Page 424 - Caldera Volcanism Analysis, Modelling and Response

Hydrothermal Fluid Circulation and its Effect on Caldera Unrest 399

where S b is phase saturation (volumetric fraction occupied by phase b), f rock
porosity and q b a sink or source (if negative) of phase b. Equation (2) is non-linear
due to the non-linear relations linking phase saturation, capillary pressure and
relative permeability (Helmig, 1997). If more than one component is present
(i.e. volcanic gases, such as carbon dioxide or dissolved solid phases), Equation (2) is
k
appropriately modified, expressing the fluid mass per unit volume as r X , where
b
b
k
X represents the mass fraction of component k in phase b. As mentioned above,
b
fluid properties that explicitly appear in the mass and energy balance equations may
change significantly within hydrothermal systems. As a consequence, modelling
of hydrothermal fluid circulation also requires the definition of appropriate
equations of state which describe fluid properties at the conditions of interest.
In some cases it is possible to approximate the behaviour of fluid properties as linear
function of pressure and temperature. In other cases this is not feasible and more
accurate non-linear equations of state are required, increasing the complexity of the
numerical problem.
The energy balance equation (for phase b) is written assuming local thermal
equilibrium between solid rock and fluid. It accounts for heat transport by fluid
convection and by conduction through the porous matrix:
!
@T X @ðU b r S b Þ X
b
b
ð1 fÞr c R þ f þr u b r h b þr fl R rTg q ¼ 0 (3)
E
R
@t @t
b b
3
where r R is rock density [kg/m ] (subscript R refers to rock properties); c R is
specific heat of the rock [J/Kg 1K]; T is temperature [1K]; U b and h b are the internal
energy and enthalpy of phase b, respectively; l R is rock thermal conductivity [W/
m1K] (which depends not only on the rock, but also on thermal properties of the
permeating fluid and on its saturation), and q E represents any energy sink or source
within the system.
Due to the difficulty in the simultaneous solution of these highly non-linear
and fully coupled equations, early models were limited to simple systems (often
isothermal) with a single-phase fluid of constant properties flowing through a
homogeneous porous medium (Elder, 1967a, b). Better computational capabilities
and improved numerical techniques have since allowed solution of the coupled
energy and mass transport equations. Pioneering studies focused on hydrothermal
fluid circulation nearby cooling plutons (Cathles, 1977; Norton and Knight, 1977;
Delaney, 1982), and considered fluid density to be constant everywhere except
in the evaluation of buoyancy forces (Boussinesq approximation). Subsequent
improvement in numerical models has been supported by the geothermal industry.
A detailed overview of geothermal reservoir modelling and its development
through time is given by Pruess (1990) and O’Sullivan et al. (2001). Modelling of
heat and fluid flow through porous media is now a well-developed and highly
sophisticated research field. At present, geothermal simulators include realistic
descriptions of fluid properties and account for phase transitions and associated
latent heat effects (Pruess, 1990, 1991; Hayba and Ingebritsen, 1997). Different
features may characterise specific models, features such as the presence of additional
fluid components (non-condensable gases or dissolved salt) or sophisticated rock