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Debris mobility modelling

Empirical approach

The earliest approach to assessing debris mobility of natural terrain landslides is largely based on the experience gained from the use of empirical method in estimating runout distances of landslides on man-made slopes. Amid the growing concern on the potential hazards of natural terrain landslides in the 1990s, empirical methods, initially based on the concept of travel angle and later evolving into other enhanced formulations, have been developed and applied to assess debris mobility of natural terrain landslides.

Fig 1. Testing

(a) The Travel Angle Method Travel angle is the angle of the line connecting the head of the landslide source to the distal end of the displaced mass. The term is similar to other terms like “apparent angle of friction”, “equivalent coefficient of friction” and “average coefficient of friction” but these latter terms are derived from the line linking the centres of gravity of the landslide source and the displaced material. The travel angles of landslides tend to decrease (i.e. mobility of landslides increase) with an increase in debris volumes and that the travel angles of landslides are critically dependent on the failure mechanisms and modes of debris movement. Lo (2000) summarised the distribution of the travel angles for different types of natural terrain landslides in Hong Kong (Fig. 2).

Lau & Woods (1997) discussed one of the limitations of the travel angle method for natural terrain landslides. It is noted that the accuracy of estimating runout distance (d) by the travel angle method decreases rapidly when the slope angle at the final point of debris deposition approaches the angle of reach of the landslide. This limitation is particularly significant when the method is applied to steep natural terrain, given the same degree of change in the angles of reach (d) (Fig. 3). Furthermore, the travel angle method does not take the effects of terrain characteristics on debris mobility into consideration. For example, different terrain profiles may lead to debris having different runout behaviours, and therefore different mobility, while their respective travel angles may stay the same. The complex terrain characteristics of typical natural hillsides in Hong Kong have mean that the use of the travel angle method is subject to constraint.

(b) The Toe Slope Angle Method

The toe slope angle is another parameter that has been studied in Hong Kong in the late 1990s for assessing debris mobility of natural terrain landslides. The toe slope angle is defined as the ground slope angle at the distal end of a landslide trail. Choi et al (2003) examined the toe slope angle and the distance that landslide debris has travelled beyond 15° ground slope for more than 50 m of large (scar > 15 m wide) recent natural terrain landslides in the Natural Terrain Landslide Inventory (NTLI). The results show that only about 15% of the landslides in the NTLI reached ground with a slope angle less than 15° (or have toe slope angles smaller than 15°) and only two of them have travelled more than 50 m beyond 15° sloping ground. Based on this study, it can be observed that natural terrain landslides in Hong Kong do not usually travel beyond ground with a slope angle less than 15° and even fewer travel more than 50 m beyond 15° sloping ground. This suggests that the 50 m zone provides an adequate buffer area for debris deposition in the majority of the cases of natural terrain landslides in Hong Kong. The small portion of landslides with long runout distances is mostly associated with very mobile channelised debris flows. The findings serve as a rough indication of the typical landslide debris mobility in Hong Kong. The method has also been adopted in the screening criteria currently adopted by the GEO for screening new development sites for natural terrain hazard studies.

(c) Hybrid Method Using Travel Angle and Travel Distance

Because of the limitations of the travel angle method described above, an additional parameter, travel distance beyond 15° sloping ground, has been incorporated in the model to account for the effects of terrain characteristics along credible flow paths on debris mobility. An empirical classification of proximity zones of facility at risk of natural terrain landslide hazards based on the travel angles and the travel distances beyond 15° sloping ground of the historical landslides in the NTLI has been developed. This has been applied in the consequence model of the global quantitative risk assessment on natural terrain landslides in Hong Kong. (Wong et al 2006; Wong & Ho 2006) (Fig. 4).

The development of the empirical approaches to assessing debris mobility of natural terrain landslides has provided geotechnical practitioners in Hong Kong with useful tools in estimating probable runout distances of natural terrain landslides and potential risk to population. There is however practical limitation of the empirical methods - they cannot provide information on the runout behaviour of landslide debris in motion. Runout behaviour may include debris velocity, debris thickness and lateral spread of debris, which are critical information for systematic study of landslide behaviour, determination of debris influence zone and engineering design of landslide mitigation measures.

