 1.0,
radius = r,
radius_patch = R,
specific_storage_inner = s_1,
specific_storage_outer = s_2,
hydraulic_conductivity_inner = t_1,
hydraulic_conductivity_outer = t_2,
n_stehfest = 14L) |>
plate()
dim_time_scale = (4.0 * t_1) / (s_2 * r^2)
dim_s_scale = 4.0 * pi * t_2 / q
bh$dim_time <- bh$time * dim_time_scale
bh$dim_s <- bh$aquifer_patch * dim_s_scale
plot(dim_s~dim_time, bh, type = "l",
ylab = "drawdown (dimensionless)",
xlab = "time (dimensionless)",
log = "x",
ylim = c(0, 10))
bh = recipe(formula = formula, data = dat) |>
step_aquifer_patch(time = time,
flow_rate = q,
thickness = 1.0,
radius = r,
radius_patch = R,
specific_storage_inner = s_1,
specific_storage_outer = s_2,
hydraulic_conductivity_inner = t_1,
hydraulic_conductivity_outer = t_2 * 10,
n_stehfest = 14L) |>
plate()
dim_time_scale = (4.0 * t_2 * 10) / (s_1 * r^2)
dim_s_scale = 4.0 * pi * t_2 * 10 / q
bh$dim_time <- bh$time * dim_time_scale
bh$dim_s <- bh$aquifer_patch * dim_s_scale
points(dim_s~dim_time, bh, type = "l")
bh = recipe(formula = formula, data = dat) |>
step_aquifer_patch(time = time,
flow_rate = q,
thickness = 1.0,
radius = r,
radius_patch = R,
specific_storage_inner = s_1,
specific_storage_outer = s_2,
hydraulic_conductivity_inner = t_1 * 10,
hydraulic_conductivity_outer = t_2,
n_stehfest = 14L) |>
plate()
dim_time_scale = (4.0 * t_2) / (s_1 * r^2)
dim_s_scale = (4.0 * pi * t_2) / q
bh$dim_time <- bh$time * dim_time_scale
bh$dim_s <- bh$aquifer_patch * dim_s_scale
points(dim_s~dim_time, bh, type = "l")
Papadopulos-Cooper wellbore storage(Papadopulos-Cooper, 1967)
dat <- data.frame(time = 10^seq(-5, 2, length.out = 100),
flow_rate = rep(10, 100))
formula <- as.formula(flow_rate~time)
pc = recipe(formula = formula, data = dat) |>
step_aquifer_wellbore_storage(time = time,
radius_casing = 0.3,
radius_well = 0.3,
radius = 10,
flow_rate = 10.0,
hydraulic_conductivity = 10.0,
specific_storage = 1e-4) |>
plate()
th <- recipe(time~flow_rate, dat) |>
step_aquifer_grf(time = time,
flow_rate = flow_rate,
thickness = 1.0,
radius = 10,
specific_storage = 1e-4,
hydraulic_conductivity = 10,
flow_dimension = 2) |>
plate()
plot(aquifer_wellbore_storage~time, pc,
type = "l", lwd = 3,
log = "x",
ylab = "drawdown",
xlab = "elapsed time")
points(aquifer_grf~time, th, type = "l", col = "darkgrey", lty = "dashed")
Jacob-Lohman constant drawdown (Jacob-Lohman, 1952)
time <- 10^seq(-5, 2, 0.1)
form <- formula(time~.)
dat <- data.frame(time = time)
jl = recipe(formula = form, data = dat) |>
step_aquifer_constant_drawdown(time = time,
drawdown = 10,
thickness = 10,
radius_well = 0.15,
specific_storage = 1e-6,
hydraulic_conductivity = 1) |>
plate()
plot(aquifer_constant_drawdown~time, jl, type = "l",
ylab = "flow rate",
xlab = "elapsed time",
log = "x")
References
Barker, J.A., A generalized radial flow model for hydraulic tests in fractured rock. Water Resour. Res., 24 (1988), pp. 1796-1804, 10.1029/WR024i010p01796
Barker, J.A., and R. Herbert, 1982: Pumping tests in patchy aquifers, Ground Water, vol. 20, No. 2, pp. 150-155.
