ORIGINAL PAPER

Estimation of Pore Water Pressure of Soil Using GeneticProgramming

Ankit Garg • Akhil Garg • K. Tai • S. Sreedeep

Received: 27 October 2013 / Accepted: 16 April 2014

� Springer International Publishing Switzerland 2014

Abstract Soil–water characteristic curve (SWCC) is

one of the input components required for conducting

the transient seepage analysis in unsaturated soil for

estimating pore water pressure (PWP). SWCC is

usually defined by saturated volumetric water content

(hs), residual water content (RWC) and air entry value

(AEV). Mathematical model of PWP could be useful

to unearth the important SWCC components and the

physics behind it. Based on authors’ knowledge, rarely

any mathematical models describing the relationship

between PWP and SWCC components are found. In

the present work, an evolutionary approach, namely,

multi-gene genetic programming (MGGP) has been

applied to formulate the relationship between the PWP

profile along soil depth and input variables for SWCC

(hs, RWC and AEV) for a given duration of ponding.

The PWP predicted using the MGGP model has been

compared with those generated using finite element

simulations. The results indicate that the MGGP

model is able to extrapolate the PWP satisfactory

along the soil depth for a given set of boundary

conditions. Based on the given AEV and saturated

water content, the PWP along the depth can be

determined from the newly developed MGGP model,

which will be useful for design and analysis of slopes

and landfill covers.

Keywords MGGP � SWCC � Pore water

pressure � Genetic programming � Evolutionary

approach � Artificial intelligence

1 Introduction

Transient seepage analyses in the unsaturated soil

involve the numerical solution of Richards equation

(Richards 1931), which is often used to determine the

variation of pore water pressure profile (PWPP) with

respect to the two factors: position and time. Such

modelling is useful for analysing rainfall induced

slope failures, design of landfill covers, and also water

uptake by vegetation (Biddle 1998; Blight 2005). Soil

water characteristic curve (SWCC) and permeability

function (PF) are the two input components required

for the computation of PWPP using the unsaturated

seepage modeling. SWCC represents the ability of soil

to hold water which can be also defined as the variation

between soil suction and water content. Saturated

volumetric water content (hs), residual volumetric

water content (RWC) and air entry value (AE) are the

parameters that govern the SWCC. On the other hand,

PF is the variation between unsaturated permeability

and soil suction/water content. It is noted from the

A. Garg (&) � S. Sreedeep

Department of Civil Engineering, Indian Institute of

Technology Guwahati, Guwahati, Assam, India

e-mail: garg1988@gmail.com

A. Garg � K. Tai

School of Mechanical and Aerospace Engineering,

Nanyang Technological University, 50 Nanyang Ave,

Singapore 639798, Singapore

123

Geotech Geol Eng

DOI 10.1007/s10706-014-9755-6

literature that SWCC is usually determined with the

help of several direct and indirect method for

measuring SWCC (Sreedeep 2006; Sreedeep and

Singh, 2011).

Owing to the difficulty in estimating the PF, it is

estimated indirectly from SWCC (Van Genuchten

1980). There are different factors: range and type of

soil suction, procedure adopted for suction measure-

ment, hysteresis, experimental errors etc., which

influences the uniqueness of SWCC (Shah et al.

2006; Sreedeep and Singh 2010; Malaya and Sreedeep

2010). Such variations in SWCC would also influence

the estimation procedure of PF and PWPP. Therefore,

the present study first quantifies the sensitiveness of

SWCC variation on PWPP. For this purpose, the

solution of Richards equation has been performed

using commercial available numerical program

SEEP/W (Geo-Slope 2007). Apart from the numerical

solution, several novel approaches of soft computing

methods such as hybridizing differential evolution

algorithm with receptor editing property of immune

system (Yildiz 2012a, b, 2013a), artificial bee colony

algorithm with Taguchi’s method (Yildiz 2013b, c),

differential algorithm with Taguchi’s method (Yildiz

2013d), cuckoo search algorithm (CS) (Yildiz 2013e)

and immune algorithm with hill climbing local search

algorithm (Yildiz 2009a, b) can also be used to

optimize the performance characteristics of the soil.

