Analysis of Parameter Sensitivity to Enhance the Detection of Leaks in Sealed Landfills

– It was shown in a previous article that the presence of leaks in sealed landfills due to confinement failures can be assessed measuring surface moisture and relating it to leaks at the bottom of the landifll through a regularised inversion algorithm based on Richard’s equation with a piecewise linear boundary condition. Under the assumption of absence of leaks as the null hypothesis, the algorithm provides the value of the relevant F-statistic as a function of the accuracy of soil moisture measurements and of physical and meteorological parameters. In this presentation we take into account the uncertainties of the parameters and estimate the corresponding ranges of the F-values by evaluating the derivatives of the F-statistic with respect to the parameters.


Introduction
Waste management relies worldwide on landfill disposal for over two thirds of the waste generated [1] either through some form of landifill management or open dumping. Low-income countries presently using open dumping are expected to extend the use of controlled landfills, which may prove to be locally more appropriate solutions with respect to other advanced solutions such as recycling, composting, and incineration. Hence, the increasing role of sealed landfills and the necessity of a better modeling strategy for their monitoring and control.
The presence of leaks in landfill liners is generally monitored using wells below the water table and sensors in the vadose zone [2]. In addition to instrument malfunctioning of both sensors and wells (which can give rise to data misinterpretation), sensors in the vadose zone may fail to detect a narrow plume close to the liner [3], whereas sensors located in deeper regions could fail to detect the plume due to an impeding layer that diverts percolating water horizontally [4]. It was shown in a previous article [5] that the presence of leaks in sealed landfills due to confinement failures can be assessed measuring surface moisture and relating it to leaks at the bottom of the landifll through a regularised inversion algorithm based on Richard's equation with a piecewise linear boundary condition. Under the assumption of absence of leaks as the null hypothesis, the algorithm provides the value of the relevant F-statistic as a function of the accuracy of soil moisture measurements and of physical and meteorological parameters.
In this paper we discuss how the uncertainties of the parameters can affect the value of the F-statistic and consequently the reliability of the model in the assessment of leaks. The paper is organized as follows: in the next section a previously developed inverse model will be presented. The sensitivity analysis which is the main goal of this paper will be consider in the following section, whereas the significance of the algorithm will be examined in the Conclusions.

Percolation model
The one-dimensional pressure-head form of Richards' equation is given by [6] ( ) ( ) ( ) where ψ is the pressure head, ( ) K ψ is the hydraulic conductivity ( ) is the differential water capacity and θ is the volumetric water content, S is water uptake (which is negative when considering landfills because it gives the water generated by wastes) and z is the vertical coordinate pointing upward. The initial condition ( ) 0 z ψ and boundary conditions at z=0 (the bottom of the landfill) and at z=L (at the surface of the landfill) are to be provided.
The flux at the surface ( )  is equal to the amount of water released by evapotranspiration per unit surface minus the rate of rainfall infiltration amount of rain, i.e.
If there are no leaks, the condition at the bottom is given by ( ) where the function ( ) are the soil pore-size distribution, the hydraulic conductivity, the water content at saturation, the residual water content, and the soil moisture diffusivity, respectively. The number of intervals that can be considered to increase the accuracy of the approximation is limited by the necessity of preventing the problem from becoming ill-posed.
Suitable regularization techniques are discussed in [5]. This solution can be rewritten as where ( ) ( ) ( ) Rewriting the flux at the bottom and N-i-1 degrees of freedom. The absence of leaks at all the measured times N is given by the condition to the fluxes being zero or negative and consequently by the statistic Thus, the calculated value of the statistic can provide the required probability for the absence of leaks at the bottom of the landfill.

Sensitivity analysis
While the analysis of experimental errors carried out in [5] shows that the significance of the test is roughly consistent with the accuracy of present day instrumentation, the influence of the parameters π contained in the model (i.e. The following steps have been employed to evaluate (8).
(a) Compute the derivatives of the matrices by differentiating each element with respect to the parameters. Indeed, the differentiation of a matrix with respect to a scalar is given by a matrix whose elements are the derivatives of the elements of the original matrix , u