A new mathematical model for horizontal wells with variable density perforation completion in bottom water reservoirs
A new mathematical model for horizontal wells with variable density perforation completion in bottom water reservoirs
Dian-Fa Du 0 1 2
Yan-Yan Wang 0 1 2
Yan-Wu Zhao 0 1 2
Pu-Sen Sui 0 1 2
Xi Xia 0 1 2
0 Shengli Production Plant, Shengli Oilfield Branch Company , Sinopec, Dongying 257051, Shandong , China
1 Sinopec Petroleum Exploration and Production Research Institute , Sinopec, Beijing 100083 , China
2 School of Petroleum Engineering, China University of Petroleum , Qingdao 266580, Shandong , China
Horizontal wells are commonly used in bottom water reservoirs, which can increase contact area between wellbores and reservoirs. There are many completion methods used to control cresting, among which variable density perforation is an effective one. It is difficult to evaluate well productivity and to analyze inflow profiles of horizontal wells with quantities of unevenly distributed perforations, which are characterized by different parameters. In this paper, fluid flow in each wellbore perforation, as well as the reservoir, was analyzed. A comprehensive model, coupling the fluid flow in the reservoir and the wellbore pressure drawdown, was developed based on potential functions and solved using the numerical discrete method. Then, a bottom water cresting model was established on the basis of the piston-like displacement principle. Finally, bottom water cresting parameters and factors influencing inflow profile were analyzed. A more systematic optimization method was proposed by introducing the concept of cumulative free-water production, which could maintain a balance (or then a balance is achieved) between stabilizing oil production and controlling bottom water cresting. Results show that the inflow profile is affected by the perforation distribution. Wells with denser perforation density at the toe end and thinner density at the heel end may obtain low production, but the water breakthrough time is delayed. Taking cumulative free-water production as a parameter to evaluate perforation strategies is advisable in bottom water reservoirs.
Bottom water reservoirs perforation completion; Inflow profile; Cumulative free-water production
& Dian-Fa Du
& Yan-Yan Wang
Bottom water reservoirs are widely distributed on earth and
hold a large proportion of oil reserves (Islam 1993). Taking
China for example, there exist a large number of bottom
water reservoirs, most of which are developed using
horizontal wells. Compared with vertical wells, the producing
sections of horizontal wells have direct contact with oil
reservoirs, which not only reduces the producing pressure
drawdown, but also ensures bottom water flowing into the
wellbore more smoothly in a form of ??pushing upward??
(Besson and Aquitaine 1990; Dou et al. 1999; Permadi
et al. 1996; Zhao et al. 2006). Owing to these advantages, it
can effectively control bottom water cresting. The need of
economic and effective development of bottom water
reservoirs leads to the appearance of many types of
completion methods, such as barefoot well completion, slotted
screen well completion and perforation completion
(Ouyang and Huang 2005). Recently, partial completion,
variable density perforation completion and other new
completion methods have been put forward to further
control bottom water cresting (Goode and Wilkinson 1991;
Sognesand et al. 1994). By accurately finding out the water
production interval of horizontal wells, adopting plugging
strategies or properly adjusting the bottom water inflow
profile, these techniques can effectively prolong the life of
production wells. And among all these techniques,
perforation completion, including variable density perforation
and selectively perforated completion, plays a critical role
in alleviating water cresting (Pang et al. 2012).
Previously, scholars put more emphasis on the
productivity evaluation of horizontal wells (Dikken 1990; Novy
1995; Penmatcha et al. 1998) and bottom water cresting
(Permadi et al. 1995; Wibowo et al. 2004; Chaperon 1986).
The published papers mostly focused on horizontal wells
with open-hole completion. There is little research into
horizontal wells with variable density perforation
completion, and the ones that exist turned out to be very
problematic: (1) The method for open-hole completion
horizontal wells was used ignoring the fluid flow in
perforation tunnels in these studies (Landman and Goldthorpe
1991; Yuan et al. 1996; Zhou et al. 2002). Then, a model
describing the damage zone must be introduced to
characterize the influence of perforation holes
(Umnuayponwiwat and Ozkan 2000; Muskat and Wycokoff 2013).