Butler, J.J., 1988: Pumping tests in nonuniform aquifers – The radially symmetric case, Journal of Hydrology, Vol. 101, pp. 15-30.
Hantush, M.S. and C.E. Jacob, 1955. Non-steady radial flow in an infinite leaky aquifer, Am. Geophys. Union Trans., vol. 36, no. 1, pp. 95-100.
Heilweil, V.M. and Hsieh, P.A., 2006. Determining anisotropic transmissivity using a simplified Papadopulos method. Groundwater, 44(5), pp.749-753.
Jacob, C.E. and S.W. Lohman, 1952. Nonsteady flow to a well of constant drawdown in an extensive aquifer, Trans. Am. Geophys. Union, vol. 33, pp. 559-569.
Papadopulos, I.S., 1965. Nonsteady flow to a well in an infinite anisotropic aquifer.
Papadopulos, I.S. and H.H. Cooper, 1967. Drawdown in a well of large diameter, Water Resources Research, vol. 3, no. 1, pp. 241-244.
Prodanoff, J.H.A., Mansur, W.J. and Mascarenhas, F.C.B., 2006. Numerical evaluation of Theis and Hantush-Jacob well functions. Journal of hydrology, 318(1-4), pp.173-183.
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knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )
library(hydrorecipes)
Generalized radial flow model (Barker, 1988)
time <- 1:2000 flow_rate <- c(rep(0.001, 500), rep(0.002, 500), rep(0.0, 1000)) dat <- data.frame(time, flow_rate) # radial (flow_dimension = 2 Theis) grf <- recipe(time~flow_rate, dat) |> step_aquifer_grf(time = time, flow_rate = flow_rate, thickness = 1.0, radius = 20, specific_storage = 1e-5, hydraulic_conductivity = 1e-3, flow_dimension = 2) |> plate() plot(aquifer_grf~time, grf, type = "l", ylab = "drawdown", xlab = "")
Anisotropic Theis flow model (Papadopulos, 1965)
time <- 1:2000 flow_rate <- c(rep(0.001, 500), rep(0.002, 500), rep(0.0, 1000)) dat <- data.frame(time, flow_rate) # along major direction theis_aniso_x <- recipe(time~flow_rate, dat) |> step_aquifer_theis_aniso(time = time, flow_rate = flow_rate, thickness = 1.0, distance_x = 20, distance_y = 0, specific_storage = 1e-5, hydraulic_conductivity_major = 1.0e-3, hydraulic_conductivity_minor = 1.0e-4, major_axis_angle = 0.0) |> plate() # along minor direction theis_aniso_y <- recipe(time~flow_rate, dat) |> step_aquifer_theis_aniso(time = time, flow_rate = flow_rate, thickness = 1.0, distance_x = 0, distance_y = 20, specific_storage = 1e-5, hydraulic_conductivity_major = 1.0e-3, hydraulic_conductivity_minor = 1.0e-4, major_axis_angle = 0.0) |> plate() # midway between minor and major theis_aniso_xy <- recipe(time~flow_rate, dat) |> step_aquifer_theis_aniso(time = time, flow_rate = flow_rate, thickness = 1.0, distance_x = 20 / (sqrt(2)), distance_y = 20 / (sqrt(2)), specific_storage = 1e-5, hydraulic_conductivity_major = 1.0e-3, hydraulic_conductivity_minor = 1.0e-4, major_axis_angle = 0.0) |> plate() # isotropic higher K theis_aniso_iso_high <- recipe(time~flow_rate, dat) |> step_aquifer_theis_aniso(time = time, flow_rate = flow_rate, thickness = 1.0, distance_x = 20, distance_y = 0, specific_storage = 1e-5, hydraulic_conductivity_major = 1.0e-3, hydraulic_conductivity_minor = 1.0e-3, major_axis_angle = 0.0) |> plate() # isotropic lower K theis_aniso_iso_low <- recipe(time~flow_rate, dat) |> step_aquifer_theis_aniso(time = time, flow_rate = flow_rate, thickness = 1.0, distance_x = 20, distance_y = 0, specific_storage = 1e-5, hydraulic_conductivity_major = 1.0e-4, hydraulic_conductivity_minor = 1.0e-4, major_axis_angle = 0.0) |> plate() plot(aquifer_theis_aniso~time, theis_aniso_iso_low, type = "l", ylab = "drawdown", xlab = "", lty = 2, col = "red") points(aquifer_theis_aniso~time, theis_aniso_iso_high, type = "l", ylab = "drawdown", xlab = "", lty = 2) points(aquifer_theis_aniso~time, theis_aniso_x, type = "l", ylab = "drawdown", xlab = "") points(aquifer_theis_aniso~time, theis_aniso_y, type = "l", ylab = "drawdown", xlab = "", col = "red") points(aquifer_theis_aniso~time, theis_aniso_xy, type = "l", ylab = "drawdown", xlab = "", col = "blue") # Early time plot(aquifer_theis_aniso~time, theis_aniso_iso_low, type = "l", ylab = "drawdown", xlab = "", lty = 2, xlim = c(0, 50), ylim = c(0, 1), col = "red") points(aquifer_theis_aniso~time, theis_aniso_iso_high, type = "l", ylab = "drawdown", xlab = "", lty = 2) points(aquifer_theis_aniso~time, theis_aniso_x, type = "l", ylab = "drawdown", xlab = "") points(aquifer_theis_aniso~time, theis_aniso_y, type = "l", ylab = "drawdown", xlab = "", col = "red") points(aquifer_theis_aniso~time, theis_aniso_xy, type = "l", ylab = "drawdown", xlab = "", col = "blue")
Hantush-Jacob leaky aquifer (Hantush-Jacob, 1955)
time <- 1:2000 flow_rate <- c(rep(0.001, 500), rep(0.002, 500), rep(0.0, 1000)) # high dat <- data.frame(time = 1:2000, flow_rate = flow_rate) hj <- recipe(flow_rate~time, dat) |> step_aquifer_leaky(time, flow_rate, leakage = 100, radius = 100, storativity = 1e-6, transmissivity = 1e-4) |> plate() plot(aquifer_leaky~time, hj, type = "l", ylab = "drawdown", xlab = "")
Barker-Herbert radially symmetric patch (Barker-Herbert, 1982)
Figure 2
formula <- as.formula(flow_rate~time) q <- 0.05 t_1 <- 1.0 t_2 <- 1.0 s_1 <- 1e-5 s_2 <- 1e-7 r <- 10 R <- 100 time <- 10^seq(-4, 4, 0.01) dat <- data.frame(time = time) bh = recipe(formula = formula, data = dat) |> step_aquifer_patch(time = time, flow_rate = q, thickness = 1.0, radius = r, radius_patch = R, specific_storage_inner = s_1, specific_storage_outer = s_2, hydraulic_conductivity_inner = t_1, hydraulic_conductivity_outer = t_2, n_stehfest = 12L) |> plate() dim_time_scale = (4.0 * t_1) / (s_1 * r^2) dim_s_scale = 4.0 * pi * t_1 / q bh$dim_time <- bh$time * dim_time_scale bh$dim_s <- bh$aquifer_patch * dim_s_scale plot(dim_s~dim_time, bh, type = "l", ylab = "drawdown (dimensionless)", xlab = "time (dimensionless)", log = "x", ylim = c(0, 50)) bh = recipe(formula = formula, data = dat) |> step_aquifer_patch(time = time, flow_rate = q, thickness = 1.0, radius = r, radius_patch = R, specific_storage_inner = s_1, specific_storage_outer = s_2, hydraulic_conductivity_inner = t_1, hydraulic_conductivity_outer = t_2 * 10, n_stehfest = 12L) |> plate() dim_time_scale = (4.0 * t_1) / (s_1 * r^2) dim_s_scale = 4.0 * pi * t_1 / q bh$dim_time <- bh$time * dim_time_scale bh$dim_s <- bh$aquifer_patch * dim_s_scale points(dim_s~dim_time, bh, type = "l") t_1 <- t_1 * 10.0 bh = recipe(formula = formula, data = dat) |> step_aquifer_patch(time = time, flow_rate = q, thickness = 1.0, radius = r, radius_patch = R, specific_storage_inner = s_1, specific_storage_outer = s_2, hydraulic_conductivity_inner = t_1, hydraulic_conductivity_outer = t_2, n_stehfest = 12L) |> plate() dim_time_scale = (4.0 * t_1) / (s_1 * r^2) dim_s_scale = (4.0 * pi * t_1) / q bh$dim_time <- bh$time * dim_time_scale bh$dim_s <- bh$aquifer_patch * dim_s_scale points(dim_s~dim_time, bh, type = "l")
