Computational study of seismic waves in homogeneous dynamicporosity media with thermal and fluid relaxation: Gauging Biot theory

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110,, doi: /2004jb003347, 2005 Computational study of seismic waves in homogeneous dynamicporosity media with thermal and fluid relaxation: Gauging Biot theory G. Quiroga-Goode 1 and S. Jiménez-Hernández Instituto de Investigación en Ingeniería, Universidad Autónoma de Tamaulipas, Tampico, México M. A. Pérez-Flores Geofísica Aplicada, Centro de Investigacion Científica y de Educación Superior de Ensenada, Ensenada, México R. Padilla-Hernández Instituto de Investigación en Ingeniería, Universidad Autónoma de Tamaulipas, Tampico, México Received 28 July 2004; revised 14 January 2005; accepted 18 February 2005; published 8 July [1] The attenuation effects predicted by Hickey s poroelastic theory (Hpt) are quantified by means of seismic modeling in an unbounded, homogeneous, isotropic porous model fully saturated with water and with oil. The numerical results are compared to those predicted by Biot poroelastic theory (Bpt). As opposed to Bpt, Hpt accounts for thermomechanical coupling and viscous fluid relaxation and adequately models the transient fluctuations of porosity and mass densities as the wave compresses and dilates the porous medium during propagation. Despite all these theoretical improvements over Bpt, the numerical results show that both theories produce remarkably similar waveforms. Without considering thermal relaxation, Hpt produces less than 1% higher-amplitude attenuation and velocity dispersion than Bpt. The major contrasts correspond to the slow P wave. Thermomechanical coupling affects the fast P wave: seismic amplitudes are 1% smaller and some dispersion for the oil-permeated case can be observed. It produces no effects on the fast S or the slow P wave. Therefore these numerical experiments appear to substantiate what Biot assumed at the outset of his theoretical developments: that the effects of transient oscillations of porosity during wave propagation are negligible in terms of velocity dispersion and amplitude attenuation. Furthermore it is confirmed that in a homogeneous porous medium, the combined dissipation mechanisms mentioned are not adequate to explain the total amount of energy dissipation observed in the field or laboratory. Citation: Quiroga-Goode, G., S. Jiménez-Hernández, M. A. Pérez-Flores, and R. Padilla-Hernández (2005), Computational study of seismic waves in homogeneous dynamic-porosity media with thermal and fluid relaxation: Gauging Biot theory, J. Geophys. Res., 110,, doi: /2004jb Introduction [2] Analyzing sedimentary materials as a fluid-saturated porous system may prove an advantage for petrophysical imaging because microstructural parameters such as pore pressure, fluid viscosity, porosity, permeability, etc. can be taken into account that are not considered explicitly in single-phase theories, i.e., elasticity or viscoelasticity. The added feature is that resolution attributes are increased as the dynamics of porous media couples the propagation of seismic waves with local diffusion of viscous fluids. This means that while seismic waves can only resolve gross 1 Formerly at Instituto Mexicano del Petróleo, México D.F., México. Copyright 2005 by the American Geophysical Union /05/2004JB003347$09.00 macroscopic geological features, i.e., in the order of tens of meters within the surface exploration frequency band ( Hz), slow P wave phenomena [Biot, 1962], which is associated to viscous flow of fluids, can span down to the microscopic scale [Carcione and Quiroga-Goode, 1995]. [3] The slow P waves are difficult to measure unless the sensing device is close to the vicinity where they are generated, i.e., at the source location [Carcione and Quiroga- Goode, 1995] and in the neighborhood of material heterogeneities [Quiroga-Goode and Carcione, 1997a, 1997b], although it appears that they can be remotely detected by a dual-wave approach [Quiroga-Goode, 2001]. [4] It is generally acknowledged that fluids play an important role in the attenuation and dispersion of seismic waves [O Connell and Budiansky, 1977]. However, identifying the precise set of relaxation mechanisms is still the subject of experimental and theoretical research. Viscous 1of15

