SPE

Society of Petroleum Engineers

SPE 18300

Pressure Transient Analysis and Inflow Performance for

Horizontal Wells
by F.J. Kuchuk and P.A. Goode, Schiumberger-Doll Research; B.W. Brice and D.W. Sh~rrard,
Standard Alaska Production Co.; and R.K.M. Thambynayagam, Schlumberger Well Services
SPE Members

Copyright 1988, Society of Petroleum Engineers

This paper was prepared for presentation at the 63rd Annual Technical Conference and Exhibition of the Society of Petroleum Engineers held in
Houston, TX, October 2-5, 1988.

This paper was selected for presentation by an SPE Program Committee following review of information .contained in an a~stract sUbmit~ed by the
author(s). Contents of the paper, as presented, have not been reviewed by the Society ?f Petroleum Englnee.rs and ~re s~bJect to correction by the
author(s). The material, as presented, does not necessarily reflect any position of ~he Society of Pe~roleum Engineers, Its.offlcers, or ~e~bers. Pape~s
presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleu~ Engln~ers. Permission to copy IS
restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstra?t should contain conspicuous acknowledQment of
where and by whom the paper is presented. Write Publications Manager, SPE, P.O. Box 833836, Richardson, TX 75083-3836. Telex, 730989 SPEDAL.

Abstract due to the fact that most horizontal wells will exhibit partial
In recent years, pressure transient behavior and inflow perfor- penetration effects, even when they are fully perforated. This
mance of horizontal wells have received considerable attention due fact has already been observed by many authors 1 - ll , and specific
to the increase in horizontal drilling. In this paper we develop an methods have been proposed to identify How regimes and their
interpretation methodology for horizontal well pressure transient durations under ideal conditions. However, it has not been shown
testing. This methodology is then applied to the interpretation how to extend the identification of How regimes and their usage
of an actual horizontal well test performed in Prudhoe Bay. The to the interpretation of real pressure transient tests.
complex geometry associated with horizontal wells makes inter- The purpose of this paper is to provide a methodology for the
pretation of well tests a difficult task. Uniquely determining the interpretation of well test data from horizontal wells. To achieve
system parameters from short time (typical times for vertical well this goal, an analysis of synthetic and real well test data are pre-
testing) pressure tests is not possible. Combining drawdown and sented. In addition to an interpretation methodology new analyt-
buildup tests, with downhole Howrate measurement is critical for ical formulas for inflow performance of horizontal wells with and
proper interpretation. without the effect of a gas cap or aquifer are also presented.
We also provide a solution for the Inflow Performance of a hor- Mathematical Models
izontal well completed in a rectangular drainage volume, where
the well can be of arbitrary length and completed at any location Solutions with and without Gas Cap or Aquifer
within the drainage volume.
In a horizontal well, there will be considerable wellbore volume (50
Introduction to 100 bbls) below the tool, even when a downhole shut-in device
is used or the downhole How rate is measured. The storage effect
Now forward solutions to the diffusivity equation for horizontal due to this volume typically last longer than that for a vertical
well geometry with varied boundary conditions are available in well in the same formation because the anisotropy reduces the
the literature1 - 11 • Moreover, Reiss 12 and Sherrard13 presented effective permeability at early times to ..jkhkv. For a horizontal
performance and production data from several horizontal wells well, the downhole pressure at any point in the wellbore (from
and mentioned interpretation of well test data. the surface to sandface) is given as
The interpretation of well test data from horizontal wells is a
much more difficult task than its vertical counterpart. This diffi-
culty stems from: 1. the main search direction of the parameters Pwbj(t) = Pi - it qD(T).dP~j(t - T)dT, (1)
usually does not coincide with the depositional environment, 2.
where .dYwj is the constant How rate, q, response of the system
the three dimensional nature of the How geometry and lack of
including the effect wellbore volume below the measuring point
radial symmetry, and 3. more parameters (information) to be where the downhole How rate, qm, is acquired with the downhole
obtained. In addition to these difficulties, zonal variation of ver-
pressure, Pwbj. The dimensionless How rate qD = qm/q.
tical permeability and shale distribution will make interpretation
intricate. The Laplace transform of .dYwj is given as 14,15

A well defined How period comparable to that of the infinite acting

radial How period (free from storage and boundary effects) of a (2)
vertical well is not apparent for horizontal wells. This is largely

383
2 Pressure Transient Analysis and Inflow Performance for Horizontal Wells SPE 18300

where q is the constant flow rate, C is the wellbore storage coef- Distinct flow periods may occur due to the transient pressure test-
ficient due to the volume below the measuring point and AplI/ is ing of horizontal wells. The first will be a period of radial flow
given by around the well, where there is negligible effect from the ends of
the well and no effect from the boundaries. If there is a vertical
barrier to flow near the well and the influence of that boundary
(3) is felt before any significant effect from the ends of the well, a pe-
where 5(t) is the Dirac delta function, and AplI the pressure drop riod of hemi radial flow will develop. Following, this if the well is
due to skin. very long compared to the formation thickness, a period of linear
flow may develop in the case of two noflow boundaries. Hone
Ap/ is the constant flow rate response of a horizontal well, shown of the boundaries is a constant pressure boundary the linear flow
in Fig. 1, which is completed in an infinite anisotropic medium period will not be observed and a steady state pressure will be
bounded above and below by horizontal planes. The permeabil- achieved, which is controlled by the pressure at the constant pres-
ities in the horizontal and vertical directions are denoted by kh' sure boundary. In the case of two noflow boundaries a final flow
and le", respectively. The flow of a slightly compressible fluid of period of radial flow in the plane of the formation may develop,
constant compressibility and viscosity is assumed throughout the similar to that observed for vertical wells.
medium. Gravity effects are neglected. Two types of top and bot-
tom boundary conditions are considered. In the first case, both Inflow Performance
the top and the bottom boundaries have no-flow conditions. In
the second case, one of the boundaries is at constant pressure Constant Pressure Boundary
(the system either has a gas cap at the top boundary or an ac-
The Inflow Performance (IP) of a well which is influenced by the
tive aquifer at the bottom boundary), while the other is a no-flow
presence of gas cap (constant pressure boundary) does not depend
boundary as before. For convenience we shall refer to the first
on its drainage area. The equation for the steady state pressure
model as the "no-flow boundary model" and to the second as the
for a well influenced by a constant pressure boundary was devel-
"constant pressure boundary model." The notation of this paper oped in Ref. 10 and can be written in terms of IP as
assumes that in the latter model the constant pressure boundary
is at the top (the gas cap case), but the formulae may be readily IP= q (9)
adapted for the case of an aquifer at the bottom. <p> -Pw/

The Laplace transform of the constant rate response of the for- where
mation , Aft" is given bylO
< P > -pw/

aft/{.) = ~ [F{U) + 2 t, F{et) co. j~Z co. j'lrh"" ] (4)

where ej =0"+(j~'/t)2, for the no-flow boundary case, and and < P > is the average reservoir pressure. Equation 10 is only
valid when ..ffl"" < 5, if this condition is not satisfied the full
form of the equation, involving the second integral of the modified
Bessel function K o, must be used (see Ref. 10).
where ej=0"+«(j-~PillL/~)2,for the constant pressure bound- N oflow Boundary
ary case.
A general solution for a horizontal well of arbitrary length pro-
In both Eqs. 4 and 5 the pressure is evaluated at d~cing from a closed rectangular region of arbitrary aspect ratio
Z = Zw + !f (1 + v'B and with the well placed at any location within the box (see Fig. 2) is
not available in the literature 1,8. The formulas given in the liter-
oo dusin2 U
F(v) ==
lo U2 vu 2 +V
, (6) ature are for special cases, thin reservoir (no appreciable pressure
drop in the vertical direction), the well crosses the entire drainage
volume (infinite well length), or the well is located in the center
(7) of an elliptic drainage volume. H a field, or an area of a field,
is completed with horizontal wells the assumptions of long wells.
and and thin reservoirs are unnecessarily restrictive.
L~r/Jp.Ct
t7 = 0.000263kh s. (8) In Appendix A we have derived a general solution for the pseudo
Eqs. 1 with a chosen downhole flow rate schedule, qm, will' be steady state pressure drop of a horizontal well producing from a
used to generate downhole pressure data with skin and wellbore rectangular region. We have made no assumption about the rel-
storage (due to the volume below the rate measuring point) for a ative length and position of the well or the formation thickness.
horizontal well with and without the gas cap effect. The only assumption necessary is that the the distance from the
well to any lateral boundary must be large relative to the dis-
Since we are using the line source approximation, Eqs. 4 and 5 tance from the well to the top and bottom of the reservoir. This
are valid only when tD/r~D > 25 (see Ref. 10 for the definitions). assumption will almost always be satisfied in practice, unless the
This is an important limitation because for large anisotropy, it vertical permeability is small, which would make the reservoir a
may take more· than several seconds to satisfy this condition. poor candidate for development with horizontal wells. The dif-
ference between the wellbore pressure and the average reservoir

384
SPE 18300 F.J. Kuchuk, P.A. Goode, B.W. Brice, D.W. Sherrard and R.K.M. Thambynayagam 3

pressure, which can be used to calculate IP using Eq. 9, is written the constant flow rate case are available in the literature1 - 10, it
as is well known that in a real well, the constant rate flow condi-

