U. S. Geological Survey

 

A Comparative Analysis of Hazard Models for Predicting Debris Flows in Madison County, Virginia

 

 

By

Meghan M. Morrissey, Gerald F. Wieczorek, and Benjamin A. Morgan

 

 

 

 

Open-File Report 01-0067

 

2001

 

 

 

This report is preliminary and has not been reviewed for conformity with U.S. Geological Survey editorial standards nor with the North American Stratigraphic Code. Any use of trade names in this publication is for descriptive purposes only and does not imply endorsement by the U.S. Geological Survey.

 

 

U.S. Department of the Interior

U.S. Geological Survey

 

 

 

Meghan M. Morrissey, Colorado School of Mines, Golden CO

Gerald F. Wieczorek, U.S. Geological Survey, Reston VA

Ben A. Morgan, U.S. Geological Survey, Reston VA


 

Abstract

 

During the rainstorm of June 27, 1995, roughly 330-750 mm of rain fell within a sixteen-hour period, initiating floods and over 600 debris flows in a small area (130 km2) of Madison County, Virginia.  Field studies showed that the majority (70%) of these debris flows initiated with a thickness of 0.5 to 3.0 m in colluvium on slopes from 17 o to 41 o (Wieczorek et al., 2000).  This paper evaluated and compared the approaches of SINMAP, LISA, and Iverson's (2000) transient response model for slope stability analysis by applying each model to the landslide data from Madison County.  Of these three stability models, only Iverson's transient response model evaluated stability conditions as a function of time and depth.  Iverson’s model would be the preferred method of the three models to evaluate landslide hazards on a regional scale in areas prone to rain-induced landslides as it considers both the transient and spatial response of pore pressure in its calculation of slope stability. The stability calculation used in SINMAP and LISA is similar and utilizes probability distribution functions for certain parameters. Unlike SINMAP that only considers soil cohesion, internal friction angle and rainfall-rate distributions, LISA allows the use of distributed data for all parameters, so it is the preferred model to evaluate slope stability over SINMAP.  Results from all three models suggested similar soil and hydrologic properties for triggering the landslides that occurred during the 1995 storm in Madison County, Virginia.  The colluvium probably had cohesion of less than 2KPa. The root-soil system is above the failure plane and consequently root strength and tree surcharge had negligible effect on slope stability.  The result that the final location of the water table was near the ground surface is supported by the water budget analysis of the rainstorm conducted by Smith et al. (1996).

Introduction

The Appalachian Mountains in the eastern United States can experience intense rainfall during storms in the summer and autumn months.  These storms have caused hundreds of debris flows in concentrated areas such as the Little River Basin of western Virginia in 1949 (Hack and Goodlett, 1960), in Nelson County, Virginia, in 1969 (Williams and Guy, 1973), the Upper Potomac and Cheat River Basins of Virginia and West Virginia in 1985 (Jacobson, 1993), and Madison County, Virginia in 1995 (Wieczorek et al., 2000).  As rain infiltrates into the soil, it increases the pore-fluid pressure within the soil, which in turn reduces the shear strength of the soil.  In slope-stability analyses such as SINMAP (Pack et al., 1999); SHALSTAB (Montgomery and Dietrich, 1994); LISA (Hammond et al., 1992) and that developed by Iverson (2000), soil properties are used to calculate a factor of safety based on an infinite slope stability analysis.  Although these slope-stability analyses all consider the same stabilizing forces, they differ in the method in which pore pressures are calculated.  The first three models (SINMAP, SHALSTAB, and LISA) assume steady state, saturated flow parallel to the slide surface and use Darcy's law for estimating the spatial distribution of pore pressures (except in LISA which requires only the water table depth).  Iverson's (2000) model considers transient unsaturated flow and uses the Richards equation for estimating soil pore pressure response at depth.   The objective of this paper is to evaluate the approaches of SINMAP, LISA, and Iverson's transient response model for slope stability analysis and compare the results of these analyses with the locations of over 600 debris flows that occurred in Madison County, Virginia in 1995.