2 attenuation in Biot theory is taken into account in the relative motion between the porous matrix and the saturating fluids. It is induced by the pressure differences arising among the peaks and troughs of the wave. For homogeneous materials, the magnitude and behavior of the predicted dissipation are not realistic for typical sedimentary porous rocks [White, 1975; Bourbie et al., 1987]. [5] This attenuation scenario is substantially changed when heterogeneities are considered because now local flows are induced, additionally to the global ones, in response to the pressure gradients generated by the wave among heterogeneous regions of varying compressibilities. These range from lithological variations (e.g., mixtures of sands and clays) with a single fluid saturating all the pores to a single uniform lithology saturated in mesoscopic patches by two immiscible fluids (e.g., air and water). The mechanism is termed mesoscopic flow [Pride et al., 2004; Carcione et al., 2000, 2003; Helle et al., 2003]. It is also called macroscopic squirt flow [Quiroga-Goode, 2002a] because it is analogous to but an upscaled version of the more widely known squirt flow attenuation mechanism [Dvorkin et al., 1994] that occurs at broken contacts and/ or microcracks in the grains [Norris, 1993; Gurevich et al., 1997; Gelinsky and Shapiro, 1997]. It has been demonstrated that these wave-induced mesoscopic flows represent a major source of intrinsic dissipation in the surface seismic frequency band [Pride et al., 2004; Carcione et al., 2003; Helle et al., 2003]. They are predicted naturally in Biot heterogeneous formulation, i.e., without making ad hoc assumptions in the derivation of the theory [Quiroga-Goode and Carcione, 1997a, 1997b; Quiroga- Goode, 2002a; Pride et al., 2004; Carcione et al., 2003; Helle et al., 2003]. In the presence of sharp angular discontinuities, they are seen to produce vortexes [Quiroga-Goode, 2002a] that appear to induce permanent residual deformation [Quiroga-Goode, 2002b]. [6] It is not yet clear whether there could be other equally important attenuation mechanisms affecting seismic waves, since there are other sources of dissipation, also related to fluids, that are not considered in Biot theory but that may also contribute to the overall dissipation of seismic energy. For instance, at the microscopic level, the electrokinetic dissipation associated with electrochemical double layers which build up in narrow pores of sediments [Pride and Morgan, 1991; Leurer, 1997], viscous dissipation of momentum within the pore fluid, thermomechanical coupling, transient oscillations of porosity, squirt flow [Pride et al., 2004; de la Cruz and Spanos, 1989; Hickey, 1994; Sahay, 2001; Sahay et al., 2001] which could make up for additional attenuation. It appears, however, that the squirt flow is incapable of inducing significant losses (10 2 < Q 1 < 10 1, Q represents the quality factor) within the seismic band of frequencies, i.e., Hz [Pride et al., 2004]. [7] The main focus of this work is to determine if by including additional attenuation mechanisms in a perfectly homogeneous porous medium, as proposed in Hpt, realistic levels of seismic dissipation can be reproduced. To achieve this goal, a different approach is followed. Instead of deriving analytical expressions to compute velocity dispersion and attenuation as functions of frequency, Hickey s nonlinear system of partial differential equations (PDE) that couple seismic wave propagation with fluid flow, thermomechanical coupling and viscous fluid relaxation is numerically solved. This allows the study of material deformations in different ways, including the analysis of the transient oscillations of porosity and mass densities. The numerical simulation of seismic wave propagation through porous media is carried out based on Hickey s modifications and improvements [Hickey, 1994; Hickey et al., 1995] over de la Cruz and Spanos theory [de la Cruz and Spanos, 1985; 1989, de la Cruz et al., 1993]. It incorporates several of the attenuation mechanisms mentioned above: viscous losses within fluids, solid-fluid inertial coupling and heat relaxation, in addition to the potential effects associated to the transient fluctuations of porosity [Sahay, 1996; Sahay et al., 2001; Sahay, 2001]. [8] The dynamics of porous media is quantified in this work from a different perspective than that provided by the large body of work associated to Biot. Here it is considered Biot theory as a guideline and thus comparisons are made for two representative geological scenarios. Despite that de la Cruz and Spanos theory has remained controversial, as explained below, the numerical results obtained in this work show that Hickey s modifications to de la Cruz and Spanos theory [Hickey et al., 1995] produce remarkably similar waveforms compared to Bpt. [9] In the early stages of the development of the de la Cruz and Spanos theory [de la Cruz and Spanos, 1985] it was assumed that the effective shear modulus could be replaced with that of the solid and that the effective bulk modulus could be substituted with Wood s modulus [Pride et al., 1992]. However, the de la Cruz and Spanos theory was later modified [Hickey, 1994; Hickey et al., 1995]. Furthermore, the theory was also criticized arguing that as shear stresses cannot significantly transmit through fluids, one should not expect to observe a slow shear wave in sedimentary materials [Pride et al., 1992]. However, it should be born in mind that the primary interest in considering fluid viscosity is because a portion of seismic momentum is dissipated within the fluid, as opposed to only detecting the slow shear wave. On the other hand, it should be considered that additional energy losses can result from conversion into the slow S wave at discontinuities; for multiple heterogeneities that could add up to substantial wave attenuation, just as in the case of wave-induced mesoscopic flows related to the slow P wave conversion [Pride et al., 2004]. [10] In section 2, the governing equations are rewritten as a first-order system. In section 3 it is solved numerically by an explicit mesh method in the time domain. In section 4 they are contrasted with the results obtained based on Biot theory. The conclusions are presented in section Governing Equations [11] Through volume averaging microscopic solid and fluid motions along with suitable boundary conditions at the pore scale, de la Cruz and Spanos [1985, 1989] research group, Hickey [1994], and Hickey et al. [1995] derived a system of macroscopic equations that describes seismic wave propagation in porous homogeneous isotropic media and its couplings to fluid flow. The theory assumes that the 2of15

3 fluid, Newtonian, viscous and compressible, is fully saturating a permeable matrix. Unconnected porosity is neglected. As the system of PDE was derived through a homogenization procedure, the wavelengths under consideration must be at least an order of magnitude larger than V 1/3 (where V corresponds to the averaging volume), which in turn must be at least an order of magnitude larger than the characteristic pore/grain dimension. As mentioned previously, the theory incorporates several sources of attenuation. First, as fluids create and experience shear-restoring forces due to viscosity, dissipation of momentum within the fluid is produced. The second, which is also related to fluid viscosity, is the well-known solid-fluid inertial coupling, which is also present in Biot theory. It gives rise to fluid diffusion at low frequencies and turns asymptotically into a slow propagating mode when viscous forces are overcome by inertial forces at higher frequencies, or at any frequency when the fluid is nonviscous. [12] On qualitative basis, it would appear that coupling heat flow to seismic wave propagation would not contribute to significant energy losses since particle velocities as well as displacements are assumed small, therefore the rate of heat generation is of second order. However, mechanical compression brings forth a temperature change which, being proportional to pressure variation, is a first-order quantity and must therefore be included in a first-order account [de la Cruz and Spanos, 1989]. The resulting heat losses may cause additional attenuation. Also, owing to thermal expansion, the temperature change must react back on the mechanical motion. Under these conditions, it is important to bring the thermomechanical coupling into the picture [de la Cruz and Spanos, 1989]. These grain boundary heat exchanges are included in Hickey s formulation by volume averaging the heat equation and the equations of motion for the solid and fluid at the microscopic level [de la Cruz and Spanos, 1989; de la Cruz et al., 1993; Hickey et al., 1995]. [13] A seismic disturbance propagating through porous materials is expected to induce infinitesimal fluctuations of porosity and density as waves bring about medium deformation through compressions and dilatations. These oscillations may account for an additional source of attenuation, even under linear stressing [Sahay, 2001]. In the case of Hickey s field equations, porosity and mass densities are allowed to play an explicit dynamical role. Their temporal-spatial behavior is characterized by a set of PDE. In Bpt, on the other hand, it was postulated an elastic energy potential, which forms the foundation of his theory of poroelasticity, but the dynamic porosity is explicitly absent [de la Cruz et al., 1993]. As a result, the formulation is restricted to a type of processes where variation in porosity during deformation is related only to the difference in the pressures of the two phases, omitting those processes in which the variation of porosity is also dependent on the total sum of pressures of the two phases [Sahay, 2001]. [14] The main outcome is that velocity dispersion and amplitude attenuation appear to be underestimated [Sahay, 2001]. Many research works followed attempting to compensate for this [Dvorkin and Nur, 1993; Norris, 1993; Carcione and Quiroga-Goode, 1996; Gurevich et al., 1997; Gelinsky and Shapiro, 1997]. [15] It appears, however, that simply by considering porosity in the same footing as mass densities and strain tensors, is sufficient to account for the levels of seismic attenuation as measured experimentally in laboratories. The de la Cruz and Spanos equations were reformulated by Sahay [2001] into a coupled system of PDE representing the natural vibrations of the porous medium, i.e., the center-ofmass velocity and the internal field. He obtained an analytical solution in the frequency domain and computed synthetic waveforms. Such waveforms show strong amplitude attenuation and velocity dispersion. One of the goals in the present work is to corroborate, by a different approach, the attenuation effects of dynamic porosity. [16] The set of macroscopic nonlinear relations that expresses the coupling among seismic waves, local fluid motion and heat conduction [Hickey et al., 1995] can be recast as Equations of motion r s t v s ¼ K s ½rru ð s Þ frh a s rt s Š 1 K m f h 2 o fvi þ fr 12 t vi þ fm M r 2 u s þrðru s Þ=3 r f t v f ¼ rp f þ m F r 2 v f þ ðx f þ m F =3Þrðrv f Þ þ 1 x h f r o t h 1 ð1 h h o Þm f m M fm 1 s 1 o r 2 1 v s þrðrv s Þ=3 þ K m Fh o v i 1 r h 12 o t vi ; ð2þ Equations of continuity 1=r o s 1=r o f Pressure equations 1 K s 1 K f Heat equations t r s f t h þrv s ¼ 0 t r f þ 1 h o t h þrv f ¼ 0; t p s ¼ rv s þ f t h þ a s t T s t p f ¼ rv f 1 h o t h þ a f t T f ; r s c s v t T s ¼ T o K s a s f t h rv s þ fk s M r2 T s k s =k f k f M h ok f r 2 T f þ fg ð Tf T s Þ r f c f p t T f T o a f t p f 1 k f M h r2 T f k f =k s k s M 1 h ð oþk s r 2 T s o 1 gðt f T s Þ; ð8þ h o ð1þ ð3þ ð4þ ð5þ ð6þ ð7þ 3of15

