- Email: [email protected]
Effect of heterogeneity scale on chemical grouting in porous media
891
Effect o f Heterogeneity Scale on Chemical Grouting in Porous M e d i a Tirupati Bolisetti* and Stanley Reitsma Civil and Environmental Engineering, University of Windsor, Windsor, N9B 3P4, Canada Performance of chemical grouting as a containment technology is investigated through a numerical modeling study. A new grouting model that couples MODFLOW [1 ] and RT3D [2] combined new modules for the gelling process is presented. The present study is aimed at understanding the role of heterogeneity on grout curtain performance. The paper addresses two important issues related to heterogeneity and grid resolution: (1) relative influence of small-scale heterogeneity and large-scale heterogeneity, and (2) effect of numerical grid resolution on predicted curtain performance. Monte-Carlo simulations with statistically equivalent heterogeneous hydraulic conductivity (K) fields are conducted to assess the uncertainty of the grout performance. Grid resolution adequacy is determined by generation of heterogeneous K-fields at a coarse resolution and completing simulations using grid resolution equal to the K-field resolution and then refining that grid up to five times. Comparison of results at different grid resolution provides estimates of increased error associated with grid coarsening. Using the optimal grid resolution, effect of large-scale and small-scale heterogeneity is explored using simulations that incorporate primary and secondary heterogeneity each with different correlation lengths and variability and comparing to simulations having only large-scale heterogeneity. Grout curtain performance is assessed by simulation of grout injection, determination of post-grouted conductivity field, and assessment of overall grout curtain hydraulic conductivity using a flow model. 1. INTRODUCTION Grouting is a process in which a material is injected into a soil or rock formation to change the physical characteristics of the formation [3]. Grouts are classified as particulate grouts and chemical grouts based on the material used in grouting. Cemetitious, soil-bentonite slurry walls, etc., are called particulate grouts, where as silicates, acrylamides, etc., are called chemical grouts. Chemical grouting, the focus of this study, is the process of injecting a chemically reactive solution that behaves as a fluid but reacts after a predetermined time to form a solid, semisolid, or gel. The solid or gel reduces the hydraulic conductivity and contains the flow through the pores. The grouted barriers should be continuous and should not have any preferential flow channels to effectively contain the wastes. Pearlman [4] raised two concerns with the subsurface barriers: uncertainty about hydraulic conductivity of the installed barrier and barrier continuity. Heterogeneity is the main reason for discontinuous and non-uniform formation of the barrier. Hence there is a strong need to study the grouting process and grout curtain formation in heterogeneous material. Grouted barrier performance in arresting the flow in heterogeneous porous media is the focus of the present study. *On study leave from Regional Research Laboratory(CSIR),Bhopal, India
892 The main objectives of the present study are: (i) to determine the grid resolution adequacy, and (ii) to assess the effect of inclusion or exclusion of small-scale heterogeneities along with large-scale heterogeneities for the silicate grouted barriers. The performance is assessed using the post-grouted hydraulic conductivity field. Monte-Carlo simulations, with statistically equivalent heterogeneous hydraulic conductivity (K) fields, are conducted to assess the uncertainty of the grout performance. Log-normal K-fields with different correlation lengths and variability are generated using sequential gaussian simulation (SGSim) software [5]. A newly developed model is presented. MODFLOW is used to determine the flow field and RT3D to simulate the injection process, and determine gel age and gel viscosity. Gel viscosity is indirectly incorporated in MODFLOW by changing the effective hydraulic conductivity in each cell based on gel age and concentration. The paper presents some of the preliminary results of the numerical study. 2. MODELING BARRIER WALLS: REVIEW
