Quantitative Fracture Modeling on Low-permeability Sandstone ...
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Quantitative Fracture Modeling on Lowpermeability Sandstone Reservoirs by Stress Field Qinghua Chen School of Geosciences, China University of Petroleum, Qingdao City, Shandong Province 266580, China Email: [email protected]
Yan Liu1,2 1
School of Geosciences, China University of Petroleum, Qingdao City, Shandong Province 266580, China 2 Shandong Shengli Vocational College, Dongying City, Shandong Province 257097 Email: [email protected]
Jianwei Feng School of Geosciences, China University of Petroleum, Qingdao City, Shandong Province 266580, China
Abstract—A new method is used to quantify the relationships between stress field and primary fracture characters (aperture, density) in low-permeability sandstone reservoirs based on laboratory experiments and mathematical development, through which, quantitative fracture modeling is set up. Using this modeling, we can load it into the stress analysis software (Ansys 10.0) to acquire 3D palaeo stress field and current-stress field after selecting appropriate mechanical parameters and stress loading. Index Terms—Low-permeability, Quantitative Modeling, Stress Field, Fracture, Ansys
I.
INTRODUCTION
Low-permeability sandstone reservoirs are generally defined as oil reservoirs with a gas permeability ranging from 0.1 ×10-3μm2 to 50 ×10-3μm2, whose canonical characteristics are low textural and compositional maturity, high digenesis, low-porosity and permeability, developed fractures [1, 2] and strong heterogeneity (Lingzhi Jiang, 2004). Additionally, reservoirs with such characters are brittle, easily to generate fractures in tectonism. Therefore, researchers usually finite element stress simulation to predict this kind of fractures(Mingsheng Yuan, 2000), after this, fracture development directions are presented, and based on the measure of stress, the fracture intensity index is interpreted and evaluated by strain and strain energy qualitatively(e.g. J.E., Warren et al., 1963; H. Kazemi et al.,1969; A., Deswaan., 1976; D. Bourdet, J. A. et al., 1984). Nevertheless, to this day, quantitative prediction in any position of structure still is a problem that remains to be solved, and there is no reliable solution to it (e.g. Z.-X., Chen, 1989; G.V., Chilingarian et al.,1992; Nelson. R. A et al.,1977; M., Chen et al.,1998). Domestic scholars have ever tried to apply data of strain energy density and fracture probability for numerical fitting, then further © 2013 ACADEMY PUBLISHER doi:10.4304/jnw.8.8.1890-1898
through which to predict fracture density. Huizhen Song etc made a test on limestone samples of buried hill reservoir of Ordovician in Lunnan area of Tarim basin, fitting quantitative formulas between strain energy density and fracture volume density. However, Zhongyi Ding etc thought that there was certain limitation in simple fitting by strain energy density or cracking value, and proposed his combination idea of the two characters, using the least-squares algorithm to do the fitting in order to get the final prediction, as a result the error did not exceed 25 per cent. As to quantitative method for fracture aperture calculation [3, 13], there is still no any report (Xinsheng Zhou, 2003). Most often the reason for this is the absence of effective reasonable mechanical model for a long time (e.g. Murray GH, 1968; Wang Ren et al., 1979 ). As is well known, structural fracture often appears as various shapes, such as tensional fractures, conjugate shear fractures and shear opening fractures falling in between, so in practice the tensional fracture criteria and shear fracture criteria should be applied comprehensively (Shipeng wen, 1996). Generally, in numerical modeling of current tectonic stress field, Griffith Criterion is utilized for the calculation of tensional fracture, and Coulumb-Mohr Criteria for shear fracture (e.g. Chen Bo et al., 1998; Chengxuan Tan et al., 1999). Thus, the paper here attempts to set up this kind of model through test and Theory deduction, to get stress field distribution and main fracture characters, such as aperture and density, and then to instruct quantitative fracture predication with stress field simulation. II.
PRELIMINARIES
Test results summed up by the predecessors show that, under slow uniaxial compression, there will generate crossing angle less than 45°between the shear fractures
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and compression direction; on the contrary, under rapid uniaxial compression, there will occur tensional fracture parallel to compression direction, in that way, rapid loading causes to brittle fracture, slow loading causes to ductile fracture (Kezheng Lu, 1996). As has been already demonstrated, underground rock deformation progressed during the long geological time continuously, marked by slow loading and creep behavior, therefore this study has designed the uniaxial compression tests. According to the standards of International Association of Rock Mechanics (IARM) on uniaxial compression tests (Chuanjin Zhanget al., 2002), in which rock samples were processed into cylinders with diameter 50mm, height 25mm [4], and compression test was carried on with permanent strain speed 2.0 105 s 1 , synchronously obtaining the various breakage shape charts of all rock samples as follows (Fig. 1).
Figure 1.
Rupture drawing of some test samples
Sample S101-DC3, S101-DA4 and S101-DA2 performance for shear opening fractures, sample S103DA3,S101-DB3 and S105 performed for shear fracture, with a little extension. Among them, sample S101-DB3’s rupture trace is characterized by a series of tension cracks to joint and run through into a macroscopical shear zone. Results show that the angle of rupture is less than 45°, in conformity with the Mohr-Coulomb criterion [8]. Next, the enveloping curve will be extracted from drawing mohr stress circle by experimental results (Fig. 2). Here 1 , 2 respectively is internal frictional angle in different confining pressure, k1 , k 2 respectively is the straight-line segment slope of Mohr-Coulomb curve, that is internal friction coefficient, and 0 is cut-off point of confining pressure, whose value is 5MPa. After calculation, it can be concluded as follows.
Figure 2.
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relatively large, the situation is opposite, in this way, the failure mode will change from extention fracture to shear fracture, even compresso-shear fracture. The numerical value is: If 0 3 0 5MPa , 1 51.83 (angle of rupture) 1 45 / 2 19.09 If 3 0 5MPa , 2 41 (angle of rupture)
2 45 / 2 24.5 . From above analysis it is not difficult to see that: under state of compression, two-step Mohr-Coulomb criterion is applicable to the low-permeability sandstone’s breaking situation. But under state of extension, it will be replaced by Griffith criterion, because the above criterion is inapplicable. At this time, for offwhite fine sandstone, the criterion will be selected as following: (1) If 3 0 ,the Mohr-Coulomb criterion will be applied, that is 1 3 3 C0 cos 1 sin 2 2 where, if 3 0 5MPa , 2 41 , angle of rupture 2 45 / 2 24.5 ; If 0 0 5MPa , 1 51.83 , angle of rupture 1 45 / 2 19.09 , cohesion
C0 6.53MPa as a actual value. (2) If 3 0 , Griffith criterion is used, next, we will take discussion according to the situation If (1 3 3 ) 0 , the fracture criteria is (1 2 )2 ( 2 3 )2 ( 3 1 ) 2 24 T ( 1 2 3 )
1 3 2( 1 3 ) If (1 3 3 ) 0 , fracture criteria is simplified to 3 T , 0 . cos 2
The basic idea of physical test is firstly to select a large number of core samples with common lithology and good mechanical properties, which can be processed into cylindrical specimen complying with test conditions, and reached various stress state by loading cycle, secondly the core samples will be bought out from testing machine and abraded into thin slice for optical observation and statistics, finally the test results are analyzed.
Two-step Mohr-Coulomb curve in Shishen100 block
When the confining pressure is less than a value, the internal frictional angle is quite large, and angle of rupture 45 / 2 relatively small, thus extension fracture predominantes. When confining pressure is © 2013 ACADEMY PUBLISHER
,
Figure 3.
