Equivalent properties of a jointed Biot material

Int. J. Rock Mech. Min. Sci. & Geomech Abstr. Vol.30, No.7, pp. 1151-1157, 1903 Printed in Great Britain

0148-9062/93 $6.00 + 0.00 Pergamon Press Ltd

Equivalent Properties of a Jointed B iot Material W. G. PARISEAU~-

In many practical cases, rock mass structural and material discontinuities are too numerous to be accounted for on a one-to-one basis, so that an equivalent properties approach is necessary. All existing equivalent properties formulations are based on the assumption of a representative volume element (RVE) that severely restricts application to engineering design. Application of a non-RVE (NRVE) procedure to coupled, poroelastic constitutive equations is described in this paper. The results are in excellent agreement with constraints intrinsic to Blot's material law and with previous NRVE results obtained in application to Hooke's law in linear elasticity and to Darcy's law in seepage analysis. Results for the classic one-dimensional consolidation problem of a layered isotropic material are in reasonable agreement with results for the same sample characterized by equivalent homogeneous, anisotropic properties.

INTRODUCTION This paper describes an equivalent materials approach to porous, jointed rock masses. The approach taken does not depend on the usual assumption of a representative volume element (RVE) at the rock mass scale. It has the significant advantage of practical utility in numerical modeling where dimensional restrictions between joint spacing, excavation size, and element dimension often preclude RVE-based models. In numerical analysis, the RVE should be no larger than the smallest element, cell or grid point spacing in the considered region, otherwise rational assignment of material properties is not possible. Previous results for dry elastic moduli and hydraulic properties apply. They constrain the properties derived for the coupled case where deformation and flow are linked. The NRVE (non-RVE) analytical approximation of equivalent properties for jointed, multi-component Biot materials are consistent with the imposed constraints. The approach seems worthy of further research and should be helpful in design of excavations in porous, jointed rock BIOT MATERIALS A Biot material is a poro-elastic continuum representation of media composed of fluid and solid phases. The fluid occupies connected void space in a sample that is large compared with void space and solid grain dimensions. If grains and voids are of millimeter size, then a representative volume element of the poro-elastic continuum has linear dimensions of centimeters. The continuum constitutive relations, first presented by M.A. Biot [1], couple the solid and fluid phase responses. ]'University of Utah, 315 WBB, Salt Lake City, Utah 84112-1183, U.S.A. 1151

In a Biot material, elastic strains {~} and fluid content ~" are linearly related to stresses {a} and excess pore pressure 7r. In the anisotropic case,

({;>) __ [t ,J {c'>l where {} denotes a 6xl column matrix, [ ] denotes a 6x6 matrix, {}t and [ ]t are transposes. All quantities in (1) are referenced to dimensions of the continuum. When experimentally determined, overall sample dimensions are used. If V and A are volumes and areas of the continuum, then the overall volume and area are given by V = V , + V f and A = As + Af where the subscripts refer to solid and fluid phases, respectively. Porosity (n) of the saturated material is defined by n=Vf/V. Generally the surface porosity (A/A) is taken to be equal to n (the volume porosity). The primes in (1) imply undrained quantities. When g'=0, no change in fluid content occurs, and [C'] is seen as a matrix of undrained elastic constants. The parameters {c'} couple fluid and porous solid response; the scalar c' is related to fluid and solid compressibilities. The inverted form of (i) is

-{s}'

s J ~ ~r t

The material properties in the coefficient matrix of (2) are the drained material properties. When drainage is unimpeded, no excess pore pressure develops; 7r is zero, and [S] is seen as a matrix of drained elastic constants. These are just the constants that would be determined, say, in a triaxial compressive test on an unjacketed test specimen or in a test of a dry sample. Although the overall 7x7 coefficient matrices in (1) and (2) are mutual inverses, the matrices [C'] and [S] are not. The drained

