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This article was published in an Elsevier journal. The attached copy
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Pressure transient analysis for long homogeneous
reservoirs using TDS technique
Freddy Humberto Escobara,, Yuly Andrea Hernndez b, Claudia Marcela Hernndez c
a Universidad Surcolombiana, Av. Pastrana Cra. 1, Neiva, Huila, Colombiab Hocol S.A., Cra. 7 No 114-43, Floor 16, Bogota, Colombia
c Weatherford, Cra. 7 No 81-90, Neiva, Huila, Colombia
Received 11 July 2006; received in revised form 3 November 2006; accepted 19 November 2006
Abstract
A significant number of well pressure tests are conducted in long, narrow reservoirs with close and open extreme boundaries. It
is desirable not only to appropriately identify these types of systems but also to develop an adequate and practical interpretation
technique to determine their parameters and size, when possible. An accurate understanding of how the reservoir produces and the
magnitude of producible reserves can lead to competent decisions and adequate reservoir management.
So far, studies found for identification and determination of parameters for such systems are conducted by conventional
techniques (semilog analysis) and semilog and loglog type-curve matching of pressure versus time. Type-curve matching is
basically a trial-and-error procedure which may provide inaccurate results. Besides, a limitation in the number of type curves plays
a negative role.
In this paper, a detailed analysis of pressure derivative behavior for a vertical well in linear reservoirs with open and closed
extreme boundaries is presented for the case of constant rate production. We studied independently each flow regime, especially the
linear flow regime since it is the most characteristic fingerprint of these systems. We found that when the well is located at one of
the extremes of the reservoir, a single linear flow regime develops once radial flow and/or wellbore storage effects have ended.
When the well is located at a given distance from both extreme boundaries, the pressure derivative permits the identification of two
linear flows toward the well and it has been called that dual-linear flow regime. This is characterized by an increment of the
intercept of the 1/2-slope line from 0.5 to with a consequent transition between these two straight lines. The identification of
intersection points, lines, and characteristic slopes allows us to develop an interpretation technique without employing type-curve
matching. This technique uses analytical equations to determine such reservoir parameters as permeability, skin, well location and
reservoir limits for both gas and oil linear reservoirs. The proposed technique was successfully verified by interpreting both field
and synthetic pressure tests for gas and oil reservoirs. 2006 Elsevier B.V. All rights reserved.
Keywords: Radial flow; Parabolic flow; Dual linear flow; Single linear flow; Permeability; Well test analysis
1. Introduction
A representative number of buildup and drawdown
pressure tests are conducted in narrow, long reservoirs,
limited reservoirs or both. This specific type of
Journal of Petroleum Science and Engineering 58 (2007) 68 82
www.elsevier.com/locate/petrol
Corresponding author.
E-mail addresses: [email protected] (F.H. Escobar),
[email protected] (Y.A. Hernndez),
[email protected] (C.M. Hernndez).
0920-4105/$ - see front matter 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.petrol.2006.11.010
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geometries, normally referred as channelized systems
such as fluvial and deep sea fans, requires its proper
identification and the determination of reservoir para-
meters and dimensions when possible. Reservoir
management strongly depends upon the appropriate
estimation of reservoir parameters.
Many of the reservoirs in Colombia have been found
in the Magdalena River Valley Basin. Several of these
display channel behavior due to, probably, fluvial
deposition. There are many other geological possibilities
that describe channel flow coupled with two or more
porous and permeable structures; to name some of them,
it is possible to find long and narrow reservoirs in deltaic
or turbiditic environments, elongated facies of compos-
ite porous media and faulted reservoirs. In this particular
case, the reservoir is limited by two faults so that an
elongated reservoir geometry is formed.Miller (1966) presented the first solution to water
influx in a linear aquifer. It was followed by another
investigation by Nutakki and Mattar (1982) for infinite
channel reservoirs using a vertical fracture approach
with a pseudoskin factor. Ehlig-Economides and
Economides (1985) presented an analytical solution
for linear flow to a constant-planar source solution in
drawdown tests. In 1996, Raghavan and Shu, pre-
sented a method to estimate average pressure when
radial flow conditions are nonexistent for linear and
bilinear flow regimes which can be applicable tochannel reservoirs. Massonet et al. (1993) presented
the results of flow simulations in geological complex
channelized reservoirs. Their well test analysis was
performed via pure flow simulation and no proposed
interpretation technique was presented. Wong et al.
(1986) presented new type curves to interpret pressure
transient analysis for rectangular reservoirs. They use
type-curve matching and conventional techniques on
actual field data.
A modern technique known as Tiab's Direct
Synthesis technique (Tiab, 1995) employs the pressure
and pressure derivative curves to interpret pressure
buildup and drawdown tests without using type-curve
matching. Because of its simplicity and practicality, this
technique has been extended here to analyze pressure
behavior in channelized reservoirs.