Compression test curve of porous racks
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Through collecting predecessor’s data [5-7] and actual sample tests on Shishen block 100, it is concluded that micro-fractures at the course of compression have some evolutionary characteristics as following: (1) In the compact segment (OA of stress-strain curve, see Fig. 3), the micro-fractures of porous rock samples are compressed to cause density, fracture aperture and permeability to loss or diminish dramatically. (2) In the AB segment of stress-strain curve, the curve presents proximate straight line, at this point, rock samples occur elastic deformation with slope be elastic modulus. (3) From the BC segment in Fig. 3, we can see that the stress-strain curve no longer maintain a straight line, a significant increase in radial strain begins, known as expansion.
exists here. However, types Ia, II and III are all belong to compressional stress types. In order to establish the relationship between stress, strain and fracture aperture, density under the complex stress condition, we selected an element (REV) to take analysis [10-12], while adhering to a moderate simplification policy we make some hypothesis. (1) This element is so small enough that it can be easily cut through thoroughly. (2) If the randomly distributed micro-fractures in the element to be eliminated, we think the fractures with permeability could not exist before not exposed or subjected to stress. (3) Although the confused permutation among crystal grains will cause brittle rock to show approximate isotropic as a whole, as long as the mineral composition, the cementation and the diagenesis vary within limits, it will be considered as homogeneous isotropic medium. (4) The element is supposed as a parallel epipedon with with sides L1, L2, L3 (Fig. 5), here we specify the compression stress positive and 1 2 3 , thus the
1 corresponding with the side L1, the 2 corresponding with the side L2, the 3 t corresponding with the side L3.
Figure 4.
Uniaxial stress-strain curves of core samples of well S103
(4) While stress is increased to the peak stress (C point), the overall macroscopic cracking occurs, with specimen damaging, stress decreasing and strain increasing significantly. (5) Breakdown of sandstone is the result of intragranular micro-fractures, as well as intergranular micro-fractures constantly development, assemble, even connect to each other [9]. (6) At the base of Xia Jixiang (Jixiang Xia, 1982), Jaeger, Cook and this laboratory results, we can further generalize modification stage of stress and fractures into Fig. 4. III.
If fractures generate under loading (stress) at this time, the fracture plane normal certainly will distribute within principal plane of 1 3 , and long axis direction of fracture will form an angle with maximum principal stress ( 45
, according to Mohr-Coulomb 2 strength theory) or parallel to intermediate stress, therefore, fracture morphology can be equivalent as following (Fig. 5).
PROPOSED SCHEME
As well known, crustal stress is the main factor to generate fractures in underground rock mass, that in most areas is characterized by disharmonious triaxial state of stress with the horizontal one as principal, among the three usually two are horizontal, one approximately vertical, finally to develop three classed or four types of stress state. The major principal stress of Class I distributes along vertical direction, namely H h z ; the minor principal stress of class II distributes along vertical direction, namely H z h ; the middle principal stress of class III distributes along vertical direction, namely H z h . Among them, class I also can be further divided into types Ia and Ib, the former is marked by v H h 0 , the later by
v 0 , and at least one of H and h is less than 0, or they synchronously equal to 0, that is tensional stress © 2013 ACADEMY PUBLISHER
A Figure 5.
B
Element volume characterizing relationship between fracture parameters and stress
A: Equivalent fracture shape inner element under
1 2 3 coordinate system;
B: Transection perpendicular to 2 , namely 1 3 plane According to Elasticity Theory(B.R Loren et al. 1985) there only agglomerates elastic strain energy in solid material when it deforms, and the energy size can be measured by strain energy density, that is
1 2
(11 2 2 3 3 )
(1)
Here, elastic strain also subjects to Hooke's law (Weixiang Wang, 1984), that is
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1
1 1 ( 2 3 ) E
(2)
2
1 2 (1 3 ) E
(3)
3
1 3 (1 2 ) E
(4)
Combing Eqs. (1), (2), (3) with Eqs. (1), we get strain energy density expressed by principal stress at once, that 1 is [12 2 2 32 2 ( 1 2 2 3 1 3 )] 2E According to Maximum Strain Energy Density Theory [14], when release rate of elastic strain energy agglomerates in the brittle material equals to the energy demanded for generating fractures per unit volume of element, the brittle material will break. Theory and practice have proved that not only the sandstone, fine sandstone but also the sandstone comprising calcareous all will show strong brittle fracture character. For these brittle rocks, when the 3-D stress condition achieves its resistance to bursting strength [15], brittle fracture will occurs with strain energy to release, part of which to offset the surface energy for new fractures, meanwhile rest of which to give off by elastic waves, that is the cross-section energy mentioned in above references (Barton. N et al., 1977). Nevertheless, to fracture, the released elastic wave is so weak that it can be neglected, so we develop the following formula by energy conservation principle.
Dvf
Sf V
f J
(5)
calculated through physical concept and conversion to obtain a and b; another way is through experimental data, curve fitting, mathematical methods to calculate the coefficients In most cases, rock in earth crust is under 3-D compression stress state, such as stress Ia, II and III. Because there exists much difference in mechanical property of rock between under compression stress and tensile stress, here we must discuss respectively. A. Relationship between Stress-strain and Fracture Volume Density Under Uniaxial Compression As stated above, the stress 0.85 c acts as a key value for the beginning of new fractures produced in physical test and at this point the corresponding strain density energy is so associated with e in concept, in that case, let us assume e the strain density energy of 0.85 c to be substituted into the formula to get J , then a and b also can be acquired. Based on the results of rock mechanics test we had average uniaxial compressive strength of gray finemedian grained sandstone reservoir in Shishen block 100, such as c 42.69MPa , 0.85 c 36.28MPa , after calculation to get e 1.133 103 J / m3 . If we turn Eqs. (6) into J
- e Dvf
, the J will be acquired.
Then the average of J is also obtained, that is , then together with J 1087.35J / m2 e 1.133 103 J / m3 are be substituted into the Eqs. (6) , we quickly get the relationship of strain density and fracture volume density
Dvf 9.2 104 104.2
where Dvf is defined as volume density per unit volume (REV); f is strain energy density for new fractures; V is characterization of the element (REV); S is surface area of new fractures; J is the required energy for fractures per unit area, i.e., cross-section energy of fracture (note: here the cross-section is different from the one by intermolecular attraction based on theoretical deduction, and far less than the theoretical value). If f in formula (5) is regarded as the residual strain
(7)
In order to testify the effectiveness of e , we use method of curve fitting to get a and b for comparison with the first approach (Fig. 6). The results are as follows:
Dvf 8.04 104 86.128
(8)
where, the correlation coefficient is 0. 995.
energy density derived from current strain energy density per unit volume subtracting elastic strain energy density for new fractures ( e ), then we have:
Dvf
Sf V
f J
- e J
1 e a b (6) J J
where is the strain energy under current stress; e is the elastic strain energy for new fractures, and a
b-
e
1 , J Figure 6.
is the coefficient, it can be derived by two
J methods. One way is based on the values of J , e
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Relationship between strain energy density and fracture volume density of specimens
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f e
B. Relationship between Rock’S Stress-strain and Fracture Volume Density under Complex 3-D Compression Stress State Although the quantitative relationship has been described in detail above, in practice, rock always is in complex 3-D compression state [21] to cause the formula lacking of transferability. Nevertheless, there still exist much similarity between the curves of triaxial compression test and uniaxial compression test, such as when axial stress arrives 0.85 max , precursory fractures will begin forming, whose strain energy is the elastic strain energy ( e ) for overcoming fractures [16].
