of the permeability of cementitious materials. G. Mayer, F. Jacobs and F,H. Wittmann. Institute for Building Materials, ETH Ziirich, Switzerland. Received 20 July ...
Nuclear Engineering and Design 138 (1992) 171-177 North-Holland
171
Experimental determination and numerical simulation of the permeability of cementitious materials G. Mayer, F. Jacobs a n d F,H. W i t t m a n n Institute for Building Materials, ETH Ziirich, Switzerland Received 20 July 1992
Safety analysis of a repository requires a detailed numerical study of the coupled transport processes of gas and water in the repository. Experimental studies were carried out to measure the transport parameters of different types of normal concrete. The capillary pressure curve, the gas and the water permeabilities were detemined in separate experiments. Especially the influence of the water content of the samples was under investigation. Additionally coupled gas and water flow experiments were carried out and numerically simulated with the measured transport parameters. With the relative permeability curve and the experimental determined pore size distribution it is possible to describe the coupled transport of gas and water through specimens.
I. Introduction
2. Transport equations
In a repository for low and intermediate level radioactive waste the consequences of the formation of gas must be considered [1,2]. It is assumed that mainly hydrogen is formed by corrosion of the waste packages and microbiological degradation of organic materials. From this point of view two consequences are possible: First, a rising gas pressure can damage the construction. Second, contaminated water can be drained out of the repository. For that reason a detailed knowledge of the transport parameters of the host rock, the liner and the backfill material of the waste canisters is necessary. The transport parameters are the gas and water permeability and the capillary pressure as function of the water content. The purpose of this project is (i) to derive each transport parameter in a seperate laboratory experiment, (ii) validate the numerical model by the experimental derived transport parameters and (iii) predict the coupled transport processes of gas and water in the repository and the surrounding hostrock by a numerical simulation. In this study step (i) and (ii) are presented for normal concrete.
To describe transport processes in a repository in a realistic way, the coupled transport of two components here hydrogen and water, where each component can exist in two phases (gas and liquid) must be taken into account [3,4]. First of all, the continuity equation must be fulfilled. For each component the mass balance equation of a finite volume V, surrounded by its surface area Fn can be written in integrated form.
K ~ [water, hydrogen]; M(~): mass per volume of component K, F¢~): flux density of component K, q(~): source term of component K. Each component K exists in two phases (/3 = liquid, gas) where Pt3 is the density of phase /3 and X (~ is the mass fraction of component K present in phase/3. With the porosity ~b and the saturation Se of phase/3 the mass accumulation term for each component K is
u~=ep Correspondence to: Mr. Gerhard Mayer, Swiss Federal Institute of Technology, Ziirich Hfnggerberg, CH-8093 Ziirich, Switzerland.
E
sopox~~.
(2)
/3= liq,gas The saturation Se is the fraction of pore volume,
0 0 2 9 - 5 4 9 3 / 9 2 / $ 0 5 . 0 0 © 1992 - Elsevier Science P u b l i s h e r s B.V. All rights reserved
172
G. Mayer et al.
/
Permeability of cementitious materials
occupied by phase /3. The total mass flux of each component ,¢ is the sum over the two phases: F'")=
E
functions for various materials can be found. Most of them have the form of a polynom and the coefficients must be fitted according to experimental data. In a partial saturated porous media the liquid and gaseous phases share the pore space. A pressure difference between the phases, a so called capillary pressure Pcao exists. It can be described as a function of the saturation of the liquid phase.
(3)
F~ ")"
B = liq,gas
The flux of each phase is described by Darcy's law: (4)
V~") = - k kr'o pt3X(K)( VPO - p~g ) .