2. Mathematical formulation
Combining the line-source solution of a well having
a radius of zero inside an infinite reservoir and the
superposition principle method of images we
obtained the pressure behavior for a well inside a
rectangular drainage area with close, open or mixed
extreme boundaries and we assume that long and narrow
reservoirs approach to rectangular geometry systems.
This method can be described as a procedure for
distribution of sources and sinks in a porous medium
having no-flow or constant-pressure boundaries. Our
study considers a rectangular reservoir with the nearparallel boundaries always closed and the extreme
boundaries are either open or close to flow.
Fig. 1. Pressure derivative behavior for a well (a) both extreme boundaries are close, (b) close near boundary and open far boundary.
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Based upon characteristic behaviors found on the
pressure and pressure derivative plot, in order to
describe the flow regimes, we classified the reservoirs
into two groups:
(1) The well is near the no-flow boundary. This group
includes rectangular reservoirs with either (a) both
extreme boundaries closed or (b) one extreme is
closed and the far extreme is open. The pressure
behavior is the same until the pressure transient
reaches the far boundary. Once the dual-linear
flow vanishes, for the first case the pseudosteady-
state flow is developed and, for the second case,
steady-state flow regime develops. See Fig. 1.
(2) The well is near the constant-pressure boundary.
This group includes rectangular reservoirs inwhich (a) both extreme boundaries have constant
pressure boundaries, and (b) the near boundary is
closed and the far boundary is opened. The
pressure behavior is the same until the pressure
wave has arrived to the far extreme lateral side of
the reservoir. In both cases, the simultaneous
effect of the open boundary and the linear-flow
regime produces a negative one-half slope which
does not correspond to either hemispherical or
spherical flow regimes, as normally expected.
This flow regime has been called parabolic flow
(Escobar et al., 2005).
The pressure and pressure derivative behavior of wells
inside rectangular reservoir has been treated by several
authors (Nutakki and Mattar, 1982; Wong et al., 1986).
Therefore, we will devote our attention to the philosophy
of the Tiab's Direct Synthesis, TDS, Technique (Tiab,
1995) to develop an interpretation technique using
characteristics points, slopes and lines found on the
pressure and pressure derivative plot to obtain analytical
equations for estimation of reservoir parameters.
3. Basic equations
Let us define dimensional quantities. Starting with
dimensional time:
tD 0:0002637kt/lctr2w
1:a
tDA 0:0002637kt/lctA
1:b
tDL tD
W2D1:c
Dimensionless reservoir width and well position:
WD YErw
2:a
XD 2bxXE
2:b
YD 2byYE
2:c
Dimensionless pressure and pressure derivative:
PD kh141:2qlB
DP 3:a
tDTPV
D kh
141:2qlBtTDPV 3:b
4. Characteristic lines and points
Many wells have been observed to display long-term
linear flow. Linear flow can be detected by a 1/2-slope
line in a loglog pressure of the reciprocal of flow rate
versus time. El-Banbi and Wattenberger (1998) pre-
sented the linear reservoir analytical solution for the
constant pressure production case, as follows:
1
qD 2k ffiffiffiffiffiffiffiffiffiktDAp 4:aThis article, however, is not focused on the constant
pressure case. However, the equations for the TDS
technique for linear reservoirs can be easily derived
following the same methodology as for the constant rateproduction case.
Linear-flow regime is also observed when the well is
located at one of the reservoir extremes as depicted in
Fig. 1. This is governed by the following constant rate
equation:
PD 2kffiffiffiffiffiffi
tDLp SL 2k
ffiffiffiffitD
pWD
SL 4:b
From observation of the above relationship, the
constant in Eqs. (4.a) (4.b) and (4.c) (Ehlig-Economidesand Economides, 1985) is not correct. As long as the
well is located far away one of the extreme boundaries,
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dual-linear flow is developed before the linear flow. In
this case, the governing equation is:
PD 2 ffiffiffiffiffiffiffiffiffiktDLp SDL 2 ffiffiffiffiffiffiffiktDpWD SDL 4:c
Suffices L and DL in Eqs. (4.b) and (4.c) stand for
linear and dual-linear flow regimes. Pressure derivatives
for Eqs. (4.b) and (4.c) are, respectively, developed in
this study as:
tDTPVDL k
ffiffiffiffitD
pWD
5:a
tDTPVDDL ffiffiffiffiffiffiffiktD
pWD
5:b
Plugging Eqs. (1.a), (2.a) and (3.b) into Eqs. (5.a) and
(5.b) yields:
ffiffiffik
pYE 7:2034qB
htTDPVL
ffiffiffiffiffiffiffiffiffiffiDtLl
/ct
s6:a
ffiffiffik
pYE 4:064qB
htTDPVDL
ffiffiffiffiffiffiffiffiffiffiffiffiDtDLl
/ct
s6:b
Eq. (6.b) was also presented by Nutakki and Mattar
(1982). When the pressure derivative value is read at the
time, t=1 h, Eqs. (6.a) and (6.b) will then become:
ffiffiffikp YE 7:2034qB
htTDPVL1ffiffiffiffiffiffiffil/ct
r 7:a
ffiffiffik
pYE 4:064qB
htTDPVDL1
ffiffiffiffiffiffiffil
/ct
r7:b
The skin factor caused by the convergence from
radial flow into linear flow is determined by dividing the
dimensionless pressure equation by the dimensionless
pressure derivative equation. The same procedure isperformed for the skin factor due to the convergence
from either lineal to dual-linear or lineal to radial flow.