Some scholars utilized the numerical simulation method to
discuss the impact of selective perforation on the
productivity of horizontal wells and built a single-phase flow
variable density perforation model for horizontal wells by
two filtration zones (Li et al. 2010). Since the seepage
resistance needs to be considered more precisely,
especially in the middle and later periods of the oilfield
development, the existing results are somewhat inaccurate.
(2) Conventional simplified models cannot analyze
formation pressure thoroughly and predict bottom water
cresting. Furthermore, a non-uniformly distributed bottom
water inflow profile along the wellbore was obtained
without considering the wellbore pressure drop (Guo et al.
1992). (3) In order to optimize completion parameters for
horizontal wells, oil production is usually viewed as the
only objective function. It is reasonable for horizontal wells
in conventional reservoirs. However, it is not accurate for
horizontal wells located in bottom water reservoirs because
of ignoring bottom water cresting, which decreases the
effective production period of wells (Luo et al. 2015).
In this paper, based on the precise consideration of the
fluid flow in each perforation, the flow behavior in
perforations, wellbores, as well as reservoirs, was analyzed.
Coupling the fluid flow in reservoirs and wellbore pressure
drop, a comprehensive model, which can be used to
evaluate productivity of horizontal wells, was developed based
on potential functions and solved using the numerical
discrete method. Then, a model describing bottom water
cresting was established on the basis of the piston-like
displacement principle. Finally, both the bottom water
cresting behavior and the factors influencing inflow profile
were analyzed using the developed model, and a more
systematic optimization method was proposed by
introducing the concept of the cumulative free-water
production, which could realize a balance between stabilizing oil
production and controlling bottom water coning.
2 Productivity analysis for horizontal wells with variable density perforation completion
For horizontal wells with perforation completed in bottom
water reservoirs, formation fluids firstly flow into
perforation holes before converging in the horizontal wellbore.
Under this circumstance, the effect of perforation holes is
similar to that of short producing branches in inclined
horizontal wells (Holmes et al. 1998). The real producing
part for horizontal wells should be quantities of perforation
holes. Therefore, the perforation holes can be regarded as
??source-sink?? term, and this kind of problem can be solved
with a source function.
2.1 Analysis of fluid flow near wellbore regions A model was built to describe the flow of formation fluid near horizontal wellbores, and the assumptions for this model are as follows:
The formation is homogeneous with a uniform
The horizontal permeability meets the following
basic relationship: Kx = Ky = Kh, and the vertical
permeability is Kz = Kv. The target reservoir is
infinite in the horizontal plane;
The single-phase fluid flowing in the reservoir is
incompressible and the fluid flow obeys Darcy?s law;
The wellbore is horizontal, in which the perforations
are unevenly distributed;
The perforating direction is perpendicular to the
wellbore, and the lengths as well as radius of all the
perforation tunnels are the same.
Acoordinate system is established as shown in Fig. 1, in
which the heel end of the wellbore is M0 (x0, y0, z0). The
horizontal part of the wellbore (total length L, m) is divided
into N segments. Therefore, the length of each segment is
Different segments have different characteristic
parameters: the perforation density, np(i); the perforation
depth, lp; bore diameter, Dp; phase angle x; and the initial
perforation angle x0. The heel end is selected as the
origin of this coordinate, and the x direction is parallel
with the wellbore. The coordinates of any point (x, y, z) in
the jth perforation tunnel in the ith segment are as
sin hk sin ak ? np?i? sin hi sin ai ? t lp sin ci j sin vi j
where hk is the angle between the kth segment and the
vertical direction, k ? 1; 2; . . .; i 1; ak is the angle
between the x-axis and the projection of the kth segment
in the horizontal plane, k ? 1; 2; . . .; i 1. hi is the angle
between the ith segment and the vertical direction; ai is
the angle between the x-axis and the projection of the ith
segment in the horizontal plane; ci j is the angle between
the jth perforation in the ith producing part of the
horizontal well and the z-axis; vi j is the angle between the
xaxis and the projection in the xy plane of the jth
perforation in the ith segment of the horizontal well; in this
paper, when all the perforation is perpendicular to the
horizontal wellbore, then vi,j = p/2.