ecap(Sliq) = Pgas -- eliq" k is the absolute permeability of the porous material which is independent from the fluid, k~,~ is the relative permeability of phase/3, ~b is the viscosity of phase/3, and g the gravitational acceleration. The product k kr, ~ is called the effective permeability ken ~ of phase /3. The relative permeability kr, ~ is a normalized function of the saturation S t. The knowledge of the characteristic functions k~,~ for each phase in a porous material is necessary to describe the coupled transport of both phases. If a sample is fully saturated by one phase then the relative permeability for this phase is equal to one. Hence the relative permeability for the other phase is zero. In the literature [5,6,7,3] different
(5)
For instant, this pressure is the reason for capillary suction of water into a dry sample of concrete. If the sample is fully water saturated, the capillary pressure is zero and suction takes not place. In the literature [3] different functions for different materials are known. Looking at the pore level of the material, the Laplace equation of capillarity is responsible for the pressure difference. Pcao =
-
(6)
2y/rm"
r m is the mean curvature and 3' the surface tension of the interface between the phases. The mean curvature
GAS
WATER
,-q
= H
I
1 ' ×~× H r-
~
NORMAL CO CRE x W/C 0, 7 o W/C O, 6 W/C 0,5 W/C 0,4
E
J~ r-
~
1 ~
['/
O× j
I
WATER
50 SATURATION
100 [%]
WATER
50 SATURATION
I00 [%]
Fig. 1. Permeability of normal concrete made with different water/cement ratios, measured at different water saturations with (a) gas and (b) water.
G. Mayer et al. / Permeability of cementitious materials is affected by the pore size and the pore shape. In a capillary tube of diameter D equation (6) becomes P~ap = - 4 3 ' cos a / D ,
direction of flow. The compressible gas phase is treated as an ideal gas. Thus integration of (3) gives:
Q( Po)l
(7) keff,gas
where a is the interracial contact angle. This equation is known as the Washburn equation. To sum up it can be said that the description of the coupled transport of gas and liquid through a porous material requires (i) a detailed knowledge of the gas and liquid permeability as a function of the saturation, (ii) the capillary pressure curve in the whole range of the liquid saturation and (iii) the determination of the porosity.
3. E x p e r i m e n t s
The effective permeability to gas and liquid can be calculated from a uniaxiai flow experiments on cylindrical samples. Applying a constant pressure difference ( P 1 - P0) at the end faces of the samples the corresponding volume flux Q through the sample can be measured [8]. Neglecting gravitation in (4) the effective permeability for a noncompressible fluid, here water, can be derived by integration:
Ql
1
keff'water = l/water A P1 - P0"
(8)
l is the length of the sample in the direction of flow and A the cross sectional area perpendicular to the
173
= '/'/gas
A
2Po p? _ p2"
(9)
In each derivation it is assumed, that both the effective permeability and the liquid saturation are constant everywhere in the sample. This assumption is required to treat each experiment as a one component and one phase experiment. Experiments were carried out on normal concrete specimens with water/cement ratios from 0.4 to 0.7 [8]. The normal concrete specimens were stored for 28 days in water. Afterwards they were stored at 35%, 70%, 90% relative humidity up to the 91st day. First gas permeability and second water permeability were measured. The degree of water saturation was calculated from the weight loss after drying at 105°C. In fig. l a the measured effective permeabilities of normal concrete for gas are shown. The permeability strongly depends on the water content of the sample. With increasing water saturation the permeability to gas decreases. The influence of the water/cement ratio is small compared to the influence of the water saturation. If the water saturation exceeds a certain limit, gas flow can only be measured if the inlet pressure is higher than the threshold pressure. The corresponding gas saturation is called the residual gas saturation Sgas,r~s. This phenomenon can be explained by the porous structure of the material. Above a certain water content, there is no continuous pathway for gas to flow through the concrete. Hence, some of the pores filled
Table 1 Transport parameters for coupled flow of hydrogen and water through normal concrete Required material parameters for TOUGH absolute gas permeability k~as absolute water permeability kliq relative gas permeability kr,gas relative water permeability kr,li q
capillary pressure curve Pcap
Chosen parameters for normal concrete
krgas = ( 1 - Sl*q)2(1- S . 2 ' k ' _ c.4 a liq ] r,liq - °liq Sli*q = (Sii q - Sliq,res)/(1
kgas kliq
1.0 × 1.0- t6 m 2 1.2 × 10-18 m2
Sliq,res Sgas,res
30% a 18%
P,
1.0× 10s Pa
P,
1.2 × 106 Pa
- Sliq,re s - Sgas,res)
[ 1 - Sjiq ~ l/~
1.3 porosity ~b a Estimated.