Modeling of silicate grouting as a containment technology has been conducted by several researchers. Finsterle et al. [6] presented modeling strategy of grouting and proposed equations for gelling. Moridis et al. [7] studied the problem of grouting and compared standard engineering design with optimization design for their performance in arresting contaminant release. Tachavises and Benson [8] numerically studied the effect of defects of predefined sizes in cutoff walls. Sullivan et al. [9] monitored a soil/cement barrier emplaced around a buried drum at Hanford geotechnical test facility using perfluorocarbon tracer (PFT). They analyzed the tracer data through numerical modeling to determine the integrity of the barrier. They observed small scale breaches of few centimeters size. The uncertainty through a Monte-Carlo analysis can provide some insight into the problem. In Monte-Carlo simulation approach, large number of independent conductivity realizations are generated and the flow and transport equations for each of them is solved. 3. MATHEMATICAL MODEL In the present study, chemical grouting of sodium silicate solution into porous media is simulated by coupling MODFLOW and RT3D. MODFLOW is a three-dimensional groundwater flow simulator and RT3D is a modular computer code for Reactive multi-species Transport in 3-Dimensional groundwater aquifers. Using velocity fields generated by MODFLOW, transport of grout is solved using RT3D. The velocity field is then updated at regular intervals using MODFLOW and transport computation is repeated. When the grout is injected, the viscosity changes with concentration of grout and age of grout. In order to make RT3D suitable for this application, two reaction modules are added to RT3D. One module (GROUT_AGE) keeps track of grout age. The other (GROUT_GEL) calculates grout viscosity and modifies the hydraulic conductivity. MODFLOW is called from RT3D periodically to update the flow field using the changing K-field. The details on governing equations and solution techniques related to MODFLOW and RT3D are given in their respective references. 3.1 Gelling module (GROUT_GEL) Gel time is the interval between initial mixing of the grout components and formation of the gel. Gel time can be adjusted by varying the composition of grout. As the grout gels,
893
viscosity increases. For some grouts like silicates the viscosity gradually changes, whereas for acrylamides the viscosity remains fairly constant until the gel time is reached and then increases rapidly. The viscosity increase with time and concentration is described by gel time curve [6, 7]. The gel viscosity [6, 7] is given as
lage, = )', + r2 exp(y3t)
(1)
where flgel is gel viscosity; t is the time (s) and ~,~, ~'2 and 3'3 are fitting parameters. The parameters, 7~, ~'2 and % of 2.0, 1.0 and 0.00659 respectively are used here and are representative of a silicate gel with gel time of 15 minutes. During injection silicate grout mixes with water. The mixing process reduces the viscosity of the solution and may be described by linear and power law mixing rules [6, 7]. The linear mixing rule is given by gel IUl = X l /lgel "-I-(1
-
(2)
X S )/~w
where pt is the mixture viscosity;/.t~ is water viscosity; and Xz gel is concentration of gel. However in the present study the equation is modified to prevent gelling below a certain minimum concentration level and is given as
p, =
fl gel ( x ge` -- Xmin )
(1- Xmin)
+ (1 - Xtg~' + Xmin )i/,/w
(3)
where Xmi, is minimum concentration below which gelling does not occur. We use Xmi, equal to 5%. Having calculated the grout viscosity in each grid, the effective hydraulic conductivity, ggrouted, is calculated as g original Kgrouted = ~ (4) /-tt
3.2 Grout Age(GROUT_AGE) Grout viscosity increases with time as per Eq.1. Hence the age of the grout needs to be tracked. This is implemented as a second reaction module in RT3D. The following equation is proposed for the age of gel. 0(grout age) -- exp I . Ot
.
.
.-10-9.
l(max((X~el--Xmi
.
n *0.9)
(5) 10-41/3
This equation lets the gel age only when the concentration is higher than Xmi, and water does not age. The equation is introduced because without it grout injection into aged water resulted in premature gelling.
