Scientific Journal of Earth Science June 2014, Volume 4, Issue 2, PP.37-54
Three-dimensional Pore-network Cracks Transient Hydrofacturing-liquefaction in Tight Sandstone under Seismic Wave & Electromagneto-thermo-elastic Fields Yongbing Li1,2,3, Bojing Zhu1,2† 1
Key Laboratory of Computational Geodynamics of Chinese Academy of Sciences, Beijing 100049, China
2
College of Earth Science, University of Chinese Academy of Sciences, Beijing 100049, China
3
Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China
Email addresses: cynosureorion@ucas.ac.cn; bojingzhu@gmail.com†; bojing.zhu@durham.ac.uk (B.J. Zhu†).
Abstract This contribution presents a new theory with lattice Boltzmann & finite element & hypersingular integral equation (LB-FE-HIE) to explore the three-dimensional pore-network cracks transient hydrofacturing-liquefaction in tight sandstone (3D PC-TH-TS) under seismic waves & electro-magneto-thermo-elastic fields through intricate theoretical analysis and numerical simulations. First, the 3D PN-TH-TS problem is reduced to solving a set of coupled LB-FE-HIEs by using the Green functions and distribution functions, in which the unknown functions are the extended pore-network cracks displacement discontinuities. Then, the behavior of the extended displacement discontinuities at the pore-network cracks surface terminating at the solid-liquid film interface are analyzed through extended dynamic hypersingular integral main-part analysis method of LB-FE-HIEs. Closed formed solutions for the extended singular dynamic stress, the extended dynamic stress intensity factor and the extended dynamic energy release rate near the dislocations interface are provided. Last, the tight sandstone sample from the Ordos Basin Triassic formation is selected, the fluid-solid coupled digital rock physical modeling is established, and the simulations of 3D PC-TH-TS process is presented on the parallel CPU-GPU platform. The hydrofacturing-liquefaction varying with the amplitude, frequency and constituting time of earthquake wave is obtained. The relationship between the tight sandstone pore-network cracks transient propagation-evolution and the maximum tight sandstone fracturing-liquefaction stress criteria is explored. Keywords: Hydrofracturing-liquefaction, Tight Sandstone, Lattice Boltzmann Method, Finite Element Method, Hypersingular Integral Equation, Seismic Wave & Electro-magneto-thermo-elastic Fields, Parallel CPU-GPU Technology
1 INTRODUCTION Hydrofracturing-liquefaction mechanism is the basic theory for understanding the in-situ stress measurement, the petroleum-gas (nature gas and shale gas)-geothermic develop, and the earthquake evaluation and mechanism exploration. With the complex and challenge of this issue, the mechanism of dynamic Hydrofracturing-liquefaction is still not clear even a lot of fundamental and landmark achievements have been obtained in this field since the last century 60’s. The oscillation singularity of dynamic stress wave as well as stress oscillation singularity index of pore-network cracks transient Hydrofracturing-liquefaction (PC-TH) fluid pore-solid sketch interface under seismic wave & electro-magneto-thermo-elastic fields (SW&EMTE) makes it much more difficult than the unsaturated cased of the ordinary cracks, for in this process, including fluid transient undrained process, fluid transient compression and expansion process, physical properties of tight stone varying with PC-TH propagation process and tight stone transient failure with the liquefaction process (the liquefaction process time is around 10E-5s). - 37 http://www.j-es.org
The mode I stress intensity factor for a constant velocity semi-infinite crack moving in a fluid-saturated porous medium with finite height is calculate[1]. The linear elastic solid with cavities containing nonviscous compressible fluid is explored[2], and the abnormally high fluid pressure in the cavities affects is analyzed. The mode I crack in an elastic fluid-saturated porous solids is studied[3], and relatively closed-form asymptotic solution near the crack tip is obtained. Extended horizontal cracks in a vertical column of saturated sand is explored [4], and the theory is good qualitative agreement with the experimental findings. The interaction of a normally incident time-harmonic longitudinal plane wave with a circular crack imbedded in a porous medium is considered [5], and the problem is formulated in dual integral equations for the Hankel transform of the wave field. The formation mechanism of “water film� (or crack) in saturated sand is analyzed theoretically and numerically[6]. The cracks in fluid-saturated two-phase medium is explored [7] by finite element method. A circular crack imbedded in a porous medium under a normally incident time-harmonic longitudinal plane wave is studied by second kind single Fredholm integral equation[8]. A finite element algorithm is presented for the numerical modeling of cohesive fracture in a partially saturated porous media[9]. The saturated cracks in 4% porosity Fontainebleau sandstone fro an effective mean pressure ranging from 2 to 95MPa under high ultrasonic frequencies, Vp/Vs, ratio is explored [10]. The narrow distribution of cavities permeability model is devised combining the hydraulic radium and percolation concepts [11]. But this problem is far from solved for complex relationship between fluid states, physical properties of tight sandstone and ultra high temperature-pressure (UHTP). There are four differences between classical porous medium and tight sandstone. First, the pore/void size is located at atom-molecular-pico-nano scale level, the fluid viscous (as function of UHTP), the effects of boundary layer, and the unsteady fluid flow (eddy flow and turbulent flow) can not be neglected; Second, when the P-T conditions are high enough, the water role in the rock/mineral includes freesupercritical-constitutional state, the fluid flow particles are composed of four components [H2O, H+, (OH)-, (H3O)+]. Third, the micro pore is composed of four types [multi-grain gap, polycrystalline space, crystal space and crystal internal space] and the deformation of the micro-structure had to be considered; Last, the fluid flow permeability and diffusion include intermolecular collisions and diffusion (Fick’s laws of diffusion), molecular collisions with interface (Knudsen diffusion), molecular and interfacial adhesive and viscous flow (Darcy and Forchheimer flow). With the scale decrease, the surface stress component became domain, the effect of the body stress component reducing, and the classical N-S equation is no longer applies. In this work, based on the previous work on saturated dislocation transient propagation-evolution in olivine structure under ultra-high temperature and pressure (UHTP)[12], and general solutions of extended displacement (elastic displacement, electrical potential, magnetic potential and thermal potential) given in the previous work[13, 14], the three-dimensional pore-network crack transient Hydrofracturing-liquefaction in tight sandstone (3D PC-TH-TS) under seismic waves & electro-magneto-thermo-elastic fields is studied by using lattice Boltzmann & finite element & hypersingular integral equation (LB-FE-HIE) on parallel GPU-CPU environment. First, the 3D PN-TH-TS problem is reduced to solving a set of LB-FE-HIEs coupled with extended boundary integral equations by using the Green functions and distribution functions, in which the unknown functions are the extended pore-network cracks displacement discontinuities. Then, the behavior of the extended displacement discontinuities around the pore-network cracks surface terminating at the solid-liquid film interface are analyzed by the extended hypersingular integral main-part analysis method of LB-FE-HIEs. Analytical solutions for the extended singular dynamic stresses, the extended dynamic stress intensity factors and the extended dynamic energy release rate near the dislocations front are provided. Thrid, the tight sandstone sample from the Ordos Basin Triassic formation is selected and different tomography resolution data is obtained by X-ray CT digital technology, the fluidsolid coupled porous medium physical modeling is established, and the relatively Hydrofracturing-liquefaction numerical modelling of tight sandstone is obtained for the first time. Last, The simulations of 3D PC-TH-TS process is presented, the hydrofacturing-liquefaction varying with the amplitude, frequency and time of earthquake wave is obtained. The relationship between the tight sandstone pore-network cracks transient propagation-evolution and the maximum tight sandstone fracturing-liquefaction stress criteria is explored.