1 ( 11 2 2 3 3 ) 2 0.852 2C0 cos (1 sin ) 3
(14)
225.2(1 sin ) 0.85 ( 2 3 ) 112.6
If the minimum stress 3 is known, here certainly has 3 0MPa ( 3 0MPa representing tensile stress state), by the rock failure criteria we will calculate the minimum value of maximum principal stress, namely
1min
2C cos (1 sin ) 3 0 1 sin
Figure 7.
(9)
The calculated 1 is not the actual maximum principal stress, but the failure stress under 3 acting, then for well discrimination we replace 1 by p
p
2C0 cos (1 sin ) 3 1 sin
Relationship between fracture surface energy and confining pressure
(10)
1 , if 2 Under random stress state ij 3 3 0MPa , inserting ij in Eqs. (10), and 1 p , the stress state will conform to conditions of rock failure, then according to elastic theory to get e and adding Eqs. (1) and Eqs. (2) we get
1 1 2 2 (11) 1 0.85 p (0.85 p 2 ( 2 3 )) 2E
Without question, under condition of confining pressure, obviously the pressure will stop fracture producing in certain extent, thus accumulative energy not only needs to overcome cohesive strength between molecules, but also to overcome confining pressure to tear rock finally, at the same time, the fracture surface energy also increases. Now assume a default fracture model (Fig. 7), with aperture b, area S and confining pressure 3 , then the work ( 3 ) for fracture is
Dvf
f e
2C cos (1 sin ) 3 E 112.6 0 1 sin
(13)
Put Eqs. (10) and Eqs. (13) into Eqs. (12), then it be simplified to © 2013 ACADEMY PUBLISHER
J / s 3b
(16)
We insert Eqs. (14), Eqs. (15)and Eqs. (16) into Eqs. (6) to get fracture volume density formula：
Combining Eqs. (6) and Eqs. (10) we have
Although not described the influencing factors on rock’s parameters in detail above, we indeed have acquired the relationship between elastic modulus and confining pressure, that is
(15)
Because of
e p (0.85) p (0.85) 0.85 p 0.85 p
1 ( 11 2 2 3 3 ) (12) 2 1 0.85 p (0.85 p 2 ( 2 3 )) 2E
3bS
f J
1 ( 11 2 2 3 3 ) 2( J 0 J )
0.852 2C0 cos (1 sin ) 3
(17)
225.2(1 sin )( J 0 J )
0.85 ( 2 3 ) 112.6( J 0 J ) where J 0 is surface energy when confining pressure equals to 0, unit:J/m2; J is the additional surface energy by confining pressure 3 ; is Poisson's ratio. Based on above, Eqs. (17) is equivalent to
Dvf
f J
(11 2 2 3 3 ) 0.852 11 2( J 0 J ) 2( J 0 J )
(18)
To verify the validity of above formula or modeling, we use Eqs. (18) to calculate fracture volume density and
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compare with actual values, the raw data mainly comes from the triaxial test results of sandstone by Tao Zhenyu (Zhenyu Tao,1996) and the compression test under confining pressure of sandstone by Institute of Geom. C. Relationship between Fracture Linear Density and Volume Density Here we take discussion at different situations about their quantitative relationship, and define fracture linear density as the numbers per unit length. a. Relationship between them as fracture surface forms an angle to the principal stress
Figure 9.
Relationship between true fracture space and sighted space in 1 3 section
As shown in Fig. 9,when fracture surface is parallel to principal stress 1 , there is 0 , obviously
S f L 3 Dlf L1.L2 Equivalent fracture model of 1 3 section
Figure 8.
while 0
perpendicular to 2 , and let condition of fractures equivalent to uniformly-spaced arrangement (Fig. 8). Now we calculate total surface area of fracture by getting total length of fracture body firstly, secondly multiply total surface area L2. As shown in Fig. 10, the fracture can be divided into two parts [22], one is along direction of L1 ,the other is along direction of L2. Note: Since we have omitted all the minor details in the deducting process, then in section 1 3 the total length of fracture body will be
Lt L1t L3t
L Dlf sin L1 2cos
L3 Dlf cos L3 2
2sin
(19)
Consequently total surface area of fracture body is calculated as following
S f L2 Lt L2
L12 Dlf sin L1 2cos
L32 Dlf cos L3 2sin
Put Eqs. (20) into the formula Dvf
Sf V
(20)
, where
2 Dvf L1 L3 sin cos L1 sin L3 cos L12 sin 2 L23 cos2
and V L1 L2 L3 , then
Dlf Dvf
(23)
D Relationship between stress-strain and fracture aperture To supply aperture with b, space with d, and L1 L3 L , clearly there is
Dlf
1 d
(24)
While the angle between fracture surface and maximum principal stress is greater than zero, the Fig. 11 characterizes their relationship. In the direction of 1 , fracture sighted space d1 d / sin , sighted aperture b1 b / sin , thereby
Dlf 1
1 Dlf sin d1
(25)
In the direction of 3 , fracture sighted space d3 d / cos , sighted space b3 b / cos , then
Dlf 3
1 Dlf cos d
(26)
in the direction of 1 and 3 ,Combing Eqs. (25) and Eqs. (21)
b. Relationship between them as fracture surface forms an zero angle to the principal stress
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V
where Dlf 1 and Dlf 3 is the fracture sighted linear density
V L1 L2 L3 , and simplified to Dlf
Sf
simplified to
In the REV element, fracture body is parallel to intermediate principal stress, fracture strike has an angle with 1 . So as to we make section 1 3
2 1
Insert it into formula Dvf
(22)
(26), we get In the direction of 1 ,
Dlf 1b1 Dlf sin b / sin Dlf b In the direction of 3 ,
(27)
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Dlf 3b3 Dlf cos b / cos Dlf b
(28)
It is thus clear that the product of fracture aperture and linear density in direction of 1 and 3 is invariable.
Dlf 3b3 Dlf cos b / cos Dlf b
(29)
Next, under condition of principal strain, fracture aperture is decided by the tensile strain. For strong brittle sandstone, we define a new parameter f as fracture strain, non-dimensional, and think that fractures will begin to produce once tensile strain exceeds some value, after this, the tensile strain is mainly caused by the cracking of micro-fractures [18], hence we set up the relative formulas:
Dlf 3 ( L0 L) b3 f L0
(30)
L f L0
(31)
Figure 10.
Maximum principal stress 1 isoline space diagram by Ansys
and
Combing Eqs. (30) and Eqs. (31) can obtain
Dlf 3b3
f 1 f
(32)
Since 1>> f , inserting Eqs. (29) in Eqs. (32) and simplified to b
f Dlf
Figure 11.
Intermediate principal stress 2 isoline space diagram by Ansys
, where f | | | 0 | ; b is
fracture aperture; f is fracture strain; Dlf is fracture linear density; is the tensile strain under current stress; 0 is the maximum tensile strain in elastic deformation stage corresponding to the point of fracture initializing [17]. IV.
EXPERIMENTAL RESULTS
Take Shishen Block100 as an example, through uniaxial compression test we have got 0 ＝ 11.40 104 , J＝1087.35J/m2, 0 , hence combing Eqs. (8) gives: Dlf Dvf 8.04 104 86.128 After above study to obtain: f | | | e | 11.40 104 b Dlf Dlf 8.04 104 86.128 Using above eqs, loaded relate data to ANSYS, we got three principal stress isoline space diagram(see Fig. 10,11,12).This fully indicates that replacing e by strain energy density corresponding with 0.85 c is correct, based on this principle, just a few tests can work the quantitative relationship between stress-strain and fracture volume density out. The results are so close to the observed data under microscope that the relative error is less than 10%, which verify the validity of the approach [19, 20].
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Figure 12.
