Page 423 - Caldera Volcanism Analysis, Modelling and Response

398 Micol Todesco

possible to neglect the details of local variations and to assume that averaged
quantities are representative of the overall macroscopic behaviour over that volume.
The macroscopic approach to porous media flow is based on Darcy’s law, an
empirical equation derived experimentally in 1856, which describes the steady flow
of liquid water through a vertical column of sand. Darcy’s law relates the specific
water discharge (also known as filtration or Darcy velocity) to the pressure gradients
across the sand’s column through a coefficient known as hydraulic conductivity
[m/s], which depends on both fluid and rock properties. Darcy’s law has been tested
over a wide range of conditions and in most cases theoretical results are consistent
with experimental data. Exceptions are found under extreme conditions, in case of
very high (or very low) flow velocity when inertial (or interfacial) forces that are
neglected in the empirical formulation become relevant. These extreme conditions
are very rare in natural systems and Darcy’s law has been successfully used to set up
the water mass balance equation in countless hydrological applications. Detailed
discussion of Darcy’s law, its limitations and its derivation from the Navier–Stokes
equation can be found in de Marsily (1986), Dullien (1992), Helmig (1997),
Ingebritsen and Sanford (1998). Here we shall only briefly overview some of its
basic aspects, following Helmig (1997).
Hydrothermal fluid circulation requires a slightly more complex formulation
than common groundwater problems. Complexities mostly arise from the wide
range of temperatures characterising hydrothermal systems, where water can exist as
a liquid, a gas, a two-phase mixture or a supercritical fluid. Liquid water and steam
are characterised by very different thermal and transport properties, and as the
fluid propagates through different regions its properties may change considerably,
affecting fluid motion. At the same time fluid flow contributes significantly to
heat transport, affecting temperature distribution. Physical models of hydrothermal
circulation therefore need to solve the fully coupled equations of both mass and
energy balance.
For multi-phase flow problems, Darcy’s law can be rewritten to explicitly state
the dependence of the filtration velocity, u b [m/s] on fluid properties:

k b
u b ¼ k ðrp þ r gÞ (1)
b
b
m b
2
where k is the intrinsic rock permeability [m ] and expresses the resistance opposed
by porous media to fluid flow along different directions. It depends on rock porosity
and pore connectivity. k b is the relative permeability of phase b; it expresses the
interference between phases and it ranges from 0 to 1 as a function of gas saturation.
m b is viscosity [Pa s]; p b pressure [Pa] (accounting for the effects of capillary force)
3
and r b density [kg/m ] (where subscript b refers to fluid phase). g is the
2
gravitational acceleration [m/s ].
Based on the above multi-phase version of Darcy’s law, the mass balance
equation for phase b can be written in differential form as:

@ðS b fr Þ
b
þr ðr u b Þ r q ¼ 0 (2)
b
b b
@t