Seismic Fragility Formulations for Segmented Buried Pipeline Systems Including the Impact of Differential Ground Subsidence
Omar Pineda-Porras * Los Alamos National Laboratory P.O. Box 1663, MS K557, D-4 Los Alamos, NM 87545 USA
Mario Ordaz Instituto de Ingeniería, Edificio Fernando Hiriart Ciudad Universitaria, UNAM, a.p. 70-472, Coyoacán 04510, Distrito Federal, México
Abstract Though differential ground subsidence (DGS) impacts the seismic response of segmented buried pipelines and increases their vulnerability, there are no fragility formulations that estimate repair rates under such conditions found in the literature. Although physical models that estimate pipeline seismic damage considering other cases of permanent ground subsidence (e.g., faulting, tectonic uplift, liquefaction, and landslides) have been extensively reported, this is not the case for DGS. The refinement of the study of two important phenomena in Mexico City—the 1985 Michoacan earthquake and the sinking of the city due to ground subsidence—has contributed to the analysis of the interrelation of pipeline damage, ground motion intensity, and DGS. From the analysis of the 122-cm (48-inch) diameter pipeline network of the Mexico City water system, fragility formulations for segmented buried pipeline systems for two DGS levels are proposed. The novel parameter PGV²/PGA (composite parameter in terms of peak ground
*
Corresponding author. Phone: (505) 665-6997, email:
[email protected]
velocity [PGV] and peak ground acceleration [PGA]) has been used as a seismic parameter in these formulations because in previous studies it has shown better correlation to pipeline damage than PGV alone. By comparing the proposed fragility formulations, it is concluded that a change in the DGS level (from low-medium to high) could increase the pipeline repair rates (number of repairs per kilometer) by factors ranging from 1.3 to 2.0, with a higher seismic intensity corresponding to a lower factor. Keywords: Ground Subsidence, Segmented Pipelines, Pipeline Fragility Functions, Mexico City.
Introduction Though differential ground subsidence (DGS) impacts the seismic response of segmented buried pipelines and increases their vulnerability, there are no fragility formulations that estimate repair rates under such conditions found in the literature. Although physical models that estimate pipeline seismic damage considering other cases of permanent ground displacement (e.g., faulting, tectonic uplift, liquefaction, and landslides) have been extensively reported, this is not the case for DGS. In this paper, we analyze the interrelation of pipeline damage, ground motion intensity, and DGS for the case of the 122-cm (48-inch) diameter pipeline network of the Mexico City’s Water System (MCWS). It is important to mention that the method of analyzing the impact of DGS in the seismic damage of buried pipelines proposed in this paper is valid only for rigid pipelines that have bell and spigot joints; other pipeline types might require other physical models to account for the effects of DGS in the seismic damage analysis.
Case Study: The 122-cm (48-inch) Diameter Pipeline Network of the Mexico City’s Water System
Based on previous work studying two important phenomena in Mexico City—the 1985 Michoacan earthquake and the sinking of the city due to ground subsidence—it was possible to quantify the impact of
DGS in the seismic damage through the computation of two fragility formulations for two DGS levels. In this context, “fragility” means pipeline susceptibility to damage due to seismic wave propagation. The novel parameter PGV²/PGA (a composite parameter in terms of peak ground velocity [PGV] and peak ground acceleration [PGA]) has been used as a seismic parameter in these formulations because it has shown better correlation to pipeline damage than PGV alone (Pineda and Ordaz, 2007). A comprehensive discussion of the relationship between seismic parameters (e.g., PGA, PGV, and PGV²/PGA) and pipeline seismic damage, and a discussion of buried pipeline seismic damage estimation in Mexico City can be found in the recent work of Pineda and Najafi (2010).
Mexico City’s population of over 20 million people demands a consistent water flow of 60 m³/sec, of which 40 m³/sec are obtained from the underground aquifers located beneath the city. The continuous water pumping over the last few decades has caused a decrease in the water level and, therefore, cumulative ground subsidence. According to Illades et al. (1998), the subsidence rate through the city ranges from 5 cm to more than 35 cm per year; the greatest levels of cumulative ground subsidence, found in the downtown zone, range from 8 m to 10 m due to the consolidation process over the period from 1891 to 1995. Other researchers have studied the relationship between groundwater flow and ground subsidence; for example, Rivera et al. (1991) created a non-linear model of the interaction between soil compaction and total subsidence on a multi-layered system through a simultaneous numerical solution of the groundwater flow equation and a one-dimensional consolidation equation. In another study, CisnerosIturbe and Dominguez-Mora (2005) reported the impact of the over-exploitation of the aquifers beneath the city on the DGS levels; they observed that the over-exploitation could reach levels close to the 100% of the recharge capacity of the aquifers.
