Combustion, Explosion, and Shock Waves, Vol. 38, No. 3, pp. 346 351, 2002
Pressure Relaxation in a Hole Surrounded
by a Porous and Permeable Rock
V. Sh. Shagapov,1 G. Ya. Khusainova,1 UDC 622.276.031
I. G. Khusainov,1 and R. N. Khafizov1
Translated from Fizika Goreniya i Vzryva, Vol. 38, No. 3, pp. 106 112, May June, 2002.
Original article submitted May 29, 2001.
The problem of explosion in a hole surrounded by a porous medium is considered.
Integral equations describing pressure relaxation in the hole due to gas filtration
into the surrounding porous space are obtained within the framework of plane one-
dimensional, radial, and spherical formulations. An analysis of numerical solutions of
these equations shows that the characteristic time of pressure relaxation in the hole
depends weakly on its initial value.
Key words: pressure relaxation, explosion in a hole, porous medium, filtration of
explosion products, bottom-hole cleaning.
INTRODUCTION 1. BASIC EQUATIONS
We assume that in the initial state (t < 0), the
Long-term operation of gas and oil wells leads
gas pressure in the entire porous bed around the hole
to clogging of the critical area of formations by solid
is constant and equal to p0, and the hole (a crack or a
deposits (for example, wax or asphalt resinous mate-
cylindrical or a spherical region) is filled with an explo-
rial), which ultimately decreases the well discharge.
sive. At the time t = 0, an explosion occurs, the hole
Technologies using explosion energy are among the
is filled with explosion products, and the hole pressure
most effective methods of bottom-hole cleaning. High-
reaches the value of pe instantaneously. Next, the hole
temperature explosion products, penetrating deep into
pressure decreases to the value of p0 due to filtration of
porous rock, can melt deposits of heavy hydrocarbon
the explosion products.
systems, thus raising the efficiency of cleaning of porous
In describing these processes, we assume that the
systems.
porous skeleton is incompressible and homogeneous and
In addition, information obtained during explosion
the viscosity of the gas does not depend on temperature
pressure relaxation due to filtration of explosion prod-
and pressure. Using the above-stated assumptions, we
ucts into the surrounding porous medium can be used
write the equation of conservation of mass for the hole
to control the collecting characteristics of the near-well
gas, the nonlinear equation of piezoconduction, and the
zone. In particular, from the relaxation rate of hole
Darcy law for gas filtration in the porous and permeable
pressure, it is possible to determine the porosity, per-
rock around this hole as follows:
meability, and fracturing of a bed. Necessary estimates
dÁ n + 1
for technological calculations can be obtained by solv-
= - Áv ; (1.1)
ing plane one-dimensional, radial, and spherical prob- dt a r=a
lems. In particular, filtration in the porous bed around
"p(1) k 1 " "p(1)
a well can be analyzed in a radial formulation, and the
= rnp(1) ,
"t µm rn "r "r
processes in cracks can be studied by solving a plane
(1.2)
one-dimensional problem.
k "p(1)
v(1) = - , a < r < ",
µ "r
1
Sterlitamak State Pedagogical Institute,
Sterlitamak 453118; [e-mail].
346 0010-5082/02/3803-0346 $27.00 © 2002 Plenum Publishing Corporation
Pressure Relaxation in a Hole Surrounded by a Porous and Permeable Rock 347
t
Here a is the radius of the hole, m and k are the porosity
p kł p(t ) - p0
" dt , j = 1,
ln = - "
and permeability, Á and µ are the density and viscosity
pe aµ Ä„Ç - t
t
of the gas, and p(1) and v(1) are the pressure and ve-
0
(2.2)
locity around the hole; the subscripts n = 0, 1, and 2
t
correspond to the plane one-dimensional, radially sym-
kł (p(t ))2 - p2
0
" dt , j = 2.