2 BASIC EQUATIONS Here summation from 1 to 3(1 to 6) over repeated lowercase (uppercase) subscripts is assumed, and a subscript - 38 http://www.j-es.org
comma denotes the partial differentiation with respect to the coordinates. The ultralow permeability tight sandstone (UPTS) consists of six constituents, i.e. the particles of solid skeleton (the 1st component), bound liquid film (the 2nd component), static pore-liquid (the 3rd component, free state water), dynamic pore-liquid I (the four component, free state water), dynamic pore-liquid II (the five component, supercritical state water) and dynamic pore-liquid III (the 6th component, constitutional state water [H+, (OH)-, (H3O)+]. The governing equations and constitutive relations of UPTS under EMTE field can be expressed as [15, 16], AL ijL, j BL iJ ,i f J 0
1 (1 ) P 2 3P AL P 4 5P P 6
(1)
L 1 L2 L3 L4 L5 L6
(1 ) 1 2 U ( xi , t )1i 5 N N2 1 N1 U ( xi , t )( N 1)i U ( xi , t ) Ni N 1 5 P U ( x , t ) N 2 N N2 1 N1 U ( xi , t )( N 1)i U ( xi , t ) Ni i 2i 2 2 P 5 2 1 1 U ( xi , t )3i N 3 N N N U ( xi , t )( N 1)i U ( xi , t ) Ni BL 3 3 P U ( x , t ) 5 2 1 1 U ( x , t ) N 4 N N N i ( N 1)i U ( xi , t ) Ni i 4i 4 4 P U ( x , t ) 2 1 1 U ( x , t ) U ( x , t ) i 5i 5 5 5 i 6i i 5i 5 5 P U ( x , t ) 2 1 1U ( x , t ) i 6i 6 6 6 i 6i 6 6
L 1 L2 L3 L4 L5 L6
U ( xi , t )i1 , U ( xi , t )i 2 , U ( xi , t )i 3 , U ( xi , t )i 4 , U ( xi , t )i 5 and U ( xi , t )i 6 denote the components of displacements of drained porous solid frame, the components of displacements of drained bound liquid film, denote the components of displacements of the static pore-liquid, the components of displacements of the dynamic pore-liquid I, the components of displacements of the dynamic pore-liquid II, and the components of displacements of the dynamic pore-liquid III respectively; L , NL , , NL and L denote density, viscosity, intrinsic permeability, relative
permeability and volume fractions of L th component of ultra-low permeability porous rock respectively; , , 2P ,
3P , 4P , 5P and 6P represent macro porosity of ultra-low permeability rock, fraction occupied by bound liquid film, the connected porosity of bound liquid film, the connected porosity of static pore-liquid part, the connected porosity of dynamic pore-liquid I, II and III respectively. In addition, the extended stress displacement matrix tensor and the extended body load vector, iJ and f J , are defined respectively as
iJ
ij J j 1, 2,3 D J 4 i J 5 Bi i J 6
f j J j 1, 2,3 f J 4 fJ e J 5 f m f J 6
(2)
The combined constitutive equation is written as
iJ
EiJKl Z Kl
(3)
where the EMTE constant matrix tensor and the extended strain field matrix tensor, EiJKl and Z Kl , take the form C EiJKl iJKl 0
iJ 0
0 iJ
Z Kl [U Ks ,l
,l ]T
(4)
the extended elastic stiffness tensor, the extended thermal stress constants tensor and the extend elastic displacement vector, CiJKl , iJ and U K , are defined as - 39 http://www.j-es.org
CiJKl
cijkl e lij eikl d lij dikl gil il il
J , K 1, 2,3 J 1, 2,3; K 4 J 4; K 1, 2,3 J 1, 2,3; K 5 J 5; K 1, 2,3 J 4; K 5orJ 5; K 4 J,K 4 J,K 5
ij J 1, 2,3; K 6 iJ il J 4; K 6 J 5; K 6 il
uis K 1, 2,3 s K 4 s U K 4s K 5 5 6s K 6
(5)
ij , Di , Bi , i , f i , f e , f m , f , cijkl , elij , dlij , il , gil , il , ij , il , il , ij , uis , 4s , 5s and 6s are defined in [14]. The drained porous solid frame and the drained bound liquid film have the same displacement and pressure; The static poreliquid, the dynamic pore-liquid I have the same displacement; The pressure of saturated porous solid frame is the sum of static pore-liquid, dynamic pore-liquid I, dynamic pore-liquid II and dynamic pore-liquid III components pressure. The seismic wave displacement in ultra-low permeability porous rock can be written as U ( xi , t )iL 1 ( xi , t )iL,i ijk 2 ( xi , t ) jL,k
where E (2L x1 x2 x3 sin 1L cos 2L ) E (2L x1 x2 x3 cos 1L sin 2L ) 3i ( xi , t )iL Ai E (t ) E (2L x1 x2 x3 cos 1L cos 2L ) E (2L x1 x2 x3 sin 1L sin 2L ) E (2L x3 cos ( x3 , x2 ) pL ) E (2L x3 cos ( x3 , x1 ) pL ) E (L x1 sin ( x3 , x2 )pL ) E (L x2 sin ( x3 , x1 )pL ) E (2L x1 x2 sin 1L cos 2L ) E (2L x1 x2 cos 1L sin 2L ) E (2L x1 x2 cos 1L cos 2 L ) E (2L x1 x2 sin 1L sin 2 L )
1L
( x3 , x2 ) pL ( x3 , x1 ) pL 1 ( x3 , x2 ) pL ( x3 , x1 ) pL 2
, 2L , L , E () ei () ( x , x ) ( x , x ) / vL 1 1 3 2 sL 3 1 sL P L ( x , x ) ( x , x ) 2 / v 2 3 2 sL 3 1 sL S
v1P , vP2 , vP3 , vP4 , vP5 and vP6 denote the P-wave velocity in drained porous solid frame, drained bound liquid film, static pore-liquid, dynamic pore-liquid I, dynamic pore-liquid II and dynamic pore-liquid III respectively; v1S , vS2 , vS3 , vS4 , vS5 and vS6 denote the P-wave velocity in drained porous solid frame, drained bound liquid film, static pore-liquid, dynamic pore-liquid I, dynamic pore-liquid II and dynamic pore-liquid III respectively; ( x3 , x2 ) pL , ( x3 , x1 ) pL ,
( x3 , x2 )sL and ( x3 , x1 )sL denote the P-wave incidence angle between x3 and x2 axes, the P-wave incidence angle between x3 and x1 axes, the S-wave incidence angle between x3 and x2 axes and the S-wave incidence angle between
x3 and x1 axes in the L th component of ultra-low permeability porous rock respectively. If we defined that solid skeleton component and liquid components are parallel and subjected to the same strain, the time-dependent rigidity modulus components GL of ultra-low permeability porous rocks can be defined as GL (1 f )G0 f (vsL L2 ),t
f , , L2 and G0 represent macro-porosity of ultra-low permeability porous rocks, fraction occupied by bound liquid film, non-dimensional parameter (linked fluid viscosity, bulk modulus, density and permeability) of L th component of ultra-low permeability porous rock, and rigidity modulus of solid grains.
3 LB-FE-HIE FOR 3D PC-TH-TS UNDER SEISMIC WAVE & ELECTRO-MAGNETOTHERMO-ELASTIC FIELDS 3.1 Lattice Boltzmann & finite element (LB-FE) coupled modelling It is shown in the Fig 1, The LB-FE coupled modeling for tight sand porous media is established. The finite element method is using to analyze solid sketch part of the porous tight sand (the strain/strain rate-stress/stress rate-temporal - 40 http://www.j-es.org
spatial solid sketch structure-macro strength) (Fig1D, 1E); D3Q27 lattice Boltzmann method is using to analyze fluid pore part of the porous tight sand ( fluid stress-viscous-density-time-bulk strain) (Fig 1F); LB-FE coupled modeling can deal with interface stress-strain between solid sketch and fluid pore (surface of pore-network cracks) (Fig 1C). In our work, the three-dimensional 20 node arbitrary hexahedral element (3D-20 Hexahedral element) is defined for the solid sketch part of the porous tight sand, and the three-dimensional 27 particle arbitrary hexahedral node (D3Q27) is defined for the fluid pore part of the porous tight sand; if the element is located at the surface of the pore-network cracks, then we defined that each D3Q27 node has the same physical meaning as one arbitrary node of 3D-20 Hexahedra element (Fig 1G,1H and 1I), the distribution function of lattice Boltzmann and the shape function in finite element can be linked through intricate theoretical formulation, and the LB-FE coupled fluid pore-solid sketch porous modeling can be established, the theoretical derivation can be written as follows,
FIG 1. THE LATTICE BOLTZMANN & FINITE ELEMENT FLUID-SOLID COUPLED MODELLING FIG 1A. CROSS-SECTION OF ROCK CT SCAN DIGITAL DATA, FIG 1B. CROSS-SECTION OF DIGITAL ROCK, FIG 1C. LBM & FEM FLUID-SOLID COUPLED MODELING; FIG 1D, SKETCH OF 20 NODES FEM CELL, FIG 1E. SKETCH OF 20 NODES ISOPARAMETRIC FEM CELL, FIG 1F SKETCH OF D3Q27 LBM GRID POINT; FIG 1G. MICRO STRUCTURE OF TIGHT ROCK (SCALE I), FIG 1H. MICRO STRUCTURE OF TIGHT ROCK (SCALE II), FIG 1I MICRO STRUCTURE OF TIGHT ROCK (SCALE III)
The equation of shape function for 3D-20 Hexahedra isoparametric element is defined as Nk 3 3 (1 xik xi ) xik xi 2 k 1 ~ 8 i 1 i 1 3 2 k 9 ~ 12 2 i 1 (1 x1 ) 2 2(1 x2 )(1 xik x1 )(1 x3k x3 ) k 13 ~ 16 2(1 x 2 )(1 x x )(1 x x ) k 16 ~ 20 3 1k 1 3k 3
(6)
The relatively stiffness matrix for 3D-20 Hexahedra isoparametric element is defined as T
K 1 1 1 B D B J dx1 dx2dx3 1
1
1
where - 41 http://www.j-es.org
(7)
BK
N K 1 0 0 N K 2 0 N K 3
0
0
N K 2
0
0
N K 3
N K 1
0
N K 1
N K 2
0
N K 1
E (1 v) D (1 v)(1 2v) 1 A1 A2 A 1 A 1 1 A2 A1 1 0 0 0 0 0 0 0 0 0 v 1 2v A1 A2 1 v 2(1 v)
0
0
0 0
0 0
A2 0
0 A2
0
0
J 0 1 x1 0 0 1 x 0 2 1 0 x A2 3
2 x1 2 x2 1 x3
3 x1 3 x2 1 x3
The extended equivalent stress and tension for 3D-20 Hexahedra isoparametric element is defined as
Rvolume 1 1 1 N p J dx1dx2 dx3 1
1
1
T
Rsurface A
x1x2
N p x 1 dA T
(8) (9)
1
Take the distribution function of D3Q27 LB node[17] into the shape function of 3D-20 Hexahedra element, the LBFE modeling can be established to analyze the fluid-solid coupled porous structure problem.