Minimum principal stress 3 isoline space diagram by Ansys
It is shown by results that relative error of 94.11% samples are less than 15%, proving the validity of the model. On the other side, as long with the increasing of 3 , f decreases drastically, this appearance is inconsistent with actual situation. V.
CONCLUSION
As above we firstly have discussed the relationships between stress-strain and fracture aperture, density of rocks and set up relative theoretical model [23], then examined the effectiveness through true test data. Secondly we have discussed the relationships between them under the condition mingled with tensile stress, comprehensively described as follows: (1) If there is no tensile stress, for sandstone material the Mohr-Coulomb criterion will be applied, that is 1 3 3 C0 cos 1 sin 2 2
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Hence for fine sandstone reservoirs in Shishen block100, if all the conditions conform to above formula, there definitely will produce fractures suitable for following computation process [24, 25]: 2 2 2 1 1 2 3 2 ( 1 2 3 ) w f w we 2 E 0.852 p 2 2 ( 2 3 )0.85 p 2C0 cos (1 sin ) 3 p 1 sin E E 0 p D wf vf J J J 0 J J 0 3b 2 Dvf L1 L3 sin cos L1 sin L3 cos Dlf L12 sin 2 L23 cos 2 Dlf b 3 0 1 0 ( 3 (0.85 p 2 ) E Here the meanings of parameters are same as above. Because rock uniaxial compression being a special case in triaxial compression, we take Shishen block100 as an example to express the relationship between stress-strain and fracture arameters: Dvf 9.2 104 w 104.2 Dlf 0.5428Dvf 1.2215 4 Dlf b f | 3 | | e || | 11.40 10 (2) If there have tensile stress, to sandstone material the Griffith criterion is used, that is When 3 0 and (1 3 3 ) 0 , the applied failure criterion is (1 2 )2 ( 2 3 )2 ( 3 1 ) 2
24 T ( 1 2 3 )
cos 2
1 3 2( 1 3 )
When (1 3 3 ) 0 , failure criterion is simplified to
3 T , 0 Based on this, the relationship between stress-strain and fracture parameters is expressed by 1 1 2 w f w we 2 ( 11 2 2 3 3 ) 2 E t wf Dvf J J J 0 J J 0 3b 2 Dvf L1 L3 sin cos L1 sin L3 cos Dlf orDlf D vf L12 sin 2 L23 cos 2 Dlf b 3 0 0 t E
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Take Shishen block100 as an example, in the middle of the third Shahejie formation there have J 0 1087.35J / m2 , 0 4.90 104 , E 4.73GPa . If
(1 3 3 ) 0
arccos ( 1 3 ) / 2( 1 3 ) 2
,
then
,
Thereby take the formula 2 Dvf L1 L3 sin cos L1 sin L3 cos to calculate Dlf L12 sin 2 L23 cos2 the fracture linear density. If (1 3 3 ) 0 , and 0 , then the fracture linear density definitely equals to volume density, that is Dlf Dvf . REFERENCES [1] Lingzhi Jiang, Jiayu Gu, and Bincheng Guo, “Characteristics and formation mechanism of lowpermeability clastic reservoir of oil and gas-bearing basins in China,” Acta Sedimentologica Sinica, vol. 22, pp. 13-18, 2004. [2] Mingsheng Yuan, Mao Pan, Hengmao Dong et al. “Exploration on low-permeability fractured reservoirs,” JPT, Beijing, vol. 5, pp. 3-14, 2000. [3] Xingui Zhou, Chengjie Cao, Jiayin Yuan, “The status and prospects of quantitative structural fracture prediction of reservoirs and research of oil and gas seepage law,” Advances in Earth Science, vol. 18, pp. 398-404, 2003. [4] G. I., Barenblatt, I. P., Zheltov, and N., Kochina, “Basic concepts in the theory of seepage of homogeneneous liquids in fissured rocks,” Prikl. Mat. Mekh., vol.24, pp. 852-864, 1960. [5] J. E., Warren and P. J., “RootThe Behavior of naturally fractured reservoirs,” J. Soc. Pet. Eng., vol. 2, pp. 245-255, 1963. [6] H. Kazemi, “Pressure transient analysis of naturally fractured reservoirs with uniform fracture distribution,” Soc. Pet. Eng. J., vol. 146, pp. 451-161, 1969. [7] A., Deswaan, “Analytical solutions for determining naturally fractured reservoir properties by well testing,” Soc. Pet. Eng. J. vol 1, pp. 117-122, 1976. [8] D. Bourdet, J. A., Ayoub and Y. M, “Pirard. Use of pressure derivative in well test interpretation,” SPE 12777, Cal. Regional Meeting, pp. 1-17, 1984. [9] Z.- X.,Chen, “Transient flow of slightly compressible fluid through double-porosity, double-permeability systems-a state-of-the-art review,” Transport in Porous Media, vol. 4, pp. 147-184, 1989. [10] M., Chen, M., Bai, “Modeling Stress-dependent permeability for anisotropic fractured porous rocks,” International Journal of Rock Mechanics and Mining Science, vol.351, pp. 1113-1119, 1998. [11] Nelson. R. A and Handin, J. w, “Experimental Study of Fracture Permeability in Porous Rocks,” AAPG Bulletin, vol. 61, pp. 227-236, 1977. [12] Ren Wang, Zhongyi Ding, Youquan Yin et al. ,“Solid Mechanics,” Geological Publishing House, Beijing, vol. 82, pp. 124-141, 1979. [13] Murray GH, “Quantitative fracture study,” Sanish pool, Mckenzie County, North Dakota. AAPG Bulletin, vol.52, pp. 57-65, 1968. [14] Wen Shipeng, Li Detong, “Simulation technology on structural fractures of reservoirs,” Journal of China
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University of Petroleum (Edition of Natural Science), vol. 20, pp. 17-24, 1996. Bo Chen, Chonglu Tian, “To establish and research of threee-dimensional geological and mathematical model for quantitative prediction of structural fracture,” Acta Petrolei Sinica, vol. 19, pp. 50-54, 1998. Huizhen Song, “A attempt of quantitative prediction of natural crack on brittle rock reservoir,” Journal of Geomechanics, vol. 5, pp. 76-84, 1999. Chengxuan Tang, Lianjie Wang, “An approach to the application of 3_D tectonic stress field numerical simulation in structural fissure analysis of the oil_gas_bearing basin,” Acta Geoscientia Sinica, vol. 20, pp. 392-394, 1999. B. R Loren, T. R. Willshaw, “Brittle fracture mechanics of solids,” Seismological Press, Beijing, vol. 5, pp. 5-25, 1985. Barton. N, Choubey. V, “The shear strength of rock and rock joints in theory and practice,” Rock Mechanics, vol. 10, pp. 1-54, 1977. Wills-Richards. J, Watanabe. K, Takahashi. H, “Progress toward a stochastic rock mechanics model of engineered geothermal systems,” J Geophys Res, vol. 101, pp. 481496, 1996. Jing. Z, Wills-Richards. J, Watanabe. K, Hashida. T, “A new 3-D stochastic model for HDR geothermal reservoir in fractured crystalline rock,” Fourth International HDR Forum. Strasbourg. France, pp. 28-30, 1998.