metric, and spherical problems, respectively. p = pe - "
2aµ Ä„Ç
t - t
For the process considered, the initial and bound-
0
ary conditions for Eq. (1.2) can be written as
For further analysis, these equations are conveniently
written in dimensionless form
p(1) = p0 (t = 0, r > a);
Ä
P Å‚m P (Ä ) - 1
(1.3)
p(1) = p(t), v(1) = v (t > 0, r = a). ln = - " " dÄ , j = 1,
Pe Ä„ - Ä
Ä
0
(2.3)
A general analytical solution of the nonlinear piezocon-
Ä
duction equation for gas filtration (1.2) subject to con-
1 Å‚m (P (Ä ))2 - 1
ditions (1.3) was not obtained. Therefore, below we use
P = Pe - " " dÄ , j = 2,
2 Ä„
Ä - Ä
this equation in a linearized approximation:
0
where
"(p(1))j 1 " "(p(1))j
= Ç rn , (1.4)
p
"t rn "r "r
P = , t = Ät", t = Ä t",
p0
kp0
(2.4)
Ç = .
µm a2 a2µm pe
t" = = , Pe = .
Ç kp0 p0
Here Ç is the piezoconductivity and the values of the ex-
Equations (2.3) can be transformed by replacing
ponent j = 1 and 2 correspond to ordinary linearization
the dimensionless time Ä by the new dimensionless vari-
and Leibenzon linearization [2], respectively.
able Ä:
Ü
The relation between the current density and pres-
Ä„
sure in the hole is written as
Ä = Ä. (2.5)
Ü
Å‚2m2
Å‚
p Á
= , (1.5)
pe Áe
In this case, Ä is related to the dimensional time t by
Ü
where Å‚ is the polytropic exponent and pe and Áe are
Ä„t" Ä„a2µ
Ü Ü
t = t"Ä, t" = = . (2.6)
Ü
the pressure and density in the hole after explosion.
Å‚2m2 Å‚2mkp0
Then, Eqs. (2.3) become
2. PLANE ONE-DIMENSIONAL PROBLEM
Ä
Ü
(n = 0, r = x)
P P (Ä ) - 1
Ü
ln = - " dÄ , j = 1,
Ü
Pe
Applying the Duhamel principle, we write the solu- Ä - Ä
Ü Ü
0
tion of Eq. (1.4) subject to conditions (1.3) in the form (2.7)
[3]
Ä
Ü
1 (P (Ä ))2 - 1
Ü
t
P = Pe - " dÄ , j = 2.
Ü
"U(x - a, t - t )
2
Ä - Ä
Ü Ü
(p(1))j = ((p(t ))j - pj ) dt ,
0
0
"t
0
(2.1)
From (2.7) it follows that the variable Ä is self-similar
Ü
x - a
and the solution of this equation depends only on the
U(x - a, t - t ) = 1 - Åš ,
2 Ç(t - t )
dimensionless initial hole pressure Pe.
Figure 1 shows results of numerical solution of Eqs.
Ä…
2 (2.7). Here and below, numerical results are obtained
"
Åš(Ä…) = exp(-¾2)d¾.
by the method described in [4]. The algorithm used
Ä„
0
for the calculations was tested on the exact solutions
Using this solution and (1.1), we obtain the integral of the Abel equation. In this and other figures, thick
equation describing the evolution of hole pressure: curves correspond to ordinary linearization (j = 1) and
348 Shagapov, Khusainov, Khusainova, and Khafizov
t
p kÅ‚ Ç(t - t )
ln = - Õ p(t ) - p0 dt ,
pe a2µ a2
0
j = 1,
(3.2)
t
kÅ‚ Ç(t - t )
p = pe - Õ (p(t ))2 - p2 dt ,
0
2a2µ a2
0
j = 2,
"
8 exp(-Su2) du
Õ(S) = .
2 2
Ä„2 J0 (u) + Y0 (u) u
0
Here J0(u) and Y0(u) are zero-order Bessel functions of
the first and second kind.
