S. Oladyshkina∗ · M. Panfilovb
Hydrogen penetration in water through porous
medium: application to a radioactive waste storage
site
Stuttgart, January 2011
a
Institute of Hydraulic Engineering (LH2), University of Stuttgart,
Pfaffenwaldring 61, 70569 Stuttgart, Germany
[email protected]
www.iws.uni-stuttgart.de
b
Laboratory LEMTA, UMR 7563, ENSG INPL,
2 avenue de la Fort de Haye, BP 160, 54501, Vandoeuvre-les-Nancy, France
http://www.lemta.fr
∗
Notice: The presented research work was performed in the laboratory LEMTA, UMR 7563, France.
Abstract Hydrogen penetration in water through porous medium was analyzed in the paper. A
two-phase compositional model approach was considered. The first part of the work deals with the
thermodynamic analysis of the hydrogen-water system. The thermodynamic model was calibrated using the experimental data of hydrogen solubility in water. The phases densities, viscosities and phase
concentrations were presented in an analytical form. Moreover, the domain of validity of analytical laws
- such as Henry’s, Raoult’s and Kelvin’s laws - for the estimation of phase properties was presented for
the analyzed system. The second part deals with two-phase hydrodynamic behaviors. An analytical
solution for the non-compressible flow was constructed. In general case the influence of relative permeabilities on the flow regimes was analyzed numerically. The notion pseudo saturation was introduced
to define phases appearance. Actually, mobile gas created a time displaced front relatively slower than
mobile gas flow. Diffusion becomes really important for low mobile gas case, as the penetration accelerates for the large range of saturation. In contrast to this, the mass exchange phenomena has a
small influence on the flow type. Thus, the regimes of hydrogen penetration in liquid was shown really
sensitive to the relative permeability form.
Keywords hydrogen-water · two-phase flow · gas diffusion · thermodynamic · solubility · relative
permeability
Preprint Series
Stuttgart Research Centre for Simulation Technology (SRC SimTech)
SimTech – Cluster of Excellence
Pfaffenwaldring 7a
70569 Stuttgart
[email protected]
www.simtech.uni-stuttgart.de
Issue No. 2011-1
2
S. Oladyshkin, M. Panfilov
1 Introduction
This work deals with the study of the gas migration around storage of radioactive waste. Important
quantities of hydrogen (H2 ) are produced due to radiolysis and corrosion. Other gases, such as CH4 ,
C2 H6 , C2 H2 , etc., as well can be produced , however quantities of such gases are very small in comparison to H2 . The document [1] provides detailed information about the mechanisms of hydrogen
generation around a radioactive waste storage. The particular properties of hydrogen is small size of
molecule , this why H2 has a strong ability of transport in porous medium even with almost impermeable properties. The danger consists in transport of radioactive elements on the hydrogen molecule
from the waste deposit. Let us notice, that the water is present usually in the porous medium around
the storage. That is why the analysis of gas transfer (actually hydrogen transfer) in porous medium
saturated by water is a non-trivial problem and demands better understanding of transport processes.
The current paper contains research initiated in scope of MoMaS project and based on data of
French National Radioactive Waste Management Agency (ANDRA), see [1]. Modeling of gas transport around a radioactive waste storage asks for very large time scale simulations, and as consequence,
demands the high computational power. That why question about model complexity describing the
flow in porous medium and possible assumptions are still open. In this paper we consider full multiphase compositional model containing gas and liquid phases, which consist from two main components:
hydrogen and water. These components can be present in all phases. In this paper we provide clarification about thermodynamics model complexity, and we illustrate errors of simplified thermodynamics
relations in comparison to full compositional formulation, see the section 3. The section 4 demonstrates
influence of compositional effects on hydrogen transport in porous medium. Also we analyze the influence of relative permeability on the regimes of hydrogen penetration using the presented compositional
approach.
2 Hydrogen-water compositional model of two-phase flow in porous medium
We will consider a compositional mixture composed of 2 chemical components, H2 and H2 O, capable
of forming two thermodynamic phases, gas and liquid, separated from one another by an interface.
The permanent mass exchange of hydrogen and water components between liquid and gas phases was
determined by thermodynamic conditions. We will suppose that there is no chemical reactions are
observed, but each component may be dissolved in both phases.
2.1 Compositional two-phase flow model
We shall consider a fine dispersed system which is in local equilibrium. According to the phenological
approach we will examine gas, liquid and solid as three interpenetrating continuums, Nikolaevsky et
al. [10], Nigmatulin [9], Sedov [17]. As accepted in underground thermo and hydrodynamics, we
consider an isothermal process, as the overall mass of earth rocks surrounding the examined reservoir
plays the role of a huge calorimeter with a huge integral calorific capacity which maintains a constant
temperature.