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[22] Chen. Z, Naryan. S. P, Yang. Z et al., “An experimental investigation of hydraulic behavior of fractures and joints in granitic rock,” International Journal of Rock Mechanics and Mining Sciences, vol. 7, pp. 2061-2071, 2000. [23] William W. Guo, Mark Looi, A Framework of TrustEnergy Balanced Procedure for Cluster Head Selection in Wireless Sensor Networks, Vol. 7, No. 10, Journal of Networks, 1592-1599, 2012 [24] Yan Zhao, Hexin Chen, Shigang Wang, Moncef Gabbouj, An Improved Method of Detecting Edge Direction for Spatial Error Concealment, Journal of Multimedia, Vol. 7, No. 3, pp. 262-268, 2012 [25] Shuang Xu, Jifeng Ding, Palmprint Image Processing and Linear Discriminant Analysis Method, Journal of Multimedia, Vol. 7, No. 3, pp. 269-276, 2013
Qinghua Chen, male, born in November 1958, Shan County, Shandong Province, Professor, Doctoral Tutor. Mainly engaged in the study of structure geology analysis ,the basin structural characteristics, the law of oil and gas distribution, reservoir architecture analysis. Yan Liu, male, born in November 1972, Associate professor. Mainly engaed in the sturcture stress analysis .
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Quantitative Fracture Modeling on Lowpermeability Sandstone Reservoirs by Stress Field Qinghua Chen School of Geosciences, China University of Petroleum, Qingdao City, Shandong Province 266580, China Email: [email protected]
Yan Liu1,2 1
School of Geosciences, China University of Petroleum, Qingdao City, Shandong Province 266580, China 2 Shandong Shengli Vocational College, Dongying City, Shandong Province 257097 Email: [email protected]
Jianwei Feng School of Geosciences, China University of Petroleum, Qingdao City, Shandong Province 266580, China
Abstract—A new method is used to quantify the relationships between stress field and primary fracture characters (aperture, density) in low-permeability sandstone reservoirs based on laboratory experiments and mathematical development, through which, quantitative fracture modeling is set up. Using this modeling, we can load it into the stress analysis software (Ansys 10.0) to acquire 3D palaeo stress field and current-stress field after selecting appropriate mechanical parameters and stress loading. Index Terms—Low-permeability, Quantitative Modeling, Stress Field, Fracture, Ansys
I.
INTRODUCTION
Low-permeability sandstone reservoirs are generally defined as oil reservoirs with a gas permeability ranging from 0.1 ×10-3μm2 to 50 ×10-3μm2, whose canonical characteristics are low textural and compositional maturity, high digenesis, low-porosity and permeability, developed fractures [1, 2] and strong heterogeneity (Lingzhi Jiang, 2004). Additionally, reservoirs with such characters are brittle, easily to generate fractures in tectonism. Therefore, researchers usually finite element stress simulation to predict this kind of fractures(Mingsheng Yuan, 2000), after this, fracture development directions are presented, and based on the measure of stress, the fracture intensity index is interpreted and evaluated by strain and strain energy qualitatively(e.g. J.E., Warren et al., 1963; H. Kazemi et al.,1969; A., Deswaan., 1976; D. Bourdet, J. A. et al., 1984). Nevertheless, to this day, quantitative prediction in any position of structure still is a problem that remains to be solved, and there is no reliable solution to it (e.g. Z.-X., Chen, 1989; G.V., Chilingarian et al.,1992; Nelson. R. A et al.,1977; M., Chen et al.,1998). Domestic scholars have ever tried to apply data of strain energy density and fracture probability for numerical fitting, then further © 2013 ACADEMY PUBLISHER doi:10.4304/jnw.8.8.1890-1898
through which to predict fracture density. Huizhen Song etc made a test on limestone samples of buried hill reservoir of Ordovician in Lunnan area of Tarim basin, fitting quantitative formulas between strain energy density and fracture volume density. However, Zhongyi Ding etc thought that there was certain limitation in simple fitting by strain energy density or cracking value, and proposed his combination idea of the two characters, using the least-squares algorithm to do the fitting in order to get the final prediction, as a result the error did not exceed 25 per cent. As to quantitative method for fracture aperture calculation [3, 13], there is still no any report (Xinsheng Zhou, 2003). Most often the reason for this is the absence of effective reasonable mechanical model for a long time (e.g. Murray GH, 1968; Wang Ren et al., 1979 ). As is well known, structural fracture often appears as various shapes, such as tensional fractures, conjugate shear fractures and shear opening fractures falling in between, so in practice the tensional fracture criteria and shear fracture criteria should be applied comprehensively (Shipeng wen, 1996). Generally, in numerical modeling of current tectonic stress field, Griffith Criterion is utilized for the calculation of tensional fracture, and Coulumb-Mohr Criteria for shear fracture (e.g. Chen Bo et al., 1998; Chengxuan Tan et al., 1999). Thus, the paper here attempts to set up this kind of model through test and Theory deduction, to get stress field distribution and main fracture characters, such as aperture and density, and then to instruct quantitative fracture predication with stress field simulation. II.
PRELIMINARIES
Test results summed up by the predecessors show that, under slow uniaxial compression, there will generate crossing angle less than 45°between the shear fractures
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and compression direction; on the contrary, under rapid uniaxial compression, there will occur tensional fracture parallel to compression direction, in that way, rapid loading causes to brittle fracture, slow loading causes to ductile fracture (Kezheng Lu, 1996). As has been already demonstrated, underground rock deformation progressed during the long geological time continuously, marked by slow loading and creep behavior, therefore this study has designed the uniaxial compression tests. According to the standards of International Association of Rock Mechanics (IARM) on uniaxial compression tests (Chuanjin Zhanget al., 2002), in which rock samples were processed into cylinders with diameter 50mm, height 25mm [4], and compression test was carried on with permanent strain speed 2.0 105 s 1 , synchronously obtaining the various breakage shape charts of all rock samples as follows (Fig. 1).
Figure 1.
Rupture drawing of some test samples
Sample S101-DC3, S101-DA4 and S101-DA2 performance for shear opening fractures, sample S103DA3,S101-DB3 and S105 performed for shear fracture, with a little extension. Among them, sample S101-DB3’s rupture trace is characterized by a series of tension cracks to joint and run through into a macroscopical shear zone. Results show that the angle of rupture is less than 45°, in conformity with the Mohr-Coulomb criterion [8]. Next, the enveloping curve will be extracted from drawing mohr stress circle by experimental results (Fig. 2). Here 1 , 2 respectively is internal frictional angle in different confining pressure, k1 , k 2 respectively is the straight-line segment slope of Mohr-Coulomb curve, that is internal friction coefficient, and 0 is cut-off point of confining pressure, whose value is 5MPa. After calculation, it can be concluded as follows.
Figure 2.
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relatively large, the situation is opposite, in this way, the failure mode will change from extention fracture to shear fracture, even compresso-shear fracture. The numerical value is: If 0 3 0 5MPa , 1 51.83 (angle of rupture) 1 45 / 2 19.09 If 3 0 5MPa , 2 41 (angle of rupture)
2 45 / 2 24.5 . From above analysis it is not difficult to see that: under state of compression, two-step Mohr-Coulomb criterion is applicable to the low-permeability sandstone’s breaking situation. But under state of extension, it will be replaced by Griffith criterion, because the above criterion is inapplicable. At this time, for offwhite fine sandstone, the criterion will be selected as following: (1) If 3 0 ,the Mohr-Coulomb criterion will be applied, that is 1 3 3 C0 cos 1 sin 2 2 where, if 3 0 5MPa , 2 41 , angle of rupture 2 45 / 2 24.5 ; If 0 0 5MPa , 1 51.83 , angle of rupture 1 45 / 2 19.09 , cohesion
C0 6.53MPa as a actual value. (2) If 3 0 , Griffith criterion is used, next, we will take discussion according to the situation If (1 3 3 ) 0 , the fracture criteria is (1 2 )2 ( 2 3 )2 ( 3 1 ) 2 24 T ( 1 2 3 )
1 3 2( 1 3 ) If (1 3 3 ) 0 , fracture criteria is simplified to 3 T , 0 . cos 2
The basic idea of physical test is firstly to select a large number of core samples with common lithology and good mechanical properties, which can be processed into cylindrical specimen complying with test conditions, and reached various stress state by loading cycle, secondly the core samples will be bought out from testing machine and abraded into thin slice for optical observation and statistics, finally the test results are analyzed.
