The BIOSCREEN-AT, provided in {ref}Karanovic et al. (2007)<refbs> Model provides an exact analytical Solution to the BIOSCREEN Model. The model provides a three-dimensional solution for transport of dissolved contaminants, incorporating natural attenuation processes. As shown in {numref}bscreen_f1 , the source is given as a patch specified-concentration boundary condition.
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The BIOSCREEN-AT model concept
Karanovic et al. (2007) Provide the following expression for an exact analytical solution:
$$ c(x,y,z, t) = C_0 \frac{x}{8\sqrt{\pi D_x'}}\exp(-\gamma t)\times\ \times \int\limits_0^t\frac{1}{\xi^{3/2}}\exp\Bigg{(\gamma-\lambda_{EFF})\xi- \frac{(x-v'\xi)^2}{4D_x'\xi}\Bigg}\times\ \times\Bigg[\text{erfc}\Bigg{\frac{y-\frac{w}{2}}{2\sqrt{D_y'\xi}} \Bigg}-\text{erfc}\Bigg{\frac{y+\frac{w}{2}}{2\sqrt{D_y'\xi}} \Bigg}\Bigg]\times \ \times \Bigg[\text{erfc}\Bigg{\frac{z-H}{2\sqrt{D_z'\xi}}\Bigg}- \text{erfc}\Bigg{\frac{z+H}{2\sqrt{D_z'\xi}} \Bigg}\Bigg]\text{d}\xi $$(eqbio2007)
ANTON Pls. ADD UNITS
$c_0$ = initial source concentration (mg/L)<br/>
$D'x$ = dispersion coefficient divided by retardation ($R$) factor $D_x/R$<br/>
$D'y$ = dispersion coefficient divided by retardation factor $D_y/R$<br/>
$D'z$ = dispersion coefficient divided by retardation factor $D_z/R$<br/>
$\gamma$ = source decay coefficient <br/>
$\lambda_{eff}$ = effective first order decay coefficient<br/>
$W$ = source width (m)<br/>
$H$ = source depth (m)
The model is based on the following assumptions:
- Aquifer extends semi-infinite in
$x$ -direction, infinite in$y$ -direction and extends from water table to relatively large depth. - Groundwater flow is steady and one dimensional.
- The solute undergoes equilibrium sorption and first-order transformation reactions.
- The aquifer is homogeneous and
$L_{max}$ is always found on the center line.
Threshold Concentration- The evaluation of the analytical solution shows, that the contaminant concentration
Time - The BIOSCREEN analytical solution is a transient solution, which means that the state of the contaminant plume is dependent of time. The interaction of plume growing effects and attenuation processes will often result in a state of equilibrium in which the plume keeps a constant size. It is not possible to define a global valid period after which a steady state is reached, since this dependes heavily on the hydrogeologic conditions. However, a simple approximation can be used based on the decay within the plume
Longitudinal Dispersivity - is an aquifer parameter which effects the intensity of mixing in the liquid phase along the
Horizontal Transverse Dispersivity - effects the mixing intensity of the liquid phase along the
Vertical Transverse Dispersivity - effects the mixing intensity of the liquid phase along the
Effective Diffusion Coefficient - this parameter defines particle movement and therefore the effect of plume growth based on Brownian molecular motion. For most scenarios it has a negligible role. But for cases with very low groundwater velocities and long time periods it gains on impact on the result.
Retardation Factor - the ratardation factor quantifies natural attenuation processes such as adsorption and desorption of particles over time in the soil matrix. Very high ratardation factors result in smaller plumes due to stronger attenuation of the contamination.
Source Decay Coefficient - describes the degredation of the source and therefore the decrease of the contaminant concentration within the source. This can be the result of biological degredation of the contaminant or other forms of decay.
Decay Coefficient -
The concentration contour for BIOSCREEN-AT (eq. {eq}eqbio2007) includes an integral that has to be numerically solved for obtaining
CAST uses Gauss-Legendre Quadrature to solve for eqbio2007. The Gauss-Legendre Quadrature method requires specifying the number of sample points (called Gauss points), based on which corresponding specific weights are obtained. The weights are then used to obtain the solution. Generally, higher the Gauss points, the higher the accuracy of the solution. However, higher sample points also means a large number of computations. In CAST Python codes are developed very identical to the FORTRAN code provided in {ref}Karanovic et al. (2007)<refbs> to solve {eq}eqbio2007) (using Gauss-Legendre Quadrature method). Python library Scipy provides functions for obtaining the roots and weights required by the Gauss-Legendre Quadrature method.
