.. _formulation-of-toil-ims: The Mathematical Formulation of TOIL_IMS ======================================== The mathematical model of the TOIL_IMS module implemented in PFLOTRAN describes the oil/water non-isothermal immiscible fluid flows by a system of three equations (3 degrees of freedom, DOFs): a molar balance equation :eq:molar_balance_toil for each phase, which can be easily recast as mass balance, an energy equation :eq:eng_balance_toil which assumes that the two phases and the rock are in thermal equilibrium and neglects the kinetic and potential energy. Fluxes are modeled using a multi-phase Darcy's formula :eq:darcy_toil. See below the equation system .. math:: \frac{\partial}{\partial t} \phi \left(s_\alpha\eta_\alpha\right) + \nabla\cdot\left(\mathbf{q_\alpha}\eta_\alpha\right) = Q_i :label: molar_balance_toil .. math:: \frac{\partial}{\partial t} \left[ \phi \sum_{\alpha}{s_\alpha\eta_\alpha U_\alpha} + (1-\phi)\rho_r c_r T \right] + \nabla \cdot \sum_{\alpha}{\left[ q_\alpha\eta_\alpha H_\alpha + \kappa \nabla T \right]} = Q_e :label: eng_balance_toil .. math:: q_\alpha = \frac{K k_\alpha}{\mu_\alpha} \nabla(P_\alpha - \rho_\alpha g z) :label: darcy_toil Where: * :math:\eta_\alpha is the molar density of phase :math:\alpha; * :math:H_{\alpha} is the enthalphy of phase :math:\alpha; * :math:U_{\alpha} is the internal energy of phase :math:\alpha; * :math:s_{\alpha} is the phase saturation of phase :math:\alpha; * :math:K is the saturated media permeability; * :math:k_\alpha is the relative permeability of phase :math:\alpha; * :math:\mu_\alpha is the viscosity of phase :math:\alpha; * :math:P_\alpha is the pressure of phase :math:\alpha; * :math:\rho_\alpha is the mass density of phase :math:\alpha; * :math:g is the gravitational constant; * :math:z is the height coordinate. The system of equations is discretised in space with a finite volume scheme that uses a two-point flux formula, and in time with a first order fully implicit scheme. The equations are linearised using the Newton-Raphson approach. An adaptive time stepping scheme is adopted. The resulting system features three equations in three unknowns (primary variables) that are chosen as follows: Oil Pressure, Oil Saturation and Temperature.