Published in Vadose Zone Journal 3:917-925 (2004)
© 2004 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
SPECIAL SECTION: RESEARCH ADVANCES IN VADOSE ZONE HYDROLOGY THROUGH SIMULATIONS WITH THE TOUGH CODES
Physical and Numerical Model of Colloidal Silica Injection for Passive Site Stabilization
Patricia M. Gallaghera,* and
Stefan Finsterleb
a Drexel University, Department of Civil, Architectural and Environmental Engineering, 3141 Chestnut Street, Philadelphia, PA
b Earth Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA
* Corresponding author (pmg{at}drexel.edu)
Received 17 November 2003.
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ABSTRACT
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Passive site stabilization is a new technology proposed for nondisruptive mitigation of liquefaction risk at developed sites susceptible to liquefaction. This technology is based on the concept of slow injection of stabilizing materials at the edge of a site and delivery of the stabilizer to the target location by using the natural or augmented groundwater flow. In this research, a box model was used to investigate the ability to uniformly deliver colloidal silica stabilizer to loose sands using low-head injection and extraction wells. Five injection wells and two extraction wells were used to deliver stabilizer in a generally uniform pattern to the loose sand formation. Numerical modeling was used to identify the key parameters affecting stabilizer migration and to determine their effective values for the box experiment. In our modeling approach, the stabilizer is treated as a miscible fluid, the viscosity of which is a function of time and the concentration of stabilizer in the pore water. Inverse modeling techniques are employed to reproduce data from the laboratory experiment for the determination of soil and stabilizer properties. While the details of the stabilizer distribution were difficult to reproduce with the simplified conceptual model we used, the overall system behavior was well captured, providing confidence that numerical simulation is a useful tool for designing centrifuge model tests, pilot tests, and eventually field stabilizer-injection projects.
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INTRODUCTION
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LIQUEFACTION IS A PHENOMENON marked by a rapid and dramatic loss of soil strength, which can occur in loose, saturated sand deposits subjected to earthquake motions. Certain types of sand deposits, hydraulic fills, and mine-tailing dams are particularly susceptible to liquefaction. The onset of liquefaction is usually sudden and dramatic and can result in large deformations and settlements, the floating of buried structures, or loss of foundation support. These settlements can disrupt the ground surface and vadose zone. At sites susceptible to liquefaction, the simplest way to mitigate the liquefaction risk is to densify the soil. If soil densification is impossible because of site constraints, grouting or underpinning is typically used to protect structures against the effects of liquefaction. In the case of grouting, the typical method is to inject cement or chemical grout under pressure through closely spaced boreholes. However, the need for closely spaced boreholes can limit the applicability of typical grouting methods at developed sites.
Passive site stabilization is a new technology proposed for nondisruptive mitigation of liquefaction risk at developed sites susceptible to liquefaction (Gallagher et al., 2002; Gallagher and Koch, 2003). It is based on slow injection of stabilizing materials at the up-gradient edge of a site, with subsequent delivery of the stabilizer to the target location via groundwater flow (Fig. 1)
. Low-head extraction wells may be used to adjust the groundwater flow pattern for targeted delivery of the stabilizer to the entire site. Relying on low-gradient stabilizer delivery instead of the traditional high-pressure injection of grout requires that the stabilizer have a low initial viscosity, a long induction period (during which the viscosity stays fairly low), and long gel times (on the order of 50–100 d).
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Fig. 1. Schematic showing concept of stabilizer delivery to target area by means of natural and controlled groundwater flow.
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Dilute colloidal silica (5% w/w) was selected as the stabilizer (Gallagher, 2000; Gallagher and Mitchell, 2001). Colloidal silica is an aqueous suspension of microscopic silica particles (7–22 nm) produced from saturated solutions of silicic acid. In dilute solutions, colloidal silica has a low initial viscosity of about 1.5 x 10–3 Pa s (1.5 cP; water = 1 cP), a long induction period, and long controllable gel times of up to a few months. Colloidal silica is also nontoxic, biologically and chemically inert, and has excellent durability characteristics, making it an excellent candidate for a stabilizer. The factors controlling the gel time of colloidal silica include the silica solids content, the pH, and the ionic strength of the diluted colloidal silica solution. For a given silica content, the gel time can be altered by lowering the pH and changing the ionic strength of the dilute colloidal silica solution. In field applications, gel times will be formulated using water and soil obtained from the candidate site. If the pore water has a high ionic strength or the soil has a large number of exchangeable cations, preflushing may be required (Persoff et al., 1994).