<p >
_ _ 70.6qp, [411'" L y
p1O! - kh h Lz 3
(!_ +y~)
Yw
Ly L2y
tion is very difficult to achieve. At early times, wellbore storage
(even with a shut-in device or downhole flow rate measurement)
will mask the well defined characteristic features of the pressure

+ L~
11'"
2L2 ~ 1 .... 2 (
LJ -':z
10 n=l n
1+r,;t)
h
+L- fih(S + )]
10"
-k Z Sm (11)
response of a horizontal well. At late times, because of multiple
boundaries with irregular shapes, it will be difficult to construct
a complete model for a given horizontal well as the transient trav-
where els away from the wellbore. Moreover, a slight decline of the flow
rate as a function of time will smooth the sharp features of the
pressure behavior, which usually carries information about the
system. Nevertheless, it is useful to know the constant rate be-
havior of horizontal wells in order to further our understanding of
the pressure response of a well with other effects such as wellbore
storage, unknown boundaries, etc.
2e +e aL , a (L,-Ylll) +eaYllI 2n1l'"
e= ----1-_-e- ";'"'----
aL , a = -T;;;' As stated above, interpretation of well test data from a horizontal.
well presents two problems which must be solved simultaneously.
and
The first is the definition (diagnosis) of the system; that is, iden-
.... 2. ( L 10 ) :1: 10 )
':z = -sm n1l'"-L cos n1l'"-L .
(

n z z
tification of the presence of the boundaries with and without in-
dependent information. Geological and geophysical methods may
This equation may be written as
provide clues about the presence and location of the boundaries,
< P >-p1o!= and whether those boundaries are due to no-flow barriers, such
as shales, faults, etc. However, these sources may not reveal the
70.6qp, [F (Y1O L y L 10 :1:10 ) + ~ ~S ] (12) interference boundaries between the wells. The second problem
khh L y ' L z ' L z ' L 10 L 10 VIi;; Z •
is the estimation of the distance to the boundaries from the the
Typical values for the function F have been tabulated in Table 1. wellbore and other unknown reservoir parameters (permeabilities,
skin, etc).
Figs. 3 through 5 illustrate how the inflow performance is af-
fected by various parameters. One can see from Fig. 3 that the The identification of flow regimes during a transient well test for
IP increases almost linearly with increasing well length, not log- a horizontal well can be invaluable when used in conjunction with
arithmically as previously reported by Karcer 8. The reason for deconvolved pressure (the system influence function). The tran-
sition and duration of each flow period can disclose fine details of
this is that most of the pressure drop (in any geometry for hori-
the system, particularly if it is used with proper displaying tech-
zontal wells) occurs near the well where there is essentially radial
niques (log-log, semilog, pressure derivative, etc), each of which
flow around the well in the yz plane. This pressure drop is re-
may exhibit particular features of the system.
flected in the Sz term of Eq. 11. The pressure drop near the well
will decrease linearly as the well length increases. If the formation Displaying techniques can be a powerful identification tool par-
is extremely thin, (or the vertical permeability very high) that is ticularly if the effect of the wellbore storage is minimized by mea-
the well is analogous to a fully penetrating vertical fracture, then suring downhole flow rate or using a downhole shut-in tool. The
the pressure drop will decrease logarithmically as the well length interpretation of a given horizontal well test can be enhanced con-
increases. However, in most practical cases the productivity of a siderably if geological and geophysical methods can provide the
horizontal well will increase approximately linearly as the length presence and location of the boundaries.
of the well increases.
For the identification and estimation, the conventionallogarith-
It is reasonable to think that if S z was set equal to 0 that Eq. mic convolution, deconvolution, and nonlinear least-squares esti-
11 would give the pseudo steady state pressure drop for a fully mation methods may be used. Since these methods have been
penetrating vertical fracture. This is not in fact the case because presented in detail in the literature we will not discuss them here.
we have employed pressure averaging which is accurate for hori- However, their application to a horizontal well test example, in
zontal wells where the radius is small compared to the well length conjunction with the solutions presented earlier, is demonstrated
but will cause errors for fractures where the fracture width is not in the following section.
necessarily small compared to the fracture length.
Application
In Fig. 4 we have shown the effect of the vertical location of the
well. It can be observed, as would be expected, that the well In the previous section the constant flow rate behavior of the sys-
performs best when it is placed in the center of the formation. tem was presented. In this section, we attempt to analyze well
test data from a hypothetical well, where we use Eq. 1 to gener-
Because of the geometry involved with horizontal wells it can ate the pressure response. Horizontal and vertical permeabilities,
take hundreds or even thousands of hours to reach pseudo steady distances to the top and bottom boundaries, and fluid and rock
state flow. This must be recognized when planning a well test properties for this example are given in Table 2. The reservoir is
to determine IP, it may in fact (particularly in low permeability assumed to be of infinite extent but is bounded by no-flow or con-
formations) be impractical to perform a test to determine IP. stant pressure boundaries at the top and bottom. Although an
analytical model is employed for the computation of the downhole
Interpreta~ion
pressure, P1Ob! from Eq. 1, a downhole flow rate based on realistic
Although analytic solutions and their characteristic behavior for

385
4 Pressure Transient Analysis and InfloW' Performance for HorizontalWells SPE 18300

wellbore flowing conditions is used. As stated above, the additive stant rate behavior) that the time interval of 0.1 and 75 hrs is the
storage effect below the rate measuring point is also included in transition between the first and the intermediate time r,adial flow
the model. periods which corresponds to the semilog period of 0.6-70 hours,
given in Fig. 8. Note also from Fig. 7 that at the time interval of
a. Drawdown Test
o and 0.15 hrs, the]l/ curve is flat, indicating the first radial flow
1. Well between two No-flow Boundaries period. For the time interval of 1 to 3 hrs the ]1/ curve becomes
almost flat because of the second radial flow period. Although it
The hypothetical well is put on production at a constant surface is not exact, this would normally be taken as an indication of a
rate of 10,000 B /D with an initial reservoir pressure of 4400 psi. second radial flow period.
The downhole· pressure and flow rate are recorded for 72 hours.
A 72-hour test for a vertical well would normally be a very long The nonlinear least-squares estimation (automated type-curve or
test (assuming the same kh) ; this duration, however, may not be history matching) technique is used in order to improve the above
very long for horizontal wells. It is possible that for new wells estimates or at least try to answer some of the uncertainties of
either cost or operational restrictions may make it impractical to the semilog analysis. The model pressure is computed using Eq.
run well tests this long. If the downhole flow rate is measured it is 2 (which assumes constant flow rate), that is the constant well-
probable, although it may not be desirable, that the test time be bore storage model for a no-flow boundary reservoir model with
shortened. After 7053 hours of production, the well was shut-in a known formation thickness, h, and standoff, zw. The match be-
for a 72-hour buildup test. tween measured and computed pressure obtained from the nonlin-
ear estimation was not satisfactory and the residual was large. Of
The downhole pressure and flow rate for the drawdown test is course, this was expected because it is usually difficult to simulate
shown in Fig. 6. Fig. 7 presents a log-log plot of the derivatives of the downhole pressure with a constant storage model, particularly
Pwb! (measured pressure from Eq. 1), Pw/ (the wellborepressure at early times. As stated above, for the estimation of the verti-
with the effects of wellbore storage and skin from Eq. 2), and P/ cal and horizontal permeabilities simultaneously, it is essential to
(the formation pressure from Eq. 3). Of course in a real test, Pw/ simulate the downhole pressure accurately at early time. Never-
and P/ are usually unavailable. However, they will guide, us for theless, the estimates obtained· from the nonlinear least squares
the understanding of the interpretation methods used here. Note estimation are kh of 319.0 md, k" of 4.0 md, 8 m ' of 10.0, and C of
that the pressure derivative of Pwb/ is not smooth at certain times,. 0.29 bbl/psi. Interestingly enough, the estimated skin is accurate.
This is not due to differentiation technique, but rather to sm.a11 The geometric mean of the permeabilities (v'khkl1 = 35.7 md) is
rate fluctuations, particularly at late times. One therefore has to close to the actual value of 31.6 md. An accurate estimation of
be careful when smoothing the pressure derivative for drawdown parameters would have been possible if the wellbore storage had
tests. been constant. Thus, this suggests that the least-squares residual
is insensitive to variations in kh and k~, but is sensitive to v'khk".
When the P'wb/ curve is compared with the pi of the system given
in Fig. 7, it is clear that the derivative behavior of the downhole In order to minimize the effect of the flow rate variation during the
pressure is dominated by the flow rate variations at early times. drawdown test, the measured downhole rate data is used for the
At late times, it is also affected by small fluctuations of flow rate. estimation of reservoir parameters. Fig. 9 shows the logarithmic
This is very coIrimon for drawdown tests and is mainly due to convolution plot of the normalized pressure as '8. function of the
the multiphase flow in the production string. The semilog plot, logarithmic convolution time. It is interesting to observe that
Fig. 8, exhibits a long semilog straight line period (0.6-70 hrs) this plot looks like a semilog plot with wellbore storage effect at
which yields k = 80.5 md if the formation thickness of 100 ft is early times; this is expected due to the volume of the horizontal
used. The permeability would be 8.0 md if the length of the well section of the well below the tool, which is included in the model.
is used. A skin, St ,where St is the total skin which may include As shown in Fig. 9, the logarithmic convolution plot exhibits
various geometric effects and is given by two straight lines. The slope of the second straight line is twice