894 4. NUMERICAL CASE STUDY A two-dimensional horizontal test problem having dimensions of 420 cm x 240 cm is used to study the issues discussed above. Grid resolution tests are conducted to identify the level of refinement needed to adequately model the grouting process. Having identified the grid resolution, the effect of correlation scale and variability are investigated. Effect of considering only primary (large-scale) and, primary and secondary (small-scale) heterogeneities with respect to grout barrier performance, are also investigated. 4.1 Grid resolution test
The domain is discretized into three levels of grids viz., 42x24 grids (10 cm x 10 cm), 84x48 grids (5 c m x 5 cm) and 210x120 (2 cm x 2 cm) grids. One hundred conductivity fields are generated for Monte-Carlo simulations using SGSim program at the coarsest grid (42x 24). The same coarse conductivity field, is mapped on to the fine grids. A correlation length of 100 cm (equivalent to 10 grids on 42x24 grid), mean hydraulic conductivity of 10.2 cm/s and a variability of 0.2 is adopted. A staggered injection [10] as shown in Fig.la is chosen. The injection wells are shown in blocks and numbers near the injection site indicate the sequence of injection. The injection wells are implemented as a general head boundary, under a pressure head of 600 cm. The area of the injection wells is kept same for all grids. Area of injection is reduced to half on the bottom and top boundaries to maintain symmetry. Individual wells are injected for 15 minutes following the respective time sequence shown in Figure 1 (i.e., 1,2,3,4, which would extend the wall below 10, 5, etc.). The additional time allowed for wells 4, 5, 6, 7, etc., simulated continued extension of the wall and allowed gelling of injected grout before infilling. Total simulation time extends simulated well beyond the last injection to allow the grout to completely gel. Hydraulic conductivity of the grout barrier is determined at end of the grout simulation using MODFLOW. A separate model is set up using the post-grouted K-fields to obtain flux through the grout walls for each of 100 realizations. 4.2 Correlation scale test
In the present investigation, effect of correlation length and variability on grout performance is investigated on the 84x48 grid. For this study, conductivity fields are generated using correlation lengths ()~) of 0.0 m, 1.0 m and 2.0 m. To study the combined effect of large-scale and small-scale heterogeneity, conductivity fields are generated with small scale correlation length ()~s) of 0.2 m super imposed on large scale correlation length ()~l) of 1.0 m. Similarly, to study the effect of variability, three conductivity scenarios are generated with a standard deviation of 0.00 (homogeneous case), 0.25 and 0.50 for the correlation length of 1.0 m. The mean is kept at 1.0xl0 -2 cm/s. Results are presented as Keffective calculated by assuming wall thickness in 1.2 m. The ensemble statistics of Keffective from grouted K-fields over 100 realizations are presented in Table 1. As an illustration of effect of variability, grout concentration distribution in homogeneous K-field and a K-field with a standard deviation of 0.5 is as shown in Fig. 2.
895
=
I
D
!I
=
::!
D 2
~
(a) 10 cm x 10 cm grid Flux: 6.25 xl0 "6 cm3/s Keffective:4.56xl 0 "8 cm/s
,.
1~~~
!,i..i '.b ...
.
,:-~.
~
.......
t.:.
,ig
... I.
,'~ .,~,....~~
9
0.05
0.10
-
.
':
(b) 5 cm x 5 cm grid Flux" 5.40 xl 0 .6 cm3/s Keffective" 3.94xl 0 .8 cm/s
.~.~,"
(c) 2 cm x 2 cm grid Flux" 5.75 xl 0 "6 cm3/s Keffective" 4.19x 10 s cm/s
~..
0.15
0.20
0.25
0.30
Figure 1. Grout concentration distribution for different grid resolutions for a selected realization. Blocks indicate injection wells and numbers indicate the sequence of injection. Concentration is in fraction by volume. Injection concentration is 30 %
896
(a) Homogeneous K-field Flux: 8.75xl 0-6 cm3/s Keffective:6.38xl0-Scm/s
,....
,~:,,~
.
'
~,
~-.
:...~
~
(b)Standard Deviation-0.5 Flux" 12.81x 10-6 cm3/s Keffective:9.34xl0"8 cm/s
''
......
' ....
:~,',:.~,~:,::,:~;'~|
'.'i'
.. ;:....';t:'.~ ,": <: .........
~,.,. "...... '~....... i................... ~,~.:~,~,,.,.,.,M ,, ~ ....... I
0.05
0.10
0.15
0.20
: . . . . . . . . . . .
.
I
l
0.25
0.30
I
Fig.2: Grout concentration distribution in (a) homogeneous and (b) variable conductivity fields simulated on 84x48 grid (i.e., 5 c m x 5 cm grid block). Concentration is in % by volume. Table 1" Ensemble Statistics of Keg-ect~ve(x 10 .8 cm/s) from grouted K-fields on 84 x 48 grid Standard Deviation of K" 0.25 )~=0m )~=1 m Z = 2 m
Zl=l m )~s= 0.2 m
Correlation Length of K: 1.0 m cy = 0.0 or=0.2 ~ = 0 . 2 5 Homogeneous
or=0.5
Mean
2.829
2.953
3.201
2.895
6.380
2.888
2.953
S.D.
