A realization of the uncertainty uniformity principle in a grouting model
A method of a realization of the uncertainty uniformity principle in grouting model calculations is described.
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Інститут кібернетики ім. В.М. Глушкова НАН України
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irk-123456789-488772013-09-06T03:05:36Z A realization of the uncertainty uniformity principle in a grouting model Demchuk, M. Saiyouri, N. A method of a realization of the uncertainty uniformity principle in grouting model calculations is described. Описаний спосіб реалізації принципу рівномірності похибки в розрахунках згідно моделі нагнітання. 2012 Article A realization of the uncertainty uniformity principle in a grouting model / M. Demchuk, N. Saiyouri // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2012. — Вип. 7. — С. 77-92. — Бібліогр.: 20 назв. — англ. XXXX-0059 http://dspace.nbuv.gov.ua/handle/123456789/48877 624.048-033.26:621.651 en Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки Інститут кібернетики ім. В.М. Глушкова НАН України |
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A method of a realization of the uncertainty uniformity principle in grouting model calculations is described. |
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Article |
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Demchuk, M. Saiyouri, N. |
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Demchuk, M. Saiyouri, N. A realization of the uncertainty uniformity principle in a grouting model Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки |
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Demchuk, M. Saiyouri, N. |
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Demchuk, M. |
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A realization of the uncertainty uniformity principle in a grouting model |
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A realization of the uncertainty uniformity principle in a grouting model |
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A realization of the uncertainty uniformity principle in a grouting model |
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A realization of the uncertainty uniformity principle in a grouting model |
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A realization of the uncertainty uniformity principle in a grouting model |
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realization of the uncertainty uniformity principle in a grouting model |
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Інститут кібернетики ім. В.М. Глушкова НАН України |
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2012 |
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A realization of the uncertainty uniformity principle in a grouting model / M. Demchuk, N. Saiyouri // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2012. — Вип. 7. — С. 77-92. — Бібліогр.: 20 назв. — англ. |
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Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки |
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2025-07-04T09:38:50Z |
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| fulltext |
Серія: Фізико-математичні науки. Випуск 7
77
UDC 624.048-033.26:621.651
Mykola Demchuk*, Master of Science,
Nadia Saiyouri**, Doctor of Ecole Centrale de Paris
*National University of Water Management and Natural Resources Use, Rivne,
**Research Institute of Civil and Mechanical Engineering (GeM),
Nantes, France
A REALIZATION OF THE UNCERTAINTY UNIFORMITY
PRINCIPLE IN A GROUTING MODEL
A method of a realization of the uncertainty uniformity prin-
ciple in grouting model calculations is described.
Key words: method error, numerical solution analysis, uncer-
tainty uniformity principle, high gradient region.
Introduction. Before creations of foundations on weak soil plots, to
make better elastic properties of soil tracts the grouting operations are per-
formed. In this technique a cement grout is injected under high pressure (10—
30 bars) into a dry porous medium or water saturated one. The aim is reached
after the grout hardening. The infiltrate is composed of particles that can be
responsible for the phenomenon called the deep bed filtration or the depth
filtration. Depending on particle and pore throat sizes two limiting cases are
distinguished in the description of this phenomenon. In the first case, large
particles are trapped by small pore throats. Whereas in the second case, small
particles deposit over large pore bodies and large pore throats that result in a
gradual reduction of pore throat sizes [19, p. 1637].
Grouting is rather costly and time consuming. Its regime is deter-
mined by the concentration distribution evolution [12, p. 1195]. Therefore,
a calculation of this evolution using mathematical modeling is important.