1152

ROCK MECHANICS IN THE 1990s

and undrained elastic constants are related through the coupling parameters. The meaning of the poro-etastic constants in a soil mechanics context is discussed by Biot [1] and Blot and Willis [2]. Their interpretation is intimately linked to the concept of effective stress (a'ij), originally proposed by Terzaghi [3]. In the isotropic case, a'~i = % - c~'~ij. The meaning of ot has been the subject of much discussion, is still somewhat controversial and linked to the interpretation of 7r. Common assumptions in soil mechanics are that the solid grains are in point contact, the pore fluid is incompressible and et = 1. In rock mechanics a value less than one may be appropriate. Simon and others [4] state that n < o t < 1. A mixed formulation for the coupled constitutive equations used in many numerical analyses is --

=,

The elements of the 7x7 coefficient matrix in (3a) may be called the drained moduli, coupling parameters and compressibility coefficient, [C], {c} and c, respectively. They are related to the drained compliances [S], undrained moduli [C'], coupling properties {c'} and c'. Thus [C]

=

[S]", {c}

=

{c'}/c', c = l i e '

(3b,c,d)

But also [c] = [c']-{c'}{c'y/c',

{s}

= [s]{c},

s = c + {c}'{s}

(3e,f,g)

According to Simon and others [4], as e' ~ oo, c ~ 0 and {c}t --, (1 1 1 0 0 0). Use of (3b-g) shows that (1) and (2) are indeed mutual inverses. The constitutive relations (1), (2) (3), and (4) below, should also hold for equivalent properties. In this regard, (3b-g) provide important constraints on any procedure for obtaining equivalent properties. In particular, (3b) and (3e) show that [C] can be obtained in two ways: directly by inversion of the 6x6 drained elastic compliances (3b), or first,by inversion of the 7x7 drained properties matrix, and then by (3e). The two paths are independent. Fluid flow in any ease is described by Darey's law. Thus, {¢v} -- -[k]{h}

(4)

where {w} is the fluid displacement relative to the solid, the dot denotes time derivative, [k] is a 3x3 matrix of seepage coefficients, and {h} is the gradient of the hydraulic potential. In subscript notation, ~ = ~r,i - 71 where i = 1,2,3 and the Vi are components of the fluid body force per unit volume o f fluid (specific weight of fluid). The positive direction of the body forces are opposite to the coordinate directions. The displacement of tlae fluid relative to the solid is wi=(UruO where Ui and ut are the

displacements of the fluid and solid skeleton, respectively. If the solid is rigid, then ut = 0. A nominal, seepage velocity oa can be defined as nwl, then qffi.q~ and is the specific discharge or volume flow rate per unit total area normal to the flow direction specified by n~. The strains in a Biot material are related to the solid displacements in the usual way: 2¢~j = u~ + uj,i where the comma denotes spatial differentiation. The fluid content ~" = wi.i. All displacements are macroscopic, that is, averaged over an RVE relative to grain scale as are the quantities in (1,2,3,4). Stress, deformation and flow in Blot materials are coupled only during transients between steady, equilibrium states. Application of load in the form of a step function results in an instantaneous, undrained response. As time passes, the excess pore pressure diminishes, if drainage is allowed. In the final, equilibrium state, the response is drained (drainage allowed). In general, the response of a Biot material ranges between the undrained and drained material descriptions. Numerous coupled, computational models of Blot materials have evolved that range from steady state to transient descriptions. Extensions to elastic-plastic and partially saturated behavior have been made and to double porosity models as well. Many applications have been made in soil mechanics where three-dimensional consolidation and subsidence caused by drawdown of the water table is an important problem. In rock mechanics, permeability of porous, intact samples is relatively insensitive to deformation, so seepage and deformation analyses can be done separately. However, seepage in jointed rock masses is likely to be coupled to deformation because of tim cubic law and consequent sensitivity of fluid discharge to normal and shear loading of joints [5,6]. If it were possible to represent all "joints" in a field scale rock mass in a design analysis, then there would be no need for equivalent properties. A rock mass cube 1000 m on edge in a region of three orthogonaljoint sets spaced a generous 10 m apart where an excavation 100 m in size is planned contains a million "joints'. Billaux and others [7] state an astounding 5 million fractures were generated in a 68 cubic meter volume from mapping statistics in consideration of seepage in three-dimensional fracture networks. There is no realistic possibility for directly modeling the joints in such cases. Moreover, the size of representative volume elements (say, unit cell size of 10m) is beyond testing in standard laboratory fashion. An approximate theory is therefore needed for the estimation of equivalent material properties. EQUIVALENT JOINTED R O C K MASS PROPERTIES The equivalent properties approach seeks to replace a representative volume of the actual, heterogeneous material by a fictitious, homogeneous material of the same size that behaves on average like the original material. Solutions to boundary value problems involving the heterogeneous sample would supply the requisite information [8]. In the