0.066
0.357
0.984
0.211
-
0.241
0.357
1.181
Minimum Maximum Median
2.698
2.406
2.297
2.443
-
2.472
2.406
2.523
3.011
4.681
8.896
3.449
-
3.938
4.681
9.340
2.815
2.924
2.975
2.895
-
2.858
2.924
3.588
3.908
)~= Correlation length, suffix 1 and s indicate large-scale and small-scale; c~=standard deviation of hydraulic conductivity field; S.D.: Ensemble standard deviation of Keffec,ve
897 5. DISCUSSION
Effective conductivities of the grouted barrier is considered as a metric of performance. The fluxes obtained for 210x120 grid are assumed to be the best possible estimates. The fluxes for the different grid resolutions indicate that the 42x24 grid normally yielded higher fluxes. The fluxes obtained for 84x48 grid are on average within 10 % of fluxes obtained for 210x120 grid. Therefore, 84x48 grid is selected for the heterogeneity studies to achieve the similar accuracy of 210x120 grid with significantly less computational effort. Concentration distribution in Fig.1 is presented on the discretized domain rather than contours for better visualization of fronts. As expected, significant numerical dispersion is observed in the coarsest grid resulting in overlapping of the grout bulbs. However in the fine grids, seams are clearly seen. The grout in both the fine grids is found to move farther from the injection well towards boundaries compared to the coarsest grid. Convergence problems have been observed for the MODFLOW simulation on post-grouted conductivity fields for the fine grid resolution which necessitated higher number of iterations in MODFLOW. The statistics presented in Table 1 demonstrate that the Keffective of the grouted barriers is increasing with increase in correlation length and variability. When the small-scale correlation length of 0.20 m is incorporated along with the large scale correlation length of 1.0 m, the fluxes, and hence Keffec,ve, obtained are between the corresponding values obtained for 0.0 m and that obtained for 1.0 m. The small-scale heterogeneity results in more randomness and therefore results approach those from the random field. The ensemble averages stabilized for the correlation lengths of 0.0 m, 1.0 m and combined correlation length in about 40 realizations. Whereas for the case of correlation length of 2.0 m, the ensemble averages fluctuate until around 80 realizations. Thus more realizations are necessary for Monte-Carlo simulation for higher correlation length and larger standard deviation. A similar trend is observed for the increasing standard deviation in K-field. Figure 2 presents the concentration plot for homogeneous case and standard deviation of 0.5. In the homogeneous case, the grout bulbs formed as near circles and the grout bulbs connect. Due to this, the centerline wells could not inject as much grout as the wells did in the case of heterogeneous case. Also, it may be observed that there are larger seams between the grout bulbs in heterogeneous case, resulting in higher effective conductivity. 6. CONCLUSIONS A model approach for simulating chemical grouting, developed by combining RT3D and MODFLOW, is presented. Reaction modules, GROUT_AGE and GROUT_GEL, are developed to track the age of the grout and determine the reduced hydraulic conductivity due to increase in gel viscosity. Grid of 84 x 48 is an optimal resolution between accuracy and computational effort for the present numerical modeling study. Fine grid resolution resulted in better grout barrier compared to that of coarse grid resolution. Conductivity fields generated using higher correlation lengths and higher standard deviations resulted in higher fluxes which means variability is affecting the grout barrier formation. ACKNOWLEDGEMENTS The first author thanks the Director, RRL and CSIR for sanctioning study leave. The funding for this research has been provided through National Science and Engineering Research
898 Council grant and Ontario Graduate Scholarship for Science and Technology and is gratefully acknowledged. REFERENCES
1. M.D.McDonald and A.W.Harbaugh, A modular three-dimensional finite difference flow model, Techniques in Water Resources Investigations of the U.S. Geological Survey, Book 6., (1996) 586. 2. T.P. Clement, RT3D, A modular computer code for simulating Reactive multi-species Transport in 3-Dimensional groundwater aquifers, Pacific Northwest National Laboratory, Richland, WA, USA, 1998. 3. American Society of Civil Engineers, Chemical grouting, Technical engineering and design guides as adopted from Army Corps of Engrs., No 24, 1997. 4. L.Pearlman, Subsurface containment and monitoring systems: Barriers and beyond, OSWER, USEPA, Washington, http://www.epa.or~swertiol/remed.htm, 1999 5. C.Deutsh and A.Journel, GSLIB: Geostatistical software library and users guide, 2nd Edition, Oxford Press, NY, 1998. 6. S.Finsterle, S., C.M.Oldenburg, A.L.James, K.Pruess, and G.J.Moridis, Intl Containment Tech. Conf. and Exhibition, Feb 9-12, 1997, pp.438. 7. G.J.Moridis, S.Finsterle, J.Heiser, Water Resour. Res., 35 (1999) 2937. 8. C.Tachavises and C.H.Benson, In situ remediation of the geoenvironment, ASCE Geotechnical special publication no. 71, Reston, VA (1997) 168. 9. T.M.Sullivan, J.Heiser, A.Gard and G.Senum, J. of Environ. Engg., 124 (1998) 490. 10. R.H.Karol, Chemical grouting, Marcel and Dekker Inc., NY, 1990.