A grouting model is only a simplified version of a real system containing a
porous medium and an infiltrate. It is constructed to simulate the excita-
tion-response relations of the modeled system [11, p. 11]. In article [3,
p. 61—63] it was argued that under condition
8 LV t a (1)
where V is the fluid particle velocity, La is the coefficient of the longitu-
dinal dispersion of the cement concentration in the fluid phase during an
injection of the grout in the water saturated porous medium, and t is the
injection duration, the peculiarities of a solute propagation in a porous
medium can be neglected, and grouting can be modeled by a problem with
a free moving boundary (1) is usually satisfied in fieldworks [3, p. 72; 4,
p. 123]. Therefore, if a grouting model does not belong to the class of
© Mykola Demchuk, Nadia Saiyouri, 2012
Математичне та комп’ютерне моделювання
78
problems with free moving boundaries, then it can be referred to the class
of problems about pollution propagations. Articles [5, p. 46; 12, p. 1215—
1216; 15, p. 5—6; 16, p. 230—233] deal with modeling of the standard
laboratory test described in the next section. In several papers [1, p. 675; 2,
p. 269; 3, p. 63; 13, p. 670—671; 14, p. 1042—1043; 18, p. 166; 20,
p. 211—212] a cement grout propagation in a porous medium is examined
in configurations more realistic during in situ grouting. In part of them [14,
p. 1048; 18, p. 166] it is indicated that the deep bed filtration is not a do-
minant phenomenon.
Existing models of a cement grout propagation in a porous medium
use different sets of simplifying assumptions. In models [5, p. 48—50; 14,
p. 1040—1041; 15, p. 2—3; 16, p. 228—230; 18, p. 165—168; 20,
p. 212—214] the ground skeleton is regarded as absolutely rigid, while in
models [1, p. 675—685; 2, p. 269—271; 3, p. 63—65; 12, p. 1211—1215;
13, p. 669—670] it is assumed to be deformable. In models [1, p. 675—
685; 2, p. 269—271; 3, p. 63—65; 20, p. 212—214] peculiarities of a so-
lute propagation in a porous medium are neglected, in model [18, p. 165—
168] the hydromechanical dispersion and diffusion are neglected, and in
models [15, p. 2—3; 5, p. 48—50] the deep bed filtration is not taken into
account. In models [14, p. 1040—1041; 16, p. 228—230] the first limiting
case is assumed in the depth filtration description, while in models [12,
p. 121—1215; 13, p. 669—670; 18, p. 165—168] the second one is as-
sumed. A comparison of model calculations with laboratory measurements
and calculation consistency checks justify the set of assumptions used in a
model formulation. Amounts of information they provide depend upon
values of uncertainties in compared quantities [5, p. 46—47]. There are
three types of numerical calculation errors: the error due to uncertainties in
input data, the round off error, and the error of a calculation method [8,
p. 11]. If input data are fixed, then the error of the first type is zero. If the
finite difference scheme according to which the calculations are performed
is conditionally stable, then the round off error is negligible. This property
of the scheme is usually verified during a numerical solution analysis.
Models [1, p. 675—685; 2, p. 269—271; 3, p. 63—65], and [20, p. 212—
214] belong to the class of problems with free moving boundaries. Condi-
tion (1) implies high injection pressure values that are difficult to achieve
in laboratory conditions. Therefore, calculations according to models of
this type have not been compared with laboratory measurements yet. In
each case considered in papers [2, p. 282—286; 3, p. 69—74], and [20,
p. 218—221] the error of a numerical method of a calculation of the final
injection front position was estimated using the assumption that a contri-
bution of the uncertainty in this position due to the uncertainty in the
Серія: Фізико-математичні науки. Випуск 7
79
choice of a method of a free surface interpolation on each time layer to this
error is negligible. In paper [4, p. 128—129] it is shown that results of
calculations presented in [2, p. 282—286; 3, p. 69—74], and [20, p. 218—
221] are consistent. Since these calculations were performed on scanty
grids, this consistency check provides a small amount of information. In
article [1, p. 675—685] the model solutions were found analytically. Since
in the models belonging to the class of problems about pollution propaga-
tions [5, p. 48—50; 12, p. 1219—1222; 13, p. 672; 14, p. 1045—1046; 15,
p. 6; 16, p. 238; 18, p. 169—171] the sought functions contain high gra-
dient regions, according to the uncertainty uniformity principle [10, p. 35]
to estimate the approximation errors properly in these models calculations
on time layers should be performed on non-uniform grids with smaller
space increments inside these regions and larger ones outside them. Posi-
tions of these high gradient regions are changing with time and not known
in advance. Therefore, in these models calculation errors are not estimated.