The study further demonstrates the use of an

evolutionary approach, namely, multi-gene program-

ming (MGGP) for generating PWPP in unsaturated soil

for a given boundary value problem and important

SWCC parameters such as RWC, AEV, slope of SWCC

and hs. The method is well known for producing models

which represents the explicit relationship between the

input and output process parameters. The advantage of

MGGP method is that it can predict PWPP directly

without the need to perform numerical solution of highly

non- linear Richard’s equation. The PWPP obtained

using the MGGP model has been compared with those

obtained from the numerical solution.

2 Finite Element Modeling (FEM)

for the Generation of Data

A commercial finite element program, SEEP/W (Geo-

Slope 2007) is used to conduct the transient seepage

analysis by solving the Richards Equation. As shown

in Fig. 1, the problem domain consists of a cylindrical

column of homogenous sandy silt. The diameter and

height of the column are 3 and 6 m, respectively. One

dimensional transient unsaturated seepage analysis

was performed for this soil column by specifying the

following boundary conditions.

1. At the top of the column, total head causing

seepage is 15 m by considering bottom of the

column as reference datum. This corresponds to a

ponding condition of water of 9 m above the top

surface of the column.

2. At bottom of the column, pressure head is zero,

which specifies water table condition.

3. The sides of the column are specified as no flow

boundary condition.

4. Effect of evaporation has not been considered in

this modelling.

5. Initial condition is based on the PWP condition

which varies from the groundwater table.

The seepage modelling results are obtained in terms of

parameters such as PWP as a function of both depth.

The parametric variation with time has been obtained

for a point at 1 m depth from the top surface of the soil.

This is mainly because the variations in parameter due

to unsaturated seepage will be more prominent at a

shallow depth and hence the results obtained would

clearly help to understand the sensitivity of SWCC

better.

The reference SWCC (SW) for the present soil has

been obtained from the literature (Sreedeep and Singh

2011). Further, the statistical variation of SW has been

obtained as depicted in Fig. 2a, b, c. It can be noted

Soil

Water table

9 m

6 m

3 m

Seepage Figure not to scale

Depth

Fig. 1 Soil column modeled in this study

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from the figure that varied SWCCs have been obtained

by systematically varying the parameters hs and AEV.

SW1 and SW2 are vertical variation of SW by

changing hs while AEV remaining constant. SW3

and SW4 represent horizontal variation in SW by

changing AEV, while hs remaining constant. SW5 and

SW6 represent combined vertical and horizontal

variation by changing both AEV and hs. For a

particular set of SWCC variation, the RWC and slope

change is negligible. For clarity, the important

parameters of SWCC such as hs, AEV, RWC and

slope are listed in Table 1. These statistically varied

SWCCs have been used to model the seepage situation

as discussed before. This would help to understand the

influence of SWCC variation on unsaturated seepage

behaviour. It must be noted that the difference in

seepage results, if any, would be mostly attributed to

the change in AEV and hs of SWCC, as discussed

above. The saturated hydraulic conductivity ksat

considered in this study is 10-5 m/sec. The transient

seepage modelling with respect to different SWCC

variations for a homogenous unsaturated sandy silt

layer has been obtained with the help of SEEP/W

software (Geo-Slope 2007). Results of the transient

seepage analysis of unsaturated sandy silt layer in

terms of variation of parameters such as PWPP at end

of 1,000 s of duration of ponding is presented as

follows.

As shown in Figs. 3, 4 and 5, there is marginal

variation in PWP versus depth response in the seepage

zone (1 m) for SWCC variations considered in this

study. In general, PWP reduces to a negative value

close to 1 m depth indicating that the seeping water

front has just reached this depth within the prescribed

time. This observation is consistent with that of hvariation with depth. From 1 m depth, PWP further

increases and approaches to zero at the bottom of the

soil column, which is defined as water table boundary.