Use of colloidal silica for stabilizing sands has been investigated by Yonekura and Kaga (1992), Persoff et al. (1999), Gallagher and Mitchell (2002), and Liao et al. (2003). Yonekura and Kaga (1992) proposed colloidal silica as a replacement for the most commonly used chemical grout, sodium silicate. Persoff et al. (1999) investigated the effect of dilution on the strength and hydraulic conductivity of sand treated with colloidal silica. Gallagher and Mitchell (2002) studied the liquefaction resistance of clean, unconsolidated sands treated with colloidal silica in percentages that varied from 5 to 20% (w/w). Liao et al. (2003) also studied the liquefaction resistance of sand stabilized with colloidal silica.
Injection of a water-based stabilizer into the sand formation leads to a system consisting of three separate phases: (i) the solid grains, (ii) a noncondensible gas, and (iii) an aqueous phase of variable stabilizer concentration. After some time, the gelation process is initiated, increasing the viscosity of the water–stabilizer mixture, which turns into a non-Newtonian, visco-elastic fluid that eventually solidifies. By the time the stabilizer is completely gelled, the sand formation has a lower porosity, a new pore structure, reduced permeability, increased strength, and greater resistance to liquefaction. Loose sands treated with colloidal silica stabilizer had significantly higher deformation resistance to cyclic loading than untreated sands (Gallagher and Mitchell, 2002; Pamuk et al., 2002; Gallagher et al., 2002).
The primary feasibility issue remaining before implementation is uniform delivery of the stabilizer to the target location. Several aspects need to be investigated, including (i) the ability of the colloidal silica solution to permeate the porous material in a uniform manner, (ii) the potential of incomplete coverage as a result of soil heterogeneities and flow instabilities, and (iii) the control of the groundwater flow pattern by means of injection and extraction wells. With the box model experiments discussed below we examined the first aspect by looking at the permeation process on a small scale. In addition, we used the data of these laboratory experiments to test a numerical model, which can then be employed to design and optimize stabilizer delivery and groundwater control schemes in the field. For example, the results will be used to design centrifuge model tests in which the stabilizer will be delivered in-flight using a robot (Gallagher and Koch, 2003; Pamuk et al., 2002). Experimental and numerical investigations of stabilizer delivery at large scales are beyond the scope of this study.
A small box model (76 by 30.5 cm, 26.5 cm high) was constructed to investigate the delivery of dilute colloidal silica stabilizer to loose sand deposits under small gradients imposed by injection and extraction wells. Five injection and two extraction wells were used to deliver about 1.5 pore volumes of 5% (w/w) DuPont Ludox-SM (DuPont, Wilmington, DE) colloidal silica to the sand for approximately 10 h. During delivery, the concentration profile across the model was monitored. The purpose of this work was to identify the key parameters affecting colloidal silica delivery during a box model experiment. With this in mind, numerical simulations were performed to obtain insight into the complex migration behavior of a gelling liquid, and to determine effective formation and fluid parameters through a formal inversion process using iTOUGH2 (Finsterle, 1999).
Numerical models designed to simulate gel injection for enhanced oil recovery have been presented by Scott et al. (1985), Hortes (1986), Todd (1990), and Kim and Corapcioglu (2002). The main emphasis in this previous research was on the kinetic models of the gelation process. For example, Todd (1990) combined transport equations for 10 components with models of gelation kinetics, deposition, compaction, and filtration of gel aggregates, which leads to a decrease in porosity and an increase in flow resistance. This model has been successful in qualitatively reproducing the data from one-dimensional laboratory gel displacement experiments performed by McCool (1988). However, the model requires a large number of parameters that are unknown and difficult to determine. We used a special module of the iTOUGH2 numerical simulation and optimization code (Finsterle et al., 1994), in which gelation of the water–stabilizer mixture is approximated by a simple gel-time curve and mixing rule. The parameters of these empirical functions can either be measured in the laboratory or estimated by inverse modeling, as discussed below.