For the kernel Õ(S) of the integral equations (3.2),
the following asymptotic expansions hold [5]:
Fig. 1. Relaxation of the dimensionless pressure for
2 1 S 1
"
Õ(S) = + 1 - + S · · · , S 1,
its various initial values in the plane one-dimensional
2 Ä„ 4
Ä„S
problem: thick curves correspond to ordinary lin-
(3.3)
earization (j = 1) and thin curves refer to Leibenzon
4 4“
linearization (j = 2).
Õ(S) = - - · · · , S 1,
ln(4S) - 2“ [ln(4S) - 2“]2
“ = 0.57722 . . .
thin curves to Leibenzon linearization (j = 2). As fol-
(“ is an Euler constant).
lows from Fig. 1, a description of the filtration pro-
In the dimensionless variables introduced similarly
cess using the equation linearized after Leibenzon un-
to Sec. 2, Eqs. (3.2) become
derestimates the rate of hole pressure relaxation. The
Ä
times of pressure reduction to a certain intermediate
P Å‚m
value P (1 < P < Pe) obtained by this two methods
ln = - " Õ(Ä - Ä )(P (Ä ) - 1) dÄ ,
Pe Ä„
of allowing for gas filtration differ by less than a fac-
0
j = 1,
tor of three. In addition, from the plots, it is evident
Ä
(3.4)
that the initial hole pressure has a weak effect on the
Å‚m
P = Pe - " Õ(Ä - Ä ) (P (Ä ))2 - 1 dÄ ,
characteristic dimensionless time of hole pressure relax-
2 Ä„
0
ation (value Ä H" 10 is obtained from the ordinary lin-
Ü
j = 2.
earized equations). Therefore, in the case of a hole with
Ü
Ü Ü
plane-parallel walls, the characteristic time t = 10t" In this case, the choice of a new variable Ä for the di-
mensionless time in Eqs. (3.4) does not allow us to
Ü
(t" = Ä„a2µ/Å‚2mkp0) determines the order of magni- "
 hide the factor Å‚m/ Ä„. Consequently, in contrast
tude of the complete pressure relaxation time.
to the plane one-dimensional case, the problem is not
self-similar.
Retaining only the first term in the first formula of
3. RADIAL PROBLEM (n = 1)
(3.3), we obtain the equation for pressure relaxation in
the initial stage, where the initial depth of the filtration
In the case (n = 1), the solution of (1.4) subject
zone near the hole walls does not exceed the radius of
to conditions (1.3) can be written in the form (2.1). In
the hole. Using the self-similar variable from (2.5) for
this case,
the dimensionless time, for the initial stage of pressure
"
2 Çu2(t - t )
relaxation, we have
U(r, t - t ) = 1 + exp -
Ä„ a2
Ä
Ü
0
P P (Ä ) - 1
Ü
ln = -2 " dÄ , j = 1,
Ü
J0(ur/a)Y0(u) - J0(u)Y0(ur/a) du Pe
Ä - Ä
Ü Ü
× . (3.1)
0
2 2 (3.5)
J0 (u) + Y0 (u) u
Ä
Ü
Using (3.1) and allowing for (1.1), we obtain the equa-
(P (Ä ))2 - 1
Ü
tion describing the evolution of pressure in a cylindrical
P = Pe - " dÄ , j = 2.
Ü
Ä - Ä
Ü Ü
hole:
0
Pressure Relaxation in a Hole Surrounded by a Porous and Permeable Rock 349
As follows from Fig. 3a, as the porosity changes from
m = 0.1 to 0.4, the relaxation time increases by a factor
of about three or four. However, because the charac-
teristic time from (2.7) is in inverse proportion to the
porosity, the dimensional relaxation time depends very
weakly on m.
Figure 3b shows the evolution of pressure for m =
0.1 for various values of the dimensionless pressure
(Pe = 10 and 5). It is evident that the initial hole
pressure has a weak effect on pressure relaxation time.
