EFFECT OF STORAGE CAPACITY ON VERTICAL DRAIN PERFORMANCE IN LIQUEFIABLE SAND DEPOSITS

Transcription

1 EFFECT OF STORAGE CAPACITY ON VERTICAL DRAIN PERFORMANCE IN LIQUEFIABLE SAND DEPOSITS Juan M. Pestana 1, M. ASCE Christopher E. Hunt 2, Student M. ASCE R. Robert Goughnour 3, M. ASCE Ann M. Kammerer 2, Student M. ASCE ABSTRACT The use of vertical drains to improve the performance of potentially liquefiable ground during earthquakes has received increased attention over the last two decades. This paper briefly describes the formulation of a finite element code, FEQDrain, developed to analyze the development of excess pore pressure in a layered soil profile, accounting for vertical and horizontal drainage. The code includes equations describing a vertical drain with a non-constant equivalent hydraulic conductivity which more accurately describes the flow properties of perforated pipes and wick drains, and head losses due to horizontal flow into the drain. It can also model the presence of a reservoir directly connected to the drain, allowing the accumulation of the water discharged by the drain element as well as head losses in the reservoir itself. In contrast, most analyses used in practice assume the water level in the drain is at the ground surface and therefore all flow into the drain escapes. The accumulation of water within the drain, however, can lead to a significant retardation of flow into the drain, causing an increase in the predicted pore pressures developed during the earthquake. INTRODUCTION Over the past several decades, research into seismically induced liquefaction, largely in the United States and Japan, has followed several different paths. Early investigations focused on the causes of liquefaction and determination of the 1 Assistant Professor, Civil and Environmental Engineering Dept, University of California, Berkeley, CA Graduate Student Researcher, Civil and Environmental Engineering Dept., University of California, Berkeley, CA Principal, Geotechnics America, Inc., P.O. Box 2324, Peachtree City, GA Pestana, et al.

2 susceptibility of soils and sites to this phenomenon (Seed & Peacock, 1971; Seed & Idriss, 1971; Lee & Albaisa, 1974). In latter years, while research has continued in these areas, there has been much more focus on quantifying the effects of liquefaction on sites and structures. In addition to the research into causes and effects of seismically induced liquefaction, many researchers and practitioners have attempted to develop field techniques for remediation of susceptible sites. These methods generally involve densification of the soil to reduce its ability to generate excess pore pressures, or installation of drains to dissipate much of the excess pore pressure generated during a seismic event, or, in some instances, a combination of the two techniques (Baez & Martin, 1992). Along with these field techniques, analytical tools have been developed to aid in the design of remediation schemes. Two of the most widely used analytical tools for designing remediation schemes using drainage are a computer code called LARF by Seed and Booker (1976), and another by Onoue (1988), both of which were also used to develop simplified design charts. These codes attempt to analyze the effect of gravel drains on a liquefiable soil system. LARF, probably the most widely used of these tools, is a finite element code which analyzes a single elevation within a liquefiable soil profile. It examines radial drainage towards a perfect, infinitely permeable drain at the center of the system, and assumes that any flow of excess pore pressure that reaches that drain has exited the system. Onoue s experiments (Onoue, 1987) and finite difference code (Onoue, 1988) challenge the perfect drain concept of LARF, adding well resistance as a key component of the analysis, and stating that by ignoring this component, LARF yields unconservative results. The analysis in this paper is performed using a new finite element computer code, FEQDrain (Pestana et al., 1997), which incorporates and improves upon the capabilities of the previous ones by Seed and Booker and by Onoue. In addition, FEQDrain provides the capability to analyze remediation with newer composite drainage products, which typically feature a perforated geopipe for transmission of water and a geotextile fabric to prevent clogging of the drain. A key component of FEQDrain, which will be explored in this paper, is its ability to look at storage of the flow entering the drain. This storage appears as a rise in the water level within the drain and acts as an additional static head that retards subsequent flow into the drain. It has been found that this additional head can have a significant effect on the ability of the drain to reduce the excess pore pressures within the liquefiable layer. FEQDrain ANALYTICAL METHOD The basic equations FEQDrain uses to analyze the generation and dissipation of pore water pressure are the same ones used by Seed and Booker (1976), Booker, Rahman and Seed (1976), and Onoue (1988), with modifications in the treatment of boundary conditions and drain elements. The flow of pore water is governed by Darcy s Law as 2 Pestana, et al.