Such an increase in PWP with depth may be attributed

to a marginal increase in h with depth.

2.1 Observations from the FEM Analysis

The influence of the three types of SWCC variations of

a sandy silt (vertical, horizontal and combined vertical

and horizontal), on unsaturated seepage modeling

results were investigated. The results indicate that the

vertical variation in SWCC due to change in saturated

volumetric water content influences seepage modeling

results considerably. For the range of variation in AEV

(horizontal variation in SWCC) considered, there is

not much influence on the seepage modeling results.

Further investigations are required for higher varia-

tions in AEV. It is observed that maximum variation in

seepage modeling results is obtained when there is

combined vertical and horizontal variation in SWCCs.

It is also worth noting that maximum % variation in

seepage parameters is almost proportional to the

percentage variation in SWCCs. The result indicate

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

θ

Suction (kPa)

SW SW1 SW2

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

SW

SW3

SW4

θ

Suction (Kpa)

0.01 0.1 1 10 100 1000

1E-3 0.01 0.1 1 10 100 1000 10000

1E-3 0.01 0.1 1 10 100 1000 10000

0.000.050.100.150.200.250.300.350.400.450.500.550.60

SW

SW5

SW6

θ

Suction (kPa)

(a)

(b)

(c)

Fig. 2 a Details of SW, SW1 and SW2. b Details of SW, SW3

and SW4. c Details of SW, SW5, SW6

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that the experts dealing with the unsaturated soil need

to be more cautious in using the appropriate SWCC for

the seepage modeling for obtaining the realistic

results.

2.2 Data Preparation for Training MGGP Model

168 sets of data samples are generated from FEM as

discussed. Five input variables (AEV (x1), RWC (x2),

hs (x3), slope (x4), depth (x5)) and output variables

(PWP (y)) is discussed. First 120 samples are chosen

as a set of training data with the remaining as a set of

test samples. Training samples include the measure-

ment values of PWP at RWC values of 0.04, 0.05 and

0.06, whereas, the testing samples include the

measurement values of PWP at the RWC of 0.07

and 0.09. The test data samples are used for testing the

extrapolation ability of the MGGP model while only

the training is used for formulating the model.

3 Multi-gene Genetic Programming

To have an idea about the working of the evolutionary

approach, MGGP, firstly the GP is discussed. Based on

the collected experimental data, GP evolves the

models. These models are generated automatically

without any pre-definition of the structure of the model

(Koza 1996). Mechanism of GP is same as that of

genetic algorithms (GAs). Only difference between

them is that the latter evolves solutions represented by

Table 1 Details of SWCC variations used in this study

SWCC hs AEV (kPa) RWC Slope (linear) % variation w.r.t to. SW

hs AEV RWC Slope (Linear)

SW 0.35 0.695 0.05 0.020 0 0 0 0

SW1 0.40 0.695 0.06 0.023 14 0 0 15

SW2 0.45 0.695 0.07 0.026 29 0 0 30

SW3 0.35 1.110 0.06 0.012 0 60 0 -35

SW4 0.35 1.530 0.06 0.009 0 120 0 -55

SW5 0.45 0.903 0.07 0.020 29 30 0 0

SW6 0.55 1.112 0.09 0.019 57 60 0 -5

0 1 2 3 4 5 6 7 8-80

-60

-40

-20

0

20

40

60

80

SW SW1 SW2

PW

P (

kPa)

Depth (m)

Fig. 3 Variation of PWP with depth for SW1 and SW2

0 1 2 3 4 5 6 7 8-80

-60

-40

-20

0

20

40

60

80

SW SW3 SW4

PW

P (

kPa)

Depth (m)

Fig. 4 Variation of PWP with depth for SW3 and SW4

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strings of fixed length in real or binary form, whereas,

the former evolves models represented by tree struc-

tures of different sizes (Garg et al. 2014a, b).

