9.2.0.14     Scalar Transport Equation in Incompressible Miscible Multi-Phase- Flows

 

cmi_QDCA

 

 

            Element_name = cmi_QDCA,  etc…

                  < scalar diffusion material data >

                  < body force data >

                  < connectivity data >

 

 

            The element is used for solution of the following scalar mass balance equation in miscible incompressible flows:

 

 

                                               

 

where  = volumetric concentration of the invading fluid;  = velocity of the fluid mixture (typically obtained by solving Darcy’s equation per Section 9.2.0.12);  = diffusion-dispersion tensor;  = porosity of the medium;  = injection/extraction volumetric flow rate;  is either the specified concentration of the injected fluid at injection wells or  is the resident concentration at production wells.  One degree of freedom is assigned to each node for the concentration.

 

 

 

 

9.2.0.15     Immiscible Multi-Phase Flow Equation

 

QDC_MFLOW

 

 

            Element_name = QDC_mflow,  etc…

                  < scalar diffusion material data >

                  < body force data >

                  < connectivity data >

                  < field output data >

 

 

            The element is used for solution of the following mass balance equations in multi-phase fluid flow thru porous media:

 

where

 

 

*  = fluid phase number (=1, …, number_of_phases)

 = volumetric flux of phase  (volume per total area per unit time)

 = fluid (seepage) velocity of phase

 = solid velocity

 = porosity (volume of voids per unit volume)

 = degree of saturation (ratio volume of fluid to volume of voids);

 = fluid phase mass density

 = fluid viscosity

 = relative permeability

intrinsic absolute permeability (units [L2])

 = fluid pressure

body force (per unit mass)

 = source (or sink) of mass to phase (units: [M/L3/T]) =

 

            One degree of freedom is assigned to the fluid pressure for each phase.

 

 

            For two-phase flows , the two coupled equations can be written as follows:

 

 

           

 

where

= fluid compressibility

 

 = total permeability (units [L2])

 

 = material derivative

 

            The degree of water saturation  is a nonlinear (multi-valued) function of  capillary pressure, shown schematically in Figure 9.2.0.15a, where  residual water saturation and  with  residual oil saturation.  The relative permeability coefficient  for each fluid phase is a nonlinear function of the water saturation shown schematically in Figure 9.2.0.15b (see Section 10.15 for further details).

 

 

 

 

                    (a)  Typical  Curves                                   (b)  Typical  Curves

 

Figure 9.2.0.15.1  Typical Material Curves

 

 

 

 

 

 

9.2.0.16     Pressure Equation in Incompressible Immiscible Multi-Phase Flows

 

            The element is used for solution of the following pressure equation in incompressible immiscible multi-phase flows:

 

                                 Lagrangian porosity

 

where the total flux:

 

                              

 

and:

 

 

                                        

 

 

and the global pressure P (Chavent [1981, 1984]) for 2 phases:

 

                                                       

 

where

 

                       

 

The phase fluxes can be computed as:

 

     capillary pressure

 

*  = fluid phase number (=1, …, number_of_phases)

 = volumetric flux of phase  (volume per total area per unit time)

Lagrangian porosity (volume of voids per unit volume)

 = Eulerian porosity

 = degree of saturation (ratio volume of fluid to volume of voids);

 = fluid phase mass density

 = fluid viscosity

 = relative permeability

= intrinsic absolute permeability (units [L2])

 = fluid pressure

= body force (per unit mass)

 = source (or sink) of mass to phase (units: [M/L3/T]) =

 = material derivative

 

9.2.0.16.1  Pressure Formulation:

 

                  There is one degree of freedom assigned to the global pressure .

 

QDCP_PRESSURE

 

 

            Element_name = QDCP_PRESSURE,  etc…

                  < scalar diffusion material data >

                  < body force data >

                  < connectivity data >

 

 

 

9.2.0.16.2  Mixed Formulation:

 

            In the mixed formulation both the global pressure P and the total fluxare treated as unknowns.