Two-step Mohr-Coulomb curve in Shishen100 block
When the confining pressure is less than a value, the internal frictional angle is quite large, and angle of rupture 45 / 2 relatively small, thus extension fracture predominantes. When confining pressure is © 2013 ACADEMY PUBLISHER
,
Figure 3.
Compression test curve of porous racks
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Through collecting predecessor’s data [5-7] and actual sample tests on Shishen block 100, it is concluded that micro-fractures at the course of compression have some evolutionary characteristics as following: (1) In the compact segment (OA of stress-strain curve, see Fig. 3), the micro-fractures of porous rock samples are compressed to cause density, fracture aperture and permeability to loss or diminish dramatically. (2) In the AB segment of stress-strain curve, the curve presents proximate straight line, at this point, rock samples occur elastic deformation with slope be elastic modulus. (3) From the BC segment in Fig. 3, we can see that the stress-strain curve no longer maintain a straight line, a significant increase in radial strain begins, known as expansion.
exists here. However, types Ia, II and III are all belong to compressional stress types. In order to establish the relationship between stress, strain and fracture aperture, density under the complex stress condition, we selected an element (REV) to take analysis [10-12], while adhering to a moderate simplification policy we make some hypothesis. (1) This element is so small enough that it can be easily cut through thoroughly. (2) If the randomly distributed micro-fractures in the element to be eliminated, we think the fractures with permeability could not exist before not exposed or subjected to stress. (3) Although the confused permutation among crystal grains will cause brittle rock to show approximate isotropic as a whole, as long as the mineral composition, the cementation and the diagenesis vary within limits, it will be considered as homogeneous isotropic medium. (4) The element is supposed as a parallel epipedon with with sides L1, L2, L3 (Fig. 5), here we specify the compression stress positive and 1 2 3 , thus the
1 corresponding with the side L1, the 2 corresponding with the side L2, the 3 t corresponding with the side L3.
Figure 4.
Uniaxial stress-strain curves of core samples of well S103
(4) While stress is increased to the peak stress (C point), the overall macroscopic cracking occurs, with specimen damaging, stress decreasing and strain increasing significantly. (5) Breakdown of sandstone is the result of intragranular micro-fractures, as well as intergranular micro-fractures constantly development, assemble, even connect to each other [9]. (6) At the base of Xia Jixiang (Jixiang Xia, 1982), Jaeger, Cook and this laboratory results, we can further generalize modification stage of stress and fractures into Fig. 4. III.
If fractures generate under loading (stress) at this time, the fracture plane normal certainly will distribute within principal plane of 1 3 , and long axis direction of fracture will form an angle with maximum principal stress ( 45
, according to Mohr-Coulomb 2 strength theory) or parallel to intermediate stress, therefore, fracture morphology can be equivalent as following (Fig. 5).
PROPOSED SCHEME
As well known, crustal stress is the main factor to generate fractures in underground rock mass, that in most areas is characterized by disharmonious triaxial state of stress with the horizontal one as principal, among the three usually two are horizontal, one approximately vertical, finally to develop three classed or four types of stress state. The major principal stress of Class I distributes along vertical direction, namely H h z ; the minor principal stress of class II distributes along vertical direction, namely H z h ; the middle principal stress of class III distributes along vertical direction, namely H z h . Among them, class I also can be further divided into types Ia and Ib, the former is marked by v H h 0 , the later by
v 0 , and at least one of H and h is less than 0, or they synchronously equal to 0, that is tensional stress © 2013 ACADEMY PUBLISHER
A Figure 5.
B
Element volume characterizing relationship between fracture parameters and stress
A: Equivalent fracture shape inner element under
1 2 3 coordinate system;
B: Transection perpendicular to 2 , namely 1 3 plane According to Elasticity Theory(B.R Loren et al. 1985) there only agglomerates elastic strain energy in solid material when it deforms, and the energy size can be measured by strain energy density, that is
1 2
(11 2 2 3 3 )
(1)
Here, elastic strain also subjects to Hooke's law (Weixiang Wang, 1984), that is
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1
1 1 ( 2 3 ) E
(2)
2
1 2 (1 3 ) E
(3)
3
1 3 (1 2 ) E
(4)
Combing Eqs. (1), (2), (3) with Eqs. (1), we get strain energy density expressed by principal stress at once, that 1 is [12 2 2 32 2 ( 1 2 2 3 1 3 )] 2E According to Maximum Strain Energy Density Theory [14], when release rate of elastic strain energy agglomerates in the brittle material equals to the energy demanded for generating fractures per unit volume of element, the brittle material will break. Theory and practice have proved that not only the sandstone, fine sandstone but also the sandstone comprising calcareous all will show strong brittle fracture character. For these brittle rocks, when the 3-D stress condition achieves its resistance to bursting strength [15], brittle fracture will occurs with strain energy to release, part of which to offset the surface energy for new fractures, meanwhile rest of which to give off by elastic waves, that is the cross-section energy mentioned in above references (Barton. N et al., 1977). Nevertheless, to fracture, the released elastic wave is so weak that it can be neglected, so we develop the following formula by energy conservation principle.
Dvf
Sf V
f J
(5)
calculated through physical concept and conversion to obtain a and b; another way is through experimental data, curve fitting, mathematical methods to calculate the coefficients In most cases, rock in earth crust is under 3-D compression stress state, such as stress Ia, II and III. Because there exists much difference in mechanical property of rock between under compression stress and tensile stress, here we must discuss respectively. A. Relationship between Stress-strain and Fracture Volume Density Under Uniaxial Compression As stated above, the stress 0.85 c acts as a key value for the beginning of new fractures produced in physical test and at this point the corresponding strain density energy is so associated with e in concept, in that case, let us assume e the strain density energy of 0.85 c to be substituted into the formula to get J , then a and b also can be acquired. Based on the results of rock mechanics test we had average uniaxial compressive strength of gray finemedian grained sandstone reservoir in Shishen block 100, such as c 42.69MPa , 0.85 c 36.28MPa , after calculation to get e 1.133 103 J / m3 . If we turn Eqs. (6) into J
- e Dvf
, the J will be acquired.
Then the average of J is also obtained, that is , then together with J 1087.35J / m2 e 1.133 103 J / m3 are be substituted into the Eqs. (6) , we quickly get the relationship of strain density and fracture volume density
Dvf 9.2 104 104.2
where Dvf is defined as volume density per unit volume (REV); f is strain energy density for new fractures; V is characterization of the element (REV); S is surface area of new fractures; J is the required energy for fractures per unit area, i.e., cross-section energy of fracture (note: here the cross-section is different from the one by intermolecular attraction based on theoretical deduction, and far less than the theoretical value). If f in formula (5) is regarded as the residual strain
(7)
In order to testify the effectiveness of e , we use method of curve fitting to get a and b for comparison with the first approach (Fig. 6). The results are as follows:
Dvf 8.04 104 86.128
(8)
where, the correlation coefficient is 0. 995.
energy density derived from current strain energy density per unit volume subtracting elastic strain energy density for new fractures ( e ), then we have:
Dvf
Sf V
f J
- e J
1 e a b (6) J J
where is the strain energy under current stress; e is the elastic strain energy for new fractures, and a
b-
e
1 , J Figure 6.
is the coefficient, it can be derived by two
J methods. One way is based on the values of J , e
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Relationship between strain energy density and fracture volume density of specimens
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f e
B. Relationship between Rock’S Stress-strain and Fracture Volume Density under Complex 3-D Compression Stress State Although the quantitative relationship has been described in detail above, in practice, rock always is in complex 3-D compression state [21] to cause the formula lacking of transferability. Nevertheless, there still exist much similarity between the curves of triaxial compression test and uniaxial compression test, such as when axial stress arrives 0.85 max , precursory fractures will begin forming, whose strain energy is the elastic strain energy ( e ) for overcoming fractures [16].
