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
Trang 1Copyright 2006, Society of Petroleum Engineers
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Abstract
The storage capacity ratio, ω, measures the flow capacitance
of the secondary porosity and the interporosity flow
parameter, λ, is related to the heterogeneity scale of the
system Currently, both parameters λ and ω 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 ω 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 λ
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
Trang 2dolomite, 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, ω and λ, were
introduced by Warren and Root3 to characterize naturally
fractured reservoirs These dimensionless parameters λ and ω
are mathematically expressed as3:
m t f t
f t t
t
f
t
c c
c c
c
) ( ) (
) ( )
(
)
(
φ φ
φ φ
φ
ω
+
=
2
2
m
w
f
m
x
r
k
k
α
λ=
……… (2) The geometry parameter, α, is defined as:
)
2
(
= n n
α
……… (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
-t Ei
-t Ei t
=
P D D+ + ⎜⎜⎛− D ⎟⎟⎞− ⎜⎜⎛− D ⎟⎟⎞ +
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
) 1 ( ) 1 ( 80908 0
ln
2
1
ω
λ ω
ω
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟⎟
⎞
⎜⎜
⎛−
⎟⎟
⎞
⎜⎜
⎛−
×
) 1 ( exp ) 1 ( exp 1 2
1 '
ω ω
λ ω
λ
-t +
-t
-= P
The second pressure derivative of the dimensionless pressure equation is:
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟⎟
⎞
⎜⎜
⎛
−
−
⎟⎟
⎞
⎜⎜
⎛
−
−
−
×
) 1 ( exp ) 1 ( exp 1 ) 1 ( 2 )' ' (
ω
λ ω
ω
λ ω ω
λ
-t
-t
= P
(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:
m
qB
kh=162.6 oμ……… (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 << (kh)f then kh can be approximated by (kh)f
In Type 3 reservoirs, both straight lines are related to fracture flow and matrix flow, the product kh in Eq 7 is therefore equivalent to (kh)t
2 Skin Factor
The skin factor is obtained using conventional technique, i.e.:
⎦
⎤
⎢
⎢
⎣
⎡
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
− Δ
=
+
23 3 log
) ( 1513 1
2 1
w m f t
hr
r c
k m
P s
μ
( P)1hr is taken from the second straight line
3 Fracture Storage Capacity Ratio
The vertical distance between the two semilog straight lines, δP, may be used to estimate3
the storage capacity ratio, ω:
⎟
⎠
⎞
⎜
⎝
⎛−
=
m P
δ
ω exp 2.303 ……… (9)
or
m
P /
10 δ
ω= − ……… (10)
In Type 4 naturally fractured reservoirs the value of is close to unity The sugar cube model is not realistic in Type 4
Trang 3fractured 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
(tD×PD’)’ = 0 The dimensionless time at which this minimum
point occurs is given by the following expression4, 5, 6
⎥
⎦
⎤
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
=
ω λ
ln
min
D
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:
⎥
⎦
⎤
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
=
ω λ
ln
inf
D
The dimensionless time is defined as:
2
inf inf
) (
0002637
0
w m f t
D
r c
t k t
μ
φ +
Where tinf = tmin Combining Eqs 12 and 13 and solving
for λ, yields a new relationship for the interporosity flow
parameter:
⎥
⎦
⎤
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
ω ω μ φ
λ 3792( ) ln 1
inf
2
t
k
r
c t f m w
……… (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 )
( inf
p
H
t
t ……… (15)
Where (HT) is the Horner time (tp+ t)/ t or the effective
Horner time tp t/(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 ω 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 ω 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:
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛ + +
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ Δ
Δ +
−
=
ω
ω
1 log log
t
t t m P
Extrapolating the first straight line to a Horner time of unity, i.e (tp+ t)/ t = 1, where P ws =P FF1, then the storage capacity ratio can be calculated from:
m P P
m P P
FF i
FF i
/ ) (
/ ) (
1 1
10 1
10
−
−
−
=
P FF1 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 P FF1 will always be greater than (by a value equal
to p) the average pressure, P i and P*, since normally the
second parallel line is used to estimate these three pressure
values If the initial reservoir pressure P i 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
Trang 4blocks, the inflection point is at equal distance between the
two parallel lines Therefore
m
P1inf
2
10
Δ
−
=
ω ……… ……… (18)
Where:
P1inf (= 0.5 P) 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 :
ω ω μ
φ
013185
0
)
(
1
2
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
EL
w m f
t
kt
r c
……… ……… (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:
⎦
⎤
⎢
⎢
⎣
⎡
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
− Δ
=
+
23 3 log
) ( ) (
1513
.
