As developed by Bourdet and Gringarten (1980) and expanded by Bourdet et al. (1984) to include the pressure-derivative approach, this type curve is built in the same way as that for the pseudosteady-state inter- porosity flow. As shown in Figure 17-10, the pressure behavior is defined by three component curves: (CDe2s)f, β\, and (CDe2s)f+m. The authors defined β\as the interporosity dimensionless group and given by:
where the parameter δ is the shape coefficient with assigned values as given below:
δ = 1.0508, for spherical blocks δ = 1.8914, for slab matrix blocks
As the first fissure flow is short lived with transient interporosity flow models, the (CDe2s)fcurves are not seen in practice and therefore have not been included in the derivative curves. The dual-porosity derivative response starts on the derivative of a β\transition curve and then follows a late transition curve labeled λ(CD)f+m/(1 − ω)2until it reaches the total system regime on the 0.5 line.
Bourdet (1985) points out that the pressure-derivative responses during the transition flow regime are very different between the two double- porosity models. With the transient interporosity flow solutions, the tran- sition starts from early time and does not drop to a very low level. With pseudosteady-state interporosity flow, the transition starts later and the shape of the depression is much more pronounced. There is no lower limit for the depth of the depression when the flow from the matrix to the
β δ
λ
\= ⎡( )
⎣
⎢⎢
⎤
⎦
⎥⎥
− +
C eD s
f m e s
2 2
fissures follows the pseudosteady-state model, whereas for the inter- porosity transient flow, the depth of the depression does not exceed 0.25.
In general, the matching procedure and reservoir parameters estimation as applied to the type curve of Figure 17-10 are summarized by the fol- lowing steps:
Step 1.Using the actual well test data, calculate the pressure difference Δp and the pressure-derivative plotting functions for every pres- sure test point.
For drawdown tests
The pressure difference Δp =pi−pwf The derivative function
For buildup tests
The pressure difference Δp =pws−pwf@Δt=0 The derivative function
Step 2. On a tracing paper with the same size log cycles as in Figure 17-10, plot the data of Step 1 as a function of flowing time t for drawdown tests or versus equivalent time Δtefor buildup tests.
Step 3.Place the actual two sets of plots, i.e., Δp and derivative plots, on Figure 17-9 or Figure 17-10 and force a simultaneous match of the two plots to Gringarten–Bourdet type curves. Read the matched derivative curve [λ(CD)f+m/(1 − ω)2]M.
Step 4.Choose any point and read its coordinates on both Figures 17-9 and 17-10 to give:
(Δp, pD)MPand (t or Δte, tD/CD)MP
Step 5.With the match still maintained, read the values of the curves labeled (CDe2s) which match the initial segment of the curve [(CDe2s)f]M and the final segment [(CDe2s)f+m]M of the data curve.
Δ Δ Δ Δ
Δ
Δ t p t t t Δ
t
d p d t
e
\= ⎛ P+
⎝ ⎞
⎠ ( ) ( )
⎡
⎣⎢ ⎤
⎦⎥ t p t d p
Δ \= − d t( )Δ
( )
⎛
⎝⎜
⎞
⎠⎟
Step 6.Calculate the well and reservoir parameters from the following relationships:
• (17-5)
(17-6)
(17-7)
(17-8)
(17-9)
(17-10) Selection of the better solution between the pseudosteady-state and the transient interporosity flow is generally straightforward; with the pseudo- steady-state model, the drop in the derivative during transition is a func- tion of the transition duration. Long transition regimes, corresponding to small ω values, produce derivative levels much lower than the practical 0.25 limit of the transient solution.
The pressure-buildup data given by Bourdet et al. (1984) and reported by Sabet (1991) are used below as an example to illustrate the use of the pressure-derivative type curves.