Ground subsidence in Mexico City is mainly observed in the lake zone, which is characterized by a sequence of stratified soft clays with high water content that show greater dynamic amplification under seismic wave propagation than the city’s –transition and hill zones (e.g., Singh et al. 1988, and Singh and Ordaz 1993). The transition zone is formed by alluvial sandy and silty layers with occasional intervals of clay layers, and the hill zone is composed by a surface layer of lava flows or volcanic tuffs (e.g., Edmunds et al. 2002 and Vera et al.1995). Because the 122-cm (48-inch) diameter pipeline network of the MCWS is mainly located in the lake zone, ground subsidence has had a long-term effect on the seismic response of this pipeline system increasing its vulnerability to seismic wave propagation.
One of the most comprehensive studies of potable water management in Mexico City is the “Plan Maestro de Agua Potable del Distrito Federal” (DGCOH 1997) funded by the Construction and Hydraulic Operations Department of the Federal District (Dirección General de Construcción y Operación Hidráulica; DGCOH), which from January 1, 2003, was fused with the Commission of Federal District’s Water (Comisión de Aguas del Distrito Federal) to form the current institution in charge of the management of potable water in Mexico City—Sistema de Aguas de la Ciudad de México (SACM). One of the contributions of the DGCOH’s study was the measurement of the subsidence levels through the city for the period from 1983 to 1992. This subsidence data was then employed to compute a mean yearly ground subsidence map for Mexico City (Figure 1), which is employed in this study to quantify the impact of differential subsidence in the 1985 damage scenario for the MCWS’ 122-cm (48-inch) diameter pipeline system.
Pipeline Fragility Formulations, Including the Impact of Differential Ground Subsidence
Before calculating a fragility model, we must assess how DGS could impact the seismic response of buried pipelines. It is important to note that the proposed analysis method is applicable only to rigid segmented pipelines with bell-spigot joints. Figure 2 illustrates how a pipeline system may become more vulnerable due to DGS. Two pipes, with lengths L1 and L2 (segments A-B and B-C, respectively), connected by a bell-spigot joint (see detail in point B), are horizontally buried at a constant depth H0. After DGS alters their original location (points A', B', and C'), the pipes now have slopes J 1 and J 2 , and variable depth. By comparing the original and final states of the pipe connection (details of points B and B', respectively), one can see that the bending of the two-pipe system reduces the contact area of the gasket; this observation shows how DGS can affect the seismic response capacity of the connection point. Figure 2 shows an extreme case of the impact of DGS in the system, its purpose is illustrative.
Clearly, the parameter related to the system vulnerability is J , which is defined as the difference of the slope of the segments (Equation 1). In Equations 2 and 3, the slopes of both segments are defined in terms of the sinking level of the segment ends ( H A , H B , and H C ) and lengths ( L1 and L2 ). Assuming that both segments have the same length L (Equation 4), J is then defined in terms of H A , H B , H C , and
L (Equation 5). Two cases in which J is zero are: if the sinking is uniform ( H A after the sinking both pipes have the same slope ( J 1
HB
H C ), and if
J 2 ). In either case, the vulnerability of the two-
pipe system is not affected by DGS. It is important to note that the vulnerability of a multi-segment system could be affected by DGS if J is other than zero in at least one joint. In Equations 1 to 6, J , J 1 ,
and J 2 , are non-dimensional variables; H A , H B , H C , L , L1 , and L2 , must be expressed with the same units of length.
J
J1
J2
L1
J
J 2 J1
[1]
HB HA L1
[2]
HC H B L2 L2
L
HC 2H B H A L
[3]
[4]
[5]
The described model assumes the following conditions: 1) pipes have a very high flexural rigidity, thus the effects of ground subsidence in negligible; 2) there is no relative displacement between soil and pipes, so the impact of ground subsidence is directly concentrated in the segment connection; and 3) (considering the last point) J 1 and J 2 are small enough that the pipes are not lengthened by the effect of DGS.
