Well planning demands a knowledge of the pressures required to initiate a fracture into a format] Fracture gradient calculations, as they are termed, are essential in minimizing or avoiding lost circulation problems and in selecting proper casing seat depths.
Theoretical Determination
A number of theoretical and field-developed equations have been used to approximate fracture gradients (Figure 1-14). Many of these are suitable for immediate application in a given area, while require a hindsight approach based on density (or other) logging measurements taken after the well been drilled.
Calculation procedures for these areas rely on cither a history of the field or geological structure, or on field determinations utilizing leak-off sts or logging methods.
Hubbert and Willis Method
Hubbert and Willis explored the variables involved in initiating a fracture in formation. According to the authors, the fracture gradient is a function of overburden stress, formation pressure, and a relationship between the horizontal and vertical stresses. They believed this stress relationship to be in the range of 1/3 to 1/2 of the total overburden. Therefore fracture gradient
23

determination would be as follows:
P/Z min
= 1/3 [(S z
/ Z) + (2p / Z)]
P/Z min
= 1/2 [ 1 + (p / z)]
P = fracture pressure, psi
Z = depth, ft
S z
= overburden at depth Z, psi
P = pore pressure, psi
If an overburden stress gradient (Sz) of 1 psi/ft is assumed;
P/Z = 1/3 [ 1 + (2p / Z)],
These procedures can be used in a graphical form for a quick solution. In Fig.
2-1, enter the ordinate with the mud weight required to balance the formation.
With a horizontal line, intersect the formation pressure gradient line and construct a vertical line from this point to the minimum and maximum fracture gradients. Read the fracture mud weight from the ordinate. From the example in Figure2-1, the fracture mud weight for a 12.0-lb/gal equivalent formation pressure could range from 14.4 to 15.
In these equations, Hubbert and Willis assumed that the stress relationships and the overburden gradients were constant for all depths. Since this has been proven untrue in most cases, subsequent methods have attempted to account for one or both of these variables more accurately.
24

Figure 2-1 Graphical determination of fracture gradients as proposed by
Hubbert and Willis
Matthews and Kelly Method
In realizing that the cohesiveness of the rock matrix is usually related to the matrix stress and varies only with the degree of compaction, Matthews and
Kelly developed the following equation for calculating fracture gradients in sedimentary formations:
F = (P/D) + (K i
 / D)
P = formation pressure at the point of interest, psi
D = depth of interest, ft
 = matrix stress at the point of interest, psi
25

K
II
= matrix stress coefficient for the depth at which the value of  would be normal matrix stress, dimensionless
F = fracture gradient at the point of interest, psi/ft
The matrix stress coefficient relates the actual matrix stress conditions of the formation to the conditions of matrix stress if the formation were compacted normally. The authors believed that the conditions necessary for fracturing the formation would then be similar to those for the normally compacted formation. The stress coefficient vs. depth is presented in Fig. 2-2.
Matthews and Kelly believed that the coefficient would vary with different geological conditions. Substituting actual field data of breakdown pressures into above equation and solving for Ki obtained the values shown. The procedure for calculating fracture gradients using the Matthews and Kelly technique:
1. Obtain formation fluid pressure, P. This can be measured by drill stem tests, kick data. logs, or another satisfactory method.
2. Obtain the matrix stress by using above equation and assuming a gradient of
1.0 psi/ft for the overburden.
 = S - P
26

Figure 2-2 Matrix stress coefficients of Matthews and Kelly
3. Determine the depth, D i
, for which the matrix stress, a, would be the normal value. Assume that the overburden pressure is 1.0 psi/ft. From this it follows that:
0.535 D i
=  from which the value of D i
, can be found.
4. Use the value of D i
, apply it to Fig. 1-15 to obtain the corresponding value of
27