After replacing the dimensionless quantities in these
results and solving for the skin factor we obtain:
SL D
PLtTDPVL 2
134:743YE
ffiffiffiffiffiffiffiffiffiffiktL/lct
s 8:a
SDL DPDLtTDPVDL2
1
19:601YE
ffiffiffiffiffiffiffiffiffiffiktDL
/lct
s8:b
where PL and (tP)L are the pressure and pressure
derivative values read on the linear flow regime during
any convenient time, tL. Similar for the dual linear case.
The total skin factor results as the summation of the
linear, dual-linear and mechanical (from radial flow)skin factors.
As observed in Figs. 2 and 4, parabolic flow,
characterized by a slope of1/2 of pressure derivative
curve, develops as a result of the simultaneous effect of an
open boundary near the well and the expected linear flow
regime along the far lateral side of the reservoir. The
reader should refer to Escobar et al. (2005) for a better
understanding of this particular behavior. The pressure
and pressure derivative governing equations for this flow
regime, respectively, are:
PD WDXD2 XEYE
2t0:5D SPB 9:a
tDTPV
D WD
2XD2 XE
YE
2t0:5D 9:b
Dividing Eqs. (9.a) by (9.b) and replacing the
dimensionless parameters, we can obtain an equation
for the parabolic skin factor using the pressure and
pressure derivative values read at any convenient time onthe minus one-half-slope line:
SPB DPPBtTDPVPB 2
123:16b2x
YE
ffiffiffiffiffiffiffiffiffiffi/lct
ktPB
s10:a
Replacing the dimensionless parameters into Eq. (9.b)
will result in a relationship to obtain either permeability,
reservoir width or well location, as preferred:
k1:5YE
b2x 17; 390 qlB
htTDPVPB
!/lct
tPB
!0:
5 10:b
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where (tP)PB is the pressure derivative during
parabolic flow read at any convenient time, tPB.
5. Characteristic lines and points
5.1. Intersection of dual-linear, linear and radial lines
with pseudosteady-state line
For long producing times in a closed reservoir the
pressure derivative is characterized by a unit-slope line
which governing equation is:
tDTPVDpss 2kTtDA 11
The intersections of this line with the linear and dual-
linear pressure derivative lines allow us to obtain an
expression to estimate the reservoir area. See Fig. 3.
Therefore, combination of Eqs. (5.a) and (5.b) with Eq.
(11) will provide the following equations once the
dimensionless quantities are replaced:
A ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ktDLPSSi Y2E
301:77/lct
s12:a
A ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ktLPSSi Y2E948:047/lct
s12:b
As expressed by Tiab (1995), the intersection of the
radial and the pseudosteady-state flow lines takes place at:
tDARPi 14k
12:c
After replacing the dimensionless parameters, it yields:
A ktRPSSi301:77/lct
12:d
5.2. Intersection of radial-flow line with either linear or
dual-linear flow lines
This intersection point provides an equation to
estimate the reservoir width. The intersection point
between the radial-flow line and the linear-flow line,
tRLi, is unique. See Fig. 3. At that point the dimensionlesspressure derivative takes a value of one half when the
well is centered regarding the reservoir's parallel
boundaries, otherwise, the pressure derivative value is
one and two horizontal lines may be observed. Based on
this observation will result:
tDTPVDDL ffiffiffiffiffiffiffiktD
pWD
0:5 13:a
tDTPVDL k
ffiffiffiffitDp
WD 0:5 13:b
Fig. 2. Pressure derivative behavior for a well (a) both extreme boundaries are open, (b) open near boundary and close far boundary.
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Replacing the dimensionless parameters into Eqs.
(13.a) and (13.b) will give:
YE 0:05756ffiffiffiffiffiffiffiffiffiffiffiffi
ktRDLi/lct
s14:a
YE 0:1020ffiffiffiffiffiffiffiffiffiffi
ktRLi/lct
s14:b
When two horizontal lines are seen, the reader ought
to replace in Eqs. (14.a) and (14.b) the constants
0.05756 and 0.1020 by 0.02978 and 0.051, respectively,
Similarly for Eq. (2.8), Tiab (1995), the constant 70.6
should be changed to 141.2.