13%
174
G. Mayer et al. / Permeability of cementitious materials"
with water must be emptied to get a continuous pathway through the sample. This coupled gas and water flow experiments will be discussed in more detail and simulated in the next section. We found that the gas permeability of normal concrete can be described with a relative permeability curve, proposed by Corey [3]. kr,gas = ( 1 - - Sliq) . 2(1 - S,*q2)
(10)
The normalized liquid saturation Sji* is calculated by (Sliq
* -
Sliq I
-- Sliq,res)
(1 --Sli q .... --Sg ...... )
,
(11)
where Sliqae~ and Sg~s,,e~ are the residual water and gas saturation. In table 1 the parameters are listed which are necessary to describe the coupled transport of hydrogen and water in normal concrete. The water permeability was measured on nearly totally water saturated samples (fig. lb). Experiments on partial saturated samples are much more difficult, because capillary suction must be prevented by a gas pressure. For the liquid phase Corey [6] proposed: kr,liq -- S*41iq-
(12)
So far no measurements have been performed to confirm this assumption for normal concrete. The measured water permeability for normal concrete is more
742
"--_ "'~
-..
U3
cO
% £ f.i c~
g= CC3 Ld Ca-
C:2 C:2
WATER
50 SATURATION
10()
[~)
Fig. 2. Capillary pressure of normal concrete for hydrogen/water calculated from the measured pore size distribution with mercury intrusion porosimetry.
than two orders of magnitude lower than the gas permeability (table 1). A swelling of hydration particles in contact with water, chemical reactions or a blocking of pores due to advection could be the reason. The capillary pressure can be measured with the porous diaphragm method (Dullien [9], Corey [4]), but it is very time consuming for dense materials like concrete. As a first approximation, the capillary pressure is only a function of the porous structure of the material and the surface tension between gaseous and liquid phase. Using the pore size distribution, measured with mercury intrusion porosimetry, the capillary pressure P c a p function for hydrogen/water shown in fig. 2 can be calculated by using the Washburn equation (7). y is the surface tension of water and a is the interfacial contact angle. The approximated capillary pressure function can be described in a form proposed by Narasimhan [10].
Pcap( Sliq) = pi _ pj
/
--~
~
°,iq/ Sliq ]
\l/v .
(13)
The parameters Pi, P~ and v for normal concrete are fitted from the experimental curve (table 1).
4. C o m p u t e r
simulations
In the following chapter applications of computer simulations are presented. The system of partial differential equations (eqs. (1)-(13)) describing the coupled gas and liquid transport are solved with the finite volume (integrated finite difference) method, implemented in the F O R T R A N code T O U G H [3]. Minor modifications were made to the code to implement both absolut permeabilities to gas and water. One dimensional coupled gas and water flow experiments through partial water saturated concrete analog to the laboratory experiments were simulated. For such a simulation the transport parameters, the geometry of the sample, the initial conditions and the boundary conditions must be fixed. The transport parameters are listed in table 1. Assuming that the choosen functions of the relative permeabilities and the capillary pressure are correct, eight transport parameters are needed for input to characterize the porous material with respect to the coupled transport of hydrogen and water. The geometry of the cylindrical concrete samples (diameter: 15 cm, length: 6 cm) was modelled in a one-dimensional mesh of 30 elements, each with the transport parameters determined in the experiments (table 1).
G. Mayer et al. / Permeability of cementitious materials The initial conditions of the samples are fixed choosing an initial water saturation and an initial gas pressure of the sample. In each simulation the initial gas pressure in the sample is, analog to the storage conditions fixed to atmospheric pressure. Different initial water saturations of the samples were simulated. A constant pressure at the inlet P1 and a constant pressure P0 at the outlet of the sample can be simulated as follows. A very large volume of the boundary elements ensures that their thermodynamic variables (pressure) remain constant over the simulated time. First of all the gas flow through a sample with an initial water saturation of 84% is simulated. The initial gas saturation (1 - Sliq) is below the residual gassaturation Sga~,~s and initially prohibits a flow of gas. A gas flow through the sample can only be measured, if the applied pressure difference exceeds the threshold pressure. The first situation is simulated for a pressure difference of five bars, Six bars are applied at the inlet, one bar (atmosperic pressure) at the outlet. In figure 3a it is shown that the gas flux first decreases and after a time of 5000 s the
175
(a) L
_9
"
m
on
T-~
CN ~-
± 100
10000
TIME
[5]
Ib) tt~ ca
kl
,