ROCK MECHANICS IN THE 1990s elastic domain the classical Voigt and Reuss estimates based on the assumptions of a uniformly strained and stressed sample provide bounds to the equivalent moduli [9]. Bounds to equivalent elastic properties are closely related to energy minimization theorems in elasticity theory, and a Biot material admits an energy potential. However, a uniform strain field in a heterogeneous material violates stress equilibrium, while a uniform stress field violates strain compatibility when the two fields are associated through Hooke's law. (Interface slip and separation are ruled out as inelastic phenomena.) A procedure for improved bounds to the elastic properties of composite materials is described by Hashin and Shtrikman [10]. In this regard, there is an extensive literature concerning estimation of properties of manufactured composites that unfortunately has limited applicability to rock mass mechanics. In rock mechanics as in composite mechanics, all equivalent material properties models assume the existence of an RVE. A unit cell is an RVE, so models based on periodic structures which define a unit cell are included in the category of RVE-based models. Microplane models and similar ubiquitous joint models simply ignore the RVE question by (tacitly) assuming an infinitesimal size RVE. In jointed rock mass mechanics, the RVE must be large relative to joint spacing, that is, S < < s where S is joint spacing and s is RVE dimensions. In numerical analysis, say, by the finite element method, the elements near the excavation should be much smaller than the excavation size in order to obtain accurate numerical information, that is, d < < D where d is the element size near the excavation and D is the excavation size. The RVE can be no larger than the smallest element in the mesh in order to assign meaningful element properties, so that s < d and S < < D. These scale relations are seldom satisfied in practice. A non-RVE (NRVE) based equivalent properties approach to elastic properties of jointed rock masses is outlined by Pariseau and Moon [11]; details are presented by Moon [12]. An elastic-plastic theory is described by Pariseau [13]. The approach was extended to the equivalent elastic properties of cable-bolted rock masses [14,15], and later generalized to estimation of seepage properties of jointed rock masses [16]. The NRVE (nervy?) approach is an analytical approximation that is independent of the considered volume. RESULTS Accuracy of the NRVE analytical approximation can be assessed by direct comparison of results with those obtained from numerical solution to the appropriate boundary value problem. An appropriate boundary value problem is usually one that simulates an ordinary laboratory property test procedure where the boundary conditions are uniform, e.g. compression testing for Young's modulus. Figure 1 shows results from past application to the 6x6 elastic properties estimation problem for the jointed rock test volume shown in Fig. 2. RHMS30:7-Q

1153

The test volume contains two joints that intersect at a constant 30 degree angle; inclination of the joint angle bisector was varied from the horizontal to the vertical. The NRVE results are in almost exact agreement with the results of numerical solution to the traction boundary value problem. The results in Fig. 1 confirm the tensorial nature of the NRVE estimates since they are symmetric and follow the tensor transformation law with rotation of the reference axes. In this regard, symmetry is not imposed in the NRVE process, but is used as a check on the results. Symmetry is also described by reciprocity relations for anisotropic media which are satisfied by the NRVE results. Figure 3 shows a test volume containing four joints. The results of past application to the 3x3 hydraulic properties estimation problem are shown in Fig. 4. The results indicate a slightly anisotropic medium with respect to seepage. The range of the diagonal values are contained within the bars shown in Fig. 4. The approximation is adequate for most rock engineering purposes. Figure 5 shows the results when applied to the 6x6 elastic properties estimation problem. The results are in excellent agreement with detailed numerical solutions over a considerable range of contrast between intact and joint Young's moduli. Also shown in Fig. 5 are the classic Voigt and Reuss bounds and the Hashin-Shtrikman upper and lower bounds [12]. The NRVE results are best over the entire range of rock to joint modulus contrast. The drained elastic portion of results obtained from application of the NRVE procedure to the 7x7 Blot material properties problem for the test volumes in Figs. 2 and 3 are also shown in Figs. 1 and 4. The 7x7 results show that the NRVE procedure produces tensorial elastic properties as a subset of the total Biot properties and satisfies the important consistency constraint discussed previously. Indeed, the results are identical to those obtained in consideration of the 6x6 elastic compliance matrix only. Relations between the 7x7 Biot matrices of equivalent drained compliances (D'), undrained moduli (B') and drained moduli (E*) are identical in form to (1), (2) and (3a); the constraints (3b-g) also apply. Since c' becomes indefinitely large when the solid grains and pore fluid are considered incompressible, numerical inversion of (D) in the homogeneous case leads to unreliable results. The inversion should produce B , = 0 in (B). The same problem may arise with (D'). However, when the properties (E') are desired (mixed formulation for numerical analysis), then (3f, g) may be used to obtain {c*} and c*, and (3b) to obtain (C'). Table 1 shows the organization of the 7x7 mixed Biot properties matrix, and Table 2 shows the NRVE results obtained for a test volume containing a single, flat joint with modulus and permeability contrasts of 10 and a joint volume fraction of 0.1.