 

QDCP_MIXED

 

 

            Element_name = QDCP_MIXED,  etc…

                  < scalar diffusion material data >

                  < body force data >

                  < connectivity data >

 

 

 

 

 

9.2.0.17     Saturation Equation in Incompressible Immiscible Multi-Phase Flows

 

cmi_QDCS

 

 

            Element_name = cmi_QDCS,  etc…

                  < scalar diffusion material data >

                  < body force data >

                  < connectivity data >

 

 

            The element is used for solution of the saturation equation in incompressible immiscible multi-phase flows:

 

where= total flux (typically obtained by solving the appropriate pressure equation; see Section 9.2.0.16).

 

 

*  = fluid phase number (=1, …, number_of_phases)

 = volumetric flux of phase  (volume per total area per unit time)

 Lagrangian porosity

 = Eulerian porosity (volume of voids per unit volume)

 = degree of saturation (ratio volume of fluid to volume of voids);

 = fluid phase mass density

 = fluid viscosity

 = relative permeability

= intrinsic absolute permeability (units [L2])

 = mobility

 = fluid pressure

= body force (per unit mass)

 = source (or sink) of mass to phase (units: [M/L3/T]) =

 = material derivative

 

One degree of freedom is assigned to each node for the saturation.

 

 

 

 

9.2.0.18     Pressure Equation in Compressible Immiscible Compositional Multi-Phase Flows

 

            The element is used for solution of the pressure equation in compressible immiscible compositional multi-phase flows:

 

                       

 

                                     

 

where the total mass flux:

 

           

 

and the global pressure P (Chavent [1986]) for 2 phases:

 

                                           

 

where

 

                                   

 

The phase mass fluxes can be computed as:

 

 

*  = fluid phase number (=1, …, number_of_phases)

 = volumetric flux of phase  (volume per total area per unit time)

Lagrangian porosity

 = Eulerian porosity (volume of voids per unit volume)

 = degree of saturation (ratio volume of fluid to volume of voids);

 = fluid phase mass density

 = fluid viscosity

 = relative permeability

 = intrinsic absolute permeability (units [L2])

 = mobility

 = fluid pressure

= body force (per unit mass)

 = source (or sink) of mass to phase (units: [M/L3/T]) =

 = material derivative

 

            Two formulations of the equation are available: (1) a primal formulation for which the total pressure is the unknown; and (2) a mixed formulation for which both the total pressure and the total mass flux are treated as unknowns.

 

9.2.0.18.1  Pressure Formulation:

 

            The equation is obtained by substituting the total mass flux in the conservation equation.  There is one degree of freedom assigned to each node for the total pressure.

 

cmi_QDCP_PRESSURE

 

 

            Element_name = cmi_QDCP_pressure,  etc…

                  < scalar diffusion material data >

                  < body force data >

                  < connectivity data >

 

 

 

 

 

9.2.0.18.2  Mixed Formulation:

 

            Both the total pressure and the mass flux are treated as unknowns.  There are (NSD+1) degrees of freedom assigned to each node for the total mass flux and the total pressure.

 

cmi_QDCP_MIXED

 

 

            Element_name = cmi_QDCP_mixed,  etc…

                  < scalar diffusion material data >

                  < body force data >

                  < connectivity data >

 

 

 

 

 

9.2.0.19     Mole/Mass Transport Equation in Compressible Immiscible Compositional Multi-Phase Flows

 

            The element is used for solution of the following mass-balance equation(s) in compressible immiscible compositional multi-phase flows:

 

 

where = flux of phase  (typically obtained by solving the appropriate pressure equation; see Section 9.2.0.18);

 

             = mole/mass fraction of component k in the mixture

             = mole/mass fraction of component k in phase

             = mass of component k

*  = fluid phase number (=1, …, number_of_phases)

 = volumetric flux of phase  (volume per total area per unit time)

 = Lagrangian porosity

 = Eulerian porosity (volume of voids per unit volume)

 = degree of saturation (ratio volume of fluid to volume of voids);

 = fluid phase mass density

 = diffusion/dispersion coefficient matrix (units {L2/T]

= body force (per unit mass)

 = source (or sink) of mass to phase (units: [M/L3/T]) =

 = source (or sink) to component  = (units [M/L3/T])

 = material derivative

 

            One degree of freedom is assigned to each node for the mole or mass fraction  for which the equation is solved.

 

 

cmi_QDCZ

 

 

            Element_name = cmi_QDCZ,  etc…

                  < scalar diffusion material data >

                  < body force data >

                  < connectivity data >

 

 

 

 

 

 

 

Notes . .