Instead, in papers [13, p. 675—676; 14, p. 1042], to prove that the chosen
numerical methods are precise enough, the numerical solutions of bench-
mark problems obtained by these methods were compared with their pre-
cise analytical solutions. In paper [18, p. 168] in the limiting case of a
small variation of porosity due to the depth filtration the analytical solu-
tion of the model was obtained. To justify the numerical procedure
used in [18, p. 169—171] this analytical solution was compared with
numerical solutions of the model [18, p. 166—168] valid for signifi-
cant as well as small variations of porosity due to the deep bed filtra-
tion. The numerical methods used in papers [12, p. 1224; 16, p. 235]
were validated in earlier articles. In article [15, p. 5], to circumvent
numerical oscillations appearing near relatively sharp concentration
fronts, the space and time increments were chosen to satisfy Courant
and Peclet number restrictions. Calculations according to model [5,
p. 48—50] have not been presented yet.
The aim of this work is to develop a method of a proper treatment of
high gradient regions in sought functions during numerical calculations
according to model [5, p. 48—50].
Uniaxial injection test. The injection tests are performed in vertical col-
umns of height 0,75 m and 0,08 m in diameter filled with Loire sand (0,70 m).
Loire sand characteristics are available in [17, p. 1—7]. The sand is put in
place in the column by layers and a compaction method is used to get a fixed
density. The injection is performed at a constant pumping rate
31,5 6 impq E m s after the sand has been saturated with water. The
column is open at the top. The material properties are given in Table 1.
Математичне та комп’ютерне моделювання
80
Three pressure transducers are positioned along the column and one
at the bottom of the column (injection point). They are spaced out 0,2 m
apart. The whole concentration field is not experimentally available, and
only the grout front position can be obtained. It is detected by the acous-
tic emission (AE) using transducers coupled with three wave guides
which cross the sand column. They are horizontally placed at the same
height as the pressure transducers. An increase of the acoustic activity is
observed when the grout reaches the wave guide. Consequently, the
grout front position is able to be detected at different times and locations
in the sand column. The technique of AE is validated by performing in-
jection tests in transparent columns. In this case, the grout propagation is
visually followed, to ensure that the AE detection really occurred when
the grout arrived at the wave guide. During injection tests the volume
and the weight of injected grout, the pumping pressure, and the pumping
rate are also recorded.
Table 1
Material properties
Porosity 0,335
Grout density 1370 3
kg m
Grout viscosity 3
2, 9 10
Pa s
Intrinsic permeability 11
1,17 10
2
m
Diffusion coefficient 10
1, 0 10
2
m s
Longitudinal dispersion coefficient 2
2, 0 10
m
Compressibility coefficient 8
3, 0 10
1
Pa
Water viscosity 3
1,1 10
Pa s
Figures (1a), (1b), and (1c) show the acoustic activity recorded by
AE versus time for three transducers c1, c2, and c3 associated with the
wave guides. Transducers c1, c2, and c3 detect the grout front at moments
of time 1 100 st , 2 250 st , and 3 400 st respectively. Since they are
spaced out 0,2 m apart, the fluid particle velocity can be estimated as
30, 2 150 1,33 10 V m s m s . (2)
Using data from Table 1 one can estimate 8 11La V s . Comparing
this value with 1t , 2t , and 3t we can conclude that condition (1) is
not satisfied. Figure (1d) shows the evolution of the experimental injection
pressure with time. It exhibits oscillations due to vibrations of the pump
during the injection.
Серія: Фізико-математичні науки. Випуск 7
81
Fig. 1. (a), (b), (c) Grout propagation followed by transducers c1, c2, c3.
(d) Evolution of the experimental injection pressure with time
A grouting model belonging to the class of problems about pollu-
tion propagations. We assume that the coordinate origin is chosen at the
injection point and that the coordinate axis is directed upward. In this work
we use the following notations. is a small positive parameter, 0 and 1
are large positive parameters. They are introduced in the model to conform
initial conditions with boundary conditions, and their values will be obtained
later in the result of the analysis of the numerical solutions. ,I L is
an interval (a set of points x : ,x L ). 0,TI I T is a rectangle
(a set of points ,x t : x I , 0,t T ), ,1TA is the bottom side of TI
(a set of points , t : 0,t T ), ,2TA is the top side of TI (a set of
points , L t : 0,t T ), ,p x t and ,c x t denote the pore fluid pres-
sure and the grout concentration respectively. K is the permeability of the
medium, m is the medium porosity, is the fluid phase density, g is the
acceleration of free fall, 1 and impc are the grout viscosity and the grout
concentration inside the injector respectively, 2 is the water viscosity. La
is the longitudinal dispersion coefficient, *D is the diffusion coefficient,
Математичне та комп’ютерне моделювання
82
1 2 2impc c , *
h LD a V D , V K p x g m .
