 Open Access
 Total Downloads : 318
 Authors : John Lander Ichenwo, Adewale Dosunmu
 Paper ID : IJERTV6IS040191
 Volume & Issue : Volume 06, Issue 04 (April 2017)
 Published (First Online): 11042017
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Estimation of Horizontal Stresses and Stress Directions from Inversion of Leakoff Test DataA Computer Approach
1John Lander Ichenwo and 2Adewale Dosunmu
1,2Department of Petroleum Engineering, University of Port Harcourt,
Port Harcourt, Nigeria
Abstract – Insitu stresses play the most important role in borehole stability during drilling operation. Problems pose by Uncertainties in the measurement of stress during drilling are enormous in the petroleum industry. In this paper a handy computer tool for estimating the magnitude and direction of horizontal matrix stresses was developed. This tool is based on the inversion model and uses data from a Leak off Test (LOT) together with overburden stress, pore pressure and well orientation data.
Index Terms: Insitu Stress, Leak off data, extended leak off test, Density Logs.
SYMBOLS and Notations
VSPs Vertical Seismic Profiles (VSPs LOT Leak off Test
LOP Leak off Pressure
XLOT extended leak off Test
Average mass density
() Density as a function of vertical
. Depth
w Density of water
zw Water depth
Pwf Bottom Hole Flowing Pressure
h Depth
g Acceleration due to gravity

INTRODUCTION
Accurate prediction of the insitu stresses) is important in the petroleum industry. Knowledge of the insitu stress has implications for not only drilling safety and well design but also the costs of extraction of hydrocarbons. It is generally accepted that hydraulic fracturing is the most accurate method to measure stress at deep hole [12]. The magnitude and direction of the horizontal insitu stresses can be estimated from leakoff data using inversion method.
Hubbert and Willis (1957) provided the most important effort in the interpretation of hydraulic fracturing mechanism, by using the theory of elasticity to reach the conclusion that the direction of the induced hydraulic fracture and the pressures recorded during borehole pressurization are directly related to the principal in situ stresses, this followed the successful use of hydraulic fracture as stimulation technique in the 1940s. Hydraulic fracturing has now become one of the key methods for rock stress estimation as suggested by the International Society for Rock Mechanics (ISRM). Fairhurst was among the first
to advocate the use of hydraulic fracturing for in situ stress determination.
Kirsch (1989) presented solutions from which the basic equations describing the stress distribution around a horizontal, vertical and inclined wellbore may be derived. It is generally believed that a fracture initiates when the maximum tensile stress induced at any point around the wellbore exceeds the tensile strength of the formation at that point. When this occurs, the resulting fracture on the wellbore wall will have an orientation that is perpendicular to the direction of the most tensile principal stress. The pressure of wellbore failure is given by equation 1.3. [10]
The technique for inverting results from a minimum of two leak off tests at different well inclinations and azimuths, gives an estimate of both horizontal stress (maximum and minimum) magnitudes and directions. However, the published technique suffers from the assumption that shear stresses are neglected. The magnitude and direction of the horizontal insitu stresses can be estimated from leakoff data using inversion method. [12] The method makes use of the fracture equation which is derived from the Kirsch equations and stress transformation equation. But the original method was inaccurate, because it ignores shear stress, and proposed an improved method. However found that the improved inversion also contained large uncertainties, in part due to the inaccuracy of the LOTs and suggested the use of multiple techniques to determine the in situ stress.
The in situ stress state can be determined from multiple fracturing data and induced fractures from image logs. A solution can be obtained with a minimum of two data sets. However, using an inversion technique, a solution can be obtained with any number of Datasets, as the solution is over determined [7].

