Mathematical modeling techniques in the engineering of landfill sites.

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

2
Typical Landfill Site
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).

4
STEADY-STATE UNSATURATED MOISTUREDISTRIBUTIONS
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 N2 system of equations with 2 boundary
  conditions.

5
STEADY-STATE UNSATURATED MOISTUREDISTRIBUTIONS
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.

6
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
7
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!).

8
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

9
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.

10
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 (Richards Law, Daltons 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.

11
Thank you!