f x is the function that tends to the function
0, 0,
1 2, 0,
1, 0,
if x
x if x
if x
(3)
with [6, p. 677]. S is the area of the interface between the injec-
tor and the porous medium, L is the tube length,
,0 1 2 0 21L f L ,
0,0 1 2 0 21 f ,
,0 1 0 1 1 ,01L Lx L
V V f x x df x dx
, 0 impV q S m .
In recent paper [5, p. 48—50] the mathematical model of the labora-
tory experiment described in the previous section with boundary condi-
tions conforming to initial ones has been formulated. In it the ground ske-
leton is regarded as absolutely rigid and the deep bed filtration is not taken
into account. This model is the following system of two partial differential
equations valid for such values x and t that , Tx t I
hm c t mV c x mD c x x , (4)
0pm p t V p x K p x g x (5)
with such initial conditions valid if 0t and x I
1 0 1 1p g L x mV x f x K , (6)
01 impc f x c (7)
and boundary conditions
1 0 1 11p x mV f x x df x dx K g , (8)
01impс c f (9)
where ,1, Tx t A and
1 0 1 1p mV L f L K , (10)
0 ,0 * *imp L L Lc x c df x dx a V D a V D (11)
where ,2, Tx t A .
In this model at any moment of time t such that 00 t L V the in-
jection front position is determined by high gradient regions of cement
concentration and pore fluid pressure space distributions. The positions of
Серія: Фізико-математичні науки. Випуск 7
83
these regions are changing with time and are not known in advance.
Therefore, in this work we are looking for each numerical space distribu-
tion evolution in the form of the sum of two functions. The first one is
modeling the evolution of the respective distribution high gradient region,
while the second one does not have such region. Although condition (1) is
not satisfied according to estimations made in the second section, the solu-
tion of the injection test model with a free moving boundary can provide a
good start for the model function construction.
A grouting model with a free moving boundary. In this section we
develop a mathematical model with a free moving boundary of the standard
laboratory test described in the second section for the injection duration T
such that 0T L V . We assume that the grout penetrates the water saturated
sand filling the tube a distance 0 0x V t from the injection point by the mo-
ment of time t . We chose the upward direction as positive direction of 3x
axis and place the coordinate origin in the center of the bottom tube side. We
assume that by the moment of time t the sand in the bottom part of the tube
3 00 x V t is saturated with cement grout, while the sand in the upper part
of the tube 0 3V t x L is saturated with water. Since the solution of this
model is used for a model function construction, we assume that water and
grout densities are identical and equal . In this section we use the following
notations. is a circle (a set of points 1 2, x x : 2 2
1 2x x R ) where R is
the tube radius. 1 00, Q t V t is a cylinder (a set of points
1 2 3, , x x x : 1 2, x x , 3 00, x V t ), 1S is the bottom base of 1Q t
(a set of points 1 2 3, , x x x : 1 2, x x , 3 0x ), 2S is the side surface of
1Q t (a set of points 1 2 3, , x x x : 2 2 2
1 2x x R , 3 00, x V t ),
2 0 ,Q t V t L is a cylinder (a set of points 1 2 3, , x x x :
1 2, x x , 3 0 ,x V t L ), 3S is the top base of 1Q t or the bottom base
of 2Q t (a set of points 1 2 3, , x x x : 1 2, x x , 3 0x V t ), 4S is the
side surface of 2Q t (a set of points 1 2 3, , x x x : 2 2 2
1 2x x R ,
3 0 , x V t L ), 5S is the top base of 2Q t (a set of points 1 2 3, , x x x :
1 2, x x , 3x L ). The ground skeleton is regarded as absolutely rigid.
