Analytical and Numerical Solutions are affected by: Differences in conceptual model (confined vs unconfined) Dimensionality (1D vs 2D) Numerical Solutions also affected by: Grid Spacing (4000 ft vs 1000 ft) Island Recharge Problem

Well Bottom 4 rows

R  x  y yy xx b Recharge goes in over an area (  x  y) and comes out through an area (b  y) or rate out is R  x / b. 1D flow

R  x  y yy xx b Recharge goes in over an area (  x  y) and comes out through an area (b  y). 1D flow 2D flow Recharge goes in over an area (  y  x) and comes out through areas (b  y) and (b  x) or through a total area of b(  x+  y). yy xx R  x  y

unconfined h(x) = R (L 2 – x 2 ) / 2T “confined” h (x) =  [R (L 2 - x 2 )/K] + (h L ) 2 h = h o at x = 0h o = R L 2 / 2T at x = 0; h = b + h o R = 2 Kb h o / L 2 R = (2 Kb h o / L 2 ) + (h o 2 K / L 2 ) h L = b  (b + h o ) 2 = [R L 2 /K] + b 2 &  To maintain the same head (h o ) at the groundwater divide as in the confined system, the 1D unconfined system requires that recharge rate, R, be augmented by the term shown in blue. b L 0 Manipulating analytical solutions

Well Bottom 4 rows -R = Q well / {(  x  y)/4} = Q well / (a 2 /4) = 4 Q well / a 2 Pumping treated as a diffuse sink.

R  x  y Q Distributed source (L 3 /T) Point source (L 3 /T) yy xx Finite difference models simulate all sources/sinks as distributed sources/sinks; finite element models simulate all sources/sinks as point sources/sinks.

r = a rere Using the Thiem eqn., we find that r e = 0.208a Use eqn. 5.1 or 5.7 in A&W to correct the head at sink nodes. Sink node (i, j) (i+1, j) r e is the radial distance from the node where head is equal to the average head in the cell, h i,j h i,j is the average head in the cell.

(Gauss-Seidel Formula for Laplace Equation) SOR Formula Relaxation factor = 1 Gauss-Seidel < 1 under-relaxation >1 over-relaxation SOR where, for example,

SOR solution for confined Island Recharge Problem The Gauss-Seidel formula for the confined Poisson equation where

Inflow = Outflow +  S Recharge Discharge Transient Water Balance Eqn.

General governing equation for transient, heterogeneous, and anisotropic conditions Specific Storage S s =  V / (  x  y  z  h)

Figures taken from Hornberger et al. (1998) Unconfined aquifer Specific yield Confined aquifer Storativity S = V / A  h S = S s b b hh hh

Law of Mass Balance + Darcy’s Law = Governing Equation for Groundwater Flow div q = - S s (  h  t) (Law of Mass Balance) q = - K grad h (Darcy’s Law) div (K grad h) = S s (  h  t) (S s = S /  z)

Figures in slide 13 are taken from: Hornberger et al., Elements of Physical Hydrology, The Johns Hopkins Press, Baltimore, 302 p.