K i
.
6. Use the value of Dj, and apply it to Fig. 4-2 to obtain the corresponding value of K,.
Example 2-1:
Casing was set on a Texas Gulf Coast well at 7,200 ft. It was estimated that formation pressure was equivalent to 11 lb/gal mud. What is the fracture gradient immediately below the casing seat? Use the Matthews and Kelly procedure.
Solution:
1. P = (11.0 lb/gal) (0.052) (7,200 ft) = 4,118 psi
2.  = S - P = 7,200 - 4,118 = 3082 psi
3. Depth equivalent; 0.535 D i
= 
D i
=  / 0.535
where, 0.535 psi/ft is the rock matrix stress,
4. From Fig. 2-2, K i
= 0.695
F = (P / D) + (K i
 / D)
F = (4118 / 7200) + (0.695 x
3082 / 7200)
F = 0.869 psi/ft = 16.7 ppg
28

A graphical solution to the Matthews and Kelly technique is p. rented in
Fig. 2-3. Note that the curved lines on the graph represent actual formation pressures and not mud weight in use. Unfortunately, these are often erroneously interchanged. To solve for fracture gradients enter at the desired depth and read horizontally until the actual formation pressure line is intersected. Plot a vertical line from this point and read the fracture gradient in pounds per gallon.
Figure 2-3 Graphical determinations of fracture gradients using
Matthews and Kelly approach
Eaton Model
Eaton extended the concepts presented by Matthews and Kelly to introduce Poisson's ratio into the expression for the fracture pressure gradient:
29

F = [((S - P) / D) (  / (1 -  )] + P / D
P = well bore pressure, psi
D = depth, ft
S = overburden stress, psi
 = Poisson's ratio
F = fracture gradient, psi/ft
Eaton assumed that both overburden stress and Poisson's ratio were variable with depth. Using actual field fracture data and log-derived values, he prepared graphs illustrating these variables. Using a suitable choice for each variable, the monograph prepared by Eaton. (Fig. 2-4) can be used to calculate a fracture gradient. A graphical presentation for the Eaton approach provides a quick solution. The chart (Fig. 2-5) is used in the same manner as the Matthews and Kelly chart.
30

Figure 2-4 Variable overburden stress by Eaton
31

Figure 2-5 Variable Poisson’s ratio with depth as proposed by
Eaton
6 Nomograph determinations of fracture gradients as proposed by
Figure2-
32

Eaton Figure 2-7
Graphical determinations of fracture gradients using the Eaton approach Christman Model Fracture gradient determination procedures assume that overburden stress consist of rock matrix stress and formation fluid stress.
Offshore, the water has no rock matrix. Fracture gradients are lower when compared to land at equivalent depths. In shallow water, the reduction in,
33

fracture gradient is almost insignificant. As water depth increases, fracture gradient declines. Christman has accounted for the effect of water depth in his total overburden gradient equation.
G ob
=1/D (0.44
D w
+  b
D mi
)G ob
= total overburden gradient, psi/ft D = depth below datum, ft D w
= water depth, ft  b
= average bulk density, g/cc D mi
= depth below mud lineThe effect of water depth on fracture gradients can be seen in Fig. 2-8. Other procedure has been developed to calculate deepwater fracture gradients. It utilizes the techniques established by Christman and the data collection by Eaton. The following example illustrates the procedure for a well drilled in 1000 ft of water.
Example2-2: In the illustration in Fig. 2-8, what is the effective fracture gradient at the casing seat? Solution:1.Convert the water depth to an equivalent section of formation: 1000 ft x 0.465 psi/ft = 465 psi 2. From Eaton overburden stress chart in Fig. 2.4 the stress gradient at 4000 ft equals 0.89 si/ft.
465 psi / 0.89 psi/ft = 522 ft,
34

equivalent Figure 2-8
Nomograph determinations of fracture gradients by Eaton3.Calculate and convert the apparent fracture gradient to actual fracture gradient.
522 ft +
3000 ft = 3522 ft 4. From Eaton fracture gradient chart, the gradient at 3522 ft is 13.92 ppg , or,Fracture Pressure = 0.052 x 13.92 x 3522 ft = 2549 psi
Field Determinations of Fracture Gradients The most common procedure used for the field dertermination of fracture gradient is the leak off tes (pressure
35