St = 1.151 [(Pi - Pw/(lhr») -log

m
A
'f'JLCtTw
+ 3.23] , (13)
that of the first one. This can be taken as an indication of a
top or bottom no-flow bounda:.ry. Note that the start of the first
straight line is the same as the start for the semilog straight line
of -2.9 is also calculated from the same semilog straight line. St given in Fig. 8. Let us pretend that this is a real well tes~ and
is the skin which is obtained from a semilog analysis which ig- we do not know the flow regimes for the calculation of reservoir
nores the geometry of the well (i.e. the well is considered to be a parameters' from these straight lines. They can be due to the
fully penetrating vertical well). The actual skin formulas with the first and second early time radial flow or the second early and
effect of geometry and the relevant slopes for each of the flow pe- intermediate time radial flow periods. The doubling of the slope
riods are given in ApP€,"-1jx B. From this plot, one cannot deduce suggests that these two straight lines can be due to the first and
whether this is a horizo: ",;>ermeability, kh, or a geometric mean second early time radial flow periods. However, Fig. 7 indicates
of the horizontal and v ],1 permeabilities (v'khk,,), or whether that the transition time between the first and second early time
the skin is due to damv"",; or geometry. If the corresponding flow radial flow periods is usually short compared to the t:ransition time
regime during this straight line period is identified, We would then between the second and the intermediate time radial flow periods.
know what the appropriate length to be used is, and what the esti- It can be deduced from the above arguments that two straight
mated permeability and skin are. The important point is whether lines from the logarithmic convolution correspond to the second
this semilog straight line is due to the early time first or second early and intermediate time radial f:l.ow periods. Thus, from the
radial flow periods, or to the intermediate time radial flow period. slope of the first straight line, one can compute v'khk" = 44.9
Unfortunately, the drawdown pressure data and its derivative do md andSt = 5.3 using the horizontal length of well. From the
not give any clues about these flow periods. For this example, we, slope of the second one, the permeability, and skin are calculated
of course, know from the ]1/ curve in Fig. 7, (which is the con-

386
SPE 18300 F.J. Kuchuk, P.A. Goode, B.W. Brice, D.W. Sherrard and R.K.M. Thambynayagam 5

as kh = 113.8 md , and St = -1.1 using the formation thickness. interpreted as the presence of a constant pressure boundary unless
From above estimates, the vertical permeability is k1J = 17.7 md the derivative becomes zero.
and the damage skin Sm = 5.4.
The semilog plot, as shown in Fig. 12, exhibits a semilog straight
Although logarithmic convolution indicates the presence of a no- line which yields kh = 82 md and St = -2.84. Mter 8 hours,
flow boundary and the possible existence of a second radial flow although essentially flat, the drawdown pressure oscillates due to
period, this period is not fully developed, as is shown in Fig. 7. the late time flow rate fluctuations. The vertical permeability, k 1J ,
Thus, the slope of the second straight line from the logarithmic is estimated as 18.0 md from Eq. 26 in Ref. 10 and the time at
convolution time would be less than the actual slope, and the which the pressure curve becomes flat (this is very approximate
estimated v'khk1J would be higher than the actual value. For a because of the flow rate fluctuations). As expected, the nonlinear
real well however, this information is unavailable at the beginning least-squares estimation with a constant wellbore storage (with-
of the interpretation. out flow rate measurement) did not yield the correct reservoir
parameters because the wellbore storage is not constant, due to
Fig. 10 presents a semilog plot of the deconvolved delta pressure the downhole flow rate variations, as in the previous case.
(the constant rate behavior of the well, including the effect of
the additional storage volume below the tool), which yields two Fig. 13 presents a logarithmic convolution plot of the normalized
semilog straight lines. From the slope of the first straight line pressure as a function of the logarithmic convolution time. The
which starts after 0.06 hours, one can compute v'khk1J = 26.05 logarithmic convolution plot also exhibits only one straight line
md and St = 8.0 using the horizontal well length. As can be seen period after which the curve becomes flat because of the constant
from this plot the first semilog straight line starts at about 0.06 pressure boundary. The time at which the curve becomes flat
hours, that is much earlier than the onset of the second radial can also be used for the estimation of vertical permeability. The
flow period from the logarithmic plot, Fig. 9. Thus, one can straight line in Fig. 13 yields .jkhk1J = 43.9 md and St = 5.2
deduce that the first line is due to first radial flow period. The using the length of the well. It is assumed that the straight line
second straight line in Fig. 10 is possibly due to the intermediate is due to the second radial flow period, but not fully developed,
time radial flow because there is a long transition period between as explained above. The deconvolved pressure for this case will
these periods. However, as can be seen from Fig. 7, the first be the same as that is given in Fig. 10, but it will become flat at
straight line is the transition period between the first and second late times.
radial flow periods. From the slope of the second straight line,
the permeabllity and skin are kh = 112.0 md and St = 0.0, using The nonlinear least-squares estimation using the measured down-
the formation thickness. These estimated values are close to the hole pressure and flow rate data, Eq. 1 with one no-fl()w and one
actual values of the parameters. It should be noted that there is constant pressure boundary, is also performed for this case. It is
a long transition period between the two straight lines. assumed that formation thickness, h, and standoff, Zw are known.
We are able to estimate parameters accurately. The nonlinear
The last method used to analyze this drawdown test is the non- estimation is almost essential for the constant pressure case be-
linear least-squares estimation with the measured downhole flow cause the intermediate time semilog straight line will not evolve,
rate data. For the computation of the model pressure behavior, and therefore cannot be used to obtain the horizontal permeabil-
Eq. 1 with a no-flow boundary reservoir. model and a known for- ity.
mation thickness, h, and standoff, Zw are used. Using measured
downhole flow rate, we are able to estimate reservoir parameters b. Buildup Test
accurately. We performed some experiments to determine, for
1. Well between two No-flow Boundaries
this particular case, how long a drawdown test should be run in
order to estimate parameters uniquely. From Fig. 7 it can be Fig. 14 presents the buildup pressure and after-flow rate during
seen that the effect of the farthest boundary starts at a time of the buildup test which was started after 7053 hours of produc-
4 hours. It is worth mentioning that if one of the boundaries is tion for the well discussed above. Due to the large volume of
close to the wellbore and the anisotropy ratio is not large, then the well, the measurable after-flow rate period is approximately
the estimation of the horizontal and vertical permeabilities can 4 hours. The log-log pressure and pressure derivative plots are
be a problem if the well test data does not go beyond the effect given in Fig. 15. The pressure derivative is smooth after 7 hours
of the farthest boundary. It seems reasonable to conclude from and indicates that it is approaching a radial flow period. The
the interpretation of the drawdown test that the downhole flow horizontal permeability obtained from the derivative plot is 107.0
rate measurements will play an important role in the estimation md if we assume that the flattening of the derivative is due to the
of the reservoir parameters with horizontal well geometry. More- intermediate radial flow period.
over, the nonlinear estimation (type-curve matching) will be the
integral PJ.l't for the interpretation of well test data from horizon- The Hotner plot of the buildup test, Fig. 16, exhibits a semilog
tal wells. straight line between 6.0 and 72 hours, and yields a permeability, .
kh = 120 md and skin of St = -0.6, where for Horner analysis
2. "VeIl between a No-flow and Constant Pressure Bound-
ary St = 1.151 ((Pwdhr - PwaOhr) -loglo ~ + 3.23) (14)
m 'l"J.tCtTw
For this case, the same downhole flow rate data given in Fig. 5 The extrapolated pressure ,P*, calculated from the semilog stra-
is used to compute the downhole pressure from Eq. 1, assuming ight line shown in Fig. 16 is 4355.8 psi, which is 44 psi lower
a constant pressure boundary at the top and a no-flow boundary than the initial pressure prior to the drawdown test. As can be
at the bottom. Fig. 11 presents a log-log plot of the downhole seen from the expanded portion of the Horner plot, also given in
pressure and its derivative with respect to In t. The plot shows Fig. 16 (small figure at the bottom), the curve bends upward
the effect of the constant pressure boundary. As shown in Fig. even at late times which implies that the semilog straight is not
7 of Ref. 10, for horizontal wells a decreasing slope cannot be