3. 2 LB-FE-HIE for 3D PC-TH-TS under seismic wave & electro-magneto-thermo-elastic fields Using Micro XCT-400 CT system, the high resolution data are obtained, the 3D virtual digital rock are reestablished, and the LB-FE-HIE physical model are constructed; The threshold is arrange from 78~100, the interior geometry structure at low level scales translate from the pix RGB color into the cell-mesh-grid model, the initial/boundary conditions, the physical/chemical parameters of rock and water, and the low level scales geometry pore-solid structure can be defined. Fig 2. shows the PC-TD-TS LB-FE-HHIE coupled modeling, the diameter and the length of the core samples are equal to 5cm and 10cm respectively; the resolution of cross-section and the interval between cross-sections are equal to 10µm. The general model was formulated by utilizing an automatic local amplification grid technique, and the initial LB-FE-HIE physical model is equal to 102410241024 pixels, which does not include the extended variables. Applying function in references [18, 19] to the initial virtual digital rock modelling, we can obtained the random LB-FE-HIE modeling (decreasing the computational scale in parallel CPU-GPU environment). The pore/void size is located at atom-molecular-pico-nano scale level, the fluid viscous (as function of pressuretemperature), the effects of boundary layer, and the unsteady fluid flow (eddy flow and turbulent flow) had to be considered. The water role in the rock includes free state, supercritical state and constitutional state, the fluid flow particles are composed of four components [H2O, H+, (OH)-, (H3O)+]. The micro pore is composed of four kinds [multi-grain gap, polycrystalline clearance, crystal space and crystal internal clearance] and the deformation of the micro-structure had to be considered. The fluid flow permeability and diffusion include intermolecular collisions and diffusion (Fick’s laws of diffusion), molecular collisions with interface (Knudsen diffusion), molecular and interfacial adhesive and viscous flow (Darcy flow and Forchheimer flow). With the scale decrease, the surface stress component became domain, the effect of the body stress component reducing, and the classical N-S equation is no longer applies. The more detailed introduction can be found elsewhere in the literature[20-23]. - 42 http://www.j-es.org
A
B
D
C
FIG 2. THE TIGHT SANDSTONE TRANSIENT HYDROFRACTURING-LIQUEFACTION LATTICE BOLTZMANN & FINITE ELEMENT FLUID-SOLID COUPLED MODELLING (FIG 2A. SAMPLE OF TIGHT SANDSTONE, FIG 2B. VIRTUAL DIGITAL ROCK SAMPLE OF TIGHT SANDSTONE; FIG 2C, A CIRCLE DRILLING IN THE TIGHT SANDSTONE, FIG 1D. SKETCH OF TRANSIENT HYDROFRACTURING-LIQUEFACTION IN THE TIGHT SANDSTONE)
Consider the tight sandstone containing a random three-dimensional transient hydrofacturing-liquefaction crack as shown in Fig.2D. A fixed global rectangular Cartesian system xi (i=1,2,3) is chosen. Assume that the pore-network crack S (S
S ) is subjected to remote the seismic wave mechanical loads p j ( P, Q, t ) , the seismic wave electrical
loads q( P, Q, t ) , the seismic magnetic loads b( P, Q, t ) and the thermal loads ( P, Q, t ) , respectively. The local rectangular Cartesian system i are chosen, the stochastic flaws are assumed to be in the 1 2i plane and normal to the 3 axis. Using the EMTE form of the Somigliana identity, the extended displacement vector, U I ( p) , at interior point p( x1 , x2 , x3 ) is expressed as
U I ( p) (U IJ ( p, q)TJ (q) TIJ ( p, q)U J (q))ds (q) S
(TIJ ( p, q)U J (q) U IJ ( p, q)TJ (q))ds(q) U IJ ( p, q) f J (q)ds (q)
(10)
where is the domain occupied by the EMTE-CMCs, is the external boundary, TJ (q) is the extended elastic tractions on boundaries, U IJ ( p, q) and TIJ ( p, q) are the fundamental solutions. Using constitutive Equation (10), the corresponding extended stresses tensor, IJ , is expressed as
IJ ( p) SKiJ ( p, q)U K (q)ds(q) ( DKiJ ( p, q)TK (q) S KiJ ( p, q)U K (q))ds(q) DKiJ ( p, q) f K (q)ds(q)
S
(10)
The kernels functions, S KiJ and DKiJ , are as follows: SKiJ ( p, q) EiJMnTMK ,n ( p, q)
DKiJ ( p, q) EiJMnU MK ,n ( p, q)
The hypersingular integral equations can be obtained as 5 5 2 c44 D0 s02 ( 3r, r, ) ( 3r, r, ) i 1 i2ti2 3r, i 1 33 si2ti 5 6 2 2 ˆ ds K u u ) i 1 i ti m3 um,1 6 3 4 r r S
(11)
(12)
K i ui ds p S
n 3 i 1 t 5
5
m t i i
6 m 3
um,2
5 5 5 r, 5 Ai ti2 i1 mt t 33 r, i 1 si2 i im i 1 n 3 i 1 i i u un u6 ds 2 3 4 r r r S
K mi ui ds pm S
- 43 http://www.j-es.org
(13)
5 ( 3r, r, ) 5 Ai 3 ti2 33 r, i 1 Ai si2 i 33 i6 i 1 u u6 ds K 6i ui ds p6 i 1 A t u r2 r4 S S 5
2 i 3 i ,
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(14)
ˆ
where pii , p4i (qi ) , p5i (bi ) and p5i ( i ) can obtained from the solution for the loads of the solids. The material ˆ
constants, tim , im and the kernels functions K iji are given in Appendix A.
4 DYNAMIC STRESS, DYNAMIC STRESS INTENSITY FACTORS AND ENERGY RELEASE RATE NEAR THE DISLOCATIONS FRONT In order to investigate the singularity of the dislocations front, consider a local coordinate system defined as x1 x2 x3 . The x1 -axis is the tangent line of the dislocation front at point q0 , x2 -axis is the internal normal line in the dislocation plane, and x3 -axis is the normal of the dislocation. Then, the extended displacement discontinuities of the dislocation surface near a dislocation front point q0 can be assumed as ˆ
ˆ
ˆ
ˆ
ˆ
[uii (q) i (q) i (q) i (q)] gki (q0 )2k
0 R e(k ) 1
(15)
where gk (q0 ) and k represent non-zero constants and singular index, respectively. The dynamic stress intensity factors are defined as ˆ
ˆ
i K1i lim 2r 33 (r , ) r 0
ˆ
ˆ
0
ˆ
K4i lim 2rD3i (r, ) r 0
ˆ
i , K2i lim 2r 32 (r , ) r 0
ˆ
0
ˆ
0
ˆ
, K5i lim 2rB3i (r, ) r 0
ˆ
i , K3i lim 2r 31 ( r, ) r 0
ˆ
0
(16)
0
ˆ
, K6i lim 2r3i (r, ) r 0
(17)
0
where r is the distance from point p to the dislocation front point q0 , the dynamic stress around the dislocation front can be expressed as follows ˆ
ˆ
13i c442 D0 s02
23i i 1 ˆ
5
rri
32 2rri sin
(2cos
1
i
1
3i n i 1 in si [ ˆ
5
(ti2 g 2i cot i cos 1
2 A [ rri cos ˆ
5
Ai rri
[cos
i 2
rr0
3sin 2
4
i 2
0
(18)
2
cos
i
3 2 1 sin 2 i cos i ) 2 332 sin 1 i 2 4 2 r ri 2
(19)
i
ˆ
rri
cos
5 5 ˆ 3cos cos i )] i2 cos i (t2 g 2 4 tim g mi )} 2 2 2 2 m 3
i 33
3i i 1
i
ˆ
{g6i 33 [1533 si2 cot i (3cos
g1i
i 2
i 2
6cot
5
i
( g cot i t g ) iˆ 2
m 3
m i
iˆ m
(32 433 si2 ) m 2 tim g mi ˆ
5
i
5
ˆ
ˆ
t g mi ) g6i si i rri cot i cos
m m 3 i
ˆ
si2 33i g6i
cos
rri
i 2
i 2
)]
(20) ], n 3 5
ˆ
432ti2 g 2i si2 i 3 (232 cos i 2 332 sin 1 i ) 3 2 r ri 2
(21)
3 32 1 15rri g s cot i (3cos 6cos sin sin 2 i cos i )(2 si2 33 )] 4 2 2 4 2 rri sin i iˆ 2 6 32 i i
ˆ
ˆ
i The extended singular stresses, 3i n [ 33
i
2
ˆ
2
ˆ
D3i
B3i ] , can be rewritten as ˆ
ˆ
13i K3i cos ˆ
ˆ
ˆ
ˆ
i ˆ ni 3 [ 23 33i D3i
ˆ
ˆ
B3i 3i
i
ˆ
0 2
ˆ
(22)
/ 2rr0 ˆ
ˆ
ˆ
] ( K1i f n1 K2i f n2 K4i f n4 K5i f n5 K6i f n6 ) / 2r , n 2 6
(23)
where the coefficient functions f IJ are listed in [14]. Other extended singular stresses near point q0 can also be ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
obtained by use of above method. K 2i is coupled with K 3i and K 6i , K1i is coupled with K 4i , K 5i and K 6i .The energy release rate can be obtained and expressed as followings, - 44 http://www.j-es.org
K n2 a32n a12n a22n a42n a52n a62n dW 2 2rS dV E 2 2r E W Us Uk U f
(13) (14)
Where W , U s , U k , U f , V , S , DKiJ , S KiJ , TK , U K , TIJ , U IM , U K , f K , EkJMn , E , , E , K n and ain represent extended change in elastic strain energy, change of surface energy (to create new slip surface area), kinetic seismic energy, friction energy (heat), volume of the slip, extended strain energy intensity function, slip rate kernel function, state kernel function, state force solution, slip rate solution, state force basic solution, slip rate basic solution, slip rate, extended body force, extended elastic modules, elastic modules, shear elastic modules, equivalent elastic modules, extended stress intensity factor and physical character parameters, respectively.