Example 17-2
Table 17-1 shows the pressure-buildup and pressure-derivative data for a naturally fractured reservoir. The following flow and reservoir data are also given:
λ λ
ω
= ( ) ω
( − )
⎡
⎣⎢ ⎤
⎦⎥ ( − )
+ ( )
+
C
C
D f m
M D f m
1
1
2
2
s
C e C
D s
f m M D f m
= [ ( ) ]
( )
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
+ +
0 5
2
. ln
C C
D f m c hr
t w
( )+ =
0 8926
2
. φ
C k h t
C C
f MP
D D MP
=⎡
⎣⎢ ⎤
⎦⎥ ( )
( )
0 000295. μ
Δ
k h QB p
p md ft
f
D MP
= ⎛
⎝⎜
⎞ 141 2. μ ⎠⎟ ,
Δ w =[ ( ) ]
( )
[ + ]
C e C e
D s
f m M
D s
M 2
2
Table 17-1
Pressure Buildup Test, Naturally Fractured Reservoir (After Sabet, 1991)
Dt (hour) Dpws(psi) Slope (psi/hour) (psi)
0.00000E+00 0.000 3180.10
3.48888E−03 11.095 14,547.22 1727.68 8.56
9.04446E−03 20.693 5612.17 847.26 11.65
1.46000E−02 25.400 3477.03 486.90 9.74
2.01555E−02 28.105 2518.92 337.14 8.31
2.57111E−02 29.978 1974.86 257.22 7.64
3.12666E−02 31.407 1624.14 196.56 7.10
3.68222E−02 32.499 1379.24 459.66 6.56
4.23777E−02 33.386 1198.56 127.80 6.10
4.79333E−02 34.096 1059.76 107.28 5.64
5.90444E−02 35.288 860.52 83.25 5.63
7.01555E−02 36.213 724.39 69.48 5.36
8.12666E−02 36.985 625.49 65.97 5.51
9.23777E−02 37.718 550.38 55.07 5.60
0.10349 38.330 491.39 48.83 5.39
0.12571 39.415 404.71 43.65 5.83
0.14793 40.385 344.07 37.16 5.99
0.17016 41.211 299.25 34.38 6.11
0.19238 41.975 264.80 29.93 6.21
0.21460 42.640 237.49 28.85 6.33
0.23682 43.281 215.30 30.96 7.12
0.25904 43.969 196.92 25.78 7.39
0.28127 44.542 181.43 24.44 7.10
0.30349 45.085 168.22 25.79 7.67
0.32571 45.658 156.81 20.63 7.61
0.38127 46.804 134.11 18.58 7.53
0.43628 47.836 117.18 17.19 7.88
0.49298 48.791 104.07 16.36 8.34
0.54793 49.700 93.62 15.14 8.72
0.60349 50.541 85.09 12.50 8.44
0.66460 51.305 77.36 12.68 8.48
0.71460 51.939 72.02 11.70 8.83
0.77015 52.589 66.90 11.14 8.93
0.82571 53.208 62.46 10.58 9.11
0.88127 53.796 58.59 10.87 9.62
0.93682 54.400 55.17 8.53 9.26
0.99238 54.874 52.14 10.32 9.54
1.04790 55.447 49.43 7.70 9.64
1.10350 55.875 46.99 8.73 9.26
1.21460 56.845 42.78 7.57 10.14
D D
p t t
t
p p
\ +
t t
t
p+D D
Table 17-1 Continued
Dt (hour) Dpws(psi) Slope (psi/hour) (psi)
1.32570 57.686 39.28 5.91 9.17
1.43680 58.343 36.32 6.40 9.10
1.54790 59.054 33.79 6.05 9.93
1.65900 59.726 31.59 5.57 9.95
1.77020 60.345 29.67 5.44 10.08
1.88130 60.949 27.98 4.74 9.93
1.99240 61.476 26.47 4.67 9.75
2.10350 61.995 25.13 4.34 9.87
2.21460 62.477 23.92 3.99 9.62
2.43680 63.363 21.83 3.68 9.79
2.69240 64.303 19.85 3.06a 9.55b
2.91460 64.983 18.41 3.16 9.59
3.13680 65.686 17.18 2.44 9.34
3.35900 66.229 16.11 19.72 39.68
Adapted from Bourdet et al. (1984).
a(64.983 -64.303)/(2.9146 -2.69240) =3.08.
b[(3.68 ∏3.06)/2 ¥19.85 ¥2.692402/50.75 =9.55.
D D
p t t
t
p p
\ +
t t
t
p+D D
Q =960 STB/day Bo=1.28 bbl/STB ct=1 ×10−5psi−1
φ =0.007 μ =1 cp rw=0.29 ft
h =36 ft
It is reported that the well was opened to flow at a rate of 2,952 STB/day for 1.33 hours, shut-in for 0.31 hour, opened again at the same rate for 5.05 hours, closed for 0.39 hours, opened for 31.13 hours at the rate of 960 STB/day, and then shut-in for the pressure-buildup test.
Analyze the buildup data and determine the well and reservoir para- meters assuming transient interporosity flow.
Solution
Step 1.Calculate the flowing time tpas follows:
Step 2.Confirm the double-porosity behavior by constructing Horner’s (1967) plot as shown in Figure 17-11. The graph shows the two parallel straight lines confirming the dual-porosity system.
tp=( )(24 2030) = hours 960 50 75. total oil produced N
STB
= P = [ + ]
+
2952
4 1 33 5 05 960
24 31 13 2030
. .
. ⯝
Figure 17-11. The data of the Horner plot. (After Sabet, 1991.)
Step 3.Using the same grid system of Figure 17-10, plot the actual pres- sure derivativeversus shut-in time, as shown in Figure 17-12a, and Δpwsversus time, as shown in Figure 17-12b. The 45° line shows that the test was slightly affected by the wellbore storage.
Step 4.Overlay the pressure difference and pressure-derivative plots over the transient interporosity type curve, as shown in Figure 17-13, to give the following matching parameters:
•
•
•
• [(CD e2s)f]M=33.4
• [(CD e2s)f +m]M=0.6
Step 5.Calculate the well and reservoir parameters by applying Equations 17-5 through 17-10, to give:
Kazemi (1969) points out that if the vertical separation between the two parallel slopes Δp is less than 100 psi, the calculation of ω by Equation 17-2 will produce a significant error in its values. Figure 17-11 shows that Δp is about 11 psi, and Equation 17-2 gives an erroneous valueof:
ω =10−(Δp/m)=10−(11/12)=0.316
• k h QB p
p md ft
f
D MP
= ⎛
⎝⎜
⎞
⎠⎟ = ( )( )( )( )
=
141 2 141 2 960 1 1 28 0 053 9196
. μ . . .
Δ ω =[ ( ) ]
( )
[ + ] = =
C e C e
D s
f m M D
s f 2
2
0 6
33 4. 0 018
. .
λ ω CD f m
M
( )
( − )
⎡
⎣⎢ ⎤
⎦⎥ =
+
1 2 0 03. t C
t
D D
Δ MP
⎡
⎣⎢
⎤
⎦⎥ =270 p
p
D
Δ MP
⎡
⎣⎢ ⎤
⎦⎥ =0 053.
Figure 17-13.Type-curve matching. (Copyright ©1984World Oil, Bourdet et al., 1984.)
•
•
•
HYDRAULICALLY FRACTURED WELLS
Many wells, particularly wells in tight (low-permeability) formations, require hydraulic fracturing to be commercially viable. Interpretation of pressure transient data in hydraulically fractured wells is important for evaluating the success in fracture treatments and predicting the future per- formance of these types of wells. A well-documented and comprehensive report that overviews theories, design methods, and materials used in a hydraulic fracture treatment was published by the Environmental Protec- tion Agency in June 2004 (EPA, 2004). Hydraulic fracturing is the process of pumping a fluid into a wellbore at an injection rate that is too high for the formation to accept in a radial flow pattern. As the resistance to flow in the formation increases, the pressure in the wellbore increases to a value that exceeds the breakdown pressure of the formation that is open to the wellbore. Once the formation “breaks down,” a crack or fracture is formed, and the injected fluid begins moving down the fracture. In most formations, a single, vertical fracture that propagates in two directions from the wellbore is created. These fracture “wings” are 180° apart and are normally assumed to be identical in shape and size at any point in time. In naturally fractured or cleated formations, such as gas shales or coal seams, it is possible that multiple fractures can be created and prop- agated during a hydraulic fracture treatment.
Fluid that does not contain any propping agent, often called “pad,” is injected to create a fracture that grows up, out, and down; therefore, the fluid creates a fracture that is wide enough to accept a propping agent.
λ λ
ω
= ( ) ω
( − )
⎡
⎣⎢ ⎤
⎦⎥ ( − )
( ) =( )⎡⎣⎢( − ) ⎤
⎦⎥ = ×
+
+
C −
C
D f m
M D f m
1
1 0 03 1 0 018
4216 6 86 10
2
2 2
. . 6
. s
C e C
D s
f m M D f m
= [ ( ) ]
( )
⎡
⎣
⎢⎢
⎤
⎦
⎥⎥= ⎡
⎣⎢ ⎤
⎦⎥= −
+ +
0 5 0 5 0 6
4216 4 4
2
. ln . ln .
