Coupled stress and permeability behavior in fractured media and its application

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1 Pacific Rocks 2000, Girard, Liebman, Breeds 8 Doe (eds)o 2000 Balkema, Rotterdam, ISBN Coupled stress and permeability behavior in fractured media and its application Jincai Zhang School of Petroleum and Geological Engineering, The University of Oklahoma, Norman, Okla., USA Jianxue Wang China Coal Research Institute, Beijing, People's Republic of China ABSTRACT: The coupling between stress and permeability in fractured rock media is important for many engineering problems. For inst,ance, in the petroleum industry, the near-field fluid pressure and associated effective stress change due t,o oil production in fractured formations, which lead to change in fracture aperture, variation in formation permeability. For problems in mining engineering, the stress field is perturbed and subsequent redistributed after mining. The fracture aperture changes due to the stress changes control the permeability and water flow into the mined area. However, permeability is usually considered to be independent of the state of stress, which may result in erroneous prediction. This work forcuses on the analysis of coupling between permeability and stress variations. An analytical relationship coupling the permeability with stress is proposed which is used in the development of a finite element method. Applying this model, the permeability changes due to underground mining is examined. 1 INTRODUCTION The fractured rock mass can be described as a composition of series of blocks of intact rock separated by natural fractures. Fluid flow through rock mass is determined both by the properties of the rock matrix and the fractures. The rock matrix lisually has such low permeability that the fractures are dominant and fluid flow occurs mainly through fractures. Conceptualization of flow in a single fracture can be given through parallel plate analog where a fracture is idealized as a planar opening having a constant aperture. The magnitude of permeability of a single fracture, parallel to its plane, is given by the permeability of a set of parallel fract,ire can be written by The permeability change in the fracture set due to the aperture change can be expressed as follows (Zhang and Zhang, 1998) where, k is the permeability change due to the aperture change Au, and ko = b3/12s. With changing normal stress, the aperture, the relative contact surface area and the degree of as- perity contact of a fractlre are changed. Thus, flow of fluid through fractures and their permeabilities where, b is the effective fracture aperture, and kf are directly dated to stress and resuitant deformais the fracture permeability. tion. A lot of in-sit11 and laboratory experiment re The permeability of a set of parallel fracture is sldts have shown the flow through fractured rock is constant in every direction parallel to the set and highly sensitive to changes in effective stress. Varits magnitude is given by ious authors have proposed many models to de scribe the variation of fracture permeability with stress. The model presented by Snow (1968) had fol- (2) lowing form where, k, is the set permeability, k, is the rock matrix permeability and s is the mean fracture spack, b2 k = ko + ( T)( - PO) (5) ing. 0 The permeability of the rock matrix can be neg- where k is the permeability of horizontal fractures ligible in comparison with the term (%kf), Hence, at pressure or stress p, ko is the permeability at

2 initial pressure Po, k" is the normal stiffness of the fracture. Jones (1975) proposed an empirical equation for the fracture permeability k of Carbonate rocks O"h J3 k = ko[log(-) 0" where ko is the initial permeability, 0" is the effective stress, and O"h is the effective stress when k = ko Through many field well pumping tests in boreholes at various depths in fissured formation have shown that permeability with stress variation had following empirical forms (Louis,1974) (6) k = ko exp( -QO"e) (7) where O"e is the effective stress, which can be expressed as: O"e = "(H - p in which, is the overburden density, H is the depth of the location, p is the fluid pressure, and Q is a coefficient. Kranz et al (1979) presented the following relationship among the confining stress(o"c), water pressure(p) and permeability for whole and jointed granite: b k ex -(O"c - -p) a where b and a are experimental constants. Walsh (1981) obtained the following empirical relationship derived from laboratory test data: k = ko[l - (holn(w 0"0 where 0"0 is the initial effective stress, and is a constant related to the fracture geometry Walsh and Brace (1984) showed that permeability, elevated to a certain power n (0<n< 1/ 3) should be proportional to the log of 0" c (8) (9) k n = (A log o"c + B) (10) where, A and B are constants. Gale (1988) gave a relationship between the permeability and stress for both induced and natural fractures in gneissic granite k f = (3O"Q (11 ) where, Q and (3 are experimental constants. Bai and Elsworth presented the relationship between strain and permeabili ty change /::;'k = ko[l + /::;.E(b + ;t 1 J3 (12) where, /::;.E is strain change, E is Young's modulus. Zhang and Liu (1999) proposed the following relationship between three-dimensional stress and permeability in fractmed media k z ko {exp( _ O"n ) _ bok" i;[/::;.o"x - v(/::;.o"y + /::;.O"z))}3 ( 13) where, O"n is the normal stress of the fracture. Based on this approach, further study will be given in present paper. 2 PERMEABILITY-STRESS RELATION IN FRACTURED MEDIA In order to determine stress-dependent permeability in fractured media, both the fracture aperture variation and the rock matrix deformation need to be considered. To derive the permeability and stress relation, an idealized three-dimensional regularly spaced fracture-matrix system is assumed as illustrated in Figure 1. 60,,: J 6q. /' matrix Kn,1 /' b Figme 1: stresses Coupled fracture-matrix system with The change of total displacement along x direction is t.he sum of the fracture displacement change and the matrix displacement change. /::;.Utx = /::;.ufx + /::;.u,.x (14) where, /::;'Utx, /::;.ufx and /::;.u rx are the displacements of total fracture-matrix system, fracture and rock mat.rix, respectively. The displacement across the fracture can be obtained by /::;.Ufx = /::;.Utx - /::;.urx (15) The above equation can be expressed as the strain form /::;'li'fx = (sx + bx)/::;.etx - Sx/::;.Erx (16) where, Sx and b x are the fracture spacing and apertme along x-direction, respectively. /::;. Etx and /::;.Erx are the total strain and matrix strain, respectively. The total strain along x-direction can be obtained according to the Hooke's law 1 /::;.Etx = E [/::;.O"x - v(/::;.o"y + /::;.O"z) J (17) mx 812

