OPM - Reference Documentation for ewoms

Compositional Multi-Phase Model Directly Solving Non-linear Complementarity Problems. More...

Collaboration diagram for NCP:

Files
file	ncpproperties.hh
	Declares the properties required for the NCP compositional multi-phase model.

Classes
class	Ewoms::NcpBoundaryRateVector< TypeTag >
	Implements a boundary vector for the fully implicit compositional multi-phase NCP model. More...

class	Ewoms::NcpExtensiveQuantities< TypeTag >
	This template class represents the extensive quantities of the compositional NCP model. More...

struct	Ewoms::NcpIndices< TypeTag, PVOffset >
	The primary variable and equation indices for the compositional multi-phase NCP model. More...

class	Ewoms::NcpIntensiveQuantities< TypeTag >
	Contains the quantities which are are constant within a finite volume in the compositional multi-phase NCP model. More...

class	Ewoms::NcpLocalResidual< TypeTag >
	Details needed to calculate the local residual in the compositional multi-phase NCP-model . More...

class	Ewoms::NcpModel< TypeTag >
	A compositional multi-phase model based on non-linear complementarity functions. More...

class	Ewoms::NcpNewtonMethod< TypeTag >
	A Newton solver specific to the NCP model. More...

class	Ewoms::NcpPrimaryVariables< TypeTag >
	Represents the primary variables used by the compositional multi-phase NCP model. More...

class	Ewoms::NcpRateVector< TypeTag >
	Implements a vector representing mass, molar or volumetric rates. More...

Detailed Description

Compositional Multi-Phase Model Directly Solving Non-linear Complementarity Problems.

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

SET_TYPE_PROP(MyProblemTypeTag, FluxModule, Ewoms::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