According to the classical theory of flow in porous
media (Cheng 2011), the fluid flow at any point in an
infinite reservoir obeys the Laplace equation. So we have:
Fig. 1 Schematic diagram near the horizontal wellbore in a bottom
oo2xU2 ? oo2yU2 ? oo2zU02 ? 0 ?3?
where U ? pKhKv lop, m2/s; lo is the oil viscosity,
mPa s; and also the corresponding parameters of the
l0p ? lp b2 cos ci2j ? 2 sin ci2j;
b sin c
sin c0ij ? qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiiffijffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
b2 cos ci2j ? b12 sin ci2j
In order to simplify the solution procedure, one
assumption that the flow rates of all perforations in each
segment are equal, is proposed (Wang et al. 2006):
where qp is the flow rate of each perforation at the ith
segment, m3/s; qra?i? is the flow rate of the ith segment, m /
s; Dx is the length of the ith segment, which is equal to L/N,
m; and np?i? is the perforation density of the ith segment,
Compared with the length of the horizontal wellbore, the
perforation is rather short. Therefore, it can be regarded as
an infinitesimal line source. After integrating the point sink
solution over the perforation direction, the pressure
response of any perforation hole in the formation is
obtained. For example, the potential at point M(x, y, z0)
caused by the jth perforation in the ith segment can be
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiaffi?ffiffiffi?ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds ? Cij
4plpLnp?i? ?xp x?2 ? ?yp y?2 ? ?z0p z0?2
The integration of Eq. (5) is as follows:
Uij?x; y; z0? ?
Nqra?i? ln r1ij ? r2ij ? l0p
4plpLnp?i? r1ij ? r2ij l0p ? Cij
where qr(i, j) is the flow rate of the jth perforation tunnel in
the ith segment, m3/s; r is the distance between the source
point Mp?x; y; z0? and the target point M?x; y; z0?; Cij is an
integration constant; r1ij is the distance between the heel
end and the target point; r2ij is the distance between the toe
end and the target point, and they observe the following
r1ij ? ?xp?i; j; 0? x 2 ? ?yp?i; j; 0? y 2 ? ?z0p?i; j; 0? z0 2
?xp?i; j; 1? x 2 ? ?yp?i; j; 1? y 2 ? ?z0p?i; j; 1? z0 2
where xp(i, j, 0), yp(i, j, 0), zp(i, j, 0) and xp(i, j, 1), yp(i, j, 1),
zp(i, j, 1) are the coordinates of the left and right ends of the
jth perforation in the ith producing part.
Based on the mirror image reflection and superposition
principle, the potential of the jth perforation tunnel for
the ith production segment at point Mp?x; y; z0? is
U?x; y; z0? ?
nij?2h0 ? 4nh0 z0pij?i; j; 0?;
2h0 ? 4nh0 z0pij?i; j; 1?; x; y; z0?
? nij?4nh0 ? z0pij?i; j; 0?; 4nh0 ? z0pij?i; j; 1?; x ; y; z0?
nij? 2h0 ? 4nh0 ? z0pij?i; j; 0? ; 2h0
? 4nh0 ? z0pij?i; j; 1?; x ; y ; z0?
z0pij?i; j; 0?; 4nh0
z0pij?i; j; 1?; x ; y ; z0? ? Ci0
nij?e0; e1; x; y; z0? ? ln
According to the superposition principle, the potential at
any point of the infinite formation created by all the
perforation tunnels of the horizontal well is:
Uij?x; y; z0? ?
nij 2h0 ? 4nh0 z0pij?i; j; 0?; 2h0 ? 4nh0
z0pij?i; j; 1?; x; y; z0 ? nij 4nh0 ? z0pij?i; j; 0?; 4nh0
?z0pij?i; j; 1?; x; y; z0
2h0 ? 4nh0 ? z0pij?i; j; 0?; 2h0
?4nh0 ? z0pij?i; j; 1?; x; y; z0
where pe is the boundary pressure, MPa.