4. SPHERICAL PROBLEM (n = 2)
As in the previous cases, for pressure distribution in
the porous bed around a hole, the solution of Eq. (1.4)
can also be written in the form (2.1), and
a r - a
U(r, t - t ) = 1 - Åš ,
Fig. 2. Curve of the kern of the integral equation ob-
r
2 Ç(t - t )
tained numerically (thick curve) and the curves calcu-
(4.1)
lated by asymptotic formulas (thin curves): curves 1
Ä…
4 correspond to the number of terms (beginning from
2
"
the first) used in the first formula of (3.3) and curve 5 Åš(Ä…) = exp(-¾2) d¾.
Ä„
corresponds to calculation by the second formula of
0
(3.3).
Using this solution, from Eq. (1.1) we obtain
t
p 3kł 1 1
ln = - +
These equations differ from (2.7) in a factor of two larger
pe µa a
Ä„Ç(t - t )
coefficient on the right side. Therefore, for a cylindri- 0
cal hole, the pressure relaxation rate is higher. This is
× p(t ) - p0 dt , j = 1,
explained by the fact that in this case, the specific sur-
(4.2)
face area through which the gas is filtered is larger than
t
that for a hole with plane-parallel walls. In addition,
3kł 1 1
from the form of (2.7) (this equation does not contain
p = pe - +
2µa a
Ä„Ç(t - t )
Å‚ and m) it follows that the initial stage of the pressure
0
relaxation process is self-similar.
Figure 2 gives a curve of the function Õ(S) obtained
× (p(t ))2 - p2 dt , j = 2.
0
numerically [5], and curves calculated by asymptotic
formulas. It is obvious that the best approximation In the dimensionless variables, Eqs. (4.2) reduce to
for 0 < S < 10 is obtained by asymptotic expansion form
of Õ(S) for small values of S (S 1) in which the first Ä
Ü
P Ä„ 1
two terms are retained. Therefore, in further numerical
ln = -3 + "
Pe Å‚m - Ä
Ü
calculations using the integral equations (3.4) for Õ(S), Ä Ü
0
we used this two-term approximation.
Figure 3a illustrates the effect of the porosity m on
× P (Ä ) - 1 dÄ , j = 1,
Ü Ü
the solutions of (3.4) (for m = 0.1 and 0.4). In the ini-
(4.3)
tial stage of the process (Ä < 10-3), all curves are close.
Ü
Ä
Ü
In the interval 10-4 Ä 1, the curves are slightly dif-
Ü
3 Ä„ 1
P = Pe - + " P (Ä )2 - 1 dÄ ,
Ü Ü
ferent. With further increase in the dimensionless time
2 Å‚m - Ä
Ä Ü
Ü
(Ä 1), all curves merge again. From this it also follows 0
Ü
that the characteristic dimensionless relaxation time is
j = 2.
in the range 0.2 < Ä < 1. In addition, it is obvious that
Ü
"
the lower the porosity, the faster (for smaller values of For the initial stage of the process ( Ä Å‚m/Ä„), from
Ü
the dimensionless time Ä) the hole pressure relaxation. (4.3) we obtain the equations
Ü
350 Shagapov, Khusainov, Khusainova, and Khafizov
Fig. 3. Relaxation of dimensionless pressure in a cylindrical hole for various porosities (a) and various initial
pressures (b): thick curves refer to ordinary linearization (j = 1) and thin curves refer to Leibenzon linearization
(j = 2).
Fig. 4. Relaxation of dimensionless pressure in a spherical hole for various porosities (a) and various initial
pressures (b): thick curves refer to ordinary linearization (j = 1) and thin curves refer to Leibenzon linearization
(j = 2).
Ä
Ü
P - 1 Pe - 1 3Ä„
= exp - Ä , j = 1,
Ü
P P (Ä ) - 1
Ü
P Pe Å‚m
ln = -3 " dÄ , j = 1,
Ü
Pe
Ä - Ä
Ü Ü
(4.5)
0
(4.4)
P - 1 Pe - 1 3Ä„
= exp - Ä , j = 2.