Lumped mass approach

Lumped mass model treats landslide debris as a non-deformable mass block.  The equation of motion of the mass block can be derived from the Newton’s Second Law.

(a)  Equation of motions

Let’s consider a ball rolling down a frictionless inclined plan (Fig. 1).  The net force (F) acting on the ball is mg sinq.  Therefore, with Newton’s Second Law (F = ma; m is the mass of the ball and a is the acceleration of the ball), we have

Once the acceleration of the ball (a) is determined, we can calculate the velocity and displacement of the ball at any time step.

The equation of motion of a lumped mass block is very similar to the above scenario but we need to consider one additional force rather than the body force (i.e. mg sinq  ).  Fig. 2 depicts the forces acting on the lumped mass block.  In addition to the body force, friction at the base of the lumped mass should be considered as shown in the figure.

The net force (F) acting on the lumped mass block is mg sinq  - Basal Friction.  If we assume that basal friction is a Coulomb type friction (i.e. the friction is linearly proportional to the normal force), the basal friction can be expressed as

Basal Friction = mg cosq tanf; where f = friction angle

Following Newton’s Second Law, we can determine the acceleration (a) of the lumped mass block:

With the acceleration (a), we are able to determine the velocity and displacement of the lumped mass block.  However, slope inclination (q) and friction angle (f) may vary along the landslide trail.  In order to determine the velocity and displacement along the landslide trail, we adopt the concept of time marching.

(b)  Concept of time marching

As slope inclination (q) and friction angle (f) vary along the landslide trail, we calculate the velocity and displacement of the lumped mass block in a piecewise manner.  We first determine the acceleration of the lumped mass using the equation a = g (sinq - cosq tanf) based on the values of q  and f  at the initial location of the lumped mass.  We then calculate the velocity and displacement over a very small time duration (say, Dt) based on the acceleration (see also Fig. 3).

(c)  Limitations of lumped mass model

With  lumped mass model and the concept of time marching , we are able to determine  the  velocity  and displacement of a lumped mass block at any given time after the motion of the mass block starts.  The application of lumped mass model for calculation of landslide debris runout is attractive because it is simple and computationally efficient.  However, there are several limitations:

(i)  the calculation does not consider debris thickness, which is needed for design of mitigation measures;

(ii)  we have no idea about the likely debris influence zone (see Fig. 4).

Dynamic Model DAN

Beginning with the principle of lumped mass model (see also the second module “Lumped Mass Model”), Hungr (1995) proposed a numerical model for analysing dynamics of landslide debris. The numerical model has been coded into a Window-based program called “DAN-W” (see also the user manual of DAN-W, HGR (2010)).

(a) Hungr’s idea

Hungr (1995) proposed to divide the landslide debris into several slices. Each of the slices can be regarded as a lumped mass block. We can then work out the net force acting of the slices. Following the principles of lumped mass model, we calculate the velocity and displacement of the slices. Fig. 1 shows the idea for two slices.

Having calculated the new positions of all slices, we then compare the separation distance between slices with the previous separation distance. If the separation distance is shorter than before, the thickness of debris at that particular location would be greater. In contrast, if the separation distance is larger than before, the thickness of debris at that particular location would be smaller. Fig. 2 illustrates this concept. At time step i, the separation distance of the slices of red and green cross is si. If si is larger than si+1 i.e. the separation distance at time step i +1, thickness debris in between the slices of red and green crosses would be smaller at time step i+1 than the time step i.

The above introduces the concept of how DAN-W determines changes in debris thickness at different time steps. However, it is a simplified version which does not consider the width of the landslide trail. In addition to the separation distance, the width also plays a role in determining the debris thickness. We shall return to this point in the next page “Dynamic Model DMM”.

(b) Key parameters used in the model

The equation of motion of the landslide debris slice governs the calculation of the debris mobility. Its accuracy relies on the net force acting on the slice. Fig. 3 depicts the forces acting on a slice (Hungr (1995) assumed that the cross-section of debris slice is rectangular in shape). The forces include basal friction, body force and the internal force. The basal friction and body force have been introduced in lumped mass model. The internal force is acting on the interface between slices which is akin to lateral earth pressure.