3 u u u T g { } [ k] = m v γ w t t where is the gradient operator, [k] is the matrix of permeability coefficients assuming isotropic conditions in a horizontal plane, u is the excess pore pressure, γ w is the unit weight of water, m v is the coefficient of volume compressibility (assumed isotropic), and u g is the excess pore pressure generated by cycling shear stresses due to earthquake loading. The generation term can be expressed in terms of the number of stress cycles in an earthquake, N, as u t g u g N = N t Widely used pore pressure generation curves (Seed et al., 1975b) are described by the equation u g 2 N arcsin σ ' π = (3) o N l where σ o is the initial mean effective stress, N l is the number of uniform stress cycles that will cause liquefaction if no drainage is provided, and θ is an empirical constant dependent on soil type and test conditions which is taken as 0.7 for a wide range of soils (Seed et al., 1975b). Taking the derivative of equation (3) yields u g σ ' o 1 π u = where X = (4) 2θ 1 N θπn l sin ( X )cos( X ) 2 σ ' o By representing the cyclic loading of an earthquake as a series of uniform stress cycles, N eq, over a duration t d (Seed et al., 1975a), the last part of equation (2) can be represented as N N eq N = (0 < t t d ) and = 0 (t > t d ) (5) t t t d Drain treatment in FEQDrain is handled in one of four different ways. In the first case, there is no drain, thus allowing the site to be analyzed prior to remediation. The second method uses a perfect drain, similar to LARF, in which excess pore pressures below the water table within the drain are uniform. Thus, if the water level in the drain starts out at the ground surface, the excess pore pressures in the drain will always be zero. If however, the water level is below the ground surface, water can accumulate within the drain, leading to a uniform rise in the excess pore pressures throughout the drain, thus retarding subsequent entry of water. The third method follows an Onoue-type analysis in which the drain is represented by a soil element with both horizontal and vertical hydraulic conductivities which can be set independently of the soil outside the drain. Thus, a very high permeability channel can be created. As in the perfect drain method, FEQDrain has the additional capability for allowing water to accumulate within the drain itself. 1 2θ (1) (2) 3 Pestana, et al.

4 The last drain type in FEQDrain was created to model newer composite drains consisting of a perforated geopipe, frequently wrapped in a geofabric to prevent clogging. For this analysis, the head loss in the drain is calculated in two parts. First, there is a lateral component calculated as 2 Q Q h = + (6) 2 Aorf 2g A surf ψ where h is the head loss, Q is the flow across the pipe, A orf is the area of the orifices through which Q flows, A surf is the surface area of the geofabric through which Q flows, and ψ is the permittivity of the geofabric. Second, there is a vertical component of head loss for the water that has gotten into the drain and is flowing through it, computed using a form of Manning s Equation as h c = c1( Q) 2 c or h = z c1( Q) 2 (7) z where z is the horizontal distance over which the head loss is being calculated, and c1 and c2 are parameters specific to the composite drain being used. DESCRIPTION OF THE CASES TO BE STUDIED Figure 1 shows a schematic of the system being analyzed. A two-layer soil system s d 1 m Low Permeability Clay DWL GWT 5 m Liquefiable Sand Deposit Drain Impermeable Base Figure 1: Schematic of System Being Analyzed 4 Pestana, et al.

5 with a very low permeability clay overlying the liquefiable sand layer was chosen. The parameter s is the diameter of the drain, d is the diameter of the tributary area the drain is acting upon, GWT is the depth to the ground water table, and DWL is the depth to the water level within the drain. Analysis was performed on the profile in Figure 1 to compare cases with and without storage capabilities, with differing amounts of storage, with different constant hydraulic conductivity (Onoue-type) drain resistances, with different sized reservoirs in the upper layer, and over several different spacing ratios. In order that each case be comparable, the initial effective stresses in the liquefiable layer were set equal by adding a constant value of overburden at the surface. This however, caused the initial effective stresses in the upper layer to be unequal. For this reason, the upper layer was effectively disconnected from the lower layer by giving it very low hydraulic conductivities and by giving it zero potential for generation of pore pressures. In effect, this prevents FEQDrain from analyzing the vertical migration of pore pressures through the upper soil layer, and forces all flow through the lower layer and the drain. PERFECT DRAIN WITHOUT STORAGE CAPACITY Figure 2 shows average pore pressure ratios, Ru z (max), versus cycle ratio, r N. Ru z is the average pore pressure ratio at a depth z in the liquefiable layer. In all of the cases run for this paper, as the upper soil layer acted as a barrier to continuing vertical 1.0 Ru z (max) s/d = 10 s/d = 8 s/d = 7 s/d = 6 s/d = 5 s/d = r N Figure 2: Pore Pressure Ratio Variation with Drain Spacing (no storage) 5 Pestana, et al.

