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A new method of modelling the rock micro-fracturing process in double-torsion experiments using neural networks

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INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS
Int. J. Numer. Anal. Meth. Geomech., 23, 905}923 (1999)
A NEW METHOD OF MODELLING THE ROCK MICROFRACTURING PROCESS IN DOUBLE-TORSION
EXPERIMENTS USING NEURAL NETWORKS
XIA-TING FENG* AND MASAHIRO SETO
Department of Mining Engineering, Northeastern University, Shenyang 110006, People+s Republic of China
National Institute for Resources and Environment, Tsukuba 305, Japan
SUMMARY
Microfracturing of rock is a complicated damage evolution process. Inaccurate prediction of microfracturing behaviours suggests a need for the development of a better modelling method. Analysis of acoustic
emission (AE) measurements in double-torsion tests indicates that micro-fracturing behaviours during the
loading stage have fractal time structures. This fractal behaviour can be described by C(t)Jt", where D is
the correlation exponent, t is the time and C(t) is the correlation integral. Furthermore, by utilizing measured
AE data, a new method has been developed to model the AE behaviours of micro-fracturing in rock, in air,
and following soaking in water and in a chemical solution of DTAB. The neutral models NN(10,21,2) and
NN(10,20,2) were found to describe reasonably well the AE behaviours of micro-fracturing in rock under air
and DTAB conditions, and water conditions, respectively. The cumulative AE events and the cumulative AE
counts predicted by the neural models agreed well with those measured in experiments. Copyright 1999
John Wiley & Sons, Ltd.
Key words: Rock microfracturing; self-similarity; fractal; neural network; double torsion; acoustic
emission
INTRODUCTION
Accurate recognition and prediction of the AE behaviours of micro-fracturing in rock are
important tasks in earthquake prediction and rockbust prevention, and in the "eld of sub-critical
cracking nuclear waste disposal. Acoustic emission (AE), which is produced by micro-crack
generation and growth, is a ubiquitous phenomenon associated with brittle fracture. It has
provided a wealth of information regarding the failure process of rock. Some important
advances have been made in modelling change of AE behaviours during rock micro-fracturing
process. For example, some researchers obtained the Omori's formula as
n(t)"K/(t#c)N
(1)
* Correspondence to: X.-T. Feng, College of Resources and Civil Engineering, Northeastern University, Shenyang 110006,
People's Republic of China
Contract grant sponsor: Japanese Government ITIT; National Natural Science Foundation of China: contract grant
number: no. 59604001
CCC 0363}9061/99/090905}19$17.50
Copyright 1999 John Wiley & Sons, Ltd.
Received 18 July 1998
Revised 31 August 1998
906
X.-T. FENG AND M. SETO
where n(t) is the frequency of aftershocks per unit time at time t after the main shock (t"0), and
K, c and p are constants.
The constants K, c and p in equation (1) are determined either by trial and error or through an
optimization procedure. Their values vary with di!erent testing conditions (creep, relaxation, etc.)
and depend on whether one is describing the overall AE rate or individual burst sequences. Also,
several researchers have observed a close correlation between inelastic strain and AE.
The existing description of the rock micro-fracturing process are all based on mathematical
functions. Rock has very complicated mechanical characteristics. Its mechanical behaviour is the
consequence of numerous factors relating to the fracturing processes. It is di$cult to include these
factors, especially the geological ones, in a traditional mathematical model. It is, therefore,
extremely hard or even impossible to describe mathematically the fracturing process in rock.
With mathematical modelling alone, a reasonable predictive capability is lacking for rock.
Unlike conventional modelling methods, which are mathematical expressions to approximate
the experimentally observed behaviour of the rock micro-fracturing process, neural networks
o!er a fundamentally di!erent approach to modelling the intrinsic phenomena in rock fracturing.