For perforated completion, the perforation tunnels
directly contact the formation. Therefore, the flow potential
of some point, which is just located in the perforation, can be
obtained using Eq. (15). In this case, there are two points
involved: the target point and the source point (perforation
point). To get rid of singularity phenomenon, the central hole
in the wall is chosen as the jth perforation?s target point when
calculating the distance between two perforation points.
Only calculating the pressure of one central point for the
segment and regarding it as the pressure of the whole
segment will result in deviation when analyzing the
segment?s pressure of a horizontal wellbore. In this paper,
taking the average over all the perforations? pressure in the
same segment and using the average value as the
representative pressure of that segment, the pressure of the ith
segment is as follows:
where pwf(i) is the flow pressure for all the perorations in
the ith segment, MPa.
The boundary pressure at the oil?water interface is
assumed to be constant. Therefore, the potential at any
point of the formation may be expressed as:
Uij?x; y; z0? ? Ue
Combining the definition of potential, the pressure at
any point of the formation can be given as follows:
Nloffiffiffiffiffiffiffiffiffiffi XN qra?i? NLXnp?i?
4plpLpKhKv i?1 np?i? j?1
There are 2N variables required to be calculated by
analyzing pressure distribution along the wellbore: pwf(i)
and qra(i), (i = 1, 2, ?, N). However, it only contains
N equations in the flow model [Eq. (15)]. Therefore, one
more model is needed to describe pressure drop along the
wellbore (Li et al. 1996, 2006).
2.2 Wellbore pressure drop model
For perforated horizontal wells, the main idea to develop a
wellbore pressure drop model is to divide the horizontal
wellbore into several segments, and each segment is
subdivided into several smaller parts that only include one
perforation (Su and Gudmundsson 1994).
According to the analysis of the pressure drop in a
wellbore, the total pressure loss can be written as:
dpwf ?i? ? dpfric?i? ? dpacc?i? ? dpmix?i? ? dpG?i?
where dpfric(i) is the friction loss of the ith segment, MPa;
dpacc(i) is the acceleration loss of the ith segment, MPa;
dpmix(i) is the mixing loss of the ith segment, MPa; and
dpG(i) is the gravity loss of the ith segment, MPa.
It should be noted that, in the previous section, a
mechanical field described by the flow model is
established based on the potential function. Therefore, when
the potential function is used to deal with flow problems,
the pressure loss caused by viscous force has been
considered. As we all know, the mathematical expression of
fluid potential is Kp/l, where K is the formation
permeability, p represents pressure, while l means the fluid
viscosity, and it is used to describe the viscous force,
which will lead to pressure loss along the perforation. So
it means that the viscous force has been taken into
consideration in the first model. In other words, when
building wellbore pressure here, only four kinds of
pressure loss should be calculated.
The calculation method of frictional loss and
acceleration loss between two perforation tunnels is expressed,
Dpfric ? 1:34
where q is the liquid density, kg/m3; D is the wellbore
diameter, m; qL(i,j) is the flow rate along the wellbore, m3/
d; vs?i; j? is the average velocity of the jth perforation in the
ith segment, m/s; and ffric?i; j? is the friction coefficient of
the jth perforation in the ith segment.
In Eq. (19), one parameter, called the friction
coefficient, is introduced. The calculation of the friction
coefficient is dependent on the Reynolds number. If the Reynolds
number is less than or equal to 2000, the flow is laminar;
otherwise, it is turbulent flow. And the expression for the
Reynolds number is as follows:
Re ? 7:3682844
where Re is the Reynolds number; Q is the axial flow rate
along the wellbore, m3/d; l is the viscosity of the flowing
fluid, mPa s; and r is the wellbore radius, m.