387
6 Pressure Transient Analysis and Inflow Performance for Horizontal Wells SPE 18300

a fully developed one. Thus, the permeability calculated from the standoff because the after-flow period, during which the flow
the Homer semilog straight line is slightly higher than the aetual rate is measurable for buildup test, could be too short for some
permeability and p* is lower. For vertical wells, this situation is cases. The linearized sensitivity analysis method indicates that
normally reversed. min{zw, (h-z.wH is strongly correlated with k". Furthermore, the·
identification and estimation may become a problem if the stand-
Fig. 17 shows the logarithmic convolution plot which exhibits a
off is either too small, or too large (less than 5 ft and greater
straight line period (it may be due to the second early radial flow
than 500 ft, depending on the anisotropy ratio). Thus, a priori
period) between 0.2 and 4 hours, following which the after-flow
information about the standoff is essential for these cases.
rate is too small to be measured in practice. Using a length of
1000-ft for the well, the straight line yields v'khk" = 41.7 md The identification and estimation of the formation thickness, h,
and St = 4.5. This calculated permeability is higher than the is possible from measured downhole pressure and flow rate data
actual v'khk'fj' The vertical permeability is obtained from ...,IliiJi'; if the standoff distance, min{z.w, (h - z.w)}, is known. The prob-
of the logarithmic convolution and kh of the Horner analysis as lem of estimating the formation thickness and standoff along with
k" = 14.5 md. other reservoir parameters could in principle be solved simulta-
neously with a constraint of h > max{zw, (h ~ zw)}. However, h
Nonlinear estimation and deconvolution were also performed for
is strongly correlated with kll and Zw, or (h - zw). The complex-
this buildup· test data.. The estimates from the nonlinear esti-
ity of the simultaneous estimation is avoided if the estimation is
mation are accurate. The results from the deconvolution are pre-
done in two stages: first, estimate all parameters but not h and
sented in Fig. 18. It should be noted that the permeability, v'khk"
second, estimate h while holding the other parameters fixed. This
is lower than the actual value.
procedure should be iterated until a satisfactory match is found.
2. Well between a No-flow and Constant Pressure Bound-
As stated above, for horizontal wells, the most important param-
aries
eters to be determined are vertical and horizontal permeabilities,
The after-flow rate shown in Fig. 14, is used to compute the skin factor, and reservoir pressure. Although it is a linearpa-
buildup pressure for this case. As can be seen from Fig. 19, rameter, the determination of reservoir pressure is, in principle,
the constant pressure boundary dominates the downhole pressure more difficult than other parameters. If the producing period
after 5 hours. The logarithmic convolution and deconvolution is long, so that the start of the intermediate radial flow period
plots for this case are the same as the previous buildup case, at is realized, the extrapolated pressure can be obtained from the
early times. The Horner plot, as shown in Fig. 19, becomes intermediate Horner semilog straight line. Otherwise, it can be
flat before reaching the intermediate semilog straight line period. estimated simultaneously with other reservoir parameters if the
Thus, it is not useful for obtaining the reservoir parameters. As Horner semilog straight line is not available.
in the previous case, we are able to estimate reservoir parameters
from the nonlinear estimation methods using measured downhole For the determination of average reservoir pressure (which could
pressure and flow rate. then be used in Eq. 11 to calculate the IP), first, an accurate
model is required for the system being tested. Second, the poros-
Discussion ity, compressibility, and thickness product (¢Cth) should be known
accurately. But, (¢Cth) cannot be uniquely determined from tran-
To this point, we have been concerned with estimating parameters sient data from a single well18 unless the sandface flow rate is
from perfect measurements. Let us suppose that measured down- measured precisely19. Conventionally porosity, 4>, and thickness,
hole pressure and flow rate contains measurement error. Guiluit h, are determined from open-hole logs and the total compressibil-
and Horne16 and Shah et al. 17 addressed this problem in connec- ity is obtai,ned from the fluid and rock properties. For horizontal
tion with transient well testing. The linearized sensitivity analysis wells, the determination of lithology can be a problem since the
method suggested by Shah et al. 17 is used to determine the prob- well is usually completed horizontally in a single geological zone
able error in the estimates. Because of the space limitation, only unless, necessary open-hole logs are acquired from the vertical
main results from the linearized sensitivity analysis are discussed. section of the well.
\
The horizontal permeability, kh and vertical permeability, k", are For producing horizontal wells, the unknown initial pressure dis-
strongly correlated, particularly if the test is short. The skin is tribution also will affect the estimation of reservoir pressure. On
correlated to both kh and k"j for large measurement errors, it is the other hand, in a given system, a transient pulse from the well-
however, correlated to kh more than k". If the wellbore storage bore can be created in such a way that the effect of the unknown
due to the horizontal section of the well is constant, then the initial pressure distribution can be minimized for the estimation of
storage coefficient can be estimated accurately. On the other permeability and skin factor which governsthe early time pressure
hand, if the wellbore storage is not constant, it has an important behavior of the system. In other words, the wellbore pressure of a
effect on the estimation of the reservoir parameters, particularly horizontal well is a strong function of the permeability, anisotropy
k", which is a most sensitive parameter to measurement errors. and skin factor, particularly during the early transient period.
It is natural to ask whether or not distances to the boundaries Field Example
from the wellbore (zw and h) can be estimated simultaneously
with other parameters. In general, it is possible to identify a\ This well test example for Well JX-2 in the Prudhoe Bay field was
no-flow or a constant pressure boundary and to estimate the discussed in detail by Sherrard et al13 • The well was drilled in
standoff (zw or h - zw) uniquely, provided the downhole flow an oil zone of the Ivishak formation and is approximately 120-130
rate is measured accurately or the wellbore storage is constant ft (the exact location is not known) below the gas cap and 5-10
and known. Drawdown tests are preferable for the estimation of ft above a continuous shale zone as shown in Fig. 20. Sherrard
et al13 also presented performance, production, and buildup test

388
SPE 18300 F.J. Kuchuk, P.A. Goode, B.W. Brice, D.W•. Sherrard and R.K.M. Thambynayagam 'T

data for this well. 3. Wellbore storage,

The horizontal section of the well is 1,576 ft in length and is 4. The location of the gas cap,
completed with a 4.5 in slotted liner as also shown in Fig. 20.
The fluid and formation properties for this well are given in Table 5. The horizontal (completed) length of the well.
2. The average reservoir pressure at the time of test was under As we know from the analysis of the synthetic data, the estimation
the bubble-point pressure. However, the producing GOR was the of these parameters may not be unique. ill other words, many
same as the solution GOR. Thus, a reduction of oil permeability combinations of these parameters may produce nearly the same
caused by liberated solution gas is therefore to be expected in the buildup behavior.
reservoir. Moreover, the effect of the permeability reduction due
to the liberated solution gas, in the near-wellbore vicinity may Fig. 22 shows several history matches, Curves A, B, and C, of
appear as a positive skin. the derivative of the 24-hour measured buildup pressure with the
model response. This figure also shows that none of the mat~hes
As described by Sherrard et al13 , and shown in Fig. 21, following are good, as well as indicating mis-matches between the measured
six approximately 12-hour flow tests during which only surface data and the model response. Three sets of the estimates from
flow rates are measured, JX-2 was shut-in for a 24-hour buildup each history match are presented in Table 3. The underlined itaUc
test. The pressure was below the bubble point pressure of the numbers in Table 3 denote parameters that are held fixed at the
reservoir during the flow periods and subsequent buildup test. values given in the table during the history matching process. The
Fig. 22 presents the log-log derivative plot which does not indicate match shown by Curve A in Fig. 22 which has the smallest sum of
any radial flow period. We have also taken the derivative of the the squares of the residuals, yields a vertical permeability of kv =
buildup pressure with respect to the Horner and multirate time 50.0 md which is twice as large as the horizontal permeability,
functions, since the multirate test precedes the buildup. As can be kv = 25.0 md. This is almost impossible because the expected
seen from Fig. 22, after 8-9 hours, the pressure derivative starts horizontal permeability for the Ivishak formation from cores and
declining much faster than in the previous hours. In other words, well tests of nearby wells is about kh = 100.0 md. To test this
a continuous flattening of the pressure took place, as also shown in possibility, the horizontal permeability is held fixed at kh = 100.0
Fig. 23. Thus, it may be taken as a gas cap effect. However, The md for the match shown by Curve C, which is also unsatisfactory.
pressure flattening or the decline of the pressure derivative as a Of course, this does not imply that kh is different than 100md. As
function of time for horizontal wells can be somewhat misleading shown in Table 3, the well length, 2L1I1 , is also held fixed at the
(unless it becomes exactly flat) since some combinations of well- reported drilled length. Note that for all matches, the distance
reservoir parameters could give a similar appearance, as in Fig. from the wellbore to the no-flow bottom boundary, Z1l1' is held
7 in Ref. 10. Nevertheless, the vertical permeability is calculated fixed at 10.0 ft and the formation thickness are estimated for the
from Eq. 17 in Ref. 10 using t.jfJe (the time to feel the gas cap location of the gas cap that is h - %w. We have also taken the rate
effect) of 8 hours as 28.6 md if a distance of 135 ft from the well variation into account for the history match, it does not however,
to the gas cap is used and 6.6 md if 65 ft is used. improve the results.

