A simple approximation for seismic attenuation and dispersion in a fluid-saturated porous rock with aligned fractures

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1 A simple approximation or seismic attenuation and dispersion in a luid-saturated porous rock with alined ractures Robert Galvin 1,, Julianna Toms 1, and Boris Gurevich 1, 1 Curtin University o Technoloy, CSIRO Division o Petroleum Resources Summary Physical properties o many reservoir rocks can be modelled usin the concept o poroelasticity. Many reservoir rocks, in addition to the network o pores, contain larer ractures or cracks. Galvin and Gurevich (6) solved the sinle scatterin problem or a crack in a poroelastic medium and then estimated the eective properties or a distribution o cracks. However this problem requires the solution o a Fredholm interal equation o the nd kind which in eneral has no analytical solution or intermediate requencies. We propose a simple analytical approximation o this solution usin the branchin unction approach. Quantitative comparison shows ood areement between the two solutions. Our analytical solution exhibits a relaxation peak at a requency where the luid diusion lenth is o the order o the crack diameter. The diusion lenth is proportional to ω / (where ω is requency) and is usually much smaller than the wavelenth o the normal compressional or shear wave. This shows that the presence o cracks in a luid-saturated porous medium can cause siniicant attenuation and dispersion at very low requencies, well beore the onset o elastic (Rayleih) scatterin. Introduction Physical properties o many reservoir rocks can be modelled usin the concept o poroelasticity. A poroelastic material consists o an elastic rame permeated by an interconnected pore space illed with a Newtonian luid (Biot 196). Many reservoir rocks, in addition to the network o pores, contain larer ractures or cracks. A common method o detectin such cracks is based on the use o elastic wave scatterin. Attenuation and dispersion o the passin wave due to a distribution o cracks can be estimated usin multiple-scatterin theory. This approach requires an understandin o how an elastic wave interacts with a sinle crack. For luid-saturated reservoir rocks this interaction diers rom the correspondin elastic scatterin, as it involves low o the pore luid between the crack and the host medium, induced by the passin wave. This eect is particularly siniicant or thin cracks, as their hih compliance (compared to that o the relatively sti pores) causes the luid to low in and out o the crack durin rareaction and compression wave cycles. Galvin and Gurevich (6) solved the sinle scatterin problem or a crack in a poroelastic medium and then estimated the eective properties or a distribution o cracks. However this problem requires the solution o a Fredholm interal equation o the nd kind which in eneral has no analytical solution and must be solved numerically or every requency. There are asymptotic analytical solutions, however, in the limits o hih and low requency. A model involvin numerical solution o an interal equation is cumbersome or practical purposes and especially or any inversion procedure. Thereore some analytical approximation o this solution is desirable. In this paper we derive such an approximate solution usin a branchin unction approach. This approach consists in connectin low- and hih requency asymptotic solutions by a unction that ensures that the result is physically consistent or all requencies. The branchin unction approach has been utilized in many dierent applications. Johnson et al (1987) employed this approach to build a model or dynamic permeability and tortuosity in luid saturated porous rock subjected to a mechanical wave. Models or attenuation and dispersion due to patchy saturation (Johnson, 1) and double porosity and dual permeability (Pride and Berryman, 3,a,b) have also utilized a branchin unction. Pride et al (1987) have investiated the eect o replacin dierent branchin unctions in Johnson et al. (1987) model or dynamic permeability, whilst Toms et al (6) compared Johnson s (1) branchin