5.3. Intersection between the parabolic-flow line with
either dual-linear or radial-flow lines
Parabolic flow only takes place when the well is
near a extreme constant pressure boundary. These
intersection points, Figs. 2 and 4, allow for the
estimation of bx. Equating Eq. (9.b) with (5.b) will
result in Eq. (15) after replacing the dimensionless
quantities:
bx 165:41
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiktDLPBi/lct
s15
Equating Eq. (9.b) to 0.5 and plugging the dimen-
sionless parameters will yield:
bx
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiYE
246:32 ffiffiffiffiffiffiffiffiffiffiffiktRPBi/lcts
vuut 16
5.4. Intersection between the 1-slope line with either
dual-linear, radial or parabolic lines
When the two extreme sides of a rectangular
reservoir are constant-pressure boundaries we assume
that a 1-slope line follows the parabolic flow line, See
Fig. 2. The governing equation for this line is:
tDTPV
D W2Dk
2 X1:5
D XE
YE 3
t1D 17
For the mixed boundary case, once the parabolic line
vanishes, the derivative rises up, and then, falls down. We
assume that a 1-slope line could be drawn on this last
curve. Its governing equation is:
tDTPV
D W2Dk
X1:5D XE
YE
3t1D 18
By equating Eqs. (17) and (18) with Eqs. (5.b), (9.b)
and the dimensionless value of the pressure derivative
during the infinite-acting period, (tD
PD)= 0.5, thefollowing relationships are obtained once the respective
dimensionless quantities are replaced.
Fig. 3. Pseudosteady-state line intersection with either linear, dual-linear or radial flow lines.
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5.4.1. Open boundaries
Intersection of1-slope and dual-linear flow lines:
X3E 1
1:426
109
ktSS1DLi/lct
31
b3x
19:a
Intersection of1-slope and radial flow lines:
X3E 1
4:72 106
ktSS1Ri/lct
2Y2Eb3x
19:b
Intersection of1-slope and parabolic lines:
X3E 1
77:9
ktSS1PBi/lct
bx 19:c
Suffix SS1 stands for the first1-slope line observed.
5.4.2. Mixed boundaries (well near the constantpressure boundary), Fig. 2
Intersection of1-slope and dual-linear lines:
X3E 1
1:42 1010
ktSS2DLi/lct
31
b3x
20:a
Intersection of1-slope and radial lines:
X3E 1
4:66 107
ktSS2Ri/lct
2Y2Eb3x
20:b
Intersection of1-slope and parabolic lines:
X3E 1
768:4
ktSS2PBi/lct
bx 20:c
Suffix SS2 stands for the second 1-slope line
observed.
5.5. Inflection point between dual linear and linear flow
lines
Dual-linear and linear flows take place when the well is
off-centered with respect the reservoir's extreme lateral
sides. The distance from the well to the near boundary can
be estimated from the inflection point during the transition
period between dual-linear and linear-flow lines. See
Fig. 5. For this point, the following equations are given:
tDTPVDF k ffiffiffikp
2WDTt0:5D 21
tDF W2DXDffiffiffi
k
p XEYE
222
Combining Eqs. (21), (22), (1.a) and (3.b), and
solving for the well position, yields:
bx ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi
1
156:17
YE
X0:5E
kh
qlB
tTDPVF
s23:a
bx ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ktF
5448:2/lct
s23:b
Fig. 4. Intersection of the parabolic flow line with either radial or dual-linear lines.
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5.6. Maximum points
5.6.1. Well near the constant pressure boundary
At later times, the pressure derivative curve displays
two maximum points when the reservoir has mixed
boundaries and the well is near the open one. The first
maximum takes place when dual-linear flow ends and the
parabolic flow follows. The second maximum point is
formed once the parabolic line ends and the no-flow
boundary has been reached by the transient wave. The
constant-pressure effect still dominates the test. When
both extreme sides of the reservoirs are open to flow the
pressure derivative behaves in a similar way as for the
mixed boundary case. However, no second maximum
point is observed. See Fig. 2. Equations of the maximum
points are used to estimate reservoir area and well location.
5.6.1.1. First maximum point (change from dual-linearto parabolic-flow regime), Fig. 2.
tDTPVDX1 2
3
ffiffiffik
pWD
t0:5DX1 24:a
XE
YE 2
3
ffiffiffik
pWDXD
t0:5DX1 24:b
XE
YE
ffiffiffikp
XD
tDTPVDX1 24:c
5.6.1.2. Second maximum point (end of parabolic-flow
line and start of steady-state line), Fig. 2.
tDTPVDX2 ffiffiffik
pWD
X2Dt0:5DX2 25:a
XE
YE k
2WD
t0:5DX2 25:b
XE
YE
ffiffiffik
p2X2D
tDTPVDX2 25:c
Plugging the dimensionless quantities into Eqs. (24.
b), (24.c), (25.b) and (25.c) and solving for well position,
bx, or reservoir length, XE, respectively, we have:
bx 158:8 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ktX1/lct
s 26:a
bx khYEtTDPVX1
159:327qlB26:b
XE 637:3 b2
x
YE
qlB
kh
1
tTDPVX2
26:c
XE 139:2
ktX2
/lct
0:
5 26:d
Fig. 5. Location of inflection points between the transition period of dual linear and linear flow regimes.