Although a straight line is drawn between 9 and 24 hours, the Fig. 24 presents two history matches, Curves D and E, of the
Horner plot, Fig. 23, does not really exhibit a well defined semilog derivative of the 6-hour measured buildup pressure with the model
straight line period (the derivative plot, Fig. 22 also indicates the response. Note that the matches are good up to one hour. The
same character). The permeability and skin calculated from the estimates from each history match are also presented in Table
Homer plot are k = 200.5 md and S = 5.0, if the formation 3. The horizontal permeability of kh = 1000.0 md estimated
.thickness is used. The extrapolated pressure ,P*, calculated from from the match given by Curve D is again unlikely for the Ivishak
the Horner semilog straight line shown in Fig. 23 is 3804.4 psi, formation. The match for the Curve F given in Table 3 is not
which is 40.6 psi higher than the initial average pressure prior to plotted because it is very similar to Curve E. Note that the esti-
the multirate test. mated wellbore storage coefficient, C, is almost the same for all
matches.
A production logging survey was run for this well. Although not
conclusive, it indicated that the bottom section of the well was not. Fig. 21 presents the history matches of the measured multirate
contributing to the flow. Another problem was the uncertainty in pressure data with the model response (from Eq. 1 using the
the exact location of the gas cap. surface flow rate data). Although the match looks satisfactory,
many other combinations of parameters yielded a similar match.
As we stated above in general the interpretation of well test data The estimates from this match are also presented in Table 3. Fig.
from horizontal wells is a much more difficult task than its ver- 25 presents the multirate plot using the surface flow rate and the
tical counterpart. Our difficulty in interpreting the buildup test logarithmic approximation for the model. Although there is no
from JX-2 is compounded by: 1) changing wellbore storage effect, justification to have radial flow earlier than 12 hours, ~ is
2) an ambiguity on the location of the gas cap, and 3) ambiguity 44.2 md from the slope of the straight line, if h=145 ft is used. H
on the length of the well. In addition to these problems, the pro- kh is assumed to be 100 md, then kv would be 4.4 md, which is
ducing time was short and disturbed by the six flow tests during different to 0.6 md determined from the nonlinear estimation. The
which the flow rates were not exactly constant and they were mea- skin also differs from that obtained from the nonlinear estimation.
sured only at the surface. Nevertheless, the nonlinear estimation
method (type-curve matching) is used to estimate: Table 3 and matches shown in Fig. 21, 22, a.nd 24 reveals a few
interesting points:
1. Horizontal and vertical permeabilities,
1. The estimates are non-unique and associated with gross mis-
2. Damage skin,

389
8 Pressure Transient Analysis and Inflow Performance for Horizontal Wells SPE 18300

matches. This may also suggest that perhaps the model used for It is shown that the inflow performance of a horizontal well is
the estimation is not accurate. affected by the various parameters. The IP increases almost lin-
early with increasing well length, not logarithmically as previously
2. When certain parameters are held fixed at some values a trade reported by Karcer 8. For the case of extremely thin (or the ver-
off between estimates is evident. This trade of is particularly true tical permeability very high) formations, the well is analogous to
between horizontal and vertical permeabilities. a fully penetrating vertical fracture, then the pressure drop will
3. The measure of the uniqueness of the estimates from of a match decrease logarithmically as the well length increase. However,
cannot be judged by simple inspecting the sum of the squares in most practical cases the productivity of a horizontal well will
of the residuals and the plot of measured data with the model increase approximately linearly as the length of the well increases.
response unless the uniqueness of each parameter is investigated
beforehand. As pointed out by Shah et aliT, a linearized sensitivity
analysis may provide an indication for uniqueness of the estimates. Nomenclature
Conclusions
Ct total compressibility, atm- I [Pa- I ].
Although the intermediate radial flow period for horizontal wells C wellbore storage coefficient, RBpsi- 1 [m3 pa- I ].
is comparable with the infinite acting period for vertical wells, the h reservoir thickness, ft [m].
lack of its evolution in a reasonable testing time makes interpreta- k permeability, md [m2 ].
tion difficult. Large anisotropy ratio and the existence of multiple L length, ft [m].
boundaries with unknown distances to the wellbore increase the Image sum over y.
m
complexity of the interpretation problems.
n Fourier transform variable.
In this paper we illustrate that using only pressure data, par- p pressure, psi [Pal.
ticularly for wells with a constant pressure boundary, has major pi dp/dt, psi/hr [Pas-I].
drawbacks which prevent the determination of the reservoir sys- p Laplace transform of p.
tem under testing, and its parameters. Different interpretation p finite Fourier cosine transform of p in z.
methods should be combined to improve the uniqueness problem; q volumetric flow rate, STB/D [m3 s- 1 ].
two techniques must be considered. First, important features of
q' dq/dt, STB/D/hr [m3 s- 2 ].
the system should be obtained from direct methods, such as log-
r radius, ft [m].
log, derivative, semilog, logarithmic convolution. etc. Large dif-
ferences from different techniques suggests the possibility of an 8 dimensionless Laplace transform variable.
incorrect model which could be improved by using other informa- S skin.
tion or just a process of elimination. Second, a robust· nonlinear t time, hours [s].
estimation method should be employed to refine the previous es- :II coordinate, ft [m]
timates. It appears, however, that the accuracy and resolution y coordinate, ft [m]
of the estimates from early time data (which could be hours) are z coordinate, ft [m]
limited unless downhole pressure and flow rate are measured accu-
rately. Furthermore, drawdown and buildup data tend to provide Greek Letters
almost the same reservoir information at early times. At late
times, however, they diverge from each other (see Ref. 5) after '1 0.577215... (Euler's constant).
the intermediate radial flow period (infinite acting period for hor-
izontal wells) and they may provide complementary information
a difference.
about the system. Thus, the entire transient response, drawdown JL viscosity, cp [pa s] .
and buildup ofthe system is required to construct the model and 1r 3.14159265•.•.
estimate its parameters. q, porosity, fraction.

As you may remember the pressure derivative (Fig. 18) and

Subscripts
semilog (Fig. 19) plots for the buildup tests provided useful infor-
mation for the onset of the intermediate radial flow period while
the same information could not be obtained from the same plots D dimensionless.
for the drawdown test because of the flow rate fluctuations at late h horizontal.
times. Nevertheless,it should be emphasized that a drawd01l1n fol- i initial.
lowed by a buildup test, or ",isa ",ersa, will produced satisfactory m mechanical (damage).
results for the estimation of reservoir parameters for horizontal 8f sandface.
wells if downhole ftowrate and pressure are measured. S8 steady state.
New formulas for Infiow Performance (IP) of horizontal wells with t total.
and without gas cap or aquifer effects are presented. These solu- 11 vertical.
tions can be applied to a horizontal well of arbitrary length pro- w well.
ducing from a closed rectangular region of arbitrary aspect ratio wbf wellbore (bottom hole).
where the well is placed at any location within the region. Un- wf flowing pressure (drawdown).
like most existing formulas, no assumptions are made about the wfss steady state flowing pressure.
relative length and position of the well or the formation thickness. wfOhr extrapolated flowing pressure at 0 hrs.

390
SPE 18300 F.J. Kuchuk, P.A. Goode, B.W. Brice, D.W. Sherrard and R.K.M. Thambynayagam 9

wf1hr extrapolated flowing pressure at 1 hr. 17413 presented at the 1988 SPE California Regional Meet-
ws1hr shutin pressure, extrapolated to 1hr. ing, Long Beach, Ca., March 23-25.
wsohr final flowing pressure.
coordinate indicator. [11] King, G.R. and Ertekin, T.: "Comparative Evaluation of
y coordinate indicator. Vertical and Horizontal Drainage Wells for the Degasification
of Coal Seams", SPE Reservoir Engineering, May 1988, 720-
z coordinate indicator.
734.

[12] Reiss, H.L.:"Production From Horizontal Mter 5 Years," J.

Acknowledgments Pet. Tech. (Nov. 1987) 1411-1416.
The authors are grateful to Schlumberger-Doll Research and Stan- [13] Sherrard, D.W., Brice, B. W. and MacDonald, D.G.: "Ap-
dard Alaska Production Co. for permission to present this paper. plication of Horizontal Wells at Prudhoe Bay," J. Pet. Tech.
We are grateful to D.J. Wilkinson for his valuable comments. The (Nov. 1987) 1417-1425.
interpretation of the Prudhoe Bay field example is that of the au-
thors and does not necessarily reflect the opinion of any of the [14] van Everdingen, A. F., and Hurst, W.: "The Application
Prudhoe Bay owners. of the Laplace Transformation to Flow Problems," Trans.,
AIME (1949), Vol. 198, 171-176.

[15] Agarwal, R., Al-Hussainy, R., and Ramey, H. J., Jr.: "An In-
References vestigation of Wellbore Storage and Skin Effect in Unsteady
Liquid Flow: I. Analytical Treatment," Soc. Pet. Eng. J.
[1] Giger, F. M., Reiss, L. H. and Jourdan, A. P.:"Reservoir (Sept. 1970) 279-290; Trans., AIME, 249.
Engineering Aspects of Horizontal Drilling," SPE 13024 pre-
[16] Guillot, A.Y. and Home, R.N.:"Using Simultaneous Flow
sented at the 1984 SPE Annual Technical Conference and
Rate and Pressure Measurements to Improve Analysis of
Exhibition, Houston Texas, Sept. 16-19.
Well Tests," SPEFE, 1 no. 3: 217-226.
[2] Vicanek, J.:"Oil Production with Horizontal Wells," Erdoel-
Gas. Vol. 101, No.3, March 1985, 85-90. [17] Shah, P.C., Karakas, M., Kuchuk, F.J. and Ayestaran,
L.:"Estimationofthe Permeabilities and Skin Factors in Lay-
[3] Giger, F. M.:"Horizontal Wells Production Techniques in ered Reservoirs Using Downhole Rate and Pressure Data,"
Heterogeneous Reservoirs," SPE 13710 presented at the SPE 14131, presented at the SPE International Meeting in
1985 Middle East Oil Technical Conference and Exhibition, Beijing, March 17-20, 1986.
Bahrain, March 11-14.
[18] Dogru, A.H. and Steinfeld, J.H., "Design of Well Tests to
[4] Daviau, F., Mouronval, G., Bourdarot,. G. and Curutchet, P.:
Determine the Properties of Stratified Reservoirs," Proceed-
"Pressure Analysis for Horizontal Wells," SPE 14251 pre-
ings of 5th SPE Symposium on Reservoir Simulation, Den-
sented at the 1985 SPE Annual Technical Conference and
ver, CO, February 1-2, 1979.
Exhibition, Las Vegas, Sept. 22-25.
[19] Kuchuk, F.J.:"Gladfelter Deconvolution," SPE 16377 pre-
[5] Goode, P.A. and Thambynayagam, R.K.M.: "Pressure
sented at the 1987 SPE California Regional Meeting, Ven-
Drawdown and Buildup Analysis for Horizontal Wells in
tura, Ca., April 8-10.
Anisotropic Media," SPE Formation Evaluation, Dec. 1987,
683-697. [20] Wilkinson, D.J. and Hammond, P.:"A Perturbation Theo-
[6] Clonts, M.D. and Ramey, H.J., Jr.: "Pressure Transient rem for Mixed Boundary Value Problems in Pressure Tran-
Analysis for Wells with Horizontal Drainholes," SPE 15116 sient Testing," Schlumberger-Doll Research report, 1987.
presented at the 1986 SPE California Regional Meeting, [21] Carslaw,H.S. and Jaeger, J.C.: Conduction of Heat in Solids
Oakland, Ca., April 2-4.
Oxford University Press,New York(1959).
[7] Joshi, S.D."Augmentation of Well Productivity Using
[22] Dietz, D.N.:"Determination of Average Reservoir Pressure
Slanted and Horizontal Wells," SPE 15375 presented at the
From Build-Up Surveys," Trans.,AIME(1965), 955-959.
1986 SPE Annual Technical Conference and Exhibition, New
Orleans, Oct. 5-8. [23] Matthews, C.S., Brons, F. and Hazebroek, P.:"A Method
[8] Karcher, B.J., Giger, F.M. and Combe, J:"Some Practical for Determination of Average Pressure in a Bounded Reser-
Formulas to Predict Horizontal Well Behavior," SPE 15430 voir" ,Trans.,AIME(1954) 201,182.
presented at the 1986 SPE Annual Technical Conference and
Exhibition, New Orleans, Oct. 5-8. Appendix A