Therefore, at any moment of time t such that 00 t L V the head function
1 1 1 2 3 3, , , ,h p x x x t g x (12)
Математичне та комп’ютерне моделювання
84
where 1p is the grout pressure satisfies at any point 1 2 3, , x x x such that
1 2 3 1, , x x x Q t the following equation [7, p. 38]
2 2 2 2 2 2
1 1 1 1 2 1 3 0h h x h x h x (13)
and satisfies on the boundary of 1Q t the following conditions:
1
1 01
1n
S
m Vh
K g
,
2
1
2
0
n
S
h
,
3
1 01
3n
S
m Vh
K g
. (14)
In equations (14) ni
is external with respect to the interior of 1Q t and
normal to iS unit vector where 1,3i . At the same moment of time t the
head function
2 2 1 2 3 3, , ,h p x x x t g x (15)
where 2p is the water pressure satisfies at any point 1 2 3, , x x x such that
1 2 3 2, , x x x Q t the following equation [7, p. 38]
2 2 2 2 2 2
2 2 1 2 2 2 3 0h h x h x h x (16)
and satisfies on the boundary of 2Q t the following conditions:
3
2 02
3l S
m Vh
K g
,
4
2
4
0
l
S
h
,
5
2 02
5l S
m Vh
K g
,
5S
h L . (17)
In equations (17) li
is external with respect to the interior of 2Q t and
normal to iS unit vector where 3,5i . In addition, we assume that at any
moment of time t the pore fluid pressure is continuous on 3S
3 3
1 3 2 3S S
h x h x . (18)
Definition 1. The classical solution of the system of the boundary
problems (13), (14), (16)–(18) is a pair of functions 1 1 2 3, , ,h x x x t and
2 1 2 3, , ,h x x x t with the following properties.
1. If 1 2 3 1, , x x x Q t 1 2 3 2, , x x x Q t , then all derivatives of
function 1 1 2 3, , ,h x x x t 2 1 2 3, , ,h x x x t with respect to variables 1x ,
2x , and 3x up to the second order at any fixed t such that
00 t L V are continues.
Серія: Фізико-математичні науки. Випуск 7
85
2. If 1 2 3 1, , x x x Q t 1 2 3 2, , x x x Q t , then function 1 1 2 3, , ,h x x x t
2 1 2 3, , ,h x x x t and all its first derivatives with respect to variables
1x , 2x , and 3x at any fixed t such that 00 t L V are continues.
3. If 1 2 3 1, , x x x Q t 1 2 3 2, , x x x Q t , then function 1 1 2 3, , ,h x x x t
2 1 2 3, , ,h x x x t satisfies equation (13) ((16)) at any fixed t such that
00 t L V .
4. At any fixed t such that 00 t L V function 1 1 2 3, , ,h x x x t satis-
fies the first equation, the second one, and the third one of equations
(14) on 1S , 2S , and 3S respectively.
5. At any fixed t such that 00 t L V function 2 1 2 3, , ,h x x x t satis-
fies the first equation and the second one of equations (17) on 3S and
4S respectively.
6. At any fixed t such that 00 t L V function 2 1 2 3, , ,h x x x t satis-
fies the third equation and the fourth one of equations (17) on 5S .
7. At any fixed t such that 00 t L V functions 1 2 3, , ,ih x x x t where
1,2i satisfy equation (18) on 3S .
Theorem 1. If the classical solution of boundary problem system
(13), (14), (16)–(18) exists, then it is unique.
Proof. Let us assume that there are two classical solutions of boundary
problem system (13), (14), (16)—(18) 1 2 3, , ,ih x x x t and 1 2 3, , ,ih x x x t
where 1,2i . To prove that 1 2 3, , ,ih x x x t 1 2 3, , ,ih x x x t where
1, 2i , we denote their difference in the following way
1 2 3 1 2 3, , , , , ,i i iw h x x x t h x x x t , 1,2i . (19)
If two functions 1 2 3, ,u x x x , 1 2 3, ,v x x x , and all their first derivatives
are continuous on iQ t , and all their second derivatives are continuous on
iQ t where 1i or 2i , then the following Green’s formula holds [9, p. 291]
3
1
i i i
j j
jQ t Q t t
u vd u x v x d u v n d
. (20)
Here i t is the piecewise-smooth boundary of iQ t , 1 2 3d dx dx dx
is a volume element, d is an area element. Substituting iw where 1i
or 2i in (20) instead of u and v , we get
Математичне та комп’ютерне моделювання
86
3 2
1
i i i
i i i j i i
jQ t Q t t
w w d w x d w w n d
. (21)
Since 1 1 0h h in 1Q t and 2 2 0h h in 2Q t , the left hand
side of equation (21) equals zero. Since the normal derivative of 1w is
equal to zero on surface iS where 1,3i , and the normal derivative of
2w is equal to zero on surface jS where 3,5j , the surface integral in
the right hand side of equation (21) is equal to zero. Therefore, from equa-
tion (21) it follows that
2 2 2
1 2 3 0
i
i i i
Q t
w x w x w x d .