.
C C
D f m c hr
t w
( )+ =0 8926 =( ) ( (0 8936 0 01× − )( ) ( )( ) ) =
0 07 1 10 36 90 29 4216
2 5 2
. . .
. .
φ
C k h t
C C bbl psi
f MP
D D MP
=⎡
⎣⎢ ⎤
⎦⎥ ( )
( ) =( ( )( ) )( )=
0 000295 0 000295 9196
1 0 270 0 01
. .
. .
μ
Δ
The purpose of the propping agent is to “prop open” the fracture once the pumping operation ceases, the pressure in the fracture decreases, and the fracture closes. In deep reservoirs, we use human-made ceramic beads to prop open the fracture. In shallow reservoirs, sand is normally used as the propping agent. The sand used as a propping agent in shallow reservoirs, such as coal seams, is mined from certain quarries in the United States.
Silica sand is a natural product and will not lead to any environmental concerns that would affect the United States Safe Drinking Water Act.
In general, hydraulic fracture treatments are used to increase the pro- ductivity index of a producing well or the injectivity index of an injection well. The productivity index defines the volumes of oil or gas that can be produced at a given pressure differential between the reservoir and the wellbore. The injectivity index refers to how much fluid can be injected into an injection well at a given pressure differential. The EPA (2004) report lists different applications for hydraulic fracturing, such as:
•Increasing the flow rate of oil and/or gas from low-permeability reservoirs
•Increasing the flow rate of oil and/or gas from wells that have been damaged
•Connecting the natural fractures and/or cleats in a formation to the wellbore
•Decreasing the pressure drop around the well to minimize sand production
•Decreasing the pressure drop around the well to minimize problems with asphaltine and/or paraffin deposition
•Increasing the area of drainage or the amount of formation in contact with the wellbore
•Connecting the full vertical extent of a reservoir to a slanted or hori- zontal well
A low-permeability reservoir has a high resistance to fluid flow. In many formations, chemical and/or physical processes alter the structure of a reservoir rock over geologic time. Sometimes, these diagenetic processes restrict the openings in the rock and reduce the ability of fluids to flow through the rock. Low-permeability rocks are normally excellent candi- dates for stimulation by hydraulic fracturing.
Regardless of permeability, a reservoir rock can be damaged when a well is drilled through the reservoir and when casing is set and cemented in place. Damage occurs because drilling and/or completion fluids leak
into the reservoir and plug up the pores and pore throats. When the pores are plugged, permeability is reduced, and the fluid flow in this damaged portion of the reservoir may be substantially reduced. Damage can be severe in naturally fractured reservoirs, such as coal seams. To stimulate damaged reservoirs, a short, conductive hydraulic fracture is often the desired solution.
The success or failure of a hydraulic fracture treatment often depends on the quality of the candidate well selected for the treatment. Choosing an excellent candidate for stimulation often ensures success, while choos- ing a poor candidate normally results in economic failure. To select the best candidate for stimulation, the design engineer must consider many variables. The most critical parameters for hydraulic fracturing are:
•Formation permeability
•In situstress distribution
•Reservoir fluid viscosity
•Skin factor
•Reservoir pressure
•Reservoir depth
If the skin factor is positive, the reservoir is damaged and could possi- bly be an excellent candidate for stimulation.
The best candidate wells for hydraulic fracturing treatments will have a substantial volume of oil and gas-in-place, and will have a need to increase the productivity index. Such reservoirs will have
•a thick pay zone and
•medium to high pressure, and will either be
•a low-permeability zone or a zone that has been damaged (high skin factor).
Hydraulic fracturing theory and design has been developed by other engineering disciplines. However, certain aspects, such as poroelastic theory, are unique to porous, permeable underground formations. The most important parameters are: Poisson’s ratio; Young’s modulus; and in situstress.
Poisson’s ratio(v), named after Simeon Poisson, is defined as the ratio of the relative contraction strain (transverse strain) divided by the relative extension strain (or axial strain).