Neural networks have the capability to avoid some of the intrinsic shortcomings of mathematical
modelling techniques. Neural networks have learning capabilities, and use massively parallel
simple computational units for knowledge representation and information processing. Their
unique learning capabilities can be used to analyse complex non-linear relationships.\ Learning with neural networks is possible for highly non-linear associations even in the presence of
noise and uncertainty. Such learning capabilities encourage some researchers to investigate the
applicability of neural networks to geomechanical problems.\
This paper reports on AE behaviours of Oshima granites in double-torsion tests as measured in
the laboratory. The specimens were placed in air or soaked in water or in a chemical solution of
dodecyl trimethyl ammonium bromide (DTAB). Analysis of the inherent characteristics of AE
data has uncovered self-similarity and nested structures. Considering these inherent characteristics and the measured data, the behaviours of micro-fracturing in rocks in the loading stage
under each soaking condition (air, water and DTAB) has been modelled using neural networks.
Each model has been tested by AE data for other specimens that were not used to build the model.
EXPERIMENTAL METHOD AND RESULTS
Fracture experiments were carried out on Oshima granites under double-torsion conditions at
the National Institute for Resources and Environment. A notch was cut along the central line
parallel to the long axis of the plate specimen (Figure 1). Up to the distribution of weak planes
contained in specimen, three types of specimens have been investigated (Figure 1(b)}(d)).
The specimens were loaded by a hydraulic testing machine under servo-control. The loading
rates were set to values of 0)002, 0)0008, 0)001, 0)02, 0)03, 0)05, 0)06, 0)08, 0)1 and 0)2 mm/min. The
maximum load was set to 70}90 per cent of the strength of the specimen. After the load reaches its
maximum, each specimen was allowed to undergo a relaxation deformation for about 1}2 h.
Displacement was obtained by averaging signals from two displacement transducers attached to
the upper platen. During the AE measurement, the specimens were subjected to the di!erent
soaking conditions (air, water or DTAB).
The AE signals were measured by two PAC nano-30 sensors mounted on the surface of the
specimen, 1)5 cm far from loading point (Figure 1). The frequency was selected to range from 200
to 1200 kHz for channel 1 to record AE events with higher frequency and from 20 to 200 kHz for
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech., 23, 905}923 (1999)
ROCK MICRO-FRACTURING PROCESS
907
Figure 1. (a) Specimen used in double torsion test; the distance of two loading points is 8 mm; there is a notch with 1 mm
width and 1 mm length on the central line of the plate. (b) Type A specimen: rift plane is parallel to the notch and vertical
to the plate plane, grain plane is vertical to plate plane and the notch. (c) Type B specimen: rift plane is parallel to the notch
and the plate plane, hardware plane is vertical to plate lane and is parallel to the notch, grain plane is vertical to plate
plane and notch, (d) Type C specimen: rift plane is vertical to the notch and the plate plane, grain plane is vertical to plate
plane and is parallel to the notch
channel 2 to record AE events with lower frequency. AE parameters, including rise time,
amplitude, event rate, count rate, energy and duration, were recorded by a MISTRAS-2001
system. Signals from the AE sensors were ampli"ed by a pre-ampli"er with 450 dB gain, passed
through a band pass frequency "lter (50}1200 kHz) and a post-ampli"er of 40 dB gain inside the
system. The amplitude threshold was set to be 45dB.
As example, Figures 2}4 shows the time distributions of the AE event rate, the cumulative AE
events, the AE count rate and the cumulative AE counts of type B specimens during the rock
micro-fracturing process exposed to air, water and DTAB, respectively. From the experimental
results, we found that during the loading stage AE became more active as the load increased.
Initially, when the load was small (e.g. less than 4 kg), no AE occurred.
SELF-SIMILARITY OF ROCK MICRO-FRACTURING PROCESSES
The fundamental observation from the measured AE data is that most rock microfracturing
processes have the inherent properties of self-similarity. The self-similarity can be investigated
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech., 23, 905}923 (1999)
908
X.-T. FENG AND M. SETO
Figure 2. Time behaviour of AE of sample no. 7 under air condition: (a) time distribution of AE event rate and cumulative
AE events; (b) time distribution of AE count rate and cumulative counts
using fractal theory. Studies} have already indicated that the spatial distribution of the
fracturing process from the small scale (i.e. micro-cracking on the scale of centimeters) to the large
scale (i.e. an earthquake on the scale of kilometers) has a fractal behaviour. One can ask whether
the distribution of the micro-fracturing process of rock in time dimension also has a fractal
behaviour.