4 Table 1. Phase Velocities for a Berea Sandstone Type of Rock a FLUID Fast S Slow S Fast P Slow P 1 Slow P 2 Slow P 3 Water Bitumen a Velocities are in m s 1. At 0 1 khz. Porosity equation t h ¼ d srv s d f rv f ; where f = 1/(1 h o ). The parameter h o represents static porosity while h its dynamic counterpart. Vector v i, as defined below, represents the internal motion of the porous medium [Sahay, 1996]. The operator / t corresponds to the partial derivative with respect to time and the spatial differential operator r corresponds to nabla (divergencer, gradient r and Laplacianr 2 ). Superscripts or subscripts s and f stand for the solid and fluid phases of the porous medium, respectively. The dynamic variables correspond to r s and r f for densities, T s and T f for temperatures, u s for solid particle displacement vector, v s and v f for the velocity vectors, p s and p f for pressures and h for dynamic porosity. Their corresponding static (constant) values are denoted with subscript or superscript o, which indicate unperturbed : r o s, r o f and h o, respectively. The temperature and pressure ambient values correspond to T o and p o. Permeability is given by K and r 12 measures the intercomponent force arising from the relative acceleration between the solid and fluid phases. The parameters k s and k f f s play the role of heat conductivities, and k M and k M are their corresponding macroscopic values. The new phenomenological macroscopic parameters are the bulk shear modulus m M and the bulk fluid viscosity x f. On the other hand, the corresponding microscopic counterparts correspond the solid shear modulus m s and the fluid viscosity is m f. The solid and fluid thermal expansion coefficients are a s and a f, respectively, and c s p and c f v correspond to the specific heat constants of the solid and fluid. The bulk modulus of the solid and fluid correspond to K s and K f, respectively. The stagnant effective thermal conductivity is k eff. The parameter g describes the average heat transfer between phases. The dimensionless parameters d s and d s, which correspond to the solid and fluid compliances, relate the solid and fluid volume fluctuations to dynamic porosity h. ð9þ They are empirical constants whose values depend on the process considered. For low-frequency seismic propagation, they can be defined as [Hickey, 1994] d s ¼ h o K f K bc h o K bp z 1 ð10þ d f ¼ h o K bp ðfk s K bc Þ z 1 ; ð11þ where z = h o K bp (K f K s ) + K f K bc. The constants K bp and K bc correspond to the drained compressibility (bulk modulus) and pseudobulk compressibility, respectively. [17] It should be noted that porosity equation (9) is similar in form to equation 53 of Pride et al. [1992] that characterizes the fractional change in pore volume. In particular, Zimmermann [1991] derived a porosity equation from Biot theory, t h ¼ C h t P c a h t P F ; ð12þ but the porosity effective stress coefficient [Carcione, 2001] a h ¼ h 0= þ K f ðh 0 1=BÞð1=K D 1=K U Þ=B 1: h 0 =K D ð1=k D 1=K U Þ=B ð13þ This can be shown by substituting Skempton s coefficient B=aM/K U.P c corresponds to the confining P c =(1 h o )P s h o P f and P f to the pore fluid pressure. The fluid modulus 1/M = (a h o )/K M + h o /K f where a = 1 K D /K m corresponds to the Biot-Willis parameter. The drained modulus corresponds to K D. C h can also be written as C h = (1 h o )K 1 D K 1 m or C h = (a h o )K 1 D, with K m being the bulk modulus of the grain. Relation (12) can be used to compute Biot predictions of Table A1. Material Properties of the Porous Medium a Physical Parameter Solid Units Physical Parameter Water Oil Units h o 0.25 o r s 2650 kg m 3 o r f kg m 3 K s Pa K f Pa m s Pa m f Pa s s c v 755 J (kg C) 1 f c p J (kg C) 1 a s C 1 a f C 1 k s 8.4 W (m C) 1 k f W (m C) W (m C) 1 f k M 2/3 h o k f 2/3 h o k f W(m C) 1 k M s T o 20 C r kg m 3 K m 2 g W(m 3 C) 1 m M Pa x F Pa s K bc Pa k eff W (m C) 1 a The material properties of a porous material similar to Berea sandstone saturated with water and light oil [Hickey, 1994]. 4of15