Lateral pressure

Hungr (1995) calculated the lateral earth pressure within the landslide debris by multiplying the overburden pressure with a pressure coefficient. The pressure coefficients corresponding to ‘active’ and ‘passive’ states are selected based on the separation distance between debris slices. For example, when we calculate the lateral pressure acting at the interface between the slices of red and green cross at time step i+1, we will compare si and si+1. If si > si+1, this means that the two slices are getting closer together. In this case, the lateral earth pressure coefficient corresponding to ‘passive’ condition would be applied. In contrast, if si < si+1, an ‘active’ earth pressure coefficient would be used. The ‘active’ and ‘passive’ earth pressure coefficients could be in a function of basal frictional angle (f) and internal friction angle of debris (a) (Pudasaini & Hutter, 2007):

DAN Equation.png

In addition to ‘active’ and ‘passive’ conditions, ‘at-rest’ condition may come into play. This happens when si = si+1 . The value of ko could be taken as 1.0 or other appropriate value.

Basal friction

When friction rheology is adopted, shear stress to resist the debris motion would be calculated based on the effective stress (s’) at the base of the debris slice. DAN-W works with total stress and apparent friction angle (fa) in the related calculations as illustrated below:

When Voellmy rheology is adopted, a turbulence term is included in the calculation of the basal shear stress as expressed below:

(c) Calibration

Hungr (1998) and Anyotte & Hungr (1998) reported back analysis of about 20 debris flows and open hillside failures in Hong Kong. They used the numerical model presented by Hungr (1995) in the analysis. Lo (2000) reported the range of rheological parameters (i.e. fa and x) that could be appropriate for forward predictions of the mobility of landslide debris. The work of back analysis have been continued because more landslide runout data are collected from time to time.  Wong & Kwan (2006) presented the data of back analysis of some 60 mobile debris flows in Hong Kong. The value of ru adopted in their work is 0.5 and Ka = 0.8, Kp = 2.5 and Ko = 1.0 are used.  Single set of voellmy parameters over the whole landslide trail was used in each of the back analyses.  More recent publications and guidance on rheological parameters can be found in TGN 29, TGN 34, TGN 38 and Kwan et al (2013).

Dynamic Model DMM

Cross-section of drainage lines in natural hillsides is very often trapezoid in shape.  Fig. 1 shows a typical drainage line in Hong Kong.  The photograph was taken in Tung Chung after a major debris flow event.  The cross-sectional shape that we often encounter in hillsides does not fit in with the assumption of DAN, in which the cross-section of landslide trail is assumed to be rectangular in shape.  Kwan & Sun (2006) revised DAN’s formulations so that the dynamic model considers a trapezoid landslide trail.  The revised formulations have been programmed in the computer program 2d-DMM, which has two versions that are commonly used, namely the C# Version (Version 2.0) and the Spreadsheet Versions (Versions 1.1 and 1.2).

(a)  Why bother?

Suppose we are going to carry out debris mobility analysis for a 200 m3 landslide scenario.  The landslide trail profile and rheological parameter are presented in Fig. 2.  Field inspection confirms that cross-section of the landslide trail is trapezoid in shape.  The base width of the cross-section is 3 m and the banks of the landslide trail is inclined as 45o(see Fig. 3), and the geometry does not vary along the trail.  If we carry out the debris mobility analysis assuming that the cross-section is rectangular in shape, what is the width that should be specified in the analysis?  On the basis that the base width is 3 m, the width we specified for the analysis should be larger than 3 m.  Should it be 6m? or 9m? or…

To answer the question, we have carried out debris mobility analysis using DAN-W (Release 10) and 2d-DMM (Version 1.2).  We used DAN-W to undertake two analyses.  In the first analysis, we assumed that the width is 6 m and in the second analysis, the width is assumed to be 9 m.  In the 2d-DMM (Version 1.2) run, we input the exact geometry of the cross-section for analysis.  The table below shows the results, which indicate that the width specified in DAN-W could influence the calculated runout distance and debris velocity.  The greater the width the shorter the runout distance.  In addition, debris runs slower with a greater debris width.