The fractal dimension of the distribution in time is obtained from a correlation integral
concept. Consider a time distribution of AE events in time range (t , t ) (Figures 2}4). The total
number of AE events in this time range can then be counted and is denoted by N. The number of
AE event pairs (p , p ) with time scale t (t (t(t ) is counted as N(t) (t (t(t ). The
G H
correlation integrals C(t) for the AE event distributions (p , p , 2 , p ) were then calculated by
,
2
C(t)"
N(t) (t (t(t )
N(N!1)
(2)
Thus, we can obtain a set of data C(t ) associated with di!erent time scale t . From fractal
G
G
geometry, there exists a relation between C(t ) and t in the form
G
G
C(t)Jt"
(3)
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech., 23, 905}923 (1999)
ROCK MICRO-FRACTURING PROCESS
909
Figure 3. Time behaviour of AE of sample no. 11 under DTAB condition: (a) time distribution of AE event rate and
cumulative events; (b) time distribution of AE count rate and cumulative counts
where D is a kind of fractal dimension called the correlation exponent that gives the lower limit
of the Hausdo! dimension.
The correlation integrals C(t ) versus the time scale t for the AE event distribution during the
G
G
rock micro-fracturing process is plotted on a double-logarithmic scale. The results show that the
actual data fall on straight lines (Figure 5). The coe$cients of determination are greater than 0)99.
The statistical signi"cance indicate that the distributions of acoustics emission events in time
dimension during the rock micro-fracturing process have fractal structures.
The fractal dimensions estimates from the slope for each specimen during the loading (constant
displacement rate) stage are shown in Table I. Here, time scale t was set to be 1, 2, 3, 2 , 64 s,
respectively. It can be seen from the calculated fractal dimensions that micro-fracturing in time in
the loading stage is self-similar.
NEURAL NETWORK MODELLING METHODOLOGY
The learning capabilities allow neural networks to be trained directly with the results of
experiments. Intrinsic rock fracturing behaviour in experimental data is &memorized' in the
connection weights of the network. If the training data contain the relevant information on rock
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech., 23, 905}923 (1999)
910
X.-T. FENG AND M. SETO
Figure 4. Time behaviour of AE of sample no. 19 under water condition: (a) time distribution of AE event rate and
cumulative events; (b) time distribution of AE count rate and cumulative counts
Figure 5. Correlation integrals versus time
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech., 23, 905}923 (1999)
911
ROCK MICRO-FRACTURING PROCESS
Table I. Sample data and fractal dimensions
No.
Plane type
Soaking
condition
Loading rate
(mm/min)
Maximum
load
(kg)
Fractal
dimension
D
Coe$cient
of determination
1
2
3
4
5
6
7*
8R
9*
10R
C
C
B
B
B
A
B
A
A
A
Air
Air
Air
Air
Air
Air
Air
Air
Air
Air
0)05
0)05
0)1
0)05
0)1
0)1
0)1
0)1
0)05
0)01
19)75
21)9
18)5
18)95
20)55
19)4
20)3
19)0
19)30
16)1
0)96
0)96
0)81
0)94
0)88
0)79
0)95
0)85
0)96
0)96
0)9999
0)9998
0)99
0)999
0)999
0)9871
0)9998
0)9932
0)9995
0)9999
11
12*
13
14*
15R
16
B
B
C
C
C
A
DTAB
DTAB
DTAB
DTAB
DTAB
DTAB
0)1
0)1
0)1
0)1
0)1
0)1
17)1
17)1
15)25
15)25
16)41
13)4
0)87
0)89
0)96
0)96
0)94
0)98
0)9960
0)9977
0)9992
0)9991
0)9991
0)9995
17R
18
19
20*
21
A
C
B
B
C
Water
Water
Water
Water
Water
0)1
0)1
0)1
0)2
0)2
10)15
16)8
17)85
10)5
10)5
0)93
0)88
0)96
0)97
0)98
0)998
0)9911
0)9992
0)999
0)9987
* Experimental data from these tests were used to check &over-training' of the network, called &over-training' check cases
R Experimental data from these tests were used for &true' prediction, called prediction cases
fracturing behaviours, then the trained neural network can generalize from its training data to
new cases.