Effect o f Heterogeneity Scale on Chemical Grouting in Porous M e d i a Tirupati Bolisetti* and Stanley Reitsma Civil and Environmental Engineering, University of Windsor, Windsor, N9B 3P4, Canada Performance of chemical grouting as a containment technology is investigated through a numerical modeling study. A new grouting model that couples MODFLOW [1 ] and RT3D [2] combined new modules for the gelling process is presented. The present study is aimed at understanding the role of heterogeneity on grout curtain performance. The paper addresses two important issues related to heterogeneity and grid resolution: (1) relative influence of small-scale heterogeneity and large-scale heterogeneity, and (2) effect of numerical grid resolution on predicted curtain performance. Monte-Carlo simulations with statistically equivalent heterogeneous hydraulic conductivity (K) fields are conducted to assess the uncertainty of the grout performance. Grid resolution adequacy is determined by generation of heterogeneous K-fields at a coarse resolution and completing simulations using grid resolution equal to the K-field resolution and then refining that grid up to five times. Comparison of results at different grid resolution provides estimates of increased error associated with grid coarsening. Using the optimal grid resolution, effect of large-scale and small-scale heterogeneity is explored using simulations that incorporate primary and secondary heterogeneity each with different correlation lengths and variability and comparing to simulations having only large-scale heterogeneity. Grout curtain performance is assessed by simulation of grout injection, determination of post-grouted conductivity field, and assessment of overall grout curtain hydraulic conductivity using a flow model. 1. INTRODUCTION Grouting is a process in which a material is injected into a soil or rock formation to change the physical characteristics of the formation [3]. Grouts are classified as particulate grouts and chemical grouts based on the material used in grouting. Cemetitious, soil-bentonite slurry walls, etc., are called particulate grouts, where as silicates, acrylamides, etc., are called chemical grouts. Chemical grouting, the focus of this study, is the process of injecting a chemically reactive solution that behaves as a fluid but reacts after a predetermined time to form a solid, semisolid, or gel. The solid or gel reduces the hydraulic conductivity and contains the flow through the pores. The grouted barriers should be continuous and should not have any preferential flow channels to effectively contain the wastes. Pearlman [4] raised two concerns with the subsurface barriers: uncertainty about hydraulic conductivity of the installed barrier and barrier continuity. Heterogeneity is the main reason for discontinuous and non-uniform formation of the barrier. Hence there is a strong need to study the grouting process and grout curtain formation in heterogeneous material. Grouted barrier performance in arresting the flow in heterogeneous porous media is the focus of the present study. *On study leave from Regional Research Laboratory(CSIR),Bhopal, India
892 The main objectives of the present study are: (i) to determine the grid resolution adequacy, and (ii) to assess the effect of inclusion or exclusion of small-scale heterogeneities along with large-scale heterogeneities for the silicate grouted barriers. The performance is assessed using the post-grouted hydraulic conductivity field. Monte-Carlo simulations, with statistically equivalent heterogeneous hydraulic conductivity (K) fields, are conducted to assess the uncertainty of the grout performance. Log-normal K-fields with different correlation lengths and variability are generated using sequential gaussian simulation (SGSim) software [5]. A newly developed model is presented. MODFLOW is used to determine the flow field and RT3D to simulate the injection process, and determine gel age and gel viscosity. Gel viscosity is indirectly incorporated in MODFLOW by changing the effective hydraulic conductivity in each cell based on gel age and concentration. The paper presents some of the preliminary results of the numerical study. 2. MODELING BARRIER WALLS: REVIEW