6 flow, the maximum value of Ru z, Ru z (max), occurred at the interface between the upper and lower soil layers. The cycle ratio, r N, is a ratio of the number of uniform stress cycles in an earthquake versus the number of uniform stress cycles required to cause liquefaction in the soil layer in question under undrained conditions. The purpose of extending the tests to very large, and likely unreasonable, values of r N was to generate enough pore pressure to show the full effect of the drain. There may be specific soils that reach these same levels of pore pressure ratio in fewer cycles, and thus at smaller values of r N. The main purpose in examining the perfect drain case without storage is to show the general trends that occur with pore pressure generation and dissipation. At tight drain spacings (s/d = 4, 5 & 6), where the tributary area given by the parameter s is not too large, the pore pressure ratio rises steeply at first, but reaches an equilibrium point quickly as well, where the pore pressure being generated is equivalent to the amount being dissipated. At this point, the curve levels out into a flat line. At very wide drain spacings (s/d = 8 & 10) pore pressures ratios rise very steeply and continue to rise as there the tributary area is large enough to continue generating more pore pressure than the drain can handle. Thus, the soil liquefies. The average value may be lower than 1.0 as the soil close to the drain will still be much lower than 1.0 while, the soil at the edge of the tributary will almost definitely be completely liquefied. The intermediate drain spacing (s/d = 7) rises steeply at first and appears to be bending over toward an equilibrium value similar to the smaller drain spacings. However, there are two factors that cause the curve to steepen again. First, the generation curve is nonlinear, and as the soil approaches a liquefied state, pore pressure is generated more quickly. Second, following the work of Lee and Albaisa (1974) and Seed et al. (1975b), FEQDrain implements a nonlinear coefficient of volumetric compressibility which increases rapidly at high values of pore pressure ratio, thus increasing the storage capacity within the soil itself, leading to lower dissipation of pore pressures. PERFECT DRAIN WITH STORAGE CAPACITY If the water table in the system is not at the ground surface, there is room within the drain for water to rise. This rise of water in the drain distributes itself as an increase in static head throughout the drain, thus creating a back pressure that prevents pore pressures from dissipating into the drain as easily as before. Figure 3 demonstrates the, in some cases drastic, effect that allowing water to accumulate in the drain has on the system. For all storage cases shown, the water level in the drain began at a 1 meter depth. Once again, the problem can be divided into three zones, depending on the drain spacing of the system. At the small drain spacing (s/d = 4), the rise in the water level in the drain forces a steady rise in the pore pressure ratios in the drain compared to the case with no storage. The curve does not reach an equilibrium point because the water level is continuing to rise throughout the whole earthquake, thus adding more resistance continually. In the case of s/d = 6, where the previous case reached an 6 Pestana, et al.

7 1.0 Ru z (max) s/d = 10 s/d = 10 (stor) s/d = 8 s/d = 8 (stor) s/d = 6 s/d = 6 (stor) s/d = 4 s/d = 4 (stor) r N Figure 3: Pore Pressure Ratio Variation with Drain Spacing and Storage equilibrium point early on, the rise in the water table increased the pore pressures enough to push the curve into the nonlinear range which eventually led to full liquefaction. Finally, at the high values of drain spacing (s/d = 8 & 10), as the pore pressure ratio is already rising very quickly, the increased resistance to flow does not have as much of an effect. Still, in the steepest portion of the curve for s/d = 8, there is a 10% increase in pore pressure ratio. DRAIN WITH STORAGE AND CONSTANT Kx & Ky RESISTANCE Research by Onoue (1988) showed the effect of including drain resistance in the form of constant values of k x and k y in the drain. He demonstrated that ignoring this resistance was unconservative, even when k x and k y in the drain were 400 times larger than the hydraulic conductivity in the soil, when Seed and Booker (1976) said the drain need only be 200 times more permeable than the soil for an essentially free draining condition to occur. Figure 4 shows a combined plot of storage and several different levels of drain resistance, given by the ratio k d /k s, where k d is the hydraulic conductivity of the drain and k s is the hydraulic conductivity of the soil. The plots are only for a drain spacing of s/d = 5 with water level in the drain beginning at a 1 meter depth. The data displays the relative scale of importance between storage and drain resistance. It is immediately obvious that neither of them can be ignored in the analysis. The case of infinite k d /k s is equivalent to zero resistance. For the drain 7 Pestana, et al.

