XVIII International Conference on Water Resources
CMWR 2010
J. Carrera (Ed)
CIMNE, Barcelona 2010
COMPOSITIONAL FORMULATION FOR MULTI PHASE FLOW
REACTIVE TRANSPORT IN POROUS MEDIA: AN APPROACH THAT
SIMPLIFIES COUPLING BETWEEN PHENOMENA.
Pablo Gamazo*, Maarten W. Saaltink* and Jesús Carrera†
*
Hydrogeology Group, Technical University of Catalonia (UPC)
C/Jordi Girona 1-3, 08034 Barcelona, Spain
e-mail: pablo.gamazo@upc.es, maarten.saaltink@upc.edu, web page: http://www.h2ogeo.upc.edu/
†
IDAEA, Institute of Environmental Assessment and Water Research, Spanish Research Council
(CSIC)
C/Jordi Girona 18-26, 08034 Barcelona, Spain
e-mail: jcrgeo@cid.csic.es, web page: http://www.idaea.csic.es/
Key words:
formulation
reactive transport, multiphase flow, coupling phenomena, compositional
Summary. In this work we discuss coupling difficulties that arises when solving phase and
species conservation equations. We present a compositional formulation that avoids these
difficulties, and allows solving multiphase reactive transport problems for a generic number
of species and reactions. The formulation is illustrated by an example.
1
INTRODUCTION
Reactive transport modeling involves the simultaneous study of the conservation of several
phenomena. Most general formulations include: conservation of phases (solid and fluids),
conservation of species, momentum conservation and energy conservation7. Several variables
in these equations are related by constitutive laws that couple these phenomena, increasing the
difficulty to solve them. Variables like phase density, viscosity and saturation degree are
present in all equations and may be affected by all the states variables of the system1.
Reactions relating species form different phases (heterogeneous reaction) also coupled species
and phases conservation phenomenon by producing source/sink terms that will affect both.
Depending on the individual characteristics of the problem to be solved, there are several
ways of dealing with the coupling of these phenomena. For example when changes in linking
variables are relatively small during a time step, phenomena can be solved independently and
changes in variables may be considered time-laged, or even neglected6,2. In case were these
simplification cannot be made (e.g., density driven flow, production of gaseous CO2 and CH4,
dry scenarios with evaporation or precipitation of hydrated minerals) interaction between
phenomena must be considerd. This implies a significant increment in calculation time and, if
all phenomena are solved simultaneously, an increase in size and complexity of the system to
be solved.
Pablo Gamazo, Maarten W. Saaltink and Jesús Carrera
In this work we focus on the interaction between the phase conservation and the species
conservation and we present a compositional formulation that avoids coupling problems.
Momentum conservation is considered by Darcy’s law and is directly substituted in the other
conservation equations. Coupling with energy conservation is not discussed.
2 REACTIVE TRANSPORT COMPOSITIONAL FORMULATION
2.1 Vantages of compositional formulation
Most of the reactive transport codes, at our knowledge, consider both conservation of
phases and conservation of species. This may be conflictive, because by definition the sum of
all conservation equations of species belonging to a phase must be equal to the conservation
equation of the phase. Thus, in order to obtain a linearly independent system of equation one
species conservation equation should be removed for each phase. When considering phase
conservation another difficulty appears which is the phase sink/sources due to heterogeneous
reactions in equilibrium. The fact that there is no explicit expression for equilibrium reaction
rates increases the complexity of the system to be solved.
A compositional formulation eliminates these conflicts by setting out a combination of
species conservation equations, avoiding phases4, and eliminating all equilibrium reaction
rates terms. Excluding phase conservation equations also facilitates coupling with momentum
or energy conservation by reducing the kinds of phenomena to solve.
Composition formulation has been used to solve multiphase problems in aquifers and oil
reservoirs but normally for a small number of species. In this work we present a general
formulation that can be applied to an undefined number of species.
2.2 Proposed formulation
We start the formulation introducing a general conservation equation for specie i that
belongs to phase 
Ne

 ci  L ci   Se j ,i  re j  fi
(1)
t
j 1
Where  is the volumetric content of phase  , ci is mol concentration of specie i
( mol i / m3 ),. L () is the transport operator, Ne is the number equilibrium equations
Se j ,i is the stoichiometric coefficient of reaction j for specie i , re j is the reaction rate for


 
reaction j and f i is a source/sink term that contains contributions of kinetic reactions and
boundary exchanges of specie i . The L 
 operator consists of

L  ci      ci q     jD ,i
q  
Kkr

 p   g 
2

; jD ,i  D   ci
(2)