1
2 1
1
w m f t FF
i hr
r c
k m
P P P
s
μ
or
⎦
⎤
⎢
⎢
⎣
⎡
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
− Δ
− Δ
=
+
23 3 log
2 ) (
1513
.
1
2 inf
1 1
w m f t
hr
r c
k m
P P
s
μ
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, λ, and ω from
q = 125 STB/D h = 17 ft
tp = 1200 hr φ = 13.0%
pwf = 211.20 psia rw = 0.30 ft
µ = 1.72 cp B=1.054 RB/STB
ct =7.19×10-6
psi-1 Solution
The following data are read from Figure 3:
tinf = 0.63 hr ΔP1inf = 33 psi
P1hr = 497 psi m=35.67 psi/cycle
tEL1 = 0.012 hr
From Equation 7:
md
) 17 )(
67 35 (
) 72 1 )(
054 1 )(
125 ( 6
= From Equation 21 the storage capacity ratio is:
014 0
10 35 67
) 33 ( 2
=
= −
ω Using equation 1, we can calculate (φct)f:
8 6
10 3 1 014 0 1
014 0 ) 10 19 7 )(
13 0 ( ) (
1 ) ( ) (
−
⎠
⎞
⎜
⎝
⎛
−
×
=
⎟
⎠
⎞
⎜
⎝
⎛
−
=
f t
m t f t
c
c c
φ
ω
ω φ
φ
From equation 21 the skin factor is:
89 0
23 3 ) 3 0 )(
72 1 )(
10 3 1 10 19 7 13 0 (
7 60 log
67 35 ) 33 2 8 285 ( 1513 1
2 8
6
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+
⎟
⎞
⎜
⎛
× +
×
×
−
×
−
s s
From Equation 14, the interporosity flow parameter is:
7
2 8
6
10 7 8
014 0
1 ln 014 0 )
63 0 )(
7 60 (
) 3 0 )(
72 1 )(
10 3 1 10 19 7 13 0 ( 3792
−
−
−
×
=
⎥
⎤
⎢
⎡
⎟
⎜
×
×
× +
×
×
= λ λ
From Equation 19:
7
2 8
6
10 1 2
014 0 ) 014 0 1 ( )
012 0 )(
7 60 )(
013185 0 (
) 3 0 )(
72 1 )(
10 3 1 10 19 7 13 0 (
−
−
−
×
=
−
×
⎟⎟
⎞
⎜⎜
=
λ λ
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:
m
P2inf
2
10
Δ
−
=
ω ………… ……… (22)
P2inf (= 0.5 p) 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 :
Trang 5) 1 ( 10
27
5
)
(
2 5
2
ω μ
φ
⎠
⎞
⎜
⎜
⎝
⎛
×
SL
w m f
t
kt
r c
…… ………… …… (23)
EXAMPLE 2
Given the build up test data in Table 2 and the following
formation and fluid properties, estimate formation
permeability, skin factor, λ, and ω
q = 125 STB/D h = 17 ft
tp = 1200 hr φ = 13.0%
pwf = 211.20 psia rw = 0.30 ft
µ = 1.72 cp B=1.054 RB/STB
ct =7.19×10-6
psi-1
Solution
The following data are read from Figure 4:
tinf = 3.05 hr ΔP2inf = 24 psi
P1hr = 419 psi m=30 psi/cycle
tSL2 =55 hr
From Equation 7:
md
) 30
)(
17
(
) 72 1 )(
054
.