8 1.0 Ru z (max) No Storage k d /k s = Infinity k d /k s = 1000 k d /k s = 750 k d /k s = 500 k d /k s = 250 k d /k s = 100 k d /k s = 50 k d /k s = r N Figure 4: Pore Pressure Ratio with Storage and Varying Drain Resistance (s/d = 5) spacing shown, the pore pressure ratio rises gradually to approximately 0.7 by the end of the test and is beginning to curve up into the steep nonlinear portion. The addition of even a very small amount of resistance (k d /k s = 1000) is sufficient to drive the soil to liquefaction by the end of the test. There is a steady increase in pore pressure ratio as the resistance goes up. The case with k d /k s = 250, which is close to Seed and Booker s cutoff ratio of 200, shows nearly equal influence of storage and drain resistance. Thus, as one would expect, the addition of drain storage takes a situation which Onoue already pointed out as being unconservative, and shows it to be even more critical. In addition, for a larger tributary area, DRAIN WITH VARIATION OF INITIAL WATER DEPTH All of the previous cases involving storage have looked at an initial water depth of 1 meter in the drain. In some cases, there is sufficient water from the earthquake to fill the drain up to the surface, at which point the pore pressure ratio curve then tends to find an equilibrium value. Figure 5 looks at the case where the starting water level in the drain is raised to several values. This has the effect of reducing the amount of storage that can occur within the drain, thus putting a cap on the amount of resistance the drain can feel due to the rise in the water table. As the initial water level rises, the series of pore pressure curves generated are bracketed by the maximum values with water at 1 meter and the zero storage case with water at the ground surface. As the earthquake progresses, all of the curves with initial water levels below the surface follow the same curve. When the water level in any given case reaches the 8 Pestana, et al.

9 DWL = 1.0m DWL = 0.8m DWL = 0.6m DWL = 0.4m DWL = 0.2m DWL = 0.0m Ru z (max) r N Figure 5: Pore Pressure Ratio with Varying Initial Drain Water Levels (s/d = 5) ground surface, there is no more increase in drain resistance and the pore pressure ratio curve seeks out an equilibrium level. Thus, the cases that have less room for water to rise in yield a lower maximum pore pressure ratio. Once again, the curves in Figure 5 are for s/d = 5 and a larger tributary area will generate more pore pressures, leading to curves that may become nonlinear with effects that are not nearly as regular as shown here. DRAIN WITH PRESENCE OF RESERVOIR OF VARYING SIZE Geotechnics America, Inc., has taken a novel approach to reducing both the resistance to flow into a drain, but also reducing the rise in the water level within the drain itself. By using a composite drainage product consisting of a perforated geopipe surrounded by a filter fabric to prevent clogging of the orifices, they are using a vertical drain that, while it has resistance to radial flow into the drain, has very low resistance to vertical flow within the drain as compared to the typical gravel drains used in practice. Additionally, because the pipes are not filled with gravel, they can achieve the same flow and storage capacities with a smaller cross sectional area. Geotechnics America, Inc. has also developed a technique by which they auger a hole larger than the diameter of the drain near the ground surface and fill it with crushed rock or gravel. This acts as a reservoir for flow rising out of the drain during an earthquake, and at the same time, if the equivalent area of the reservoir is larger than that of the drain, provides a means for reducing the rise in water level within the drain as the same volume of water is now spread out over a larger area. 9 Pestana, et al.

