Mathematical Model a set of equations that describes

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

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

Mathematical Model of the Toth Problem h = c x + zo Laplace Equation 2 D, steady state

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)

Toth Problem Math Model z h = c x + zo x Analytical Solution continuous solution Numerical Solution discrete solution

Toth Problem Math Model h = c x + zo z x Analytical Solution Numerical Solution h(x, z) = zo + cs/2 – 4 cs/ 2 … (eqn. 2. 1 in W&A) z x continuous solution discrete solution

Toth Problem Math Model h = c x + zo z x Analytical Solution Numerical Solution h(x, z) = zo + cs/2 – 4 cs/ 2 … (eqn. 2. 1 in W&A) z hi, j = (hi+1, j + hi-1, j + hi, j+1 + hi, j-1)/4 x continuous solution discrete solution

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

div q = 0 Steady state mass balance q = - K grad h q Darcy’s law equipotential line grad h Isotropic q grad h Anisotropic

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

Governing Eqn. for Topo. Drive Steady-state, heterogeneous, anisotropic

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

Kxx Kxy Kxz K= Kyx Kyy Kyz Kzx Kzy Kzz Principal components of K