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MATERIALS AND METHODS
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The box model used for the experiment has three compartments, a central chamber for sand placement and two outer reservoirs for groundwater control. A photograph of the box model is shown in Fig. 2
. The model is constructed of 10 mm thick (3/8") Plexiglas with internal dimensions of 76 by 30.5 cm and a height of 26.5 cm (30 by 12 by 10.5"). The flow length through the sand is 46 cm (18"), and each water reservoir is 15 cm (6") long. Screens with a No. 200 mesh size are used between the water and soil compartments to prevent soil loss from the central chamber into the reservoirs. In Fig. 2, the left and right sides of the tank are the upstream and downstream chambers, respectively. Sampling ports in the sand compartment, visible in Fig. 2, are used to extract fluid samples across the soil profile to facilitate measuring the changes in the pore fluid chemistry as the colloidal silica is delivered to the formation. The side of the sand compartment can be removed, so the treated sand can be excavated for visual inspection and strength testing.
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Fig. 2. Box model experiment for colloidal silica delivery study. (a) Side view; (b) plan view. Colloidal silica progression after 3 h. Flow is from left to right.
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The model was filled by pluviating the sand to a height of 20 cm (7.9") at a relative density of 40% (Koch, 2003). Given the measured void ratios under very loose and dense conditions of 0.83 and 0.52, respectively, this corresponds to a porosity of approximately 42%. After sand placement, the upstream reservoir was filled with water to saturate the sand. After saturation, an overall gradient of 0.02 was established using the constant-head overflow ports in each reservoir chamber.
After the overall flow gradient was established, the stabilizer was introduced to the formation using delivery wells. Five wells were constructed of 19-mm (3/4") PVC pipe with three 6-mm diameter (1/4") injection ports screened with nylon. The ports were arranged in one vertical column at depths of 2.5, 4.5, and 6.5 cm below the sand surface. The wells were installed by gently pushing them into the sand deposit at 5-cm intervals with the ports in the downstream direction. The wells were located 15 cm from the upstream edge. A distribution bay (8 by 30.4 cm, 5.1 cm high [3.2 by 12 by 2"]) was placed on top of the wells to maintain a constant supply of colloidal silica to the wells.
During colloidal silica delivery, a constant head of 20 cm (7.9") from the bottom of the tank was maintained in the delivery wells. This excess head resulted in stabilizer movement in both the upstream and downstream directions. During a period of 10 h, 16.5 L (approximately 1.5 pore volumes) of 5% (w/w) colloidal silica solution (0.1 M, pH 6.3) were delivered to the formation. The ionic strength of the solution was adjusted using NaCl so that the viscosity of the gel increased by a factor of 10 within approximately 10 h after mixing (looking ahead to Fig. 6). The colloidal silica solution was colored with red food dye so investigators could visually determine the colloidal silica advancement on the top and sides of the model, as shown in Fig. 2.
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Fig. 6. Measured and estimated gel-time curves. Symbols represent viscosity measurements made on stabilizer samples taken during the sandbox experiment. The solid line is an exponential gel-time curve with parameters determined by inverse modeling of the sandbox experiment; the dashed lines represent the 95% error band calculated using linear uncertainty propagation analysis. (1 cP corresponds to a dynamic viscosity of 0.001 Pa s.)
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Two extraction wells were used to withdraw fluid from the sand formation at a rate of 12 mL min–1. The extraction wells were constructed of 19-mm (3/4") PVC pipe with nine 6-mm-diameter (1/4") ports screened with nylon. The ports were uniformly distributed along the length of the well, starting at a depth of 2.5 cm below the sand surface. The wells were installed adjacent to the downstream edge of the model at equally spaced intervals, with the extraction ports facing upstream.