To describe a hydrogen-water flow in porous medium we will use the general compositional model
which is composed in this case of 2 equations of mass balance for each chemical component, H2 and
H2 O, and two equations of momentum balance for each phase, gas and liquid, see Nigmatulin [9],
Sedov [17], Coats [4] and Aziz and Settari [2]:
∂ (k)
(1a)
φ
ρl cl s+ρg c(k)
g [1−s]
∂t (k)
(k)
(k)
, k = 1, 2
+div ρl cl Vl − Jl + ρg c(k)
g Vg − Jg
Vl = −
Kkl
gradPl ,
μl
Vg = −
Kkg
gradPg
μg
(1b)
where superscript ”k” refers to k-th chemical component (H2 denoted as w or H2 O denoted as h),
(k)
(k)
indexes g and l to gas and liquid; φ is porosity; ρ is the phase density; cg or cl is the mass
Hydrogen penetration in water through porous medium
3
concentration of k-th component in gas or liquid; s is the volume saturation of pores by liquid; kg and
kl are the relative permeabilities; μ is the phase viscosity; K is the absolute permeability; P is the
phase pressures, V is the Darcy velocity.
The momentum balance equations (1b) are written in the form of the Darcy law for each phase,
thus we neglect the momentum exchange at phase transitions. The phase viscosity is a given function
of pressure and phase composition. The absolute permeability K and porosity φ are given functions of
space coordinates. The relative permeabilities structure may be arbitrary, in particular they may be
examined as functions of velocity as in [7], but not only of saturation.
Due to the capillarity, a phase pressure difference appears in the compositional model:
Pg − Pl = Pc (s)
(2)
Pc - is the given function of the effective capillary pressure. Let us note, that we will us the notation
P for the gas pressure (Pg ) below.
The diffusive flow of dissolved hydrogen can be expressed by applying Ficks Law:
Jlh = ρl Dlh gradchl
(3)
Jgh = ρg Dgh gradchg
(4)
here the diffusion coefficients, [1], Dlh and Dgh are:
Dlh = Dl0 s
Dgh
=
Dg0 (1
μ0l
,
μl
P0
− s) ,
P
Dl0 = 1.57 · 10−14
Dg0
D0 φ Pat
= 2
θ P0
φT
θ2 μ0l
T
T0
1.75
θ is tourtosity and T0 = 303K, Pat = 1.01 · 105 P a, D0 = 9.5 · 10−5 m2 /s; P0 - characteristic values of
the pressure.
The system of 4 equations (1) is written with respect to pressure P , saturation s, velocities Vl
and Vg . Thus, for two components, h and w, the system (1) does not contains the independent concentrations. This means that, all concentrations and phase densities are determined from additional
thermodynamic relationships. On other words, for a mixture consisting of two chemical components
are possible to totally split the general compositional model into a thermodynamic system and a hydrodynamic system, which may be resolved independently of one another. Let us note, that such spilling
not possible for general class of multicompositional system, however can be performed for a particular
case along streamlines [11].
2.2 Thermodynamic closure relationships
The closure relationships for system (1) describe the local equilibrium thermodynamic behavior. They
consist of 2 equilibrium equations in terms of chemical potentials (5a) for chemical component h and
w, two equations of phase state (5b) and two normalizing equations for concentrations (5c):
h
h w
νgh P, chg , cw
(5a)
g = νl P, cg , cg ,
w
h w
w
h w
νg P, cg , cg = νl P, cg , cg ,
ρg = ρg P, chg , cw
(5b)
g ,
h w
ρl = ρl P, cg , cg ,
chg + cw
g = 1,
chl
+
cw
l
(5c)
=1
where νih and νiw are the chemical potential of components h and w in ith phase (i = g, l). Equations
(k)
of state (5b) are written in general form. Functions νg (P, ...) are given. Various versions of these
4
S. Oladyshkin, M. Panfilov
functions may be found in [5], [3]. The detail description of thermodynamic system will be presented
below.
h w
The system (5a) – (5c) includes 6 equations with respect to 7 variables: P , ρg , ρl , chg , cw
g , cl , cl . The
difference between the number of variables and the number of equations, v = 7 − 6 = 1, is called the
thermodynamic variance and determines the number of independent variables. We select the pressure
as an independent variable. As a result the phase densities and concentrations depend only on the
pressure P , which constitute the Gibbs rule of phase.
2.3 Relative permeabilities models
The flow of two-phase system in porous medium depend strongly on the petrophysic closure relations.
We will analyze the influence of the relative permeability on the regimes of hydrogen penetration in
water in porous medium. In this paper we shall consider two principal model of gas penetration: the
model of mobile gas and the model of low mobile gas.
Mobile gas model:
kg (s) = 1 − sα ,
(6a)
β
kl (s) = γ(s − sr )
(6b)
kg (s) = (1 − s)α ,
(7a)
Low mobile gas model:
β
kl (s) = γ(s − sr )
(7b)
3 Hydrogen-water thermodynamic behaviuors
In the previous section we have shown that the hydrodynamics and the thermodynamics may be split
for the hydrogen-water mixture. As the result of such a splitting we obtain a hydrodynamic model (1)
consisted of two equations (for two variables pressure and saturation) and a thermodynamic model (5).