Though J can be used to quantify DGS, it has an important disadvantage for the computation of pipeline fragilities: it must be calculated at each pipe joint. On the contrary, for the generation of fragility functions, seismic intensity parameters, such as PGV, PGV²/PGA, are generally computed at the midpoint
of each individual pipe to define the mean seismic intensity level of the segment (Pineda and Ordaz 2003; Pineda and Ordaz 2007). To estimate DGS at the midpoint of each segment, Equation 6 can be used to define a new subsidence-related parameter ( J r ), which is the slope of each pipe segment. Though J r is based on a simpler model than that expressed by Equation 5, the results described in this section show that J r can be used to represent DGS in the pipeline fragility analysis.
Jr
HB HA L
[6]
The MCWS 122-cm (48-inch) diameter pipeline network, formed by 323 km of concrete segmented pipes, was severely affected by the 1985 earthquake, causing extensive damage and a total count of 95 repairs reported by Ayala and O’Rourke (1989). The damage scenario shows that two-thirds of the damage was located at pipe joints; the observed failure types include lateral crushing of pipes and crushing and unplugging of joints. Repair techniques included encasing broken pipes and joints with split pipe segments and replacement of pipes and/or joints. In their report, Ayala and O’Rourke conclude that ground subsidence could have influenced the MCWS seismic response during the 1985 earthquake. The fragility formulations proposed in this paper demonstrate that influence. It is important to note that the proposed model to quantify the effect of DGS in the seismic damage of the MCWS’ 122-cm (48-inch) diameter pipeline network does not account for soil-pipe interactions; hence Equation 6 is employed to quantify the DGS level related to every pipe segment from actual ground subsidence measurements through Mexico City, as explained below.
For this work, the MCWS 122-cm (48-inch) diameter concrete pipeline system was divided into 50m or shorter segments; this is consistent with previous studies (e.g., Pineda and Ordaz 2003) showing that 50m length limits provide sufficient accuracy for buried pipeline damage analyses in Mexico City. Then we computed J r for every segment (Equation 6), employing the ground subsidence levels at each segment
end, H A and H B , and the segment length ( L ). To obtain H A and H B for each segment, the mean yearly ground subsidence map proposed by the DGCOH (1997) was digitalized and recorded in a threedimensional array [X, Y, H]; where X and Y represent the longitude and latitude, respectively, and H represents the mean yearly ground subsidence level. X, Y, and H were obtained for a grid with cell size of 100 x 100 m through the city area. Using the endpoint coordinates of every pipe segment and a linear interpolation, H A and H B were estimated. Because H A and H B are in units of cm/year, J r has units of 1/year. J r values depend not only on the subsidence rate levels (Figure 1) but also on the orientation of each pipe segment. Hence, the identification of the most exposed pipes to the DGS effects may not be clearly noted in Figure 1.
The values of PGV²/PGA for each segment were computed through a linear interpolation using the coordinates of each segment centroid and the PGV²/PGA map for Mexico City proposed by Pineda and Ordaz (2007) (Figure 3).
In Figure 4, the relationship between J r and PGV²/PGA for the 7,444 pipe segments (of 50 m or shorter) of the MCWS 122-cm (48-inch) diameter pipeline network is shown. This figure was obtained by dividing the MCWS 122-cm (48-inch) network into two groups: the first group is composed of all the pipeline segments with J r values less than or equal to 0.001 1/year (low-medium DGS) and the second group contains the rest of the segments (high DGS). The horizontal dashed line, corresponding to
Jr
0.001 1/year, divides the pipeline segments into the two DGS levels; the vertical dashed lines
separate the pipe segments in four seismic intensity groups (in terms of PGV²/PGA). The repair rates (RR) for each DGS and PGV²/PGA levels are computed by dividing the number of repairs by the pipe lengths (within each dashed-line rectangle); these results are shown in Table 1. The PGV2/PGA values were
calculated as the average value for each respective seismic intensity group. For example, in the first seismic intensity zone, the PGV2/PGA value is 5 cm, the average between 0 cm and 10 cm; in the second seismic intensity zone, the PGV2/PGA value is 15 cm, the average between 10 cm and 20 cm; and so on.
