Finite Difference Method

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Presentation transcript:

Finite Difference Method

Derivation of 3-D GW Flow Equation from Darcy’s Law z y Mass In - Mass Out = Change in Storage

Steady State Flow Replace qx, qy, and qz with Darcy using Kx, Ky, and Kz Divide out constant , and assume Kx= Ky= Kz = K

Transient Flow h is related to q through the soil water characteristic curve

Linear Second-order PDEs Linear second-order PDEs are of the form where A - H are functions of x and y only Elliptic PDEs: B2 - AC < 0 (steady state equations) Parabolic PDEs: B2 - AC = 0 (transfer equations) Hyperbolic PDEs: B2 - AC > 0 (wave equations)

Difference vs Differential

Formulas for 1st, 2nd Derivatives

y x Discretization of the solution domain Vertical (j index) . . . 5 4 Vertical (j index) 3 2 1 1 2 3 4 5 . . . N-2 N-1 N x Horizontal (i index)

(i,j+1) (i,j) (i-1,j) (i+1,j) (i,j-1)

Type 1 (Dirichlet) Boundary Condition M M-1 M-2 . . . 5 4 3 2 1 1 2 3 4 5 . . . N-2 N-1 N Variable Specified on Boundary

Type 2 (Neumann) Boundary Condition . . . 5 4 3 2 1 1 2 3 4 5 . . . N-2 N-1 N Derivative of Variable Specified on Boundary (usually zero)

Variable Conductivity Steady State Flow Variable Conductivity K2 K1 Dx Dx Dx

K2 K1

Steady State Flow

Steady State Flow Steady State Flow Single Conductivity