In addition, we also introduced a criterion when
calculating the mixing loss between two perforation holes. The
specific calculation method for frictional loss and mixing
losses is listed in Table 1. In Table 1, qc is the critical rate,
m3/d; e is the surface roughness, mm.
When calculating the mixing loss, one parameter, Dpper,
is introduced. It is the frictional loss showing up after
perforating and can be figured out using Eq. (19).
The friction loss, acceleration loss and the mixing loss
of the ith segment are listed below:
Table 1 Calculation method for friction coefficient and mixing loss between two perforation holes
dpfric?i? ? 1:0862
If the horizontal well is inclined, the gravity loss is
non-ignorable, and the wellbore pressure drop model
In this paper, an iterative method is used to solve the
above-mentioned model, and the iterative process is shown
in Fig. 2.
? qgDx cos hi
2.3 Coupling model
When a horizontal well begins to produce reservoir fluids,
the fluids in the perforations connect the wellbore and oil
reservoir together. Therefore, perforation is considered as
an infinitesimal linear sink, which directly contacts
reservoir and wellbore. Meanwhile, pressure responses are
generated in the whole reservoir. The generated pressure
responses near the perforations are associated with oil
inflow in the radial direction of the horizontal wellbore,
which can be calculated by utilizing the reservoir flow
model. Since the pressure in perforation holes is relevant to
the wellbore pressure, a coupling relationship exists
between the reservoir flow model and the wellbore pressure
According to the reservoir flow model, a model is
developed to calculate steady-state productivity of the
horizontal well with variable density perforation, in which
the pressure drop along the horizontal wellbore is
3 Analysis of inflow profiles in bottom water reservoirs
The bottom water rises fastest in the vertical plane of the
horizontal wellbore (Cheng et al. 1994). In other words, the
bottom water will firstly break through into the wellbore in
this plane due to the highest pressure gradients in this profile.
Therefore, a complicated 3-D problem can be turned into a
2-D problem in the xz profile where the wellbore lies. The
formation between the wellbore and the oil?water interface
is discretized according to the division of the horizontal
wellbore in the reservoir flow model. And the total number of
grids in the vertical direction is nz, which is shown in Fig. 3.
The rise of bottom water is treated as a piston-like
flooding process. That is to say, there is an obvious
interface between the oil zone and the water zone. The oil?
water contact moves upward to the horizontal wellbore in
the vertical direction. Once it reaches any point of the
wellbore, water breakthrough occurs there. According to
the material balance theory (Xiong et al. 2013), we have:
pwf?i? ? pwf ?i 1? ? 0:5?Dpwf ?i 1? ? Dpwf? i?
where pwf(i) is the wellbore pressure in the ith segment,
MPa; N is the number of divided segments along the
wellbore; and /kj is a function corresponding to the
horizontal wellbore as well as the oil?water interface.
Swc udxdydz ? vwdxdydt
Based on Darcy?s law, the water rise velocity is as
Input formation and well completion parameters
Divide segments and establish the coefficient matrix A
pwf0 = pwf1
Solve the matrix equation and get the flow rate q(i) for each segment
Fig. 2 Flowchart for solving the coupling model
Fig. 3 Schematic diagram of the physical model for bottom water cresting
Output production and pressure
where Sw(i, k ? 1) is the water saturation of the (i, k ? 1)
grid at time t in the longitudinal profile; Swc is the connate
water saturation; Krw is the relative permeability to water;
and lw is the water viscosity, mPa s.
According to the results of grid discretization, the
vertical pressure gradients between any two contiguous grids
are as follows:
Dp?i; k? ? p?i; k ? 1? p?i; k?
Dp?i; k? ? pwf ?i? p?i; k?
