Green-Ampt half-spherical pressure flow

This function models saturated flow from a wetted half-spherical surface into a soil profile. Flow is driven by pressure only and gravity is ignored. This might mimic a situation of subsurface recharge where the pressure head gradient is significantly larger than the elevation head gradient. Otherwise, the assumptions are the same as in the Green-Ampt equation. The equation includes \(r_f\), which is the radius to the edge of the front from the center of the sphere. The equation to calculate the time required to achieve some quantum of recharge is:

$$t = \frac{\Delta \theta}{K_{sat} (h_b - h_0)} \Bigg[ \frac{r_f^3}{3 r_b} - \frac{r_f^2}{2} + \frac{r_b^2}{6} \Bigg]$$

It is straightforward to convert between cumulative infiltration \(F_c\) and \(r_f\):

$$F_s = \Delta \theta \pi \frac{2}{3}\Big( r_f^3 - r_b^3 \Big)$$

and

$$r_f = \Big( \frac{3 F_s}{2 \Delta \theta \pi} + r_b^3 \Big)^{1/3}$$

Usage

get_greenampt_hsphere_numerical(
  theta_0,
  theta_s,
  Ksat,
  h_b,
  h_0,
  r_b,
  times,
  F_units = "ft^3"
)

Arguments

theta_0

Soil volumetric water content prior to event

theta_s

Soil porosity

Ksat

Saturated hydraulic conductivity

h_b

Hydraulic head at soil surface boundary

h_0

Hydraulic head in soil prior to event

r_b

radius from the centroid to the free water--soil boundary

times

times at which to calculate cumulative infiltration

F_units

character indicating the volumetric (L^3) `units` for the output

Value

Returns the cumulative amount of volumetric infiltration occurs through the half sphere.

Examples

library(units)
r_b <- set_units(2, "ft") # length
theta_0 <- 0.2 # unitless
theta_s <- 0.35 # unitless
F_s <- set_units(c(1, 5, 10, 20), "ft^3") # units of length^2
Ksat <- set_units(0.2, "cm/h") # length / time
h_b <- set_units(6, "ft") # hydraulic head (length)
h_0 <- set_units(-10, "cm") # hydraulic head (length)
times <- get_greenampt_hsphere_time(theta_0, theta_s, F_s, Ksat, h_b, h_0, r_b)
F_s_calc <- get_greenampt_hsphere_numerical(theta_0, theta_s, Ksat, h_b, h_0, r_b, times)
F_s_calc
#> Units: [ft^3]
#> [1]  1.000002  4.999985  9.999978 20.000015