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

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 2D, 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 h = c x + zo z Math Model x Analytical Solution Numerical Solution continuous solution discrete solution

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

Toth Problem h = c x + zo z Math Model x Analytical Solution Numerical Solution h(x,z) = zo + cs/2 – 4cs/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 = - Kgrad h Darcy’s law

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

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

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

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

Kxx Kxy Kxz KyxKyy Kyz Kzx KzyKzz K = Principal components of K

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