unction model or patchy saturation aainst an exact analytical solution, and showed that the approach is reasonably accurate at intermediate requencies. In this paper we apply the branchin unction approach to the low and hih requency asymptotes o dispersion and attenuation in porous rocks with alined cracks to et an approximate solution or all intermediate requencies. Galvin and Gurevich (6) presented a theoretical study o the problem o the interaction o a plane lonitudinal elastic wave in a poroelastic medium with an open circular oblate spheroidal crack o radius a and small thickness b<< a placed perpendicular to the direction o wave propaation. They restricted the analysis to so-called mesoscopic cracks whose radius is small compared to the wavelenth π /k1 o the normal compressional wave, but lare compared to the individual pore size. Crack thickness b was assumed smaller than the luid diusion lenth SEG/San Antonio 7 Annual Meetin 1679

2 (wavelenth o Biot s slow wave). In elastic media (say, in a dry porous material) such cracks would have very little eect on wave propaation. However in porous media there may be a siniicant eect due to luid diusion in and out o the crack (as the luid diusion lenth is much smaller than the wavelenth). Sinle Scatterin The poroelastic sinle scatterin problem can be solved in the same ashion as the correspondin problem or elasticity (Robertson, 1967). Consider an incident plane normal (ast) compressional wave propaatin in a porous medium with porosity φ alon the z -axis o the cylindrical co-ordinate system with the axial solid in displacement uz = uexp( ik1z), where k1 is the wavenumber (time dependency exp( iωt) is assumed). We obtain the secondary (scattered) ield ur () resultin rom interaction o the incident wave with the crack occupyin the circle r a in the plane z =. The total ield is T in thereore u ( r) = u z e z + u( r), where e z is a unit vector in the z -direction. We assume that the crack is in hydraulic communication with the surroundin pore space. Both the scattered and total ields must each satisy Biot s (196) equations o poroelasticity in the semi-ininite poroelastic medium z. The distribution o displacements and stresses in the neihborhood o the crack is the same as that produced in a semi-ininite porous medium z when its ree surace z = is subject to the ollowin boundary conditions: σ rz =, r < (1) u z =, a< r < () w z =, a< r < (3) uz + wz =, r < a (4) ( α ) σ zz + p = ik H M u r < a 1, (5) where σ ij is a component o the total stress tensor, p is the luid pressure, wz is the z -component o the so-called relative luid displacement, M and H are poroelastic material constants related to the bulk moduli o the luid K, solid K, and dry skeleton K by the Gassmann equations M = ( α φ)/ K + φ/ K and H = K + 4 μ/3+ α M, and α = 1 K / K. The eneral solution o the equations o motion in cylindrical coordinates can be obtained by representin the our axial and radial components uz, ur, wz, and wr o the solid and relative luid displacements in the orm o an inverse Hankel transorm with respect to the radial coordinate r. Then the boundary conditions yield a pair o dual interal equations or the unknown wave amplitudes (in the requency-wavenumber domain). As shown by Noble (1963), such interal equations are equivalent to a sinle Fredholm equation o the second kind in the unknown amplitude unction: 1 B( x) + R(, x y) F( y) B( y) dy= ps() x, (6) π where sin ax ( y) sin ax ( + y) Rxy (, ) =, (7) x y x+ y sinax axcosax S( x) =, (8) π x ( α ) p = ik1 H M u, μ ( K 4 μ/3) = +, μ is the shear modulus o the solid skeleton, q = y k, k is the wavenumber o Biot s slow wave and ( α ) α ( α ) y k yq k + αy F( y) = M. (9) H ( ) yq k For requencies much smaller than Biot s characteristic requency ωb = φη κρ, Biot s slow wave has a diusionlike character and its wavenumber (inverse o the luid diusion lenth) is proportional to the square root o requency. A Random Distribution o Alined Cracks The theory presented above can be used to estimate attenuation and dispersion o an elastic wave propaatin in a medium with a random distribution o alined cracks. This can be done by usin a Foldy-type approximation o multiple scatterin (Waterman and Truell 1961) which expresses the eective wavenumber k or the medium with cracks in terms o the number o scatterers per unit volume n and the ar-ield orward scatterin amplitude () or a sinle scatterer. For a small concentration o alined cracks the scatterin theorem takes the orm k ( ) π n k (1) k 1 SEG/San Antonio 7 Annual Meetin 168