The independent thermodynamic model for the hydrogen-water mixture consists of two equilibrium
equations, two equations of phase state and two normalizing equations for concentrations. In this
section we will describe the mathematical model for h − w two-phase system through thermodynamic
relations.
3.1 Phase equilibrium conditions
The equality of chemical potentials of each component in all the co-existing phases at equilibrium and
described by the equation (5a). The chemical potentials for gas νgh , νgw and liquid νlh , νlw phases of
components h and w can be defined as:
h h
h
P, T, chi , cw
(8)
νih = νip
i + RT ln fi ci ,
w
w w
νiw = νip
P, T, chi , cw
i + RT ln (fi ci ) , i = g, l
where R is the universal gas constant; T and P are the reservoir temperature and pressure; fih and
h
w
and νip
are the chemical potentials of the ideal gas:
fiw are the fugacities in phase i; νip
h
(9)
= Ghp (P, T ) + RT ln chi ,
νip
w
w
w
νip = Gp (P, T ) + RT ln (ci ) , i = g, l
herein Ghp (P, T ) and Gw
p (P, T ) are the chemical potential of the pure components h and w correspondently.
Hydrogen penetration in water through porous medium
5
Therefore, using (8) and (9), the phase equilibrium conditions (5a) lead to a balance between the
fugacities:
h
h w
(10)
fgh P, T, chg , cw
g = fl P, T, cl , cl ,
w
h w
w
h w
fg P, T, cg , cg = fl P, T, cl , cl
The fugacities will be introduced through the equation of state for gas-liquid system, section 3.2,
and will be described in section 3.3.
3.2 Equations of state for the gas-liquid systems
Equations of state are basically developed for pure components [16], [18], but may be applied to
multicomponent systems by using some mixing rules which determine the averaged parameters for
a mixture [14], [12]. The mixing rules take into account the prevailing forces between molecules of
different substances which form the mixture.
The equations of state like Peng-Robinson is usually formulated with respect to the z-factor z
(compressibility) as an implicit function of P , T and concentrations:
zi3 − (1 − Bi )zi2 + (Ai − 3Bi2 − 2Bi )z
−(Ai Bi − Bi2 − Bi3 ) = 0, i=g, l
(11)
where zi is the z-factor of i-th phase; coefficients Ai , Bi depend on pressure P , temperature T and
mixture composition:
Ai = a¯i
P
R2 T 2
,
P
,
Bi = b¯i
RT
i=g, l
(12)
Parameters a¯i and b¯i describe the mixture composition and the mixing rules:
h,w
a¯i = xhi xhi ah,h + xhi xw
i a
h w,h
w h,w
+xw
+ xw
,
i xi a
i xi a
√
kj
kj
k
j
a = a a (1 − δ ), k, j = h, w,
b¯i = xh bh + xw bw , i=g, l
i
i
kj
where δ is known as a binary interaction parameter [5]; xhi and xw
i are the mole fractions of components h and w in the i-th phase (gas or liquid).
Parameters ak and bk for k-th chemical component (h or w) are defined by the following equations:
k
R2Tcr
a = 0.4274800232
k
Pcr
k
2
1+m
k
1−
2
mk = 0.37464 + 1.5422ω k − 0.26992ω k ,
RT k
bk = 0.086640350 kcr , k=h, w
Pcr
T
k
Tcr
2
,
(13a)
(13b)
(13c)
k
k
where Tcr
and Pcr
are the critical fluid temperature and pressure for component k (h, w); ω k is the
acentric Pitcer factor.
The cubic equation (11) for the z-factor may be solved for liquid and vapor phases. Generally three
solutions of equation (11) are obtained. Finally, the smallest root corresponds to liquid, while the
largest root describes vapor.
6
S. Oladyshkin, M. Panfilov
3.3 Fugacities of chemical components
Let us introduce the fugacities to describe the attraction/repulsion between molecules in the hydrogenwater mixture.
The ratio of the fugacity to the pressure is called the fugacity factor φhi , φw
i :
φhi =
fih
,
P xhi
φw
i =
fiw
,
P xw
i
i = g, l
(14)
The fugacity factors are calculated using following relation:
bh
ln φhi = ¯ (zi − 1) − ln(zi − Bi ) −
bi
2 xhi ah,h + xhi ah,w
Ai
bh
√
− ¯
a¯i
bi
2 2Bi
√
zi + ( 2 + 1)Bi
√
· ln
,
zi − ( 2 − 1)Bi
(15)
bw
ln (φw
i ) = ¯ (zi − 1) − ln(zi − Bi ) −
bi
w,h
w,w
2 xw
+ xw
Ai
bw
i a
i a
√
− ¯
a¯i
bi
2 2Bi
√
zi + ( 2 + 1)Bi
√
· ln
zi − ( 2 − 1)Bi
where zi , Ai , Bi , a¯i , b¯i and akj are defined by equations above (i=g, l).
