Page 94 - Physical Principles of Sedimentary Basin Analysis

76 Linear elasticity and continuum mechanics

where u = (u x , u y ) is the Darcy flux. The flux is proportional to the gradient of a potential
p according to Darcy’s law
k
u =− ∇ p. (3.201)
μ
A constant permeability k and viscosity μ give the 2D Laplacian equation for the potential
2
2
∂ p ∂ p
2
∇ p = + = 0. (3.202)
∂x 2 ∂y 2
A general solution to the continuity equation (3.200)is
∂ ∂
u x = and u y =− (3.203)
∂y ∂x
which is a solution for any function = (x, y). The function is related to the potential
by Darcy’s law (3.201), which gives the Cauchy–Riemann equations
∂ k ∂p ∂ k ∂p
=− and = . (3.204)
∂y μ ∂x ∂x μ ∂y
The factor k/μ does not belong to the Cauchy–Riemann equations and is normally left out.
If the Laplace equation is solved for the potential, and the Darcy velocities are known, it is
possible to obtain the function by integration, since it follows from (3.203) that

= u x dy or =− u y dx. (3.205)
The stream function is constant along each streamline, which is a property of the stream
function that follows from
∂ ∂
d = dx + dy =−u x dx + u y dy = 0 (3.206)
∂x ∂y
because for streamlines we have
dx dy
= . (3.207)
u x u y
The streamlines are also orthogonal to the iso-potential curves (in the case when the flow
field is given by the gradient of a potential), because the Cauchy–Riemann equations give
that
∇ p ·∇ = 0. (3.208)
The gradient ∇ p is normal to the iso-potential curves and the gradient ∇ is normal to the
streamlines. The streamlines and the iso-potential curves are therefore orthogonal.
Another direct consequence of the Cauchy–Riemann equations is that the stream
function is also a solution of the Laplace equation
2
2
∂ ∂
+ = 0. (3.209)
∂x 2 ∂y 2