The 6-m-wide DAN-W analysis gives results similar to the 2d-DMM (Version 1.2) run.  Before we give an explanation, let’s look into how DAN calculates the thickness of debris.  In the previous page “Dynamic Model DAN”, we explained that DAN calculates debris thickness based on the separation distance between debris slices. If the separation distance is getting shorter, the thickness of debris at that particular location would be greater.  In contrast, if the separation distance is getter larger, the thickness of debris at that particular location would be smaller.  Indeed, the separation distance is one of the parameters considered by DAN in determining the debris thickness.  Other parameter is also considered by DAN.  In addition to the separation distance, DAN considers the width of trail specified in determining debris thickness.  Fig. 4 illustrates this idea.  At time step i, the separation distance of the slices of red and green cross is si.  At that particular location if the specified width is w1, then the debris volume between the red and green crosses would be si hi w1.  At time step i+1, the separation distance becomes si+1 and the specified width at that location is w2.  The debris thickness at time i+1 can be determined based on conservation of mass (assumed no entrainment, no deposition and the debris slices are connected with each others).  That is hi+1 = (si hi w1)/(si+1 w2).

The terms hi w1 and hi+1 w2 above are indeed the cross-sectional area of the debris (the cross-section is assumed to be rectangular in shape).  In theory, we can specify the cross-sectional area as a function of debris thickness (h) for analysis (i.e. Volume = si A(hi); if we assume that cross-sectional area is trapezoid in shape, function A(h) can be expressed as

0.5h(2w+h(cotqL+cotqR)), where w is the base width, qL and qR are the side angles of the trapezoid cross-section).  2d-DMM (Version 1.2) adopts this idea for determination of debris thickness.  When w, qL and qR are specified for the analysis, 2d-DMM (Version 1.2) solves the equation

Volume = si 0.5h0.5hi(2w+hi (cotqL+cotqR)) to obtain hi based on the principle of conservation of volume.

Let’s go back to the example discussed above.  The relationship between cross-section area and debris depth is a non-linear function if trapezoid shape is assumed.  If rectangular shape is assumed, cross-section area and debris depth is then in a linear relationship.  Fig. 5 shows the relationship.  The initial thickness of debris is in the range of 0.3 m to 3 m (see Fig. 6).  For debris thickess < 3 m, Fig. 5 indicates that the function of width assumed to be 6 m would be more similar to the trapezoid shape function.  Therefore, the 6-m-wide DAN-W analysis gives results similar to the 2d-DMM (Version 1.2) run.  The findings for 2d-DMM (Version 1.2)​ is equally applicable to 2d-DMM (Version 2.0), since the two versions of the 2d-DMM adopt the same calculation algorithm. ​

Frequently Asked Questions of Using 2d-DMM (Versio​n 2.0)

Input Parameters

1. What are the values of Ka, Kp and Ko to be used?

2. What is the difference between "​Pore pressure ratio for earth pressure" and "Pore pressure ratio for basal friction"? What are they for?

3. What are the rheological parameters for forward predictions of the runout and velocity of open hillside failures (OHF), channelised debris flows (CDF) and failures within topographic depression catchments (TDF)?

4. Landslide debris travels on a broad and planar slope before it enters into drainage line. Should I use OHF parameters in the broad and planar slope portion of runout path and CDF/TDF parameters in the drainage line portion for predictive purpose?

5. How many number of x-y co-ordinates should be entered for specifying the flow path?

6. Entrainment is defined in term of volume/unit time in the programme. However, the hazard assessment may only give you a single value of entrainment volume. How can this value in volume be converted to volume/unit time?

Results Interpretation

7. What value of velocity of a mass block can be used to determine the time when it comes to rest?

8. The predicted debris thickness is very small. Is the model still applicable?

Other

9. How many number of boundary blocks should be entered?

10. The analysis starts without any problem but then 2d-DMM (Version 2.0) stops running. What are the possible causes?

Input Parameters

1. What are the values of Ka, Kp and Ko to be used?

Mobile landslides (including OHF, CDF and TDF) had been back analysed using 2d-DMM (Version 2.0).  In the back calculations, constant values of Ka, Kp and Ko were used.  The value of Ko was taken as 1.0.  The equation developed by Saverage & Hatter was used to estimate the values of Ka and Kp.  The equation is introduced in the ''Mobility Model DAN'' page.  According to the equation, the values of Ka and Kp are dependent on the bulk friction angle (f) and basal friction angle (fb) of landslide debris.  It can be shown that for a typical range of f and fb (20o <f< 35o and 10o <fb< 30o), the average values of Ka and Kp are 0.8 and 2.5 respectively.  These values of Ka, Kp and Ko could be adopted for debris mobility analysis (see figure below).

q1.jpg

2.  What is the difference between "​Pore pressure ratio for earth pressure" and "Pore pressure ratio for basal friction"?  What are they for?