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5.6.2. Well near the no-flow boundary
When a rectangular reservoir has mixed boundaries
and the well is near the no-flow boundary, the pressure
derivative displays a maximum point once the constant
pressure boundary is felt as shown in Fig. 1. Thegoverning equation for this maximum point is:
XE
YE k
1:5
4
1
WD
t0:5DX3 27
Substituting Eqs. (1.a) and (2.a) into Eq. (27) and
solving for the reservoir length, XE, gives:
XE 144:24
ktX3
/lct 0:5
28
5.7. Other relationships
If two radial flow lines are observed, the distance
from the well to the closest boundary can be found by
reading the end-time of the radial flow regime, tre, and
using the equation below taken from Ispas and Tiab
(1999).
by ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:000422tre/lcts 29All the necessary equations for gas flow in vertical
wells are presented in Appendix A, with the same
equation numbers given above for oil wells.
6. Step-by-step procedures
6.1. Case I. Well near a no-flow boundary (all flow
regimes are observed)
The following procedure applies to rectangular
reservoirs where the well is located near the close
boundary and the far boundary is either open or close to
flow. The test lasts long enough so that radial, dual-
linear, linear and either pseudosteady-state or steady-
state flow regimes are well defined.
Step 1 PlotPand tP versus time on a loglog plot
Step 2 Draw the infinite acting behavior, dual-linear
and linear flow lines. If given the case, draw the
pseudosteady-state line. Read the value of(tP)r. Care must be taken when the radial
flow line has already arrived to one of the
parallel boundaries because a wrong reading
may double the permeability value.
Step 3 Calculate k using Eq. (2.8) by Tiab (1995).
Step 4 Choose any convenient time on the dual-linear
and linear flow lines and read tDL, (tP)DL,PDL and tL, (tP)L, PL, respectively.
Step 5 Determine k0.5YE using either Eq. (6.a) or
Eq. (6.b).
Step 6 Using the permeability value from Step 3, find
the reservoir width, YE, from the value ofk0.5YE
estimated in Step 5.
Step 7 Read the intersection time of the radial line with
both the dual-linear and linear flow lines: tRDLiand tRLi
Step 8 Verify the reservoir width value, YE, using Eqs.
(14.a) and (14.b).Step 9 Far close boundary. Read the intersection time
between the pseudosteady-state and radial, dual-
linear and linear lines: tRPi, tDLPi, and tRLPi,
respectively. Calculate the reservoir area using
Eqs. (12.a), (12.b) and/or (12.d).
Step 10 Far constant pressure boundary. Once the linear
flow line vanishes and the flow boundary acts, a
maximum point on the pressure derivative is
seen. Read the coordinates of this point: tX3,
(tP)X3, and find XE using Eq. (28).
Step 11 Calculate the radial skin factor, s, from Eq. (2.34)
by Tiab (1995). Find the skin factors due to linear
and dual-linear flow regimes using Eqs. (8.a) and
(8.b). The total skin factor results from adding
these three skin factors.
Step 12 Read the pressure derivative value, (tP)F,
of the inflection point during the transition
between dual-linear and linear flow lines. Find
the distance from the well to the near
boundary or well location, bx, using Eqs.
(23.a) and (23.b).
6.2. Case II. Well near a constant-pressure boundary(all flow regimes are observed)
The procedure below corresponds to a well inside a
rectangular reservoir with its two extreme boundaries
open to flow or the near boundary is open and the far
boundary is closed to flow. The test lasts long enough so
that radial, dual-linear, parabolic and either one or two
maximum points are observed.
Step 1 Same as Step 1, Case I.
Step 2 Draw the infinite acting behavior, dual-linearand the parabolic flow lines. Read the value of
(tP)r. Care must be taken when the radial
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flow line has already arrived to one of the
parallel boundaries because the wrong reading
may double the permeability value.
Step 3 Same as Step 3, Case I.
Step 4 Choose any convenient time on the dual linearflow line and read tDL, (tP)DL, PDL.
Step 5 Determine k0.5YE using Eq. (6.b).
Step 6 Same as Step 6, Case I
Step 7 Read the intersection time of the radial line with
the dual linear: tRDLi.
Step 8 Verify the YE value using Eq. (14.a).
Step 9 Select any convenient time, tPB, on the parabolic
flow line and read (tP)PB and PPB.
Calculate (k1.5/bx2) with Eq. (10.b). Either k or
bx can be verified. Also estimate the parabolic skin
factor, sPB, using Eq. (10.a).Step 10 Read the intersection time of the parabolic flow
and both dual linear and radial flow lines: tPBDLiand tPBRi. Find the distance from the well to the
near extreme boundary, bx, using Eqs. (15) and
(16).
