The continuum approach in a grouting model
Отримано значення максимального розміру пори, при якому континуальний підхід все ще можна застосовувати в моделюванні поширення цементу в насиченому піску під час цементації, що не руйнує структуру ґрунту....
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Інститут проблем математичних машин і систем НАН України
2014
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| Cite this: | The continuum approach in a grouting model / M.B. Demchuk, N. Saiyouri // Математичні машини і системи. — 2014. — № 2. — 113-116. — Бібліогр.: 14 назв. — англ. |
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irk-123456789-843892015-07-07T03:02:31Z The continuum approach in a grouting model Demchuk, M.B. Saiyouri, N. Моделювання і управління Отримано значення максимального розміру пори, при якому континуальний підхід все ще можна застосовувати в моделюванні поширення цементу в насиченому піску під час цементації, що не руйнує структуру ґрунту. Получено значение максимального размера поры, при котором континуальный подход всё ещё можно применять в моделировании распространения цемента в насыщенном песке при цементации, которая не разрушает структуру грунта. The value of the maximal pore size whereby the continuum approach can still be adopted for modeling cement grout propagation in saturated sand during permeation grouting is obtained. 2014 Article The continuum approach in a grouting model / M.B. Demchuk, N. Saiyouri // Математичні машини і системи. — 2014. — № 2. — 113-116. — Бібліогр.: 14 назв. — англ. 1028-9763 http://dspace.nbuv.gov.ua/handle/123456789/84389 624.048–033.26:621.651 en Математичні машини і системи Інститут проблем математичних машин і систем НАН України |
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Моделювання і управління Моделювання і управління |
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Моделювання і управління Моделювання і управління Demchuk, M.B. Saiyouri, N. The continuum approach in a grouting model Математичні машини і системи |
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Отримано значення максимального розміру пори, при якому континуальний підхід все ще можна застосовувати в моделюванні поширення цементу в насиченому піску під час цементації, що не руйнує структуру ґрунту. |
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Demchuk, M.B. Saiyouri, N. |
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Demchuk, M.B. Saiyouri, N. |
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Demchuk, M.B. |
| title |
The continuum approach in a grouting model |
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The continuum approach in a grouting model |
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The continuum approach in a grouting model |
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The continuum approach in a grouting model |
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The continuum approach in a grouting model |
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continuum approach in a grouting model |
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Інститут проблем математичних машин і систем НАН України |
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2014 |
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Моделювання і управління |
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The continuum approach in a grouting model / M.B. Demchuk, N. Saiyouri // Математичні машини і системи. — 2014. — № 2. — 113-116. — Бібліогр.: 14 назв. — англ. |
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Математичні машини і системи |
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AT demchukmb thecontinuumapproachinagroutingmodel AT saiyourin thecontinuumapproachinagroutingmodel AT demchukmb continuumapproachinagroutingmodel AT saiyourin continuumapproachinagroutingmodel |
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© Demchuk M., Saiyouri N., 2014 113
ISSN 1028-9763. Математичні машини і системи, 2014, № 2
UDC 624.048–033.26:621.651
M. DEMCHUK*, N. SAIYOURI**
THE CONTINUUM APPROACH IN A GROUTING MODEL
* National University of Water Management and Nature Resources Use, Rivne, Ukraine
** Institute of Mechanics and Engineering (I2M), Bordeaux, France
Анотація. Отримано значення максимального розміру пори, при якому континуальний підхід все
ще можна застосовувати в моделюванні поширення цементу в насиченому піску під час цемента-
ції, що не руйнує структуру ґрунту.
Ключові слова: континуальний підхід, елементарний об'єм, цементація ґрунту, похибка числового
розрахунку, ущільнення ґрунту.
Аннотация. Получено значение максимального размера поры, при котором континуальный под-
ход всё ещё можно применять в моделировании распространения цемента в насыщенном песке
при цементации, которая не разрушает структуру грунта.
Ключевые слова: континуальный подход, элементарный объём, цементация грунта, погрешность
численного расчёта, уплотнение грунта.
Abstract. The value of the maximal pore size at which the continuum approach can still be adopted for
modelling cement grout propagation in saturated sand during permeation grouting is obtained.
Keywords: continuum approach, elementary volume, permeation grouting, numerical calculation error,
soil compaction.
1. Introduction
Before tunnelling construction in a weak soil, permeation grouting is used to increase the soil
stand up time. Since chemical grouts are hazardous to environment [1], а cement grout is used as an
infiltrate in this technique. A regime of permeation grouting is determined by the evolution of the
cement concentration distribution in space [2]. Therefore, mathematical modelling of this evolu-
tion is important. The сement grout consists of particles. If they are large enough, they can get
trapped in small pore throats. Otherwise, they can only deposit on walls of these throats and pores
[3]. Therefore, the mathematical description of cement grout propagation in a porous medium is
cumbersome. In [1, 2], to shed light on various issues that arise during the construction of this
description a standard laboratory test is modelled. In this test, cement grout is injected in a vertical
tube opened at the top and filled with water saturated sand. The injection point is at the bottom of
the tube. In the papers [4, 5], to do the same, problem set ups that correspond to in situ grouting
are considered. In [1], the injection pressure reaches 8 bars and it is assumed that at such the pres-
sure the structure of the grouted sand is not ruined. The fact that results of numerical calculations
according to the model [1] of permeation grouting coincide with the results of respective labora-
tory measurements verifies this assumption. Moreover, during grouting of this type the injection
pressure can be as high as 12 bars [6]. In [7], Demchuk argues that permeation grouting per-
formed at such the values of injection pressure can be modelled by a problem with a free moving
boundary. Demchuk [8] presents grouting models of this class and shows that the continuum ap-
proach is properly adopted in them. As for the models [1, 2] and [4, 5], they can be used only
when a size of an elementary volume over which the averaging is performed in the continuum
approach is much smaller than a characteristic length of a domain in which modelling is per-
formed. The greater uncertainties in parameters that characterize a porous medium are, the
smaller the elementary volume is [9]. Since an inherent error is a part of a total calculation error,
maximal allowed uncertainty in a parameter that characterizes soil is determined by the total cal-
culation error. In [1, 2] and [4, 5], errors of numerical calculations were not estimated because the
solutions have regions of high gradients which positions depend on time and are not known in
114 ISSN 1028-9763. Математичні машини і системи, 2014, № 2
advance. One of the main drawbacks of the models [1, 2] and [4, 5] is that calculations according
to them require significant computer resources. It can be explained by the fact that they are sys-
tems of differential equations supplemented with boundary and initial conditions that do not con-
form to each other [10]. Demchuk [10] presents the model of the standard laboratory test [11], in
which this drawback is absent. Demchuk and Saiyouri [11] propose the method of uncertainty
uniformity principle realization in calculations according to the model [10]. Demchuk [12] esti-
mates the errors of these calculations. The aim of this work is to check whether the continuum
approach was properly adopted in [12]. Since productivity of tunnelling construction in the weak
soil highly depends on quality of stabilization of this soil, this research responds to the urgent prob-
lem in the time of the global recession.