Representation of a non-linear model for AE behaviours during rock micro-fracturing process
Considering the self-similar structures of AE data describes in the previous section, the intrinsic
relationship between the AE behaviours *CC and *CE at a time interval *t (or load increment
H
H
*P) and their previous behaviours CC , CC , 2 , CC , CE , CE , 2 , CE
can be
H\
H\
H\I
H\
H\
H\I
represented by
f : RLPRK
HI
(*CC , *CE )"f (CC , CC , 2, CC , CE , CE , 2, CE , P , P , 2,
H
H
HI
H\
H\
H\I
H\
H\
H\I H\ H\
;P , *P ) ( j"k#1, k#2, 2, )
(4)
H\I
H
where f is the inherent relationship implicated in measured AE data. P , P , 2, P
are
HI
H\ H\
H\I
loads to produce cumulative AE counts CC , CC , 2, CC , and cumulative AE events
H\
H\
H\I
CE , CE , 2, CE , respectively. *P is the load increment to produce AE count increment
H\
H\
H\I
H
*CC and AE event increment *CE . k is number of history points.
H
H
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech., 23, 905}923 (1999)
912
X.-T. FENG AND M. SETO
The goal of modelling of AE behaviours is to "nd a non-linear model NN(n, h , h , 2) to
represent the f in equation (4) as
HI
NN (n, h , h , 2) : RLPRK
(*CC) , *CE) )"NN (n, h , h , 2) (CC , CC , 2, CC , CE , CE , 2
H
H
H\
H\
H\I
H\
H\
CE , P , P , 2, P , *P ) ( j"k#1, k#2, 2 ) (5)
H\I H\ H\
H\I
H
where *CC) , *CE) are computed values that approximate the measured data, *CC and *CE ,
H
H
H
H
respectively. NN (n, h , h , 2) is a neural network (Figure 6), n is the number of input nodes
(n"3k#1), h is the number of nodes on the "rst hidden layer F , and h is the number of nodes
F
on the second hidden layer F .
The neural model represented by equations (5) is distributed in parallel within the network. The
important characteristic for this distributed representation is that data processing is quick
enough to extrapolate predictions of AE behaviours in the rock-micro-fracturing process.
Improved BP learning algorithm
The model NN (n, h , h , 2) can be obtained by using BP algorithm to train neural network. In
order to overcome the &over-training' problem of BP algorithm, an improved learning algorithm
is thus suggested in this study. This improved algorithm uses the minimum-error principle for the
extrapolated predictions of testing cases*instead of the minimum-error criterion for learning
Figure 6. A feedforward neural network representing AE behaviours during the rock microfracturing process. When the
model is used for prediction of novel cases, the predicted AE event increase and AE count increase will be feed back to
input of the model for the next prediction (at k"3)
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech., 23, 905}923 (1999)
ROCK MICRO-FRACTURING PROCESS
913
sample predictions*to determine whether or not the neural network learning should "nish.
Meanwhile, the parameters k, h , h and the learning iterations i are also determined properly by
using this algorithm.
The function used to evaluate the accuracy of predictions for testing cases is
1 K
E (i)" (yL !y )
J
JH
JH
2
H
1 +
E (i)
(6)
E
(i)"
J
M
J
where yL ( (i) is the predicted output for the jth variable of the lth case data set that is not used to
JH
train the network, y is the desired output of the jth variable of the lth case data set and M is the
JH
number of testing samples to be predicted.