Modeling of silicate grouting as a containment technology has been conducted by several researchers. Finsterle et al. [6] presented modeling strategy of grouting and proposed equations for gelling. Moridis et al. [7] studied the problem of grouting and compared standard engineering design with optimization design for their performance in arresting contaminant release. Tachavises and Benson [8] numerically studied the effect of defects of predefined sizes in cutoff walls. Sullivan et al. [9] monitored a soil/cement barrier emplaced around a buried drum at Hanford geotechnical test facility using perfluorocarbon tracer (PFT). They analyzed the tracer data through numerical modeling to determine the integrity of the barrier. They observed small scale breaches of few centimeters size. The uncertainty through a Monte-Carlo analysis can provide some insight into the problem. In Monte-Carlo simulation approach, large number of independent conductivity realizations are generated and the flow and transport equations for each of them is solved. 3. MATHEMATICAL MODEL In the present study, chemical grouting of sodium silicate solution into porous media is simulated by coupling MODFLOW and RT3D. MODFLOW is a three-dimensional groundwater flow simulator and RT3D is a modular computer code for Reactive multi-species Transport in 3-Dimensional groundwater aquifers. Using velocity fields generated by MODFLOW, transport of grout is solved using RT3D. The velocity field is then updated at regular intervals using MODFLOW and transport computation is repeated. When the grout is injected, the viscosity changes with concentration of grout and age of grout. In order to make RT3D suitable for this application, two reaction modules are added to RT3D. One module (GROUT_AGE) keeps track of grout age. The other (GROUT_GEL) calculates grout viscosity and modifies the hydraulic conductivity. MODFLOW is called from RT3D periodically to update the flow field using the changing K-field. The details on governing equations and solution techniques related to MODFLOW and RT3D are given in their respective references. 3.1 Gelling module (GROUT_GEL) Gel time is the interval between initial mixing of the grout components and formation of the gel. Gel time can be adjusted by varying the composition of grout. As the grout gels,
893
viscosity increases. For some grouts like silicates the viscosity gradually changes, whereas for acrylamides the viscosity remains fairly constant until the gel time is reached and then increases rapidly. The viscosity increase with time and concentration is described by gel time curve [6, 7]. The gel viscosity [6, 7] is given as
lage, = )', + r2 exp(y3t)
(1)
where flgel is gel viscosity; t is the time (s) and ~,~, ~'2 and 3'3 are fitting parameters. The parameters, 7~, ~'2 and % of 2.0, 1.0 and 0.00659 respectively are used here and are representative of a silicate gel with gel time of 15 minutes. During injection silicate grout mixes with water. The mixing process reduces the viscosity of the solution and may be described by linear and power law mixing rules [6, 7]. The linear mixing rule is given by gel IUl = X l /lgel "-I-(1
-
(2)
X S )/~w
where pt is the mixture viscosity;/.t~ is water viscosity; and Xz gel is concentration of gel. However in the present study the equation is modified to prevent gelling below a certain minimum concentration level and is given as
p, =
fl gel ( x ge` -- Xmin )
(1- Xmin)
+ (1 - Xtg~' + Xmin )i/,/w
(3)
where Xmi, is minimum concentration below which gelling does not occur. We use Xmi, equal to 5%. Having calculated the grout viscosity in each grid, the effective hydraulic conductivity, ggrouted, is calculated as g original Kgrouted = ~ (4) /-tt
3.2 Grout Age(GROUT_AGE) Grout viscosity increases with time as per Eq.1. Hence the age of the grout needs to be tracked. This is implemented as a second reaction module in RT3D. The following equation is proposed for the age of gel. 0(grout age) -- exp I . Ot
.
.
.-10-9.
l(max((X~el--Xmi
.
n *0.9)
(5) 10-41/3
This equation lets the gel age only when the concentration is higher than Xmi, and water does not age. The equation is introduced because without it grout injection into aged water resulted in premature gelling.