The J r
0.001 1/year boundary between the two DGS levels (low-medium and high) was selected as a
reference value for calculating the proposed fragility relationships. Though the full range of values for the MCWS is 0-0.006 1/year (Figure 4), the boundary value ( J r
0.001 ) provides a conservative definition
for the high DGS level since two-thirds of the pipe damage caused by the 1985 earthquake is related to that DGS level ( J r values higher than 0.001 1/year). Since J r has units of 1/year, to analyze total DGS, this amount must be multiplied for the time in years. For example, J r
0.001 1/year for a 244-cm (8-ft)
length pipe segment corresponds to 4.83 cm of DGS between both pipe ends over a period of 20 years. For each group, RR - PGV²/PGA data points were calculated for four ground motion intensity levels (Table 1 and Figure 4); and for each DGS fragility relationship (low-medium and high), a linear functional form was adopted to compute Equations 7 and 8.
Originally, the linear fitting of the J r - PGV²/PGA data points for 15, 25, and 35 cm (Table 1 and Figure 5), for low-medium and high DGS levels, showed slopes equal to 0.0547 and 0.061, respectively. This implies that both lines have very close slopes. To simplify the fragility functions, the same slope was assumed in both lines. Thus, the combined fitting of both lines, for low-medium and high DGS, was recalculated with the condition of sharing the same slope parameter (0.058 in Equations 7 and 8) resulting in simpler fragility curves with coefficient of determination around 0.98. In Equations 7, 8, and 9, RR and PGV2/PGA are in units of rep/km and cm, respectively.
Fragility Relationship for Low-Medium DGS
RR
° ® ° ¯
0 0.134
if PGV 2 / PGA 1.25 cm if 1.25 d PGV 2 / PGA 14.4 cm
0.058 ( PGV 2 / PGA)-0.7
if PGV 2 / PGA t 14.4 cm
[7]
Fragility Relationship for High DGS
RR
° ® ° ¯
0
if PGV 2 / PGA 1.25 cm
0.2 0.058 ( PGV 2 / PGA)-0.42
if 1.25 d PGV 2 / PGA 10.7 cm if PGV 2 / PGA t 10.7 cm
[8]
Fragility Relationship for the 122-cm (48-inch) pipeline network without considering DGS (Pineda 2006)
RR
° ® ° ¯
0 0.162 0.058 ( PGV 2 / PGA)-0.534
if PGV 2 / PGA 1.91cm if 1.91 d PGV 2 / PGA 11.82 cm if PGV 2 / PGA t 11.82 cm
[9]
The proposed fragility relationship has three parts where RR can be zero, constant, or linearly dependent on PGV²/PGA. This three-zone functional form is also observed in the fragility function for 122-cm (48inch) diameter concrete pipes (Equation 9) proposed by Pineda and Ordaz (2006), which does not consider the effects of DGS. The no-damage zone is defined for seismic intensity levels not associated with pipeline damage seen in the 1985 earthquake. A likely explanation of the constant-damage zone, observed in several fragility models, is the presumably about-to-fail precondition of some pipe segments due to the combined effects earthquakes that occurred prior to the 1985 event and the increased pipe vulnerability because of the DGS impact. Currently, there is not information to conduct a separate analysis of the participation of these effects..
The proposed fragility formulations for low-medium and high DGS (Figure 5) fall below and above the fragility function, respectively, calculated for the entire 122-cm (48-inch) diameter pipeline system (Pineda and Ordaz 2006). Consequently, this comparison suggests that the 2006 fragility function implicitly considers the impact of an intermediate DGS level, between the low-medium and high DGS levels defined in this study.
From Equations 7 and 8, and Table 1, we conclude that a change in the DGS level (from low-medium to high) could increase the pipeline repair rates (repairs per kilometer) by factors ranging from 1.3 to 2.0; with the higher seismic intensity level corresponding to a lower factor. For example, for PGV²/PGA values of 35 cm, RR increases from 1.310 rep/km (RR for low-medium DGS) to 1.664 rep/km (RR for high DGS), which increases the factor to 1.27 (§ 7KHKLJKHVWIDFWRULVIRXQGIRUPGV²/PGA values of 15 cm, where RR increases from 0.698 to 0.945 rep/km.
Conclusions
The generation of the proposed fragility functions was possible thanks to the refined PGA, PGV, and ground subsidence maps for the Valley of Mexico, developed mainly over the past two decades by many researchers and engineers. We believe that the availability of information on the above-mentioned parameters for any place through Mexico City and, in particular, for the MCWS locations, was crucial to the generation of the fragility formulations.