3 where, Em::c is the Young's modulus of the rockmass in x-direction. The matrix strain along the x-direction may be written by 1 L\erx = Er [L\O"x - v(l\o"y + L\O"z)) (18) where, Er is the Young's modulus of the rock matrix... Substituting Eqs. (17) and (18) into (16), one can obtain the change of the fracture aperture due to the stress variation 8,g 6 14 a.. '" 2 Kn = MPaim Er: 1000 MPa 5:1.0m, b=lmm L\u/x.(8X + bx E""" [L\O"x - v(l\o"y + L\O"z)) Stress increment (MPa) Figure 2: Calculated stress-permeability relation The Young's modulus of the rockmass Emx related to the properties of the intact rock matrix and the fractures ('.an be {"xpressed as follows where, ""= is the fracture normal stiffness in x direction. Substituting Eq. (20) into Eq. (19), the fracture displacement in x-direction can be obtained 1 bx bx ) L\u/x ( k nx k nx8x Er [L\O"x - v(l\o"y + L\O"z)] (21) The fracture permeability in the direction perpendicular to x-direction may be calculated directly from the parallel plate analo defined in Eq. (3). With reference to Eqs. (21) and (4), the change of fracture permeability inay be eva1uated as kz " koz{1-(kb++e). nx x "'nxsx r [L\O"x v(l\o"y + L\O"z)jp (22) where, koz and kz are the permeabilities along z direction before and after stress change, respectively. The generalized permeability-stress relation may be written according to the above equation 111 kk k ok {l (k "b" + -k" "+-E) nl 1 n"st r [L\O"; - v( L\O"j + (23) o ro ro m m w m E Distance frompanel center (rri) Figure 3: panel The partial FEM mesh for a where, kok and kz are the permeabilit,ies along k direction before and after stress changes, i = x, y, Z j j = y, z, Xj i z, x, Y, i =f j =f k. " For one-dimensional stress state or the normal stress L\O"x is dominant, Eq. (22) can be simplified as "1 1 1 kz = koz{l-(-k b + -k + E" )L\O"xP(24) nx x nx8 x r Ac:cording to the above equation, when k nx 104MPa, Er = 103MPa, 8x = 1m, bx := 1O-3 m, the stress-permeability relation can be plotted as Figure 2. It can be found that the permeability increases as tensile stress increasing and compressive stress decreasing, which has similar trend with experimental results, such as Gale,

4 <U 12' <.> <= ::l '" "0. ' <U ;;- <U..c: E- -60 h=16m h=26m ' h=36m h=45m,g 15 I! :s l o Distance along mining direction (m) Figure 6: Vertical permeability change over coal seam alone: mining direction 80 o w 00 W 1001WI100 The distance from the center (m) Figure 4: Contour of vertical permeability ratio around mining panel c::::-n". Only one set of fracture is exalnined in above analysis. When three parallel fracture sets exist in x-direction, y-direction and z-direction, respectively, the permeability along z-direction may be written as kz ( )[6. -k b + /,. + -E 0'", nx:t n.nxsx r 1 1 -v(6.o'y + 6.O'z)] - (/,. b "'ny Y - v(6.o'x + 6.O'z)j}3 (25) s I.g I j <1.O==-==> O.S Panel width=ioom where, By and by are the hacture spacing and aperture along y-direction, respectively, k ny is the fracture normal stiffness in y-direction. In generalized form, the permeability change for three fracture sets can be written as kk III kok {I - (-;::--;:- + + E) [[l.o'i 1 1 -v(6.o' + [l.o'k) J knjs j E) [[l.o'j V([l.O'i + [l.o'k)]} 3 FINITE ELEMENT MODELLING o loolwllm1w The distance from the panel center (m) 5: Contour of horizonal permeability ratio around mining 3.1 Governing equation For linear elastic fluid saturated media, by substituting the strain-stress and pore pressure relation into an equilibrium relation, the governing equation for the solid phase can be obtained (Bm & 1994). 814