Step 11 Use this step whether the parabolic flow line is
not seen or to verify results from Step 10. Read
the coordinates of the first maximum point (end
of dual linear flow line and start of the
parabolic flow line): tX1 and (tP)X1. Esti-
mate the well location, bx, from Eqs. (26.a) and
(26.b).
Step 12 Far no-flow boundary. Read the coordinates of
the second maximum point: tX2 and (tP)X2and calculate the reservoir length, XE, using
Eqs. (26.c) and (26.d). If this maximum point is
not clearly observed, it is recommended to
estimate XE using Eqs. (20.a), (20.b), and/or
(20.c) using the intersection of the 1-slope
line with the dual-linear flow, radial flow and
parabolic flow lines: (tSS2PBi , tSS2DLi and
(tP)SS2Ri). Since XE and YE are known the
reservoir size can be calculated.Step 13 Far flow boundary. Read the intersection point
of the 1-slope line (seen after reaching the
open boundary) and dual-linear and the para-
bolic flow lines: tSS1PBi and tSS1DLi, and
estimate the reservoir length, XE, using Eqs.
(19.a) and (19.c). When the dual linear flow is
not present (for small XE/YE ratios) XE can be
estimated, using Eq. (19.b), from the intersec-
tion of the 1-slope line and the radial flow line
(tP)SS1Ri and tSS1Ri.
Step 14 Same as Step 11, Case I. Be aware that sL doesnot exit for this case.
Step 15 Same as Step 12, Case I.
6.3. Case III. It is suspected that wellbore storage
masks the radial-flow regime
Step 1 Same as Step 1, Case I.
Step 2 Read any point on the unit-slope line duringwellbore storage. Find wellbore storage, C,
using Eq. (2.3) by Tiab (1995). Read the
coordinates of the maximum point (peak) on
the pressure derivative curve during wellbore
storage: tx and (tp)x and estimate wellbore,
C, using correlation (2.22b) from Tiab (1995). C
from Eq. (2.3) and C from correlation (2.22b)
should be close meaning that the maximum
point was properly chosen, otherwise, pick a
new maximum point. Then, find k using
correlation (2.22.a) and solve for (t
P)r fromEq. (2.8), Tiab (1995). Then, draw the radial
flow line and read the intersection time of this
line with the unit-slope line, ti, Then, estimate
the radial flow skin factor, s, using correlations
2.27 or 2.28 from by Tiab (1995).
Step 3 Continue the procedure from Step 4 of either
Case I or Case II, depending on the given
situation.
7. Field examples
The proposed technique was successfully applied to
field cases and synthetic data. Because of space
constraints only two field cases are presented.
7.1. Field case 1
A pressure drawdown test was run in a well in a
channelized reservoir in the Colombian Eastern Planes
basin. Reservoir and well parameters are given in
Table 1 and pressure data is given in Fig. 6. Determine
reservoir permeability, reservoir dimensions and well
location.
Table 1
Reservoir and well parameters for field examples
Parameter Value
Field case 1 Field case 2
q (BPD) 1400 740
h (ft) 14 13.1
ct (psi1) 9 106 14.1106
rw (ft) 0.51 0.188
(%) 24 15B (bbl/STB) 1.07 1.255
(cp) 3.5 0.6
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7.1.1. Solution
The following parameters were read from Fig. 6:
tTDPVr 60 psi; DPr 122:424 psi;tr 0:498 h; tDL 2 h; tTDPVDL 105:81 psi;DPDL 265:942 psi; tRDLi 0:7 h;
tPB 10:157 h; tTD
P
V
PB 132:873 psi;DPPB 458:466 psi; tPBDLi 6 h; tPBRi 50 h;tSS1Dli 7:5 h; tSS1Ri 24 h; tSS2PB 12 h
Permeability is estimated with Eq. (2.8) from by Tiab
(1995) to be 440.7 md. Reservoir width values of 352.4
and 367.7 ft were obtained with Eqs. (6.b), (10a) and
(10b), respectively. The well location, bx, of 283.7 ft
was estimated with Eq. (10.b) and, then, it was verified
with Eqs. (15) and (16), by giving values of 285.9 and
283.9 ft, respectively.
Once the parabolic line vanishes, the pressurederivative rises up before falling down. We conclude
that the far boundary is closed. The maximum point is not
clearly observed. Therefore, we utilize the intersection of
the 1-slope with the dual-linear, parabolic and radial
lines. The reservoir length is found with Eqs. (20.a)
(637.2 ft), (20.b) (628.3 ft) and (20.c) (637.1 ft). Notice
the good agreement among the results.
The mechanical skin factor is estimated using Eq.
(2.34), by Tiab (1995), to be 4.9. The dual-linear flow
regime skin factor is estimated with Eq. (8.b), as 0.4 and
Eq. (10.a) gives a parabolic skin factor, sPB, of 6.07.Therefore, the total skin factor st=s +sDL+sPB=4.9+
0.4+6.07=1.57. This field example was solved using
non-linear regression (simulation) with a commercial
software for well test interpretation. The results are:
k 440:1 md s 4:8 YE 263 ftXE 600 ft bx 260 ft
Even though, the simulation was not very accurate,
the simulated results match closely with the values
estimated using the proposed technique. Needless to say,that the simulation does not take into account the
parabolic skin factor, in such a case, the total skin factor
(4.9+0.4)=4.5 which closely agrees with the results
from non-linear regression analysis.