The Cl– concentration of the pore fluid was monitored during the course of stabilizer delivery. Pore fluid samples were extracted from the sampling ports at times of 0.75, 2.75, 5, 8.25, and 9.75 h after delivery began. Chloride is considered to be a conservative tracer and is used to determine the relative concentration of colloidal silica present in the pore fluid. After delivery was completed, the model was cured for 14 d and then excavated into six block samples. The six block samples were carved into smaller samples for unconfined compression testing. Twenty-one unconfined compression tests were performed. Unconfined compressive strength ranged from a low of 16 kPa (2.3 psi) to a high of 61 kPa (8.9 psi). These values are in general agreement with unconfined compressive strength of samples tested at a known concentration of 5% (w/w) colloidal silica, which showed an average baseline strength of 32 kPa (4.7 psi) (Gallagher and Mitchell, 2002). Similar results were obtained by Persoff et al. (1999). The cyclic deformation resistance of sand treated by 5% (w/w) colloidal silica was shown to be sufficient to effectively mitigate the liquefaction risk. Given the consistent geotechnical behavior of the samples from the box model and those reported by Gallagher and Mitchell (2002), it was concluded that fairly uniform and sufficient coverage was obtained by low-gradient stabilizer delivery.
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MODELING APPROACH
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The multiphase flow and transport simulator TOUGH2 (Pruess, 1991a, 1991b) was extended by Finsterle et al. (1994) to model stabilizer injection and gelation based on the following two major assumptions.- The chemical process of gelation is not explicitly modeled. Instead, we calculate the viscosity of the aqueous phase as a function of stabilizer concentration and time. The viscosity of pure stabilizer as a function of time is represented by a gel-time curve. Mixture viscosity varies with the concentration of stabilizer in the aqueous phase and is described by a mixing rule.
- Initially, the stabilizer is treated as a miscible aqueous solution and therefore does not form a separate phase. After completion of the gelling process, we assume that the gel (which is a fluid of very high viscosity) solidifies instantaneously. By doing so, the porosity is reduced. The new porous medium thus has a lower permeability and different characteristic curves in the region affected by the stabilizer. The transition of the stabilizer from a highly viscous fluid to a solid part of the matrix is described by the solidification model.
In the model, the pore space is occupied by two fluids: the gaseous phase, consisting of air and water vapor, and the liquid phase, consisting of water, stabilizer, and dissolved air. The stabilizer and water volumes are assumed to be additive, which results in a mixture density of the liquid phase 
l given by
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where w is water density, gel is the density of the stabilizer, and Xgell is the mass fraction of stabilizer in the liquid phase. The viscosity of the liquid phase depends on stabilizer concentration and time. The increase in viscosity of the pure stabilizer (i.e., Xgell = 1) as a function of time is described by a parameterized gel-time curve, which can be fitted to laboratory data. Measured gel-time curves suggest the use of an exponential function of the form
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where t is time, and a1, a2, and a3 are fitting parameters. After injection, the stabilizer suspension becomes diluted due to mixing with pore water. Todd (1990) proposed a power-law mixing rule to calculate the viscosity of the liquid phase, µl, as a function of stabilizer mass fraction:
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Todd (1990) suggested using a quarter-power mixing rule (i.e., b = 0.25). It is understood that the gel-time curve and mixing rule are only meaningful as long as the stabilizer is miscible with water and as long as the stabilizer–water mixture behaves like a Newtonian fluid. However, movement of the stabilizer in the subsurface becomes insignificant at the time these assumptions become invalid; thus, the simplified model is considered reasonable for practical applications.
Because stabilizer injection for soil stabilization is not primarily concerned with the hydraulic properties of the treated region, we do not present the solidification model here. Details can be found in Finsterle et al. (1994). The gel-time curve and mixing rule described above were incorporated into the nonisothermal multiphase flow simulator TOUGH2 (Pruess, 1991a, 1991b; Finsterle et al., 1994) and linked to the iTOUGH2 (Finsterle, 1999) optimization code.
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SIMULATION OF SANDBOX EXPERIMENT
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A simplified, two-dimensional model of the stabilizer injection experiment described here was developed and calibrated against the observed Cl–, to examine the relative importance of soil and fluid parameters affecting flow and gelation processes.