Thus, the fugacity factor of a component depends on the composition and the z-factor of the mixture
k
k
(zi ), on the additional functions (a¯i and b¯i ), on the properties of the pure components (Tcr
, Pcr
, and
ω k ), and on the cross parameters δ kj .
3.4 Thermodynamic closure relations
To close the thermodynamic system we will introduce the relation below.
h
w
Let us introduce a relation between the mole fractions xhi , xw
i and the mass concentrations ci , ci :
mh
,
V
chi =
cw
i =
mw
V
(16)
where V is the volume; mh and mw are the mass of the chemical components h and w, i.e. is the
product of the mole number nk and the molar mass M k :
mh = n h M h ,
mw = n w M w
(17)
Therefore:
nh =
chi V
,
Mh
nw =
cw
i V
Mw
(18)
On the other hand, the mole fractions xhi and xw
i in phase i is the ratio of the mole number for
components h and w to the total number of moles:
xhi =
nh
,
n
xw
i =
nw
n
(19)
Hydrogen penetration in water through porous medium
7
Taking into account the fact that the total number of moles is the sum of mole numbers for each
component, n = nh + nw , we obtain:
xhi =
chi
Mh
cw
chi
i
+
Mh
Mw
,
cw
i
Mw
xw
i =
cw
chi
i
+
Mh
Mw
,
i = g, l
(20)
We also need to use the inverse relation for the concentrations chi and cw
i :
chi =
xhi M h
,
w
xhi M h + xw
i M
cw
i =
w
xw
i M
, i = g, l
w
xhi M h + xw
i M
(21)
The fluid densities can be calculated as:
ρi =
P Mi
,
zRT
i = g, l
(22)
w
where Mi is the molar mass for i-th phase, Mi = xhi M h + xw
i M .
Fig. 1 Calibration of H2 solubility in H2 O: non-correlated model (dashed curve) and correlated model (solid
curve)
3.5 Calibrated thermodynamic model for h-w system
The described mathematical model is based on the compositional approach depended on numbers of a
binary interaction and pseudo individual parameters. Thus, such a theoretical model, which describes
the phase behaviors, needs to be calibrated with experimental data.
Usual thermodynamic conditions for the underground storage of radioactive wastes are the pressure
about 50 bars and the temperature about 20o C. For the calibration we will consider the large pressure
range for the hydrogen-water system, but constant temperature. This constant approximation due to
the small temperature variation from 19.6o C up to 23.3o C around the storage.
The calibration of mathematical model for compositional thermodynamics is based on the fitting
of experimental data. The most often representation of experimental data are known in terms of
the hydrogen solubility in the water: Pray, Schweickert and Minnich [15], Wiebe and Gaddy [19],
(author?) [20], Morrison and Billet [8]. The calibration of PVT-model consists in corrections of
the binary interaction parameters and the pseudo individual parameters. Actually, applying of the
8
S. Oladyshkin, M. Panfilov
phase equilibrium law for the gases as h, He and N e requires the adaptation of the critical constant
of temperature Tc and pressure Pc . For the hydrogen the modification of such pseudo individual
parameters are obtained in [6]:
Tc =
43.6
,
1 + 10.8/T
Pc =
20.2
1 + 21.9/T
Here temperature T is measured in K and pressure P is measured in atm. On the other hand, the
research of the binary interaction parameter reminds the iterative process of the root search for an
nonlinear system.
The calibrated theoretical model is shown on Figure 1 by solid line. Unfortunately, the noncalibrated theoretical model can increase the error and be totally non-realistic (dashed curves on
Figure 1). Finally, the binary interaction parameter for h − w mixture δ h,w is −0.121. The individual
parameters for calibrated PVT are model presented in the Table 1.
Component
Molar Weight
Critical Presssure (bar)
Critical Temperature (K)
Omega A
Omega B
Acentric Factor
Parachors
V Critical (m3 /kg-mole)
Z Critical
Boil Temperature (K)
h
2.016
18.82
42.08
0.45724
0.077796
-0.218
34.0
0.065
0.34965
20.3
w
18.015
220.48
647.3
0.45724
0.077796
0.344
53.1
0.22942
0.056
373.2
Table 1 Individual parameters
3.6 Polynomial form for the thermodynamic variables
In this section we propose the illustrations of hydrogen-water thermodynamic behaviors using the
calibrated data in the range of pressure from 1 up to 100 bars.