The procedure of the improved BP algorithm is described as follows:
Step 1: Collect a measured data set of AE behaviours of the rock micro-fracturing process. The
data set is divided into two groups. One group is (x , y ) (p"1,2, 2, N), used for training the
N N
network to obtain a non-linear model of AE behaviours, while the other group is (x , y )
J J
(l"1, 2, 2, M), used to test the model, where M is number of learning samples. Follow Steps 2}7
to obtain the model NN (n, h , h , 2).
Step 2: Let h "n to 2n#1 and h "0 to 2n#1 carry out Steps 3}6.
Step 3: Use the standard BP algorithm to carry forward calculation of the network to obtain an
initial model for AE behaviours as
(*CC)
(i), *CE)
(i))"NN (n, h , h , 2) (CC
, CC
, , CC
,CE
,
NH
NH
NH\
NH\ 2
NH\I
NH\
CE
, , CE
,P
,P
, ,P
, *P ) ( j"k#1, k#2, 2, p"1,2, 2, N)
NH\ 2
NH\I NH\ NH\ 2 NH\I
NH
where i is the learning iteration.
(7)
Step 4: Use NN (n, h , h , 2) in Equation (7) to compute the extrapolated prediction for testing
cases l at current learning iteration i as
(*CC) (i), *CE) (i))"NN (n, h , h , 2) (CC
, CC
, , CC
,CE
,
JH
JH
JH\
JH\ 2
JH\I
JH\
CE
, , CE
,P
,P
, ,P
, *P ) (j"k#1, k#2,2, l"1,2, 2, M)
(8)
JH\ 2
JH\I JH\ JH\ 2 JH\I
JH
Step 5: Compute the system error of the extrapolated prediction of the learning model as
E (i)" +(*CC) !*CC )#(*CE) !*CE ),
J
JH
JH
JH
JH
1 +
E
(i)"
E (i)
(9)
J
IFF‚
M
J
Step 6: Use the standard delta rule to adjust the connection weight and thresholds to obtain
a more reasonable representation of NN (n, h , h , 2) at given values of k, h and h .
Step 7: Find
E"min ( min +Ek, h ,h (i),)
k,h , h 0)i)I
(10)
where I is the allowable maximum iteration of the system learning.
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech., 23, 905}923 (1999)
914
X.-T. FENG AND M. SETO
The model NN (n, h , h , 2) corresponding to E is thus an optimum non-linear modelling of
the AE behaviours. The model can be used for extrapolating predictions of AE behaviours of new
cases.
Prediction
The learned NN (n, h , h , 2) can be used to obtain &true' prediction of the testing cases. It builds
the whole curve of cumulative AE counts and cumulative AE events from t"0 (or P"0)
condition. It uses feed back *CC) , *CE) to calculate CC)
, CE)
(for example, CC) "
H
H
H>
H>
H
CC #*CC) , CE) "CE #*CE) and use for next increment CC)
"CC
#*CC)
,
H\
H
H
H\
H
H>G
H\>G
H>G
CE) "CE)
#*CE) ) (i"1, 2, 2 ). Thus, a prediction algorithm can be built as follows:
H>G
H\>G
H>G
For the "rst increment prediction ( j"k#1):
(*CC) , *CE) )"NN (n, h , h , 2) (CC , CC , 2, CC ,CE ,CE
H
H
H\
H\
H\I
H\
H\,
,
CE
,
P
,
P
,
,
P
,
*P
)
2
H\I H\ H\ 2 H\I
H
For the second increment prediction (j"k#2):
(*CC) , *CE) )"NN (n, h , h , 2) (CC)
, CC , 2, CC ,CE)
,CE
H
H
H\
H\
H\I
H\
H\,
,
CE
,
P
,
P
,
,
P
,
*P
)
2
H\I H\ H\ 2 H\I
H
where
CC) "CC #*CC) , CE) "CE #*CE)
H
H\
H
H
H\
H
$
( j"k#2)
For the kth increment prediction ( j"2k):
(*CC) , *CE) )"NN (n, h , h , 2) (CC)
, , CC)
, CC ,CE)
, , CE)
H
H
H\ 2
H\I>
H\I
H\ 2
H\I>,
CE , P , P , 2, P , *P )
H\I H\ H\
H\I
H
where
CC) "CC)
#*CC) , CE) "CE)
#*CE)
H
H\
H
H
H\
H
CC)
"CC #*CC)
, CE)
"CE #*CE)
H\I>
H\I
H\I>
H\I>
H\I
H\I>
For the (k#1)th or next increment prediction ( j"2k#1, 2);
( j"2k)
(*CC) , *CE) )"NN (n, h , h , 2) (CC)
, CC)
, , CC)