Figure 3
formula <- as.formula(flow_rate~time) t_1 <- 1.0 t_2 <- 1.0 s_1 <- 1e-5 s_2 <- 1e-5 bh = recipe(formula = formula, data = dat) |> step_aquifer_patch(time = time, flow_rate = q, thickness = 1.0, radius = r, radius_patch = R, specific_storage_inner = s_1, specific_storage_outer = s_2, hydraulic_conductivity_inner = t_1, hydraulic_conductivity_outer = t_2, n_stehfest = 12L) |> plate() dim_time_scale = (4.0 * t_1) / (s_1 * r^2) dim_s_scale = 4.0 * pi * t_1 / q bh$dim_time <- bh$time * dim_time_scale bh$dim_s <- bh$aquifer_patch * dim_s_scale plot(dim_s~dim_time, bh, type = "l", ylab = "drawdown (dimensionless)", xlab = "time (dimensionless)", log = "x", ylim = c(0, 50)) bh = recipe(formula = formula, data = dat) |> step_aquifer_patch(time = time, flow_rate = q, thickness = 1.0, radius = r, radius_patch = R, specific_storage_inner = s_1, specific_storage_outer = s_2, hydraulic_conductivity_inner = t_1, hydraulic_conductivity_outer = t_2 * 10, n_stehfest = 12L) |> plate() dim_time_scale = (4.0 * t_1) / (s_1 * r^2) dim_s_scale = 4.0 * pi * t_1 / q bh$dim_time <- bh$time * dim_time_scale bh$dim_s <- bh$aquifer_patch * dim_s_scale points(dim_s~dim_time, bh, type = "l") t_1 <- t_1 * 10.0 bh = recipe(formula = formula, data = dat) |> step_aquifer_patch(time = time, flow_rate = q, thickness = 1.0, radius = r, radius_patch = R, specific_storage_inner = s_1, specific_storage_outer = s_2, hydraulic_conductivity_inner = t_1, hydraulic_conductivity_outer = t_2, n_stehfest = 12L) |> plate() dim_time_scale = (4.0 * t_1) / (s_1 * r^2) dim_s_scale = (4.0 * pi * t_1) / q bh$dim_time <- bh$time * dim_time_scale bh$dim_s <- bh$aquifer_patch * dim_s_scale points(dim_s~dim_time, bh, type = "l")
Figure 4
formula <- as.formula(flow_rate~time) q <- 0.05 t_1 <- 1.0 t_2 <- 1.0 s_1 <- 1e-5 s_2 <- 1e-5 r <- 100 R <- 90 bh = recipe(formula = formula, data = dat) |> step_aquifer_patch(time = time, flow_rate = q, thickness = 1.0, radius = r, radius_patch = R, specific_storage_inner = s_1, specific_storage_outer = s_2, hydraulic_conductivity_inner = t_1, hydraulic_conductivity_outer = t_2, n_stehfest = 14L) |> plate() dim_time_scale = (4.0 * t_1) / (s_2 * r^2) dim_s_scale = 4.0 * pi * t_2 / q bh$dim_time <- bh$time * dim_time_scale bh$dim_s <- bh$aquifer_patch * dim_s_scale plot(dim_s~dim_time, bh, type = "l", ylab = "drawdown (dimensionless)", xlab = "time (dimensionless)", log = "x", ylim = c(0, 10)) bh = recipe(formula = formula, data = dat) |> step_aquifer_patch(time = time, flow_rate = q, thickness = 1.0, radius = r, radius_patch = R, specific_storage_inner = s_1, specific_storage_outer = s_2, hydraulic_conductivity_inner = t_1, hydraulic_conductivity_outer = t_2 * 10, n_stehfest = 14L) |> plate() dim_time_scale = (4.0 * t_2 * 10) / (s_1 * r^2) dim_s_scale = 4.0 * pi * t_2 * 