The domain of dimensions X x Y x Z = 46 x 30.5 x 27 cm was discretized into a two-dimensional model of uniform gridblocks of size 
X x Y x Z = 1.0 x 30.5 x 1.0 cm (Fig. 3)
, with boundary gridblocks attached to the side and top boundaries. In this two-dimensional model, the five injection and two extraction wells, which are aligned in the Y direction, cannot be discretized individually. However, the volumes, cross-sectional areas, and hydraulic properties of the gridblocks containing the wells were adjusted to properly represent the wells and the ports, and extra connections in the X direction were added to allow fluid flow around the wells. This approximate representation of the three-dimensional physical system is considered appropriate for the purpose of this study, but may lead to some modeling errors when comparing simulated and observed Cl– concentrations.
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Fig. 3. Computational grid, injection and extraction wells, and location of Cl– concentration sampling points A1 through C6.
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Hydrostatic pressure profiles were prescribed on the left and right sides of the model domain, with atmospheric conditions fixed where the water level was maintained in the upstream and downstream reservoirs (i.e., at depths of –2.0 and –3.0 cm from the sand surface, respectively). Atmospheric boundary conditions were applied at the top of the model. A no-flow boundary condition was imposed across the bottom. Given these boundary conditions, the model was run to steady state to obtain the initial background groundwater flow with a head gradient of 0.02. A thin unsaturated zone develops between the water table and the sand surface.
After steady-state conditions were reached, stabilizer was supplied to the distribution bay at a constant pressure of 0.101 MPa (1 atm) prescribed at the sand surface. The distribution bay is connected to five injection wells, which are represented in the model by a single column of highly conductive gridblocks. Each well has three downstream-facing outlet ports at depths of 2.5, 4.5, and 6.5 cm from the sand surface; the diameter of each opening is 0.6 cm. Liquid was pumped from the two extraction wells (again represented by a single column of gridblocks) at a total rate of 12 mL min–1. Each extraction well has nine upstream-facing ports, which are uniformly distributed over the entire length of the well.
The viscosity of the injected stabilizer increases according to the gel-time curve given by Eq. [2]. Density and viscosity of the fluid mixture in each point of the model domain are calculated as functions of stabilizer concentration according to Eq. [1] and [3], respectively. Initial values for the anisotropic permeability of the sand as well as the fluid properties of the stabilizer (i.e., the parameters of the gel-time curve, mixing rule, and stabilizer density) were specified, as shown in Table 1. These parameters were then automatically updated by matching the calculated stabilizer mass fraction to Cl– concentrations measured at 12 locations within the sandbox (see discussion below). Porosity and the unsaturated hydraulic properties of the sand were fixed at the values given in Table 1.
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Table 1. Initial guess, best estimate, and estimation uncertainty of model parameters used to simulate stabilizer injection experiment.
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Before discussing the inverse modeling results, we describe the system behavior as calculated with the calibrated model. The distribution of liquid viscosity (Fig. 4)
shows both the location of the stabilizer and its gelation state as a function of time. Stabilizer supplied to the distribution bay has to displace the water that is initially in the well, before entering the sandbox through the uppermost port (Fig. 4a). Despite the ports facing downstream, the applied overpressure pushes the stabilizer toward the upstream reservoir. The stabilizer–water mixture flows from the injection wells in response to pressure, viscosity, and gravity (Fig. 4b). As indicated by the flow direction vectors shown in Fig. 4, liquid exits into both the upstream and downstream reservoirs as a result of the imposed injection pressure. Specifically, the pumping rate in the extraction well is initially smaller than the background flow rate and the increased flux from the stabilizer injection (i.e., some of the stabilizer bypasses the extraction wells). However, at later times, as the stabilizer viscosity increases rapidly between 6 h (Fig. 4c) and 10 h (Fig. 4d; see also gel-time curve in Fig. 6 below), the imposed injection pressure dissipates within a short distance of the treated area. As a result, the flow direction in the bottom part of the sandbox is reversed, with fresh water flowing from the upstream to the downstream reservoir, creating a channel with much reduced stabilizer coverage. Because flow is greatly reduced in the upper, high-viscosity part of the model domain, the extraction wells start to draw water from the downstream reservoir rather than the upstream side of the sandbox.
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Fig. 4. Calculated distribution of viscosity and liquid flow direction 5 min and 2, 6, and 10 h after beginning of stabilizer injection. (1 cP corresponds to a dynamic viscosity of 0.001 Pa s.)
