Conceptual Model A descriptive representation of a groundwater system that incorporates an interpretation of the geological & hydrological conditions. Generally includes information about the water budget.

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 Specified head (1st type or Neumann) constant head Specified flux (2nd type or Dirichlet) no flux Initial Conditions (for transient conditions)

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 z x Analytical Solution Numerical Solution h = c x + zo Mathematical model x Analytical Solution Numerical Solution continuous solution discrete solution

Toth Problem z x z x Analytical Solution Numerical Solution h = c x + zo Mathematical 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 z x z x Analytical Solution Numerical Solution h = c x + zo Mathematical 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

Hinge line Add a water balance & compute water balance error Example of spreadsheet formula

OUT IN Q= KIA Hinge line OUT – IN = 0

Hinge line Add a water balance & compute water balance error

Q = KIA=K(h/z)(x)(1) A x=z  Q = K h z x z x 1 m

Mesh centered grid: area needed in water balance (x/2) x No Flow Boundary x (x/2) water table nodes

x=z  Q = K h

Block centered grid: area needed in water balance No flow boundary x x water table nodes

K as a Tensor

div q = 0 q = - K grad h Steady state mass balance eqn. Darcy’s law z equipotential line grad h q grad h x Isotropic Anisotropic Kx = Kz Kx  Kz

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

(homogeneous and isotropic conditions) div q = 0 steady state mass balance eqn. q = - K grad h Darcy’s law Assume K = a constant (homogeneous and isotropic conditions) Laplace Equation

Governing Eqn. for TopoDrive 2D, steady-state, heterogeneous, anisotropic

global local z z’ x’ x Kxx Kxy Kxz Kyx Kyy Kyz Kzx Kzy Kzz K’x 0 0 bedding planes x’  x Kxx Kxy Kxz Kyx Kyy Kyz Kzx Kzy Kzz K’x 0 0 0 K’y 0 0 0 K’z

q = - K grad h Kxx 0 0 0 Kyy 0 0 0 Kzz qx qy qz = -

q = - K grad h Kxx Kxy Kxz Kyx Kyy Kyz Kzx Kzy Kzz K = K is a tensor with 9 components Kxx ,Kyy, Kzz are the principal components of K

q = - K grad h Kxx Kxy Kxz Kyx Kyy Kyz Kzx Kzy Kzz qx qy qz = -

This is the form of the governing equation used in MODFLOW.

global local z z’ x’ x Kxx Kxy Kxz Kyx Kyy Kyz Kzx Kzy Kzz K’x 0 0 bedding planes x’  x Kxx Kxy Kxz Kyx Kyy Kyz Kzx Kzy Kzz K’x 0 0 0 K’y 0 0 0 K’z

z local global z’ grad h q q’ x’ Kz’=0 x  Assume that there is no flow across impermeable bedding planes z local global z’ grad h q q’ x’ Kz’=0  x

[K] = [R]-1 [K’] [R] global local z z’ x’ q q’ x Kxx Kxy Kxz bedding planes x’ q q’  x Kxx Kxy Kxz Kyx Kyy Kyz Kzx Kzy Kzz K’x 0 0 0 K’y 0 0 0 K’z [K] = [R]-1 [K’] [R]