2. Elementary Volume Size Estimation
Estimating a size of an elementary volume Bear and Bachmat assume that this volume is a cube
divided into N 3 equal cubic parts [9]. Demchuk [8] shows that if the uncertainty in the porosity
m is equal to δm , then the minimal characteristic size s of the elementary volume can be esti-
mated as
dNs ⋅= (1)
where
( ) ( ) ( ) ( )( ) 321114 2
0 mmmmmlnmd
~
d δ−−⋅δ−⋅−⋅⋅= , (2)
0
~
d is the average diameter of a pore, and N is the solution of the following equation:
( ) ( ) ( )
3 3
02 3 2 13 6
1 1,
0,32 1 pq
N N
h d m
p q p q
m m m N e N− ⋅ −
= = ≠
⋅ δ = ⋅ − +
∑ ∑
ɶ
(3)
where pqh is the distance between the centres of the above mentioned cubic parts with numbers
p and q .
3. Results of numerical calculations
A total error can be estimated as the square root of the sum of squares of errors from different
sources [13]. Therefore, to estimate a minimal size of an elementary volume according to Eq. (1),
in what follows we assume that maximal allowed uncertainty in porosity is determined by the con-
dition that uncertainty in a calculated value due to uncertainty in the porosity is three times
smaller than the total error of the calculation of this value. In the standard laboratory test [11], the
porosity of the sand in the tube has such the value 0,335m= and the injection front is detected at
the moments of time 1001 =t sec, 2502 =t sec, and 4003 =t sec at distances from the injection
point respectively equal to 0,2 m, 0,4 m, and 0,6 m. The respective values of the cement con-
centration in the fluid phase calculated in [12] coincide within the limits of the total calculation
error bars. Performing the numerical analysis similar to the one presented in [12], we obtain that if
the uncertainty in the porosity 1mδ is equal to 0,014, then the uncertainties in these concentration
values due to the uncertainty in the value of the porosity are three times smaller than the respective
total calculation errors obtained in [12]. The dependence of the injection pressure upon the time
calculated in [12] coincides with the one measured in [11] within the error bar limits. Performing
the numerical analysis similar to the one presented in [12], we obtain that if the uncertainty in the
porosity has such the value
2 0,0463mδ = , then the uncertainty in the injection pressure due to the un-
certainty in the value of the porosity is three times smaller than the total error of the respective
ISSN 1028-9763. Математичні машини і системи, 2014, № 2 115
calculation obtained in [12]. Since 2 1m mδ > δ , in what follows we assume that the maximal al-
lowed uncertainty in the value of m is equal to 1mδ . Substituting 1mδ for mδ in Eqs. (2) and (3)
we obtain that
03,48 , 15 16d d N= ⋅ < <ɶ . (4)
The 1-dimensional model used in [11, 12] is derived from the 3-dimensional one in [10].
Therefore, the characteristic length of the domain in which the numerical modelling [12] is per-
formed is equal to the diameter of the tube which is the following: 0 0,08l = m [11]. We assume
that the continuum approach is properly adopted in [12] if 0 10s l< where s is given by Eq.
(1). Therefore, it follows from (1) and (4) that the continuum approach is properly adopted in [12]
if 4
0 1,5 10d −< ⋅ɶ m. Pores in sand are mesopores. Their diameters range from 51,0 10−⋅ m to
31,0 10−⋅ m [14]. Since in the laboratory test [11] the sand in the tube was compacted, we can
assert that it is likely that the continuum approach was properly adopted in [12].
4. Conclusion
The degrees of uncertainties in the diameters of the tubes in the standard laboratory tests [1, 2],
and [11] are approximately the same. Demchuk [12] shows that the main contributions to the errors
of calculated values come from uncertainties in these values due to uncertainties in the diameter of
the tube and concludes that in the recent research [1, 2] as well as in [12] the comparisons of
model calculations with laboratory measurements provide small amounts of information. Sizes of
pores in sand range from 51,0 10−⋅ m to 31,0 10−⋅ m [14]. In this work, we have shown that the
continuum approach was properly adopted in [12] only if the average pore size in the compacted
sand was smaller 41,5 10−⋅ m. Thus, we can conclude that to improve the quality of the comparisons
of model calculations with laboratory measurements in the recent research [1, 2], and [12] it is
necessary not only to increase the accuracy of the measurements of the diameters of the tubes but
also to ensure strong enough compactions of the sands.
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