5 NUMERICAL SIMULATION AND DISCUSSION A tight sandstone core samples from the Ordos Basin Triassic formation of China which include multi pore-network cracks is selected, the rock depth, the diameter, the length, the density, the confining pressure, the pore stress and the temperature are defined as 862.76~864.36m, 25.4mm, 25~30mm, 2.359~2.426g/cm3, 0~200MPa, 0~10MPa, 35~45 C respectively, the more detail parameters of the tight sandstone core sample are shown in Tab.1. The diagonal component of permeability tensor is given in the appendix B. TAB.1. MINERAL COMPOSITION, POROSITY AND MATRIX DENSITY OF THE TIGHT SANDSTONE SAMPLES
Sample Number S1
S2
S3
Direction X1 Y1 Z1 X2 Y2 Z2 X3 Y3 Z3
Quarzt 24.34 23.05 28.66 33.33 27.93 23.42 34.05 29.14 26.36
Mineral composition(volume percent) Feldspar Rock debris Mica Heavy placer mineral 63.05 8.17 3.73 0.71 63.16 9.98 3.63 0.18 61.36 7.95 0.16 1.87 57.43 9.24 61.39 10.68 56.32 12.08 8.18 57.93 7.83 0.19 58.48 11.43 0.95 58.53 13.56 1.55
Porosity (%) 8.216 6.744 6.992 6.252 6.822 7.649 7.017 7.333 6.918
Density (g/cm3) 2.368 2.359 2.379 2.426 2.384 2.379 2.374 2.379 2.382
5.1 Hydrofacturing-liquefaction process varying with the frequency of earthquake wave
FIG 3. THE TRANSIENT HYDROFRACTURING-LIQUEFACTION PROCESS AS FUNCTION OF FREQUENCY OF EARTHQUAKE WAVE[FIG3A. FREQUENCY IS 10HZ, FIG3B. FREQUENCY IS 50HZ, FIG3C. FREQUENCY IS 100HZ; FIG3D. FREQUENCY IS 200HZ, FIG3E.FREQUENCY IS 250HZ, FIG3F. FREQUENCY IS 300HZ; FIG3G. FREQUENCY IS 400HZ, FIG3H. FREQUENCY IS 450HZ, FIG3I. FREQUENCY IS 500HZ]
The pore-network cracks transient hydrofacturing-liquefaction as function of the frequency of the earthquake wave is shown in Fig. 3. At the fixed amplitude and constituting time, the dynamic fluid stress act on the tight sandstone as electromagnetic wave and stress wave, at the range of 10Hz~100Hz, the surface of pore-network cracks do not appear liquefaction phenomena, even the surface of pore-net-work cracks began propagation and hydrofacturing phenomena appears; with the frequency increasing, when it reach to 200Hz~300Hz, the hydrofacturing phenomena - 45 http://www.j-es.org
is domain and some pore-network cracks zone of the tight sandstone appear hydrofacturing-liquefaction phenomena, and the constituting time and amplitude decreasing with the frequency increased, the constituting time is around 10E-5s and no draining phenomena appears during hydrofacturing-liquefaction process.. When the frequency is lower than 10Hz, the dynamic fluid stress act on the tight sandstone as quasi-static pore hydrostatic stress, and accompanying draining phenomena in the hydrofacturing failure progess; the maximum tight sandstone factruing-liquefaction stress is the criteria which leading the hydrofacturing failure. In general, the low frequency part of the earthquake wave act on the tight sandstone as static/quasi-static style, when the maximum principle stress is reach to the tight sandstone strength criteria, the pore-network cracks begin propagation and accompanying with draining process, no liquefaction appearing for do not reach the maximum micro tight sandstone structure strength limit; the dynamic stress released sharply with the distance between the surface of cracks, we defined it as crack surface frequency earthquake wave or near-field (around the crack surface) frequency earthquake wave. The high frequency part of the earthquake wave act on the tight sandstone as transient electromagnetic wave and stress wave, the constituting time is very short and no draining accompanied in the whole process; the high frequency energy reach the maximum micro tight sandstone structure strength limit and destroy the tight micro porenetwork structure in 10E-5s (Non-connected pore throat destruction); when the frequency is exceed to 400Hz, the electromagnetic wave can leading pore stress oscillation sharply and make the transient liquefaction appear in the whole tight sandstone zone, without draining and hydrofacturing process.
FIGURE 4. DYNAMIC STRESS IN X DIRECTION AS FUNCTION OF FREQUENCY AT UNDRAINED STATE UNDER EARTHQUAKE WAVE[FIG4A. FREQUENCY IS 10HZ, FIG4B. FREQUENCY IS 50HZ, FIG4C. FREQUENCY IS 100HZ; FIG4D. FREQUENCY IS 200HZ, FIG4E. FREQUENCY IS 250HZ, FIG4F. FREQUENCY IS 300HZ; FIG4G. FREQUENCY IS 400HZ, FIG4H. FREQUENCY IS 450HZ, FIG4I. FREQUENCY IS 500HZ]
FIGURE 5. DYNAMIC STRESS IN Y DIRECTION AS FUNCTION OF FREQUENCY AT UNDRAINED STATE UNDER EARTHQUAKE WAVE[FIG5A. FREQUENCY IS 10HZ, FIG5B. FREQUENCY IS 50HZ, FIG5C. FREQUENCY IS 100HZ; FIG5D. FREQUENCY IS 200HZ, FIG5E. FREQUENCY IS 250HZ, FIG5F. FREQUENCY IS 300HZ; FIG5G. FREQUENCY IS 400HZ, FIG5H. FREQUENCY IS 450HZ, FIG5I. FREQUENCY IS 500HZ]
- 46 http://www.j-es.org
FIGURE 6. THE TRANSIENT LIQUEFACTION PROCESS AS FUNCTION OF FREQUENCY OF THE EARTHQUAKE WAVE[FIG6A. FREQUENCY IS 500HZ, FIG6B FREQUENCY IS 750HZ, FIG6C. FREQUENCY IS 1000HZ, FIG6D FREQUENCY IS 1250HZ]
The dynamic stress in x and y direction as function of frequency at undrained state are shown in the Fig.4 and Fig.5, the low frequency dynamic stress destroy the structure closed to the surface of the pore-network cracks, the high frequency dynamic stress destroy the structure far from the pore-network cracks surface. The low frequency earthquake wave have little energy than frequency part, with the distance (far from the front of the crack) increased, their effect to the structure of the tight sandstone decreased sharply; while the high frequency carry major energy of the earthquake wave, their effect distance to the tight sandstone can reach the whole zone. All these results are consisted with the observation data in tight sandstone liquefaction process in nature earthquake[25-28]. The transient liquefaction process as function of frequency of the earthquake wave are plotted in Fig.6, when the frequency of the earthquake is higher than 1250Hz, the electromagnetic wave energy make the fluid volume in the three-dimensional pore-network cracks changed sharply, let the dynamic stress in the three-dimensional porenetwork cracks surface (solid sketch interface) exceed the solid sketch part strength criteria in 10E-5s, the liquefaction in the micro tight sandstone zone appears in 10E-5s and propagates to the whole domain zone of the tight sandstone in 10E-5s, only seldom higher strength components in the tight sandstone can keep macro-structure (the white zone in the Fig 6D).
