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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.

69F.H. Escobar et al. / Journal of Petroleum Science and Engineering 58 (2007) 6882


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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,

70 F.H. Escobar et al. / Journal of Petroleum Science and Engineering 58 (2007) 6882


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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



6/16


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.



7/16


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.



9/16


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.



10/16


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



11/16


flow line has already arrived to one of the

parallel boundaries because the wrong reading

may double the permeability value.


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.


6.3. Case III. It is suspected that wellbore storage

masks the radial-flow regime


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



12/16


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.



13/16


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.



14/16


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



15/16


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



16/16


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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