5. Diffusion-convection transport in cell experiments with rotational flow
5.5. C ELL EXPERIMENT RESULTS OF CRACKED BRICKS
5.5.3. Comparison between measured and simulated data
5.5.2.3.2 Reservoir A and B
Also here the liquid calculations are calculated assuming stationary conditions and constant heights in the reservoirs over the small time step. The salt balance is implemented with an iterative scheme following Cranck-Nicolson.
5.5.2.3.3 Schematic program plan
3dim mesh: nex.ney.nez = ne Ci, CA,CB, hA, hB, Δt
1 I. Calculating: pi, Qv,ijat time t
, 6
0 1 i j
v j
Q =
∑
= Qvij =K A grad pij. ij( )
ij +K Aij. .ijρij1 II. Calculating: Ci,t+Δt gravity 1) explicit step: salt-fluxes
a) diffusion expl
b) convection expl
=>
6
, ,exp ,
1
'. . ( )
D i l ij ij t ij
j
Q D A grad C
=
=
∑
−6 , ,
, ,exp ,
1 vol t ij.
C i l t ij
j ij
Q Q C
= ρ
=
∑
2) implicit step: salt-fluxes a) diffusion impl
b) convection impl
=>
(
, ,exp , ,exp)
, ,1 ,
D i l C i l
i t t i t
por
Q Q t
C C
+Δ V
+ Δ
= +
( )
( )( )
(
, , , , , ,exp , ,exp)
, , 1 ,
D i impl C i impl 1 D i l C i l
i t t k i t
por
Q Q Q Q t
C C
V
α α
+Δ +
+ + − + Δ
= +
6
, , , ,
1
'. . ( )
D i impl ij ij t t ij k
j
Q D A grad C+Δ
=
=
∑
−6 , ,
, , , ,
1
.
vol t ij
C i impl t t ij k
j ij
Q Q C
ρ +Δ
=
=
∑
( )
, 2
, , 1 , ,
1 2 , 1 t t k ne
i t t k i t t k
i ne
i i
C C
C
ε
+Δ
+Δ + +Δ
=
=
−
∑ <
∑
3) check convergence
NOT OK
OK!
k< 40 k> 40 t = t+Δt ; Δt= Δt x1.2
Δt = Δt x0.8 ; k=1
k=k+1
t
t+Δt
3) check convergence 2) implicit step: salt-fluxes 1) explicit step: salt-fluxes
First the evolution in time of the concentration in the reservoirs is discussed. The simulations of the concentration in the reservoirs for both tested cracked samples (Fig.
5.27) show a good agreement with the measured data. However an extra simulation shows clearly that the concentration change in the reservoirs is mainly determined by fast convective transport through the crack (Fig. 5.32). When comparing the concentrations in the reservoirs using an isolated crack (crack width of 212 µm, Table 5.3) (no presence of material), nearly the same result is found as in the previous case with presence of material. In short, all the complex patterns of fluxes in the matrix have almost no influence on the measured concentrations in the reservoirs. This leads to the conclusion that the crack is dominating the whole transport process.
0 0.2 0.4 0.6 0.8 1
0 5 10 15
Time (days)
Concentration (M)
sim cracked brick D+C sim isolated crack D+C
Fig. 5.32 Simulated concentrations in both cells for a cracked brick (thin line) and the comparison when the same crack is isolated (bold gray line)
Therefore in a second phase to learn more about the slow transport in brick material, the measured concentrations inside the brick matrix (Fig. 5.29) are compared with the simulations (Fig. 5.34, Fig. 5.35). To facilitate the comparison the measured data are redrawn in Fig. 5.33. Fig. 5.34 shows the simulated concentration distribution for the three investigated surfaces using the same colourbar as in Fig. 5.33. To make the comparison with Fig. 5.33, dots are plotted in the middle of the cut pieces. The main patterns discovered in the measured distributions are found in the simulations. There is a pattern due to rotational flow in the first surface and the dominating diffusion fronts from left and right in the last surface; also the retardation in the middle (between the two reservoirs) at the top of the sample, due to the slow diffusion transport process into the brick, is visible in all three surfaces. Nevertheless, still a discrepancy between the measured and the simulated concentrations remains. The simulated concentrations are higher than the measured ones.
Running new simulations with a lower permeability value shows that convection does not play an important role inside the brick. Reducing the diffusion coefficient on the other
hand results in different concentration distributions. The best fitting results are obtained by reducing the diffusion coefficient to half of the value (Fig. 5.35).
Vertical surface (1) next to the cracked surface
Vertical surface (2) in the middle
Vertical surface (3) farthest away from the crack
Fig. 5.33 Concentrations (mol/m³) of the solution in the pores based on the chemical analysis of the brick pieces from the cracked brick with a crack width of 266 µm
Vertical surface (1) next to the cracked surface
Vertical surface (2) in the middle
Vertical surface (3) farthest away from the crack
Fig. 5.34 Simulated concentration distributions (mol/m³).
Vertical surface (1) next to the cracked surface
Vertical surface (2) in the middle
Vertical surface (3) farthest away from the crack
Fig. 5.35 Simulated concentration distributions (mol/m³) after dividing the diffusion coefficient by a factor of two.
A number of uncertainties can be mentioned to explain the deviation between the measured and the simulated values. First, the experimental analysis introduces deviations. It takes some time to stop the test, remove the sample and saw it. This delay can result in a loss of solution due to migration of salts towards the surfaces.
During the drying process salt can also be lost. That might explain the lower values of the chemical analysis. Further, a global porosity for the brick sample is determined and assumed to be constant, where in reality different local porosity is present. The same remark can be made about the tortuosity. These pore variations have consequently an influence on permeability and diffusion coefficient distribution over the sample, which is now considered to be homogenous. Moreover, as mentioned in chapter 2 the concentration dependency of the diffusion coefficient in free solution, which is the basis of the calculation of the diffusion coefficient in the brick material, can show some deviation [BUC-00]. Additionally according to Pivonka [PIV-04], an increase of viscosity of the layered water network along the pore surface could be the reason for the differences between measured and simulated data.
One can wonder why the dependence of the viscosity on the salt concentration is not taken into account in the simulations. The reason is that the variation of the viscosity in function of the salt concentration is negligible (Fig. 5.36). However, Pinvonka [PIV-04]
mentioned that the viscosity phenomenon itself could be important. Nevertheless no adequate information is available on the viscosity dependence on the layering of water along the pores in function of the liquid velocity. More research on this point is needed.
0 0.0005 0.001 0.0015 0.002 0.0025
0 1 2 3 4 5
Concentration NaCl (M)
Viscosity [Pa.s]
20°C 25°C
Fig. 5.36 Viscosity of NaCl in function of the concentration [LOB-84]