   

During the June 27, 1995, rainstorm in Madison County, Virginia, from 330 to 750 mm of rain fell within a 16-hour period in areas affected by debris flows (Wieczorek et al., 2000). During the 5 days prior to the storm, antecedent rainfall ranged from 75 to 170 mm, increasing the soil moisture. This storm initiated floods and over 600 debris flows in a small area (130 km2) of Madison County, southeast of the Shenandoah National Park.  Field studies showed that the majority (70%) of these debris flows had thicknesses of 0.5 to 3.0 m and were initiated within colluvium (Wieczorek et al., 2000) that mantled slopes.  The other debris flows either involved a mixture of colluvium, weathered rock or saprolite.   The inclination of failed slopes ranged from 17 o to 41 o with an average steepness of 30o +/- 3.7 (Morgan et al., 1997).

            A post-storm hazard assessment was conducted by Morgan et al. (1999) that delineated the principal source areas and pathways where debris flows occurred and characterized these areas based on topographic features.  Using an algorithm developed by Campbell and Chirico (1999), Morgan et al. (1999) identified source where slopes were steeper than 26o and had upslope areas less than or equal to 5,000 m2.  We discuss results from this study with those from our evaluation of slope-stability analysis of the same data set using SINMAP, LISA and Iverson's model (Iverson, 2000).

Slope-Stability Analyses

The first method of slope-stability analysis we apply to Madison County is SINMAP (Pack et al., 1999), a computer program that predicts landslide stability potential.  SINMAP (Pack et al., 1999) and SHALSTAB (Montgomery and Dietrich, 1994) are similar numerical models.   Both models use the same equation for factor of safety and Darcy's law for saturated flow within the soil to estimate the spatial distribution of pore pressures.  Saturated flow refers to flow through porous material such as soil where the pores are completely filled with water (Domenico and Schwartz, 1998).  The main difference between SINMAP and SHALSTAB is that SHALSTAB neglects the effect of soil cohesion in its calculation  (Montgomery and Dietrich, 1994) - a potentially significant factor for soils in the Blue Ridge of central Virginia.  A version of SHALSTAB does exist that includes soil cohesion but no published results are available at this time (Montgomery and Dietrich, 1994).  The factor of safety calculation (FS) in SINMAP is based on the infinite slope form of the Mohr-Coulomb failure law as expressed by the ratio of stabilizing forces (shear strength) to destabilizing forces (shear stress) on a failure plane parallel to the ground surface (Hammond et al., 1992):

                                   (1)

Where Cr and Cs are root strength and soil cohesion, respectively, D is the vertical thickness for soil depth, Dw is the vertical thickness of the phreatic layer, and g  is the unit weight for soil (s), and water (w). Angles a and f are the slope and friction angles (Fig. 1).

The vertical thickness of the phreatic layer Dw is estimated in SINMAP and SHALSTAB by Darcy's law (:  Q is discharge, T is transmissivity, L is flux length, A is cross sectional area, and i is hydraulic head gradient) for saturated flow with the assumptions that shallow lateral subsurface flow follows the topographic slope, and lateral discharge equals recharge, R, (in most cases recharge is equal to rainfall rate).  The assumption that shallow lateral subsurface flow follows topographic gradients implies that the contributing area to the flow at any point, is given by a, the specific catchment.  The specific catchment area is defined by the ratio of the area that drains into the grid cell to the contour length across the grid cell.  For steady saturated flow, lateral subsurface flow may be expressed by the transmissivity (T) along slope (Pack et al., 1999):

                 (2a)

Where, T is assumed to be constant with depth, b is the flow length, and Ks is saturated hydraulic conductivity.  The capacity for subsurface lateral flow is defined by the product of recharge and contributing area:

                           (2b)

Where, D is the vertical thickness of soil. Combining 2a and 2b yields an expression for the thickness of the water table:

                                       (3)

            Soil properties such as cohesion and transmissivity tend to vary over a range that can be one or two orders of magnitude.  For this reason, SINMAP uses a uniform probability distribution to define the range of values for rainfall, transmissivity, internal angle of friction, and soil cohesion bounded by a lower and upper limit.  Landslide locations from the June 27, 1995, storm are utilized to evaluate the predictive results from the model. 