1
)(
125
(
6
.
=
From Equation 22:
025 0
10 30
)
24
(
2
=
= −
ω
It is possible to calculate (φct)f by:
8 6
10 4 2 025 0 1
025 0 ) 10 19 7
)(
13
0
(
)
(
1 )
(
)
(
−
⎠
⎞
⎜
⎝
⎛
−
×
=
⎟
⎠
⎞
⎜
⎝
⎛
−
=
f
t
m
t
f
t
c
c
c
φ
ω
ω φ
φ
From equation 8:
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
−
× +
−
×
×
−
2 ) 3 0 )(
72 1 )(
8 10 4 2 6 10 19 7 13 0 (
25 72 log
30
8
.
207
1513
.
1
s
69
.
1
=
s
From Equation 14:
7
2 8
6
10
36
.
2
025 0
1 ln 025 0 )
05 3 )(
25 72 (
) 3 0 )(
72 1 )(
10 4 2 10 19
.
7
13
.
0
(
3792
−
−
−
×
=
⎥
⎤
⎢
⎡
⎟
⎠
⎞
⎜
⎝
⎛
×
× +
×
×
=
λ
λ
From Equation 23:
7
5
2 8
6
10 91 6
) 025 0 1 ( )
55 )(
25 72 )(
10 27 5 (
) 3 0 )(
72 1 )(
10 4 2 10 19 7 13 0 (
−
−
−
−
×
=
−
×
⎟⎟
⎞
⎜⎜
⎛
×
× +
×
×
=
λ λ
(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 Tiab’s 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:
kh
B q P
t R 70.6 μ
) ' ( ×Δ = ……… … (24)
The subscript “R” stands for radial flow The formation permeability is therefore:
R
P t h
B q k
) ' (
6 70 Δ
×
……… …… … (25)
where (t×ΔP')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:
R
P t
m=2.303( ×Δ ') ……….……… (26)
2 Skin Factor
The second radial flow line can also be used to calculate the skin factor from10:
⎤
⎢
⎢
⎣
⎡
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
− Δ
×
Δ
=
+
43 7 )
(
ln '
) ( 5
2 2
w m f t R R
R
r c
kt P
t
P s
μ
Where tR2 is any convenient time during the system’s radial flow regime (as indicated by the horizontal line on the pressure derivative curve, Figure 2) and (ΔP)R2 is the value of
ΔP 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:
Trang 6( ) ⎥⎥⎦
⎤
⎢
⎢
⎣
⎡
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
− Δ
×
Δ
=
+
43 7 1 )
(
ln '
) (
5
1
1
ω μ
φ t f m w
R R
R
r c
kt P
t
P
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 (ΔP)R1 is the value of
ΔP 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 (tx P’) versus
test time4,5 by substituting the coordinates of the minimum
point of the trough, tmin and (tx P’)min:
min min 2
' )
(
5
42
t
P t qB
r c h
o
w m f
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
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 :
⎟⎟
⎞
⎜⎜
⎛ Δ
× Δ
×
−
−
P t
) ' ( ) ' ( 1 8684
.