The dissolution of hydrogen in water show the linear behavior relative to pressure, and are presented
on Figure 2 (left plot) in terms of the hydrogen concentration in the liquid phase. On the other hand,
the phase transitions provide the non-linear behaviors of the hydrogen concentration from the pressure
in gas phase, the right plot on Figure 2. Let us note, that thermodynamic properties of the liquid such
as density and viscosity were showed pseudo constant behaviors. The gas thermodynamic properties
keep a linear behavior relatively the pressure.
Finally, the calibrated mathematical model of h − w system can be presented in the simple polynomial form for the thermodynamic variables in respect to pressure. The concentration of h in % for
the liquid and the gas phases, the phases densities in KG/M 3 and the phases viscosity in CPOISE are
presented below relative to the pressure in bar:
chl = 0.0002P + 10−5
chg
(%),
(23a)
= 0.0944 ln(P ) + 99.545 (%),
3
(23b)
ρl = 0.0089P + 858.74 (KG/M ),
ρg = 0.0817P + 0.0155 (KG/M 3 ),
(23c)
(23d)
μl = 3 · 10−5 P + 0.3258 (CP OISE),
μg = 8 · 10−9 P 2 + 10−6 P + 0.0082 (CP OISE)
(23e)
(23f)
Thus, the total splitting of hydrodynamics and thermodynamics allows to present in an analytical
form the thermodynamic properties of hydrogen-water two-phase system. In other words, the thermodynamics behaviours introduced through the coefficients in the hydrodynamic equations and hence the
general compositional model can be analysed.
Hydrogen penetration in water through porous medium
9
Fig. 2 Hydrogen concentration in Liquid (left) and in Gas (right) phases
3.7 Error estimation for simplified analytical laws
The properties of the two-phase system can be estimated using the simplified analytical laws. The
analytical expressions allowing calculating the pressure exist such as the law of ideal gas, Henry’s law,
Raoult’s law, Kelvin’s law. Unfortunately, this kind of analytical simplified laws introduces also an
error relative to real properties of the system. In this section, we propose to analyze the domain of
validity of such laws for the presented hydrogen-water two-phase system.
Primarily, the phase equilibrium analysis was started with the ideal gas law using the equation of
state of a hypothetical ideal gas. For the gaseous mixture, on empirical law was observed by Dalton
and is related to the ideal gas laws. Mathematically, the pressure of a mixture of gases can be defined
as the summation of the partial pressures of each individual component in a gas mixture:
Pg = Pgh + Pgw
(24)
where Pgh and Pgw are the water and hydrogen partial pressure in the gas phase:
Pgw =
ρw
g
RT,
Mw
Pgh =
ρhg
RT
Mh
(25)
The Henry’s law statement is that the amount of a gas dissolved by a liquid is proportional to the
pressure of the gas upon the liquid. This law was originally formulated by William Henry in 1803 for
a dilute solutions and low gas pressures:
M h H(T )Pgh = ρhl
(26)
here H(T ) is the Henry’s constant, depending only on temperature.
The Raoult’s law is based on the following formulation: the vapor pressure of a solution of a nonvolatile solute is equal to the vapor pressure of the pure solvent at this temperature multiplied by its
mole fraction. The vapor pressure of the pure solvent depends only on the temperature and therefore
is a constant:
Pgw = pw
g
ρw
st
w
M
ρw
ρh
l +
Mh l
(27)
where pw
g is the vapor pressure of the pure solvent.
Further, the capillary forces influence can be introduced using the Kelvin’s formulation:
Pgw
Pc
w
−M h
ρ
st
RT ρl
= pw
e
g
Mw h
w
ρl + h ρl
M
(28)
10
S. Oladyshkin, M. Panfilov
However, in the current paper we considered a real mixture and obviosly that the laws based on
the ideal gas approach can not be applied for a large pressure range. Thus, we propose the estimation
of the errors for described analytical laws in application to the hydrogen-water two-phase system.
Fig. 3 Absolute end relative errors of Henry’s law for the hydrogen-water two-phase system
We shall use the calibrated data obtained from the compositional approach to calculate the absolute
and relative error of Henry’s law:
h
h
h
εH
(29)
abs = M H(T )Pg − ρl ,
h
H
H
h
h
εrel = 2εabs / M H(T )Pg + ρl
Actually, the relative error of Henry’s law for the hydrogen-water two-phase system is not more than
15%, presented on Figure 3.
At the same way the errors of Raoult’s and Kelvin’s laws can be defined:
h Pc −M
ρw
w
st
w
RT ρl ,
εK
e
w
abs = Pg − pg
M h
w
ρl + h ρl
M
⎛
⎞−1
h Pc
w
−M
ρst
R ⎜ w
w
RT ρl ⎟
e
(30)
εK
⎠
w
rel = 2εabs ⎝Pg + pg
M
h
ρw
+
ρ
l
Mh l
R
where εR
abs and εrel can be calculated for Pc = 0.
