Jökull - 01.12.1977, Side 86

Unconfined Aquifer Flow with a Linearized
Free Surface Condition
GUNNAR BODVARSSON
SCHOOL OF OCEANOGRAPHY AND DEPARTMENT OF MATHEMATICS
OREGON STATE UNIVERSITY, CORVALLIS, OREGON 97331
ABSTRACT
Applying standard linearization to the free
surface boundary condition for a fluid flowing
under gravity through a homogeneous isotropic
Darcy type porous solid, a number of useful
solulions for the free surface elevation can be
obtained. Flows without and with simple sources
are discussed below.
INTRODUCTION
The theory of Darcy type fluid flow in un-
confined homogeneous aquifers is of consider-
able importance in the modeling of many
hydrological and geothermal systems. Although
the inherently non-linear free-surface condition
presents some mathematical problems, these can
often be overcome in many practical situations.
In particular, in the case of slow small surface
amplitude flow, we are able to linearize the
free-surface condition and depend entirely on
elementary potential theory to solve the flow
problems. Many flow models with relatively
deep sources can be treated adequately by such
methods. This is of particular importance in
the case of geothermal systems where the pro-
duction boreltoles are quite deep and therefore
cause only a relatively minor perturbation of
tlie free water surface. The present paper dis-
cusses a class of such flow problems wliere a
linearization of the surface condition can be
applied.
BASIC EQUATIONS
Consider a half-space of an incompressible
homogeneous ancl isotropic porous medium of
84 JÖKULL 27. ÁR
area porosity cþ permeated by a liomogeneous
incompressible gravitating fluid F of clensity p.
Let the equilibrium static free surface of the
fluid be represented by tlie horizontal plane 2-
In the state of motion the free fluid surface is
deformetl to the non-stationary surface fi. We
place a coordinate system with the origin on 2
and the z-axis vertically down. Let P = (x,y,z)
be the general field point, S = (x,y) be points
on 2 al|d t be the time. We assume that the
flow of the fluid through the porous medium
is governed by Darcy’s law,
q = -C(Vp-pg) (1)
where q (P,t) = (u,v,w) (P,t) is the mass flow
vector, C is the fluid conducticity, p (P,t) is
the total fluid pressure and g the acceleration
of gravity. It is customary to express C = k/v
where k is the pormeability of the medium
and v is the kinematic viscosity of the fluid.
Since the fluid is homogeneous incompressible
V • q = f (2)
where f (P,t) is the source density. Assuming
p = ph + p where ph is the hydrostatic pressure
and p (P,t) the flow pressure, we obtain on the
basis of (1) and (2)
—V2p = f/C, P in F. (3)
The pressure p is thus a harmonic function of
P in F.
The free surface fi is characterized by a con-
stant external pressure which can be assumed
to be zero. A linearization of the free-surface
condition can be carried out as follows. We
assume that fi is quasi-horizontal, that is, de-
viates from 2 by a vertical amplitude h (S,t)