.. _miscible_treatment:

Miscible Treatment
================================================================

A serious problem in equation of state modelling is to establish a phase identity-independent :math:`K_{rh}(S_h,S_w)`  model which becomes a straight line when we approach the critical point. Otherwise the problem is unsolvable in this region as :math:`\partial(sat)/\partial(zmf) \Rightarrow\infty` at the critical. 

A further problem, even if the system does not approach the critical point, is that, in a miscible flood (such as when |CO2| invades hydrocarbon regions) the system phase envelope can move from right to left, so that a system can change from a liquid to a gas, without going through an intermediate two-phase state. This can produce a  serious discontinuity which renders solving the non-linear equations very difficult:

.. figure:: ../manual/images/miscible_treatment_issue_exp.png
   :scale: 30 %
   :alt: ccsmooth1

The solution is to make the relative permeability and capillary pressure functions dependent on surface tension – which is reasonable, as we would expect zero capillary pressures and straight-line relative permeabilities in the case of zero interphase surface tension. Then we make the surface tension a function of the compositions of the phases. To do this we use a classical correlation for surface tension, the McLeod-Sugden expression :cite:`CT9242501167`, which is a function of the component parachors, which become a new required set of input values.

The surface tension, :math:`\sigma`, computed by :cite:`CT9242501167`, such that :math:`\sigma \Rightarrow 0` as the oil and gas phase compositions converge, is used to define a miscibility fraction: 

.. math::

    f_m=1-f_i, \; f_i= \left(\frac{\sigma}{\sigma_{ref}} \right)^P

Where :math:`\sigma_{ref}` is the surface tension in some immiscible reference state at which the rel. perms are measured. 
P is a parameter that goes from 0.15 to 0.25 (taken equal to 0.25).

While an ‘oil’ fraction can be defined as:

.. math::

    f_o=\frac{T_c}{(T_c+T_{res})}

Where :math:`T_c` and :math:`T_{res}` are the critical and reservoir temperatures, respectively.
So that :math:`T_c=T_{res}`, :math:`f_o=0.5`; if :math:`T_c>>T_{res}` (critical above res temp), then :math:`f_o \Rightarrow 1`. 

The gas fraction is simply taken as the complement of the oil fraction:

.. math::

    f_g=1-f_o

With :math:`f_o` and :math:`f_g`, the rel. perm of each of the hydrocarbon phases is expressed as follows:

.. math::

    K_{rh} = f_o \, K_{ro} + f_g \, K_{rg} 

Where 

.. math::

    K_{rg}(S_g,S_w) = \frac{(S_o \, K_{rgo}(S_g) + S_w \, K_{rgw}(S_w))}{(S_o+S_w)}

Four tables are required: :math:`K_{row}`, :math:`K_{rog}`, :math:`K _{rgo}` and :math:`K_{rgw}`. If these are not available, :math:`K_{row}` is taken as :math:`K_{rgw}`.

:math:`K_{rog}` and :math:`K_{rgo}` are computed as:

.. math::

    K_{rog} = f_i K_{rog} (\mbox{scaled table val}) + f_m \, K_{rog} (\mbox{scaled straight line value}).

Finally, the cap press pressure goes like :math:`P_{cog} \sim f_i` and vanishes with surface tension:

.. math::
    P_{cog}(\mbox{scaled}) = f_i \, P_{cog}(\mbox{table}) 

So, as :math:`f_m \Rightarrow 1`, :math:`f_i \Rightarrow 0`, criticals and cap pressure vanish. 


The result is a much smoother system in the near-critical and miscible region. The hydrocarbon phase has properties which are functions of its composition, not its phase label.