METHODS FOR DETERMINATION OF REQUIRED DATA

Vertical stress determination
The vertical stress is assumed to be equal to the weight of the overburden and can be calculated using knowledge of the rock densities. The vertical stress is determined using this formula:
= 1.0
If the density varies with depth, the vertical stress is determined by integrating the densities of the overlaying rocks as shown in the equation below:
0
= () 1.1
In offshore area, water depth (zw) is accounted for in the calculation of the vertical stress, that is;
= +
() 1.2
Density logs and check shot velocity surveys or vertical seismic profiles (VSPs) are the two main sources of density data.

Minimum Horizontal Stress Magnitude

Leakoff test (LOT), extended leakoff tests (XLOT) and minifracture tests can be used to constrain horizontal stress magnitudes. [16], [12]. The entire test involves increasing fluid pressure in the wellbore until a fracture is created at the wellbore wall. The LOT is the most commonly undertaken and the simplest of these tests. LOTs are conducted not for the purpose of making stress estimates, but in order to determine the maximum mud weight that can be used when drilling ahead. An XLOT is conducted when information on the stress tensor is of interest [18]. As the name suggests an XLOT is an extended version of LOT, using the same basic equipment, but a different test procedure. The third type of test discussed in this section is the minifracture or hydraulic fracture test, which is specifically designed to determine the horizontal stress magnitudes. LOTs can be used to estimate . XLOTs and minifracture test provide a more reliable estimate of and under certain circumstances, an estimate of .
Leakoff pressure (LOP) is defined as the point on the pressure time curve at which the pressure buildup deviate from linearity as illustrated in fig 1.0. This is interpreted as the fracture initiation pressure (Pi). The test is referred to as a formation integrity test (FIT) if no leakoff is observed,
i.e. test is stopped at predetermined pressure that does not generate a fracture.
To calculate the fracture pressure using data from wellbore fracture, the following equation will be applied [17]:
= 3 0 + 1.3
Fig 1.0 XLOT pressure versus time showing; LOP , FBP, FPP
XLOT and minifracture tests are conducted specifically for the purpose of stress determination [16]. These tests involves multiple cycles of pressurisation and de pressurisation, but use different equipment. An XLOT can be conducted in place of a LOT during drilling when better quality stress information is required,[18].
Extended leakoff test apart from the fact that it a method for measuring they can also be used to estimate .
can be determined from these tests using fracture initiation and/or reopening pressure (Hubbert and willis, 1957; Haimson and Fairhurst, 1967). The fracture initiation and/or reopening pressure depend on the stress concentration around an open hole. The minimum stress concentration around the wellbore is given by:
= 3 1.4
Tensile failure occurs when the concentration exceeds the tensile strength of the rock (in an absolute sense, tensile stresses have been defined as negative). Hence for tensilefailure of the wellbore wall:
= 3 1.5
The fracture initiation pressure (Pi, LOP) is Pw at fracture initiation, hence:
3 = 1.6
The fracture initiation pressure can be read directly from the pressure record, as can which is the fracture closure pressure (Fig 1.0). Hence eqn 1.6 can be rewritten as:
= 3 1.7
Since the initial fracturing cycle overcomes overcomes tensile rock strength, for subsequent cycles equation 1.7 can be rewritten as
= 3 1.8

Method of Breckels and van Eekelen (1982)
+0 32 = {322 2 }
The fracture gradients and lower bounds to LOPs for the US Gulf Coast is derived using the k value (ratio of horizontal to vertical effective stress) to define the relationships of how stress changes with depth. Differences only really occur in the way they determine the minimum effective stress. Following this historical review, a relationship between the minimum stress (Sh) and depth for
+{322 2}
Redefining equation (1.14) and (1.15) results to:
= +
1.15
1.16
the US Gulf Coast using fracture or "instantaneous shutin" pressure data was derived, using a data set of over 300
points from the US Gulf Coast, they mathematically fitted a curve that described the lower bound to 93% of the data [13]. The curve, a combination of a linear and powerlaw relationship, meant the magnitude of Sh could be determined solely from the depth (D):
S h = 0.0197 Du45 for D<7500 feet. 1.9
Sh = 1.167 D 4596 for D>7500feet. 1.10
More complex relationships were derived for Sh in abnormally pressured formations in the US Gulf Coast region using the depth and the magnitude of under/over pressure (actual minus normal pore pressure). Data from Venezuela and Brunei were also used to derive powerlaw relationships for minimum stress determination using a combination of depth and under/overpressure magnitude.

Method of Djurhuus and Aadnoy
The theory for determining the in situ stress state from multiple fracturing data and induced fractures from image logs is given below. The position of the fracture on the borehole wall was determined by minimization of the tangential stress [7], resulting in the equation
and in combination with a number of data sets, the two unknown horizontal in situ stresses sH and sh were determined from Eq. (1.16) using the least square method.
[19] used a similar approach, but shear stress was included. Provided sH and sh have been obtained, further determinedand from Equations. (1.11) but the backfigured values of and were not the same as the originally assumed values [30],[12] .
3 PROPOSED METHOD
In this section the method of inversion is used for estimating the magnitude of the minimum and maximum horizontal stresses and their directions. The input data for this method includes: pore pressure, overburden pressure, azimuth and inclination and data obtained from Leakoff, tests from different wells. The data are obtained from the already drilled wells and back calculation is done to determine the horizontal stress magnitudes of the field formation. As mentioned in the previous section this method is primarily based on the Kirschs wellbore failure equation given. The fracture equation is in reference to an arbitrarily chosen borehole coordinate system x, y and z and therefore, it is applicable to any wellbore orientation.
A critical look at equationn 1.16 reveals that the only
tan(2) = 2
1.11
unknown terms are and . Inserting the well geometry constants azimuth, and inclination, the square terms are
Thus the fracturing position on the borehole wall calculated from Equation (1.11) will be either = 0 or =90.
At tensile failure (assuming rock tensile strength is zero) when = 0, and >
= 3 0 + 1.12 and when = 0, and <
= 3 0 + 1.13
After substitution of the stress transformation equations, the above equations take the form
+0 + 2 = {32 22}
resolved and the equations become linear. The linearized equations can be placed in a matrix form and be solved. When many datasets are available from different leakoff tests, the equations can be represented in the following simple form:
[] = [][] 1.17Though, equation 1.17 can be solved with as many datasets as available, a minimum of two datasets are required. The more the datasets used, the better the results obtained. When many datasets are used to solve for only the two unknowns, the equation would result in an overdetermined system of linear equations. An exact solution cannot be obtained from the resolution of the overdetermined system. The error which is the difference between the measured data and the solutions obtained from the
+{32 22}
1.14
computer model built in this paper is given by:
[] = [][] [] 1.18The error obtained from the above equation is squared using the least square method. The squared error is minimized by differentiating it with respect to [] and equating the resultant equation to zero.
The maximum and minimum insitu stresses can be calculated with the following equations:
[] = {[] []}1[] [] 1.19In order to solve the right hand side (RHS) of the above equation it is important to note that not all matrices are
end
for k = 1 : m 1
for j = k + 1 : m
L(j, k) = U(j, k)/U(k, k)
U(j, k : m) = U(j, k : m)
L(j, k)U(k, k : m)
end
invertible but if a matrix is invertible then for a matrix A.
1 = 1 = 1.20
It turns out that a naive approach to finding the inverse of a matrix for solving systems of linear equation is usually inefficient. In practice other techniques such as LUP decomposition will be more numerically stable. For the model presented in this paper the LU Decomposition method is used to obtain matrix inverse for solutions to the RHS of equation 1.19. The algorithm for the LU Decomposition is given below:
Initialize U = A, L = I

