The geodynamic equation of state: What and how

Geodynamic models commonly assume equations of state as a function of pressure and temperature. This form is legitimate for homogenous materials, but it is impossible to formulate a general equation of state for a polyphase aggregate, e.g., a rock, as a function of pressure and temperature because these variables cannot distinguish all possible states of the aggregate. In consequence, the governing equations of a geodynamic model based on a pressure‐temperature equation of state are singular at the conditions of low‐order phase transformations. An equation of state as a function of specific entropy, specific volume, and chemical composition eliminates this difficulty and, additionally, leads to a robust formulation of the energy and mass conservation equations. In this formulation, energy and mass conservation furnish evolution equations for entropy and volume and the equation of state serves as an update rule for temperature and pressure. Although this formulation is straightforward, the computation of phase equilibria as a function of entropy and volume is challenging because the equations of state for individual phases are usually expressed as a function of temperature and pressure. This challenge can be met by an algorithm in which continuous equations of state are approximated by a series of discrete states: a representation that reduces the phase equilibrium problem to a linear optimization problem that is independent of the functional form used for the equations of state of individual phases. Because the efficiency of the optimization decays as an exponential function of the dimension of the function to be optimized, direct solution of the linearized optimization problem is impractical. Successive linear programming alleviates this difficulty. A pragmatic alternative to optimization as an explicit function of entropy and volume is to calculate phase relations over the range of pressure‐temperature conditions of interest. Numerical interpolation can then be used to generate tables for any thermodynamic property as a function of any choice of independent variables. Regardless of the independent variables of the governing equations, a consistent definition of pressure, and the coupling of equilibrium kinetics to deformation, is only possible if the continuity equation accounts for dilational strain.