(3)
Pablo Gamazo, Maarten W. Saaltink and Jesús Carrera
D  Ddiff   Ddisp
;
Ddisp  dt q I   dl  dt 
q  qtr
q
(4)
Were q is the Darcy flow for the  phase and jD ,i is the diffusive-dispersive mol flow,
Fickean in this case. K is the intrinsic permeability, k r and  are the relative permeability
and the viscosity of the phase, p is the phase pressure gradient,  g is the phase density
times gravity, Ddiff is the phase diffusion coefficient,  is the tortuosity, I is the identity
matrix and dl and dt are the longitudinal and transversal dispersion coefficients. The
L   operator is null for immobile species, like minerals or adsorbed species.
Conservation equations of all species can be represented in a vectorial way

 θc   L  c   Setr  re  f
t
 Ll 
 cliquid 



Were c   c gaseous  is a vector with the concentration of all species, L     L g 
c

 0
 immovil 

(5)

 
is a


vectorial operator with the corresponding phase transport operator to each specie, S e is the
stoichiometric matrix, and f is a vector with all species source/sink terms.
Besides solving species conservation equations, a reactive transport problem implies mass
action equations. The latter can be written in a matrix notation as:
Se  log a  log k  0
(6)
where S e is the stoichiometric matrix of size Ne (number of equilibrium reactions) x Ns
(number of species), k is a vector with the equilibrium constants of all (Ne) reactions ,and
a is a vector with the activity of all (Ns) species. The activity of a specie ai (or fugacity for
gaseous species) is defined as:
ai   i xi
(7)
Were  i represents the activity or fugacity coefficient, and xi the molality or partial pressure
for a liquid or gaseous specie respectively. We treat mineral species as ideal species, thus its
activity is considered equal to one.
The set of Ne mass action equations (6) allowed us to obtain an expression to calculate Ne
species molality or partial pressure (secondary species) as a function of Ns-Ne species
(primary species). As mineral activities are considered equal to one, the number of primary
species can be reduced to Ns-Ne-Nme (were Nme is the number of equilibrium minerals).
log xsec   log γ sec  S*   log x pri  log γ pri   log k '
(8)
A full description of this procedure and how S * and log k ' are calculates is given by
Saaltink et al.5.
3
Pablo Gamazo, Maarten W. Saaltink and Jesús Carrera
Equilibrium reaction rate terms re in (5) can be eliminated by linearly combining the mass
balance equations. To do so, we define the matrix U (also known as component matrix, (NsNe)xNs ) as:
U  Ste  0
(9)
Multiplying equation (5) times U ensures elimination of all equilibrium reaction rates. It
also leads to the definition of components as u  U  c , which by virtue of (9) become linear
combinations of species that are not affected by equilibrium reactions. Multiplying system (5)
times U yields to a system of size (Ns-Ne):

(10)
 U  θ c   U  L c  U  f
t
When mineral phases are present in equilibrium, Gibbs phase rule indicates that the system
looses one degree of freedom for every additional mineral phase. Therefore the definition of
components needs to be updated. There are several ways to achieve this goal, depending on
the additional properties one may wish for components (see Molins et al.3). Here we will use
the approach of Saaltink et al.5, where the initial component matrix U is multiplied by an
elimination matrix E ( (Ns-Ne-Nme)x(Ns-Ne), were Nme is the number of mineral in
equilibrium) to obtain the reduced component matrix U '  E  U (Ns-Ne-Nme)x(Ns).
Defining
 cliquid 
 0 
 0 



' 
' 
' 
(11)
ul  U   0  ; u g  U   c gaseous  ; uim  U   0  ; f '  U '  f
 0 
 0 
c





 immovil 
system (10) can be written as:



l ul    g u g   l uim   Ll  ul   L g  u g   f '
t
t
t
(12)
2.3 Solving variables and secondary variables
We set out Ns conservation equations (1) and for each we have the species concentration
as unknown. The set of Ne mass action equations (6) and the fact that the mineral activity is
considered equal to one, allowed us to obtain all species concentration, except the equilibrium
minerals, as a function of Ns-Ne-Nme primary species. This is the dimension of the vectorial
component concentration equations (12).
As the chemical unit for aqueous species concentration is molality, the liquid h2o specie
concentration becomes a constant. This gap in the unknowns can be used to include liquid
pressure as a solving variable. Gas pressure can be calculated as the sum of all gaseous
pressures, so it can be considered as a secondary variable. Thus, the liquid and gas pressure,
which normally are associated to phase conservation, are considered as primary and
secondary variables of component conservation, respectively.
4
Pablo Gamazo, Maarten W. Saaltink and Jesús Carrera
The vectorial component concentration (12) will be solved considering liquid pressure and
Ns-Ne-Nme-1 primary species as solving variables. Aqueous species volumetric
concentrations ci can be calculated using the expression:
ci  l lh2o mi
(13)
Were mi is the specie molality,  l is the phase density and lH 2O is the water mass fraction
on the liquid phase. An expression for this last variable can be obtained from the sum of all
liquid species mass fraction  li :
1