5.2 Hydrofacturing-liquefaction process varying with the amplitude of earthquake wave
FIGURE 7. THE CRACKING PROPAGATING PROCESS VARIATION AS THE AM OF THE EARTHQUAKE WAVE[FIG7A. AM OF EARTHQUAKE WAVE IS 0.01GPA, FIG7B AM OF EARTHQUAKE WAVE IS 0.03GPA, FIG7C AM OF EARTHQUAKE WAVE IS 0.05GPA; FIG7D. AM OF EARTHQUAKE WAVE IS 0.07GPA, FIG7E AM OF EARTHQUAKE WAVE IS 0.09GPA, FIG7F AM OF EARTHQUAKE WAVE IS 0.1GPA]
The pore-network cracks hydrofacturing-liquefaction process as function of the amplitude of the earthquake wave is shown in the Fig. 7, at the fixed constituting time and frequency of the earthquake condition, when the amplitude is arrange at 0.01GPa~0.03GPa, the surface of the pore-network cracks begin propagate, the hydrofracturing around the big pore-network cracks fronts appear, and the liquefaction phenomena do not happen; with the amplitude of the earthquake increased, when it reach to 0.05GPa~0.07GPa, the coupled hydrofacturing-liquefaction begin at most of - 47 http://www.j-es.org
the pore-network cracks fronts; when the amplitude of the earthquake is exceeded to 0.07GPa~0.1GPa, the porenetwork cracks propagate sharply and the hydrofacturing-liquefaction happen in the whole tight sandstone zone, only seldom components micro structure do not failure(white zone in Fig7E, 7F). The distribution of dynamic stress in x and y direction as function of amplitude of the earthquake are shown in the Fig.8 and Fig 9, respectively. When the amplitude arrange at 0.01GPa~0.03GPa, the electromagnetic wave and stress wave propagation far in the tight sandstone, for the whole tight sandstone structure remains intact, low earthquake energy consumed in the spreading process; When the amplitude of the earthquake reach to 0.05GPa~0.07GPa, with the pore-network cracks propagation, the non-connected pore throats in the tight sandstone are destroyed quickly, most of the earthquake energy are used to providing the surface energy of the pore-network cracks interface, the electromagnetic wave and stress wave propagation distance decrease with the amplitude increasing; When the amplitude of the earthquake is exceeded to 0.07GPa~0.1GPa, the vase majority non-connected pore throats in the tight sandstones are destroyed and the whole tight sandstone structure instability and failure, the propagation media of the electromagnetic wave and stress wave translate from solid (tight porous structure) to fluid (liquefaction sandstone structure), their spreading distance decreased sharply.
FIGURE 8. DYNAMIC STRESS IN X DIRECTION VARIATION AS FUNCTION OF TIME AT UNDRAINED STATE UNDER EARTHQUAKE[FIG8A. AM OF EARTHQUAKE WAVE IS 0.01GPA, FIG8B AM OF EARTHQUAKE WAVE IS 0.03GPA, FIG8C AM OF EARTHQUAKE WAVE IS 0.05GPA; FIG8D. AM OF EARTHQUAKE WAVE IS 0.07GPA, FIG8E AM OF EARTHQUAKE WAVE IS 0.09GPA, FIG8F AM OF EARTHQUAKE WAVE IS 0.1GPA]
FIGURE 9. DYNAMIC STRESS IN Y DIRECTION VARIATION AS FUNCTION OF TIME AT UNDRAINED STATE UNDER EARTHQUAKE WAVE
[FIG9A. AM OF EARTHQUAKE WAVE IS 0.01GPA, FIG9B AM OF EARTHQUAKE WAVE IS 0.03GPA, FIG9C AM OF EARTHQUAKE WAVE IS 0.05GPA;FIG9D. AM OF EARTHQUAKE WAVE IS 0.07GPA, FIG9E AM OF EARTHQUAKE WAVE IS 0.09GPA, FIG9F AM OF EARTHQUAKE WAVE IS 0.1GPA]
In general, both high frequency and amplitude of the earthquake wave are the main factors which lead to the porenetwork cracks transient Hydrofracturing-liquefaction process; high frequency of electromagnetic wave and stress wave make the porous and permeability of the tight sandstone changed sharply in 10E-5s which lead the strength criteria of the tight sand stone decrease; high amplitude of the electromagnetic wave and stress wave make the dynamic pore stress in the tight sandstone changed sharply in 10E-5s which lead the maximum principle stress increased to the macro tight sandstone strength criteria; all these leading to pore-network cracks propaagation, local - 48 http://www.j-es.org
area hydrofacturing-liquefaction, and the eventual entire region liquefaction occurred. All these results are consisted with the nature earthquake liquefaction observation data [28, 29]. At practice drilling project, with the hydro-pressure increasing, the pore stress in the tight sandstone is reach to the maximum principal stresses and the sandstone begin failure, this stage is a static/quasi-static process which including the drained propagation; once the tight sandstone begin rupture, this means that the micro pore-network cracks begin propagation, the pore stress will spreading as stress wave way, the stage is transient process and the drained propagation do not happen. The results well consisted with the practical observation data and can well explain why secondary hydrofacturing stage maximum principal stress reduced.
6 CONCLUSIONS In this work, an novel intricate theoretical analysis and numerical simulations methods (LB-FE-HIE) is proposed by the author to explore the three-dimensional pore-network cracks transient hydrofacturing-liquefaction in tight sandstone under seismic wave & electro-magneto-thermo-elastic field. The conclusion can be drawn as follows. 1. The problem is reduced to solving a set of coupled lattice Boltzmann & finite element & hypersingular integral equations by using the Green functions and distribution functions. 2. The behavior of the extended displacement discontinuities around the pore-network cracks surface terminating at the solid-liquid film interface is analyzed. Analytical solutions for the extended singular dynamic stresses, the extended dynamic stress intensity factors and the extended dynamic energy release rate near the dislocations front are provided. 3. The tight sandstone sample from the Ordos Basin Triassic formation is selected and the fluid-solid coupled digital rock physical modeling is established. The simulations of 3D PC-TH-TS process is presented, the hydrofacturingliquefaction varying with the amplitude, frequency and time of earthquake wave is obtained, the results are consisted with the nauture earthquake observation data and practical drilling project observation data. 4. The relationship between the tight sandstone pore-network cracks transient propagation-evolution and the maximum tight sandstone fracturing-liquefaction stress criteria is explored, which are useful to guide the in-situ stress measurement in practical drilling project and are helpful to understand the liquefaction mechanismu of earthquake.
ACKNOWLEDGE The work was supported by China SinoProbe-07 and the National Natural Science Foundation of China (D0408/4097409).
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Kachanov M, Tsukrov I, Shafiro B. Materials with fluid-saturated cracks and cavities: Fluid pressure polarization and effective elastic response. Int J Fracture 1995;73:R61
[3]
Loret B, Radi E. The effects of inertia on crack growth in poroelastic fluid-saturated media. J Mech Phys Solids 2001;49:995
[4]
Cheng CM, Tan QM, Peng FJ. On the mechanism of the formation of horizontal cracks in a vertical column of saturated sand. Acta Mech Sinica 2001;17:1
[5]
Galvin RJ, Gurevich B. Interaction of an elastic wave with a circular crack in a fluid-saturated porous medium. Appl Phys Lett 2006;88
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Lu XB, Zheng ZM, Wu YR. Formation mechanism of cracks in saturated sand. Acta Mech Sinica 2006;22:377
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de Borst R, Rethore J, Abellan MA. A numerical approach for arbitrary cracks in a fluid-saturated medium. Arch Appl Mech 2006;75:595
[8]
Galvin RJ, Gurevich B. Scattering of a longitudinal wave by a circular crack in a fluid-saturated porous medium. Int J Solids Struct 2007;44:7389
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Barani OR, Khoei AR, Mofid M. Modeling of cohesive crack growth in partially saturated porous media; a study on the - 49 http://www.j-es.org
permeability of cohesive fracture. Int J Fracture 2011;167:15 [10] Wang XQ, Schubnel A, Fortin J, David EC, Gueguen Y, Ge HK. High Vp/Vs ratio: Saturated cracks or anisotropy effects? Geophys Res Lett 2012;39 [11] Sarout J. Impact of pore space topology on permeability, cut-off frequencies and validity of wave propagation theories. Geophys J Int 2012;189:481 [12] B.J.Zhu, C.Liu, Y.L.Shi, X.Y.Liu. Saturated dislocations transient propagation–evolution in olivine structure under ultra highcoupled thermal-force fields. Theor Appl Fract Mec 2012;58:9 [13] Zhu BJ, Qin TY. Hypersingular integral equation method for a three-dimensional crack in anisotropic electro-magneto-elastic bimaterials. Theoretical and Applied Fracture Mechanics 2007;47:219 [14] Bojing
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Application
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hypersingular
integral
equation
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crack
in
electromagnetothermoelastic multiphase composites. Int J Solids Struct 2007;In Press, Accepted Manuscript [15] Aboudi J. Micromechanical analysis of fully coupled electro-magneto-thermo-elastic multiphase composites. Smart Materials & Structures 2001;10:867 [16] Perez-Aparicio JL, Sosa H. A continuum three-dimensional, fully coupled, dynamic, non-linear finite element formulation for magnetostrictive materials. Smart Materials & Structures 2004;13:493 [17] H.H.Cheng, Y.C.Qiao, C.Liu, Y.B.Li, B.J.Zhu, Y.L.Shi, D.S.Sun, K.Zhang, W.R.Lin. Extended hybrid pressure and velocity boundary conditions for D3Q27 lattice Boltzmann model. Applied Mathematical Modelling 2012;36:2031 [18] Ma J, Wu K, Jiang Z, Couples GD. An implementation for lattice Boltzmann simulation in low-porosity porous media. Phys Rev A 2010;E 81, 056702 [19] Jiang Z, Wu K, Couples G. The Impact of Pore Size and Pore Connectivity on Single-Phase Fluid Flow in Porous Media. Advanced Engineering Materials. 2010;doi: 10.1002/adem.201000255 [20] Zhu B, Cheng H, Qiao Y, Liu C, Shi Y, Zhang K, Sun D, Lin W. Porosity and permeability evolution and evaluation in anisotropic porosity multiscale-multiphase-multicomponent structure. Chinese Sci Bull 2012;57:320-327 [21] Dmitriev NM. Surface porosity and permeability of porous media with a periodic microstructure. fluid Dynamics 1995; 30: 64~69. [22] C.Y K, J.D F, J.L.A C. Image analysis determination of stereology based fabric tensors. Geotechnique 1998 48:515-525 [23] Koo J, Kleinstreuer C. Liquid flow in microchannels: experimental observations and computational analyses of microfluidics effects. JOURNAL OF MICROMECHANICS AND MICROENGINEERING 2003:568-579 [24] Qin TY, Tang RJ. Finite-Part Integral and Boundary-Element Method to Solve Embedded Planar Crack Problems. Int J Fracture 1993;60:373 [25] Wang CY. Liquefaction beyond the near field. Seism. Res. Lett. 2007;78:512 [26] Popescu R. Finite element assessment of the effects of seismic loading rate on soil liquefaction. Canadian Geotech. J 2002; 29:331 [27] Ghosh B, Madabhushi SPG. A numerical investigation into effects of single and multiple frequency earthquake motions. Soil Dynamics and Earthquake Engineering 2003;23:691-704 [28] Kilb.D., J.Gomberg, P.Bodin. Aftershock triggerinig by complete Coulomb stress changes. J.Geophys.Re 2002; 107: doi:10.1029/2001JB000202 [29] Ishihara K. Soil Behavior in Earthquake Geotechnics,. Oxford: Clarendon Press. 1996:350
AUTHORS Yongbing
Li (1973-), male, Doctor,
Bojing
Zhu
(1974-), Professor
male,
Doctor,
Associate Professor in Key Lab of
Associate
Computational Geodynamics of Chinese
Mechanics/Physics
Academy of Sciences, Geochemistry and
Computational Geodynamics of Chinese
Petrology
Academy of Sciences (2008.01 -2013.10).
Email: yongbingli@ucas.ac.cn.
Email: cynosureorion@ucas.ac.an
in
of Key
Fracture Lab
of
Research Associate of Rock/Mineral physical-chemical in Rock Physical lab of Durham University(2013.02-now). Email: bojing.zhu@durham.ac.uk
- 50 http://www.j-es.org
APPENDIX A 5
2 Ki i i i ( Riˆ3 (c44 D0 s02 ( 3Riˆ, Riˆ, ) ( 3Riˆ, Riˆ, ) i2ti2 3h 2i Riˆ2 ( s0 5Riˆ, Riˆ, )) i ˆ
ˆ
ˆ
i 1
5
(A.1)
(3R Riˆ, s t 3h R R (1 5s0 5h R ) 6i ) 4 iˆ
i 1
2 iˆ
2 1 33 i i
5
2 iˆ ,
2 iˆ
2 iˆ
2 iˆ
5
5
i K mi imi (( Riˆ2 Riˆ, Ai ti2 i1 3h2i Riˆ,2 Riˆ2 (1 5s0 5h 2i Riˆ2 )) m ( Riˆ3 im tit ˆ
ˆ
ˆ
ˆ
i 1
n 3 i 1
5
3h Riˆ, Riˆ (1 5R h )) nm (3R Riˆ, i s 3h R R (1 5s0 5h R ) 6 m ) 2 iˆ
2 2 iˆ iˆ i
4 3 iˆ
i 1
2 i
2 iˆ
m i
2 iˆ ,
5
2 iˆ
2 iˆ
(A.2)
2 iˆ
2 K6i i i6i ( Riˆ3 (c44 D0 s02 ( 3Riˆ, Riˆ, ) ( 3Riˆ, Riˆ, ) i2ti2 3h 2i Riˆ2 ( s0 5Riˆ, Riˆ, )) i ˆ
ˆ
ˆ
i 1
5
(A.3)
(3R Riˆ, s t 3h R R (1 5s0 5h R ) 6i ) 4 iˆ
i 1
2 iˆ
2 1 33 i i
2 iˆ ,
2 iˆ
2 iˆ
2 iˆ
5
2 2 K11i R 3 (3c44 D0 s02 R,2 R,1 (1 3R,12 ) i2ti2 3h2 R 2 ( s0 5R,12 ))1i R 3 ((1 3c44 D0 s02 R,22 ) i 1
5
5
(A.4)
