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CHAPTER-3

FRACTURE GRADIENT DETERMINATIONS

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

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determination would be as follows:

P/Z min = 1/3 [(Sz / Z) + (2p / Z)]

P/Z min = 1/2 [ 1 + (p / z)]

P = fracture pressure, psi

Z = depth, ft

Sz = 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.

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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) + (Ki / D)

P = formation pressure at the point of interest, psi

D = depth of interest, ft

= matrix stress at the point of interest, psi

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KII = 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

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Figure 2-2 Matrix stress coefficients of Matthews and Kelly

3. Determine the depth, Di, 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 Di =

from which the value of Di, can be found.

4. Use the value of Di, apply it to Fig. 1-15 to obtain the corresponding value of

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Ki.

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 Di=

Di = / 0.535

where, 0.535 psi/ft is the rock matrix stress,

4. From Fig. 2-2, Ki = 0.695

F = (P / D) + (Ki / D)

F = (4118 / 7200) + (0.695 x 3082 / 7200)

F = 0.869 psi/ft = 16.7 ppg

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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:

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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.

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Figure 2-4 Variable overburden stress by Eaton

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Figure 2-5 Variable Poisson’s ratio with depth as proposed by

Eaton Figure2-

6 Nomograph determinations of fracture gradients as proposed by

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Eaton Figure 2-7

Graphical determinations of fracture gradients using the Eaton

approachChristman ModelFracture 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,

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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.Gob =1/D (0.44 Dw + b Dmi)Gob= total

overburden gradient, psi/ftD = depth below datum, ft Dw = water depth, ftb=

average bulk density, g/cc Dmi = 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 psi2. 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,

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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 ft4. 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 GradientsThe most common procedure used

for the field dertermination of fracture gradient is the leak off tes (pressure

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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). Figure 2-9 Typical

results from a leak-off test

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

Pressure,

psi

Volume

Pumped, bbl

Pressure,

psi

37

0 0 3.5 590

1 45 4 710

1.5 125 4.5 830

2 230 5 950

2.5 350 5.5 990

3 470 6 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/ftFrac. 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?

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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

Depth, ft Resistivity,

ohm-m

Depth, ft

0.54 6000 0.80 12400

0.64 6600 0.76 12700

0.60 7600 0.58 12900

0.70 8000 0.45 13000

0.76 8400 0.36 13100

0.60 9000 0.30 13300

0.70 9500 0.28 13600

0.74 10000 0.29 13900

0.76 10300 0.27 14300

0.82 11200 0.28 14500

0.90 11600 0.29 14700

0.84 12200 0.30 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

Depth, ft Travel

Time, sec/ft

Depth, ft

170 3400 100 9800

150 5000 110 10000

142 6600 100 10200

115 7300 110 10400

124 7900 105 10600

108 8200 105 10800

114 8600 105 11100

89 9000 107 11400

95 9200 118 11600

97 9400 105 11900

101 9600 - -

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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 Form. Pressure,

ppg

Depth, ft Form. Pressure,

ppg

3000 Normal 11000 15.1

13000 13.1 17000 18.0

9000 9.6 4500 9.9

6500 9.2 10500 Normal

8000 10.2 15000 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.

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Volume Pumped,

bbl

Pressure, psi Volume Pumped,

bbl

Pressure, psi

0 0 3.5 760

1 175 4 650

2 400 4.5 740

2.5 590 5 830

3 680 - -