The relative error of Kelvin’s and Raoult’s approximations essentially increase from pressure equal
to 30 bar, Figure 4. Evidently, Kelvin’s and Raoult’s laws can not be applied for high pressure range,
even like an approximation. On the other hand, Henrys law gives more accurate analytical approximation for hydrogen-water two-phase mixture.
4 Hydrogen-Water hydrodynamic behaviuors
This section is devoted to analysis of hydrodynamic behaviors in the hydrogen-water two-phase system. The hydrodynamic model was formulated in general case of 3D hydrogen-water flow with phase
transitions, the equations (1). Influence of relative permeabilities on the flow behaviors will be analysed
in this section as well as influence of phase exchange effect and diffusion phenomena. The thermodynamics was introduced through coefficients of the hydrodynamic equations. Such thermodynamics was
shown monovariant and all thermodynamic variables can be present on an analytical form as functions
of pressure.
Hydrogen penetration in water through porous medium
11
Fig. 4 Absolute and relative errors of Raoult’s and Kelvin’s laws for the hydrogen-water two-phase system
4.1 Dimensionless form of flow problem
Let us replace the flow velocities in the equation (1) by Darcy’s law, then the compositional model
takes the following form:
∂ρ
= div ([Ψl + Ψg ] gradP ) − div (Ψl gradPc ) ,
∂t
∂ρ(k)
(k)
gradP
= div Ψl cl + Ψg c(k)
φ
g
∂t
φ
(31a)
(31b)
(k)
−div Ψl cl gradPc
P0
(k)
+div Dl0 sρl gradcl + Dg0 (1 − s)ρg gradc(k)
,
g
P
k=2
here Ψi ≡ ρi Kki /μi is the hydraulic phase conductivity (i = l, g); ρ ≡ ρl s + ρg (1 − s) is the total
(k)
(k)
density; ρ(k) ≡ ρl cl s + ρg cg (1 − s) is the total partial density of the component k.
We will introduce the characteristic values of the length, time, gas pressure and capillary pressure,
viscosities and densities are: L, t∗ , P0 , Pc , μ0g , μ0l , ρ0g and ρ0l ; and we will perform the operations div and
grad in dimensionless space coordinates. Hence we obtained the following dimensionless definitions:
p≡
P
Pc
φρi
Kρi μ0i
, ϕi ≡
, pc ≡
, ψi ≡
,
0
P0
Pc φ ρi
K ρ0i μi
ρ≡
ρ0l μ0g
Pc ρ0l
t∗ 1
≡
,
τ
≡
t/t
,
ω
≡
,ε ≡ ,
,
∗
0
0
0
ρg
ρg μl
t∗ Ca
P0
t∗ ≡
L2 μ0g φ
K P0
D0
,Pe ≡
, σ ≡ l0
0
0
K P0
Dg μg
Dg
Here the parameter t∗ is the characteristic time of perturbation propagation caused by the pressure
variation. The parameter ε is the ratio of the perturbation propagation time t∗ to the characteristic
process time t∗ , while the parameter ω is the ratio of the liquid mobility to the gas mobility.
12
S. Oladyshkin, M. Panfilov
Thus, the full compositional model (1) takes the following dimensionless form:
ε
∂
(ϕl ρs + ϕg (1 − s))
∂τ
ω
div (ψl kl gradpc ) ,
= div ([ψg kg + ωψl kl ] gradp) −
Ca
∂
(N )
)
=
ϕl ρscl + ϕg (1 − s)c(N
ε
g
∂τ
(N )
)
div ψg kg c(N
+ ωψl kl cl
gradp
g
ω
(N )
div ψl kl cl gradpc
−
Ca
ϕg (1 − s)
1
(N )
)
+ div
gradc(N
+
σωψ
sgradc
l
g
l
Pe
p
(32a)
(32b)
The first equation (32a) describes the pressure behaviors and the second one (32b) describe the saturation evolution. In fact, the saturation evolution is caused by the following effects: convection, capillary
influence and diffusion in liquid:
∂ ε ∂s
(N ) ∂p
∼
ψl kl cl
(33)
ω ∂τ
∂x
∂x
(N )
∂c
1 ∂
σ ∂
(N ) ∂pc
ψl s l
−
ψl kl cl
+
Ca ∂x
∂x
P e ∂x
∂x
To analyse the hydrogen migration around the storage of radioactive waste we will consider the 1D
flow problem on a dimensionless domain [x∗ : 1]. The hydrogen is produced due to corrosion of metal
components of the storage and can be described by given function of the flow rate q(τ ) on the source
x∗ . At the same time, the pressure at the exterior bound of the domain (x = 1) can be function of time
p∗ (τ ). We will suppose that at the initial state there are no gas in porous medium i.e. s=1. However
due to the hydrogen production on the source x = x∗ the liquid saturation equal to 0, i.e. s=0. Finally,
the formulation (32) can be completed by the following initial and boundary-value conditions:
s|τ =0 = 1,
p|x=1 = p∗ (τ ),
∂p εq(τ )
(ψg kg + ωψl kl ) =−
∂x x=xs
2
s|x=xs = 0
(34a)
(34b)
(34c)
(34d)
The hydrogen-water two-phase system correspond to the flow around the radioactive waste storage.