3R,1 R,2 t 15h R R,1 R,2 ) 2i (3R R,1 s t 3h R R (1 5s01 5h R ) 6i i 1
2 2 i i
2
2
4
i 1
2 1 33 i i
2
2 ,1
2
2
2
5
2 2 K 21i R 3 (c44 D0 s02 (1 3R,22 ) 3R,2 R,1 i2ti2 15h 2 R 2 5R,2 R,1 )1i R 3 (c44 D0 s02 3R,2 R,1 i 1
5
5
i 1
i 1
(1 3R,22 ) i2ti2 3h2 R 2 (s0 5R,22 )) 2i (3R 4 R,2 33 si2ti1 3h2 R,22 R 2 (1 5s0 22 5h 2 R 2 ) 6i 5
(A.5)
5
K31i ( R 2 R,1 Ai ti2 i1 3h2 R,12 R 2 (1 5s0 5h 2 R 2 ))1i ( R 2 R,2 Ai ti2 i1 3h 2 R,22 R 2 (1 5s0 5h 2 R 2 )) 2i i 1
i 1
5
5
5
(2 R 3 i3tit 3h 2 R ( R,1 R,2 )(1 5R 2 h 2 )) 3i (3R 4 31 R,1 i si2 i3 3h2 R,12 R 2 (1 5s0 5h 2 R 2 ) (A.6) n 3 i 1
i 1
5
3R 4 32 R,2 i si2 i3 3h 2 R,22 R 2 (1 5s0 5h 2 R 2 ) 6i i 1
5
5
K ( R R,1 Ai ti2 i1 3h 2 R,12 R 2 (1 5s0 5h 2 R 2 ))1i ( R 2 R,2 Aiti2 i1 3h 2 R,22 R 2 (1 5s0 5h 2 R 2 )) 2i 1 4i
2
i 1
5
i 1
5
5
(2 R 3 i4 tit 3h 2 R ( R,1 R,2 )(1 5R 2 h 2 )) 4i (3R 4 31 R,1 i si2 i4 3h2 R,12 R 2 (1 5s0 5h 2 R 2 ) n 3 i 1
(A.7)
i 1
5
3R 4 32 R,2 i si2 i4 3h 2 R,22 R 2 (1 5s0 5h 2 R 2 ) 6i i 1
5
5
K51i ( R 2 R,1 Ai ti2 i1 3h2 R,12 R 2 (1 5s0 5h 2 R 2 ))1i ( R 2 R,2 Ai ti2 i1 3h 2 R,22 R 2 (1 5s0 5h 2 R 2 )) 2i i 1
5
i 1
5
5
(2 R 3 i5tit 3h 2 R( R,1 R,2 )(1 5R 2 h 2 )) 5i (3R 4 31 R,1 i si2 i5 3h2 R,12 R 2 (1 5s0 5h 2 R 2 ) n 3 i 1
(A.8)
i 1
5
3R 4 32 R,2 i si2 i5 3h 2 R,22 R 2 (1 5s0 5h 2 R 2 ) 6i i 1
5
5
2 2 K 61i R 3 (c44 D0 s02 (1 3R,22 ) (1 3R,12 ) i2ti2 3h 2 R 2 ( s0 5R,2 ))1i R 3 (3c44 D0 s02 3R,1 R,2 3R,2 R,1 i2ti2 i 1
i 1
5
2 15h 2 R 2 R,2 R,1 ) R 3 (c44 D0 s02 (1 3R,12 ) (1 3R,2 R,22 ) i2ti2 3h 2 R 2 ( s0 5R,22 )) 2i (3R 4 ( R,1 R,2 ) i 1
5
33 si2ti1 ( R,12 R,22 )3h2 R 2 (1 5s0 5h 2 R 2 )) 6i i 1
- 51 http://www.j-es.org
(A.9)
r ((1 x1 )2 (2 x2 )2 )1/2 , r, ( x ) / r , Ri , ( x ) / R
(A.10)
1 2 , K1 m K2m , K1 6 K2 6 , K61i K62i Ri ((1 x1 )2 (2 x2 )2 h2i )1/2 , K K
(A.11)
ˆ) ˆ )2 )0.5 , Ri Ri zi (1 ih (A.12) R (( x1 1 )2 ( x2 2 )2 ( zi ih ˆ ˆ ˆ ˆ i i i i Kˆ 2 ( K3 ( K11 K12 ) K16 K26 (( Kˆ11 2Kˆ12 ) K2 ( K11 K12 ) K6 )) / ( K11 K12 )( K26 ( Kˆ11 2Kˆ12 ) ( K11 K12 ) K66 ) (A.13) *iˆ iˆ iˆ iˆ *iˆ iˆ iˆ iˆ *iˆ iˆ iˆ iˆ ˆ ˆ ˆ ˆ ˆ ˆ (A.14) K1 K1 K31 K2 K36 K6 K5 K5 K51 K2 K56 K6 K4 K4 K41 K2 K46 K6 ˆ ˆ ˆ ˆ i i i i (A.15) Kˆ (( K 2K )( K K ) ( K K ) K ) / (( K K ) K K ( K 2K )) iˆ i
6
11
12
2
3
11
12
6
K i 1 A t , 5
11
12
66
16
11
12
Km6 v 2 i 1 i i
2 i 3 i
5
m i
(A.16)
APPENDIX B: PERMEABILITY OF TIGHT SANDSTONE VARYING WITH PRESSURE AND TEMPERATURE (UNIT: UD) Appendix B Tab 1. Component of permeability in x-direction as function pressure (0 ~ 100 MPa) and temperature (28 ~ 50 C) Appendix B Tab 2. Component of permeability in y-direction as function of pressure temperature (28 ~ 50 C)
(0 ~ 100 MPa) and
Appendix B Tab 3. Component of permeability in z-direction as function of pressure (0 ~ 100 MPa) and temperature (28 ~ 50 C) Appendix B Tab.4.Component of permeability in z-direction as function of pressure (0 ~ 1.4GPa) APPENDIX B TAB 1. COMPONENT OF PERMEABILITY IN X-DIRECTION AS FUNCTION PRESSURE
(0 ~ 100 MPA) AND
TEMPERATURE (28 ~ 50 C)
P T 28.0 28.1 28.2 28.5 29.0 29.1 29.2 29.3 29.5 29.6 30.0 31.0 32.0 33.0 34.0 35.0 36.0 37.0 37.5 38.0 39.0 40.0 41.0 42.7 42.9 43.7 44.0
7
12
17
27
32
37
42
47
57
62
72
54.7068 54.7096 54.7124 54.7207 54.7345 54.7373 54.7401 54.7428 54.7484 54.7511 54.7622 54.7899 54.8176 54.8453 54.8730 54.9006 54.9283 54.9560 54.9699 54.9837 55.0114 55.0391 55.0668 55.1138 55.1194 55.1415 55.1498
48.0686 48.0711 48.0735 48.0808 48.0929 48.0954 48.0978 48.1002 48.1051 48.1075 48.1173 48.1416 48.1659 48.1903 48.2146 48.2389 48.2632 48.2876 48.2997 48.3119 48.3362 48.3605 48.3849 48.4262 48.4311 48.4505 48.4578
44.2138 44.2160 44.2182 44.2249 44.2361 44.2384 44.2406 44.2428 44.2473 44.2496 44.2585 44.2809 44.3033 44.3256 44.3480 44.3704 44.3928 44.4151 44.4263 44.4375 44.4599 44.4823 44.5046 44.5427 44.5472 44.5651 44.5718
39.5674 39.5694 39.5714 39.5774 39.5875 39.5895 39.5915 39.5935 39.5975 39.5995 39.6075 39.6275 39.6475 39.6676 39.6876 39.7076 39.7276 39.7476 39.7577 39.7677 39.7877 39.8077 39.8277 39.8618 39.8658 39.8818 39.8878
37.9865 37.9884 37.9903 37.9961 38.0057 38.0076 38.0096 38.0115 38.0153 38.0172 38.0249 38.0442 38.0634 38.0826 38.1018 38.1211 38.1403 38.1595 38.1691 38.1787 38.1980 38.2172 38.2364 38.2691 38.2729 38.2883 38.2941
36.6857 36.6876 36.6894 36.6950 36.7043 36.7061 36.7080 36.7098 36.7135 36.7154 36.7228 36.7414 36.7600 36.7785 36.7971 36.8157 36.8342 36.8528 36.8621 36.8714 36.8899 36.9085 36.9270 36.9586 36.9623 36.9772 36.9827
35.5865 35.5883 35.5901 35.5955 35.6045 35.6063 35.6081 35.6099 35.6135 35.6153 35.6225 35.6405 35.6585 35.6766 35.6946 35.7126 35.7306 35.7486 35.7576 35.7666 35.7846 35.8026 35.8206 35.8512 35.8548 35.8693 35.8747
34.6387 34.6405 34.6422 34.6475 34.6562 34.6580 34.6597 34.6615 34.6650 34.6668 34.6738 34.6913 34.7088 34.7264 34.7439 34.7614 34.7789 34.7965 34.8052 34.8140 34.8315 34.8491 34.8666 34.8964 34.8999 34.9139 34.9192
33.0716 33.0733 33.0749 33.0800 33.0883 33.0900 33.0917 33.0934 33.0967 33.0984 33.1051 33.1218 33.1385 33.1553 33.1720 33.1888 33.2055 33.2222 33.2306 33.2390 33.2557 33.2724 33.2892 33.3176 33.3210 33.3344 33.3394
32.4109 32.4125 32.4142 32.4191 32.4273 32.4289 32.4306 32.4322 32.4355 32.4371 32.4437 32.4601 32.4765 32.4929 32.5093 32.5257 32.5421 32.5585 32.5667 32.5749 32.5913 32.6077 32.6241 32.6520 32.6553 32.6684 32.6733
31.2684 31.2700 31.2715 31.2763 31.2842 31.2858 31.2874 31.2890 31.2921 31.2937 31.3000 31.3159 31.3317 31.3475 31.3633 31.3791 31.3950 31.4108 31.4187 31.4266 31.4424 31.4583 31.4741 31.5010 31.5042 31.5168 31.5216
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82 30.3075 30.3090 30.3105 30.3152 30.3228 30.3244 30.3259 30.3274 30.3305 30.3320 30.3382 30.3535 30.3688 30.3842 30.3995 30.4148 30.4302 30.4455 30.4532 30.4609 30.4762 30.4915 30.5069 30.5329 30.5360 30.5483 30.5529
92 29.4819 29.4834 29.4849 29.4894 29.4969 29.4984 29.4998 29.5013 29.5043 29.5058 29.5118 29.5267 29.5416 29.5565 29.5715 29.5864 29.6013 29.6162 29.6237 29.6311 29.6461 29.6610 29.6759 29.7013 29.7043 29.7162 29.7207
APPENDIX B TAB 2. COMPONENT OF PERMEABILITY IN Y-DIRECTION AS FUNCTION OF PRESSURE
(0 ~ 100 MPA) AND
TEMPERATURE (28 ~ 50 C)