Results from the factor of safety calculations are expressed by a stability index based on values of FS ranging from 0 to > 1.5.  The stability index (SI) is defined as the probability that a location is stable assuming uniform distribution of the soil parameters over their range of values. The classification is divided into 6 classes.  Classes 1-3 are for regions that according to the model should not fail with the most conservative parameters in the specified range.  These areas have SI > 1.5 and FS > 1.0.   For classes 4-5, the calculated FS is < 1.0, yet the probability of failure is less than and greater than 50%, respectively.   These two classes define a lower and upper limit for ground failure and have SI values 1.0-1.5 and 0-1.0, respectively.  Class 6 is unconditionally unstable meaning that the probability of failure within the specified range of parameters is greatest (assumed > 90% probability).   In this case, FS is < 1.0 and SI =0.

Application of SINMAP

The study area in Madison County, Virginia, shown in Figure 2 is a section of the field area studied by Morgan et al. (1999).  Topographic data used in these calculations were constructed from six 10-m Digital Elevation Models (DEMs) representing the six quadrangles shown in Figure 2.  Landslide locations and rainfall data were obtained from Morgan et al. (1997) and Wieczorek et al. (2000).  Figure 3 shows the topography and location of inventoried landslides used in the calculations.

Other data required for the calculation are soil cohesion, internal angle of friction, transmissivity and soil density.   Values for these properties are not available for the soil in Madison County.  The soil found here is considered a colluvium comprised of a clayey silt to silty sand matrix with gneissic clasts.  The Atterberg limits for a soil with these characteristics are listed in Table 1. A similar soil comprises the landslides that occurred in Nelson County in 1969 and geotechnical data are available for this soil (Auer, 1989).  As shown in Table 1, the clay contents and Atterberg limits for the two soils are comparable, which allows the use of geotechnical data for colluvium  in Nelson County, Virginia, in this study. The value for transmissivity (T) is assumed to range from 10-9 to 10-5 m2/s, typical values for clayey silts (Domenico and Schwartz, 1998).  The soil thickness is assumed to be uniform throughout the study area with a value of 2.0 m - the average measured thickness of initial debris flows (Morgan et al., 1997).   An average rate of 25-100 mm/hr is assumed for the triggering rainfall rate (R).  For the range of values for T and R, the value of R/T (using units of meters and seconds) in equation 3 is about 1-30,000.   Although R/T ranges over four orders of magnitude, this range of hydrological conditions defines the top of the water table at the ground surface throughout the study area for the June 27, 1995, storm.  A series of calculations were performed testing the range of values for internal angle of friction. The results showed that there was negligible difference between the upper and lower value.  Therefore, failure conditions will be governed by additional factors such as slope and cohesion.

Results from a series of calculations in which dimensionless cohesion varied and all other parameters remaining unchanged were found to have a significant influence of the stability calculation.   We initially evaluate the full range of assumed values for dimensionless cohesion, then limit the range in subsequent calculations with values for all other parameters unchanged as listed in Table 2.  Root strength was included in the calculation.  Field observations of exposed soil cross sections at the head scarp indicate that tree roots do not penetrate into the weathered bedrock, thus the root zone is above the failure plane.  Following the soil-root morphology nomenclature of Tsukamoto and Kusakabe (1984), these conditions are classified as Type A and have root strength (Cr) probability distributions defined by a mean value of 3 KPa and standard deviation of 2 KPa.   Assuming a value for Cr of 3 KPa and for Cs of 1-48 KPa yield a dimensionless cohesion, C, of 0.2-1.28.

Table 1:  Geotechnical properties for colluvium in Madison and Nelson Counties. n.a. indicates that data are not available for these properties.

 

Property

Madison County*

Nelson County**

Liquid Limit

26-49

18-24

Plasticity Index

4-23

2-3

Clay Content

16%

10%

Internal Angle of Friction (o)

n.a.

25-35

Soil Cohesion (KPa)

n.a.

1-48

Density (kg/m3)

n.a.

1200

Natural Soil Moisture (%)

n.a.

15-18

(*Wieczorek et al., 2000; **Auer, 1989)

There are several steps involved in SINMAP’s stability index calculation.  The first step is a pit filling correction of the topographic data.  Pits are grid elements that are surrounded by high terrain and do not drain.  A flooding approach is used to raise the elevation of the grid elements to the lowest pour point of the pit.  The next step calculates slope and flow.  The slope and flow direction associated with each grid cell are defined as the magnitude and direction of the steepest downslope vector from eight neighboring facets formed by a 3 x 3 grid cell window centered on the grid cell of interest.  The next step calculates the contributing area defined as the upslope catchment area divided by the contour length.  The upslope area of each grid is taken as its own area plus the area from upslope neighboring cells that have some fraction that drains into it.