In GP, the models are generated by combining the

elements randomly from the user-defined functional and

terminal set. Ramped half-and-half algorithm is applied

to generate the models of uniform shape and size. The

elements, specifically, basic arithmetic operations

(?, -, 9, /, etc.), occupy the functional set F. The input

variables and range of constants considered in the study

defines the terminal set T. Number of models generated

is represented by a population size. One such example of

model formed is shown in Fig. 6. After the initialization

of the models, their performance is evaluated based on

the user-defined fitness function. The fitness function

commonly used is root means square error (RMSE)

given by

RMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

PNI¼1 jGi � Aijz

N

s

� 100 ð1Þ

Where Gi is the valued predicted of ith data sample by

the MGGP model, Ai is the actual value of the ith data

sample and N is the number of training samples.

Given the fitness values of the models, models are

ranked and selected for the genetic operations such as

crossover, mutation and reproduction to form a new

population. In crossover operation, a branch of tree is

randomly selected from both the parents and swapped

between them. In mutation operation, a random node

from the tree is selected and replaced by the branch/or

the whole new generated random tree. The process of

producing new population/generation continues as

long as the termination criterion is not met. Termina-

tion criterion is set by the user and is the maximum

number of generations and the threshold error of the

model, whichever is achieved earlier.

In MGGP algorithm, each model in the evolution-

ary stage is formed from the combination of set of

genes/GP trees. There are numerous applications of

MGGP algorithm in field of engineering and finance

(Garg and Tai 2011, 2012a, b, 2013a, b, c; Garg et al.

2013a, b, c, d, e). The step-by-step procedure of

MGGP algorithm is as follows

BEGIN

Step 1: Define problem

Step 2: MGGP algorithm

Begin

2.a Define parameters such as population size,

generations, terminal set, functional set, maximum

number of genes, depth, etc.

2.b Generate initial population of genes

2.c Combine genes using least square method to

form MGGP models

2.d Evaluate performance of models based on

fitness function, namely, RMSE

2.e Apply genetic operations and form the new

population

2.f Cross-check the models performance against the

termination criterion, and if not satisfied, GOTO

Step 2.e

End;

END;

0 1 2 3 4 5 6 7 8-80

-60

-40

-20

0

20

40

60

80

SW SW5 SW6

PW

P (

kPa)

Depth (m)

Fig. 5 Variation of PWP with depth for SW5 and SW6

tanh

×

3

+

x

+

4 y

Functions

Terminals

Fig. 6 GP model: 3tanh (x) ? (4 ? y)

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3.1 Implementation of MGGP

The evolutionary search in GP for a generalised model

is highly influenced by its parameters settings. There

are few important parameters that need to be set

properly for the evolution of model of desired

generalisation ability. In the present work, trial-and-

error route is adopted to select the parameter settings.

The parameter settings selected are shown in Table 2.

The function set F consists of few non-linear mathe-

matical functions and arithmetic operators. The func-

tion set chosen comprise of many elements since this

can assist in evolutionary search of broader variety of

nonlinear mathematical models. The parameters:

population size and number of generations represent

the number of models and number of new population

formed from genetic operations respectively. The

population size and number of generations fairly

depends on the complexity of the data. Based on

literature review by Garg and Tai (2012b), the

population size and number of generations should

not be high in-case of large number of data samples to

avoid the problem of over-fitting. The parameters that

influence the size of search space and number of

models to be searched in space is the maximum

number of genes and maximum depth of the gene.

Based on trial-and-error approach and recommenda-

tions by Garg and Tai (2012b), the maximum number

of genes and maximum depth of gene is kept at 6 and 6

respectively.

MGGP method for the prediction of pore water

pressure of soil is implemented in MATLAB

R2010b using software GPTIPS (Searson et al. 2010).

This software is a new ‘‘Genetic Programming and

Symbolic Regression’’ code written based on Multigene

GP (Hinchliffe et al. 1996) for use with MATLAB.