[9] Ozkan, E., Raghavan, R. and Joshi, S.D.: "Horizontal Well We develop an analytical solution for the stabalized inflow per-
Pressure Analysis," SPE 16378 presented at the 1987 SPE formance (1P) of a horizontal well which is producing from a
California Regional Meeting, Ventura, Ca., April 8-10. closed rectangular region. This solution is an extension of the
[10] Kuchuk, F.J., Goode, P.A., Wilkinson, D.J. and Tham- work done by Goode and Thambynayagam 5 which presented the
bynayagam, R.K.M.:"Pressure Transient Behavior of Hori- solution for a horizontal well producing from a semi-infinite reser-
zontal Wells With and Without Gas Cap or Aquifer,"SPE voir. In this case we close the boundaries in y by the method

391
10 Pressure Transient Analysis and Inflow Performance for Horizontal Wells SPE 18300

of images. The simplifying assumption that the reservoir thick- Inverting the cosine transform taken wrt Z we obtain
ness is small compared to the distance from the well to any of
the boundaries allows us to develop a computationally efficient p = ~e-../iY +
{Js3/2
_1_
1r{Js n=1
f
E z cos(n1rz) e-- .+n 2 'l1'2 y
s + n 21r 2 •
(A7)
form.

If we assume that the thickness of the reservoir is small when For convenience we will split Eq. A7 into two pieces and treat
compared to the distance from the well to any of the boundaries each separately.
in Z or y then we need only consider a two dimensional prob-
lem, where the well can be considered as an infinitely conductive P=P'o+pn . (A8)
fracture which fully penetrates the formation. The fact that we
in actuality have a horizontal well can be introduced by consid- To develop a solution for the I P we need to take the large time
ering a pseudo skin, Sz, to account for the partial penetration solution (t -+ 00 ), we call· this pseudo steady state.
effect in z.
Po = -Lw
{J
lit 0
1
_e""1l
1rT
2/4T
dT . (A9)
We consider a strip of length 2Lw centered at Zw, Yw producing
from a rectangular region of dimensions L z by L y , see Fig. To place the noflow boundaries in y we image the source at
2. The following dimensionless va.riables (written in consistent + 2mL y
"""1/w Yw +2mL1I m =-00, ... ,00
units) are introduced
where the well is at Yw.
z
ZD =-
Lz
(A1)
Po=-
Lit
w
{J 0
1
-
1rT
'"
L."
00..
[e
-{y-Yu/-2mL y )2
4T +e
-(Y+Yw-2mL y )2
4T ]dT.
m=-oo
41r L w ..jk;JiYti.p (AIO)
PD= . (A2)
qp. Using the Poisson summation formula 21 we can rewrite Eq. A10
In these variables the diffusion equation is that of an isotropic as
formation with a unit diffusion coefficient: o o - m
2'l1"2,.]
Po = ~~w 1+2 L cos (~1rY) cos (m;yw). It e ~ dT
2
apD _ a pD + a 2pD (A3) 11 [ m=1 fI 11 Jo
&tD - az}, aYb' (All)
The boundary conditions are that there is no flow across any of For pseudo steady state we let t -+ 00
the boundaries, the pressure is initially constant throughout the
_ 2Lw [ . ~ 1 (m1r Y ) (m1r Yw .) L~ ]
reservoir ,and that at the well: Po - (JL
1I
1+2 ~1 m2 cos L
1I
cos m 21r 2 • ---y;;-
(A12)
ZwD - LwD ~ ZD ~ ZwD + LwD , (A4)
The I P is defined as < >q , therefore to develop the formula
P -Pwf
for I P we must integrate Eq. A13 over all positions z and y and
h
where (J = 21rLz z • divide by L z L 1I •

The actual boundary condition at the well is the infinite conduc- However, foL Y
cos (~:y) dy = 0
tivity condition that the pressure be uniform over the sandface.
2L
As discussed previously 5,10 this poses a very difficult mixed therefore, < Po >= T w L1I • (A13)
boundary value problem. Therefore the boundary condition we
solve is that given by Eq. A4, which is for a uniform flux strip,
Hence, if we let y = Yw (i. e., measure the pressure on the strip)
the pressure is then averaged 5,10,20 to get an approximate solu-
tion to the original problem. In the limit of r:;
-+ 0 this will be
an exact solution. For a detailed discussion on the advantages of
< Po > -Pow f =- 2L(Jy L z [.!:-
3
Yw +
L 1I L~
y~] •
(A14)

pressure averaging see reference 10.

Now, for PI we can get the pseudo steady state result by taking
For convenience we will drop the subscript D as all variables the limit as 8 -+ 0, therefore
will be in dimensionless form. 1 1
L -Ez cos(n1rz)
00
PI = I.i2" X
,.,1r n=1 n
We solve Eq. A3 by the application of a finite Fourier cosine
transform and a Laplace transform. Following substitution of L
00
[e-1l'nI1l-Yw-2mLyl +e-1l'nIY+flw-2mLyl]
the boundary conditions and solution of the resulting ODE we m=-oo
have (A15)
(A5) By noting that the exponential sums are geometric series the
sum over m can be done analytically, then
where 1 1
Ez = .!n lin( n1rL w ) cos( n1rzw) • (A6) <PI >-PIU1! =-/i2'
,.,1r n=1
L
00
-Ezcos(n1rz)[l+e],
n
(A16)

392
SPE 18300 F.J. Kuchuk, P.A. Goode, B.W. Brlee, D.W. Sherrard and R.K.M•.Thambynayagam 11

where In dimensional form (conli.tent units) Eq. A18 becomes

(A17)

We now need to pressure average along the source and add in

the skin resulting from the partial penetration effect in z. We
define the following dimensionless variables

where
1rrw ( l+ (k:\. (1r Zw
5%=-ln [ h
Yk;)Sln h )]
and h D = Yfk: h%
k; L z •

Dropping the D subscript for convenience the equation giving

the difference between the wellbore pressure and the average and
reservoir pressure is

< P > -pw f =_!.

{3
[2Lw Ly (!.-3
Yw
Ly
+ Y;)
L~
For simplicity of presentation A21 may be written as
(A18)
1 ~ 1 ..... 2 ( )]
+1-- -=-3L-- ~ ~~z l+e -5%y k;
{k; ,
<P>-Pwf=

zw) +~ fk: s%]

21r w y w
where
qp.
41ry'k z k y h
[F(YW L
Ly' Lz Y(k; L
k;;' L z ' L w L w Yk; .
(A22)
(A19) Typical values for the function F have been tabulated in Table
1.
This expression for the skin is valid when h < 5L w (h is di-
mensionless), which is most often the case. IT necessary the full
expression for 5% can be found in reference 10. Appendix B
This gives a result which has only a single series, instead of the In this appendix we provide the slopes and skin equations which
triple series which could be expected from placing three sets of could be used to obtain reservoir parameters from the various
boundaries around a source. Also, the slowest converging part flow regimes associated with horizontal wells. All the slopes and
of the series converges cubically, with the other terms having skins relate to the case where the flow rate is constant.
exponential convergence. Two N o-ftow Boundaries
Early Time Radial Flow Period
To check this result we can observe that if we let 5% = 0 and A plot of pressure versus the logarithm of time will yield a
L w -+ 0 then we get the I P or a fully penetrating line source straight line which has a slope of
producing from a closed rectangular region of area LzL y • We
can then rewrite this in a form appropriate for determining the m = 162.6qp. (B1)
2.jkh k tl L w
Dietz 22 shape factors and compare then with the published re-
and a damage skin
sults.

