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induce_infiltration_time <- function(d, S, Tr, Qa, Qw) {
#' Calculate the critical time at which stream transitions from gaining to losing.
#'
#' @param d distance from well to stream [L]
#' @param S aquifer storage coefficient (specific yield if unconfined; storativity if confined)
#' @param Tr aquifer transmissivity [L2/T]
#' @param Qa ambient groundwater inflow rate per unit length of stream [L2/T]
#' @param Qw well pumping rate [L3/T]
#' @details This calculates the critical time at which induced infiltration due to groundwater pumping begins,
#' based on the \link{glover} model of streamflow depletion. Derived in Chen (2003) Eq. 4.
#'
#' Assumptions:
#' \itemize{
#' \item Groundwater flow is perpendicular to stream
#' \item Horizontal flow >> vertical flow (Dupuit assumptions hold)
#' \item Homogeneous, isotropic aquifer
#' \item Constant \code{Tr}: Aquifer is confined, or if unconfined change in head is small relative to aquifer thickness
#' \item Stream is straight, infinitely long, and remains in hydraulic connection to aquifer
#' \item Constant stream stage
#' \item No changes in recharge due to pumping
#' \item No streambank storage
#' \item Constant pumping rate
#' \item Aquifer extends to infinity
#' \item Stream fully penetrates through aquifer
#' \item No streambed resistance to flow (see \link{hunt} or \link{hantush} for streambed resistance)
#' }
#' @return A numeric of \code{tc}, the critical time at which induced infiltration begins [T].
#' @references
#' Chen, X (2003). Analysis of Pumping-Induced Stream-Aquifer Interactions for Gaining Streams.
#' Journal of Hydrology 275(1): 1-11. doi:10.1016/S0022-1694(02)00320-7
#' @examples
#' # recreate Figure 2 in Chen (2003)
#' Qa <- c(0.0001, 0.0003, 0.0005, 0.0008, 0.001)
#' tc <- induce_infiltration_time(d = 575, S = 0.2, Tr = 100*15, Qa = Qa, Qw = 2727)
#' plot(x = (pi * Qa * 100 * 15 * 575 / 2727), y = tc, log = "y")
#' @export
tc <- -d * d * S / (4 * Tr * log(pi * Qa * Tr * d / Qw))
return(tc)
}
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