,CE)
,CE)
, , CE)
,
H
H
H\
H\ 2
H\I
H\
H\ 2
H\I
P , P , 2, P , *P )
H\ H\
H\I
H
CC) "CC)
#*CC) , CE) "CE)
#*CE)
( j*2k#1)
(11)
H
H\
H
H
H\
H
where *CC) , *CE) are the predicted data of AE count increment and AE event increment at load
H
H
increment *P ( j"k#1, k#2, 2), CC) , CE) are the predicted data of cumulative AE counts
H
H
H
and cumulative AE events produced at load P ( j"k#1, k#2, 2).
H
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech., 23, 905}923 (1999)
Copyright 1999 John Wiley & Sons, Ltd.
Table II. The obtained models and their prediction accuracy for testing cases
The neural
model
Squared
error sum of
learning data
No. of
testing
samples
Air
NN (10,21,2)
0)002984
7
Air
NN (10,21,2)
0)002984
8
Air
NN (10,21,2)
0)002984
9
Air
NN (10,21,2)
0)002984
10
DTAB
NN (10,21,2)
0)00918
12
DTAB
NN (10,21,2)
0)00918
15
Water
NN (10,20,2)
0)01111
17
Linear regression between
extrapolated predictions
and experimental measurements
Coe$cient
of determination
y"0)979508*x#0)161066 (for cumulative AE events)
y"0)98783*x#0)268629 (for cumulative AE counts)
y"0)927464*x!0)170074 (for cumulative AE events)
y"0)963391*x#0)777415 (for cumulative AE counts)
y"0)992471*x#0)045633 (for cumulative AE events)
y"0)99501*x!0)41465 (for cumulative AE counts)
y"1)096289*x!0)13459 (for cumulative AE events)
y"1)07123*x!1)870797 (for cumulative AE counts)
y"1)02919*x#0)0937316 (for cumulative AE events)
y"1)09421*x#0)173198 (for cumulative AE counts)
y"0)993487*x!1)82952 (for cumulative AE events)
y"1)07356*x!2)55355 (for cumulative AE counts)
0)999537
0)999812
0)999482
0)999568
0)999124
0)9998
0)999135
0)999233
0)999814
0)999499
0)985906
0)98302
y"0)916425*x!0)341482 (for cumulative AE events)
y"0)104193*x!0)485915 (for cumulative AE counts)
0)998474
0)995476
ROCK MICRO-FRACTURING PROCESS
Note. y: experimental measurements, x: predicted results of the neural model
915
Int. J. Numer. Anal. Meth. Geomech., 23, 905}923 (1999)
Test
condition
916
X.-T. FENG AND M. SETO
Figure 7. Comparison with experimental results for the double-torsion test sample no. 1 under dry condition, used in the
training of the neural network; (a) cumulative AE events; (b) cumulative AE counts
NEURAL MODELLING OF ROCK MICRO-FRACTURING PROCESSES
The methodologies described in the previous section were applied in modelling the AE behaviours of granites under air, water and DTAB conditions. The results of a series of AE tests listed in
Table I were used to obtain and test the model. Cases under each soaking condition (air, water or
DTAB) were divided randomly into three data sets. The "rst data set of tests was used in training
the neural networks to obtain the models NN (n, h , h , 2), while the second data set, identi"ed
with asterisks, was used for &over-training' check. And the third data set, identi"ed with double
asterisks, was used to test the performance of the obtained neural models is generalizing to testing
cases. Modelling was conducted separately for each soaking condition (air, water or DTAB). The
data of cumulative AE events and counts as well as their increments were taken logarithm to the
base 10, and then all data including load data in column were respectively transferred into the
interval [0, 1] before being modelled or used for prediction.
