Analytical and Numerical Solutions are affected by:

Island Recharge Problem • 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)

Bottom 4 rows Well

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

R x y R x y b y y x x 2D flow 1D 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). Recharge goes in over an area (x y) and comes out through an area (by).

Manipulating analytical solutions h(x) = R (L2 – x2) / 2T “confined” h = ho at x = 0 ho = R L2 / 2T R = 2 Kb ho / L2 h(x) = [R (L2 -x2 )/K] + (hL)2 unconfined & at x = 0; h = b + ho hL = b b  (b + ho)2 = [R L2 /K] + b2 0 L R = (2 Kb ho / L2)+ (ho2 K / L2)  To maintain the same head (ho) 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.

Bottom 4 rows Pumping treated as a diffuse sink. Well -R = Qwell / {(x  y)/4} = Qwell / (a2/4) = 4 Qwell / a2

Q R x y y x Point source (L3/T) Distributed source (L3/T) Finite difference models simulate all sources/sinks as distributed sources/sinks; finite element models simulate all sources/sinks as point sources/sinks.

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

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

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

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

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

h h b S = V / A  h S = Ss b Confinedaquifer Unconfinedaquifer Storativity Specific yield Figures taken from Hornberger et al. (1998)

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

Figures in slide 13 are taken from: Hornberger et al., 1998. Elements of Physical Hydrology, The Johns Hopkins Press, Baltimore, 302 p.

Analytical and Numerical Solutions are affected by: