Copyright 2006, Society of Petroleum Engineers This paper was prepared for presentation at the First International Oil Conference and Exhibition in Mexico held in Cancun, Mexico, 31 August2 September 2006. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

Abstract The storage capacity ratio, , measures the flow capacitance of the secondary porosity and the interporosity flow parameter, , is related to the heterogeneity scale of the system. Currently, both parameters and are obtained from well test data by using the conventional semilog analysis, type-curve matching or the TDS Technique. Warren and Root showed how the parameter can be obtained from semilog plots. However, no accurate equation is proposed in the literature for calculating fracture porosity. This paper presents an equation for the estimation of the parameter using semilog plots. A new equation for calculating the storage capacity ratio and fracture porosity from the pressure derivative is presented. The equations are applicable to both pressure buildup and pressure drawdown tests. The interpretation of these pressure tests follows closely the classification of naturally fractured reservoirs into four types, as suggested by Nelson1. The paper also discusses new procedures for interpreting pressure transient tests for three common cases: (a) the pressure test is too short to observe the early-time radial flow straight line and only the first straight line is observed, (b) the pressure test is long enough to observe the late-time radial flow straight line, but the first straight line is not observed due to inner boundary effects, such as wellbore storage and formation damage, and (c) Neither straight line is observed for the same reasons, but the trough on the pressure derivative is well defined. Analytical equations are derived in all three cases for calculating permeability, skin, storage capacity ratio and interporosity flow coefficient, without using type curve matching. In naturally fractured reservoirs, the matrix pore volume, therefore the matrix porosity is reduced as a result of large reservoir pressure drop due to oil production. This large pressure drop causes the fracture pore volume, therefore fracture porosity, to increase. This behavior is observed particularly in reservoir where matrix porosity is much greater

than fracture porosity. Fractures in reservoirs are more vertically than horizontally oriented, and the stress axis on the formation is also essentially vertical. Under these conditions, when the reservoir pressure drops, the fractures do not suffer from the stress caused by the drop. Using these principles, a new method is introduced for calculating fracture porosity from the storage capacity ratio, without assuming the total matrix compressibility is equal to the total fracture compressibility. Several numerical examples are presented for illustration purposes. Introduction Nelson1 identifies four types of naturally fractured reservoirs; based on the extent the fractures have altered the reservoir matrix porosity and permeability: In Type 1 reservoirs, fractures provide the essential reservoir storage capacity and permeability. Typical Type-1 naturally fractured reservoirs are the Amal field in Libya, Edison field California, and pre-Cambrian basement reservoirs in Eastern China. All these fields contain high fracture density. In Type 2 naturally fractured reservoirs, fractures provide the essential permeability, and the matrix provides the essential porosity, such as in the Monterey fields of California, the Spraberry reservoirs of West Texas, and Agha Jari and Haft Kel oil fields of Iran. In Type 3 naturally fractured reservoirs, the matrix has an already good primary permeability. The fractures add to the reservoir permeability and can result in considerable high flow rates, such as in Kirkuk field of Iraq, Gachsaran field of Iran, and Dukhan field of Qatar. Nelson includes Hassi Messaoud (HMD) in this list. While indeed there are several low-permeability zones in HMD that are fissured; in most zones however the evidence of fissures is not clear or unproven. In Type 4 naturally fractured reservoirs, the fractures are filled with minerals and provide no additional porosity or permeability. These types of fractures create significant reservoir anisotropy, and tend to form barriers to fluid flow and partition formations into relatively small blocks. Nelson discusses three main factors that can create reservoir anisotropy with respect to fluid flow: fractures, crossbedding and stylolite. The anisotropy in Hassi Messaoud field, for instance, appears to be the result of a non-uniform combination of all three factors with varying magnitude from zone to zone. Stylolites, just like fractures, are a secondary feature. They are defined as irregular planes of discontinuity between two rock units. Stylolites, which often have fractures associated with them, occur most frequently in limestone,

SPE 104056

Fracture Porosity of Naturally Fractured Reservoirs D. Tiab, D.P. Restrepo, and A. Igbokoyi, SPE, U. of Oklahoma

2 SPE 104056

dolomite, and sandstone formations. Mineral-filled fractures and stylolites can create strong permeability anisotropy within a reservoir. The magnitude of such permeability is extremely dependent on the measurement direction, thereby requiring multiple-well testing. Interference testing is ideal for quantifying reservoir anisotropy and heterogeneity, because they are more sensitive to directional variations of reservoir properties, such as permeability, which is the case of type 4 naturally fractured reservoirs. It is important to take this classification into consideration when interpreting a pressure transient analysis for the purpose of identifying the type of fractured reservoir and its characteristics. Each type of naturally fractured reservoir may require a different development strategy. Ershaghi2 reports that: (a) Type 1 fractured reservoirs, for instance, may exhibit sharp production decline and can develop early water and gas coning; (b) Recognizing that the reservoir is a type 2 will impact any infill drilling or the selection of improved recovery process; (c) In Type 3 reservoirs, unusual behavior during pressure maintenance by water or gas injection can be observed because of unique permeability trends. PROPERTIES OF MATRIX BLOCKS AND FRACTURES

A naturally fractured reservoir is composed of a heterogeneous system of vugs, fractures, and matrix which are randomly distributed. Such type of system is modeled by assuming that the reservoir is formed by discrete matrix block elements separated by an orthogonal system of continuous and uniform fractures which are oriented parallel to the principal axes of permeability. Two key parameters, and , were introduced by Warren and Root3 to characterize naturally fractured reservoirs. These dimensionless parameters and are mathematically expressed as3:

mtft

ft

tt

ft

ccc

cc

)()()(

)()(

+== .. (1)

2

2

m

w

f

m

xr

kk =

(2) The geometry parameter, , is defined as:

)2(4 += nn .. (3) where n is 1, 2 or 3 for the slab, matchstick and cube

models, respectively. Assuming:(a) the flow between the matrix and the

fractures is governed by the pseudo-steady state condition, but only the fractures feed the well at a constant rate, and (b) the fluid is single phase and slightly compressible, the wellbore pressure solution and the pressure derivative in an infinite-acting reservoir are given by4,5:

s-tEi

-tEit= P DDDD +

++

)1()1(

80908.0ln21

(4)

)1(exp

)1(exp1

21'

-t+

-t-= Pt DDDD

.. (5)

The second pressure derivative of the dimensionless pressure equation is:

)1(

exp)1(

exp1)1(2

)''(

-t

-t= Pt DDDD

... (6)

(A) Semilog Analysis A plot of the well pressure or pressure change (P) versus

test time on a semilog graph should yield two parallel straight line portions as shown in Figure 1. The pressure change P during a drawdown test is (Pi - Pwf). During a buildup test P = (Pws Pwf(t=0)).

1. Fracture Permeability Figure 1 shows two well defined parallel straight lines of

slope m. The slope m of the straight lines may be used to calculate the average permeability of the fractured system or the kfh product:

mqBkh o6.162= . (7)

Assuming the sugar cube model is valid and Types 1 naturally fractured reservoirs, the product kh is essentially equal to (kh)f, so the slope of either straight line can be used to determine kh.

In Type 2 naturally fractured reservoirs the first straight line is mostly related to fracture flow, and therefore the kh product in Eq 7 is essentially (kh)f. The second straight line is however related to both fracture flow and matrix flow, thus the kh product in Eq 7 reflects both (kh)m and (kh)f. In this case it is unlikely that the two straight lines will be perfectly parallel. If however (kh)m

SPE 104056 3

fractured reservoirs, since the fractures do not provide additional porosity or permeability. These reservoirs are best treated as anisotropic and analyzed accordingly.

4. Interporosity Flow coefficient A characteristic minimum point, or trough, is typically

observed on the pressure derivative plot for naturally fractured reservoirs, as shown in Figure 2. This minimum takes place at the point where the second pressure derivative equals zero (tDPD) = 0. The dimensionless time at which this minimum point occurs is given by the following expression4, 5, 6

= 1lnminDt ... (11)

On the semilog plot of well pressure versus test time, this

minimum point corresponds to the inflection point during the transition portion of the curve. Therefore, Eq. 11 can be rewritten as:

= 1lninfDt . (12)

The dimensionless time is defined as:

2inf

inf)(

0002637.0

wmftD

rc

tkt +

= . (13)

Where tinf = tmin. Combining Eqs. 12 and 13 and solving

for , yields a new relationship for the interporosity flow parameter:

= + 1ln)(3792

inf

2

tkrc wmft

.. (14) tinf can be directly read at the inflection point of the

pressure curve from a semilog plot of the flowing well pressure versus test time. For a Miller-Dyes-Hutchinson (MDH) semilog plot, i.e. shut-in well pressure (Pws) versus shut-in time (t), tinf = tinf. When using a Horner plot, the corresponding inflection (Horner) time, (HT)inf, is read and converted to inflection time using the following equation:

1)( infinf = T

p

Ht

t ... (15)

Where (HT) is the Horner time (tp+t)/t or the effective

Horner time tpt/(tp+t). The idea of estimating the interporosity flow parameter

from semilog plots is not new. Uldrich and Ershaghi7, formulated a complex and cumbersome procedure for that purpose. They introduced one equation for pressure drawdown tests which uses the coordinates of the inflection point time, the storage capacity ratio, the skin factor and a parameter read from a plot which is a function of . They also introduced

another equation for pressure buildup tests which utilizes the inflection point time, the storage capacity ratio, the dimensionless Horner production time, tD, and two parameters read from two different plots. These two graphically-obtained parameters are also function of the value. These equations have received limited applications. Bourdet and Gringarten8 suggest plotting a horizontal line through the approximate middle of the transition portion of the curve, and then use the time at which this horizontal line intersects the parallel straight lines to calculate the storativity ratio, , and the interporosity flow coefficient, . Eq. 14 offers a much simpler and analytically sound procedure for calculating from the conventional semilog analysis.

5. Short buildup Test Second Straight is not observed The interpretation of a buildup test is similar to that of a

drawdown. Generally, the second straight line is more likely to be observed than the first one, which often is masked by near wellbore effects, such as wellbore storage. In Type 3 naturally fractured system, where the matrix has a high enough permeability for the fluid to enter the wellbore both from the fracture (mostly) and the matrix, then the first straight line should last a long time, and will not be masked by inner wellbore effects. In this system, it is also possible for an unsteady state flow regime to develop in the matrix. This flow regime will appear during the transition period, i.e. after the first semilog straight line.

However pressure buildup tests often give more reliable value of the storage capacity ratio, , especially when the second parallel straight line is not observed, such as when the pressure test is too short, or the well is near a boundary. In these cases it impossible to determine p, and consequently Eq. 10 can not be used. The equation of the early time straight line can be represented by9:

++

+=

1

loglogt

ttmPP piws ..... (16)

Extrapolating the first straight line to a Horner time of

unity, i.e. (tp+t)/t = 1, where Pws=PFF1, then the storage capacity ratio can be calculated from:

mPP

mPP

FFi

FFi

/)(

/)(

1

1

10110

= .....(17) PFF1 stands for Fracture Flow pressure, since near the

wellbore, fluid flows into the well exclusively through the fractures, particularly in Types 1 and 2 naturally fractured reservoirs. PFF1 will always be greater than (by a value equal to p) the average pressure, Pi and P*, since normally the second parallel line is used to estimate these three pressure values. If the initial reservoir pressure Pi is not available, use the average reservoir pressure instead, or the false pressure P* (if it is known from another source).

The vertical distance between the two parallel semilog straight lines and passing through the inflection point is of course identified as p. For uniformly distributed matrix

4 SPE 104056

blocks, the inflection point is at equal distance between the two parallel lines. Therefore

mP inf12

10= .... (18)

Where: P1inf (= 0.5P) is the pressure drop between the 1st

semilog straight line and the inflection point along a vertical line parallel to the pressure axis.

Equation 18 is analogous to Eq. 10 for calculating the

storage capacity ratio, and therefore should yield the same results as long as the first straight line is well defined and the pressure test is run long enough to observe the trough on the pressure derivative, and therefore the inflection point on the semilog plot. The interporosity flow coefficient is then calculated from Eq. 14.

If the inflection point is difficult to determine, then read

the end-time of the first or early time straight line, tEL1, and use the following equation to estimate :

)1(013185.0

)(

1

2

= +EL

wmft

ktrc

........... (19)

If the buildup test is however too short to even observe the

trough (which provides the best evidence of a naturally fractured system), then results obtained from the interpretation of the test should at best be considered as an approximation.

The skin factor is then obtained from the following

equation:

( )

+

=+

23.3log)()(1513.1 211

wmft

FFihr

rck

mPPPs

(20)

or

( )

+

=+

23.3log2)(1513.1 2inf11

wmft

hr

rck

mPPs

. (21)

where (P)1hr is taken from the first straight line. EXAMPLE 1 Given the build up test data in Table 1 and the following

formation and fluid properties, estimate formation permeability, skin factor, , and from.

q = 125 STB/D h = 17 ft tp = 1200 hr = 13.0% pwf = 211.20 psia rw = 0.30 ft = 1.72 cp B=1.054 RB/STB ct =7.1910-6 psi-1 Solution The following data are read from Figure 3: tinf = 0.63 hr P1inf = 33 psi P1hr = 497 psi m=35.67 psi/cycle tEL1 = 0.012 hr

From Equation 7:

mdk 7.60)17)(67.35(

)72.1)(054.1)(125(6.162 == From Equation 21 the storage capacity ratio is:

014.010 67.35)33(2

== Using equation 1, we can calculate (ct)f:

86 103.1014.01

014.0)1019.7)(13.0()(

1)()(

=

=

=

ft

mtft

c

cc

From equation 21 the skin factor is:

89.0

23.3)3.0)(72.1)(103.11019.713.0(

7.60log67.35

)3328.285(1513.1 286

=

+

+=

s

s

From Equation 14, the interporosity flow parameter is:

7

286

107.8

014.01ln014.0

)63.0)(7.60()3.0)(72.1)(103.11019.713.0(3792

=

+=

From Equation 19:

7

286

101.2

014.0)014.01()012.0)(7.60)(013185.0(

)3.0)(72.1)(103.11019.713.0(

=

+=

6. Long buildup Test First Straight is not Observed Generally, the second straight line is more likely to be

observed than the first one, which often is masked by near wellbore effects, such as wellbore storage. In Type 1 and Type 2 naturally fractured systems, where the matrix permeability is negligible, the fluid flows into the wellbore exclusively through the fractures. The first straight line will probably be too short and easily masked by inner wellbore effects.

The permeability and skin factor are calculated from Eqs. 7 and 8 respectively. The following equation provides a direct and accurate method for calculating , as long as the inflection point and the second straight line are observed and the matrix blocks are uniformly distributed:

mP inf22

10= .... (22)

P2inf (= 0.5p) is the pressure drop between the 2nd

semilog straight line and the inflection point along a vertical line parallel to the pressure axis.

The interporosity flow parameters is then calculated from

Eq. 14. If the inflection point is difficult to determine, then read

the starting-time of the second semilog straight line, tSL2, and use the following equation to estimate :

SPE 104056 5

)1(1027.5

)(

25

2

= +

SL

wmft

kt

rc..... (23)

EXAMPLE 2 Given the build up test data in Table 2 and the following

formation and fluid properties, estimate formation permeability, skin factor, , and .

q = 125 STB/D h = 17 ft tp = 1200 hr = 13.0% pwf = 211.20 psia rw = 0.30 ft = 1.72 cp B=1.054 RB/STB. ct =7.1910-6 psi-1 Solution The following data are read from Figure 4: tinf = 3.05 hr P2inf = 24 psi P1hr = 419 psi m=30 psi/cycle tSL2 =55 hr

From Equation 7:

mdk 25.72)30)(17(

)72.1)(054.1)(125(6.162 ==

From Equation 22:

025.010 30)24(2

==

It is possible to calculate (ct)f by:

86 104.2025.01

025.0)1019.7)(13.0()(

1)()(

=

=

=

ft

mtft

c

cc

From equation 8:

++= 23.32)3.0)(72.1)(8104.261019.713.0(

25.72log

30

8.2071513.1s

69.1=s

From Equation 14:

7

286

1036.2

025.01ln025.0

)05.3)(25.72()3.0)(72.1)(104.21019.713.0(3792

=

+=

From Equation 23:

7

5

286

1091.6

)025.01()55)(25.72)(1027.5(

)3.0)(72.1)(104.21019.713.0(

=

+=

(B) TDS Technique In 1993 Tiab introduced a technique10 for interpreting

loglog plots of the pressure and pressure derivative curves without using type curve matching. This technique utilizes the characteristic intersection points, slopes, and beginning and ending times of various straight lines corresponding to flow regimes strictly from loglog plots of pressure and pressure derivative data. Values of these points and slopes are then inserted directly in exact, analytical solutions to obtain reservoir and well parameters. This procedure for interpreting pressure tests, which is referred to as the Tiabs Direct Synthesis (TDS) technique offers several advantages over the conventional semilog analysis and type curve matching. It has been applied to over fifty different reservoir systems11-18, and hundreds of field cases.

1. Fracture Permeability The pressure derivative portion corresponding to the

infinite acting radial flow line is a horizontal straight line. This flow regime is given by10:

khBqPt R

6.70)'( = .. (24) The subscript R stands for radial flow. The formation

permeability is therefore:

RPthBqk)'(

6.70=

.... (25) where (tP')R is obtained by extrapolating the horizontal

line to the vertical axis. In order for the conventional semilog analysis and the TDS technique to yield the same value of k, the following equation must be true:

RPtm )'(303.2 = ..... (26)

2. Skin Factor The second radial flow line can also be used to calculate

the skin factor from10:

( )

+

=+

43.7)(

ln'

)(5.0

22

2

2

wmft

R

R

R

rckt

PtP

s (27) Where tR2 is any convenient time during the systems

radial flow regime (as indicated by the horizontal line on the pressure derivative curve, Figure 2) and (P)R2 is the value of P on the pressure curve corresponding to tR2. If the test is too short or the boundary is too close to the well to observe a well defined second straight line, then the skin factor can be estimated from the early-time horizontal straight line:

6 SPE 104056

( )

+

=+

43.71)(

ln'

)(5.0

21

1

1

wmftR

R

R

rckt

PtP

s (28)

Where tR1 is any convenient time during the early-time radial flow regime (as indicated by the horizontal line on the pressure derivative curve, Figure 2) and (P)R1 is the value of P on the pressure curve corresponding to tR1.

3. Interporosity Flow Coefficient The interporosity flow parameter can also be obtained

from the loglog plot of the derivative function (txP) versus test time4,5 by substituting the coordinates of the minimum point of the trough, tmin and (txP)min:

( )min

min2 ')(5.42

tPt

qBrch

o

wmft

= + ... (29)

The advantage of Eq. 29 over Eq. 14 is that it is independent of permeability and storage capacity ratio, and the coordinates of the minimum points are easier to determine than the inflection point on the semilog plot.

4. Storage Capacity Ratio The coordinates of the minimum point of the trough can be

used to derive two equations to calculate accurately the storage capacity ratio .

Pressure derivative Coordinate: Using the pressure

derivative coordinate of the minimum point and the radial flow regime (horizontal) line, the following equation provides a direct and accurate method for calculating :

= RPt

Pt)'(

)'(18684.0 min

10 ... (30) Equation 30 is derived by observing that:

2)()( min

PPtPt R= .... (30a)

Combining Equations 30a and 26 yields: ( )

=

=

R

R

R

PtPt

PtPtPt

mP

)()(18684.0

)(303.2)()(2

min

min... (30b)

Substituting Equation 30b into Equation 10 yields Equation 30. Equation 30 assumes wellbore storage and boundary effects do not influence the trough and the infinite acting radial flow line is well defined.

In conventional analysis this ideal case displays two well defined parallel lines with the inflection point equidistant of those two lines, which means that the fractures are uniformly distributed.

Minimum Time: Using the time coordinate of the minimum point, a less direct but just as accurate value of the storage capacity ratio can be obtained when wellbore storage is present from the following equation:

minDte = ..... (31) Where the dimensionless time at the minimum point is

calculated from:

min2min )(

0002637.0t

rc

kt

wmftD

=+

... (32)

Solving explicitly for Eq. 31 yields19:

15452.6

)ln(5688.39114.2

=

SS NN . (33) Where the parameter NS is given by:

minDtS eN

= (34) Eq. 34 is obtained by assuming values of , from 0 to 0.5,

then values of = NS were plotted against . The resulting curve was curve-fitted. Note that Eq. 33 can also be used in the semilog analysis since tmin = tinf.

It is recommended that both methods be used for comparison purposes. If the radial flow regime line on the derivative curve is not well defined due to a combination of inner and/or outer boundary effects or a short test, but the minimum of the trough is well defined, then Eqs. 29 and 33 should be used to calculate, respectively, and .

EXAMPLE 3 Tiab Direct Synthesis technique is applied to Example 2. Figure 5 is plotted with data from Table 2 and the

respective pressure derivative. From Figure 5 the following data can be read: PR = 274.51 psi tR= 156.51 psi tmin= 3.05 hours (tP)min = 1.3 psi (tP)R = 13 psi te=0.018 hr Pe = 13 psi Wellbore storage coefficient is calculated by10:

psibblPtqBC

e

/1060.713018.0

24)054.1)(125(

243=

=

=

From Equation 25:

mdk 39.72)13)(17(

)054.1)(72.1)(125(6.70 == From Equation 27:

74.1

43.7)3.0)(72.1)(104.21019.713.0(

)51.156)(39.72(ln13

51.2745.0 286

=

+

+= s

s

SPE 104056 7

From Equation 29:

7

286

1002.2

05.33.1

)054.1)(125()3.0)(104.21019.713.0)(17(5.42

=

+=

From Equation 33 the storage capacity ratio can be

calculated in the presence of wellbore storage:

024.0924.05452.6

)924.0ln(5688.39114.2

1

=

=

Table 3 is a comparison of the TDS results with that of

conventional method.

FRACTURE POROSITY AND COMPRESSIBILTY Once is estimated, the fracture porosity can be estimated

if matrix porosity, m, total matrix compressibility, ctm, and total fracture compressibility, ctf, are known, as follows:

mtf

tmf c

c

= 1 .. (35) Fracture compressibility may be different from matrix

compressibility by an order of magnitude. Naturally fractured reservoirs in Kirkuk field (Iraq) and Asmari field (Iran) have fracture compressibility ranging from 4x10-4 to 4x10-5 psi-1. In Grozni field (Russia) ctf ranges from 7x10-4 to 7x10-5. In all these reservoirs ctf is 10 to 100 folds higher than ctm. Therefore the practice of assuming ctf = ctm is not acceptable.

The fracture compressibility can be estimated from the following expression9:

( ) ( )P

kkPkk

c fiffiftf

=3

/1/1 3/1.. (36)

=fik Fracture permeability at the initial reservoir pressure ip

=fk Fracture permeability at the current average reservoir pressure.

Combining Equations 35 and 36 yields19:

( )3/1)/(11 fiftmmf kk Pc = .. (37) In deep naturally fractured reservoirs, fractures and the

stress axis on the formation generally are vertically oriented. Thus when the pressure drops due to reservoir depletion, the fracture permeability reduces at a lower rate than one would expect. In Type-2 naturally fractured reservoirs, where matrix porosity is much greater than fracture porosity, as the reservoir pressure drops the matrix porosity decreases in favor of fracture porosity9. This not the case in Type-1 naturally fractured reservoirs, particularly if the matrix porosity is very low or negligible.

For fractured reservoirs and, indeed, all highly anisotropic reservoirs, the geometric mean is currently considered the most appropriate of the three most common averaging techniques (arithmetic, harmonic and geometric). Therefore, a representative average value of the effective permeability of a

naturally fractured reservoir may be obtained from the geometric mean of kmax and kmin as illustrated in Figure 6.

minmaxkkk = .. (38) where kmax = maximum permeability measured in the direction

parallel to the fracture plane (Figure 6), thus kmax kfracture kmin = minimum permeability measured in the direction

perpendicular to the fracture plane (Figure 6), thus kmin kmatrix Substituting kf and km for, respectively, kmax and kmin,

Equation 38 becomes:

mf kkk = .. (39) The fracture permeability can therefore be estimated from:

mf k

kk2

= .. (40) Where km is the matrix permeability, which is measured

from representative cores and k is the mean permeability obtained from pressure transient tests. Combining equations 36 and 40 yields:

( )

Pkk

c itf =

3/2/1 .. (41)

Where ki = average permeability obtained from a transient test run

when the reservoir pressure was at or near initial conditions Pi and

k = average permeability obtained from a transient test at the current average reservoir pressure.

PPP i = Combining Equations 41 and 35 yields19:

( )3/2)/(11 itmmf kkPc

= ........ (42)

Matrix permeability is assumed to remain constant between the two tests. Note that equations 37 and 42 are also valid for calculating fracture porosity change between two consecutive pressure transient tests, and therefore

21 PPP = . The time between the two tests must be long enough for the fractures to deform significantly in order to determine an accurate value of ctf. Table 5 shows pressure transient analysis in Cupiaga field, a naturally fractured reservoir in Colombia22. The reduction in permeability for well 1 is about 13% and the change in pressure is 344 psi from 1996 to 1997. This type of data can be used in order to estimate f from Eq. 42. Eq. 37 should yield a more accurate value of fracture porosity than Eq. 42, as the latter assumes Eq. 39 is always applicable.

Substituting the values of ffm andkk ,, into the following equation should yield approximately the same value of the effective permeability obtained from well testing20:

ffm kkk + .. (43)

8 SPE 104056

Eq. 43 should only be used for verification purposes. The fracture width or aperture may be estimated20 from

t

ff

kw 33=

........ (44)

where: fracture width = microns, permeability = mD, porosity = fraction, and storage capacity = fraction.

EXAMPLE 4 Pressure tests in the first few wells located in a naturally

fractured reservoir yielded a similar average permeability of the system of 82.5 mD. An interference test also yielded the same average reservoir permeability, which implies that fractures are uniformly distributed. The total storativity, (ct)m+f = 1x10-5 psi-1 was obtained from this interference test. Only the porosity, permeability and compressibility of the matrix could be determined from the recovered cores.

The pressure data for the well are given in Table 4. The pressure drop from the initial reservoir pressure to the current average reservoir pressure is 300 psi. The characteristics of the rock, fluid and well are given below:

q = 3000 STB/D h = 25 ft m = 10% rw = 0.4 ft = 1 cp B=1.25 RB/STB. ctm=1.3510-5 psi-1 km=0.10 mD 1 - Using conventional semilog analysis and TDS

technique, calculate the current formation permeability, storage capacity ratio, and fluid transfer coefficient

2 Estimate the three fracture properties: permeability, porosity and width.

Solution 1(a) Conventional method From Figure 7: P = 130 psi m=325 psi/cycle tinf =2.5 hrs The average permeability of the formation is estimated

from the slope of the semilog straight line. Using Equation 7 yields: ( )( )( )

( )( ) mDk 05.7525325125.130006.162 ==

Fluid storage coefficient is estimated using Equation 10:

39.010 )325/130( == The storage coefficient of 0.39 indicates that the fractures

occupy 39% of the total reservoir pore volume. The inter-porosity fluid transfer coefficient is given by

Equation 14: ( )( )( ) 5

25

1019.139.01ln39.0

5.205.75)4.0)(1(1013792

=

=

1(b) TDS technique From Figure 8, the following characteristic points are read: tmin = 2.5 hrs (tP)R = 146 psi (tP)min = 70.5 psi

Using the TDS technique, the value of k is obtained from Equation 25:

( )( )( )( )( ) mDk 53.721462525.1130006.70 ==

The inter-porosity fluid transfer coefficient is given by

Equation 29: 5

25

1028.15.2

)5.70()25.1)(3000(

)4.0)(101)(25)(5.42( =

=

Since the two parallel lines are well defined the storage coefficient is calculated from Equation 30

35.010 1465.70

18684.0 ==

The conventional semilog analysis yields similar values of

k, and as the TDS technique. The main reason for this match is that both parallel straight lines are well defined.