5 m =1.151 [ 2~Lw
162.6qp. (Pi - Pwflhr) + 3.2275
(B2)
y
In(CA) =-41r (L -Yw+ Y;) -4 f:
!. cos (mrzw)e 2 + !. (4{k;
Yt; +Y
2 log
2
4{k;+) _ Jkj;li;)] .
k;;
log (
4JJ.tctr'f"
3 Ly n=l n
(A20)
+2ln(2"..in(".z..))+ln (~L.~ -'1. For the second radial flow period, where the influence of a near
no-flow boundary has been felt while the well is still experiencing
radial fl'lw
Table 4 presents the result from Eq. A20 for the geometries
m = 162.6qJ.t
(B3)
.jkhktlLw
shown in Fig. 5. The series in this equation is exponetially and
convergent and few terms of the sum are necessary. Some of
the values of C A determined from Eq. A20 are different from
those published by Dietz, this is because Dietz calculated his
5 m =2.302 [ ~Lw(Pi - + 3.2275
162.6qJ.t Pwflhr)
(B4)
+log ((l+~;:) -log (~)]
shape factors graphically using the work of Mathews, Brons and
Hazebroek 23 •

393
12 Pressure Transient AnalT•• and Inflow Performance for Horizontal Wells SPE 18300

If a period of linear flow exilts then a plot of the square root of Table I-Value. of the fundion F~
time versus pressure will exhibit a straight line with a slope
8.128q fi!i;p.
11V1/ L.. = 0.5 , :e./ L. = 0.5
m=-- --.- (B5)
2Lw h khcPct
and
2L• ..jkh k"
~~ 0.1 0.2
i:
0.3 0.4 0.5
8m = 141.2qp. (Pi - Pw/Ohr) - S% (B6)
where
0.25 3.80 2.1,1 1.09 0.48 0.26

V~.
lih) (1rh Ztv )]
0.50 3.25 1.87 1,12 0.69 0.52
8 % = -2.303Iog h1r,"w ( 1 + SIn (B7)
[ 1.00 3.62 2.30 1.60 1.21 1.05
2.00 4.66 3.34 2.65 2.25 2.09
4.00 6.75 5.44 4.74 4.35 4.19
Finally a period of radial flow, in the plane of the formation will
develop. For this period a semi-log plot will provide a slope Ywl LJI = 0.25 , :e./ L z = 0.5
162.6qp.
m=--- (B8) 0.1 0.2 0.3 0.4 0.5
khh
and a skin
0.25 4.33 2.48 1.36 0.70 0.46

8 m =2.303 j€ v
~h
Lw [ khh (
L"-h - - - Pi - Pw/lhr)
162.6qp.
(B9)
0.50
1.00
3.89
4.47
2.42
3.13
1.58 1.10
2.41 2.00
0.92
1.83
2.00 6.23 4.91 4.22 3.83 3.67
~ log (~)
¢P.CtLw
+ 2.5267] - 8% 4.00 9.90 8.58 7.88 7.49 7.33
where ,
Yw / LJI = 0.25 a:w / L z = 0.25
8% = -2.303 log [ h1r7'W (1+ ~.SIn h
Y"kh) (1rZw )]
0.05 0.1 0.15 0.2 0.25
(B10)
_ {k;~
VIi; L w
(!. _Zw + Z~)
3 h h2
0.25 9.08 7.48 6.43 5.65 5.05
0.50 6.97 5.56 4.71 4.12 3.71
1.00 6.91 5.54 4.76 4.24 3.90
Constant Pressure Boundary
2.00 8.38 7.02 6.26 5.76 5.44
If one of the boundaries was a constant pressure boundary and a 4.00 11.97 10.61 9.85 9.36 9.04
steady state pressure was achieved at the wellbore then the skin
yw/LJI = 0.5 , a:w / L z = 0.25
JkhkvLw
374.4qJ.£ (Pi - Pw/u) 0.05 0.1 0.15 0.2 0.25

_ 2.303 log [
1r7'w
8h
(1 + Ifi)
cot (1rzw )
2h
+ (h -
Lw
zw) fk;]
Yk;
0.25
0.50
8.44
6.21
6.94
4.83
5.98
4.02
5.26
3.47
4.70
3.08
1.00 5.86 4.50 3.73 3.23 2.90
(Bll) 2.00 6.73 5.38 4.62 4.12 3.81
Table 2-Rock and fluid data. 4.00 8.82 7.46 6.71 6.21 5.89

hypoth. well JX-2

Porosity, ¢ 0.20 0.21

Viscosity, p. 0.80 1.3 cp
Compressibility, Ct 2.010 r5 4.01Of5 psir1
Wellbore radius, '"w 0.35 0.35 ft
Formation thickness, h 100 ft
Stand-off from bottom, Zw 20 10 ft
Length of the well , 2L w 1,000 1,576 ft
Horizontal Permeability, kh 100 md
Vertical Permeability, k v 10 md
Initial Pressure, Pi 4,400 3,746 psi
Production rate, q 10,000 17,037 RB/D
Production time, t p 7,053 85.5 hr
Storage coefficient, C 0.01 bb1lpsi
Spinner conversion factor 357.143 RB/RPS
394
 Table :I_Estimates of parameters for JX2.

Table 4-Comparison of Eq. A20 with the

6-hour Buildup Test Multirate Test shape factors determined by Dietz for the
Test/ 24-hour Buildup Test
Curve B Curve C Curve D Curve E Curve F geometries shown in Fig. 3 (top to bottom).
Parameter Curve A
100 1000 J.QQ !Jl!l 97
kh,md 25 87
.04 0.6 0.6 0.6
ku,md 50 16 3
1576 1576 1576 1570 1576
2L w,ft 700 557
145 144 145
h, ft 144 75 75 11£ 22.6 21.8
.1.!l. 1J1. 1J1. .1Jl. .1Jl.
zw, ft 1.!l. 1.!l. 0.0 0.0 4.86 4.52
2.1 1.2 4.0 1.5 0.6
8m 0.086 2.07 2.08
0.11 0.16 0.14 0.081 0.086
C, bbl/psi 10.8 10.8
30.9 30.9

No-Flow
or
Constant Pressure Boundary

I--L~\-L~
Horizontal
Well--f t --
z
T
Zw
--:---------

i IP=26.03
2000

4000
y
Fig. 1-Horizontal well model (after Kuchuk et al., Ret. 10).

--:--------------- 2000 Parameters

i IP=20.17
kh
h
= 100
= 100
md
ft
= 0.354 ft
4000 L. = 2000 ft
Ly = 2000 ft
= 50 ft
---.----------------- = 1.0 cp
:: 2000
I

: IP=18.32
I

Fig. 2-Horizontal well model geometry in a closed rectangular region. 4000

---,-------------
I
I
: 2000
kv/k h I
: IP=23.36
kh = 100 md 1.0 I
30.0 h = 100 ft 0.5
= 0.354 ft 4000
Q) L. = 2000 ft 0.1
0 0.05
l: Ly = 2000 ft
C
E = 50 ft IP=31.18
L. 20.0 I-' = 1.0
~ 0.01 ---.----- 2000
L. I
I
&. :I
~
0
;:;:: 10.0
.E 2000
Fig. 5-Ettect of horizontal well-reservoir geometry on IP.

O.O+-----,r-----,--~---.,r:-----
0.000 0.125 0.250 0.375 0.500
Lw/L x
Fig. 3-Ettect of horizontally completed well length and anisotropy on IP.

15.0

14.0
12 ~
s::
'Q..
(J)

13.0

12.0+----r-----,-----~---- 'I '1

0.000 0.125
10- 1 10 0 10 1
Time, hr
Fig. 4-Effect of vertical location of horizontal well on IP. Fig. 6-Downhole pressure and flow rate for the drawdown test.

395
 720

/if-0

8. o
600

<15'
> 10'
o 8.
'~ ~.
~4n
o
·t o ~
Q o + ~
p.. 480
~ 0+
~
~
o <)
+0 Q
~ + ,120
p..
+ '2
~ + 2;
<) 0
Q >
>:: 36q
Slope = 49.90 psi/cycle
P'f 8 kh = 26051.1 md-fl
- - - - - +- - k = 26.05 md
o P'wf
Q .100
S = 8.01
Lw = 500.00 fl
+ P wbf Slope, = 116.14 psi/cycle
10' ...+-.,...,...,..,..,...,..-....-....-...............,.,.--.....-........,........,..,.r----r....................,..,.-.........................l
2·iO
10° 10 1
Time, hr 180

Fig. 7-Derivatives of Pwbf> Pwf> and PI with respect to In t. 10'

Time, hr
Fig. 10-Deconvolved pressure of the drawdown test.

10'

'iii 10' .., + +

Co
~
++ +++
+ +
'wQ. .= +
+++
~
ai'
:;
en
en
4200 ..hf+t++tt-+-+'f++-f-Ht---'4-H-++++f+-..-H-t-I+H-+I-----.f---J.---J-I..-W-I
~
.c
:t
10' - ++
..+
+
Cll Co +
D:. <I +
'0 +
10· ..,
0) '0 +
r:: r:: +
'3: CO
+
o
u:: ~ +
+
~ 10-1 ..., +
+
Co +
3750 <I +
0 Pressure I
1 + Derivative
10-' -I-......,.,.,..,·ry-I--..-....,..,.........,.·I-..,........,..,...~·I-..,.... .......,.,.~·I::::;;;;;;;:;;;;:;;:;;;;::.I
10 .. 2 10.. 1 10 0 10-2 10- 1 10· 10'
Flowing time, hr Time, t, hr
Fig. 8-Semilog plot of flowing pressure of the drawdown test. Fig. 11-Delta flowing pressure and its derivative with respect to In t for the gas cap case.