SINMAP Results

The stability index (SI) distribution in the study area calculated for a dimensionless cohesion of 0.2-1.28 and the top of the water table at the soil surface is shown in Figure 4A. Under these conditions, the model predicts that 93.5% (585) of the inventoried landslides are considered unstable (all of which are defined within the lower unstable threshold, see Table 3).  The plot in Figure 4B shows the contributing area versus slope  - boxes are the observed landslides, and the crosses represent grid cells within the study area.   This plot shows that slopes between 16o and 52o have a factor of safety < 1.0 and a < 50% probability of failure.  In the second calculation, the range for cohesion (C) is reduced to 0.2-0.5 with all other properties remaining unchanged.  Under these conditions, landslides that were previously defined as stable remain stable (note the stable regions in Fig. 4A remain the same as in Fig. 4C).  The reduction of the maximum value of C increases the probability of failure for landslides previously defined as unstable (Table 3).  Figure 4D shows that ground failure on slopes between 16-25o have a < 50% probability of failure, slopes between 26o -37o have >50% probability of failure, and slopes >37o will unconditionally fail.

 

Table 2:  Values for parameters required for factor of safety calculations for each model.  n.i. denotes parameters that are not included in the model. [] denotes values used in base line calculation.

Parameters

SINMAP

LISA

Transient Response

Failure plane slope (o)

17-41

17-41

17-41 [17]

Depth to failure plane (m)

2

0-6 (ave. 2)

2

Internal angle of friction (o)

25-35

25-35

25-35 [25]

Soil Cohesion (KPa)

1-48

1-48

1-48 [4]

Root Strength (KPa)

3 (+/-2)

3 (+/-2)

n.i.

Soil Density (kg/m3)

1200

1200

1200

Tree Surcharge (kg/m3)

n.i.

0.5-1.0

n.i.

Transmissivity (m2/s)

10-9-10-5

n.i.

n.i.

Saturated Hydraulic Conductivity (m/s)

n.i.

n.i.

10-9-10-5 [10-5]

Depth to water table (m)

calc. 0

0-3 (ave. 0)

0-2 [2]

Rainfall rate (mm/hr)

25-100

n.i.

25-100 [35]

 

            In the third calculation, C has a range of 0-0.5 with all other properties the same as in the first calculation.  Under these conditions, 99.8% (625) of inventoried landslides are predicted to occur and 76% of them will unconditionally fail (Table 3).  As shown in Figure 4E, most of the regions within the study area are defined as unstable.   According to Figure 4F, slopes that are between 5o and 20o have a <50% probability of failure, between 21o and 37o have a >50% probability of failure, and >37o will fail.  In the fourth and final calculation, C ranges between 0.1-0.2.  Under these conditions, < 3% (14) of the inventoried landslides are predicted to be stable and 88.5% (554) of all inventoried landslides will unconditionally fail (Table 3).  According to Figure 4H, slopes between 10o and 14o have a < 50% probability of failure, between 15o and 18o have a > 50% probability of failure, and those that are > 18o will unconditionally fail (with an assumed > 90% probability of failure).

 

The results from the fourth calculation best predict the inventoried landslides except for fourteen landslides.  Landslides not predicted by SINMAP for this case have field measured slopes ranging between 17 o-20 o, that are greater than those calculated by SINMAP (<14o). The discrepancy between measured and calculated slopes may be attributed to resolution of the DEM, the slope calculation used in SINMAP, and the accuracy in landslide initiation locations on the DEM.

  

Table 3:  SINMAP stability index (SI) classification of the inventoried landslides (626 total) in Madison County calculated for four ranges of dimensionless cohesion (C).

 

SI class

 1  Stable

2 Moderately stable

3  Quasi-stable

4  Lower unstable

5  Upper unstable

6  Uncond. fail

C= 0.2-1.28

14

20

7

585

0

0

0.2-0.5

14

20

7

190

356

39

0-0.5

0

0

1

108

478

39

0.1-0.2

10

4

0

20

38

554

 

For those landslides predicted by the fourth calculation, the following triggering conditions are required: C is 0.1-0.2 that is equivalent to cohesion (Cr+Cs) of 2-4 KPa and the water table must be at the surface.  The small amount of cohesion required may be attributed to either the root strength of the trees or the soil.  The soil contains 16 wt.% of clay so that the cohesion may be from the soil and not necessarily from the root mass of the tree.  The suggested location of the water table is supported by results from a water budget analysis of the storm that shows that the total amount of rainfall during the storm over the Rapidan River basin as roughly equal to the cumulative discharge from the river basin (Smith et al., 1996).