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EOUIVALENT CONDUCTIVITIES COMPARISON (a) 1.0 ~, 4[ .~ Theory O FEA4 Resu/ts 2

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+ ( )j=joint, ( ) ,=rock, E,=Young's modulus, G,= shear modulus, ~, = ='i= Poisson's ratio, MC = E,/Ej.

Table 2. Equivalent Biot Properties+: Single, Flat Joint, M C = 10, 0.9099 0.1999 0.1999 0.0 0.0 0.0 0.9996

0.1999 0.9099 0.1999 0.0 0.0 0.0 0.9996

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0.0 0.0 0.0 0.9100 0.0 0.0 0.0000

0.0 0.0 0.0 0.0 0.5263 0.0 0.0000

0.0 0.0 0.0 0.0 0.0 0.5263 0.0000

+ E,=2.4(104psf), G,= 1.0(104psf), v,=~,j=0.20, MC =E,/Ei, Vj/V =0.10 (volume ratio).

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L4-1

0.0 5I 10I

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50 MODULUSCONTRAST

100

Fig. 5. Equivalent drained elastic properties as a function of modulus contrast between rock and joint for the test volume shown in Fig. 3. (a) Elastic moduli, V=Voigt bound, R=Reuss bound, HS + and HS" are Hashin-Shtrikman upper and lower bounds. (b) Poisson's ratios. Dotted line = 6.6 and 7x7 NRVE results, solid line= boundary value problem results by the finite element method. The results in Table 2 show an equivalent transversely isotropic medium with coupling parameters {c} and c very nearly equal to the original values of {c}t=(1 1 1 0 0 0) and c = 0 , corresponding to the special case of incompressible fluid and solid grains. The differences are probably caused by numerical error. Since the vector {x}t=(1 1 1 0 0 0) and the scalar c are invariant with respect to rotation of the reference axis, these are just the

1156

ROCK MECHANICS IN THE 1990s

coupling parameters associated with the equivalent properties of an inclined joint. Indeed, NRVE results and boundary value problem results are in close agreement. Table 3 shows the Biot equivalent properties for the four-joint test volume in Fig. 3. Several interesting features appear in Table 3. Weak coupling between normal strains sad shear stresses is present as is coupling between shear strains and normal stresses. The equivalent material is fully anisotropic in mechanical properties. The coupling parameters {c} and c are nearly equal to their original values. However, it is not clear whether the small, non-zero values are numerical in origin or of physical significance. Table 4 shows similar results for the four-joint test volume when the modulus contrast is 1000. In this case, the mechanical coupling between normal and shear variables is much stronger. It is also interesting to note that some of the coupling constants are negative and thus imply dilation in shear. The fluid coupling parameters {c} remain near original values, but the parameter c has obtained a large value.

surface load was applied first using a very long time step (45,000 days) to obtain an estimate of the final surface displacements, effective stresses and pore pressures, and then using a 10 day time step to obtain data for comparisons of average surface displacements, effective stresses and pore pressures. In all analy-'~es, if drainage through the top surface is prevented (undrained case), the numerical results give the known correct values of pore pressure (equal to the applied surface traction) and the effective stress (zero). When drainage is allowed, the surface load is applied over the specified time increment (10 days and 45,000 days). A finite element code was used to solve the problem. Tables 5 and 6 show that the results of the onedimensional consolidation test using equivalent homogeneous, anisotropic properties are in close agreement with the boundary value problem results from the layered, isotropie test. Results for the unjointed case are also included. The joint nearly doubles the final surface displacement.

Table 3. Equivalent Biot Properties+: 4-Joint Volume, M C = 10

Table 5. Surface Displacement+: I-D Consolidation Test

0.7444 O. 1707 0.2079 -0.0072 0.0026 1.0162

0.1609 0.7014 0.2106 -0.0060 -0.0165 0.9828

0.1899 0.2042 0.6798 0.0070 -0.0033 1.0011

-0.0072 -0.0060 0.0070 0.6505 -0.0223 0.0179

0.0026 -0.0165 -0.0033 -0.0223 0.6738 0.0388

0.0826 1.0162 -0.0248 (t~28 -0.0382 1.0311 -0.0175 (t01D 0.0057 0.(3888 0.0253-.0036

time--, At=0 analysis layered 0. equivalent 0. no-joint 0.

At=10

At=45,000

At---co

0.2271 0.7121 0.7125 0.2488 0.7120 0.7125 0.1689 0.3749 0.3750

÷ time=days, displacement=feet

E,=2.4(106psi), G,= 1.0(10t'psi), v,= vj=0.20, MC=E,/E r Table 6. Average Stresses and Pore Pressure* 1-D Consolidation Test Table 4. Equivalent Biot Properties+: 4-Joint Volume, MC = 1000 0.0301 0.0529 0.2402 -0.8959 0.3255 10.2904 1.0498

0.0404 0.0230 0.2439 -0.7435 -2.0455 -3.0877 0.9503

0.1624-0.8959 0.2159 -0.7435 0.0204 0.8640 0.8640 0.0174 -0.4130 -2.7681 -4.7454 -2.1654 1.0188 0.0664

0.3255 10.29041.0498 -2.0455 -3.0877 ( t g ~ -0.4130-4.7454 1.0188 -2.7681 -2.1654 0ff64 0.0197 0.7043 0.1409 0.7043 0.0216 0.0909 0.1409 0.0909 -1.789

The three-dimensional column in Fig. 6 was subject to one-dimensional consolidation test (drainage possible only through the top surface) first as a layered sample of two isotropic porous materials with modulus and seepage properties contrasts of 10, and then as an equivalent homogeneous, anisotropie medium. The thin layer of low elastic modulus and high hydraulic conductivity represents a single, fiat joint occupying 0.1 of the total sample volume. The high modulus, low hydraulic conductivity material occupies 0.9 of the total sample volume. The

a

stress-* analysis layered equivalent no-joint

(tr',~)"

(a'yy)"

(a'~"

(r)"

84 87 113

84 87 113

335 349 452

665 651 548

- time= 10 days, stress=psi

The agreement between the finite element boundary value problem results and the known analytic solution for ultimate surface displacement is excellent. Agreement between the layered (jointed), isotropic and equivalent anisotropic results is also excellent at!ongtimes and within a few percent at short times. Figure 6 shows the distribution of vertical effective stress and excess pore pressure when the surfacz load is applied in a 10 day time step. The averaging effect of equivalent properties is seen in the relative low effvctive stress below where the joint zone is in the layered case and the higher effective stress above.

R O C K M E C H A N I C S IN T H E 1990s

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1157

REFERENCES

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Fig. 6. Layered, isotropic medium containing a single, fiat joint with modulus and hydraulic conductivity contrasts of 10. Also equivalent homogeneous medium used for one-dimensional consolidation test of NRVE hypotheses during a transient. CONCLUSION The analytical approximation to equivalent material properties described here is a non-RVE formulation (NRVE). Comparisons of results from the NRVE hypothesis and appropriate boundary value problems in elasticity, seepage and poro-elasticity are in close agreement. If the fundamental NRVE formulation proves reliable after additional testing, then the way is open for taking additional steps towards greater realism in rock mass mechanics by introducing statistical data obtained from site characterization studies and so forth. At present, each finite element in a graded mesh, designed for high quality numerical results, can be assigned properties dictated by geologic structure within the element regardless of element size and shape. The NRVE procedure thus takes into account geologic detail by analyticallyaveraging up to a level of aggregation that can be handled by the computer. Influence tensors constructed from the equivalent properties allow for recovery of some important details such as the partitioning of flow between joints and porous media between joints, stress state within joints and so forth. Although details are beyond the page limite to this symposium contribution, the NRVE approach does seem worthy of further research and development.

Acknowledgment-This research as been supported by the Department of the Interior's Mineral Institute program administeredby the Bureau of Mines throughthe Generic Mineral TechnologyCenter in Mine SystemsDesign and GroundControl under grant numbers Gl105151, Gl115151, Gl175151 or Gl12525l.

1. Biot, M.A. General Theory of Three-Dimensional Consolidation. J. Appl. Phys. Vol 12, 155-164 (1941). 2. Biot, M.A.D.G. Willis. The Elastic Coefficients of the Theory of Consolidation. J. AppL Phys. Vol 24, 594-601 (1957). 3. Terzaghi, K. Theoretical SoilMechanics. N.Y, Wiley, (1943). 4. Simon, B.R., O.C. Zienkiewicz and D.K. Paul. An Analytical Solution for the Transient Response of Saturated Porous Elastic Solids. Int. J. Numer. AnaL Methods Geomech. Vol 8, n 4, 381-398 (1984). 5. Serafim, J.L. Influence of Interstitial Water on the Behaviour of Rock Mases. Rock Mechanics in Engineering Practice. Wiley, N.Y., pp 55-97 (1972). 6. Witherspoon, P.A., J.S.Y. Wang, K. Iwai and J.E. Gale. Validity of Cubic Law for Fluid Flow in a Deformable Rock Fracture. Water Resources Research. Vol 16, n 6, pp 1016-1024 (1980). 7. Billaux, D., J.P. Chiles, K. Hestir and J. Long. Three-Dimensional Statistical Modelling of a Fractured Rock Mass - an Example from the FanayAugeres Mine. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol 26, n 3/4, 281-299 (1989). 8. Willis, J.R. Variational and Related Methods for the Overall Properties of Composites. Advances in Applied Mechanics (Chia-Shun Yih, ed.) pp 1-78 (1981). 9. Hill, R. Elastic Properties of Reinforced Solids: Some Theoretical Principles. J. Mech. Phys. Solids. Vol 11,357-372 (1963). 10. Hashin, Z. & S. Shtrikman. A Variation Approach to the Theory of the Elastic Behavior of Multiphase Materials. J. Mech. Phys. Solids. Vol 11, 127-140 (1963). 11. Pariseau, W.G. and H. Moon. Elastic Moduli of Well-Jointed Rock Masses. Proc. Sixth Intl. Conf. Numerical Methods in Geomechanics. Balkema, pp 815-821 (1988). 12. Moon, H. Elastic Moduli of Well-jointed Rock Masses. Ph.D. Thesis. Univeristy of Utah. Salt Lake City, UT. (1987). 13. Pariseau, W.G. On the Concept of Rock Mass Plasticity. Proc. 29th U.S. Symposium on Rock Mechanics. Balkema. pp 291-302 (1988). 14. Pariseau, W.G. and F. Duan. Progress and Problems in Cable Bolt Design. Proc. 7th Annual Workshop, Generic Mineral Technology Center, Mine @stems and Ground Control. Virginia Polytechnic Institute and State Unversity, Blacksburg, Virginia, pp 23-34 (1989). 15. Duan, F. Numerical Modeling of Cable Bolt Support Systems. Ph.D. Thesis. University of Utah. Salt Lake City, UT. (1991). 16. Pariseau, W.G. Estimation of Permeability in WallJointed Rock Masses. Proc. 7th Intl. Conf. Computer Methods and Advances in Geomechanics. Balkema, pp 1589-1594 (1991).