800 4500 : I
65.43 16.24 U93 1.330 0.4592 0.2082 0.0469 jli! II1
Real time, hr
'wQ.
~
u11111 1]\ I II!
II!II
II
700 IIII11

~ ~... I
Cll- 4350
:; lmi{ i i
en
en
~ 600
'wQ.
~-
l'l'il ~> i I 111"""1 ,I ' , 111,I'
,
[
II
JI III ... I il
~ ~ J.
Q. ~ I1II II
:J 4200 1

I II 111 ~IIII·
~ en I !'
en
Cii
'0 500 ........
Cll
C- IIIIllii
1 I I I

~ I II 11II 'j.,) 1,1 ,

'0
Cll ~ O)
r:: 4050 I
.!:!
iii 400
~
.. ~
t>,,,,
'3: I1I1I1 I 1IIIil r-r, ! I

i'. ~,IIIII'
0
E ~ u:: I'
~
/, 11'1
0 Slope = -59.38 psi/cycle \, Slope = -158.56 psi/cycle
r:: kh = 21911.3 md-fl 3900 kh = 8203.75 md-fl i
cl> k = 44.7 md k = 82.04 md

~-"
300 = -2.84
1ii
a:
S
LW
= 5.3
= 500.00 ft.
Slope, = -113.06 psi/cycle K S
h

. ,
= 100.00 fl
I ,i i III1
III
II I "'nm
'I 'I
-2.4 -1.8 -1.2 -0.6 0.0 0.6 1.2 1.8 2.4 10- 2 10- 1 10 0 10 1
Logarithmic convolution time Flowing time, hr

Fig. 9-Logarlthmic convolution plot of the drawdown test. Fig. 12-Semilog plot of floWing pressure of the drawdown test for the gas cap case.

396
 700 700
33.21 6.746 1.343 0.3483 0.0610 0.0191 0.0121 0.0301 0.0820 0.2545 0.5237 1.344 3.796
'(jj Real lime, hr
Co
-----J '(jj Real lime, hr

€ ~ I Co 600 II

~~
600
€:::3
:::3
VI ~ I

~
I
VI ::lell 500 ~
ell
is. 500
is. ~&/
~
~". , ~ fe'
~~ ....
~ 400
Ui
'tl 400
Ui
'tl
-----
'tl
ell
.!::!
'.
\ -g
.!::!
300 ------
iii 300 iii
E
(; E
o
200
c Slope = -59.28 psi/cycle c
c!l kh = 21948.6 md-f1 c!l Slope = 62.65 psi/cycle
'lU 200 I kh = 20762.0 md-ft
II:
k = 43.9 md 'lU 100 k = 41.52 md
S = 5.2 II: S = 4.5
L = 500.00 ft I Lw = 500 ft.
0 I I
10_ 2 . 25 -1.50 -0.75 0.00 0.75 0.150 2.25 3.00 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0
Logarithmic convolution time
Fig. 13-Logarlthmlc convolution plot of the drawdown test for the gas cap case.
Logarithmic convolution time
Fig. 17-Logarithmlc convolution plot of the buildup test.

640..,.----,--,
il---.rTT-..,----,---.i,...,--",'Iii '-'1"""--1,'1[,Tililil
~~
i""TT!

I
!I,I,TT11TII- l i l l

.~ 560~-++1+-+i-l--i-++++!+-H+--+ i ttlft,).f~,
"",,~'---IH
, 1 I

i .-+l*ftl-h

£~ : ilL
--II III1lilt!I -~1111
I 1

480+-+-+\+, H+~--+V-t,,:~:w;=rI-H---+-TI-1i-hllhl
~ Hi, '.' i I J II I I
~
~
400 Ii ji
320-t---ir-,H,--t+1---i-H+m-n---t-,---p=hd±±±!o±==="===,=,'l1
· 1 i 111I1 i!
§ I [I Ii Slope = 50.42 psi/cycle
o~ II i,
kh = 25800.2 md-ft
240 1'1 I k = 25.80 md
S = 7.22
III :I i , Lw = 500.00 ft.

1p-2 10- 1 100 101

i II I I i III
10. 1
Time, hr
Fig. 14-Shut-in pressure and after·flow rate. Shut-in Time, hr
Fig. 18-0econvolved pressure of the bUildUp test.
, 00000000000000000 4500
27.79 9.766 3.448 1.217 0.4296 0.1515 0.0536 0.0191 0.0071

I1II Illlr ~r
lt,mi
' ,III III1 I
4350
I iii i lilll II ii' i I I II!I I

c
10· -: 'iii
Co ~ Tilt,I
1··.l:~J
I
I

I
~ i III II nw II
I
11

I
II
1111\
ilil '
I
J
\

~ 10' ~
::l
III
III
III
4200
'·1'1 jI I

Ilill
Q)
,I I1II111
IIII! \'" ,
"0
C
ell
ii
4050 I i '. I IIII
.~
,I li!11
J
<J 10'-:
'5
.c
en II II 1II
1

1 I
I'
Iliil
i, ,
I
I
3900

III II iilil i

1111I II r···.. f·. 1.1 I 1'1

1

I + Derivative
0 Pressure I 0
i I1III
I II; I II ilill I ! Ilfi, i
11

10·' --l2-...,.....,...-r--.-.............,.,.,~ ...........,..,..~---.,....,............,~:::;::::;::;:~ 10 3 10 4 10 5

1~-· 1~-1 1~' ~~I Horner time,(tp+~t)/M
Time, t, hr Fig. 19-Horner plot of the shut-in pressure for the gas cap case.
Fig. 15-0elta shut-in pressure and its derivative.

4200 '--4-r1.9~3-2'2r-.21--11.r:-:79--=-S.2'-61:---=-:3.3::':2:-6
~t7J:.68=--:-;0.9:J::39;;:6---:-0.4;;;99;;-;4----;;-0.2;;];6;;55~ 8200

4100 N +l Real time, hr I~~~~e I II~ 1~~~'~'~p~d~CI~

.~
et
4000 t--
4150,
I I ~

I
~,~.
~~f:
k

U
= 120.88 md
~ 4~~~.7B psi - -oJ:
8400

C. 8600
9 5/8" CASING 8590' TVO
SHALE

'12' ,_~ . . . .~... r," I~'':.,. ~~~

~ 3900 Q) TOP SADLEROCHT 8631'
III "0
III GAS
~
Q)

ii 3800 4100+-l-----r- . 8800 GOC 8825'

.~
'9 7" LINER 896'~ J/~~LOTTED UNER OIL
'5
.c
(J)
3700
> 9000 HOT 9045' ----------
OWC 9090'
3600 t-
9200 WATER
3500 -++-----<~_1___+_l_l_H'
'-J.-j-I--+--+-+'+-;'i-i'r++,-r-
I--!--'--+--+---l
103 10 4 2200 2600 3000 3400 3800 4200 4600 5000 5400
Horner time,(tp+~t)/~t
Departure, ft
F19. 16-Horner plot of the shut-in pressure. Fig. 20-Location and completion of JX-2.
397
 1500

rn 0
0
Po< 0
1200 0
0
a.i 0
0

rn .~
..., 0
Po< 0
cd '-.
.~ 10'
0
' .......
...a.i
;j
900 ...
Q)
0
+//
rn "d

/
rn
...Po<
Q)
...
Q)

;j
600 rn
rn
j
Q) ...0..
Q)

+/
0

300
cd
;:::
Q)
+J //
0 /
_...SJ-'-' .JJ-.---C
-------- .

0 10'
10° 10' 10-' 10- 1 10· 10'
Time, hr Time, hr
Fig. 21-History match of the measured downhole pressure and model response for the multirate test. Fig. 22-History matches of the derivative of the 24-hour shut-in pressure and model responses for the
buildup test.

3900 -r---------------------....::.....-----,
3840 .....rn
0..
3780
a.i
3720 .~
~ 10'
.~
.~ 3660 ...
Q)
"d
~ 3600 ...;::1
Q)

J: 3540 rn
rn

3480
...0..
Q)

cd
3420 ;:::
kh = 29068.8 md-ft Q)

k = 200.47 md o
3360 s = 4.98
p* = 3804.41 psi
3300 !-===:::;::==:;::::::;::::::.-.. . . . ."""T""'l~--......_-....._ ........._+_....__.........j
10° 10' 102 10-' 10°
(t. + D.t)/6.t Time, hr
Fig. 23-Horner plot of the shut-in pressure for JX-2. Fig. 24-History matches of the derivative of the 6-hour shut-in pressure and model responses for
the buildup test.

1800
B1.16 67:B4 62.06 25.26 24.7B 24.30 1.39B
Real time, hr.

.~
1600 +

~
~
1400
(L

.s
ill 1200
0
'tl
<D
.!::!
<ii
E 1000

!
'1il
IT:
Slope
kh
BOO
k
S
h,..

-2.1 -1.B -1.5 -1.2 -0.9 -0.6 -0.3 0.0 0.3

Multirate Time Function

Fig. 25-Multlrate flow-test plot for JX-2.

398