10 / q bh$dim_time <- bh$time * dim_time_scale bh$dim_s <- bh$aquifer_patch * dim_s_scale points(dim_s~dim_time, bh, type = "l") bh = recipe(formula = formula, data = dat) |> step_aquifer_patch(time = time, flow_rate = q, thickness = 1.0, radius = r, radius_patch = R, specific_storage_inner = s_1, specific_storage_outer = s_2, hydraulic_conductivity_inner = t_1 * 10, hydraulic_conductivity_outer = t_2, n_stehfest = 14L) |> plate() dim_time_scale = (4.0 * t_2) / (s_1 * r^2) dim_s_scale = (4.0 * pi * t_2) / q bh$dim_time <- bh$time * dim_time_scale bh$dim_s <- bh$aquifer_patch * dim_s_scale points(dim_s~dim_time, bh, type = "l")
Papadopulos-Cooper wellbore storage(Papadopulos-Cooper, 1967)
dat <- data.frame(time = 10^seq(-5, 2, length.out = 100), flow_rate = rep(10, 100)) formula <- as.formula(flow_rate~time) pc = recipe(formula = formula, data = dat) |> step_aquifer_wellbore_storage(time = time, radius_casing = 0.3, radius_well = 0.3, radius = 10, flow_rate = 10.0, hydraulic_conductivity = 10.0, specific_storage = 1e-4) |> plate() th <- recipe(time~flow_rate, dat) |> step_aquifer_grf(time = time, flow_rate = flow_rate, thickness = 1.0, radius = 10, specific_storage = 1e-4, hydraulic_conductivity = 10, flow_dimension = 2) |> plate() plot(aquifer_wellbore_storage~time, pc, type = "l", lwd = 3, log = "x", ylab = "drawdown", xlab = "elapsed time") points(aquifer_grf~time, th, type = "l", col = "darkgrey", lty = "dashed")
Jacob-Lohman constant drawdown (Jacob-Lohman, 1952)
time <- 10^seq(-5, 2, 0.1) form <- formula(time~.) dat <- data.frame(time = time) jl = recipe(formula = form, data = dat) |> step_aquifer_constant_drawdown(time = time, drawdown = 10, thickness = 10, radius_well = 0.15, specific_storage = 1e-6, hydraulic_conductivity = 1) |> plate() plot(aquifer_constant_drawdown~time, jl, type = "l", ylab = "flow rate", xlab = "elapsed time", log = "x")
References
Barker, J.A., A generalized radial flow model for hydraulic tests in fractured rock. Water Resour. Res., 24 (1988), pp. 1796-1804, 10.1029/WR024i010p01796
Barker, J.A., and R. Herbert, 1982: Pumping tests in patchy aquifers, Ground Water, vol. 20, No. 2, pp. 150-155.
Butler, J.J., 1988: Pumping tests in nonuniform aquifers – The radially symmetric case, Journal of Hydrology, Vol. 101, pp. 15-30.
Hantush, M.S. and C.E. Jacob, 1955. Non-steady radial flow in an infinite leaky aquifer, Am. Geophys. Union Trans., vol. 36, no. 1, pp. 95-100.
Heilweil, V.M. and Hsieh, P.A., 2006. Determining anisotropic transmissivity using a simplified Papadopulos method. Groundwater, 44(5), pp.749-753.
Jacob, C.E. and S.W. Lohman, 1952. Nonsteady flow to a well of constant drawdown in an extensive aquifer, Trans. Am. Geophys. Union, vol. 33, pp. 559-569.
Papadopulos, I.S., 1965. Nonsteady flow to a well in an infinite anisotropic aquifer.
Papadopulos, I.S. and H.H. Cooper, 1967. Drawdown in a well of large diameter, Water Resources Research, vol. 3, no. 1, pp. 241-244.
Prodanoff, J.H.A., Mansur, W.J. and Mascarenhas, F.C.B., 2006. Numerical evaluation of Theis and Hantush-Jacob well functions. Journal of hydrology, 318(1-4), pp.173-183.
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