1 ( 11 2 2 3 3 ) 2 0.852 2C0 cos (1 sin ) 3
(14)
225.2(1 sin ) 0.85 ( 2 3 ) 112.6
If the minimum stress 3 is known, here certainly has 3 0MPa ( 3 0MPa representing tensile stress state), by the rock failure criteria we will calculate the minimum value of maximum principal stress, namely
1min
2C cos (1 sin ) 3 0 1 sin
Figure 7.
(9)
The calculated 1 is not the actual maximum principal stress, but the failure stress under 3 acting, then for well discrimination we replace 1 by p
p
2C0 cos (1 sin ) 3 1 sin
Relationship between fracture surface energy and confining pressure
(10)
1 , if 2 Under random stress state ij 3 3 0MPa , inserting ij in Eqs. (10), and 1 p , the stress state will conform to conditions of rock failure, then according to elastic theory to get e and adding Eqs. (1) and Eqs. (2) we get
1 1 2 2 (11) 1 0.85 p (0.85 p 2 ( 2 3 )) 2E
Without question, under condition of confining pressure, obviously the pressure will stop fracture producing in certain extent, thus accumulative energy not only needs to overcome cohesive strength between molecules, but also to overcome confining pressure to tear rock finally, at the same time, the fracture surface energy also increases. Now assume a default fracture model (Fig. 7), with aperture b, area S and confining pressure 3 , then the work ( 3 ) for fracture is
Dvf
f e
2C cos (1 sin ) 3 E 112.6 0 1 sin
(13)
Put Eqs. (10) and Eqs. (13) into Eqs. (12), then it be simplified to © 2013 ACADEMY PUBLISHER
J / s 3b
(16)
We insert Eqs. (14), Eqs. (15)and Eqs. (16) into Eqs. (6) to get fracture volume density formula：
Combining Eqs. (6) and Eqs. (10) we have
Although not described the influencing factors on rock’s parameters in detail above, we indeed have acquired the relationship between elastic modulus and confining pressure, that is
(15)
Because of
e p (0.85) p (0.85) 0.85 p 0.85 p
1 ( 11 2 2 3 3 ) (12) 2 1 0.85 p (0.85 p 2 ( 2 3 )) 2E
3bS
f J
1 ( 11 2 2 3 3 ) 2( J 0 J )
0.852 2C0 cos (1 sin ) 3
(17)
225.2(1 sin )( J 0 J )
0.85 ( 2 3 ) 112.6( J 0 J ) where J 0 is surface energy when confining pressure equals to 0, unit:J/m2; J is the additional surface energy by confining pressure 3 ; is Poisson's ratio. Based on above, Eqs. (17) is equivalent to
Dvf
f J
(11 2 2 3 3 ) 0.852 11 2( J 0 J ) 2( J 0 J )
(18)
To verify the validity of above formula or modeling, we use Eqs. (18) to calculate fracture volume density and
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compare with actual values, the raw data mainly comes from the triaxial test results of sandstone by Tao Zhenyu (Zhenyu Tao,1996) and the compression test under confining pressure of sandstone by Institute of Geom. C. Relationship between Fracture Linear Density and Volume Density Here we take discussion at different situations about their quantitative relationship, and define fracture linear density as the numbers per unit length. a. Relationship between them as fracture surface forms an angle to the principal stress
Figure 9.
Relationship between true fracture space and sighted space in 1 3 section
As shown in Fig. 9,when fracture surface is parallel to principal stress 1 , there is 0 , obviously
S f L 3 Dlf L1.L2 Equivalent fracture model of 1 3 section
Figure 8.
while 0
perpendicular to 2 , and let condition of fractures equivalent to uniformly-spaced arrangement (Fig. 8). Now we calculate total surface area of fracture by getting total length of fracture body firstly, secondly multiply total surface area L2. As shown in Fig. 10, the fracture can be divided into two parts [22], one is along direction of L1 ,the other is along direction of L2. Note: Since we have omitted all the minor details in the deducting process, then in section 1 3 the total length of fracture body will be
Lt L1t L3t
L Dlf sin L1 2cos
L3 Dlf cos L3 2
2sin
(19)
Consequently total surface area of fracture body is calculated as following
S f L2 Lt L2
L12 Dlf sin L1 2cos
L32 Dlf cos L3 2sin
Put Eqs. (20) into the formula Dvf
Sf V
(20)
, where
2 Dvf L1 L3 sin cos L1 sin L3 cos L12 sin 2 L23 cos2
and V L1 L2 L3 , then
Dlf Dvf
(23)
D Relationship between stress-strain and fracture aperture To supply aperture with b, space with d, and L1 L3 L , clearly there is
Dlf
1 d
(24)
While the angle between fracture surface and maximum principal stress is greater than zero, the Fig. 11 characterizes their relationship. In the direction of 1 , fracture sighted space d1 d / sin , sighted aperture b1 b / sin , thereby
Dlf 1
1 Dlf sin d1
(25)
In the direction of 3 , fracture sighted space d3 d / cos , sighted space b3 b / cos , then
Dlf 3
1 Dlf cos d
(26)
in the direction of 1 and 3 ,Combing Eqs. (25) and Eqs. (21)
b. Relationship between them as fracture surface forms an zero angle to the principal stress
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V
where Dlf 1 and Dlf 3 is the fracture sighted linear density
V L1 L2 L3 , and simplified to Dlf
Sf
simplified to
In the REV element, fracture body is parallel to intermediate principal stress, fracture strike has an angle with 1 . So as to we make section 1 3
2 1
Insert it into formula Dvf
(22)
(26), we get In the direction of 1 ,
Dlf 1b1 Dlf sin b / sin Dlf b In the direction of 3 ,
(27)
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Dlf 3b3 Dlf cos b / cos Dlf b
(28)
It is thus clear that the product of fracture aperture and linear density in direction of 1 and 3 is invariable.
Dlf 3b3 Dlf cos b / cos Dlf b
(29)
Next, under condition of principal strain, fracture aperture is decided by the tensile strain. For strong brittle sandstone, we define a new parameter f as fracture strain, non-dimensional, and think that fractures will begin to produce once tensile strain exceeds some value, after this, the tensile strain is mainly caused by the cracking of micro-fractures [18], hence we set up the relative formulas:
Dlf 3 ( L0 L) b3 f L0
(30)
L f L0
(31)
Figure 10.
Maximum principal stress 1 isoline space diagram by Ansys
and
Combing Eqs. (30) and Eqs. (31) can obtain
Dlf 3b3
f 1 f
(32)
Since 1>> f , inserting Eqs. (29) in Eqs. (32) and simplified to b
f Dlf
Figure 11.
Intermediate principal stress 2 isoline space diagram by Ansys
, where f | | | 0 | ; b is
fracture aperture; f is fracture strain; Dlf is fracture linear density; is the tensile strain under current stress; 0 is the maximum tensile strain in elastic deformation stage corresponding to the point of fracture initializing [17]. IV.