RESULTS AND DISCUSSION
Snorre field in the North Sea
Three wells, P7, P8 and P9 are considered for this test. The depths of the wells range from about 0.7 to 2.4 km and are presented in Table 1.0. Data sets from the table are inputted into the model to determine the insitu stresses ad their directions. Stress values Obtained from the model are used to compute the fracture pressures used for validating the model. The process involves comparing the difference between results obtained from the model and the values from the measured data and selecting the set of results with the smallest error.
Table 4.1: Fractured data for Field case1
Data set
Well
Casing (in)
Depth (m)
(. . )
(. . )
(. . )
(o)
(o)
1
P7
18 5
8
13 3
8
9 5
8
18 5
8
13 3
8
9 5
8
18 5
8
13 3
8
9 5
8
1160
1.44
0.9767
1.8481
19.37
196.92
2
P7
1774
1.71
1.3993
1.9649
70.63
195.90
3
P7
2369
1.87
1.3814
2.0511
60.56
220.76
4
P8
756
1.39
0.9483
1.7325
8.61
167.78
5
P8
1474
1.65
1.2213
1.9151
60.26
187.65
6
P8
2321
1.83
1.3789
2.0475
43.82
129.16
7
P9
1005
1.59
0.9685
1.8087
16.88
92.77
8
P9
1503
1.62
1.2568
1.9199
36.30
85.69
9
P9
2418
1.75
1.3840
2.0548
55.09
89.13
Figure 4.1 below shows the input interface for the computer model containing inputs for 2, 5 and 8 data sets. A simulation of all data sets (1,2,3,4,5,6,7,8,9) is run for all possible combinations around the wellbore (360 degrees) to determine state of stress, based on the minimum squared error. In validating results obtained using the above data sets by computing fracture pressure from the estimated stress value, The results from the model do not match the test data as indicated by large error values, this signify that
the simulated datasets do not accurately represent the state of stress of the entire field depth. To get a better representation of the stress state of the field, simulations are done in smaller areas. For the sake of this study only three datasets (2,5,8) are used since for the three wells, these datasets occurs within the same hole section and as such provides a better representation for the state of stress in the formation.
Figure 4.1 Input Interface for the computer model.
Running the model using the selected datasets for all possible combinations around the wellbore (360 degrees) to determine state of stress, based on the minimum squared
= 1340
error, the most suitable solution (i.e the solution with the smallest error using the least square method) is selected and given as:
= 0.9213078
= 0.59729985
Squared error= 0.00005
The results given for the horizontal stresses ratio show that the maximum horizontal principal insitu stress is 0.9213078 times the overburden, the minimum horizontal stress is 0.59729985 times the overburden and the angle beta gives the direction of the maximum insitu stress with reference to the North. Figure 1.2 below shows the result interface for the computer model.
Figure 4.1: Result interface for the computer model
The interface above shows two separate tables. The first table displays the stress results and their corresponding directions for all possible combination around the wellbore (through 360 degrees). The second table shows the best
match for the stress value and stress direction based on smallest squared error. Fig 1.3 below show the validation of this matched value relative to the other values obtained from the computer model.
Figure 1.3: A plot of Versus Squared error wing direction of H
Table 1.1 Results for selected Dataset run
Squared error
50
0.838850542867032
0.81725497507815
0.0162030340225378
51
0.864589199705222
0.803347657770512
0.0155380204849594
52
0.875296571628304
0.796739813829991
0.0146337846452796
53
0.878565054892132
0.793913022063166
0.0137572571839558
54
0.878315193878513
0.792978685209166
0.0129842865911992