 

Although we cannot directly compare the stability index distribution maps calculated by SINMAP with the hazard map constructed by Morgan et al. (1999), a comparison can be made between constrained topographic conditions.  The hazard map constructed by Morgan et al. (1999) defines areas with the greatest potential for future debris-flow initiation where catchment areas upslope from principal drainages are  < 5,000 m2  based on the June 27, 1995, landslide data set.  These areas are typically found where shallow slides are triggered in colluvium on slopes > 26o.  The areas predicted to fail by SINMAP during a storm like that on June 27, 1995 include all areas with slopes > 18o with upslope catchment areas < 10,000 m2.  These areas (see Fig. 4G) include those predicted by Morgan et al. (1999; see plate 1) and all slopes above major drainages.  The size of area predicted to fail by SINMAP may be reduced by considering non-uniformities in soil properties and landslide thickness throughout the study area and a more robust pore pressure calculation.

Level I Stability Analysis (LISA)

LISA performs a probabilistic analysis based on the factor of safety, FS, calculated by Eqn. (4).  Values for each parameter in the equation are defined by probability distribution functions (PDF’s).  Results are presented in a histogram showing the distribution of the factor of safety calculated for up to 1000 different combinations of parameters using a Monte Carlo simulation (for details on this procedure see Hammond et al., 1992).  A probability for failure is also calculated from the different combinations of parameters (SINMAP calculates a probability for the best and worst conditions defined by the PDFs) in the following equation that produce a factor of safety less than or equal to 1.0:

                                   (4)

Where, Cr and Cs are root strength and soil cohesion, respectively, qo is tree surcharge, Dm is the vertical thickness for moist soil, Dw  is the thickness of the phreatic layer, gm  is the unit weight for moist soil, g sat is for saturated soil, and g w is for water. Angles a and f are the slope and friction angles (Fig. 1).  Tree surcharge is an estimate of the tree mass (biomass) that adds additional load to the failure surface where soil depths are < 2m (Hammond et al., 1992). In Madison County, slopes are highly vegetated so the effect on slope stability needs to be evaluated.  

LISA is a tool for risk assessment developed by the USDA Forest Service for areas (defined as polygons on a map) with similar topography and geology.   A factor of safety map (comprised of polygons) may be generated for a study area where details of the geology are known.  We do not attempt to construct a map for the Madison County, Virginia.  For this study, we assume the geology and soil to be uniform throughout the study area and therefore we assess the effects of slope, location of the water table, soil thickness distribution and soil strength on the factor of safety.

Application of LISA

Many of the parameters required for calculating the factor of safety, FS, with LISA are the same as those in SINMAP except that probability distribution functions  (PDF) define the range of values for each parameter in Equation 4.  As in SINMAP, a uniform PDF with the same upper and lower limits (Table 2) was used to define soil cohesion, and internal angle of friction.  Root strength is defined by a uniform PDF with a mean of 3 KPa and a standard deviation of 2 KPa as described by Tsukamoto and Kusakabe (1984) for Type A soil-root systems.  Soil density is defined by a constant value of 1200 kg/m3.  The depth to failure plane or soil depth is defined by a log normal PDF that matches the measured thickness distribution of initial landslides in Figure 6 of Wieczorek et al. (2000).  Initiation slopes that match the slopes associated with Figure 6 of Wieczorek et al. (2000) are defined by a triangular PDF with a minimum and maximum values of 17o and 41o and apex of 30o.  The depth ratio of water table to failure plane (or soil thickness) is described by a triangular PDF ranging from 0.1 to 0.9 with the apex set at 0.5.  In LISA, the pore pressure distribution is not calculated but is estimated from the thickness of the water table that is set as an initial condition. 