894 4. NUMERICAL CASE STUDY A two-dimensional horizontal test problem having dimensions of 420 cm x 240 cm is used to study the issues discussed above. Grid resolution tests are conducted to identify the level of refinement needed to adequately model the grouting process. Having identified the grid resolution, the effect of correlation scale and variability are investigated. Effect of considering only primary (large-scale) and, primary and secondary (small-scale) heterogeneities with respect to grout barrier performance, are also investigated. 4.1 Grid resolution test
The domain is discretized into three levels of grids viz., 42x24 grids (10 cm x 10 cm), 84x48 grids (5 c m x 5 cm) and 210x120 (2 cm x 2 cm) grids. One hundred conductivity fields are generated for Monte-Carlo simulations using SGSim program at the coarsest grid (42x 24). The same coarse conductivity field, is mapped on to the fine grids. A correlation length of 100 cm (equivalent to 10 grids on 42x24 grid), mean hydraulic conductivity of 10.2 cm/s and a variability of 0.2 is adopted. A staggered injection [10] as shown in Fig.la is chosen. The injection wells are shown in blocks and numbers near the injection site indicate the sequence of injection. The injection wells are implemented as a general head boundary, under a pressure head of 600 cm. The area of the injection wells is kept same for all grids. Area of injection is reduced to half on the bottom and top boundaries to maintain symmetry. Individual wells are injected for 15 minutes following the respective time sequence shown in Figure 1 (i.e., 1,2,3,4, which would extend the wall below 10, 5, etc.). The additional time allowed for wells 4, 5, 6, 7, etc., simulated continued extension of the wall and allowed gelling of injected grout before infilling. Total simulation time extends simulated well beyond the last injection to allow the grout to completely gel. Hydraulic conductivity of the grout barrier is determined at end of the grout simulation using MODFLOW. A separate model is set up using the post-grouted K-fields to obtain flux through the grout walls for each of 100 realizations. 4.2 Correlation scale test
In the present investigation, effect of correlation length and variability on grout performance is investigated on the 84x48 grid. For this study, conductivity fields are generated using correlation lengths ()~) of 0.0 m, 1.0 m and 2.0 m. To study the combined effect of large-scale and small-scale heterogeneity, conductivity fields are generated with small scale correlation length ()~s) of 0.2 m super imposed on large scale correlation length ()~l) of 1.0 m. Similarly, to study the effect of variability, three conductivity scenarios are generated with a standard deviation of 0.00 (homogeneous case), 0.25 and 0.50 for the correlation length of 1.0 m. The mean is kept at 1.0xl0 -2 cm/s. Results are presented as Keffective calculated by assuming wall thickness in 1.2 m. The ensemble statistics of Keffective from grouted K-fields over 100 realizations are presented in Table 1. As an illustration of effect of variability, grout concentration distribution in homogeneous K-field and a K-field with a standard deviation of 0.5 is as shown in Fig. 2.
895
=
I
D
!I
=
::!
D 2
~
(a) 10 cm x 10 cm grid Flux: 6.25 xl0 "6 cm3/s Keffective:4.56xl 0 "8 cm/s
,.
1~~~
!,i..i '.b ...
.
,:-~.
~
.......
t.:.
,ig
... I.
,'~ .,~,....~~
9
0.05
0.10
-
.
':
(b) 5 cm x 5 cm grid Flux" 5.40 xl 0 .6 cm3/s Keffective" 3.94xl 0 .8 cm/s
.~.~,"
(c) 2 cm x 2 cm grid Flux" 5.75 xl 0 "6 cm3/s Keffective" 4.19x 10 s cm/s
~..
0.15
0.20
0.25
0.30
Figure 1. Grout concentration distribution for different grid resolutions for a selected realization. Blocks indicate injection wells and numbers indicate the sequence of injection. Concentration is in fraction by volume. Injection concentration is 30 %
896
(a) Homogeneous K-field Flux: 8.75xl 0-6 cm3/s Keffective:6.38xl0-Scm/s
,....
,~:,,~
.
'
~,
~-.
:...~
~
(b)Standard Deviation-0.5 Flux" 12.81x 10-6 cm3/s Keffective:9.34xl0"8 cm/s
''
......
' ....
:~,',:.~,~:,::,:~;'~|
'.'i'
.. ;:....';t:'.~ ,": <: .........
~,.,. "...... '~....... i................... ~,~.:~,~,,.,.,.,M ,, ~ ....... I
0.05
0.10
0.15
0.20
: . . . . . . . . . . .
.