A number of network architectures (h and h ), iterations i, and history points k were evaluated
in developing a neural model by using the improved learning algorithm described above. With
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech., 23, 905}923 (1999)
ROCK MICRO-FRACTURING PROCESS
917
the aim to minimize the errors on the extrapolated predictions for testing cases, the best models
were recognized for each soaking condition, respectively. Table II shows the obtained neural
models for each soaking condition and the accuracy of the extrapolated predictions for testing
cases. It was found that AE behaviours in the rock micro-fracturing process under three soaking
conditions can be represented reasonably well by neural networks if there history points (i.e.
k"3) are included.
For modelling of AE behaviours in air, sample nos, 1}6 in Table I were used to train the
network. The best model was recognized to be NN(10,21,2). At 75 162 learning iterations, the
model gave the best predictions for &over-training' check cases 7 and 9. The learned model was
used to predict AE behaviours for cases 8 and 10. Some selected results are shown in Figures 7}9
to represent performance of the neural model on predicting AE behaviours of new cases. If the
neural model prediction results versus experimental results for testing cases are plotted on the
same scale (Figure 10). The actual data fall almost on straight lines. The coe$cient of determination and regression equations between cumulative AE counts and cumulative AE events,
computed from the neural models and measured experimentally, are presented in Table II. Table
Figure 8. Comparison with experimental results for the double-torsion test sample no. 7 under dry condition, used for
&over-training' check: (a) cumulative AE events; (b) cumulative AE counts
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech., 23, 905}923 (1999)
918
X.-T. FENG AND M. SETO
Figure 9. Comparison with experimental results for the double-torsion test sample no. 8. under dry condition, used
for &true' prediction: (a) cumulative AE events; (b) cumulative AE counts
II shows that the coe$cients of determination, R, are very signi"cant. As can be seen from
Figures 8 and 9 as well as Table II, the model NN (10,21,2) is able to predict the AE behaviours of
testing cases under air condition reasonably well.
Under the DTAB condition, sample nos. 11, 13 and 16 were used to build the neural network.
The best model was found as NN (10,21,2). At 11 245 learning iterations, the model gave its best
prediction for &over-training' check cases 12 and 14 (see Figure 11). The learned model was used to
predict AE behaviours in DTAB for another case 15 (see Figure 12). As can be seen from Figure
12 and Table II, the model NN (10,21,2) is also able to predict the AE behaviours of testing cases
under DTAB condition reasonably well.
In the same way, samples nos. 18, 19 and 21 were used to obtain neural AE behaviour model for
water condition. At 12,362 learning iterations, the best model NN (10, 20, 2) was recognized
accurately and gave the best prediction for &over-training' check case no. 20. As an example, the
simulation results of the trained neural network for the water test case 19 are shown in Figure 13.
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech., 23, 905}923 (1999)
ROCK MICRO-FRACTURING PROCESS
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Figure 10. Correlation between the neural model predicting resuts and experimental results for testing case 8:
(a) cumulative AE events; (b) cumulative AE counts
Figure 11. Comparison with experimental results for the double-torsion test sample no. 12 under DTAB condition, used
for &over-training' check: (a) cumulative AE events; (b) cumulative AE counts
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech., 23, 905}923 (1999)
920
X.-T. FENG AND M. SETO
Figure 12. Comparison with experimental results for the double-torsion test sample no. 15 under DTAB condition, used
for &true' prediction: (a) cumulative AE events; (b) cumulative AE counts
The model gave its generalizing prediction for another case with no. 17 is shown in Figure 14. It
can be seen from Figure 14 and Table II that, after the model NN (10,20,2) gave the simulated
results shown in Figure 13, it is able to describe the AE behaviours of micro-fracturing in rocks
soaked in water and to predict the AE behaviours of testing cases reasonably well.