An integral of a sum of squares can be equal to zero only if each of addi-
tives is equal to zero:
1 0iw x , 2 0iw x , 3 0iw x . (22)
(22) implies that i iw c in iQ t where ic is a constant and 1,2i . Since
2 0w on 5S , 2c is equal to zero. From equations (18) and (19) it follows
that on 3S the following equality holds 1 2c c . Therefore, 1c equals zero.
Thus, we proved that 0iw where 1, 2i . The theorem is proved.
Theorem 2. If 00 t L V , then the pair of functions
0 1 0 2
1 1 2 3 0 3 0 0, , , 1
mV mV
h x x x t V t x L V t V t
K g K g
, (23)
2 1 2 3 3 0 2, , ,h x x x t L x mV K g L (24)
is the classical solution of boundary problem system (13), (14), (16)—
(18). In (23) and (24) 1 2 3 1, ,x x x Q and 1 2 3 2, ,x x x Q respectively.
Proof. From equations (23) and (24) it follows that functions
1 1 2 3, , ,h x x x t and 2 1 2 3, , ,h x x x t satisfy the first two properties from
definition 1. Substituting the right hand side of equation (23) in equations
(13), (14), and (18) instead of 1 1 2 3, , ,h x x x t and the right hand side of
equation (24) in equations (16)–(18) instead of 2 1 2 3, , ,h x x x t , it is easy
to check that functions 1 1 2 3, , ,h x x x t and 2 1 2 3, , ,h x x x t satisfy proper-
ties 3—7 from definition 1. Thus, the pair of functions 1 1 2 3, , ,h x x x t and
2 1 2 3, , ,h x x x t is the classical solution of boundary problem system (13),
(14), (16)—(18). The theorem is proved.
Серія: Фізико-математичні науки. Випуск 7
87
From equations (12) and (23) it follows that if 00 t L V and
1 2 3 1, ,x x x Q t , then
1 0 2 0 1 0 3 0p L V t g mV K g mV K x V t . (25)
In its turn, from equations (15) and (24) it follows that if 00 t L V and
1 2 3 2, ,x x x Q t , then
2 2 0 3p mV K g L x . (26)
In this model it is assumed that at any moment of time t such that
00 t L V the cement concentration is equal to impc in 1Q t and
equals zero in 2Q t . Therefore, assuming 3x x the solution of this
model can be written in the form
1 0 2 1, , , ,p x t p x t x V t p x t p x t , (27)
0, 1 impc x t x V t c (28)
where 0x V t is calculated according to equation (3) and 1 ,p x t ,
2 ,p x t are calculated according to equations (25) and (26) respectively.