I
l
0.25
0.30
I
Fig.2: Grout concentration distribution in (a) homogeneous and (b) variable conductivity fields simulated on 84x48 grid (i.e., 5 c m x 5 cm grid block). Concentration is in % by volume. Table 1" Ensemble Statistics of Keg-ect~ve(x 10 .8 cm/s) from grouted K-fields on 84 x 48 grid Standard Deviation of K" 0.25 )~=0m )~=1 m Z = 2 m
Zl=l m )~s= 0.2 m
Correlation Length of K: 1.0 m cy = 0.0 or=0.2 ~ = 0 . 2 5 Homogeneous
or=0.5
Mean
2.829
2.953
3.201
2.895
6.380
2.888
2.953
S.D.
0.066
0.357
0.984
0.211
-
0.241
0.357
1.181
Minimum Maximum Median
2.698
2.406
2.297
2.443
-
2.472
2.406
2.523
3.011
4.681
8.896
3.449
-
3.938
4.681
9.340
2.815
2.924
2.975
2.895
-
2.858
2.924
3.588
3.908
)~= Correlation length, suffix 1 and s indicate large-scale and small-scale; c~=standard deviation of hydraulic conductivity field; S.D.: Ensemble standard deviation of Keffec,ve
897 5. DISCUSSION
Effective conductivities of the grouted barrier is considered as a metric of performance. The fluxes obtained for 210x120 grid are assumed to be the best possible estimates. The fluxes for the different grid resolutions indicate that the 42x24 grid normally yielded higher fluxes. The fluxes obtained for 84x48 grid are on average within 10 % of fluxes obtained for 210x120 grid. Therefore, 84x48 grid is selected for the heterogeneity studies to achieve the similar accuracy of 210x120 grid with significantly less computational effort. Concentration distribution in Fig.1 is presented on the discretized domain rather than contours for better visualization of fronts. As expected, significant numerical dispersion is observed in the coarsest grid resulting in overlapping of the grout bulbs. However in the fine grids, seams are clearly seen. The grout in both the fine grids is found to move farther from the injection well towards boundaries compared to the coarsest grid. Convergence problems have been observed for the MODFLOW simulation on post-grouted conductivity fields for the fine grid resolution which necessitated higher number of iterations in MODFLOW. The statistics presented in Table 1 demonstrate that the Keffective of the grouted barriers is increasing with increase in correlation length and variability. When the small-scale correlation length of 0.20 m is incorporated along with the large scale correlation length of 1.0 m, the fluxes, and hence Keffec,ve, obtained are between the corresponding values obtained for 0.0 m and that obtained for 1.0 m. The small-scale heterogeneity results in more randomness and therefore results approach those from the random field. The ensemble averages stabilized for the correlation lengths of 0.0 m, 1.0 m and combined correlation length in about 40 realizations. Whereas for the case of correlation length of 2.0 m, the ensemble averages fluctuate until around 80 realizations. Thus more realizations are necessary for Monte-Carlo simulation for higher correlation length and larger standard deviation. A similar trend is observed for the increasing standard deviation in K-field. Figure 2 presents the concentration plot for homogeneous case and standard deviation of 0.5. In the homogeneous case, the grout bulbs formed as near circles and the grout bulbs connect. Due to this, the centerline wells could not inject as much grout as the wells did in the case of heterogeneous case. Also, it may be observed that there are larger seams between the grout bulbs in heterogeneous case, resulting in higher effective conductivity. 6. CONCLUSIONS A model approach for simulating chemical grouting, developed by combining RT3D and MODFLOW, is presented. Reaction modules, GROUT_AGE and GROUT_GEL, are developed to track the age of the grout and determine the reduced hydraulic conductivity due to increase in gel viscosity. Grid of 84 x 48 is an optimal resolution between accuracy and computational effort for the present numerical modeling study. Fine grid resolution resulted in better grout barrier compared to that of coarse grid resolution. Conductivity fields generated using higher correlation lengths and higher standard deviations resulted in higher fluxes which means variability is affecting the grout barrier formation. ACKNOWLEDGEMENTS The first author thanks the Director, RRL and CSIR for sanctioning study leave. The funding for this research has been provided through National Science and Engineering Research
898 Council grant and Ontario Graduate Scholarship for Science and Technology and is gratefully acknowledged. REFERENCES
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