From an interrelation analysis of pipeline damage, ground motion intensity, and DGS, using reported damage caused by the 1985 Michoacan earthquake on the MCWS, seismic fragility formulations for buried pipelines for two levels of DGS are proposed. These functions follow the same damage tendency observed in previous fragility models. From the comparison of the fragilities, we conclude that the increase of the level of DGS, from low-medium to high level, may augment the pipeline repair rate by factors ranging from 1.3 to 2.0. Finally, the good fitting (coefficient of determination around 0.98) of the RR – PGV²/PGA data points with linear functions shows that PGV²/PGA can be used as argument of fragility relationships for buried pipeline fragilities, including the effects of DGS.
The computation of reliable pipeline fragility formulations including J r as a variable is practically impossible because of the large amount of information required for that analysis. Even with refined estimations of ground subsidence and PGV²/PGA through Mexico City (Figures 1 and 3, respectively), a pipeline fragility function with PGV²/PGA and J r as arguments could not be computed because the damage scenario for the 122-cm (48-inch) diameter pipeline system comprises only 95 repairs and a length of 323 km. Instead of including J r as variable in the fragility model, the pipeline system was split taking into account two levels of DGS, low-medium and high; the results are satisfactory. We used four levels of seismic intensity to compute mean repair rate values used to create the linear fragility relationships. The use of these relationships in other systems is complicated: First, the use of PGV²/PGA to relate damage with seismic intensity was tested in the soft soils of Mexico City; for stiffer soils, the convenience of its use has yet to be studied. Second, because J r is a yearly mean value of DGS and not a measure of total DGS, J r values calculated for other cases might not be comparable with those computed for the case studied in this paper.
Symbols and Acronyms The following symbols and acronyms are used in this paper: A = location of pipe joint before the effects of ground subsidence A’ = location of pipe joint after the effects of ground subsidence B = location of pipe joint before the effects of ground subsidence B’ = location of pipe joint after the effects of ground subsidence C = location of pipe joint before the effects of ground subsidence C’ = location of pipe joint after the effects of ground subsidence DGS = differential ground subsidence H = mean yearly ground subsidence level (cm/year)
H A = sinking of pipe joint located at A’ H B = sinking of pipe joint located at B’ H C = sinking of pipe joint located at C’ H0 = pipe depth before the effects of ground subsidence
L = length of pipe segment; distance between points A and B
L1 = length of pipe segment 1 L2 = length of pipe segment 2 MCWS = Mexico City’s Water System
PGA = peak ground acceleration PGV = peak ground velocity PGV²/PGA = composite ground motion parameter in terms of PGV and PGA RR = repair rate (repairs/km); number of pipe repairs per kilometer of pipeline X = longitude Y = latitude
J = variation of the slope of pipe segments 1 and 2
J r = slope of a pipe segment; and differential ground subsidence between points A and B (1/year)
J 1 = slope of pipe segment 1
J 2 = slope of pipe segment 2
ACKNOWLEDGEMENTS
This research was conducted at the Institute of Engineering of the National Autonomous University of Mexico (UNAM) during the PhD studies of the first author. The support by the Dirección General de Estudios de Posgrado at UNAM is greatly appreciated. We thank Professors Gustavo Ayala and Michael O’Rourke for their comments on the results of this study.
The writers want to thank D-4 Research Group (Energy and Infrastructure Analysis) of the Los Alamos National Laboratory (LANL) for partially sponsoring this study.
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Figure 1. Mean yearly ground subsidence map (1983-1992) and damage zones (marked with dots) after the 1985 Michoacan earthquake for the MCWS 122-cm (48-inch) diameter pipeline network (represented by dark lines)
Figure 2. Impact model of differential ground subsidence on segmented buried pipelines
Figure 3. PGV2/PGA map (Pineda and Ordaz, 2007) and damage zones (marked with dots) after the 1985 Michoacan earthquake for the MCWS 122-cm (48-inch) diameter pipeline network (represented by dark lines)
Figure 4. Relationship between J r and PGV2/PGA for the 122-cm (48-inch) diameter pipeline system and the 1985 damage scenario (repair sites)
Figure 5. Proposed fragility relationships for segmented buried pipelines including the effects of differential ground subsidence (DGS) and comparison with a previous relationship that does not include DGS (Pineda and Ordaz, 2006)