            The FS calculation in LISA considers tree surcharge (qo in Eqn. 4) with a uniform PDF with a lower and an upper limit of 0.5 and 1.0 KPa, respectively.  Tree surcharge is defined by tree weight and tree volume per acre (Hammond et al., 1992).  It may be estimated by the average tree diameter (D) and tree height (L) within the study area using the Doyle scale that defines a board foot (BDF) as: BDF=L/16*(D-4)2 (Hammond et al., 1992).  Field observations near an initial landslide site found the average tree diameter to be 21 inches and tree height to be 120 ft that yield 2,172 BDF per tree.  Assuming 3-5 pounds per BDF (Hammond et al., 1992) and that there are 4 trees per 900 ft2 in the study area, the tree surcharge within the study area ranges between 28 to 48 psf (0.5-1.0 KPa).

            Using the above values for soil, tree surcharge and topographic and hydrologic properties, LISA predicts that for 1000 different combinations of parameters, the probability of a FS< 1 is 2.0%. The same result occurs when tree surcharge is set to 0.0 KPa with all other values remaining unchanged.  Lowering the soil cohesion to 2-4 KPa and setting tree surcharge and root strength to 0.0 (dimensionless cohesion, C=0.1-0.2), increases the probability of a FS<1 to 16.0%.   For the same conditions and changing the depth ratio of water table to failure plane to a constant value of 1.0 – the top of the water table is located at the surface, yields a probability of a FS<1 of 36%.   Next, we reduced the soil cohesion to 0.5-1.0 KPa that results in a probability of a FS<1 of 62%.  By further lowering it to 0-0.5 KPa, the probability of failure is increased to 93%.  These results are similar to those from SINMAP in that for failure conditions to occur given the field observation of slopes and thicknesses for the June 27, 1995, landslides, the water table must be at the surface with a minimum soil cohesion at the time of failure.  These results also suggest that tree mass and root strength may not play a major role in the hillside stability in Madison County, Virginia, during the 1995 storm.

A final series of calculations were conducted to compare results from LISA with those from SINMAP.  In these calculations, we used initiation slopes with values between 18o and 45o, the range of calculated slopes from SINMAP that will unconditionally fail (assuming > 90% probability of failure).  If the dimensionless cohesion was set to 0.1-0.2, the depth ratio of water table to failure plane was 1.0, and the soil thickness was set to a constant value of 2.0 m, then LISA calculated a probability for a FS<1 of 88%.  This probability is essentially the same as that calculated by SINMAP for the same range of slopes.  This result demonstrates that LISA and SINMAP calculate probabilities by a similar method.  The main difference between the two models is that LISA allows the use of a constant value or a distribution of values for the governing parameters in Equation 4.

    

Iverson’s Transient Response Model

The final stability analysis involves the mathematical model developed by Iverson (2000) based on the Richards equation for unsaturated shallow groundwater flow.  This model assesses the effects of transient rainfall infiltration on the timing and locations of landslides by approximating the pore pressure response in shallow soils to individual (short term) rainstorms.  In this model, the pore pressure is calculated for vertical flow not slope parallel flow as in SINMAP, in the unsaturated zone above the water table as well as in the saturated zone (water table).  The transient response model assumes that slopes are initially wet and the catchment area (A) is much greater than the thickness (H) of the landslide (dimensionless length scale,).  For initial landslides in the 1995 storm in Madison County, Virginia, the catchment area is < 5,000 m2 and the depth to failure or depth of interest is < 3 m (Wieczorek et al., 2000) that yields 0.01.

 

            The pore pressure water head (P) in the soil for short-term rainstorms (t<<A/Do; where Do is the hydraulic diffusivity) is approximated by a simplified version of the Richards equation with the following assumptions: , such that when the Richards equation is nondimensional (Eqn. (1) in Iverson, 2000), terms of order e and e2 can be neglected.  This assumption reduces the equation to a one-dimensional equation in the vertical direction (Z). 

                                              (5a)

                                      (5b)

Where t is time, T is the rainfall duration, b = (cos2a) and a is slope steepness, d is the initial steady state water table depth, I is the infiltration rate equal to rainfall rate, k is the hydraulic conductivity and R(t*)  is the pressure head response function. 

                                                (6)

Where t* = t/(Z2/4Do) is dimensionless time and Do is the hydraulic diffusivity (assumed to equal k/soil moisture content). Dimensionless rainfall duration is expressed as T* = T/(Z2/4Do).