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The simulated system behavior described above is consistent with the observed coverage of stabilizer, as shown in Fig. 2. In addition, the simulations reveal subtle effects in the flow field induced by the gelation process, as discussed below. These effects may become very important when designing stabilizer delivery scenarios in field applications.
Sand permeability and rheologic properties of the stabilizer were estimated by calibrating the model against the total amount of injected stabilizer and Cl– concentration data measured in the sampling points of Columns 2 through 5 (Fig. 3). Stabilizer viscosity was also measured directly to provide independent data for comparison with the gel-time curve derived by inverse modeling.
To avoid introducing a bias in the estimated parameters as a result of likely systematic modeling errors and inherent instabilities, data from Columns 1 and 6 were not used for calibration. The simplified two-dimensional model is not expected to capture the three-dimensional flow field that develops in the immediate vicinity of the two extraction wells. Because the sampling points of Column 6 are very close to the plane of the extraction wells, the Cl– data measured near the back and front walls of the sandbox represent local values within the three-dimensional flow field. These concentrations are systematically different from the averaged concentrations calculated by the two-dimensional model. The sampling points of Column 1 are located between the injection well and the upstream reservoir, which is a zone exhibiting a very small pressure gradient. Consequently, the advance of the stabilizer front depends strongly on small heterogeneities in sand properties (in the actual laboratory experiment) or small changes in the input parameters (in the numerical model), making the corresponding inverse problem unstable. This instability is also reflected in the measured data, in which the stabilizer arrival time measured near the front wall of the sandbox (
3 h) is significantly different from that at the back wall (9 h). Uneven stabilizer delivery is also evident from Fig. 2b.
The weighted least-squares objective function was used as a measure of the misfit between the data and the model output. Weighting coefficients were related to the expected uncertainty in the Cl– concentration residuals, which was estimated to be 0.025 mg mL–1 based on the differences in the values measured near the front wall (Y = 0) and back wall (Y = 30.5 cm) of the sandbox. Note that this standard deviation not only reflects a measurement error, but also includes the error introduced by averaging concentrations in the two-dimensional model. This is considered appropriate as an a priori estimate of the average residual after calibration. The total amount of injected stabilizer was measured to be 16.5 L, with an assumed uncertainty of 0.1 L. Chloride was considered as a conservative tracer tracking the stabilizer. The Cl– concentration of the injected stabilizer was fixed at 0.215 mg mL–1, and the objective function was minimized by iTOUGH2 using the Levenberg–Marquardt algorithm.
Figure 5
shows the matches obtained by the calibrated model. The model seems to capture the overall spatial and temporal distribution of the colloidal silica plume reasonably well, considering the uncertainty in the measured values and the simplifying modeling assumptions. Chloride concentrations in Row C (the bottom sampling layer) proved difficult to match, with the model overpredicting stabilizer delivery upstream from the injection wells (Column 2) and underpredicting concentrations downstream (Columns 4 and 5). The relatively poor matches of data from the upstream Column 2 indicate that the model representation of the injection through the downstream facing delivery ports is oversimplified. Nevertheless, the overall system behavior is considered reasonably well captured for the purpose of this modeling study, which is to demonstrate the code's ability to capture the gelation process and its impact on the flow field and stabilizer migration.
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Fig. 5. Comparison between measured (dashed lines with symbols) and calculated (solid lines) Cl– concentration.
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The Cl– concentration data were used to estimate the sand's horizontal and vertical permeability, as well as the rheologic properties of the injected colloidal silica stabilizer. A sensitivity analysis indicates that the migration of the plume from the injection to the extraction wells is predominantly determined by the horizontal permeability and the parameters of the gel-time curve. Vertical permeability affected the responses in Row C near the bottom of the box. Stabilizer density and the exponent of the mixing rule are comparatively insignificant. (Note that stabilizer density may have a significant impact on stabilizer delivery under large-scale field conditions.) The high sensitivity of calculated Cl– concentrations to changes in the three parameters that define the exponential gel-time curve leads to accurate estimates, despite relatively strong correlations among these parameters. The resulting gel-time curve is shown as a solid line in Fig. 6
. A 95% uncertainty band is calculated based on a linear error propagation analysis.