10
Equation 30 is derived by observing that:
2 ) (
)
(t×ΔP′ R− t×ΔP′ min =δP
……… … (30a)
Combining Equations 30a and 26 yields:
⎥
⎦
⎤
⎢
⎣
⎡
′ Δ
×
′ Δ
×
−
=
′ Δ
×
′ Δ
×
−
′ Δ
×
=
R
R R
P t
P t
P t
P t P t
m
P
) (
) ( 1
8684
0
) ( 303 2
) ( ) (
2
min
min δ
………….……… (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:
min
D
t
ω
ω = − ……… ……… (31) Where the dimensionless time at the minimum point is calculated from:
min 2 min
) (
0002637
0
t r c
k t
w m f t
⎠
⎞
⎜
⎜
⎝
⎛
=
Solving explicitly for Eq 31 yields19:
1
5452 6 ) ln(
5688 3 9114 2
−
⎟⎟
⎞
⎜⎜
⎛
−
−
=
S
N
Where the parameter N S is given by:
min
D
t
N = λ ……… (34)
Eq 34 is obtained by assuming values of , from 0 to 0.5, then values of = N S 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 (t× P’)min = 1.3 psi (t× P’)R = 13 psi te=0.018 hr
Pe = 13 psi Wellbore storage coefficient is calculated by10:
psi bbl P
t qB C
e
/ 10 60 7 13
018 0 24
) 054 1 )(
125 ( 24
3
−
×
=
⎟
⎠
⎞
⎜
⎝
⎛
=
⎟
⎠
⎞
⎜
⎝
⎛ Δ
=
From Equation 25:
md
) 13 )(
17 (
) 054 1 )(
72 1 )(
125 ( 6
= From Equation 27:
74 1
43 7 ) 3 0 )(
72 1 )(
10 4 2 10 19 7 13 0 (
) 51 156 )(
39 72 ( ln
13 51 274 5
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+
⎟
⎞
⎜
⎛
× +
×
×
−
s s
Trang 7From Equation 29:
7
2 8 6
10
02
.
2
05 3 3 1 )
054 1 )(
125 (
) 3 0 )(
10 4 2 10 19 7 13 0 )(
17
(
5
.
42
−
−
−
×
=
⎥
⎦
⎤
⎢
⎣
=
λ
λ
From Equation 33 the storage capacity ratio can be
calculated in the presence of wellbore storage:
024 0 924 0 5452 6 ) 924 0 ln(
5688 3 9114
.
2
1
=
⎟⎟
⎞
⎜⎜
⎛
−
−
=
−
ω
Table 3 is a comparison of the TDS results with that of
conventional method
FRACTURE POROSITY AND COMPRESSIBILTY
Once ω is estimated, the fracture porosity can be estimated
if matrix porosity, φm, total matrix compressibility, ctm, and
total fracture compressibility, ctf, are known, as follows:
m tf
tm f
c
c φ ω
ω
⎠
⎞
⎜
⎝
⎛
−
=
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
k k P
k k
Δ
−
≈ Δ
−
=
3
/ 1 /
1 1/3
… ……… (36)
=
fi
k Fracture permeability at the initial reservoir pressure
i
p
=
f
k Fracture permeability at the current average reservoir
pressure
Combining Equations 35 and 36 yields19:
) / ( 1
tm
m
f
k k
P c
−
Δ
⎟
⎠
⎞
⎜
⎝
⎛
−
=
ω
ω φ
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 k max and k min as illustrated in Figure 6
min maxk k
k= ……… ……… (38) where
k max = maximum permeability measured in the direction parallel to the fracture plane (Figure 6), thus
k max ≈ kfracture
k min = minimum permeability measured in the direction perpendicular to the fracture plane (Figure 6), thus
k min ≈ kmatrix
Substituting k f and k m for, respectively, k max and k min, Equation 38 becomes:
m
f k k
k= ……… (39) The fracture permeability can therefore be estimated from:
m f
k
k k
2
= ……… (40)
Where k m 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:
P k k
tf Δ
−
=1 / 2/3 ……… (41) Where
k i = average permeability obtained from a transient test run when the reservoir pressure was at or near initial conditions
P i and
k = average permeability obtained from a transient test at
the current average reservoir pressure
P P
P= i− Δ
Combining Equations 41 and 35 yields19:
) / ( 1
tm m f
k k
P c
−
Δ
⎟
⎠
⎞
⎜
⎝
⎛
−
=
ω
ω φ
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
2
P
Δ The time between the two tests must be long enough for the fractures to deform significantly in order to
determine an accurate value of c tf 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 φf 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 k m,k f,andφf into the following equation should yield approximately the same value
of the effective permeability obtained from well testing20:
k k
k≈ φ ……… ……… (43)
Trang 8Eq 43 should only be used for verification purposes The
fracture width or aperture may be estimated20 from
t
f
f
k
w
ωφ
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,
(φct)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
φm = 10% rw = 0.4 ft
µ = 1 cp B=1.25 RB/STB
ctm=1.35×10-5 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:
δP = 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:
25
325
1 25 1 3000
6
.
