In jkennel/sokol: Blended water level analysis for wells

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

Goal - Estimate the relative transmissivity change or head

Transmissivity solution

Given (Sokol eq 6): The example will be a single unit where we add a new one.

$$h_w = \frac{T_1 H_1}{T_1}$$ $$h_w + \Delta h_w = \frac{T_1 H_1 + T_2 H_2}{T_1 + T_2}$$

where:

$h_w$ is the blended head for the shorter interval
$h_w + \Delta h_w$ is the blended head for the larger interval
$T_1$ is the transmissivity for the shorter interval
$T_1 + T_2$ is the transmissivity for the larger interval
$H_2$ is the head corresponding to the $T_2$ interval

Multiply denominators:

$$h_w T_1 = T_1 H_1$$ $$h_w T_1 + h_w T_2 + \Delta h_w T_1 + \Delta h_w T_2 = T_1 H_1 + T_2 H_2$$

Subtract first from second:

$$h_w T_2 + \Delta h_w T_1 + \Delta h_w T_2 = T_2 H_2$$

Put $T_2$ on one side: $$\Delta h_w T_1 = T_2 H_2 - h_w T_2 - \Delta h_w T_2$$

Rearrange: $$\frac{\Delta h_w T_1}{H_2 - h_w - \Delta h_w} = T_2$$

Can also solve for the transmissivity ratio: $$\frac{\Delta h_w}{H_2 - h_w - \Delta h_w} = \frac{T_2}{T_1}$$

Head solution

Start from the previous proof: $$h_w T_2 + \Delta h_w T_1 + \Delta h_w T_2 = T_2 H_2$$

Rearrange: $$H_2 = \frac{h_w T_2 + \Delta h_w T_1 + \Delta h_w T_2}{T_2}$$

Examples

Can we recover missing values? In this example we test if we can recover the missing value for the $10^{th}$ interval.

library(sokol)

set.seed(123)
transmissivity <- abs(rnorm(10))
head <- sort(rnorm(10))
plot_blended(transmissivity, head)
blended_1 <- estimate_blended_head(transmissivity[1:9], head[1:9])
blended_2 <- estimate_blended_head(transmissivity, head)


# estimate interval head
estimate_missing(blended_2, 
                 c(transmissivity),
                 c(head[1:9], NA_real_))
head[10]


# estimate interval transmissivity
estimate_missing(blended_2, 
                 c(transmissivity[1:9], NA_real_),
                 c(head))
transmissivity[10]


# estimate blended head
estimate_missing(NA_real_, 
                 c(transmissivity),
                 c(head))
blended_2