:class: dropdown
```python
roots = sp.special.roots_legendre(m)[0]
weights = sp.special.roots_legendre(m)[1]
Following the approach provided in {ref}Karanovic et al. (2007)<refbs> the limits of integration of eq. {eq}eqbio2007 is scaled. This is required for speeding up the computation. The codes used in CAST for this purpose can be seen by clicking the dropdown button
:class: dropdown
```python
#scaling of the integration limits for faster computation
bot = 0
top = np.sqrt(np.sqrt(t))
Tau = (roots*(top-bot)+top+bot)/2
Tau4= Tau**4
Eq. {eq}eqbio2007 provides the solution for Karanovic et al. (2007)<refbs> ) as follows:
# Fixing domain boundary to obtain Lmax
if x<=1e-6:
if y <= W/2 and y >= -W/2 and z <= z_2 and z >= z_1:
C=Co*np.exp(-ga*t)
else:
C=0
else:
a = Co*np.exp(-ga*t)*x/(8*np.sqrt(np.pi*Dxr))
After completion the above steps, i.e., transformation of limits and scaling of the domain, inetegral in eq. {eq}eqbio2007 is evaluated using the following Python code.
# evaluating the integrand and summing
xTerm = (np.exp(-(((la-ga)*Tau4)+((x-vr*Tau4)**2)/(4*Dxr*Tau4))))/(Tau**3)
yTerm = sp.special.erfc((y-W/2)/(2*np.sqrt(Dyr*Tau4))) - sp.special.erfc((y+W/2)/(2*np.sqrt(Dyr*Tau4)))
zTerm = sp.special.erfc((z-z_2)/(2*np.sqrt(Dzr*Tau4))) - sp.special.erfc((z-z_1)/(2*np.sqrt(Dzr*Tau4)))
Term = xTerm * yTerm * zTerm
Integrand = Term*(weights*(top-bot)/2)
C = a*4*sum(Integrand)
The required
After evaluating the numerical Integration, the value for Lmax is found by For that reason the roots and their specific weights of the Legendre polynomial up to a value of "m" are genereated as shown below, where the value of m is typically 60.
:class: attention
Number of Gauss points required in computation can be any number. However using 60 is recommended as quite a high level of accuracy can be quickly obtained. This is particularly important for **online** version of CAST. For the **offline** version of CAST >104 is recommended.
Using Gauss points $<15$ is not recommended.
The CAST interface of BIOSCREEN-AT is identical to that for other analytical/empirical models. The details of the interface and available user options can be found at Liedl et al. (2005). Below a very brief outline of the CAST interface specific to BIOSCREEN-AT is provided.
First all input data needs to be defined (screenshot {numref}bscreen_f2). Then a graph is genereated, that shows the result of the computation in comparison to the field data from the CAST database (screenshot {numref}bscreen_f3).
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The CAST interface for BIOSCREEN-AT data input
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The CAST interface for BIOSCREEN-AT output
Once the first computation of Generate graph button is made, the CAST interface provides interface with sliders bars ({numref}bscreen_f4) for the most sensible model parameters.
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Interactive sliders in CAST for BIOSCREEN-AT model
The sliders can be used to change the input values while simultaneously getting a feedback of how these changes effect the model result. This has the advantage that it is not required to generate a new graph which makes the effect of the parameter change more visible and clear.
Numerical computations are used to compute $L_{max}$. As such care has to be taken while changing slider values. Long computation time may be required before the solution is obtained.
The multiple computing mode can be used to visualize different own case results in one graph, while still being able to compare to the CAST database. The interface for BIOSCREEN-AT is identical to other models. User can check Liedl et al. (2005) on details of using the multiple computing interface.
Data in the interface can be inserted via the add data button or can be uploaded from a file. A sample file can be downloaded for purposes of correct formatting. It is possible to slect the desired sites to be schown in the graph.
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Multiple computing interface in CAST for BIOSCREEN-AT model
(refbs)=
Karanovic, M., Neville, C., and Andrews, C., (2007), BIOSCREEN-AT: BIOSCREEN with an Exact Analytical Solution. Vol. 45(2), pp. 242-245, Groundwater.