Users shall enter the values of "Pore pressure ratio (Ru) for earth pressure" and "Pore pressure ratio ​(Ru) for basal friction" in 2d-DMM (Version 2.0) at the Setting page (see figure below).  The meaning and recommended values of these two parameters are given below.

q2.jpg

Pore pressure ratio (Ru) for earth pressure: 2d-DMM (Version 2.0) will use this Ru to modify the internal earth pressure.  The principles of the internal pressure calculation are introduced in ''Mobility Model DAN'' page.  In the previous back calculation exercise, the average value of this Ru was taken as 0.5.

Pore pressure ratio (Ru) for basal friction: the values of this Ru  are used to calculate the apparent friction angle in the first and the second segments. 2d-DMM (Version 2.0) allows dividing runout path into two segments and accepts the use of different rheological parameter for those two segments of the runout path based on the values of Friction angle (f) specified in the Settings page (fa = (1- ru) tanf).  Users can specify value of fa in the Settings page direct but in that case the values of Pore pressure ratio (Ru) for basal friction should be input as 0.

3.  What are the rheological parameters for forward predictions of the runout and velocity of open hillside failures (OHF), channelised debris flows (CDF) and failures within topographic depression catchments (TDF)?

Friction rheology is appropriate for modeling OHF and Voellmy rheology is appropriate for modeling CDF and TDF.  TGN 29 and TGN 34 give guidance on the rheological parameters for analyses of OHF and CDF respectively. For analysis of TDF, TGN 38 should be referred to.

4.  Landslide debris travels on a broad and planar slope before it enters into drainage line.  Should I use OHF parameters in the broad and planar slope portion of runout path and CDF/TDF parameters in the drainage line portion for predictive purpose?

Mobile landslides (including over 70 OHF, 60 CDF and 40 TDF) had been back analysed using 2d-DMM (Version 2.0).  The back analyses were benchmarked with the reported runout distance and debris velocity estimated using super-elevation data, where available. In the back analyses, a single rheology model was assumed.  On this basis and for consistency, a single rheology model should be applied in debris mobility analysis.  However, if the landslide runout path involves a substantial portion of OHF before debris entering a drainage line and becoming CDF/TDF, friction rheology could be adopted for modeling the OHF and Voellmy rheology could then be applied for modeling the CDF/TDF.

5.  How many number of x-y co-ordinates should be entered for specifying the flow path?

The recommendation regarding specification of flow path using points given in Hungr (2010) is relevant:

"The input profile should be made reasonably smooth to avoid instability.  Do not use too many points and avoid details such as minor steps in the profile.  Round out abrupt slope changes.  Users should test the influence of such simplification (usually it has relatively small effect on the results, but excessive roughness could unrealistically reduce the runout).  Ideally, a slope profile should have about 15-25 input points."

In all cases, users should adopt a reasonable number of points to represent smooth slope surfaces for realistic computation results.  Besides, smooth profiles of channel width and side angles are also recommended.  The effects of any abrupt change in channel width and side angles are not modelled in 2d-DMM (Version 2.0).

6.  Entrainment is defined in term of volume/unit time in the programme. However, the hazard assessment may only give you a single value of entrainment volume. How can this value in volume be converted to volume/unit time?

Users should work out the entrainment rate (i.e. volume/unit time) by trial and error in order to match the value of entrainment volume suggested in the entrainment assessment.

Results Interpretation

7.  What value of velocity of a mass block can be used to determine the time when it comes to rest?

It is not realistic for the computer model to achieve an exactly zero velocity at the front node for determining the total runout distance.  In fact, since no numerical damping is applied in the program, velocity of the nodes will be fluctuating between very small positive and small negative values, and the node may never come to rest (i.e. the calculated velocity genuinely equals to zero). Therefore, users should set a threshold value of frontal velocity (say, less than 0.3 m/s), below which the nodes can be considered to be at rest, in order to determine the total run-out distance.