The essential properties of this system are following: average absolute permeability K = 0.00005mD,
average reservoir porosity φ = 0.15, temperature T = 298.1500K and initial pressure is 100 bars. The
thermodynamic properties of the system are used from the correlated model presented in the section
3.
4.2 Analytical solution for non-compressible flow
An analytical solution can be constructed for the case of non-compressible flow, i.e. ψl = 1, ψg = 1
and ϕl = 1, ϕg = 1. For the 1D flow, the system (32) takes the form:
∂
∂
∂p
ε
(ρs + 1 − s) =
[kg + ωkl ]
,
(35a)
∂τ
∂x
∂x
∂
∂p
∂pc
∂s
ω ∂
=ω
ερ
kl
−
kl
(35b)
∂τ
∂x
∂x
Ca ∂x
∂x
(N )
(N )
∂c
1 − s ∂cg
1 ∂
+ σωs l
+
P e ∂x
p
∂x
∂x
Hydrogen penetration in water through porous medium
13
0.2
1
0.8
Gas Saturation, sg
Gas Saturation, sg
0.15
0.6
0.1
0.4
0.05
0
0
0.2
5
10
Distance, x
15
20
0
0
1
2
3
Distance, x
4
5
Fig. 5 Analytical (solid line) and Numerical (dashed line) solutions for the non-compressible flow: left plot mobile gas; right plot - low mobile gas
Let us substitute the equation (35b) to the equation (35a), therefore:
∂
∂p
∂pc
∂s
ω ∂
=
−ε
kg
+
kl
∂τ
∂x
∂x
Ca ∂x
∂x
(N )
(N )
∂cl
1 − s ∂cg
1 ∂
+ σωs
−
P e ∂x
p
∂x
∂x
(36)
Let us add the equation (35b) divided by ρ to the last equation (36), so we obtain the following relation:
∂p
∂
ω
kg + kl
=
(37)
∂x
ρ
∂x
(N )
(N )
∂c
1 ∂
1 − s ∂cg
1
+ σωs l
1−
ρ P e ∂x
p
∂x
∂x
∂pc
1 ω ∂
− 1−
kl
ρ Ca ∂x
∂x
In the case when diffusion and capillary effects can be neglected we obtain the following pseudostationary equation for pressure:
∂p
∂
ω
kg + kl
=0
(38)
∂x
ρ
∂x
Finally, substituting the term
∂p
form the equation (39) to (36) we obtain:
∂x
⎞
⎛
∂s
ωq ∂ ⎜
kl
⎟
− 2
⎠=0
⎝
ω
∂τ
∂x
2ρ
kg + kl
ρ
(39)
This means that saturation can be presented in the form of a Buckley-Leverett equation, with flow
ωq
ω
velocity U =
and pseudo-fractional flow function F = kl /(kg + kl ). Thus, the solution to the
ρ
2ρ2
hydrogen-water two-phase problem can be presented in an analytical form for the non-compressible
flow without capillarity end diffusion.
As a first attempt of analysis we proposed to compare the numerical simulation to the analytical
solution, equation (39), for the case of non-compressible flow without mass transfer between phases.
Illustrations of several time instance for the low mobile gas and the mobile gas are presented on Figure
5. The analytical approach match very well the numerical one.
14
S. Oladyshkin, M. Panfilov
4.3 Full concentration approach and pseudo saturation
Such a type of flow as gas penetration in liquid presents a non trivial analysis. One of the most important
question is how to define the separation of the one-phase domain from the two-phase domain. Usually
the initial phase of the system is an one-phase (liquid phase), and in the analyzed problem this liquid
phase mainly consists of water. Actually, if the liquid saturation tends to 1 (not exactly 1), then
formally this means that the system is two-phase system, but physically it can be a two-phase system
as well as an one-phase system! The question is how to define this separation, what is the numerical
error of calculation? This problematic is very relevant for the penetration of mobile gas, especially in
the simple case of non-compressible flow - the left plot on Figure 5. How to interpret the fact that
the asymptotical curve tend to 0 for gas saturation? Is this a physical reality or a numerical error? To
answer on this questions the use of the classical approach for the saturation is not sufficient. In this
paper we propose to introduce a pseudo-saturation function in the system. To reach this goal we will
first introduce the notion on full concentration of the hydrogen:
ch =
ρl chl s + ρg chg (1 − s)
ρl s + ρg (1 − s)
(40)
Then the saturation for liquid phase can be written as:
s=
ρg (chg − ch )
ρl (chl − ch ) + ρg (chg − ch )
(41)
If for a fixed pressure the full concentration of hydrogen ch is less than the concentration of hydrogen
in the liquid phase chl then the system is an one-phase system (liquid phase). The situation is similar
when ch > chg , i.e. the system is one-phase system (gas phase). The system is in two-phase state for
a fixed pressure when the follow condition is valid: chl < ch < chg . However, according to the relation
(41) the saturation s is outside of the domain [0 : 1] if the full concentration of hydrogen ch in a
non-equilibrium state, i.e. outside of the domain [chl : chg ]. This gives us the opportunity to introduce
the new notion of pseudo saturation sp which have the same value as the saturation s in the two-phase
domain [0 : 1], but can also be less than 0 and more than 1 in the one-phase domains (see Figure 6).