Model function construction. During an injection a transition zone
separates the domain in which the cement concentration is equal to impc
from the zero concentration one. The concentration is monotonic conti-
nuous function inside this transition zone. If condition (1) is satisfied, then
this zone is narrow with respect to the first domain [3, p. 61–63]. In this
case, assuming that the sand column is sufficiently high, this zone can be
treated as a sharp boundary between fluid in which the concentration
equals impc and zero concentration one. As it was shown in the second
section, condition (1) is not satisfied in the modeled test. This condition is
derived, analyzing the analytical solution of the following partial differen-
tial equation valid for 0 x and 0t
2 2D c x V c x c t (29)
where D and V are positive constants with the following initial and
boundary conditions valid for 0 x and 0t :
,0 0c x , , 0c t , 00,c t c (30)
where 0c is a positive constant. Equations (29), (30) do not contain the
acceleration of free fall. Comparing equation (29) and equation (4) one can
assume (assumption #1) that the gravity force causes the shift of the injec-
tion front velocity V equal to K g . Therefore, to construct the
Математичне та комп’ютерне моделювання
88
function modeling the pore fluid pressure high gradient region evolution,
we need to substitute 1pf x V t instead of 0x V t in equation
(27) where here and bellow 1 0 effV V K g , p is a large constant,
eff is defined below. In addition, we need to substitute
1 1 2 1 1 1 ,effp L V t g mV K g mV K x V t (31)
where effV is defined below and
2 1wp mV K g L x (32)
instead of 1 ,p x t and 2 ,p x t respectively in equation (27). In its turn,
to construct the function modeling the cement concentration high gradient
region evolution, we need to substitute 1cf x V t where c is a
large constant instead of 0x V t in equation (28). To estimate the
model calculation error properly we need to use two methods to find the
numerical solution of problem (4)—(11). In the first one the problem (4)—
(11) is discretized at once, while in the second one we are looking for the
numerical solution in the form
(0) (1), , ,p x t p x t p x t , (0) (1), , , ,c x t c x t c x t
where
(0)
1 1 2 1, , , ,pp x t p x t f x V t p x t p x t , (33)
(0)
1, 1 c impс x t f x V t с . (34)
Performing calculations by the second method we assume (assumption
№2) that calculation results obtained when 1 2 2eff and
0 1 2effV V V do not sufficiently deviate from ones obtained when
1 2eff and 0 1effV V V . Functions (0) ,p x t and (0) ,c x t
are modeling high gradient region evolutions of pore fluid pressure and
cement concentration respectively. In equation (33) 1 ,p x t and 2 ,p x t
are calculated according to equations (31) and (32) respectively.
Analyzing the analytical solution of problem (29), (30) it was shown in
[3, p. 62] that the transition zone width is equal to 2 8Dt . Using data from
Table 1 and estimation (2) one can estimate h LD a V . Since 1 8 LVt a ,
2 8 LVt a , 3 8 LVt a where 1 100t s , 2 250t s , 3 400t s and, taking
into account that problem (29), (30) is a rough model of the injection test at
hand, we assume that the half of the transition zone width, which we denote
Серія: Фізико-математичні науки. Випуск 7
89
, is given by the following expression 8 La . We define the half of the
transition zone width as the solution of the following equation
1 1 r
cf e (35)
where r is such that ln(2)r , and a method of its numerical determina-
tion is described at the end of this section. Performing calculations by the
second method we assume that if the order of magnitude of the value of
p in the right hand side of equation (33) is the same as the order of mag-
nitude of the value of с in the right hand side of equation (34), then nu-
merical solution is not sensitive to the choice of the value of p (assump-
tion №3). Therefore, in what follows we assume that p с . Assuming
8 La we find the value of с solving equation (35).
To find the numerical solution of problem (4)—(11) we cover TI with
uniform grid , , , , 0, ; N M i j ix t x i h i N , 0, jt j j M
where h L N and T M .
Performing an analysis of numerical solutions of problem (4)—(11)
one can use the following measure of a difference between two space dis-
tributions of the cement concentration or the pore fluid pressure 1 ,f x t
and 2 ,f x t at a chosen moment of time t
1 2
0,
, ,max
x L
t f x t f x t
.
We denote the measure of the difference between the space distribu-
tion of the cement concentration (the pore fluid pressure) at a given mo-
ment of time obtained by the second method on grid
1 22 ,2N N and the one
obtained by the second method on grid
1 2,N N as 1
с 1
p . We denote
the measure of the difference between the space distribution of the cement
concentration (the pore fluid pressure) at a given moment of time obtained
by the first method on grid
1 22 ,2N N and the one obtained by the second
method on the same grid
1 22 ,2N N as 2
с 2
p . An approximation error
of the discretized problem depends not only on the grid increment, but also
on high order derivatives of the sought for function. For instance, substi-
tuting the central difference formula instead of a function partial derivative
in a nodal point ,i jx t of ,N M and assuming that h is sufficiently
small, the following approximate equality holds
Математичне та комп’ютерне моделювання
90
32
3
, ,, ,
,
2 6
i j i ji j i jf x h t f x h tf x t f x th
x h x
where
, ,
i
i j j
x x
f x t f x t
x x
,
3 3
3 3
, ,
i
i j j
x x
f x t f x t
x x
.