DISCUSSIONS AND CONCLUSIONS
As a time series, Scholz showed that if a rock sample is located from axial stress after the whole
sample fracture, micro-fracturing activity decays as a function of t\ in a manner similar to
typical earthquake after-shock sequences. Kagan and Knopo!} showed that the rate of
occurrence-dependent shocks increases as t\ making every earthquake a multi-shock event.
Other researchers have analysed the overall AE rate during transient or primary creep in constant
stress experiments. Lord and Koerner found that uncon"ned creep tests on anthracite coal
obeyed equation (1) with c"0 and p decreasing with increasing stress. However, from the fractal
point of view, the present study found that during the loading process, rock micro-fracturing
activity in time obeys a function of t", where D, 0)79)D)1)0, is a Hausdo! fractal dimension.
This result indicates that micro-fracturing processes in rock under air, water and DTAB have
self-similarity.
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech., 23, 905}923 (1999)
ROCK MICRO-FRACTURING PROCESS
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Figure 13. Comparison with experimental results for the double-torsion test sample no. 19 under water condition, used in
the training of the neural network: (a) cumulative AE events; (b) cumulative AE counts
In order to predict the AE behaviours in the rock micro-fracturing process for new cases, a new
method has been proposed in the present study. The methodology is to build neural models by
using measured AE data in double-torsion tests. Considering change of AE behaviours of rock
micro-fracturing under di!erent soaking conditions, the AE behaviours of rock micro-fracturing
under air, water and DTAB conditions were modelled. Considering the inherent path dependency
of the rock micro-fracturing behaviours, various numbers of history points k were evaluated. It
was found that AE behaviours in the rock micro-fracturing process under the three soaking
conditions can be represented by neural networks if three history points (i.e. k"3) were included.
The prediction results for testing cases indicating that the model NN (n, h , 2) (n"3k#1, and h
is determined reasonably for di!erent soaking condition) is able to describe reasonably well the
micro-fracturing activity under each soaking conditions. With the aim of minimizing the error on
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech., 23, 905}923 (1999)
922
X.-T. FENG AND M. SETO
Figure 14. Comparison with experimental results for the double-torsion test sample no. 17 under water condition, used
for &true' prediction: (a) cumulative AE events; (b) cumulative AE counts
the extrapolated predictions, the best neural model for air and DTAB conditions was found to be
NN (10,21,2), while NN (10,20,2) was best for water condition. The coe$cients of determination of
the model prediction results with the experimental results for testing cases are all greater than
0)983.
Network architectures, learning iterations and history points k signi"cantly in#uence on the
accuracy of prediction of the neural models. These factors can be determined reasonably well
during the model recognition process by using the improved BP algorithm. The modelling results
indicate that if the network architecture, learning iteration and parameters, and history point
k are determined reasonably, the neural network approach can give its predictions for novel cases
with high accuracy.
Further work will aim at modelling the relaxation and creep stages and "nding a practical
method to describe, predict, and evaluate the AE behaviours of the fracturing process in rock
Copyright 1999 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech., 23, 905}923 (1999)
ROCK MICRO-FRACTURING PROCESS
923
masses. An integrated intelligent system including fracture mechanical modelling, and damage
mechanical modelling will be developed.
ACKNOWLEDGEMENTS
Finanical support by the Japanese Government ITIT program (for Xia-Ting Feng) and the
National Natural Science Foundation of China (no. 59604001) is gratefully acknowledged. The
authors would like to acknowledge Prof. C. S. Desai and two anonymous reviewers for their
helpful suggestions and comments.
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Int. J. Numer. Anal. Meth. Geomech., 23, 905}923 (1999)
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