Substituting Eqs. (28) into (27) gives the rise velocity of
According to the established mathematical model, the
pressure at any grid between the oil?water contact surface
and the horizontal wellbore can be written as:
Combining with Eq. (26), the time required for the
water rising from the (k ? 1)th grid to the kth grid is
The breakthrough time at the ith segment is:
4 Case study
where u is the porosity; nz is the number of meshes in the
longitudinal direction between the wellbore and the oil?
density of each case is 2 shots/m. In Case 1, the perforation
is uniformly distributed. In Cases 2 and 3, the perforation
density at the heel end of the horizontal well is larger than
that at the toe end of the horizontal well, while the
perforation density is denser at the toe end than that at the heel
end for Cases 4?6.
The simulation results for all the six cases are shown in
Figs. 5, 6, 7, 8 and 9. Figure 5 shows the pressure
distribution along the horizontal wellbore. In fact, there exists a
pressure drop along the perforation hole. However, it has
little relationship with our research object; thus, the
relevant calculation was not carried out in this work. In order
to better identify their characteristics, the pressure
distribution curves of only three cases (Cases 1, 2 and 5) are
plotted together in Fig. 5. It can be seen that the pressure
distribution curves are steep near the heel end of the
horizontal wellbore, while relatively flat at the toe end. In
addition, the greater the pressure drop is, the denser the
perforations will be. Near the toe end, the pressure of Case
5 is higher than those of Case 1 and Case 2, indicating a
denser perforation and a greater pressure drop in this
location, while near the heel end, the difference of these
three curves is smaller. Figure 6 shows the friction and
acceleration losses (Case 1) along the wellbore,
respectively. At any point of the horizontal wellbore, the friction
loss is greater than the acceleration loss, and the former is
nearly six times as much as the latter, which means the
friction loss plays a leading role.
Figure 7 gives the flow rate distribution along the
horizontal wellbore. For the horizontal well with uniformly
distributed perforation, the flow rate at the toe end is lower
than that at the heel end, due to the influence of the
wellbore pressure drop. These six cases have the same number
of perforations, and thus, the production is also similar,
especially for Cases 1?3. For Case 3, although the
difference of the flow rate between the heel end and the toe end
is larger compared with the other five cases (Fig. 7), the
production rate of the horizontal well is slightly larger
(Fig. 8). Therefore, in order to maximize the production of
the horizontal well, a perforation scheme with a larger
perforation density at the toe end should be adopted. It
should be noted that a larger perforation density at the toe
end does not necessarily result in a higher production rate.
The reason is that this kind of perforation scheme will lead
to much lower flow rate at the toe end. In addition, Fig. 7
also shows that the flow rate distribution of Case 6 is more
uniform compared with the other five cases. Due to the
influence of the pressure drop along the wellbore and the
reduced end effect, the flow rate distribution of Case 6 has
a more uniform distribution of cresting height, which will
delay the occurrence of bottom water breakthrough.
Simulation results indicate that reducing the perforation density
at the heel end is helpful for obtaining an evenly advancing
Using the developed model, the well productivity and
water breakthrough for a horizontal well in a bottom water
drive reservoir were evaluated. Table 2 lists the bottom
water drive reservoir properties and its drilling and
The steady-state productivity of the horizontal well with
variable density perforation completion was evaluated, and
also the bottom water inflow profile was calculated. A set
of basic variable density perforation cases are designed and
are shown in Fig. 4. For simplicity, the average perforation
Table 2 Parameters for the
bottom water drive reservoir
and its drilling and completion
200 300 400
Distance to the heel end, m
Vertical permeability, lm2
Parameter 28 25 0.2
Fig. 4 Cases for variable density perforation
Fig. 6 Friction loss and acceleration loss along the horizontal
wellbore for Case 1
200 300 400
Distance to the heel end, m
Fig. 5 Pressure distribution along the horizontal wellbore
water profile on the vertical plane of the horizontal
The detailed distribution of bottom water breakthrough
time along the wellbore is shown in Fig. 9. Compared with
Case 1 (evenly distributed perforation), the redistribution
of perforating density changes both the breakthrough
location and breakthrough time simultaneously. It is
obvious that the larger perforation density means a shorter
breakthrough time. Meanwhile, the bottom water
Fig. 7 Flow rate distribution along the horizontal wellbore for
breakthrough location and time of the whole wellbore can
be calculated by the developed model, as shown in Table 3.