As shown in Fig. 6, the gel-time curve determined indirectly by matching Cl– concentration data is consistent with independently measured viscosity data (symbols), which lie within the uncertainty band. Because the parameters of the gel-time curve are obtained from information about the stabilizer plume behavior rather than from direct viscosity measurements, the good agreement demonstrates the sensitivity of the system behavior to the rheologic properties of the stabilizer and thus highlights the significance of accurately determining the gel-time curve when predicting stabilizer delivery. It also provides confidence that the approach used to simulate flow and gelation processes in the numerical model is appropriate to capture the behavior during the sandbox experiment. In practice, however, the gel-time curve is likely to be determined experimentally rather than through inverse modeling, because stabilizer viscosity can be accurately measured in a straightforward procedure.
In this inverse modeling exercise, we also determined the horizontal and vertical permeability of the sand concurrently with the stabilizer properties. The inferred anisotropy ratio (approximately a factor of 7 between the permeability in horizontal and vertical direction) is rather substantial, but not unreasonable. A sensitivity analysis reveals that the vertical permeability specifically affects the Cl– concentration at the measurement points of the bottommost Row C. Attempts to match the observed horizontal and vertical spreading of the stabilizer plume with an isotropic model failed, indicating that the model needs to be refined in a physically defensible manner.
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CONCLUSIONS
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Delivering the stabilizer to the target area for mitigation of potential earthquake damage using low-head injection and extraction wells requires a fundamental understanding of the migration of a gelling liquid through porous media. Moreover, the rheologic properties of the stabilizer, as well as the hydraulic controls through appropriate placement of injection and extraction wells, must be carefully designed under consideration of soil characteristics and geometric constraints at the site.
We performed laboratory experiments to examine the feasibility of colloidal silica stabilizer to permeate sand delivered in a low-gradient groundwater flow field. Moreover, the experimental data were used to test a numerical model that simulates flow and transport of a gelling liquid. The model was used for data inversion and sensitivity analyses, which helped identify the key factors affecting stabilizer delivery. The following conclusions can be drawn:
- On the bench scale, it is possible to uniformly deliver colloidal silica stabilizer to loose sands using low-head injection and extraction wells.
- The gelation process of the colloidal silica can be controlled such that the natural or induced groundwater flow pattern can be exploited for stabilizer emplacement.
- A numerical model has been developed for the simulation of a gelling stabilizer through saturated and unsaturated soils. The model was capable of reproducing the salient features observed during the sandbox experiments. While the details of the stabilizer distribution (as reflected in the measurements of Cl– concentrations at discrete points in space and time) were not accurately captured as a result of the simplified conceptual model (specifically its two-dimensionality, homogeneity, and representation of injection and extraction wells), the overall system behavior was well represented. This was evidenced by the overall stabilizer coverage and the estimated parameters, which matched the independently measured data gel-time curve very well.
- Inverse modeling was used to determine formation and fluid parameters affecting stabilizer delivery. The most significant parameters were permeability of the sand, its anisotropy, and the parameters of the gel-time curve, specifically the initial stabilizer viscosity and the parameter affecting the sharp increase of viscosity at late times. The gel-time curve significantly affected stabilizer plume migration as viscous forces dominated the flow of the stabilizer. Density effects were insignificant in a model of this scale. However, under low-gradient conditions at the field scale, density differences between the stabilizer and water are expected to be significant.
The combination of laboratory experiments with numerical forward and inverse modeling provides a first step in developing and testing design tools for field-scale stabilizer injection projects. The next step is to design and perform centrifuge model tests and to do pilot-scale experiments in which all the significant parameters and processes are incorporated, including stabilizer density effects and heterogeneity. A pilot-scale facility is under design at Drexel University for this purpose.
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ACKNOWLEDGMENTS
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We would like to thank Yuanzhi Lin, Waleska Mora, and Kathy Quinn for running the box model experiments, and Jens Birkholzer and Barry Freifeld for thoughtful reviews of the manuscript. The work was supported, in part, by NSF Grant no. CMS-0219987, and by the U.S. Department of Energy, under Contract DE-AC03-76SF00098. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the United States National Science Foundation.
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ABSTRACT
INTRODUCTION