2 Current properties of the fracture (a) The fracture permeability is calculated from Equation.

40:

mDkkk

mf 606,5210.0

53.72 22 === The fracture permeability at initial reservoir pressure is:

mDkk

km

ifi 062,6810.0

5.82 22 ===

(b) The fracture porosity In fractured reservoirs with deformable fractures, the

fracture compressibility changes with declining pressure. The fracture compressibility can be estimated from Equation 41:

( ) 143/2 102.5300

062,68/606,521 == psictf The compressibility ratio is:

5.381035.1102.5

5

4

==

tm

tf

cc

Thus, the fracture compressibility is more than 38.5 folds

higher than the matrix compressibility, or tmtf cc 5.38= . The fracture porosity from Equation 42 is:

%14.000139.05.381.0

35.0135.0 =

=f The total porosity of this naturally fractured reservoir is:

1014.00014.010.0 =+=+= fmt Substituting the values of ffm andkk ,, into Equation

43: mDkkk ffm 7.73606,520014.01.0 =+=+

This value is approximately the same value of the effective permeability obtained from well testing (72.53 mD). The fracture width or aperture may be estimated from Equation 44:

mmmicronsw f 212.02121014.035.033606,52 ===

SPE 104056 9

The fracture width is a useful parameter for identifying the nature of fracturing in the reservoir.

Conclusions 1. The inflection point on the semilog plot of well pressure

versus test time and the corresponding minimum point on the trough of the pressure derivative curve are unique points that can be used to characterize a naturally fractured reservoir.

2. The interporosity flow parameter can be accurately obtained from the conventional semilog analysis if the inflection point is well defined and the new proposed equation is utilized. The equation is valid for both pressure drawdown and pressure buildup tests.

3. Two new equations are introduced for accurately calculating the storage capacity ratio from the coordinates of the minimum point of the trough on the pressure derivative curve.

4. For a short test, in which the late-time straight line is not observed, the storage capacity ratio and the interporosity flow coefficient can both be calculated from the inflection point.

5. For a long test, in which the early-time straight line is not observed, due to near-wellbore effects, the storage capacity ratio can also be calculated from the inflection point.

6. A new equation is proposed for calculating fracture porosity, as a function of reservoir compressibility.

7. The practice of assuming the total compressibility of the matrix (ctm) is equal to the total compressibility of the fracture (ctf) should be avoided. From field observations, ctf is several folds higher than ctm.

Nomenclature B oil volumetric factor, rb/STB c system compressibility, psi-1 h formation thickness, ft HT Horner time, dimensionless k permeability, md m semilog slope, psi/log cycle Pws well shutin pressure, psi Pwf well flowing pressure, psi q oil flow rate, BPD rw wellbore radius, ft s skin factor tp producing time before shut-in, hrs wf Fracture width in microns Greek Symbols P vertical distance between the two semilog straight

lines, psi Geometry parameter, 1/L2 Porosity, dimensionless P1inf Pressure drop between the 1st semilog strigth line and

the inflection point, psi P2inf Pressure drop between the 2nd semilog strigth line and

the inflection point, psi t shut-in time, hrs Interporosity flow parameter, dimensionless

Viscosity, cp Storage capacity ratio, dimensionless Subscripts i initial o oil D dimensionless f fracture, fissure m matrix t total inf inflection point min minimum 1 1st semilog straight line 2 2nd semilog straight line 1hr 1 hour References 1. Nelson, R.: Geologic Analysis of Naturally Fractured

Reservoirs. Gulf Professional Publishing, 2nd Edition. 2001 2. Ershaghi, I.: Evaluation of Naturally Fractured Reservoirs.

IHRDC, PE 509, 1995. 3. Warren, J.E. and Root, P.J.: The Behavior of Naturally

Fractured Reservoirs. Soc. Pet. Eng. J. (Sept. 1963): 245-255. Trans. AIME, 228.

4. Engler, T. and Tiab, D.: Analysis of Pressure and Pressure Derivative without Type Curve Matching, 2. Naturally Fractured Reservoirs. Journal of Petr. Sci. and Eng. 15 (1996):127-138.

5. Engler, T. and Tiab, D.: Analysis of Pressure and Pressure Derivative without Type Curve Matching, 5. Horizontal Well Tests in Naturally Fractured Reservoirs. Journal of Petr. Sci. and Eng. 15 (1996); 139-151.

6. Engler, T. and Tiab, D.: Analysis of Pressure and Pressure Derivative without Type Curve Matching - 6. Horizontal Well Tests in Anisotropic Media. Journal of Petroleum Science and Engineering, Vol. 15 (Aug. 1996) N0. 2-4, 153-168.

7. Uldrich, D.O. and Ershaghi, I.: A Method for Estimating the Interporosity Flow Parameter in Naturally Fractured Reservoirs: Paper SPE 7142, Proceedings, 48th SPE-AIME Annual California Regional Meeting held in San Francisco, CA, Apr. 12-14, 1978.

8. Bourdet, D. and Gringarten. AC.: Determination of fissured volume and block size in fractured reservoirs by type-curve analysis. Paper SPE 9293. Soc. Pet. Eng., Annu. Tech. Conf., Dallas, TX, Sept. 21-24, 1980,

9. Saidi, M. A.: Reservoir Engineering of Fractured Reservoirs. Total Edition Presse, 1987.

10. Tiab, D.: "Analysis of Pressure and Pressure Derivative without Type-Curve Matching - 1. Skin and Wellbore Storage". Journal of Petroleum Science and Engr., Vol. 12, No. 3 (January, 1995) 171-181.

11. Jongkittinarukorn, K. and Tiab, D.: Analysis of Pressure and Pressure Derivative without Type Curve Matching - 6. Vertical Well in Multi-boundary Systems. Proceedings, CIM 96-52, 47th Annual Tech. Meeting, Calgary, Canada, June 10-12, 1996.

12. Jongkittinarukorn, K. and Tiab, D.: Analysis of Pressure and Pressure Derivatives without Type Curve Matching - 7. Horizontal Well in a Closed Boundary Systems, Proceedings, CIM 96-53, 47th Annual Tech. Meeting, Calgary, Canada, June 10-12, 1996.