5 Figure 1. Snapshots of the seismic wave field at t = 3 ms. P plots of the (a, c, e) horizontal and (b, d, f) vertical components of the center-of-mass velocity (v), the internal field (v i ), and fluid velocity (v f ), respectively. The wave fronts clearly observed in these plots represent the fast P and S waves, as indicated. The weak energy around the source location in Figures 1c and 1d is the summed effect of the slow P 1 and the slow S. the temporal changes of porosity once the solid and fluid pressures have been calculated. 3. Numerical Solution [18] The set of equations (1) (9) together with definition / t u s = v s can be rewritten in matrix form as a first-order system [Quiroga-Goode, 2004]: t Vy ¼ MV y ; ð14þ where propagation matrix M contains the spatial derivatives and material properties of the porous medium. The vector of unknowns corresponds to V y. The eigenvalues of M prescribe six diffusive/propagatory modes [Hickey, 1994]. Four are associated to longitudinal waves and two describe rotational motion. The first three are diffusive (defined below as slow P 1,slowP 2, and slow P 3 ) and the fourth is propagatory (fast P). The latter is a pseudocompressional wave due to its elliptical polarization [Rasolofosaon, 1991]. It is the analogous to the classic P wave of elastic theory. The slow P 2 and P 3 are of thermal origin. Of the remaining two modes representing rotational motion, one is propagatory. It is also a pseudo shear wave for its elliptical particle motion [Rasolofosaon, 1991] and is analogous to the classical S wave in elastic theory. The remaining mode is diffusive (slow S). [19] As one of the goals in this work is to quantify the attenuation effects, the numerical results are compared with those computed using Biot theory. Therefore superscript y in equation (14) is taken as H, for Hickey, or B for Biot. Biot equations are outlined in Appendix A. [20] In two spatial dimensions (x, y), Hickey s vector of unknowns can be given by h i T V H ¼ u s x ; us y ; vs x ; vs y ; vf x ; vf y ; p s; p f ; T s ; T f ; r s ; r f ; h ð15þ and Biot s h i T: V B ¼ v s x ; vs y ; vf x ; vf y ; s xx; s yy ; s xy ; s ð16þ The solid stresses are represented by s ij (i,j = x,y) whereas the effective pressure by s. The pore fluid pressure can be computed with P f = s/h. [21] The numerical solution of the dynamic system (equation (14)) is obtained with a time domain explicit pseudospectral scheme. Assuming harmonic oscillations, equation (14) is first transformed numerically into the wave 5of15

6 Figure 2. Snapshots similar to those in Figure 1 except that an area of 1 m 2 has been zoomed in around the source location to show the slow P wave. (a d) Two-dimensional plots that correspond to v x,v y,v f x, and v f y. (e h) The 3-D view plots that correspond to v i x,v i y,v f y, and p f. They depict different mechanical behavior depending upon the field variable. Figure 2g is a 3-D view of Figure 2d. number domain and the terms involving spatial derivatives are scaled with the wave number defined by the source bandwidth. The result is transformed numerically back to the space domain using fast Fourier transform. This procedure implies spectral (infinite) accuracy for those wavelengths defined within the source frequency band. The resulting system of ordinary differential equations is marched in time via a fourth-order Runge-Kutta integration method, which is specially suited for solving nonlinear equations. [22] The isotropic homogeneous porous model is characterized by the physical constants given in Table A1. In a kinematical analysis [Hickey, 1994] it was shown that the slow P and slow S wave phase speeds for the sandstone considered in this work are in the order of millimeters to meters per second (Table 1) for up to 1 khz. [23] The numerical mesh is discretized with square cells of 2 cm. As there are no perfect and efficient absorbing boundary conditions for the artificial perimeter of the model, it is enlarged sufficiently so that the wrap-around waves generated by the periodic conditions, which are imposed by the pseudospectral Fourier method, are not confused with the physical phenomena within the model. [24] The initial conditions for dynamic porosity h, mass densities r s and r f, and temperatures T s and T f are given by their corresponding static values: h o, r o s, r o f, T o s and T o f, respectively (Table A1). For both Biot and Hickey s equations, a seismic disturbance is applied to the vertical s component of the solid velocity v y in a single node to generate cylindrical waves. The source temporal variation is given by S(t) = sin(2pt/t o ) 0.5 sin(4pt/t o ), for 0 < t < 1/T o. Here, T o = 1 ms corresponds to a dominant frequency of 1 khz and a maximum of 2 khz. As will be seen below, this frequency range is below the critical frequency above which inertial forces dominate over the viscous ones and thus the diffusive mode turns into the slow propagating compressional wave. [25] Other types of sources can be applied as well, i.e., bulk source, fluid source [Carcione and Quiroga-Goode, 1996]; the difference among them being the amount of 6of15