3 Usin equation (1) one can compute eective wave velocity c( ω) = ω/rek and attenuation (reciprocal 1 quality actor) Q = Re k /Imk as unctions o requency due to luid low between the cracks and surroundin pores. The relationship between () and the solution B( y ) o the interal equation (6) is ik1 ( H αm) B( y) ( ) = lim. (11) u μh(1 ) y y For low requencies such that ka<< 1 equation (6) can be solved analytically by usin an asymptotic expression or the transer unction T( y ) in the limit k << y and expandin R( xy, ) in powers o x. This yields an expression or eective velocity in the static limit: In equation (1) ( H αm) μh( ) ε c = c c = ω/ k = H / 1 1 ( ρ) 1/. (1) is the velocity o the ast compressional wave in the porous host (crack-ree 3 porous medium) and ε = na = (3/ 4 π)( a/ b) φ c is the crack density parameter (Hudson 198) where φc = ( 4/3) πabn is the additional porosity present due to the cracks. For dry open cracks = M =, K H = K + 4 μ /3 and equation (1) simpliies to ε c = c1 1, (13) 3 ( 1 ) which coincides with the well-known expression or the velocity o compressional waves propaatin perpendicular to a system o dry open cracks in an elastic medium in the limit o low crack density (Hudson 198). Furthermore, equation (1) coincides with the expression or the compressional wave velocity obtained rom Gassmann s exact static result or the undrained elastic moduli o an anisotropic luid-saturated porous medium with low crack density (Gurevich 3). This Gassmann consistency is an important eature o the model presented here. Lowrequency attenuation is iven by Q low Q ( α ) ( α + α ) M H M 4 3 ka = 15μH1 and is proportional to requency. ka ( ) ε, (14), that is, to the irst power o In the limit o hih requencies such that crack radius is lare compared to the luid diusion lenth, but smaller than the incident wavelenth, ka 1 << 1 << k a (while crack thickness is still smaller then the diusion lenth, kb<< 1 and the requency is still smaller than Biot s characteristic requency ω B ), a similar analysis yields hih ( ) ( ) ( α ) i k1a H M =. (15) 3μMk By substitutin this expression into the dispersion equation (1) one can see that its relative contribution to the real part o the eective wavenumber vanishes in the hih requency limit, implyin that the velocity in that limit tends to the velocity in the porous crack-ree medium. This result is loical as at suiciently hih requencies the luid has no time to move between pores and cracks, and thereore the cracks behave as i they were isolated (Thomsen 1995). Note however that this result is a consequence o assumin that there is an incompressible luid occupyin the cracks and a small aspect ratio ba; the more precise validity condition is K μ >> ba. In particular, the dry case is excluded, except in the static limit (13). Attenuation at hih requencies reads Q hih ( α ) πε H M = (16) 3μMka and thus scales with ω /. Branchin Function Approximation In the previous section we have derived asymptotic solutions or attenuation and dispersion in the low- and hih-requency limits. For intermediate requencies an analytical solution does not exist and attenuation and velocity can only be obtained by solvin interal equation (6) numerically or every requency. Fiure 1 shows this numerical solution (asterisks) and the analytical low and hih requency asymptotes. In order to et a simple analytical approximation o this complicated solution, we connect the exact low- and hihrequency asymptotic solutions usin a branchin unction, which ensures causality o the solution. Followin Johnson (1) we model the dynamic saturated P wave modulus H ~ as ( ω) = ( ) ( ω), (17) H H H H b where SEG/San Antonio 7 Annual Meetin 1681

