The document describes equations for modeling groundwater flow, including Darcy's law, the continuity equation, and Laplace's equation. It presents the derivation of the 3D groundwater flow equation from Darcy's law and the continuity equation. It also describes the transient saturated flow equation and solutions to the groundwater flow equations, noting that numerical methods are generally needed to address complex boundary conditions.

1. Equations of Groundwater Flow
Description of ground water flow is based on:
Darcy’s Law
Continuity Equation – describes conservation of fluid
mass during flow through a porous medium; results in a
partial differential equation of flow.
Laplace’s Eqn - most important in math

2. Derivation of 3-D GW Flow
Equation from Darcy’s Law
−
∂
∂x
ρVx( )−
∂
∂y
ρVy( )−
∂
∂z
ρVz( )=0
Mass In - Mass Out = Change in Storage
ρVx+
∂
∂x
ρVx( )ρVx
z
y

3. Derivation of 3-D GW Flow
Equation from Darcy’s Law
∂
∂x
ρKx
∂h
∂x





+
∂
∂y
ρKy
∂h
∂y





+
∂
∂z
ρKz
∂h
∂z





=0
Replace Vx, Vy, and Vz with Darcy using Kx, Ky, and Kz
Divide out constant ρ, and assume Kx= Ky= Kz = K
∂2
h
∂x2
+
∂2
h
∂y2
+
∂2
h
∂z2
=0
∇2
h=0 calledLaplaceEqn.

4. Transient Saturated Flow
∂
∂x
Kx
∂h
∂x





+
∂
∂y
Ky
∂h
∂y





+
∂
∂z
Kz
∂h
∂z





=Ss
∂h
∂t
−
∂
∂x
ρVx( )−
∂
∂y
ρVy( )−
∂
∂z
ρVz( )=
∂
∂t
ρn( )
A change in h will produce change in ρ and n, replaced
with specific storage Ss = ρg(α + nΒ). Note, α is
the compressibility of aquifer and B is comp of water,
therefore,

5. Solutions to GW Flow Eqns.
∂2
h
∂x2
+
∂2
h
∂y2
+
∂2
h
∂z2
=0
∇2
h=0 calledLaplaceEqn.
Solutions for only a few simple problems can be
obtained directly
Generally need to apply numerical methods to address
complex boundary conditions.
h0 h1

6. Transient Saturated Flow
∂
∂x
∂h
∂x





+
∂
∂y
∂h
∂y





+
∂
∂z
∂h
∂z





=
Ss
K
∂h
∂t
Simplifying by assuming K = constant in all dimensions
And assuming that S = Ssb, and that T = Kb yields
∂2
h
∂x2
+
∂2
h
∂y2
+
∂2
h
∂z2
=
Ss
K
∂h
∂t
∇2
h=
S
T
∂h
∂t
fromJacob, Theis

7. Steady State Flow to Well
Simplifying by assuming K = constant in all dimensions
and assuming that Transmissivity T = Kb and
Q = flow rate to well at point (x,y) yields
∂2
h
∂x2
+
∂2
h
∂y2
=−
Qx,y( )
T