7 geophones are planted is highly porous. Thus a large area of the geophone spike is in contact with fluids. The center-ofmass velocity v is related to the total linear momentum and thus measures volume fluctuations via dilatations and compressions of the porous material. On the other hand, the second mode of vibration v i = v s v f represents the internal motion of the porous material and, as will be shown below, characterizes angular momentum or spin [Sahay, 1996]. Figure 3. Synthetic seismograms comparing Hickey s theory (solid line) with that of Biot (dashed line) for a set of 103 geophones placed horizontally across the center of the model. (a) Horizontal and (b) vertical components of the center-of-mass velocity (v x and v y ). The seismic responses are virtually identical at this scale. energy partitioned between the fast and slow waves. The spatial discretization step of 2 cm guarantees that viscous flows with wavelength components equal to or larger than 4 cm are accurately resolved. [26] To quantify the dynamics of porous media it should be clear that the modes of vibration within the porous framework cannot be adequately represented with their individual solid v s and fluid v f motions, as is the case of single-phase materials, i.e., elastic or viscoelastic. This is because a transient disturbance does not act on the individual solid (1 h)r s and fluid hr f masses separately, but on its effective porous mass r =(1 h)r s + hr f [Sahay, 1996]. Therefore the field variable that geophones respond to is the center-of-mass velocity [Sahay, 1996] v ¼ ½ð1 hþr s v f þ hr f v f Š=r: ð17þ Equation (17) is especially appropriate for land exploration and ocean bottom seismics since the soil layer into which 4. Numerical Results [27] The numerical solution of matrix equation (equation (14)) is computed for physical parameters similar to those of a Berea type of sandstone given in Table A1. It is first assumed that water is saturating the porous matrix. To make the simplest comparison possible between Biot and Hickey s theories, the isothermal case is initially considered. In this case, equations (7) and (8) are dropped from Hickey s equations, as well as the heat terms a s rt s from equation (1), a s / t T s from equation (5) and a f / t T f from equation (6). The purpose is to quantify the effects of viscous relaxation and transient porosity. [28] Some of the results are presented in Figure 1 as snapshots which show the seismic wave fronts after propagating several wavelengths. They represent only Hickey s isothermal solution since those of Biot are virtually the same at this spatial scale. Only the fast P and S wave fronts can be observed clearly. The elapsed time is 3 ms. There are polarity reversals along the wave fronts due to the vertical force. It was applied at the center of the model (7.5, 7.5 m) as can be deduced from Figure 1. [29] The slow P 1 and slow S phenomena are excited in all cases. However, only the slow P 1 can be seen weakly i around the source location in v x and v i y. Despite its highly dissipative character and high frequencies (1 2 khz), it has diffused approximately 1.2 m. It should be noted that the shearing of viscous fluids against the elastic matrix, as depicted in Figures 1c and 1d, is the one of the causes of attenuation with propagating distance. Given the slow S wave theoretical phase speed of 2.4 m s 1 (Table 1) it could be found approximately at 0.07 m from the source location. However, it cannot be directly detected because of its high attenuation. Nevertheless, what appears as the slow P 1 in Figures 1c and 1d is most likely the combined effect of slow P 1 and slow S. The slow P 1 can be clearly observed in Figure 2 where a small area around the source location has been zoomed in for the various field variables. The slow P 2 and P 3 are not generated for the isothermal case considered at this stage. [30] The wave fronts in Figures 1e and 1f (v f x and v f y )are actually formed by the transient water flows induced by the fast P and S waves. Although these flows are similar in magnitude to the harmonic vibrations of the elastic matrix (v s x and v s y ), a vector plot would show that their direction is not fully longitudinal and normal to the P and S wave fronts, respectively. This slight difference in polarization induces rotation or spin within the porous medium [Sahay, 2001]. The wave fronts in Figures 1c and 1d reflect this difference. [31] Figure 3 compares synthetic seismograms simulated by the two theories. They were recorded from a set of 103 7of15

8 Figure 4. Spatial traces taken across the snapshots of Figure 1. (a) Hickey s results (solid line) laid over Biot (dashed line). (b) Difference between the two theories for same field variables as in Figure 4a. (c) Similar to those in Figure 4a except that they were zoomed in around the source. geophones placed at (0, 7.5 m) and extending laterally to the edges. The dynamic responses of the horizontal (Figure 3a) and vertical (Figure 3b) components of center-of-mass velocities of Biot theory are superimposed graphically over those of Hickey. There is no shear wave in the vertical component since the source corresponds to a vertical force. The slow P1 cannot be observed in Figure 3. As can be judged, both theories produce practically identical amplitudes and phases at this temporalspatial scale. [32] More meaningful comparisons are presented in Figure 4 where cross sections of the snapshots shown in Figure 1 are depicted as spatial traces, as opposed to the temporal traces of Figure 3. Since the horizontal component of the fast S wave has zero amplitudes along a horizontal line across the source location due to the vertical force, the cross sections are taken diagonally from the snapshots (from 0.0 to m), but only for the horizontal components. On the other hand, the vertical components of the various variables displayed as spatial traces in Figure 4 are taken from the snapshots horizontally across the center of the model. Figure 4b is obtained by subtracting Bpt from Hpt spatial traces. It shows small amplitude and phase differences for the fast rotational and compressional waves that indicate a slightly stronger attenuation and velocity dispersion. These differences are larger for the S wave in v x and for the P wave in v y as expected due to the vertical source, but in neither case greater than 10 2 m s 1. Hickey s slow P 1 wave is smaller than its Biot counterpart for the internal field and pore pressure, probably due to the viscous relaxation within the fluid. [33] To determine the dynamic effects of higher viscosity, in the following numerical experiments, water is replaced with oil. The isothermal case is still being considered. Both 8of15

9 Figure 5. Spatial traces similar to those in Figure 4 except that the water is replaced with oil. Note the increase of slow P 1 amplitudes, the decrease in phase speed, and amplitude of the fast P wave relative to the case when the medium is permeated with water. theories were compared for a set of seismograms similar to those in Figure 3 but are not shown here since they are almost identical at that scale. However, the cross sections of snapshots in Figure 5 show the actual differences in amplitudes and waveforms. As seen in v x and v y higher viscosity has caused the fast P wave to propagate with lower velocity compared to the water-saturated case while the fast S wave travels slightly faster, confirming thus the kinematic predictions of Hickey [1994] as shown in Table 1. The amplitudes of both waves have increased little in Biot and Hickey s theories because the magnitude of relative flow (v x i and v y i ) is much smaller, i.e., less seismic energy is transformed into fluid flow. As a result, the amplitudes of the slow P 1 have decreased as well, and the fluid diffusion length is smaller, i.e., the slow P 1 appears much narrower. Additionally the amplitude of the P wave are slightly larger because the cylindrical spreading is smaller, i.e., the wave has traveled slightly less distance. It should be noted that in the oil-saturated case, the slow S wave has reached further, i.e., m from the source location (0.07 m for the watersaturated case). Therefore, at this scale, Biot and Hickey s theory are in good agreement. In a closer look in all plots in Figure 5b, it can be observed that considering oil, Hpt induces more dissipation on the fast P than the S waves compared to Biot theory. Also, the pressure within pore fluid has decreased slightly. [34] To quantify the effects of thermomechanical coupling upon seismic wave propagation, synthetic seismograms are compared with the isothermal case. The synthetics are identical when the saturating fluid is water whereas for oil, the amplitudes are diminished by less than 1%, but only for the fast P wave, as shown in Figure 6. The S wave remains unaffected since Hpt considers thermomechanical coupling only through compressions. The slow P 1 i is not affected with thermal relaxation, as can be seen in v x i and v y and in v f x and v f y. The slow P 2 and P 3 could not be 9of15

