1. Mathematical modeling techniques in the engineering of landfill sites.
By Rob Krausz
For BAE 558-Fluid Mechanics of Porous Media
University of Idaho
Department of Biological and Agricultural Engineering
12/4/2018
Rob Krausz - Selkirk College

2. Rob Krausz - Selkirk College
Typical Landfill Site
12/4/2018
Rob Krausz - Selkirk College

3. Overview of Presentation
PART 1: Steady-state unsaturated moisture distributions using dispersion-advection equation (Fityus, Smith, and Booker 1998).
PART 2: Finite analytic method for modelling 2-dimensional flow (Tsai, Lee, Chen, Liang, and Kuo 2000).
PART 3: Mass transfer modelling of evaporative fluxes from non-vegetated soil surfaces (Wilson, Fredlund, and Barbour 1996).
12/4/2018
Rob Krausz - Selkirk College

4. Rob Krausz - Selkirk College
STEADY-STATE UNSATURATED MOISTURE DISTRIBUTIONS USING DISPERSION-ADVECTION EQUATION (FITYUS, SMITH, AND BOOKER 1998).
Purpose: to model contaminant transport through the vadose zone beneath landfills.
Mathematical model used:
Governing transport equation is:
where: E, G, J, L, X, Y are empirical constants
Z = vertical coordinate
= Laplace-transformed concentration of contaminant
The rest of this model is very complex, and includes an N-layered soil profile that generates an N+2 system of equations with 2 boundary conditions.
12/4/2018
Rob Krausz - Selkirk College

5. Rob Krausz - Selkirk College
STEADY-STATE UNSATURATED MOISTURE DISTRIBUTIONS USING DISPERSION-ADVECTION EQUATION (FITYUS, SMITH, AND BOOKER 1998).
Conclusions:
Equilibrium moisture conditions are reached over much shorter timeframe than the transport of contaminant down through soil liner. Therefore, it is reasonable to assume constant moisture content in the vadose zone.
As moisture content drops, 1. diffusive mass flux drops, but 2. the increase in moisture in the downward direction produces an increased concentration gradient and so the rate of diffusive mass flux actually increases proportionally.
1. & 2. oppose each other, therefore the diffusive mass flux is not significantly sensitive to moisture content at the surface.
At low moisture contents, primary barrier to diffusive mass transport will be the soil in the vadose zone, while at higher moisture contents, a geomembrane at the bottom of the waste material will act as the primary barrier to diffusive mass transport.
12/4/2018
Rob Krausz - Selkirk College

6. Rob Krausz - Selkirk College
FINITE ANALYTIC METHOD FOR MODELLING 2-DIMENSIONAL FLOW (TSAI, LEE, CHEN, LIANG, AND KUO 2000).
Purpose: to solve the 2-dimensional subsurface flow and transport equations in the vadose zone beneath landfills.
Mathematical model used:
The governing equation is:
This model uses a 9-node and 5-node Finite analytical method, with a spatial weighting scheme used to evaluate the average hydraulic conductivity in the discretized element.
where:
C = solute concentration
R = retardation factor
Vx, Vz = porous velocity components of unsaturated flow
Dxx, Dzz, Dxz, Dzx = coefficients of mechanical dispersion
= first-order decay coefficient
= volumetric water content
S = solute source-sink term
12/4/2018
Rob Krausz - Selkirk College

7. Rob Krausz - Selkirk College
FINITE ANALYTIC METHOD FOR MODELLING 2-DIMENSIONAL FLOW (TSAI, LEE, CHEN, LIANG, AND KUO 2000).
Conclusions: Analysis reveals details of how migration of solute is significantly less than vertical migration.
Model provides accurate results for landfills with irregular ground surface (very important in this region!).
12/4/2018
Rob Krausz - Selkirk College

8. Rob Krausz - Selkirk College
MASS TRANSFER MODELLING OF EVAPORATIVE FLUXES FROM NON-VEGETATED SOIL SURFACES (WILSON, FREDLUND, AND BARBOUR 1996).
Purpose: to predict the evaporative fluxes from non-vegetated soil surfaces, and to establish a relationship between actual evaporation rate and total suction.
Mathematical model used:
Governing equation is: E = f(u) (es – ea)
where:
E = rate of evaporation
f(u) = transmission function based on mixing characteristics of air
es = saturation vapour at water surface temperature
ea = vapour pressure of air above water surface
Where:
AE = actual evaporation
PE = potential evaporation
= matric suction in liquid phase
G = gravity constant
Wv = molecular weight of water
R = universal gas constant
T = absolute temperature
Ha = relative humidity of air
Equation relating actual to potential evaporation is:
12/4/2018
Rob Krausz - Selkirk College

9. Rob Krausz - Selkirk College
MASS TRANSFER MODELLING OF EVAPORATIVE FLUXES FROM NON-VEGETATED SOIL SURFACES (WILSON, FREDLUND, AND BARBOUR 1996).
Conclusions:
Evaporative fluxes from unsaturated soil surfaces can be measured via easily-measured soil properties.
Total suction (matric + osmotic) appears to be a suitable state variable for predicting evaporative fluxes.
Evaporative fluxes in unsaturated landfill soil covers are significantly less than those from saturated soil covers.
12/4/2018
Rob Krausz - Selkirk College

10. Rob Krausz - Selkirk College
Closing Remarks
There is a lot of research out there that applies vadose zone fluid mechanics to solid waste management.
Mathematical modelling used is varied and highly sophisticated, although governing formulas are common (Richard’s Law, Dalton’s Equation, etc).
Many assumptions are made to facilitate solving of equations, therefore, certain cases where these assumptions are questionable will require further investigation and remodelling.
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Rob Krausz - Selkirk College

11. Rob Krausz - Selkirk College
Thank you!
12/4/2018
Rob Krausz - Selkirk College