7.2. Field case 2
Taken from Wong et al. (1986). A well is located in
the center of Alberta, Canada. It was completed in one
of extremes of a sandstone reservoir which has a linear
tendency. The oil is found to be sweet, light andundersaturated. Reservoir and well parameters are given
in Table 1 and pressure data is provided in Fig. 7. Find
below the permeability and reservoir dimensions.
7.2.1. Solution
The following information was read from Fig. 7:
tTDPVx 701:63 psi tx 2:3 hDPr 1501 psi tr 12:1 htTDPVr 23:5 psi DPDL 1561:5 psitDL 100 h tTDPVDL 43 psi tRDLi 29 htRPi 80 h tDLPi 210 h
Fig. 6. Pressure and pressure derivative loglog plot for field example 1.
78 F.H. Escobar et al. / Journal of Petroleum Science and Engineering 58 (2007) 6882
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Since the well is located in one of the reservoir's
extreme boundaries, linear flow is developed after the
end of the radial flow regime. At late time, pseudosteady-
state behavior is observed. Notice also that early pressure
data are affected by wellbore storage. Permeability of
255.6 md is estimated with Eq. (2.8), Tiab (1995). The
reservoir width, YE, is found with Eq. (7.a), as 3956 ft
and it was verified from the intersection of linear and
radial-flow lines by using Eq. (14.b) (3897.8 ft).
Reservoir areas of 26,424,124.7 ft2 and 26,697,829.8 ft2
were found from Eqs. (12.b) and (12.d), respectively. Since
the radial flow was masked by wellbore storage, we utilized
the second radial flow regime, then constants in Eq. (12.d)
and (2.8), by Tiab (1995), are doubled. The reservoir
length, XE, is solved from the area:
A XEYEXE A
YE 26; 424; 124:7
3956 6679:5 ft
A mechanical skin factor of 23.2 is estimated usingEq. 2.34, by Tiab (1995), and a skin factor of 35.5 from
linear flow is obtained from Eq. (8.1). Then, the total
skin factor is st=s +sL=23.2+35.5=58.7.
Results from Wong et al. (1986) are given as follows:
Type curves
kh 3191 md ft k 244 md s 52:14YE 4017 ft XEYE 24:21 106 ft2
Conventional analysis
kh 3321 md ft k 254 md s 26:1YE 4035:5 ft XEYE 22:55 106 ft2
8. Analysis of results
Since the developed equations were based on
analytical solutions applied to certain regions of the
pressure and pressure derivative plot, it was expected to
obtain good results from the application of them to
either field or simulated pressure data. From the above
results, we observe a good agreement with the results
provided by the Tiab's Direct Synthesis Technique with
those obtained by either the conventional method, type-
curve matching and con-linear regression (simulation)
analysis. Because, the TDS technique is essentially a
graphic method, it is important to clarify that the
application of this technique highly depends on the
quality of the pressure derivative data. Cases with very
noise pressure derivative, erratic field data or poor
pressure derivative can lead to wrong reservoir
characterization estimation.
9. Conclusions
1. The TDS technique was extended to characterize
rectangular homogeneous reservoirs (channels) for
vertical oil and gas wells. New equations were de-
veloped using intersection points and characteristic
points to estimate reservoir dimensions. In order to
verify the estimated parameter, we introduced three
new equations for reservoir area, A, six new
equations for well location, bx, four new equations
for reservoir width, YE, and six new equations for
reservoir length, XE.The technique was successfully tested with field and
synthetic examples.
Fig. 7. Pressure and pressure derivative loglog plot for field example 2.
79F.H. Escobar et al. / Journal of Petroleum Science and Engineering 58 (2007) 6882
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2. New equations are introduced to estimate skin factors
due to the convergence from radial to dual-linear,
from linear to either radial or dual-linear, and for
parabolic flow using the pressure and pressure
derivative values read from any convenient time onthese lines.
3. It is only possible to estimate the product k0.5YEwhen only linear or dual linear flow is developed.
4. Corrected versions of the governing equations of
dimensionless pressure for long and narrow reser-
voirs are presented for both linear and dual-linear
flow. For such systems, we also developed new
governing equations of dimensionless pressure and
pressure derivative for either the parabolic flow line,
the 1-slope line, the maximum points and the
inflection point found during the transition periodbetween dual-linear and linear lines.
10. Recommendation
To extend this study for areally heterogenous
linear reservoirs and for the constant pressure
solution case.