5 E 2(1 + v) Uk,ki ) + ap,i + 1-2v Similarly, substitution of Darcy's into a continuity relation,!lbsuming steady condition, gives the governing equation for the fluid phase \1(K \1p) 0 (28) and k is stress-deoendent defined by Eqs(27) tions for the behavior. o K is the hydraulic ",,'l:u>'".v of fluid, maybe represent the governing equastate coupled flow-deformation 3.2 Finite element discretization The governing equations for both solid and fluid phases may be developed into a variational statement where the displacement (u) and pore pressure are taken as the primary unknowns. The refinite element discretization is Il:iven as 1994) ( ) (s ;) ( ) (29) where area, China, the mining depth is 305m, the coal seam thickness is the water-bearing sand lies in 75m high over the coal seam, and the sand thickness is 30m. The finite element mesh is shown in Figure 3, and the strata parameters are illustrated in Table 1. The stress changes in the fractured rock masses due to coal mining can be calculated by the finite element method, then through the stress-permeability relation presented in this paper, the permeability changes around the mined panel can be determined. Figure 4 is the calculated contour of the ratio of the permeability premining and after-mining in vertical direction. It can be found that permeability increases around mined area and decreases in some areas the mined area, and the maximum in the mining center. The height 01 ity increase zone in horizontal direction is than that in vertical direction, and there are larger magnitude of permeability change in the horizontal direction (Figure 5). Along the mining direction, the vertical permeability has a significant change near the coal roof after mining (Figure 6), and only a little change when the strata is 45m high over the mined seam. Table 1: Rock parameters used in the FEM calculation. lbtdbdv 1 ATKAdV (30) (31) lbtmndv F Q 1 NfidS l NqdS (33) (34) B is the strainelasticity matrix, K is the conductivity matrix, N is a vector of element functions, A contains the derivatives of the shape functions, F is a vector of applied boundary tractions f, Q is a vector of prescribed nodal fluxes q, V is the volume of the domain, and Rc represents the influence of seepage forces on the resulting deformation field. 3.3 ::itress-devenaent vermeabilit'u around a min- 4 CONCLUSIONS Considering three dimensional stress effect on both the rock matrix and fracture, an analytical model for evaluating stress-dependent permeability in fractured rock media are proposed, in which perand fracture aperture relation is deterthr'ollll:h the parallel plate The permeaol1ity increases with the tensile stress increascompressive stress decreasing. the permeability around a coal mining penal is determined. calculated results show that the permeability has a significant change around the mining panel, which it is of importance to predict mine inflow for mining near aquifers. In order to examine the validity of the nrnnn"pd model, an application example in a COl given. The mine locates in Yanzhou coal mine 815

6 5 ACKNOWLEDGMENT This work is supported partially by the National Natural Science Fou:ndation of China under grant No and by Prof. Tianquan Liu and Dr. M. 'Bai. These are greatly acknowledged. REFERENCES Bai, M. & D. Elsworth Modeling of suosidence and stress-dependent hydraulic conductivity of intact and fractured porous media, Rock Meeh. and Rock Engng., 27, 4, Gale, J. E The effects' of fracture type (induced versus natural) on the stress-fracture closure-fracture permeability relationship, 23rd U.S. Rock Meeh. Symp. Univ. of California, Berkeley, 1982, Kranz, R. L."et a The permeability of whole and jointed Barre granite, Int: J. Rock Meeh. Min. Sci. and Geomech. Abstr.,, Jones, F. O A laboratory study of the effects of confining pressure on fracture flow and storage capacity in carbonate rocks, J. Petrol. Technol, 1975, Louis, C Rock hydraulics, in Rock Mechanics (ed. Muller L.), Springer Verlag, Vienna, Snow D. T Rock fracture spacings, openings, and porosities, J. Soil Mech. Found. Div. Proc. ASCE94, Walsh J. B Effect of pore pressure and confining pressure on fracture permeability, Int. J. Rock Meeh. Min. Sci. and Geomeeh. Abstr.) 18, 3, Walsh, J. B. and W. F. Brace The effects of pressure on porosity and the transport properties of rock, J. Geophys. 80, Zhang, J. & Y. Zhang The effects of stresses on the permeability of fractured rock masses. Chinese J. Geoteeh. Engng, 2, Zhang, J. & T. Liu The stress-sensitive hydraulic conductivity for fractured rock masses. Int. Symp. Coupled on Phenomena in Min., fj Petrol. Engny. T.Lin, & J. -C. Roegiers (eds), Hainan, China.. 816