integrity test) . In this test, blow-out preventers are closed and then the pressure is applied incrementally to the shut-in system until the formation initially accepts fluid (Figure. 2-
9).
results from a leak-off test
Figure 2-9 Typical
Example 1-9: Casing was set at 10000 ft. The operator performs a leak-off test to determine the fracture gradient at 10000 ft. If the mud weight in the well is
11.2 ppg, what is the fracture gradient at the casing seat. Solution: 1. Close the blow-out preventers (BOP) and rig up low volume output pump.2. Apply pressure to the well and record the results.
Volume
Pumped, bbl psi
Pressure, Volume
Pumped, bbl psi
Pressure,
36

0
1
1.5
2
2.5
3
0
45
125
230
350
470
3.5
4
4.5
5
5.5
6
590
710
830
950
990
1010
3. The results are plotted on Figure 2-10. It appears that the formation will begin to fracture when 950 psi is applied.4. Determine the fracture gradient.Frac. Gradient = {[(11.2 ppg) (0.052) (10000 ft)) + 950} / 10000 ftFrac.
Gradient = 6774 psi / 10000 ftFrac. Gradient = 0.6774 psi/ft Frac. Gradient =
13.02 ppg
Figure 2-10 Results of Example 1-9 Home-work 1
1. A well is drilled to 13500 ft. The entrance into the abnormal pressures at
9000 ft is caused by under compaction. Calculate the expected formation pressure at 13500 ft. Assume formation fluid and overburden stress gradients are 0.465 psi/ft and 1.0 psi/ft respectively?
37

2. Plot the data given below on a semi-log paper. Where does the entrance into the abnormal pressure occur? Use Hottman and Johnson procedure to compute formation pressure at each 1000 ft interval below the entrance into pressures?
Resistivity, ohm-m
0.54
0.64
0.60
0.70
0.76
0.60
0.70
0.74
0.76
0.82
0.90
0.84
Depth, ft
6000
6600
7600
8000
8400
9000
9500
10000
10300
11200
11600
12200
Resistivity, ohm-m
0.80
0.76
0.58
0.45
0.36
0.30
0.28
0.29
0.27
0.28
0.29
0.30
Depth, ft
12400
12700
12900
13000
13100
13300
13600
13900
14300
14500
14700
14900
3.The following sonic log was taken from a well in Oklahama. Plot the data on semi-log paper. Use Hottman and Johnson technique to calculate the formation pressure at 11900 ft.
Travel
Time,  sec/ft
170
150
142
115
124
108
114
89
95
97
101
Depth, ft
3400
5000
6600
7300
7900
8200
8600
9000
9200
9400
9600
Travel
Time,  sec/ft
100
110
100
110
105
38
105
105
107
118
105
-
Depth, ft
9800
10000
10200
10400
10600
10800
11100
11400
11600
11900
-

4. Determine the fracture gradients for the following condition. Use the methods of a)Hubbert and Willis b)Matthews and Kelly c)Eaton
Assume “Lousiana” conditions for the Matthews and Kelly calculations.
Depth, ft
3000
13000
9000
6500
8000
Form. Pressure, ppg
Normal
13.1
9.6
9.2
10.2
Depth, ft
11000
17000
4500
10500
15000
Form. Pressure, ppg
15.1
18.0
9.9
Normal
15.6
5. Prepare a graph of fracture gradients vs. depth for the methods used in the above problem (Prob. 4). Assume normal formation pressures.
6. Calculate the fracture gradient for the following set of deep water conditions.
Freeboard: 50 ft
Water Depth: 1700 ft
Casing Depth below ssea floor: 6000 ft
7. Use the following leak-off test data to determine the formation fracture gradient. Casing was set at 12000 ft and the mud weight is 13.9 ppg.
39

Volume Pumped, bbl
0
1
2
2.5
3
Pressure, psi
0
175
400
590
680
Volume Pumped, bbl
3.5
4
4.5
5
-
Pressure, psi
760
650
740
830
-
40