A compositional multi-phase model based on non-linear complementarity functions. More...

#include <dune/common/fvector.hh>
#include <opm/common/Exceptions.hpp>
#include <opm/material/common/Valgrind.hpp>
#include <opm/material/densead/Math.hpp>
#include <opm/models/ncp/ncpproperties.hh>
#include <opm/models/ncp/ncplocalresidual.hh>
#include <opm/models/ncp/ncpextensivequantities.hh>
#include <opm/models/ncp/ncpprimaryvariables.hh>
#include <opm/models/ncp/ncpboundaryratevector.hh>
#include <opm/models/ncp/ncpratevector.hh>
#include <opm/models/ncp/ncpintensivequantities.hh>
#include <opm/models/ncp/ncpnewtonmethod.hh>
#include <opm/models/ncp/ncpindices.hh>
#include <opm/models/common/diffusionmodule.hh>
#include <opm/models/common/energymodule.hh>
#include <opm/models/common/multiphasebasemodel.hh>
#include <opm/models/io/vtkcompositionmodule.hpp>
#include <opm/models/io/vtkdiffusionmodule.hpp>
#include <opm/models/io/vtkenergymodule.hpp>
#include <algorithm>
#include <cassert>
#include <cmath>
#include <memory>
#include <sstream>
#include <string>
#include <tuple>
#include <vector>

Classes
struct	Opm::Properties::TTag::NcpModel
	Define the type tag for the compositional NCP model. More...
struct	Opm::Properties::LocalResidual< TypeTag, TTag::NcpModel >
	Use the Ncp local jacobian operator for the compositional NCP model. More...
struct	Opm::Properties::NewtonMethod< TypeTag, TTag::NcpModel >
	Use the Ncp specific newton method for the compositional NCP model. More...
struct	Opm::Properties::Model< TypeTag, TTag::NcpModel >
	the Model property More...
struct	Opm::Properties::BaseProblem< TypeTag, TTag::NcpModel >
	The type of the base base class for actual problems. More...
struct	Opm::Properties::EnableEnergy< TypeTag, TTag::NcpModel >
	Disable the energy equation by default. More...
struct	Opm::Properties::EnableDiffusion< TypeTag, TTag::NcpModel >
	disable diffusion by default More...
struct	Opm::Properties::RateVector< TypeTag, TTag::NcpModel >
	the RateVector property More...
struct	Opm::Properties::BoundaryRateVector< TypeTag, TTag::NcpModel >
	the BoundaryRateVector property More...
struct	Opm::Properties::PrimaryVariables< TypeTag, TTag::NcpModel >
	the PrimaryVariables property More...
struct	Opm::Properties::IntensiveQuantities< TypeTag, TTag::NcpModel >
	the IntensiveQuantities property More...
struct	Opm::Properties::ExtensiveQuantities< TypeTag, TTag::NcpModel >
	the ExtensiveQuantities property More...
struct	Opm::Properties::Indices< TypeTag, TTag::NcpModel >
	The indices required by the compositional NCP model. More...
struct	Opm::Properties::NcpPressureBaseWeight< TypeTag, TTag::NcpModel >
	The unmodified weight for the pressure primary variable. More...
struct	Opm::Properties::NcpSaturationsBaseWeight< TypeTag, TTag::NcpModel >
	The weight for the saturation primary variables. More...
struct	Opm::Properties::NcpFugacitiesBaseWeight< TypeTag, TTag::NcpModel >
	The unmodified weight for the fugacity primary variables. More...
class	Opm::NcpModel< TypeTag >
	A compositional multi-phase model based on non-linear complementarity functions. More...

Namespaces
namespace	Opm
	This file contains a set of helper functions used by VFPProd / VFPInj.
namespace	Opm::Properties::TTag
	The generic type tag for problems using the immiscible multi-phase model.

Detailed Description

A compositional multi-phase model based on non-linear complementarity functions.

This model implements a $M$ -phase flow of a fluid mixture composed of $N$ chemical species. The phases are denoted by lower index $\alpha \in \{ 1, \dots, M \}$ . All fluid phases are mixtures of $N \geq M - 1$ chemical species which are denoted by the upper index $\kappa \in \{ 1, \dots, N \}$ .

By default, the standard multi-phase Darcy approach is used to determine the velocity, i.e.

$\‍[ \mathbf{v}_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\mathbf{grad}\, p_\alpha - \varrho_{\alpha} \mathbf{g} \right) \;, \‍]$

although the actual approach which is used can be specified via the FluxModule property. For example, the velocity model can by changed to the Forchheimer approach by

template<class TypeTag>

struct FluxModule<TypeTag, TTag::MyProblemTypeTag> { using type = ForchheimerFluxModule<TypeTag>; };

The core of the model is the conservation mass of each component by means of the equation

$\‍[ \sum_\alpha \frac{\partial\;\phi c_\alpha^\kappa S_\alpha }{\partial t} - \sum_\alpha \mathrm{div} \left\{ c_\alpha^\kappa \mathbf{v}_\alpha \right\} - q^\kappa = 0 \;. \‍]$

For the missing $M$ model assumptions, the model uses non-linear complementarity functions. These are based on the observation that if a fluid phase is not present, the sum of the mole fractions of this fluid phase is smaller than $1$ , i.e.

$\‍[ \forall \alpha: S_\alpha = 0 \implies \sum_\kappa x_\alpha^\kappa \leq 1 \‍]$

Also, if a fluid phase may be present at a given spatial location its saturation must be non-negative:

$\‍[ \forall \alpha: \sum_\kappa x_\alpha^\kappa = 1 \implies S_\alpha \geq 0 *\‍]$

Since at any given spatial location, a phase is always either present or not present, one of the strict equalities on the right hand side is always true, i.e.

$\‍[ \forall \alpha: S_\alpha \left( \sum_\kappa x_\alpha^\kappa - 1 \right) = 0 \‍]$

always holds.

These three equations constitute a non-linear complementarity problem, which can be solved using so-called non-linear complementarity functions $\Phi(a, b)$ . Such functions have the property