 

            Slope failure on an infinite slope is described by the factor of safety calculation, FS, that balances the downslope component of gravitational stress against basal frictional shear stress and pore pressure (pressure head):  FS = Ff + Fw + Fc< 1.  The components of FS are defined as:

 

                                                                              (7A)

                                                            (7B)

                                                                (7C)

Where a and f are the slope and soil friction angle, c is soil cohesion, gs is depth average soil unit weight, gw is the unit weight of water.   During a rainstorm, the factor of safety will vary as a function of depth and time reflecting the pore pressure response.  Equations 7A-7C are combined with equation 5A-5B to yield equations 29A-29B and 30 in Iverson (2000), used in this study.

 

            We evaluated the pore water pressure response to the June 1995 storm in the colluvium on an hourly and daily time scale.  Soil properties for the calculations are listed in Table 2.  A series of calculations are conducted that evaluate the effects of soil cohesion, initial water table depth and hydraulic diffusivity.  A base line calculation is made that allows for a comparative analysis of the results (values denoted by [] in Table 2).  In this calculation, the minimum friction angle is set to 25o given the fact that the soils were probably wet from the previous five days of rain.  Soil cohesion is 4 KPa, infiltration rate is 10-5 m/s (~35 mm/hr), hydraulic diffusivity is 10-4 m/s, and the slope is set to 17 o.  The initial water table is set at 2 m below the surface (the average depth of slope failure). 

Results from Transient Response Model

Figures 5A-5B show the pore water pressure response in the soil to rainfall durations of 4-16 hr and 1-4 days, respectively when the water table is initially at 2 m.  Lines in these figures are calculated from equations 5a and 5b, respectively.  The dot-dash line represents the hydrostatic pressure at time =0 s and the dotted line represents the pressure-depth curve when the top of the water table is at the surface (WT=0 m).   In Figure 5A, the pressure increases slowly at shallow depths during the 16 hr infiltration period (each colored line represents 4 hr; green = 4 hr, blue = 16 hr), but never at depths where failure occurs.  In Figure 5B, the pressure increases slowly at depths below 1.5 m during the 4 days after the infiltration period  (each colored line represents 1 day; green = 1 day, blue = 4 days).  The change in hydrostatic pressure after 4 days is not sufficient to cause slope failure over this time period as demonstrated in Figure 5C where the factor of safety remains greater than 1.  Similar results occur when the water table is raised to 1 m.  These conditions would not induce the landslides observed from the June 1995 storm in Madison County, Virginia.

 

            A series of calculations were conducted to determine conditions favorable for initiating fast moving debris flows – like the type that occurred on June 27, 1995 in Madison County, Virginia.  According to Iverson (2000), such conditions require FS < 1 at depths above the failure plane during the rainstorm (t< T or t< 16 hr).  The first set of calculations evaluates the effect of soil cohesion.  In this case, the water table is initially at 1 m depth and soil cohesion is decreased from 4 KPa to 0 KPa with all other parameters set to the standard values in Table 2.   Reducing the soil cohesion from 4 KPa to 0 KPa moves the factor of safety at depth towards favorable conditions for slope failure (FS<1) (Fig. 6A). The second set of calculations evaluates the effect of the initial depth to water table.  In this case, soil cohesion is 2 KPa (lower than the standard calculation) and the water table depth is raised from 1 m to 0.25 m below the surface.  Figure 6B demonstrates how raising the water table from 1m to 0.25 m below the surface yields conditions favorable for slope failure.  Conditions that favor the initiation of fast moving debris flows develop within 16 hr when the water table is initially at 0.25 m.  The third set of calculations evaluates the effect of hydraulic diffusivity.  In this case, soil cohesion is 2 KPa, the water table is initially at 1 m and the hydraulic diffusivity is increased from 10-4 m/s to 10-3 m/s (decreasing the diffusivity does not move FS towards failure conditions). Figure 6C shows that increasing the hydraulic diffusivity favors conditions for fast moving debris flows within 10 hr from the start of the storm as was observed by eyewitnesses during the June 27, 1995, storm (shown in Fig. 4 of Wieczorek et al., 2000).

