E3S Web of Conferences 415, 02016 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341502016

A general two phase depth integrated model considering porewater pressure and dewatering
Manuel Pastor1*, Saeid Moussavi Tayyebi1, Andrei Hernandez1, Lingang Gao1, Miguel Martín Stickle1, Pedro Navas2,Diego Manzanal2, Ángel Yagüe2 and Miguel Molinos1
1Department of Mathematics and Computers Applied to Civil and Naval Engineering, ETS Ingenieros de Caminos, Universidad

Politécnica de Madrid, Calle del Profesor Aranguren, 3, 28040 Madrid Spain
2Department of Continuum Mechanics and Theory of Structures, ETSI Caminos, Canales y Puertos, Universidad Politécnica deMadrid, Calle del Profesor Aranguren, 3, 28040 Madrid Spain

Abstract. Debris flows can be considered a type of landslide with large velocities and long run-out
distances. There are many types of debris flows, depending on the properties of the solid and fluid
components of the mixture. The triggering and propagation of debris flows can be studied using a single 3D
mathematical model. The computational cost can be very high because of their length, and depth-integrated
models provide a good combination of accuracy and cost. Both types of models can be combined in theanalysis, using 3D models for initiation and at singular points where more accuracy is wanted. As in a chain
where the strength is never higher than that of the weaker link, we have to ensure that all the models areaccurate enough in a joint model. This paper deals with a new depth-integrated model which can take into
account the changes caused by dewatering in a debris flow.

1 Introduction
Among all types of landslides, shallow landslides on
saturated materials, especially those where excess porewater pressure has developed, present large runout
distances and velocities. This is the case of flow slides
and debris flows. Regarding the former, in some cases,
the pore air pressure can raise and cause "dryliquefaction".
It is important, therefore, to describe the main
phenomena on which pore pressure depends, and how to
implement them in mathematical and numerical models.
These phenomena are:
 Erosion of pressurized saturated basal materials
 Dewatering caused by pore fluid flowing out thebody of the landslide through the basal surface orabandoning the landslide when the interaction forces aresmall (low equivalent permeability).
 Pore pressure build-up and dissipation along thevertical direction caused by changes in height, porosity,
or consolidation
All these mechanisms can be characterized bysuitable time scales which allow writing the governing
equations in non-dimensional form. We can introduce:i. The erosion time 𝑇𝑒 = ℎ/𝑒𝑟 where 𝑒𝑟 is the erosion
rate.
ii. The dewatering time (in the case of materials withhigh permeability), given by 𝑇𝑑𝑒𝑤𝑎𝑡 = ℎ/𝑘̅𝑤
where 𝑘̅𝑤 is the perm-eability (m/s).

iii. The consolidation time, given by 𝑇𝑐 = ℎ2 /𝐶𝑣
where 𝐶𝑣 is the coefficient of con-solidation(m2 s −1 ).
iv. The propagation time 𝑇𝑝𝑟𝑜𝑝 .
All these mechanisms can be characterized bysuitable time scales which allow writing the governing
The relation between these characteristic times isfundamental when choosing which model has to beused. Here we will consider only dewatering andconsolidation, as erosion has been described in Pastor etal. (2022). Our contribution will include some examples
showing the importance of the characteristic times
presented above.

2 Mathematical model
We will consider the general case of the soil column
sketched in Fig. 1, where only the lower part of thelandslide is saturated, the upper being dry. This situation
happens when (i) the initial sliding mass has thisstructure, for instance, when water infiltrates from thebasal surface into the soil, and (ii) when horizontal
velocities for water are larger than those of the solid, andmove ahead of the sliding mass (iii) when the porosityof the bed material is high and so is its permeability, as
it happens when the material flows over basal grids,
water flows out of the soil column, and (iv) when thedebris flows arrives to a permeable wall or barrier. We
have denoted by 𝛼𝑠𝑎𝑡 the fraction of the total height ℎwhich is saturated. This height will be denoted as ℎ𝑠𝑎𝑡 =𝛼𝑠𝑎𝑡 ℎ.

* Corresponding author: manuel.pastor@upm.es© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0(https://creativecommons.org/licenses/by/4.0/).

E3S Web of Conferences 415, 02016 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341502016

The general model for the saturated part is based on
the work of Biot (1955), the group of the late Prof.
Zienkiewicz at Swansea University (Zienkiewicz andShiomi (1984), Zienkiewicz et al. (1990,1999) andAnderson and Jackson (1967), and the application todepth integrated debris flows proposed by Pitman andLe (2005) and Pudasaini (2007). The model wasrecently extended to debris flows with excess porepressure by Pastor et al. (2021).
The model is based on the assumption that themixture is composed of a solid phase and a fluid phase.
The equations are (i) Balance of mass and (ii) Balanceof linear momentum for the constituents and themixture, (iii) constitutive or rheological laws describing
the material behaviour of all constituents, and (iv)kinematics relations linking velocities to rate ofdeformation tensors.