55
0.876467671185962
0.792977811549502
0.0123228929178436
56
0.873957726508797
0.793424588941485
0.0117607370663136
57
0.871241853604652
0.794069981557905
0.0112815524330996
58
0.868541180255681
0.794785006006836
0.0108702946829976
59
0.865959134648874
0.795502859638067
0.0105143873598308
60
0.863539522266663
0.796189616854101
0.0102036973362747
61
0.861295860959391
0.796829009803338
0.00993016142899174
62
0.859226503125309
0.797414328099249
0.00968737387762403
63
0.857322547062641
0.79794401505087
0.00947022720962884
64
0.855572003660176
0.798419233330325
0.00927462388895462
65
0.85396198033385
0.79884250447445
0.00909725229009785
66
0.852479799906464
0.799216945599564
0.00893541486832753
67
0.85111354270907
0.799545843607931
0.00878689677851594
68
0.849852276332249
0.799832422353013
0.00864986522865026
69
0.848686118176835
0.800079720860271
0.00852279201934713
70
0.847606211172081
0.800290535484637
0.00840439356238269
71
0.84660465728091
0.800467398573808
0.00829358411372414
72
0.845674433435153
0.800612577535167
0.00818943904559873
73
0.844809303283168
0.800728084814862
0.00809116579067924
74
0.844003731758339
0.800815693199392
0.00799808068773419
75
0.843252805874515
0.800876953172715
0.00790959039734615
76
0.842552163138437
0.800913210452232
0.00782517688053778
77
0.841897927864727
0.800925622660069
0.0077443851734731
78
0.841286655099705
0.800915174583674
0.00766681337042099
79
0.840715281581425
0.800882691773294
0.0075921043613033
80
0.840181083054676
0.800828852393633
0.00751993897125487
81
0.83968163724467
0.800754197340437
0.00745003022630603
82
0.839214791827676
0.800659138680038
0.00738211852780377
83
0.838778636795113
0.800543966488756
0.00731596756308459
84
0.838371480675102
0.800408854171129
0.00725136081456442
85
0.837991830144222
0.80025386232714
0.00718809855627019
86
0.83763837262804
0.80007894122405
0.00712599524776064
87
0.837309961550021
0.799883931909538
0.00706487725172119
88
0.837005603944095
0.799668565981487
0.00700458081431108
89
0.836724450196705
0.79943246400617
0.00694495025733587
90
0.83646578572999
0.799175132550901
0.00688583633912274
91
0.836229024479843
0.798895959769205
0.00682709474700112
92
0.836013704061767
0.798594209445437
0.00676858468887777
93
0.835819482554684
0.798269013371024
0.00671016755477996
94
0.835646136869239
0.797919361884581
0.00665170562159648
95
0.835493562703809
0.797544092362003
0.00659306077569828
96
0.835361776129654
0.797141875388294
0.00653409322873155
97
0.835250916887687
0.796711198277944
0.00647466020169253
98
0.835161253525106
0.796250345532636
0.00641461455139619
99
0.835093190552348
0.795757375729737
0.00635380331160033
100
0.835047277862053
0.795230094218146
0.00629206611825546
101
0.83502422272504
0.794666020853213
0.0062292334844844
102
0.835024904767503
0.794062351821639
0.00616512488576094
103
0.835050394443976
0.793415914380424
0.0060995466091054
104
0.835101975658534
0.792723113046474
0.0060322893115852
105
0.835181173361326
0.79197986540773
0.00596312522257964
106
0.835289787170359
0.791181525257383
0.00589180491050807
107
0.835429932355899
0.790322790146714
0.00581805351731003
108
0.835604089898374
0.789397589663598
0.00574156634185279
109
0.835815167820635
0.788398949710367
0.00566200362535858
110
0.836066576643381
0.787318826689764
0.00557898435618433
111
0.836362322677195
0.786147903690059
0.0054920788656535
112
0.83670712402852
0.784875338319548
0.0054007999283452
113
0.837106555778596
0.783488448533605
0.00530459200570599
114
0.837567232964318
0.781972318275688
0.00520281817666725
115
0.838097042997369
0.780309298511002
0.00509474417787819
116
0.838705443369512
0.778478370527241
0.00497951882344552
117
0.839403846454329
0.77645432611432
0.00485614988452616
118
0.840206121751503
0.774206701778474
0.00472347428085376
119