4 ε( H αm) H = cρ = H1 1 (18) 3μH( 1 ) and H are the saturated P wave moduli at low and hih requency limits, b is the branchin unction: b i ωτ ζ. (19) ( ω) = ζ + ζ In equation (19) ζ and τ are the shape and requency scalin parameters, iven by τ H H = HG, ζ ( H H ) 3 Our saturated P-wave modulus H ~ requencies as =. () HH TG converes at low ( ωt) H ω 1 i (1) and at hih requencies as G H ω ( ω) = H 1. () iω By comparin our exact expressions (18) and (19) with (1) and () we have M T = ( H αm ) ( 4α + 3α ) 15μH ηha ε 4 3 ( 1 ) κm K + μ (3) where the luid diusion lenth 1/ k is o the order o the crack diameter a. Note that the diusion lenth is proportional to ω / and is usually much smaller than the wavelenth o the normal compressional or shear wave. This shows that the presence o cracks in a luid-saturated porous medium can cause siniicant attenuation and dispersion at very low requencies, well beore the onset o elastic (Rayleih) scatterin. (a) (b) 1/Q Frequency 1 and 4 4πε( H αm ) κm K + μ 3 G = (4) 3μM ηh Velocity Equations (17) (), (3) and (4) ive our branchin unction solution or attenuation and dispersion at all requencies. Fiure 1 shows the comparison o this analytical approximation (solid lines) with the oriinal results based on numerical solution o the interal equation (asterisks). One can see that the branchin unction ives a ood approximation o the numerical solution. Conclusions We have presented an approximate analytical solution or attenuation and dispersion o elastic wave in a porous luidsaturated medium with alined ractures o inite size. This solution exhibits a relaxation peak at a requency = ω /π κm K + 4 μ/3 / Hηa, the requency ( ) Frequency [Hz] Fiure 1: P-wave attenuation (a) and velocity (b) vs requency: asterisks - numerical solution, and red line analytical approximation Acknowledements The authors thank the Australian Petroleum Production and Exploration Association (K. A. Richards Memorial Scholarship), CSIRO Petroleum and Curtin Reservoir Geophysics Consortium or inancial support. SEG/San Antonio 7 Annual Meetin 168

5 EDITED REFERENCES Note: This reerence list is a copy-edited version o the reerence list submitted by the author. Reerence lists or the 7 SEG Technical Proram Expanded Abstracts have been copy edited so that reerences provided with the online metadata or each paper will achieve a hih deree o linkin to cited sources that appear on the Web. REFERENCES Biot, M. A., 196, Mechanics o deormation and acoustic propaation in porous media: Journal o Applied Physics, 33, Galvin, R. J., and B. Gurevich, 6, Interaction o an elastic wave with a circular crack in a luid-saturated porous medium: Applied Physics Letters, 88, Gurevich, B., 3, Elastic properties o saturated porous rocks with alined ractures: Journal o Applied Geophysics, 54, Hudson, J. A., 198, Overall properties o a cracked solid: Mathematical Proceedins o the Cambride Philosophical Society, 88, Johnson, D. L., 1, Theory o requency dependent acoustics o patchy-saturated porous media: Journal o the Acoustical Society o America, 11, Johnson, D. L, J. Koplik, and R. Dashen, 1987, Theory o dynamic permeability and tortuosity in luid-saturated porous media: Journal o Fluid Mechanics, 176, Noble, B., 1963, The solution o Bessel unction dual interal equations by a multiplyin-actor method: Proceedins o the Cambride Philosophical Society, 59, Pride, S. R., and J. G. Berryman, 3, Linear dynamics o double-porosity and dual permeability materials. II. Fluid transport equations: Physical Review E, 68, 36,64. Robertson, I. A., 1967, Diraction o a plane lonitudinal wave by a penny-shaped crack: Proceedins o the Cambride Philosophical Society, 63, Thomsen, L., 1995, Elastic anisotropy due to alined cracks in porous rock: Geophysical Prospectin, 43, Toms, J., T. Müller, B. Gurevich, and D. L. Johnson, 6, Attenuation and dispersion in partially saturated porous rock: Random vs Periodic Models: 68th Conerence and Exhibition, EAGE. Waterman, P. C., and R. Truell, 1961, Multiple scatterin o waves: Journal o Mathematical Physics,, reerence deleted due to duplication SEG/San Antonio 7 Annual Meetin 1683