13. Tiab, D., Azzougen, A., F.H., Escobar, and S. Berumen: Analysis of Pressure Derivative Data of Finite-Conductivity Fractures by the Tiabs Direct Synthesis Technique. Paper SPE 52201. Proceedings, SPE Mid-Continent Operations Symposium, Oklahoma City, 28 31 March 1999; Proceedings

10 SPE 104056

SPE Latin American & Caribbean Petr. Engr. Conf., Caracas, Venezuela, 2123 April 1999, 17 pages.

14. Mongi, A. and Tiab, D.: Application of Tiabs Direct Synthesis Technique to Multi-rate Tests, SPE/AAPG 62607, Proceedings, Western Regional Meeting, Bartlesville, California, 19-23 June 2000.

15. Benaouda, A. and Tiab, D.: Application of Tiabs Direct Synthesis Technique to Gas Condensate Wells. Proceedings, SPE Permian Basin Conference, Texas, May 2001

16. Jokhio, S.A., Hadjaz, A. and Tiab, D.: Pressure falloff Analysis in Water Injection Wells Using the Tiabs Direct Synthesis Technique. Paper SPE 70035, Proceedings, SPE Permian Basin Conference, Midland, Texas, May 15-16, 2001.

17. Bensadok A. and Tiab, D.: Interpretation of Pressure Behavior of a Well between Two Intersecting Leaky Faults Using Tiabs Direct Synthesis (TDS) Technique. CIP2004-123, Proceedings, Canadian International Petroleum Conference, 7 10 June 2004

18. Chacon, A., Djebrouni, A. and Tiab, D.: Determining the Average Reservoir Pressure from Vertical and Horizontal Well Test Analysis Using Tiabs Direct Synthesis Technique. Paper SPE 88619, Proceedings, Asia Pacific Oil and Gas Conference and Exhibition, Perth, Australia, Oct. 18-20, 2004.

19. Tiab, D. and E.C. Donaldson: Petrophysics: theory and practice of measuring reservoir rock and fluid transport properties. Gulf professional Publications, 2nd Edition, 2004.

20. Bona, N., Radaelli, F., Ortenzi, A., De Poli, A., Pedduzi, C. and Giorgioni, M: Integrated Core Analysis for Fractured Reservoirs: Quantification of the Storage and Flow Capacity of Matrix, Vugs, and Fractures. SPERE, Aug. 2003, Vol.6, pp.226-233.

21. Stewart G. Ascharsobbi F. Well test interpretation for Naturally Fractured Reservoirs. Paper SPE 18173.

22. Giraldo L. A., Chen Her-Yuan, Teufel L. W. Field Case Study of Geomachanical Impact of Pressure Depletion in the Low-Permeability Cupiaga Gas-Condensate Reservoir. SPE 60297. SPE Rocky Mountain Regional/Low Permability Reservoirs Symposium, Denve, CO, March 12-15, 200.

SI Metric Conversion Factors bbl x 1.589873 E-01 = m3 cp x 1.0* E-03 = Pa-s ft x 3.048* E-01 = m ft2 x 9.290304 E-02 = m2 psi x 6.894757 E+00 = kPa *Conversion factor is excat.

Table 1. Pressure data for Example 1 Time

hours

Pressure

psi

P

psi

Horner Time

0.0000 211.20 0.00

0.0010 390.73 179.53 1200001.00

0.0023 404.32 193.12 521740.13

0.0040 413.00 201.80 300001.00

0.0062 419.73 208.53 193549.39

0.0090 425.39 214.19 133334.33

0.0128 430.36 219.16 93751.00

0.0176 434.81 223.61 68182.82

0.0239 438.82 227.62 50210.21

0.0320 442.43 231.23 37501.00

0.0426 445.66 234.46 28170.01

0.0564 448.48 237.28 21277.60

0.0743 450.87 239.67 16151.74

0.0976 452.84 241.64 12296.08

0.1279 454.36 243.16 9383.33

0.1673 455.46 244.26 7173.74

0.2190 456.20 245.00 5480.45

0.2850 456.65 245.45 4211.53

0.3720 456.90 245.70 3226.81

0.4840 457.03 245.83 2480.34

0.6300 457.11 245.91 1905.76

0.8200 457.18 245.98 1464.41

1.0670 457.27 246.07 1125.65

1.3890 457.39 246.19 864.93

1.8060 457.55 246.35 665.45

2.3500 457.75 246.55 511.64

3.0500 458.01 246.81 394.44

SPE 104056 11

Table 2 Pressure data for Example 2 Time hours

Pressure psi

Time hours

Pressure psi

0.0000 211.20 1.8060 456.85 0.0010 212.07 2.3500 457.47 0.0023 213.19 3.0500 457.80 0.0040 214.64 3.9700 458.15 0.0062 216.50 5.1600 458.58 0.0090 218.90 6.7100 459.14 0.0128 221.98 8.7300 459.84 0.0176 225.91 11.3500 460.73 0.0239 230.92 14.7600 461.85 0.0320 237.26 19.1800 463.23 0.0426 245.22 24.9400 464.92 0.0564 255.11 32.4200 466.95 0.0743 267.26 42.1500 469.35 0.0976 281.94 54.8000 472.11 0.1279 299.31 71.2400 475.21 0.1673 319.31 92.6100 478.57 0.2190 341.53 120.3900 482.11 0.2850 365.13 156.5100 485.71 0.3720 388.74 203.5000 489.29 0.4840 410.60 264.5000 492.77 0.6300 428.91 343.9000 496.09 0.8200 442.40 447.0000 499.23 1.0670 450.83 581.0000 502.15

1.3890 455.12 720.0000 504.36

Table 3. Results for examples 2 and 3 Parameter MDH TDS

k, md 72.25 72.39

s 1.69 1.74

2.3610-7 2.0210-7 0.025 0.024

5,700

5,800

5,900

6,000

6,100

6,200

6,300

6,400

6,500

101001,00010,000100,000

Shut

-in P

ress

ure,

psi

a

Inflection point

P

m

tH-inf

(tp + t)/t tH=Horner time, Figure 1- Semilog pressure behavior of a naturally fractured reservoir