Results from these calculations suggest that soil conditions prior to the June 27, 1995 rainstorm in Madison County, Virginia, were as follows: the water table was at least 1 m below the surface, the diffusivity was >10-4 m/s, and soil cohesion was < 2KPa.  These values for depth to water table and diffusivity are supported by the water budget analysis of Smith et al. (1996).  Under these conditions, infiltration is fast enough to raise the water table 1m within 16 hr and drop the FS below 1.0 at depths < 2 m (Figs. 7A-B).  Note in Figure 7A that the pressure head response at < 2 m approaches the hydrostatic pressure curve when WT=0m after 10 hr of rain infiltration and is coincident when FS< 1.0 at the same depth (Fig. 7B).  These results are for slopes of 17o and should be considered a conservative estimate of triggering conditions for the full range of initiation slopes (17o-41o).

Conclusions

Results from all three models suggested that soil and hydrologic properties were similar for the landslides triggered by the June 27, 1995 rainstorm in Madison County, Virginia.  The colluvium probably had cohesion less than 2KPa - the lower limit of the range in Table 1. The water table probably had reached the ground surface at the time of failure.  Root strength and tree surcharge had negligible effect on slope stability.  The root-soil system being above the failure plane may account for the lack of stability added by vegetation.  The result that the water table reached the ground surface at the time of failure was supported by the water budget analysis of the storm conducted by Smith et al. (1996).

 

Of the three models evaluated, only SINMAP produced a landslide hazard map that could be directly compared to that produced by Morgan et al. (1999).  The landslide hazard map developed by Morgan et al. (1999) is based upon a model matching topographic conditions of slope inclination and catchment size with landslide locations noted in the June 1995 storm.   In addition to better defining the topographic conditions, SINMAP suggests soil and hydrologic properties that favor the triggering of landslides during intense rainstorms.  The stability index map produced by SINMAP shows regions within the study area where the probability of failure was >90%.  These regions have slopes that coincided with field measured slope inclinations of landslides from the June 27, 1995 storm.   Many of the areas that are considered to have high failure probabilities match those predicted by Morgan et al. (1999).  The stability index map also shows many areas that did not fail during the June 27, 1995, storm.  This overprediction may be attributed to the assumption of homogeneous and isotropic soil properties, uniform landslide thickness and spatial and temporal uniformity of rainfall.  Rainfall intensities to the northeast were not as high as those where landslides were most concentrated (Wieczorek et al., 2000).  Caution must be used when using SINMAP for landslide hazard assessment because of the overprediction of areas subject to ground failure due to the assumptions inherent in the model.

 

Our analysis with LISA suggested the effect of inhomogeneous soil properties on slope stability.  The probability of failure was lowered to 32% when measured soil thickness and slope distributions from Figure 6 of Wieczorek et al. (2000) were used as opposed to the > 90% probability calculated by SINMAP for a constant thickness and uniform slope distribution.   To achieve a probability of > 90% for failure with LISA, we had to lower the soil cohesion from 2-4 KPa to 0-0.5 KPa.  Because LISA allows the use of distributed data, it is the preferred model to evaluate slope stability over SINMAP.  Results from LISA could be used to construct a hazard map, but it is not a function presently built into the model.

 

Of the three stability models used in this study, only Iverson's transient response model evaluated stability conditions as a function of time and depth.  This model demonstrated how the pressure head changes with depth during a rainstorm from which conditions of failure can be identified.  Results from this model may be used to produce a time series of landslide hazard maps showing how the factor of safety changes with depth during a rainstorm.   This model would be the preferred method of the three models to evaluate landslide hazards on a regional scale in areas prone to rain induced landslides as it considers both the transient and spatial response of pore pressure in its slope stability calculation.

 

The three models evaluated in this study may be used to assess landslide hazards in areas prone to rain-induced shallow debris flows.   The choice of model depends on the available data for the study area.  The accuracy of these models improves when soil and hydrologic properties are constrained from either in-situ or laboratory measurements instead of using tabulated values for similar soil type. Validation of a model requires post hazard field measurements such as soil failure thickness, initiation slopes, and landslide locations.  These data allow a comparison to be made between model data and actual data.  The discrepancies between data sets can be used to evaluate the assumptions of the models.  It is recommended that high resolution (10 m or less) DEMs be used with these models.  Care must be used when interpreting the results from these models given the assumptions and limitations of each model.

 

Acknowledgments

The comments and suggested improvements made by Scot Southworth and Mark Reid were greatly appreciated.  The work was conducted by Morrissey under contract to the U.S. Geological Survey Landslide Hazards Program.

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