Nomenclature
A Area, ft2
B Oil formation factor, bbl/STB
ct Compressibility, 1/psi
h Formation thickness, ft
k Permeability, md
m(P) Pseudopressure function, psi2/cp
P Pressure, psi
PD Dimensionless pressure derivative
PD Dimensionless pressure
Pi Initial reservoir pressure, psia
Pe External reservoir pressure, psia
Pwf Well flowing pressure, psi
q Flow rate, bbl/D. For gas reservoirs
the units are Mscf/DrD Dimensionless radius
re Drainage radius, ft
rw Well radius, ft
s Skin factor
st Total skin factor
T Reservoir temperature, R
t Time, h
tm(P) Pseudopressure derivative function, psi2/cp
tD Dimensionless time
Greek Change, drop
t Flow time, h
Porosity, fraction
Viscosity, cp
Suffices
app Apparent D Dimensionless
DL dual-linear
i Intersection or initial conditions
L Linear
PB Parabolic
PSS Pseudosteady
SS Steady
DLPSSi Intersection of pseudosteady-state line with
dual-linear line
LPSSi Intersection of pseudosteady-state line with
lineal lineRPi Intersection of pseudosteady-state line with
radial line
RDLi Intersection of radial line with dual lineal line
RLi Intersection of radial line with lineal line
RPBi Intersection of radial line with with the
parabolic flow line
DLPBi Intersection of dual lineal line with the
parabolic flow line
SS1 1-slope line formed when the parabolic flow
line ends and steady-state flow regime starts.
Both extreme sides of the reservoir are open
SS2 1-slope line formed when the parabolic flow
ends and steady-state flow regime starts. Well
is near the open boundary and the far boundary
is closed
SS1Ri Intersection between the radial line and the 1-
slope line (SS1)
SS1DLi Intersection of the dual linear line with the 1-
slope line (SS1)
SS1PBi Intersection of the parabolic flow with the 1-
slope line (SS1)
re End of radial flow regime
SS2Ri Intersection of radial line with
1-slope line(SS2)
SS2DLi Intersection of the dual linear line with 1-
slope line (SS2)
SS2PBi Intersection of the parabolic flow line with 1-
slope line (SS2)
o Oil
R,r radial flow
re End of radial flow
w Well
x Maximum point (peak) during wellbore storage
X1 Maximum point found between the dual linearflow and parabolic flow when the well is near
the constant pressure boundary
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X2 Maximum point found at the end of parabolic
flow and the start of steady-state flow regime
when the well is near the open boundary and
the far boundary
X3 Maximum point found at the end of the linearflow regime when the well is near the closed
boundary and the other one is open to flow
Acknowledgments
The authors gratefully acknowledge the financial
support of the Colombian Petroleum Institute, ICP, under
the mutual agreement Number 008 signed between this
institution and Universidad Surcolombiana.
Appendix A. Gas reservoirs equations
ffiffiffik
pYE 72:571qT
htTDmP VL
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitL
/lgcti
s6:a
ffiffiffik
pYE 40:94qT
htTDmP VDL
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitDL
/lgcti
s6:b
ffiffiffik
pYE 72:571qT
htTDmP VL1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
/lgcti
s7:a
ffiffiffik
pYE 40:94qT
htTDmP VDL1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
/lgcti
s7:b
SL mPLtTDmP VL
2
1
34:743YE
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiktL
/lgcti
s8:a
SL mPDLtTDmP VDL
2
1
19:601YE
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiktDL
/lgcti
s8:b
ffiffiffiffiffik3
pYE 175; 200qTb
2x
htTDmP VPB/lgcti
tPB
0:510
A ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffi
ktDLPSSiY2
E
301:77/lgcti
s12:a
A ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi
ktLPSSiY2E948:047/lgcti
s12:b
A ktRPSSi301:77/lgcti
12:d
YE 0:05756ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ktR1DLi
/lgcti
s14:a
YE 0:02878ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ktR2Li
/lgcti
s14:b
bx khYE3717:74qTX0:5E
tTDmP VF 2
23
A.1. Intersection points
bX 165:45
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiktDLPBi/lgcti
s15
bX
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiYE
246:
32
ktRPBi
/lgcti 0:5
vuut 16X3E
1
1:4256 109ktSS1DLi
/lgctibx
319:a
X3E 1
4:724 106kYEtSS1Ri/lgcti
21
bX
319:b
X
3
E 1
77:9
kbXtSS1PBi
/lgcti 19:c
X3E 1
1:426 1010ktSS2DLi/lgctibx
320:a
X3E 1
4:66 107kYEtSS1Ri/lgcti
21
bX
320:b
X3E 1768:4kbXtSS2PBi/lgcti
20:c
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A.2. Maximum points
bX 158:8
ktX1
/lgcti 0:5
26:a
bx YEkh1605:2qT
tTDmP VX1
26:b
XE 6420 b2
X
YE
qT
kh 1
tTDm
P
V
X226:c
XE 139:203
ktX2
/lgcti
0:526:d
XE 144:24
ktX3
/lgcti
0:528
by ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0:000422tre
/lgcti
s29
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