10 R/D = 1 R/D = 2 R/D = 3 R/D = 4 R/D = 5 Ru z (max) r N Figure 6: Pore Pressure Ratio with Varying Effective Reservoir Area (s/d = 5) Figure 6 shows a series of cases where the water level in the drain began at a 1 meter depth, with a reservoir modeled above 1 meter. The value R/D is the ratio of the effective cross sectional area of the reservoir to that of the drain. Thus, a value of R/D = 2 indicates a reservoir that can store twice as much water as the drain. For the case of s/d = 5 shown, a doubling of the size of the reservoir leads to a significant reduction in the maximum pore pressure ratio. This comes about as in the R/D = 1 case, by the end of the earthquake the water has risen to 0.14 meters from the ground surface. In the R/D = 2 case, the water rises only to.55 meters from the surface, just over half the rise of the first case. This translates into a much smaller static head retarding the flow of water into the drain. As the reservoir to drain spacing ratio increases, the pore pressure ratio curves converge on the case of zero storage when in effect the reservoir is the ground surface and thus is infinite in dimension. CONCLUSIONS Historically, analysis performed for the remediation of earthquake induced liquefaction through the use of drainage, most notably gravel drains, has included the assumption that water could not accumulate within the drain. This only holds for cases in which the water table is at the surface. If the initial water level in the drain is below the surface, it will rise within the drain, creating a backpressure that will retard subsequent flow out of the susceptible soil deposit. This will in turn lead to higher pore pressures within the soil. Analyses that do not take this phenomenon into account may lead to unconservative design of remediation systems. 10 Pestana, et al.

11 Both the finite element code LARF (Seed & Booker, 1976) and the finite difference code by Onoue (1988) were used to create simplified design charts. The finite element code presented herein, FEQDrain, is capable of reproducing the charts by Seed and Booker, and those by Onoue. However, with the additional ability of calculating the effect of storage capacity within the drain, the number of parameters involved in any normalizations to produce design charts capable of handling flow through drains with resistance and storage was not considered feasible. Instead, a series of representative figures were presented to serve as guidelines for the potential issues that may arise while performing this type of analysis. Any comprehensive remediation scheme involving drainage should be examined on a site specific basis. REFERENCES Baez, J.I., and Martin, G.R. (1992). Quantitative evaluation of stone column techniques for earthquake liquefaction mitigation, Proceedings, Tenth World Conference on Earthquake Engineering, Madrid, Spain, Vol. 3, Booker, J.R., Rahman, M.S., Seed, H.B. (1976). GADFLEA: A Computer Program for the Analysis of Pore Pressure Generation and Dissipation During Cyclic or Earthquake Loading, Report No. EERC 76-24, University of California, Berkeley, CA, October Lee, K.L., and Albaisa, A. (1974). Earthquake Induced Settlements in Saturated Sands, Journal of the Geotechnical Engineering Division, ASCE, Vol. 100, No. GT4, Onoue, A., Mori, N., and Takano, J. (1987). In-Situ Experiment and Analysis on Well Resistance of Gravel Drains, Soils and Foundations, JSSMFE, Vol. 27, No. 2, Onoue, A. (1988). Diagrams Considering Well Resistance for Designing Spacing Ratio of Gravel Drains, Technical Note, Soils and Foundations, JSSMFE, Vol. 28, No. 3, Pestana, J.M., Hunt, C.H., Kammerer, A.M. (1997). FEQDRAIN: A Finite Element Program for the Analysis of Pore Pressure Generation and Dissipation with a Drain, During an Earthquake, Not yet published. Seed, H.B., and Idriss, I.M. (1971). Simplified Procedure for Evaluating Soil Liquefaction Potential, Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 97, No. SM9, Seed, H.B., and Peacock, W.H. (1971). Test Procedures for Measuring Soil Liquefaction Characteristics, Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 97, No. SM8, Seed, H.B., Idriss, I.M., Makdisi, F., and Banerjee, N. (1975a). Representation of Irregular Stress Time Histories by Equivalent Uniform Stress Series in Liquefaction Analyses, Report No. EERC 75-29, University of California, Berkeley, CA, October Seed, H.B., Martin, P.P., and Lysmer, J. (1975b). The Generation and Dissipation of Pore Water Pressures During Soil Liquefaction, Report No. EERC 75-26, University of California, Berkeley, CA, August Pestana, et al.

12 Seed, H.B., and Booker, J.R. (1976). Stabilization of Potentially Liquefiable Deposits Using Gravel Drain Systems, Report No. EERC 76-10, University of California, Berkeley, CA, April 1976 KEYWORDS Liquefaction, Sands, Drainage, Finite Element Method, Pore Pressure, Generation, Dissipation, Finite Element Method, Vertical Drains, Geopipes 12 Pestana, et al.