Ü
Ä
Ü
P + 1 Pe + 1 Å‚m
3 (P (Ä ))2 - 1
Ü
P = Pe - " dÄ , j = 2.
Ü
In the dimensional variables, they are written as
2
Ä - Ä
Ü Ü
0
p - p0 pe - p0 3kp0t
"
= exp - , j = 1,
For rather large times ( Ä Å‚m/Ä„), for which a qua-
Ü
p pe a2µ
sisteady regime occurs, ignoring the second term in the (4.6)
kernels of Eqs. (4.3), we obtain the following solutions
p - p0 pe - p0 3kp0t
= exp - , j = 2.
for pressure relaxation:
p + p0 pe + p0 a2µ
Pressure Relaxation in a Hole Surrounded by a Porous and Permeable Rock 351
Figure 5 shows numerical solutions of Eqs. (4.3)
and (4.4) for Å‚ = 1.3 and m = 0.1 and a quasisteady
solution of (4.5). In the initial stage (Ä 10-4), the
Ü
results of calculations by relations (4.4) and (4.3) are
very close. In addition, although in the initial stage,
the quasisteady solution differs markedly from the solu-
tion of the general equation, this quasisteady solution is
in good agreement with the solution of (4.3) in the sub-
sequent stage of pressure relaxation (P 1). Conse-
quently, the main stage of complete pressure relaxation
in a spherical hole proceeds in a quasisteady regime.
CONCLUSIONS
The results of numerical calculations described here
show that in the plane one-dimensional, radial, and
Fig. 5. Relaxation of dimensionless pressure in a
spherical problems, the characteristic time of pressure
spherical hole calculated by the complete equation
relaxation is only slightly affected by the initial hole
(curves 1), the equation for the initial stage (curves
pressure. In addition, with change-over from the prob-
2), and the quasisteady solution (curves 3): thick
lem with plane symmetry (n = 0) to the radial and
curves refer to ordinary linearization (j = 1) and
thin curves to Leibenzon linearization (j = 2).
spherical problems, the dependence of the characteris-
tic dimensional time of pressure relaxation on poros-
ity becomes weaker. Considering gas filtration in the
In the case of a strong explosion (Pe 1), from the
porous medium, the pressure relaxation times predicted
solutions of (4.5) for rather large times and P - 1 1,
by the ordinary linearized equations and the equations
we have
linearized after Leibenzon differ by not more than a fac-
tor of three.
3Ä„
P - 1 = P exp - Ä , j = 1,
Ü
Å‚m
REFERENCES
3Ä„
P - 1 = 2(P + 1) exp - Ä , j = 2.
Ü
Å‚m
1. K. S. Basniev, I. N. Kochina, and V. M. Maksimov, Un-
derground Hydrodynamics [in Russian], Nedra, Moscow
Figure 4a illustrates the effect of the porosity m on
(1993).
the solutions of (4.3) (for m = 0.1 and 0.4). In the ini-
2. L. Leibenzon, Motion of Natural Liquids and Gases in a
tial stage of the process (Ä < 10-4) all curves coincide.
Ü
Porous Bed [in Russian], Moscow, Gostekhizdat (1947).
Figure 4b shows the effect of the initial dimensionless
3. A. N. Tikhonov and A. A. Samarskii, The Equations
hole pressure (Pe = 10 and 5) on the subsequent pro-
of Mathematical Physics [in Russian], Nauka, Moscow
cess of hole pressure relaxation. Because for a spherical
(1972).
hole, the specific surface area of filtration is larger than
4. H. Brunner, M. R. Crisci, E. Pusso, and A. Vecchio,
that for a cylindrical hole and a hole with plane parallel
 A family of methods for Abel integral equations of the
walls, the pressure relaxation in this case is faster than
second kind, J. Comput. Math., 34, No. 2, 211 219
that in processes with plane one-dimensional and radial
(1991).
geometries.
5. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in
Solids, Oxford Univ. Press, London (1959).