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

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

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

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

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

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

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

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

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

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. 12/4/2018 Rob Krausz - Selkirk College

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