10 Figure 6. Spatial traces similar to those in Figure 4 except that Hickey s isothermal and thermal cases are compared. As can be observed from Figure 6b, only the fast P wave is affected by thermomechanical coupling. observed because the elapsed time of 3 ms was not sufficient for them to develop. Nevertheless, what appears as the slow P 1 may actually be the combined effect of slow P 1,P 2,P 3 and slow S wave. [35] As mentioned previously, the major contrast in Hpt relative to Biot theory and most of the poroelastic theories is that it properly accounts for spatial fluctuations of porosity associated to the compression/rarefaction of the porous homogeneous material during propagation. This causes some attenuation of seismic energy [Sahay, 2001]. However, all of the computational experiments conducted thus far indicate that except for the slightly stronger attenuation that Hickey s theory induces via viscous and thermal relaxation, both poroelastic theories are remarkably similar. Thus, as there is no published work in the literature that had porosity fluctuations before, it would prove useful at this stage to verify whether the magnitude and dynamic behavior of porosity and mass densities predicted in Hickey s theory are physically sound. [36] Figure 7 depicts the spatial oscillations of porosity (Figure 7, top), matrix density (Figure 7, middle) and water temperature (Figure 7, bottom) in a series of snapshots obtained at 0.75, 1.5, and 3 ms, respectively. The waveforms and travel times at 3 ms are similar to the P wave fronts in Figure 1 as expected since it is only the compressional waves that produce volume fluctuations during propagation, i.e., for shear waves rv = 0. These are displayed as relative quantities (to their static counterparts) in Figure 7 since their magnitudes are extremely small. Because of this they can have positive (or negative) values, depending on whether they have increased (or decreased) relative to their constant base values (h o, r s o, r f o,t s o and T f o ). It can be observed that the polarity of the porosity waveform is reversed compared to those of temperatures and densities. However, there is a perfect correlation among these dynamic petrophysical variables as can be verified in the waveforms in Figure 7 or the cross sections in Figure 8, some of which were extracted from Figure 7. As the P wave expands the 10 of 15

11 Figure 7. Wave fronts corresponding to the harmonic fluctuations of (a c) porosity, (d f ) matrix density, and (g i) water temperature induced by the passage of a seismic disturbance as they evolve through time-space. It should be noted that they are displayed as relative quantities. Only the fast P can be observed clearly from these plots, whereas those of the slow P 1 can be seen in Figure 10. fluid-saturated porous material, it causes porosity to increase (positive, yellows and reds) and matrix density and water temperature to decrease (negative, blues) as expected. The opposite holds: as the seismic wave compresses the porous medium, porosity decreases (negative, blues) while matrix density and water temperature increase (positive, yellows and reds). Therefore the wave fronts in Figure 7 (at 3 ms) are similar to the P wave waveforms in Figure 1 since the P wave harmonic oscillations of the porous material should mimic the transient fluctuations of porosity and density. In a closer look however, it can be seen that only i i the internal field v x and v y (Figures 1 and 4) is identical to the porosity and density oscillations (Figures 7 and 8) as clearly evidenced in Figure 9. In Figure 9 the velocity plots (solid) correspond to the first 7 m of the spatial traces of Figure 4, whereas porosity (dotted) corresponds to the first 7 m of the cross sections of Figure 8. The amplitudes of the porosity waveforms have been scaled to fit those of the velocities (Figure 9). [37] That the slow P 1 also produces harmonic oscillations of porosity, mass densities and temperatures can be verified in Figure 10. These plots zoom in 1.1 m around the source location from the snapshots of Figure 7 at 1.5 ms. 5. Analysis and Conclusions [38] The main goal of this work was to investigate if realistic levels of amplitude attenuation and velocity dispersion are produced on the transient signals by including additional attenuation mechanisms in the propagation of seismic waves, as proposed in Hickey s poroelastic theory, produced compared to Biot theory. Here a different approach is followed to attain this goal. Instead of deriving analytical expressions to compute phase velocity dispersion and attenuation as functions of frequency, the nonlinear system of PDE that couple seismic wave propagation with fluid flow is solved numerically. This was done for Biot and Hickey s poroelastic equations. It allowed the study of 11 of 15

12 Figure 8. Spatial series extracted horizontally from snapshots in Figures 7c, 7f, and 7i at 3 ms, from (0 m, 7.5 m) to (15 m, 7.5 m). The relative water density and relative matrix temperature spatial series are also included. These depict clearly the magnitude of the spatial oscillations of h, r s, r f,t s, and T f about their corresponding constant values h o, r 0 s, r 0 f,t 0 s and T 0 f when the porous medium is in equilibrium. These spatial wavelets have opposite polarity about the source location at 7.5 m as they are the result of a vertical force. (a) Porosity, (b) matrix density, (c) fluid density, (d) matrix temperature, and (e) fluid temperature. material deformations in different forms, including the transient oscillations of porosity and mass densities. Furthermore, since de la Cruz and Spanos theory has been controversial due to some of the initial assumptions during its development, the numerical simulations shown in this work served as a check to verify whether Hickey s corrections and improvements to the poroelastic theory were suitable. Indeed, it produced similar synthetics compared to Biot theory. [39] The numerical results showed small differences in amplitudes and waveforms between Hpt and Bpt, in the order of 10 2 m s 1, which were related to amplitude attenuation and velocity dispersion, respectively. This was the case for water as well as for oil saturating the porous matrix. Despite that, the two theories are in remarkable agreement. [40] The major contrasts between Hpt and Bpt were seen but only for the slow P waves in both cases, for water and for oil saturation. Viscous relaxation within the fluid may have caused this difference in wave amplitudes. The slow rotational wave was not detected from the numerical results. Thermomechanical coupling affected the fast P wave in a minor degree, but had no visible effects on the slow P 1.It did not influence the fast shear wave since thermal gradients are assumed only through compressions. In real situations, however, thermomechanical coupling may affect also rotational waves. [41] The purpose of computing the temporary fluctuations of porosity and mass densities was to verify whether their temporal-spatial behavior was physically sound. Indeed, they do follow the P wave fluctuations through compressions and dilatations of the porous medium. During compression, porosity decreased about its static value while mass densities and temperatures increased. The P wave expansion of the material caused porosity to build up, while mass densities and solid and fluid temperatures diminished. The only feature that appeared to lack physical consistency was that the porosity fluctuations behaved more like internal field v i, which measures spin or angular momentum while in principle they should mimic the center-of-mass velocity v 12 of 15

