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Table 1 Description of the quantities appearing in Eqs. ( 19 )-( 20 )

From: A hydro-geophysical simulator for fluid and mechanical processes in volcanic areas

Notation

Description

Status

ϕ(x⃗)

Porosity

Assigned

\(\rho_{r}(\vec{x})\)

Density of the rock

Assigned

\(C_{r}(\vec{x})\)

Specific heat of the rock

Assigned

λ(x⃗)

Thermal conductivity of the rock

Assigned

K(x⃗)

Permeability of the rock

Assigned

ĝ

Gravity acceleration

Assigned

\(\rho_{\beta}(\vec{x},t)\)

Density of the phase β

Unknown

\(S_{\beta}(\vec{x},t)\)

Saturation of the phase β

Unknown

\(\chi^{i}_{\beta}(\vec{x},t)\)

Mass fraction of the i-th component in the phase β

Unknown

\(e_{\beta}(\vec{x},t)\)

Internal energy of the phase β

Unknown

\(h_{\beta}(\vec{x},t)\)

Enthalpy of the phase β

Unknown

\(k_{\beta}(\vec{x},t)\)

Relative permeability of the phase β

Unknown

\(\eta_{\beta}(\vec{x},t)\)

Viscosity of the phase β

Unknown

\(P_{\beta}(\vec{x},t)\)

Pressure of the phase β

Unknown

T(x⃗,t)

Temperature

Unknown

  1. Dependence on x⃗ and/or t is also highlighted in the first column. All quantities that do not depend on time are a priori assigned. The subscript β refers to the liquid (β = l) or gas (β = g) phase.