Young’s modulusis defined as “the ratio of stress to strain for uniaxial stress.” The theory used to compute fracture dimensions is based upon linear elasticity. To apply this theory, Young’s modulus of the formation is an important parameter.
The modulus of a material is a measure of the stiffness of the material.
If the modulus is large, the material is stiff. In hydraulic fracturing, a stiff rock will result in more narrow fractures. If the modulus is low, the frac- tures will be wider. The modulus of a rock is a function of the lithology, porosity, fluid type, and other variables. Typical ranges for Young’s modulus as a function of lithology are tabulated below.
σ1
σ3
σ2 σ1
σ3
σ2
Figure 17-14. Local in situstress at depth.
Lithology Young’s Modulus (psi)
Soft sandstone 2–5 ×106
Hard sandstone 6–10 ×106
Limestone 8–12 ×106
Coal 0.1–1 ×106
Shale 1–10 ×106
In situ stresses. Underground formations are confined and under stress.
Figure 17-14 illustrates the local stress state at depth for an element of formation. The stresses can be divided into the following three principal stresses:
•the vertical stress σ1,
•the maximum horizontal stress σ2, and
•the minimum horizontal stress σ3.
where σ1 > σ2 > σ3. Depending on geologic conditions, the vertical stress could also be the intermediate (σ2) or minimum stress (σ3). These stresses are normally compressive and vary in magnitude throughout the reservoir, particularly in the vertical direction (from layer to layer). The magnitude and direction of the principal stresses are important because they control:
•the pressure required to create and propagate a fracture,
•the shape and vertical extent of the fracture,
•the direction of the fracture, and
•the stresses trying to crush and/or embed the propping agent during pro- duction.
A hydraulic fracture will propagate perpendicular to the minimum prin- cipal stress (σ3). If the minimum horizontal stress is σ3, the fracture will be vertical. The minimum horizontal stress (in situ stress) profile can be calculated from the following expression:
whereσmin=the minimum horizontal stress (in situstress) ν =Poisson’s ratio
σob =overburden stress α =poroelastic constant
p =reservoir fluid pressure or pore pressure
Poisson’s ratio can be estimated from acoustic log data or from corre- lations based upon lithology. For coal seams, the value of Poisson’s ratio will range from 0.2 to 0.4. The overburden stress can be computed using density log data. Normally, the value for overburden pressure is about 1.1 psi per foot of depth. The reservoir pressure must be measured or estimated.
The poroelastic constant α ranges from 0.5 to 1.0 and is a parameter that describes the “efficiency” of the fluid pressure to counteract the total applied stress. Typically, for hydrocarbon reservoirs, αis about 7.
σmin≅ σ α α
−vv( ob− p)+ p
1
A hydraulic fracture will propagate perpendicular to the least principal stress. In some shallow formations the least principal stress is the over- burden stress; thus, the hydraulic fracture will be horizontal. In reservoirs deeper than 1,000 ft or so, the least principal stress will likely be hori- zontal; thus, the hydraulic fracture will be vertical. The azimuth orienta- tion of the vertical fracture will depend upon the azimuth of the minimum and maximum horizontal stresses.
A fracture is defined as a single crack initiated from the wellbore by hydraulic fracturing. It should be noted that fractures are different from
“fissures,” which are the formation of natural fractures. Hydraulically induced fractures are usually vertical, but can be horizontal if the forma- tion is less than approximately 3,000 ft deep. Vertical fractures are char- acterized by the following properties:
•Fracture half-length xf, in ft
•Dimensionless radius reD, where reD =re/xf
•Fracture height hf, which is often assumed equal to the formation thick- ness, in ft
•Fracture permeability kf, in md
•Fracture width wf, in ft
•Fracture conductivity FC, where FC=kfwf
The analysis of fractured well tests deals with the identification of well and reservoir variables that would have an impact on future well perfor- mance. However, fractured wells are substantially more complicated. The well-penetrating fracture has unknown geometric features, i.e., xf, wf, and hf, and unknown conductivity properties.
Gringarten et al. (1974) and Cinco and Samaniego (1981), among others, propose three transient flow models to consider when analyzing transient pressure data from vertically fractured wells. These are: (1) infi- nite-conductivity vertical fractures; (2) finite-conductivity vertical frac- tures; (3) uniform-flux fractures.