Table 4. Pressure data for Example 4 Time hours

Pws psi

P psi

(txP') psi

0 4473.0 0.093 4373.4 99.60 84.473 0.177 4299.1 173.90 133.483 0.260 4246.1 226.90 146.776 0.343 4203.6 269.40 151.595 0.427 4173.8 299.20 157.618 0.510 4139.7 333.30 150.295 0.593 4118.5 354.50 141.355 0.677 4103.5 369.50 111.676 0.760 4086.4 386.60 99.694 0.927 4075.4 397.60 95.720 1.093 4060.3 412.70 87.234 1.260 4043.1 429.90 84.384 1.427 4032.2 440.80 76.719 2.427 3997.0 476.00 70.469 3.427 3971.3 501.70 77.268 4.427 3948.3 524.70 87.168 5.427 3931.6 541.40 95.595 6.427 3917.1 555.90 108.303 7.427 3898.4 574.60 122.336 9.427 3865.3 607.70 142.426 12.43 3824.2 648.80 137.651 14.43 3804.1 668.90 136.857 20.43 3758.7 714.30

10

100

1000

10 100 1000 10000

(txP')mintmin

Minimumpoint

Time, hr

Pres

sure

and

pre

ssur

e de

rivat

ive,

psi

(txP')R2(txP')R1

tR2

(P)R2

tR1

(P)R1

10

100

1000

10 100 1000 10000

(txP')mintmin

Minimumpoint

Time, hr

Pres

sure

and

pre

ssur

e de

rivat

ive,

psi

(txP')R2(txP')R1

tR2

(P)R2

tR1

(P)R1

Figure 2 P and pressure derivative plot for a naturally fractured reservoir

12 SPE 104056

380

400

420

440

460

480

500

520

0.001 0.01 0.1 1 10

Time, hr

Shut

in p

ress

ure,

psi

P1inf = 33 psi

m = 35.67 psi/cycle

Inflection point t inf= 0.63 hr

tEL1 = 0.012 hr

P1hr =497 psi

Figure 3 Conventional MDH plot for Example 1

200

250

300

350

400

450

500

550

0.001 0.01 0.1 1 10 100 1000

Time, hr

Shut

in p

ress

ure,

psi

Inflection point t inf= 3.05 hr

P1hr = 419 psi P2inf = 24 psi

m = 30 psi/cycle

tSL2 = 55 hr

Figure 4 -Conventional MDH plot for Example 2

0.1

1

10

100

1000

0.001 0.01 0.1 1 10 100 1000

Time, hr

P a

nd (t

x P'

), ps

i

PR = 274.51 psi

tR = 156.51 hr

tmin = 3.05 hr(tP')min = 1.3 psi

(tP')R = 13 psi

te = 0.018 hr(P)e = 13 psi

Figure 5- Log-log plot for Example 3

Wf

L

h = hf

kmin

kmax

W

Figure 6- Maximum and Minimum Permeability

3600

3700

3800

3900

4000

4100

4200

4300

4400

0.01 0.1 1 10 100Time, hr

Pres

sure

, psi

Inflection point tinf = 2.5 hr

m = 325 psi/cycleP = 130 psi

Figure 7- Conventional MDH plot for Example 4

10

100

1000

0 1 10 100Time, hours

P a

nd (t

x P'

), ps

i

PR = 669 psi

tR = 14.43 hr

tmin = 2.5 hr(tP')min = 70.5 psi

(tP')R = 146psi

Figure 8- Pressure Derivative plot for Example 4

SPE 104056 13

Table 5. Pressure Transient Analysis of Selected Cupiaga wells22

Test Type

Date

P* BHFP

(psi)

Global

Skin

Particular

Types

of Skin

K (md)

h(ft)

Comments/Remarks

Well 1

Pre-Frac Test

1996

6004 3000 91.3 Mechanical 48

16.4

171

Homogenous reservoir model. The turbulence factor is quite large due to non-darcy flow (high rates) combined with the condensate banking

Well 1

Post-Frac Test

1996

6004 3200 38 Mechanical 18

16.4

171

Homogeneous reservoir model. Rate dependent Skin was observed. The effects of the condensate banking are observed in the GOR response, at higher drawdowns the GOR increased.

Well 1

Post-Frac Test

1997

5660 3410 20 14.2 171

Homogeneous reservoir model. Some drainage area is still above the dew point.

Well 1

Drainage area

Below dew-point

1998

5150 3000 5 4.3 171

Homogeneous reservoir model. Derivative curve indicates radial flow with a low value of gas effective permeability out to a radius of 800 ft followed by an increase in effective permeability further out. This is interpreted as being due to liquid condensate drop-out.

Well 1

Injector PFO

1998

5050 5380 3 Zero skin for other

than non -Darcy

11.4 171

Homogeneous reservoir model. A small negative mechanical skin is suggested possibly due to activation of fractures by injecting pressure/temperature. The well is in an under-injecting situation.

Well 2

1995

6267 3100 Total 19

Total Mechanical

0 8.35

429

Three layer model.

Well 2

1998

4600 2772 1 Includes Turbulent and Condensate

effects

0.6

993

Homogeneous reservoir model. Entire drainage area is below dew point pressure. The explanation for the reduced skin is that due to rate dependent relative permeability and pressure dependent saturation, the condensate impact on relative permeability is less close to the wellbore than deeper in the reservoir.

Well 2

Injector PFO

1999

5500? 6947 1(@100 ft) 2.9 993

(@100 ft)

Homogeneous reservoir model. P*is difficult to estimate because of the variation of Kh with radius. Given that the pressure at 100 ft radius is well above dew point, it is of some concern that Kh has not been fully restored. At a radius of 350 ft the Kh reduces below 2000 md-ft.

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