13 Figure 9. Comparison of the horizontal and vertical components of (a b) mass velocities and (c d) those of the internal field versus dynamic porosity behavior. The velocity cross sections (solid line) correspond to the first 7 m of Figure 4, while the porosity functions (dashed line) correspond to those in Figure 8. Positive (negative) quantities in these plots indicate an increase (decrease) relative to their static values. The porosity amplitudes have been scaled to match the velocities. as it responds to volume fluctuations. This seeming discrepancy can be reconciled if it is considered that effective velocity v (center of mass) is actually a time-derivative field quantity. This is why it appears as a phase-shifted version of porosity. Porosity fluctuations can be made to match the effective motion of the porous medium by integrating v or by differentiation of h. [42] The numerical results appear to corroborate what Biot had intuitively assumed from the outset of his theoretical developments: that porosity and density fluctuations are so small for seismic strains, that their effects appear to be negligible in terms of velocity dispersion and amplitude attenuation. It can thus be inferred that the amplitude attenuation in Hpt was solely due to thermomechanical coupling and viscous dissipation within the fluids. Therefore, for practical purposes, there is no need to place porosity on an equal basis as mass densities and strain tensors. However, strictly speaking, by doing so the specific case in which variation of porosity is dependent on the total sum of pressures of the two phases is excluded [Sahay, 2001]. In this case the point of physical validity of Biot theory can be brought to question. However, since assumptions are followed all the time to simplify the mathematical or physical treatment of complex problems, the results could be valid within certain bounds. The results showed that this is the case for the representative sandstone considered in this work. This work only dealt with the case of a porous siliciclastic rock as it represents a typical case for which temporary fluctuations of porosity could have affected more the propagation of seismic waves, as opposed to the tighter carbonates which have lower porosities. [43] By applying a seismic disturbance only onto the solid phase, it excited more strongly the generation of the slow P 1. Thus it served as a check to quantify if there were any substantial differences between the two theories. [44] There was some numerical noise around the source location probably due to the spatial aliasing of the slow P 2 and the slow P 3 waves, i.e., they were spatially undersampled. To get more accurate results for the slow thermal waves, the spatial sampling rate should satisfy Fourier discretization interval of m, i.e., two grid points per minimum wavelength, assuming the phase speeds described in Table 1. However, the fact that the slow P waves of thermal origin were not accurately resolved should not affect the fast P and the fast S waves. This is because for a homogeneous medium, the rate of amplitude decay during propagation of the fast modes is not controlled by the slow modes. 13 of 15

14 Figure 10. Snapshots showing that slow P 1 wave also induces fluctuations of (a b) mass densities, (c d) temperatures, and (e) porosity. These were extracted from a small area around the source location in Figure 7 (at t = 1.5 ms). (f) Wave front of the pore fluid pressure. [45] Therefore the most relevant conclusion drawn from these results is that even in the case of including additional dissipation mechanisms such as fluid and thermal relaxation, there are no critical improvements in amplitude attenuation or velocity dispersion that could mimic realistic levels of attenuation as observed experimentally in laboratory or in the field. Appendix A: Biot Field Equations [46] Biot velocity-stress formulation [Biot, 1956, 1962] for the P-SV case in two dimensions (x, y) can be written t vs ¼ R 22 t vf ¼ R 12 x s xx þ y s xy x s xy þ y s yy R 12 x s 1 K m f h 2 oð R 12 þ R 22 Þv i ; ða1þ þ R 11 y s 1 K m f h 2 oð R 11 þ R 12 Þv i ; ða2þ t s xx ¼ ðp þ QÞrV s 2N y vs y þ Q rw; ða3þ h o t s yy ¼ ðp þ QÞrV s 2N x vs x þ Q rw; ða4þ h o t s xy ¼ N x Vs y þ y Vs x ; ða5þ t s ¼ ðq þ RÞrV s þ R rw; ða6þ h o where vector w is given by w = h o v i and coefficients R ij = r ij /(r 11 r 22 r 2 12 ), (i, j = 1,2). The factor m f h 2 o /K controls the attenuation of seismic waves. The density terms r ij are related to the constant densities of the solid r o o s and fluid r f by r 11 + r 12 = fr o s and r 22 + r 12 = h o r o f. The term r 12 describes the inertial drag (as opposed to viscous) that the fluid exerts on the solid as the latter is accelerated relative to the former and vice versa. It represents the induced mass 14 of 15