=
Fluid storage coefficient is estimated using Equation 10:
39 0
10( 130 / 325 ) =
= −
ω
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:
2 5
10 19 1 39 0
1 ln 39 0 5 2 05
.
75
) 4 0 )(
1 ( 10
1
×
=
⎟⎟
⎞
⎜⎜
⎛
⎟
⎠
⎞
⎜
⎝
⎛
×
=
λ
1(b) – TDS technique
From Figure 8, the following characteristic points are read:
Δtmin = 2.5 hrs (t×ΔP’)R = 146 psi
(t×ΔP’)min = 70.5 psi
Using the TDS technique, the value of k is obtained from Equation 25:
146 25 25 1 1 3000 6
The inter-porosity fluid transfer coefficient is given by Equation 29:
5 2
5
10 28 1 5 2 ) 5 70 ( ) 25 1 )(
3000 (
) 4 0 )(
10 1 )(
25 )(
5 42
⎜⎜
=
λ Since the two parallel lines are well defined the storage coefficient ω is calculated from Equation 30
35 0
10 146
5 70 1 8684 0
=
= − ⎜⎛ − ⎟
ω
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:
mD k
k k
m
10 0 53
72 2 2
=
=
=
The fracture permeability at initial reservoir pressure is:
mD k
k k
m
i
fi 68 , 062
10 0 5
82 2 2
=
=
= (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:
3 / 2
10 2 5 300
062 , 68 / 606 , 52
c tf
The compressibility ratio is:
5 38 10 35 1 10 2 5
5
4
=
×
×
tm tf
c c
Thus, the fracture compressibility is more than 38.5 folds higher than the matrix compressibility, orc tf =38.5c tm The fracture porosity from Equation 42 is:
% 14 0 00139 0 5 38 1 0 35 0 1 35
⎠
⎞
⎜
⎝
⎛
−
=
f
φ
The total porosity of this naturally fractured reservoir is:
1014 0 0014 0 10
= +
t φ φ
Substituting the values of km, kf, and φf into Equation 43:
mD k
k
k≈ m+φf f = 0 1 + 0 0014 × 52 , 606 = 73 7 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:
mm microns
1014 0 35 0 33 606 , 52
=
=
×
×
=
Trang 9The 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
w f Fracture width in microns
Greek Symbols
δP vertical distance between the two semilog straight
lines, psi
α Geometry parameter, 1/L2
φ Porosity, dimensionless
ΔP1inf Pressure drop between the 1st semilog strigth line and
the inflection point, psi
ΔP2inf Pressure drop between the 2nd semilog strigth line and
the inflection point, psi
Δt shut-in time, hrs
λ Interporosity flow parameter, dimensionless
µ Viscosity, cp
ω Storage capacity ratio, dimensionless
Subscripts
i initial
o oil
D dimensionless
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 Tiab’s Direct Synthesis Technique” Paper SPE
52201 Proceedings, SPE Mid-Continent Operations Symposium, Oklahoma City, 28 – 31 March 1999; Proceedings
Trang 10SPE Latin American & Caribbean Petr Engr Conf., Caracas,
Venezuela, 21–23 April 1999, 17 pages
14 Mongi, A and Tiab, D.: “Application of Tiab’s 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 Tiab’s 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 Tiab’s 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 Tiab’s
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 Tiab’s 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