# HydroGeoSphere/Travel Time Probability (Transport)

## Output travel time statistics

HydroGeoSphere will perform descriptive statistics, following Eqs. (2.154) and (2.155): mean travel time, mode and standard deviation will be calculated at each node/element.

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## Integrate production zone

1. fname Name of the file which contains the list of elements that contain a mass source function and the tabulated functions. It is formatted as follows:
(a) nprodel Number of production elements. Read the following nprodel times:
i. nel, ifunc The element number and the ID (number) of the associated tabulated function.
(b) maxdatprod, delta_conv Size of the largest set of tabulated data which follows and the timestep size delta_conv for evaluating the convolution integral in
Equation 5.16. Read the following for each of the ifunc time-series:
i. ndata Number of data to read for the current time-series.
ii. time, val Time and corresponding source value.

If ifunc = 0, then ${\displaystyle m^{*}}$  corresponds to a unit and instantaneous mass input function, and thus no time-series are required in the input file.

The element nel with the maximum value of ifunc determines how many sets of time-series data must be supplied.

This option is meant to simulate a forward transport solution by means of a backward solution. It requires the problem to be backward-in-time. Integration of the backward travel time PDFs will be performed over a series of element numbers, which are input in the .np file. In a forward transport run, these elements would contain a mass source ${\displaystyle m^{*}}$ . The following equation is solved at each time-step in order to simulate the output mass flux ${\displaystyle J_{o}(t)}$  resulting from a forward transport (see [Cornaton, 2003]):

${\displaystyle J_{O}(t)=\int _{\Delta }\int _{0}^{t}g_{t}(t-u,\mathbf {x} )m^{*}(\mathbf {x} ,u)\,du\,d{\Omega }}$                          (Equation 5.16)

if ${\displaystyle m^{*}}$  is an arbitrary mass source function [M L−3 T−1], or

${\displaystyle J_{O}(t)=\int _{\Delta }g_{t}(t,\mathbf {x} ){\delta }(\mathbf {x} -\mathbf {x_{i}} )\,d{\Omega }}$                          (Equation 5.17)

if ${\displaystyle m^{*}}$  is a unit and instantaneous mass input function [L−3 T−1]. ${\displaystyle {\Delta }}$  denotes the domain of elements where ${\displaystyle m^{*}(\mathbf {x} ,t)\neq 0}$ .

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