If the sought function contains high gradient region, then its high or-
der derivatives are large in this region. Ideally, if the value of r is chosen
correctly, then functions (1) ,с x t and (1) ,p x t do not contain such re-
gions. Therefore, 1
с and 1
p are minimal at such value of r .
It follows from equations (27), (28) that across the high gradient re-
gion the value of the cement concentration in the fluid phase varies from
impc to zero and the value of the derivative of the fluid pore pressure with
respect to space coordinate varies approximately from 1 0g mV K to
2 0g mV K .
The high gradient region of the numerical solution obtained by the first
method on a uniform grid is dispersed. We give it the following explanation. If
a nodal point ix x , jt t where 0 i N and 0 j M belongs to the
high gradient region, then the calculation of the first derivative of the cement
concentration with respect to the space coordinate and the calculation of the
second derivative of the pore fluid pressure with respect to the space coordi-
nate at this point according to the following formulas
, , , 2j i j i jc x t x c x h t c x h t h ,
2 2 2, , 2 , ,j i j i j i jp x t x p x h t p x t p x h t h
give values smaller than respective real ones. Therefore, according to Taylor
expansion the numerical cement concentration function and the numerical first
derivative of the pore fluid pressure with respect to the space coordinate reach
their limiting values at a distance from the nodal point larger than real one. If
the value of r used in calculations is close to real one, then the high gradient
region of the numerical solution obtained by the second method on a uniform
grid is sharp. Therefore, in numerical experiments we need to use the value of
r at which the ratios 2 1
с с and 2 1
p p are maximal.
Conclusions. The method of the proper treatment of high gradient
regions in sought functions during numerical calculations according to
model [5, p. 48—50] is developed.
Серія: Фізико-математичні науки. Випуск 7
91
Afterwards, the analysis of numerical solutions of model [5, p. 48—
50] will be performed. The values of constants 0 , 1 , c , p , , and
r will be chosen, the calculation error will be estimated, and assumptions
№ 1—3 will be checked as a result of this analysis. The calculations ac-
cording to the models with free moving boundaries will be performed on
relatively dense grids. The investigation will be conducted to check the
hypothesis that in a model of the real scale cement grout injection in sand
(condition (1) is satisfied) the deep bed filtration can be neglected.
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Описаний спосіб реалізації принципу рівномірності похибки в ро-
зрахунках згідно моделі нагнітання.
Ключові слова: похибка методу, аналіз числових розв’язків,
принцип рівномірності похибки, область високих градієнтів.
Отримано: 02.04.2012
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/SVE <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>
/TUR <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>
/UKR <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>
/RUS <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>
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AllowImageBreaks true
/AllowTableBreaks true
/ExpandPage false
/HonorBaseURL true
/HonorRolloverEffect false
/IgnoreHTMLPageBreaks false
/IncludeHeaderFooter false
/MarginOffset [
0
0
0
0
]
/MetadataAuthor ()
/MetadataKeywords ()
/MetadataSubject ()
/MetadataTitle ()
/MetricPageSize [
0
0
]
/MetricUnit /inch
/MobileCompatible 0
/Namespace [
(Adobe)
(GoLive)
(8.0)
]
/OpenZoomToHTMLFontSize false
/PageOrientation /Portrait
/RemoveBackground false
/ShrinkContent true
/TreatColorsAs /MainMonitorColors
/UseEmbeddedProfiles false
/UseHTMLTitleAsMetadata true
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/BleedOffset [
0
0
0
0
]
/ConvertColors /ConvertToRGB
/DestinationProfileName (sRGB IEC61966-2.1)
/DestinationProfileSelector /UseName
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements true
/GenerateStructure false
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles true
/MarksOffset 6
/MarksWeight 0.250000
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /DocumentCMYK
/PageMarksFile /RomanDefault
/PreserveEditing true
/UntaggedCMYKHandling /UseDocumentProfile
/UntaggedRGBHandling /LeaveUntagged
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [600 600]
/PageSize [419.528 595.276]
>> setpagedevice
|