1       m M   1   mli M i   lh2o  1   mli M i 
(14)
i
i
 i  H 2O

 i  h2o

Gas volumetric concentration can be calculated according to the gas model used. For ideal
gas model volumetric concentration can be expressed as:
p
ci , g  i
(15)
RT
Were pi is the partial pressure of specie i , R is the universal gas constant and T the
absolute temperature. Non equilibrium minerals are always considered as primary species.
Equilibrium minerals, (which have been eliminated from the system by multiplying by U ' ),
and equilibrium reaction rates re , are calculated from equations (5) after the rest of the species
have been solved from equations (12). As the number of unknowns (equilibrium mineral
concentrations and reaction rates that is 2Nme) is lower than the number of equations (Ns),
some equations may be ignored or least squares techniques could be applied.
i
l
h2 o
l
i
l
i
h2 o
l
2.3 Formulation application
For a better understanding of the formulation, we illustrate it by setting the equations for a
2 phase system considering the following equilibrium reactions:
H 2Ol 
H 2Ol 
N 2 l 
H   OH 
K H 2Ol 
H 2O g 
K H 2O g 
N 2 g 
Ca 2  SO42  2 H 2O
K N2
(16)
gypsum K gypsum
The vector of concentrations, components and source/sink terms, and the stoichiometric
and reduced component matrix for this system are given in Table 1
5
Pablo Gamazo, Maarten W. Saaltink and Jesús Carrera
cl
cg
cim
 mH 2 O 


 mH  
m  
 OH 
 mCa 2 


l wlh 2 o   mSO42 
m 
 N 2l  
 0 


 0 
 0 


 0 


 0 
 0 


 0 
1  0 
RT  0 


p

 H 2O g  
 p

 N 2 g  
 0 


 0 


 0 
 0 


 0 
 0 


 0 
 0 


 0 
c

 gypsum 
Setr


 H 2Ol 

 H
 OH 

 Ca 2
 SO 2
 4
 N 2l 

 H 2O g 

 N 2 g 

 gypsum
r4 
1 1 0 2 
1 0 0 0

1 0 0 0

0 0 0 1 
0 0 0 1 

0 0 1 0

0 1 0 0

0 0 1 0 

0 0 0 2 
r1
r2
r3
U'

H
1
0
1
0
0
0
1
0
1
0
0
0
0
 cH 2Ol   cOH   2  cSO42 


cH   cOH 




cN 2l 




cCa2  cSO 2

4

Ca
2
H 2O l 
ul
OH

2
4
SO
N 2 l 
H 2O g 
N 2 g 
gypsum
2
0
1
0
0
0
0
0
0
0
1
0
0
0
1
0
0
1
1
0
0
0
0
f'
ug
uim
 cH 2O g  


 0 
c

 N2 g  
 0 


0
 
0
0
 
0
 f H 2Ol   fOH   2  f SO42  f H 2O g  


f H   fOH 




f N2l   f N2 g 




fCa2  f SO2

4

Table 1 - vector of concentration, components and source/sink terms, and the stoichiometric and reduce
component matrix for this system
Substituting component vectors in component conservation equation (12) leads to
 cH 2Ol   cOH   2  cSO42 
 cH 2Ol   cOH   2  cSO42 
 cH 2O g  
 cH 2O g    f H 2Ol  







 
cH   cOH 
cH   cOH 
   0 


 
 0  
l 

  g 
  Lg 
  Ll 
cN2l 
cN2l 
c 
t 
 t  cN2 g  


 N2 g   




 0 
 0  
cCa2  cSO2
cCa2  cSO2



 

4


4

6
fOH   2  f SO2  f H 2O g  
4

f H   fOH 
 (17)

f N2l   f N2 g 


fCa2  f SO2
4

Pablo Gamazo, Maarten W. Saaltink and Jesús Carrera
One possible set of solving variables are the liquid pressure pl and the molality of Ca , N 2
and H . The rest of species can by calculated by expression (8):
 mOH  
  OH    1 1




 pH 2O g  
  H 2O g    1 0
log 
  log 

p 
 
 N2 
 N2   0 0
 m 2 
  2   2 0
 SO4 
 SO4 
 mH 2O 
  H 2O    K H 2Ol  
0 





 

K
m




0 
H 2O g  




  log  H   log  H    
 (18)
mN2
 N2
1 0 

K


N





2









0 1
m

 Ca2 
 Ca2     K gypsum 

0
0
2 CONCLUSIONS
Solving reactive transport problems considering phase conservation implies dealing with
source/sink terms that depends on equilibrium reaction rates for which there are no explicit
expressions. These terms increase the complexity of the system to solve.
We have shown that for cases where Darcy’s Law is considered to represent momentum
conservation, a generic multiphase reactive transport problem can be formulated considering
only species conservation equation and no explicit phase conservation. The formulation
eliminates all equilibrium reaction rates.
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