𝑛̅ =ℎ𝑤(3)ℎ𝑤 +𝛼𝑠𝑎𝑡 ℎ𝑠

𝛼𝑠𝑎𝑡 =

1−𝑛̅ ℎ𝑤
𝑛̅

(4)

ℎ𝑠

Using the proposed approach, once a threshold valueof porosity 𝑛̅𝑚𝑖𝑛 is reached, it will keep constant, withthe saturated layer height decreasing,
𝛼𝑠𝑎𝑡 =ℎ𝑠𝑎𝑡 =1−𝑛̅𝑚𝑖𝑛 ℎ𝑤𝑛̅𝑚𝑖𝑛

1−𝑛̅𝑚𝑖𝑛 ℎ𝑤
𝑛̅𝑚𝑖𝑛(5)ℎ𝑠ℎ𝑠ℎ(6)In the limit, we see that ℎ𝑤 → 0 and ℎ𝑠𝑎𝑡 → 0 .

3 Modelling of excess pore water pressure
The finite difference equation describing the evolutionof excess pore water pressure is𝑛𝜕∆𝑝𝑤 𝑛𝜕 2 ∆𝑝𝑤
𝑥3 𝑑ℎ𝑠𝑎𝑡 𝜕∆𝑝𝑤 𝑛| = 𝐶𝑣| −|2
𝜕𝑡 𝑗
𝜕𝑥3 𝑗 ℎ𝑠𝑎𝑡 𝑑𝑡 𝜕𝑥3 𝑗−𝐸𝑚

The 3D model is integrated along depth, including
both saturated and unsaturated layers. We will introduce
for convenience three auxiliary variables ℎ𝑠 , ℎ𝑎 and ℎ𝑤
which characterize the solid, air and fluid contents in acolumn of water of height ℎ (Fig. 2).

𝑇𝑐𝑜𝑛𝑠ℎ 𝜌 g= 0 𝑤
𝑇𝑑𝑒𝑤
𝐸𝑚The relations between them are:(1)

The authors acknowledge the financial support provided by
the Inter-American Development Bank (Project number: EST1343) and the Spanish Ministry MINECO under the PLAND project (PID2019-105630GB-I00).

whereℎ𝑠 +ℎ𝑤
1−𝑛̅(1−𝛼𝑠𝑎𝑡 )(8)which does not depend on permeability and, in mostcases, tends to zero. In those cases where propagationand dewatering times are similar, consolidation happensmuch faster, and we can approximate the excess porepressure distribution along depth for that of steady state.
For instance, in the case of a dewatering column. We
can assume that the excess pore pressure varies between
zero at the surface of the saturated part and 𝜌𝑤 𝑏3 ℎ𝑠𝑎𝑡 at
the bottom - where we assume pore pressure isatmospheric. If the time of consolidation is comparable
to the time of propagation, the dewatering time will bemuch larger than both.

Fig. 2. Magnitudes characterizing the unsaturated debris flow
components.

ℎ=(7)where the partial derivative 𝜕∆𝑝𝑤 ⁄𝜕𝑡|𝑛𝑗 where the partial derivative 𝑗. 𝐶𝑣 is the coefficient of consolidation.
𝐸𝑚 the oedometric modulus. 𝑛̅ the depth averagedporosity.
The first term of the RHS characterizes consolidation, the second takes into account that the mesh is ofarbitrary lagrangian eulerian type, the third takes into
account the changes in porosity and the fourth thechanges in total stress induced by changes of height.
The boundary conditions are: (i) at the saturated-dry
interface where ∆𝑝𝑤 = 0 and (ii) at the bottom, wherethe condition is 𝑝𝑤 = 𝑝𝑎𝑡𝑚 → ∆𝑝𝑤 = 𝜌𝑤 𝑏3 𝛼𝑠𝑎𝑡 ℎ in thecase of dewatering.
This equation includes an advective term depending
on 𝑇𝑐𝑜𝑛𝑠 /𝑇𝑑𝑒𝑤 . This term can be approximated as:Fig. 1. Representative column of soil with two layers.

ℎ = ℎ𝑠 + ℎ𝑎 + ℎ𝑤

1 𝑑 (𝑠)𝑛̅𝑑 (𝑠)ℎ𝑥− 𝜌′𝑏3(1 − 3)
1−𝑛̅ 𝑑𝑡𝑑𝑡ℎ(2)

2

E3S Web of Conferences 415, 02016 (2023)
DFHM8

https://doi.org/10.1051/e3sconf/202341502016

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