2.2 Aim and Scope of the Project

This project beneÞts from the application of a one-dimensional (1D) advectiondispersion model1 of the diffusion of a contaminant dissolved in saturated soil, developed by Acar and Haider (1990), with a view to calculating the optimum liner thickness of a landÞll during its desired operational life. All the necessary calculations are made using the CONTRANS program, written in the MATLAB environment. The source code of the program can be accessed in (Bieda 2002). The program is based on the block diagram proposed by Cokca (1999) and adapted here for the purposes of this monograph.

The abovementioned one-dimensional (1D) model, developed by Acar and Haider (1990), of the diffusion of a contaminant dissolved in saturated soil takes the following form (Cokca 1999):

where:

Rd Ð retardation coefÞcient,2

Cz Ð solute concentration Ð concentration of a dissolved contaminant in the direction of the z-axis,

Dp Ð dispersion coefÞcient,3

vs Ð percolation velocity with initial condition of:

Cz ¼ 0 for z 0 and t ¼ 0; and boundary conditions of:

Cz ¼ C0 for z 0 and t > 0 and Cz ¼ 0 for z ¼ 0 and t > 0.

The program calculates the diffusion (transfer) time of a dissolved contaminant in relation to the liner thickness of a landÞll [38Ð39]. The liner consists of clay with low conductivity coefÞcient of the aquiferous layer of the tank ( 1.0 10 9 m/s) (Raport 2007; Majer et al. 2007). In a situation where the calculated diffusion time is shorter than the planned service life of a landÞll (in years), the program demands the increase of the liner thickness.

1Advection Ð the horizontal transfer of air mass properties by the velocity Þeld of the atmosphere, of the soil (different to convection, which describes the predominantly vertical movements). Dispersion Ð mixing.

2Retardation coefÞcient Ð the movement velocity of the separation surface of mass zone.

3Dispersion consists of two elements: hydronomic-mechanic dispersion and molecular diffusion.

2.2.1Constructing the Model: Defining Input Data

In order to demonstrate the simulation model using Crystal Ball software, a certain simpliÞcation has been made; namely, the adaptation of a single equation (2.5) in this simulation. This equation takes the form of:

where:

VS Ð seepage velocity (initial condition) (m/s);

TH Ð design life of landÞll (in years);

EDC Ð diffusion coefÞcient (m2/s).

This equation is part of the computer software demonstrated in the Appendix VIII, found in Bieda (2002).

The Z2 expression is part of the algorithm of the CONTRANS calculating software, presented in Fig. 2.2 (Bieda and Wajs 2002; Bieda 2004c).

VS can be described in the following form:

where:

HG Ð hydraulic gradient;

HC Ð hydraulic conductivity (m/s);

POROS Ð porosity coefÞcient.

Equation 2.7, after considering the expression (2.8), takes the following form:

where:

TH Ð design life of landÞll (in years);

EDC Ð diffusion coefÞcient (m2/s).

The numerical data and the types of probability distribution used in random estimation of the parameters of the Z2 expression that take part in MC simulation using Crystal Ball, along with the distribution parameters, are presented in Table 2.1. These are: hydraulic conductivity Ð HC, hydraulic gradient Ð HG, diffusion coefÞcient Ð EDC, the thickness of the isolation barrier Ð TH, and porosity Ð POROS. The stochastic model includes Þve random variables in its description, listed above, which can be characterised by two probability distributions: log-normal distribution and uniform distribution (Zdanowicz 2002), sometimes also known as symmetrical or rectangular. The in-depth analysis of log-normal probability distribution, assigned to hydraulic conductivity (HC), performed by Bear and Cheng, can be

Table 2.1 The values and types of probability distribution of the parameters in the Z2 expression, used in MC simulation with the help of Crystal Ball® program along with the distribution parameters

ED Ð experimental data (ED ¼ m (mean value)), mg Ð geometric mean, sg Ð geometric standard deviation

aMinimum value bMaximum value

found in their recent publication (2010). In their work the attention is drawn to numerous studies and observations in situ, which lead to log-normal distribution of the hydraulic conductivity value (Gelhar 1986, 1993; Freeze 1975; Hoeksema and Kitanidis 1985). The authors also point out to the fact that different units describing hydraulic conductivity can be found in the subject literature. Some hydrogeologists prefer to use meters per day (m/d), while certain scientists and geotechnical engineers use centimetres per second (cm/s). The International System of Units (SI) uses metres per second (m/s) as a standard unit (Bear and Cheng 2010).

The approximation of the three Z2 parameters (i.e. HC, HG, and EDC) with log-normal probability distribution and the establishment of the magnitude characterising the functions of this distribution has been performed (geometric mean mg and geometric standard deviation sg).

The two remaining parameters: the thickness of the isolation barrier Ð TH and porosity Ð POROS, have been subjected to uniform distribution. The mean values m characterising this distribution have been accepted on the basis of the analysis of the available literature data (DE), included in these publications (Raport 2007; Majer et al. 2007).

Crystal Ball® software (Bieda 2000) allows for deÞning the input parameters of a model as random data containing assumed features of probability distribution. In its professional version, the program offers 12 different types of probability distributions, including: normal, log-normal, uniform, exponential, Poisson, and Weilbull distribution (Evans and Olson 1998).

Decisioneering Inc., an American company from Denver, has developed the CB software, which is based on a model built using the functions of a spreadsheet application (Evans and Olson 1998; Crystal Ball 2010), and it utilises the development of simulation (stochastic) models.