Figure 10 illustrates the distribution of the bottom water
rising height along the wellbore for three different cases,
which is in good agreement with the results in Fig. 9. The
bottom water cresting height of Case 2 is shown in Fig. 11.
It shows that the effect of the perforation density on bottom
water cresting height becomes more serious with the
increase of time. The deformation of the water ridge is
Fig. 8 Production rate of the horizontal well for different cases
200 300 400
Distance to the heel end, m
Fig. 9 Distribution of breakthrough time for different cases
obvious, which reflects on the degree of water crest
According to the above results, the perforation
scheme with higher production (Cases 2 and 3) will
advance the time of the bottom water breakthrough and
thus have a negative effect on the development of bottom
water reservoirs using horizontal wells. In other words, the
Table 3 Bottom water breakthrough location and time
Fig. 10 Bottom water rising height for different cases (production
time, 150 days)
horizontal well with a higher production rate will have a
shorter water-free production period. So there exists a
contradiction between increasing the well production and
delaying the breakthrough time of bottom water. In order to
comprehensively evaluate the effect of both controlling
bottom water cresting and stabilizing oil production, a new
parameter, called the cumulative free-water production, is
defined as follows:
where Qo is the cumulative oil production without water,
m3; q is the rate when the horizontal well produces
steadily, m3/d; and To is the production time without water, d.
Obviously, the cumulative oil production without water
of the horizontal well could consider both the well
production and effective production time.
According to the calculation results (Table 4), the Qo of
Cases 4 and 5 is higher than those of other Cases owing to
the more uniform inflow profile. Therefore, for the
horizontal well studied in this paper, the perforation
scheme with denser perforation hole at the toe end and
sparser perforation hole at the heel end is a more
Breakthrough time at the first Breakthrough time at the point
breakthrough point, d 100 m away from the heel, d
Breakthrough time at the point
500 m away from the heel, d
Distance to the heel end, m
Fig. 11 Rising height of bottom water at different production times
Table 4 Parameters required for calculating Qo of different cases
Breakthrough location, m
In this paper, a model coupling fluid flow in reservoirs and
pressure drop along the wellbore was developed to obtain
the distribution of flow rate along the horizontal wellbore
and to determine the horizontal well productivity. In this
model, the fluid flow at each perforation hole was taken
into account, and the perforation tunnels were treated as
many infinitesimal sections. After that, a bottom water
cresting model was developed based on the piston-like
displacement principle, which could be used to calculate
the bottom water breakthrough location, breakthrough time
and the shape of water cresting at different times.
For a horizontal well with uniform density perforation
completed in a bottom water reservoir, the pressure drop
along the wellbore may significantly affect the water
breakthrough mode. Specifically, the bottom water will
firstly break through at the heel end of the wellbore.
Neglecting the end effects, water breakthrough occurs in
proper sequence from heel end to toe end. The bottom
water breakthrough at a specific position will happen
earlier if the perforation density is increased. When a
horizontal well with denser perforation density at the toe end
and sparser perforation density at the heel end, the lowest
point of the distribution curve will move upward, which
means the water breakthrough time is delayed. Under these
circumstances, the horizontal well obtains a longer
effective production period.
It is meaningful to take cumulative oil production
without water as a parameter to evaluate the perforation
strategies for horizontal wells. With this parameter, both
the effects of productivity of the horizontal well and
bottom water breakthrough time can be considered
comprehensively. However, in order to simplify the calculation
process, some assumptions have been introduced in this
model, some of which are ideal ones and different from the
actual reservoirs. Further work needs to be done in order to
extend the application of this model.
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