P T 28.0 28.2 28.3 28.5 28.7 29.0 30.0 31.0 32.0 33.0 34.0 34.4 35.0 36.0 36.5 37.0 38.0 39.0 40.0 41.0 42.0 42.6 43.0 43.6 44.0 44.5
7
12
17
22
27
32
42
52
62
72
82
92
97
19.3377 19.3398 19.3408 19.3429 19.3449 19.3480 19.3583 19.3686 19.3790 19.3893 19.3996 19.4037 19.4099 19.4202 19.4253 19.4305 19.4408 19.4511 19.4614 19.4717 19.4820 19.4882 19.4923 19.4985 19.5027 19.5078
16.4506 16.4523 16.4532 16.4549 16.4567 16.4593 16.4681 16.4769 16.4856 16.4944 16.5032 16.5067 16.5119 16.5207 16.5251 16.5295 16.5382 16.5470 16.5558 16.5645 16.5733 16.5786 16.5821 16.5873 16.5909 16.5952
14.8184 14.8199 14.8207 14.8223 14.8239 14.8263 14.8342 14.8421 14.8500 14.8579 14.8658 14.8689 14.8737 14.8816 14.8855 14.8894 14.8973 14.9052 14.9131 14.9210 14.9289 14.9337 14.9368 14.9416 14.9447 14.9487
13.7154 13.7168 13.7176 13.7190 13.7205 13.7227 13.7300 13.7373 13.7446 13.7519 13.7593 13.7622 13.7666 13.7739 13.7775 13.7812 13.7885 13.7958 13.8031 13.8104 13.8177 13.8221 13.8250 13.8294 13.8324 13.8360
12.8981 12.8995 12.9002 12.9150 12.9029 12.9050 12.9118 12.9187 12.9256 12.9325 12.9393 12.9421 12.9462 12.9531 12.9565 12.9600 12.9669 12.9737 12.9806 12.9875 12.9944 12.9985 13.0012 13.0054 13.0081 13.0115
12.2572 12.2585 12.2591 12.2604 12.2617 12.2637 12.2702 12.2768 12.2833 12.2898 12.2964 12.2990 12.3029 12.3094 12.3127 12.3160 12.3225 12.3290 12.3356 12.3421 12.3486 12.3526 12.3552 12.3591 12.3617 12.3650
11.2969 11.2981 11.2987 11.2999 11.3011 11.3029 11.3090 11.3150 11.3210 11.3270 11.3330 11.3355 11.3391 11.3451 11.3481 11.3511 11.3571 11.3632 11.3692 11.3752 11.3812 11.3848 11.3872 11.3909 11.3933 11.3963
10.5958 10.5969 10.5975 10.5986 10.5998 10.6014 10.6071 10.6127 10.6184 10.6240 10.6297 10.6319 10.6353 10.6410 10.6438 10.6466 10.6523 10.6579 10.6636 10.6692 10.6749 10.6783 10.6805 10.6839 10.6862 10.6890
10.0512 10.0523 10.0528 10.0539 10.0549 10.0565 10.0619 10.0673 10.0726 10.0780 10.0833 10.0855 10.0887 10.0940 10.0967 10.0994 10.1048 10.1101 10.1155 10.1208 10.1262 10.1294 10.1316 10.1348 10.1369 10.1396
9.6103 9.6113 9.6118 9.6128 9.6138 9.6154 9.6205 9.6256 9.6307 9.6359 9.6410 9.6430 9.6461 9.6512 9.6538 9.6564 9.6615 9.6666 9.6717 9.6769 9.6820 9.6851 9.6871 9.6902 9.6922 9.6948
9.2425 9.2435 9.2440 9.2450 9.2460 9.2475 9.2524 9.2573 9.2622 9.2672 9.2721 9.2741 9.2770 9.2819 9.2844 9.2869 9.2918 9.2967 9.3016 9.3066 9.3115 9.3145 9.3164 9.3194 9.3214 9.3238
8.9289 8.9299 8.9303 8.9313 8.9322 8.9337 8.9384 8.9432 8.9479 8.9527 8.9575 8.9594 8.9622 8.9670 8.9694 8.9717 8.9765 8.9813 8.9860 8.9908 8.9955 8.9984 9.0003 9.0032 9.0051 9.0074
8.7883 8.7892 8.7897 8.7906 8.7915 8.7930 8.7976 8.8023 8.8070 8.8117 8.8164 8.8182 8.8211 8.8257 8.8281 8.8304 8.8351 8.8398 8.8445 8.8492 8.8539 8.8567 8.8585 8.8613 8.8632 8.8656
APPENDIX B TAB 3. COMPONENT OF PERMEABILITY IN Z-DIRECTION AS FUNCTION OF PRESSURE
(0 ~ 100 MPA) AND
TEMPERATURE (28 ~ 50 C)
P T 29.5 30.1 30.3 30.3 30.4 30.5 30.8 31.0 31.2 31.3 31.6 32.0 32.4 33.0 33.2 34.0 35.0 36.0 37.0 38.0 39.0 40.0 41.0 42.0 43.0 44.0
7
12
17
22
27
32
37
42
47
52
57
62
72
82
92
6.7737 6.7760 6.7767 6.7767 6.7771 6.7775 6.7786 6.7793 6.7801 6.7805 6.7816 6.7831 6.7846 6.7868 6.7876 6.7906 6.7943 6.7981 6.8018 6.8055 6.8093 6.8130 6.8168 6.8205 6.8243 6.8280
6.3270 6.3289 6.3296 6.3296 6.3299 6.3302 6.3312 6.3318 6.3325 6.3328 6.3338 6.3351 6.3363 6.3383 6.3389 6.3415 6.3447 6.3479 6.3512 6.3544 6.3576 6.3608 6.3640 6.3672 6.3705 6.3737
5.7391 5.7408 5.7414 5.7414 5.7417 5.7420 5.7429 5.7435 5.7441 5.7444 5.7452 5.7464 5.7476 5.7493 5.7499 5.7522 5.7552 5.7581 5.7610 5.7639 5.7668 5.7698 5.7727 5.7756 5.7785 5.7814
5.3394 5.3410 5.3416 5.3416 5.3418 5.3421 5.3429 5.3435 5.3440 5.3443 5.3451 5.3462 5.3473 5.3489 5.3494 5.3516 5.3543 5.3570 5.3598 5.3625 5.3652 5.3679 5.3706 5.3733 5.3761 5.3788
5.0418 5.0434 5.0439 5.0439 5.0441 5.0444 5.0452 5.0457 5.0462 5.0464 5.0472 5.0482 5.0493 5.0508 5.0513 5.0534 5.0559 5.0585 5.0611 5.0636 5.0662 5.0688 5.0713 5.0739 5.0765 5.0790
4.8076 4.8091 4.8095 4.8095 4.8098 4.8100 4.8108 4.8113 4.8117 4.8120 4.8127 4.8137 4.8147 4.8161 4.8166 4.8186 4.8210 4.8235 4.8259 4.8284 4.8308 4.8333 4.8357 4.8382 4.8406 4.8431
4.6161 4.6175 4.6180 4.6180 4.6182 4.6184 4.6191 4.6196 4.6201 4.6203 4.6210 4.6219 4.6229 4.6243 4.6248 4.6266 4.6290 4.6313 4.6337 4.6360 4.6384 4.6407 4.6431 4.6454 4.6478 4.6501
4.4551 4.4565 4.4569 4.4569 4.4572 4.4574 4.4581 4.4585 4.4590 4.4592 4.4599 4.4608 4.4617 4.4631 4.4635 4.4653 4.4676 4.4699 4.4721 4.4744 4.4767 4.4789 4.4812 4.4835 4.4857 4.4880
4.3170 4.3183 4.3188 4.3188 4.3190 4.3192 4.3199 4.3203 4.3207 4.3210 4.3216 4.3225 4.3234 4.3247 4.3251 4.3269 4.3291 4.3313 4.3335 4.3357 4.3379 4.3401 4.3423 4.3445 4.3466 4.3488
4.1965 4.1978 4.1982 4.1982 4.1984 4.1986 4.1993 4.1997 4.2001 4.2004 4.2010 4.2018 4.2027 4.2040 4.2044 4.2061 4.2083 4.2104 4.2125 4.2147 4.2168 4.2189 4.2211 4.2232 4.2253 4.2275
4.0900 4.0913 4.0917 4.0917 4.0919 4.0921 4.0927 4.0931 4.0935 4.0938 4.0944 4.0952 4.0960 4.0973 4.0977 4.0994 4.1015 4.1035 4.1056 4.1077 4.1098 4.1119 4.1139 4.1160 4.1181 4.1202
3.9948 3.9961 3.9965 3.9965 3.9967 3.9969 3.9975 3.9979 3.9983 3.9985 3.9991 3.9999 4.0007 4.0020 4.0024 4.0040 4.0060 4.0081 4.0101 4.0121 4.0142 4.0162 4.0182 4.0202 4.0223 4.0243
3.8310 3.8322 3.8326 3.8326 3.8328 3.8330 3.8336 3.8340 3.8343 3.8345 3.8351 3.8359 3.8367 3.8379 3.8382 3.8398 3.8418 3.8437 3.8457 3.8476 3.8496 3.8515 3.8535 3.8554 3.8573 3.8593
3.6940 3.6952 3.6955 3.6955 3.6957 3.6959 3.6965 3.6969 3.6972 3.6974 3.6980 3.6987 3.6995 3.7006 3.7010 3.7025 3.7044 3.7063 3.7081 3.7100 3.7119 3.7138 3.7157 3.7175 3.7194 3.7213
3.5769 3.5780 3.5784 3.5784 3.5786 3.5787 3.5793 3.5796 3.5800 3.5802 3.5807 3.5815 3.5822 3.5833 3.5836 3.5851 3.5869 3.5887 3.5906 3.5924 3.5942 3.5960 3.5978 3.5997 3.6015 3.6033
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APPENDIX B TAB.4.COMPONENT OF PERMEABILITY IN Z-DIRECTION AS FUNCTION OF PRESSURE (0 ~ 1.4GPA)
P 0.1~0.7 0.7~1.0 1.0~1.4
Kxx (uD) 60~80 80~130 130~135
Kyy (uD) 30~50 50~90 90~98
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Kzz (uD) 10~18 18~25 25~30