15 tensor per unit volume, which is assumed to be diagonal for a homogeneous isotropic system; it is proportional to fluid density and, because of Newton s third law, it is always negative r 12 = (a 1)h o r f o where a is a geometrical quantity. For the case of isolated spherical particles a = (h o 1 +1)/2. [47] Biot poroelastic coefficients A, Q, R, N and P can be expressed in terms of Hickey s constants [de la Cruz and Spanos, 1989] as A ¼ fk s 2=3m M K s d s ; Q ¼ K f d s ; ða7þ ða8þ R ¼ K f ð1 d f Þ; ða9þ N ¼ m M ; ða10þ with P = A + 2N. In these equations, solid stresses correspond to s ij (i,j = x,y) and pore fluid pressure is given by P f = s/h o. Spatial derivatives with respect to x and y coordinate system correspond to x and y, respectively. [48] Acknowledgment. This work was partially funded by YNF Research Programme (D.01341), IMP. References Biot, M. A. (1956), Theory of propagation of elastic waves in a saturated porous solid, I, Low-Frequency range, J. Acoust. Soc. Am., 29, Biot, M. A. (1962), Mechanics of deformation and acoustic propagation in porous media, J. Appl. Mech., 33, Bourbie, T., O. Coussy, and B. Zinszer (1987), Acoustics of Porous Media, Gulf, Paris. Carcione, J. M. (2001), Wave Fields in Real Media, Elsevier, New York. Carcione, J. M., and G. Quiroga-Goode (1995), Some aspects of the physics and numerical modeling of Biot compressional waves, J. Comput. Acoust., 3, Carcione, J. M., and G. Quiroga-Goode (1996), Full frequency-range transient solution for compressional waves in a fluid-saturated viscoelastic porous medium, Geophys. Prospect., 44, Carcione, J. M., B. Gurevich, and F. Cavallini (2000), A generalized Biot- Gassmann model for the acoustic properties of shaley sandstones, Geophys. Prospect., 48, Carcione, J. M., H. B. Helle, and N. H. Pham (2003), White s model for wave propagation in partially saturated rocks: Comparison with poroelastic numerical experiments, Geophysics, 68, de la Cruz, V., and T. J. T. Spanos (1985), Seismic wave propagation in a porous medium, Geophysics, 50, de la Cruz, V., and T. J. T. Spanos (1989), Thermomechanical coupling during seismic wave propagation in a porous medium, J. Geophys. Res., 94, de la Cruz, V., P. N. Sahay, and T. J. T. Spanos (1993), Thermodynamics of porous media, Proc. R. Soc. London, Ser. A, 443, Dvorkin, J., and A. Nur (1993), Dynamic poroelasticity: A unified model with the squirt and the Biot mechanisms, Geophysics, 58, Dvorkin, J., R. Nolen-Hoeksema, and A. Nur (1994), The squirt-flow mechanism: Macroscopic description, Geophysics, 59, Gelinsky, S., and S. A. Shapiro (1997), Dynamic equivalent medium approach for thinly layered saturated sediments, Geophys. J. Int., 128, F1 F4. Gurevich, B., V. B. Zyrianov, and S. L. Lopatnikov (1997), Seismic attenuation in finely layered porous rocks: Effects of fluid flow and scattering, Geophysics, 62, Helle, H. B., N. H. Pham, and J. M. Carcione (2003), Velocity and attenuation in partially saturated rocks: Poroelastic numerical experiments, Geophys. Prospect., 51, Hickey, C. J. (1994), Mechanics of porous media, Ph.D. thesis, Univ. of Alberta, Edmonton, Alberta, Canada. Hickey, C. J., T. J. T. Spanos, and V. de la Cruz (1995), Deformation parameters of permeable media, Geophys. J. Int., 121, Leurer, K. C. (1997), Attenuation in fine-grained marine sediments: Extension of the Biot-Stoll model by the effective grain model (EGM), Geophysics, 62, Mochizuki, S. (1982), Attenuation in partially saturated rocks, J. Geophys. Res., 87, Norris, A. (1993), Low-frequency dispersion and attenuation in partially saturated rocks, J. Acoust. Soc. Am., 94, O Connell, R. J., and B. Budiansky (1977), Viscoelastic properties of fluidsaturated cracked solids, J. Geophys. Res., 82, Pride, S. R., and F. D. Morgan (1991), Electrokinetic dissipation induced by seismic-waves, Geophysics, 56, Pride, S. R., A. F. Gangi, and F. D. Morgan (1992), Deriving the equations of motion for porous isotropic media, J. Acoust. Soc. Am., 92, Pride, S. R., J. G. Berryman, and J. M. Harris (2004), Seismic attenuation due to wave-induced flow, J. Geophys. Res., 109, B01201, doi: / 2003JB Quiroga-Goode, G. (2001), Hydrocarbon detection via fluid-relaxation analysis, paper presented at the EAGE/SEG Research Workshop on Reservoir Rocks, Eur. Assoc. of Geosci. and Eng., Pau, France, 30 April to 4 May. Quiroga-Goode, G. (2002a), Dynamics of Biot squirt-flow, Acoust. Res. Lett. Online, 3, Quiroga-Goode, G. (2002b), Residual strain in Biot porous media, paper presented at 64th Conference, Eur. Assoc. of Geosci. and Eng., Florence, Italy. Quiroga-Goode, G. (2004), Computational studies of seismic waves in porous media with fluid and thermal relaxation: Gauging Biot theory, paper presented at 1st General Assembly, Eur. Geosci. Union, Nice, France, April. Quiroga-Goode, G., and J. M. Carcione (1997a), Heterogeneous modeling behavior at an interface in porous media, J. Comput. Geosci., 1, Quiroga-Goode, G., and J. M. Carcione (1997b), Wave dynamics at an interface between porous media, Boll. Geof. Teor. Appl., 38, Rasolofosaon, P. (1991), Plane acoustic waves in linear viscoelastic porous media: Energy, particle displacement and physical interpretation, J. Acoust. Soc. Am., 89, Sahay, P. N. (1996), Elastodynamics of deformable porous media, Proc. R. Soc. London, Ser. A, 452, Sahay, P. N. (2001), Dynamic Green s function for homogeneous and isotropic porous media, Geophys. J. Int., 147, Sahay, P. N., T. J. T. Spanos, and V. de la Cruz (2001), Seismic wave propagation in inhomogeneous and isotropic porous media, Geophys. J. Int., 145, White, J. E. (1975), Computed seismic speeds and attenuation in rocks with partial gas saturation, Geophysics, 47, Zimmermann, R. W. (1991), Compressibility of Sandstones, Elsevier, New York. S. Jiménez-Hernández, R. Padilla-Hernández, and G. Quiroga-Goode, Instituto de Investigación en Ingeniería, UAT, Tampico, Tamps., México, (gquirogagoodenetscape.net) M. A. Pérez-Flores, Departimento Geofísica Aplicada, CICESE, Ensenada, Baja California, México, of 15