0.841129258280011
0.771698379031067
0.00458012117706646
120
0.842194247930774
0.768883726284504
0.00442446631836897
121
0.843427277403088
0.765706104304198
0.00425457577020172
122
0.844861356469777
0.762094477876667
0.0040681373589532
123
0.846538570842917
0.75795875841556
0.00386237904185886
124
0.848513239614404
0.753183326876016
0.00363397629762787
125
0.850856395693436
0.747617928633016
0.00337895801213026
126
0.853662212622996
0.741064766380329
0.00309263820241295
127
0.857057287812815
0.733260145655415
0.00276964328723255
128
0.861214031548835
0.723848585101269
0.00240420512141005
129
0.866369581544968
0.712347519698784
0.00199113148418028
130
0.8728507256524
0.698103939995059
0.00152845640144262
131
0.881099795176144
0.680258629529337
0.00102424023496015
132
0.891674180985399
0.657786267799832
0.000513625674200632
133
0.905112436616509
0.629857155605929
0.00010070937934591
134
0.921307809268535
0.597299847015716
0.0000543225287494335
135
0.93746237490722
0.567043341766276
0.000975141183573695
136
0.944005264611821
0.560232341455631
0.00380338075971586
137
0.925957509124309
0.607022115678056
0.00873056225409314
138
0.884557231475457
0.699874833335905
0.0135425697539379
139
0.843660303528262
0.786166581745686
0.0159311613582524
140
1.99515295954198
0.472106099551516
0.113949347874849
141
2.03542020245174
0.557852274364309
0.135142733589768
142
2.06990628066773
0.639079713153741
0.157152073123614
143
2.09870559804928
0.715499104787395
0.179759057591886
144
2.12196368028863
0.78691301180499
0.20276318652632
145
2.13986597183613
0.853205358940119
0.225982879872826
146
2.15262754249968
0.914330533443066
0.249255828836673
147
2.16048390646119
0.97030260390239
0.272438734761705
148
2.16368305168408
1.0211850350029
0.295406579241589
149
2.16247869537355
1.06708115829556
0.318051555468791
150
2.15712472069145
1.1081255587418
0.340281773562967
151
2.147870709004
1.14447645551882
0.36201983375293
152
2.13495845736402
1.17630909306914
0.383201342604351
153
2.11861935925405
1.2038101129536
0.403773430166149
154
2.09907252455175
1.22717284627242
0.423693310627424
155
2.07652351933639
1.24659344753956
0.442926916127907
156
2.05116361514017
1.26226778128486
0.461447622774248
157
2.02316944869367
1.27438896993121
0.479235079536253
158
1.99270300572272
1.2831455136254
0.496274144287662
159
1.95991185494563
1.2887198980463
0.512553926529244
160
1.92492957044403
1.29128761349431
0.528066932995722
161
1.88787629164608
1.29101651682684
0.542808310127768
162
1.84885938006959
1.28806647635923
0.556775176041547
163
1.80797414065837
1.28258924823229
0.569966033942025
164
1.76530458303403
1.27472854065883
0.582380258729955
165
1.72092420436057
1.26462022972822
0.594017648707283
166
1.67489678089468
1.25239269698996
0.604878034683575

CONCLUSION
Wellbore instability problems which result to additional drilling cost are majorly due to matrix stress. Hence accurately predicting the insitu stresses in a rock formation can go a long way to solve a lot of the challenges facing the petroleum and mining industries and a whole lot of money could be saved and accidents averted. In this project, a handy tool that is easy to use to predict the horizontal principal insitu stresses was developed. The results from simulations obtained from this work demonstrated the reliability of this program to:

Estimate the magnitude of the horizontal principal matrix stresses of a rock field based on data obtained from LOT, pore pressures, overburden stresses and well directions. The model can accommodate any number of input data but a minimum of three input data is required to get a meaningful result.

The estimated magnitude of the matrix stresses can be used to calculate fracture pressures.
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