1. a set of equations that describes the physical and/or chemical processes occurring in a system. Mathematical Model

2. Governing Equation Boundary Conditions Initial conditions (for transient conditions) problems) Components of a Mathematical Model

3. Mathematical Model of the Toth Problem Laplace Equation 2D, steady state h = c x + z o

4. Types of Solutions of Mathematical Models Analytical Solutions: h= f(x,y,z,t) (example: Theis eqn., Toth 1962) Numerical Solutions Finite difference methods Finite element methods Analytic Element Methods (AEM)

5. Toth Problem z x Analytical Solution Numerical Solution continuous solutiondiscrete solution h = c x + z o Math Model

6. Toth Problem z x Analytical Solution Numerical Solution h(x,z) = z o + cs/2 – 4cs/  2  … z x continuous solutiondiscrete solution (eqn. 2.1 in W&A) h = c x + z o Math Model

7. Toth Problem z x Analytical Solution Numerical Solution h(x,z) = z o + cs/2 – 4cs/  2  … h i,j = (h i+1,j + h i-1,j + h i,j+1 + h i,j-1 )/4 z x continuous solutiondiscrete solution (eqn. 2.1 in W&A) h = c x + z o Math Model

9. div q = 0 q = - K grad h Steady state mass balance Darcy’s law Scalar 1 component MagnitudeHead (h) Vector 3 components Magnitude and direction q & grad Tensor 9 components Magnitude, direction and magnitude changing with direction Hydraulic conductivity (K)

10. div q = 0 q = - K grad h Steady state mass balance Darcy’s law grad h qequipotential line grad hq IsotropicAnisotropic

11. div q = 0 q = - K grad h Steady state mass balance Darcy’s law Assume that K = a constant Laplace Equation

12. Governing Eqn. for TopoDrive Steady-state, heterogeneous, anisotropic

13. q = - K grad h K is a tensor with 9 components

14. K = K xx K xy K xz K yx K yy K yz K zx K zy K zz Principal components of K