Physically, notions of full concentration and pseudo saturation indicate clearly state of the system, i.e.
indicate how far the one-phase system situated from the two-phases state. Thus, this approach defines
exactly separation of the one-phase domain from the two-phases, what is a very topical for mobile gas
penetration (Figure 6).
0.03
Pseudo Gas Saturation, sgp
100 years
0.02
Two Phases
0.01
0
−0.01 10 years
One Phase
−0.02
−0.03
0
5
10
Distance, x
15
Fig. 6 Pseudo gas saturation: one phase and two phases domains
20
Hydrogen penetration in water through porous medium
15
4.4 Gas penetration in liquid through the porous medium
The general compositional flow model of gas penetration in liquid through porous medium was analyzed
numerically (Fortran Code). Using the notion of pseudo saturation influence of diffusion on mobile gas
penetration in liquid was analyzed.Typical evolution of gas saturation (pseudo saturation) for the
hydrogen-water flow around radioactive storage presented on Figure 7. Influence of diffusion increases
the gas penetration with time, for example the supplementary gas penetration can exceeds the value
of 1 meter in 200 years (in low saturated domain especially). Even without diffusion the pure gas
penetration is still fast for low mobile gas. In a case of low mobile gas penetration in liquid through a
porous medium the flow has a clear front, which displaced very slow relatively of mobile gas penetration
in liquid. Evidently, the diffusion is one of essential drives of the flow for large range of saturation.
0.02
Pseudo Gas Saturation, s
p
0.015
250 years
0.01
200 years
0.005
150 years
0
−0.005
100 years
−0.01
50 years
−0.015
−0.02
0
5
10
Distance, x
15
20
Fig. 7 Gas saturation evolution for mobile gas: solid line - without diffusion, dashed line - with diffusion
Mass exchange phenomenon have a small influence on the flow type and essential influence of a
such phenomena appears through phases properties (phases compressibilities). The Impact of phase
transition effect can be illustrated by the difference between two saturations: saturation with the
presence of diffusion effect and saturation without this effect - Figure 8. It is necessary to note, that
phase transitions influence also was introduced non directly, actually through the diffusion term. Thus,
we can conclude that influence only of the direct mass exchange phenomenon have insignificant impact
on gas displacement in liquid, not more than one millimetre for the presented example on Figure 8.
−4
2
x 10
Impact (m)
1
0
−1
−2
0
5
10
15
Distance, x (m)
20
Fig. 8 Phase exchange influence: compressibility impact on the displacement of mobile gas (simulation time
is 150 years)
16
S. Oladyshkin, M. Panfilov
Summary and conclusion
Hydrogen penetration in water through porous medium was analysed in this paper. A two-phase
hydrogen-water compositional model approach was proposed. The present work consists of two principal parts:
The first part deals with the thermodynamic analysis of the hydrogen-water system. The thermodynamic model was calibrated using experimental data of hydrogen solubility in water. The phase
densities, viscosities and concentrations were presented in an analytical form. Moreover, the domain
of validity of simplified analytical laws - such as Henry’s law, Raoult’s law and Kelvin’s law - for the
estimation of phase properties was presented for the analysed system.
The second part of the work deals with hydrogen-water two-phase hydrodynamic behaviours. An
analytical solution for the non-compressible flow was constructed. An approach based on the full
concentration and the pseudo saturation was introduced to define exactly the separation of one-phase
domain from two-phase domain. The general compositional flow model of gas penetration in liquid
through porous medium was analysed numerically. The influence of relative permeabilities on the flow
regimes was analysed. Two principal models were used: the mobile gas model and the low mobile gas
model. The low mobile gas creates a front which displaces with time, but evidently not as fast as in case
of the mobile gas flow. In both cases the influence of diffusion increases with time. However, diffusion
becomes really important for the low mobile gas case, as the penetration accelerates for a large range
of saturation. In contrast to this, the direct mass exchange phenomena has a small influence on the
flow type. Thus, the regime of hydrogen penetration in liquid were shown as very sensitive to the forms
of relative permeability.
Acknowledgements The author would like to express their thanks to the research group MoMaS (Mathematical Modeling and Numerical Simulation for Nuclear Waste Management Problems) for the financial support
of this work.
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