8.  The predicted debris thickness is very small.  Is the model still applicable?

2d-DMM (Version 2.0) models landslide debris as a continuum material.  However, when the simulated debris is so thin (say, less than 0.3 m), the dynamics of discrete particles (e.g. rock blocks) in the landslide debris would control the runout behaviour of the landslide debris and the assumed continuum model is no longer valid.  Therefore, the outputs of 2d-DMM (Version 2.0) are nothing more than theoretical values when the calculated debris thickness is very small.  Designers of natural terrain mitigation measures should be aware of this limitation, and assign a nominal thickness for design purposes as appropriate.

Other

9.  How many number of boundary blocks should be entered?

2d-DMM (Version 2.0) adopts a finite difference numerical scheme in which the resolution of computed velocity, basal resistance, debris thickness, etc, along the landslide debris would normally be improved by using more blocks.  However, increasing the number of blocks would not only demand longer processing time (Hungr, 2010), but also involve a higher chance of numerical errors (e.g. excessively  fluctuating velocity and thickness due to boundary blocks overtaking each other).  A balance needs to be struck between solution resolution, processing time and numerical stability.  A range between 11 to 50 number of boundary blocks is recommended for typical design cases.

10.  The analysis starts without any problem but then 2d-DMM (Version 2.0) stops running.  What are the possible causes?

There are several possible causes.  Some common ones are as follows:

-  Incorrect input of runout profile -  users may get some clues in the runout profile shown in the Output page

-  Incorrect input of width profile -  for example, the width is unrealistically small

-  Incorrect input of the chainage of change in segments

References

Ayotte, D. & Hungr, O. (1998). Runout Analysis of Debris Flows and Debris Avalanches in Hong Kong. A Report for the Geotechnical Engineering Office, Hong Kong. University of British Columbia, Canada, 90 p.

HGR (2010). DAN-W Release 10 User Manual. O. Hungr Geotechnical Research Inc., 61p.

Hungr, O. (1998). Mobility of Landslides in Hong Kong: Pilot Analysis Using a Numerical Model. A Report for the Geotechnical Engineering Office, Hong Kong. O. Hungr Geotechnical Research Inc., Canada, 52 p.

Hungr, O. (1995). A model for the runout analysis of rapid flow slides, debris flows and avalanches. Canadian Geotechnical Journal, vol. 32, pp 610-623.

Kwan, J.S.H., Hui, T.H.H. & Ho, K.K.S. (2013). Modelling the motion of mobile debris flows in Hong Kong. Landslide Science and Practice Vol. 3, Proceedings of The Second World Landslide Forum, Italy, C. Margottini et al. (eds.), Springer-Verlag Berlin Heidelberg, pp 29-35. (in press)

Lo, D.O.K. (2000). Review of Natural Terrain Landslide Debris-resisting Barrier Design. GEO Report No. 104, Geotechnical Engineering Office, Hong Kong, 91 p.

Wong, H.N. & Kwan, J.S.H. (2006). Runout and mobility of landslides. Presentation materials for GEO Seminar on 19 May 2006.

Lau, K.C. & Woods, N.W. (1997). Review of Methods for Predicting the Travel Distance of Debris from Landslides on Natural Terrain. Technical Note No. TN 7/97, Geotechnical Engineering Office, Hong Kong, 48 p.

Lo, D.O.K. (2000). Review of Natural Terrain Landslide Debris-resisting Barrier Design. GEO Report No. 104, Geotechnical Engineering Office, Hong Kong, 91 p.

Wong, H.N. & Ho, K.K.S. (2006). Landslide risk management and slope engineering in Hong Kong. Proc., The State-of-the-Practice of Geotechnical Engineering in Taiwan and Hong Kong, Hong Kong, 101-141.

Wong, H.N., Ko, F.W.Y. & Hui, T.H.H. (2006). Assessment of Landslide Risk of Natural Hillsides in Hong Kong. GEO Report No. 191, Geotechnical Engineering Office, Hong Kong, 115 p.

Kwan, J.S.H. & Sun, H.W. (2006). An improved landslide mobility model. Canadian Geotechnical Journal, vol. 43, 531-539 pp.