EXPERIMENTAL RESULTS
Take Shishen Block100 as an example, through uniaxial compression test we have got 0 ＝ 11.40 104 , J＝1087.35J/m2, 0 , hence combing Eqs. (8) gives: Dlf Dvf 8.04 104 86.128 After above study to obtain: f | | | e | 11.40 104 b Dlf Dlf 8.04 104 86.128 Using above eqs, loaded relate data to ANSYS, we got three principal stress isoline space diagram(see Fig. 10,11,12).This fully indicates that replacing e by strain energy density corresponding with 0.85 c is correct, based on this principle, just a few tests can work the quantitative relationship between stress-strain and fracture volume density out. The results are so close to the observed data under microscope that the relative error is less than 10%, which verify the validity of the approach [19, 20].
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Figure 12.
Minimum principal stress 3 isoline space diagram by Ansys
It is shown by results that relative error of 94.11% samples are less than 15%, proving the validity of the model. On the other side, as long with the increasing of 3 , f decreases drastically, this appearance is inconsistent with actual situation. V.
CONCLUSION
As above we firstly have discussed the relationships between stress-strain and fracture aperture, density of rocks and set up relative theoretical model [23], then examined the effectiveness through true test data. Secondly we have discussed the relationships between them under the condition mingled with tensile stress, comprehensively described as follows: (1) If there is no tensile stress, for sandstone material the Mohr-Coulomb criterion will be applied, that is 1 3 3 C0 cos 1 sin 2 2
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Hence for fine sandstone reservoirs in Shishen block100, if all the conditions conform to above formula, there definitely will produce fractures suitable for following computation process [24, 25]: 2 2 2 1 1 2 3 2 ( 1 2 3 ) w f w we 2 E 0.852 p 2 2 ( 2 3 )0.85 p 2C0 cos (1 sin ) 3 p 1 sin E E 0 p D wf vf J J J 0 J J 0 3b 2 Dvf L1 L3 sin cos L1 sin L3 cos Dlf L12 sin 2 L23 cos 2 Dlf b 3 0 1 0 ( 3 (0.85 p 2 ) E Here the meanings of parameters are same as above. Because rock uniaxial compression being a special case in triaxial compression, we take Shishen block100 as an example to express the relationship between stress-strain and fracture arameters: Dvf 9.2 104 w 104.2 Dlf 0.5428Dvf 1.2215 4 Dlf b f | 3 | | e || | 11.40 10 (2) If there have tensile stress, to sandstone material the Griffith criterion is used, that is When 3 0 and (1 3 3 ) 0 , the applied failure criterion is (1 2 )2 ( 2 3 )2 ( 3 1 ) 2
24 T ( 1 2 3 )
cos 2
1 3 2( 1 3 )
When (1 3 3 ) 0 , failure criterion is simplified to
3 T , 0 Based on this, the relationship between stress-strain and fracture parameters is expressed by 1 1 2 w f w we 2 ( 11 2 2 3 3 ) 2 E t wf Dvf J J J 0 J J 0 3b 2 Dvf L1 L3 sin cos L1 sin L3 cos Dlf orDlf D vf L12 sin 2 L23 cos 2 Dlf b 3 0 0 t E
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Take Shishen block100 as an example, in the middle of the third Shahejie formation there have J 0 1087.35J / m2 , 0 4.90 104 , E 4.73GPa . If
(1 3 3 ) 0
arccos ( 1 3 ) / 2( 1 3 ) 2
,
then
,
Thereby take the formula 2 Dvf L1 L3 sin cos L1 sin L3 cos to calculate Dlf L12 sin 2 L23 cos2 the fracture linear density. If (1 3 3 ) 0 , and 0 , then the fracture linear density definitely equals to volume density, that is Dlf Dvf . REFERENCES [1] Lingzhi Jiang, Jiayu Gu, and Bincheng Guo, “Characteristics and formation mechanism of lowpermeability clastic reservoir of oil and gas-bearing basins in China,” Acta Sedimentologica Sinica, vol. 22, pp. 13-18, 2004. [2] Mingsheng Yuan, Mao Pan, Hengmao Dong et al. “Exploration on low-permeability fractured reservoirs,” JPT, Beijing, vol. 5, pp. 3-14, 2000. [3] Xingui Zhou, Chengjie Cao, Jiayin Yuan, “The status and prospects of quantitative structural fracture prediction of reservoirs and research of oil and gas seepage law,” Advances in Earth Science, vol. 18, pp. 398-404, 2003. [4] G. I., Barenblatt, I. P., Zheltov, and N., Kochina, “Basic concepts in the theory of seepage of homogeneneous liquids in fissured rocks,” Prikl. Mat. Mekh., vol.24, pp. 852-864, 1960. [5] J. E., Warren and P. J., “RootThe Behavior of naturally fractured reservoirs,” J. Soc. Pet. Eng., vol. 2, pp. 245-255, 1963. [6] H. Kazemi, “Pressure transient analysis of naturally fractured reservoirs with uniform fracture distribution,” Soc. Pet. Eng. J., vol. 146, pp. 451-161, 1969. [7] A., Deswaan, “Analytical solutions for determining naturally fractured reservoir properties by well testing,” Soc. Pet. Eng. J. vol 1, pp. 117-122, 1976. [8] D. Bourdet, J. A., Ayoub and Y. M, “Pirard. Use of pressure derivative in well test interpretation,” SPE 12777, Cal. Regional Meeting, pp. 1-17, 1984. [9] Z.- X.,Chen, “Transient flow of slightly compressible fluid through double-porosity, double-permeability systems-a state-of-the-art review,” Transport in Porous Media, vol. 4, pp. 147-184, 1989. [10] M., Chen, M., Bai, “Modeling Stress-dependent permeability for anisotropic fractured porous rocks,” International Journal of Rock Mechanics and Mining Science, vol.351, pp. 1113-1119, 1998. [11] Nelson. R. A and Handin, J. w, “Experimental Study of Fracture Permeability in Porous Rocks,” AAPG Bulletin, vol. 61, pp. 227-236, 1977. [12] Ren Wang, Zhongyi Ding, Youquan Yin et al. ,“Solid Mechanics,” Geological Publishing House, Beijing, vol. 82, pp. 124-141, 1979. [13] Murray GH, “Quantitative fracture study,” Sanish pool, Mckenzie County, North Dakota. AAPG Bulletin, vol.52, pp. 57-65, 1968. [14] Wen Shipeng, Li Detong, “Simulation technology on structural fractures of reservoirs,” Journal of China
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[22] Chen. Z, Naryan. S. P, Yang. Z et al., “An experimental investigation of hydraulic behavior of fractures and joints in granitic rock,” International Journal of Rock Mechanics and Mining Sciences, vol. 7, pp. 2061-2071, 2000. [23] William W. Guo, Mark Looi, A Framework of TrustEnergy Balanced Procedure for Cluster Head Selection in Wireless Sensor Networks, Vol. 7, No. 10, Journal of Networks, 1592-1599, 2012 [24] Yan Zhao, Hexin Chen, Shigang Wang, Moncef Gabbouj, An Improved Method of Detecting Edge Direction for Spatial Error Concealment, Journal of Multimedia, Vol. 7, No. 3, pp. 262-268, 2012 [25] Shuang Xu, Jifeng Ding, Palmprint Image Processing and Linear Discriminant Analysis Method, Journal of Multimedia, Vol. 7, No. 3, pp. 269-276, 2013
Qinghua Chen, male, born in November 1958, Shan County, Shandong Province, Professor, Doctoral Tutor. Mainly engaged in the study of structure geology analysis ,the basin structural characteristics, the law of oil and gas distribution, reservoir architecture analysis. Yan Liu, male, born in November 1972, Associate professor. Mainly engaed in the sturcture stress analysis .