MGGP method is applied to the data set obtained from

FEM analysis in Sect. 2. The best model is selected based

on minimum RMSE on training data from all runs. The

performance of the best MGGP model (see Eq. 3) on

training and testing data is discussed in Sect. 4.

MGGP ¼ 1357:0099þ 0ð�737:0697Þ� ðsinðtanhðsquareðtanhðx5ÞÞÞÞÞ þ ð�973:9708Þ� ðtanhðexpðtanhðx5ÞÞÞÞ þ ð2:8049Þ� ððexpðtanhðx3ÞÞÞÞ þ ððsquareðexpðtanhðx5ÞÞÞÞþ ððexpðtanhðx3ÞÞÞþ ððx2Þ þ ðx5ÞÞÞÞÞ þ ð�598:4632Þ� ðppower(ppower(x2,tanh(x5)),x5ÞÞþ ð0:0001063Þ � ðppower(x5,(cos(x5))

þ ðx5))) + (48:5523Þ � (sin(square(exp(tanh(x5)))

ð2Þ

4 Results and Discussion

The results obtained from the MGGP model is shown

in Figs. 7, 8, 9 on training and testing data respec-

tively. Square of the correlation coefficient (R2) and

relative error (%) between the predicted values and the

actual values of the PWP estimated are given by

R2 ¼Pn

i¼1 Ai � At

� �

Mi �Mt

� �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Pni¼1 Ai � At

� �2Pni¼1 Mi �Mt

� �2q

0

B

@

1

C

A

2

ð3Þ

Relative errorð%Þ ¼ Mi � Aij jAi

� 100 ð4Þ

where Mi and Ai are predicted and actual values

respectively, Mi and Ai are the average values of

predicted and actual respectively, and n is the number

of training samples.

Figures 7 and 8 show the performance of the

MGGP model on the training and testing data in terms

of statistical values of R2. The graph shown in Fig. 7

indicates that the MGGP model have impressively

well learned the non-linear relationship between the

input and output process variables with high R2 values.

The result of the testing phase shown in Fig. 8

indicates that the MGGP model has shown very good

generalisation ability.

Table 2 Parameter settings for MGGP

Parameters Values assigned

Runs 15

Population size 400

Number of generations 100

Tournament size 2

Max depth of tree 6

Max genes 7

Functional set (F) Multiply, plus, minus, plog,

tan, tanh, sin, cos

Terminal set (T) (x1, x2, [- 10 10])

Crossover probability rate 0.85

Reproduction probability rate 0.10

Mutation probability rate 0.05

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The box plot of relative error (%) for the MGGP

model on the training and testing data is shown in

Fig. 9. The box plot shown in Fig. 9 indicates that the

MGGP model have lower mean relative error of 2.18

and 3.22 % on training and testing data respectively,

which explains that it is able to capture the relationship

between process variables reasonably well.

5 Conclusion and Future Work

The present work highlights the importance and need

of estimating the relationship between PWP and

SWCC components of the soil. The study conducts

FEM analysis for analysing the behaviour of PWP in

respect to various parameters of SWCC. Further, the

novel MGGP method is proposed to estimate the PWP

of the soil based on the given set of input parameters.

The performance of the MGGP model is compared

against the data obtained from the FEM. The results

discussed in Sect. 4 conclude that the performance of

the MGGP model is well in agreement with the FEM

generated data. The high generalization ability of the

MGGP model is beneficial for geotechnical experts,

who are currently looking for high fidelity models that

predict the soil behaviour under uncertain input

process conditions, and therefore the additional cost

of measuring input parameters (SWCC, AEV, RWC,

slope and hs) can be avoided.

The MGGP method provides model that represents

explicit mathematical relationship (see Eq. 2) between

the input parameters and PWP, and, thus can be used

offline to extrapolate the PWP. Future work to be done

include the introduction of new complexity measure of

the MGGP model that can gives more compact and

accurate models.

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