P11 Low Frequency Modiications o Seismic Background Noise Due to Interaction with Oscillating Fluids in Porous Rocks M. Frehner* (Geological Institute, ETH Zurich), S.M. Schmalholz (Geological Institute, ETH Zurich), Y. Podladchikov (PGP, University o Oslo) & R. Holzner (Spectraseis Technologie AG, Zurich) SUMMARY Low requency spectral modiications o seismic background waves (noise) due to interaction with partially saturated porous rocks are investigated. Non-wetting luid drops entrapped in pores can oscillate with a characteristic eigenrequency. A 1D wave equation is coupled with a linear oscillator equation representing these oscillations. The resulting system o equations is solved numerically with explicit inite dierences. The background noise is reduced to its dominant requency (0.1-0.3Hz) which is presumably related to surace waves generated by ocean waves. This requency is used as the external source. The resulting incident monochromatic wave excites the pore luid which thereater oscillates with its eigenrequency. Oscillatory energy is transerred to the porous rock which leads to an amplitude decay o the luid oscillation. The elastic matrix carries the eigenrequency o the luid oscillation in addition to the external requency. Fourier spectra o the solid velocity thereore show two distinct peaks: the external requency and the eigenrequency o the luid oscillation. Interestingly, such low requency modiications o seismic noise are observed above hydrocarbon reservoirs and the presented model is considered as one possible explanation. Time evolution o the amplitude decay o the luid oscillation seems to be related to the thickness o the porous rock. EAGE 69 th Conerence & Exhibition London, UK, 11-14 June 007
Introduction The behavior o luids entrapped in a capillary tube and in idealized pores were thoroughly studied in the past (e.g. Graham and Higdon 000a,b or Dvorkin et al. 1990). In hydrocarbon industry the results o these studies were used to develop a new enhanced oil recovery technology (EOR) which is termed wave stimulation o oil production or vibratory mobilization (Beresnev and Johnson 1994). The basic idea is to excite oscillations o the entrapped oil which are strong enough or the luid to eventually leave the pore. The present work does not ocus on the artiicially induced pore luid oscillations but on the naturally occurring oscillations generated by the ever present seismic background noise. A new model is presented that couples an oscillation equation with a 1D elastic wave propagation equation. Coupling between pore luid oscillations and elastic waves Non-wetting luid drops entrapped in a porous rock can oscillate within the pores with a characteristic eigenrequency ω 0 (Hilpert et al. 006, Beresnev 006). Capillary orces act as restoring orces that drive the oscillations. For idealized pore geometry this oscillation can be approximated by a second order ordinary dierential equation. u = ω t 0 u (1) Where t is time, u the displacement o the luid out o its equilibrium position and ω 0 the eigenrequency o the linear oscillation, which is calculated rom material parameters describing the pore geometry and the luid (e.g. surace tension and density). For requently occurring physical values, ω 0 is in the low requency range (Holzner et al. 006). Wave propagation in the elastic porous rock in one dimension is described by a second order partial dierential equation. us 1 us 1 = E F ( x, t) t ( 1 φ) ρs x x + () ( 1 φ) ρs Where u s is the displacement o the solid, φ the porosity o the rock, ρ s the density o the rock material, E the Young s modulus and x the spatial coordinate. F represents any external orce acting on the elastic porous rock. To couple equation (1) and () both are written in terms o volumetric orces. The restoring orce term has to be written in terms o relative displacements and is treated as an addition orce term acting on the elastic solid. u = ω0 ( u us) t (3) us us ( 1 φ) ρs = E + ω0 ( u us) + F( x, t) t x x This system o equations is solved numerically using explicit inite dierences on a staggered grid (Virieux 1986). The applied physical parameters are given in Table 1. Energy conservation and transer In the introduced model our dierent volumetric energies can be calculated and integrated over the model. kin el E = vdx, E = 0 ( ) ω u us dx (4) kin ( 1 φ) ρs el E us Es = vsdx, Es = dx x Where E kin and E el are kinetic and elastic (strain) energies o the luid (subscript ) and the solid (subscript s), respectively. Using two relecting boundary conditions and omitting the external orce, the total energy in the model, i.e. the sum o all our energies stays EAGE 69 th Conerence & Exhibition London, UK, 11-14 June 007
constant over time. In the simulation shown (Figure 1) a model size o 40m and a Gaussian initial condition or the solid velocity was applied. The initial condition inluences the time evolution o the dierent energies in that sense that at the initiation o the simulation only kinetic energy o the solid is present. Ater that, the energy is distributed between solid and luid and transerred back and orth between the two media (thick black and thick red line). This transer happens in the same order o magnitude as the total energy o the system. This suggests that the energy transer rom the pore luid oscillations to the solid elastic medium is strong enough to inluence the spectral content o the elastic waves. Table 1: Parameters used in numerical simulations Figure 1: Time evolution o dierent energies in the model. The total energy, i.e. the sum o all energies (green line) stays constant over time. External source and model setup The Fourier spectrum o a typical measurement o ambient seismic background waves (noise) shows a very distinct peak at around 0.1-0.3Hz (let gray bar in Figure ). This high energy spectral peak is a global eature that can be measured everywhere in the world. It is presumably related to seismic surace waves generated by ocean waves (e.g. Aki and Richards 1980). In this study the seismic background noise is reduced to this dominant peak. The external source term in equation (3) becomes F ( xt, ) = A0 ( x) sin( Ω t) (5) with A0 ( x) = 0 orx xsource A0 ( x) = 1 orx= xsource (6) where Ω=1.89 (= 0.3Hz π). Figure : Field measurements o ambient seismic background noise. One measurement above (red) and one nearby (blue) the location o a producing subsurace hydrocarbon reservoir. Spectraseis campaign: Mososro, Mexico, 00 Figure 3: Model setup or numerical simulations consists o two nonrelection boundary conditions, three receivers R 1 -R 3 and one source S EAGE 69 th Conerence & Exhibition London, UK, 11-14 June 007
The external source is applied only at one point x source in the model. The model setup (Figure 3) or the ollowing numerical simulations consists o two non-relecting boundary conditions (Ionescu and Igel 003), three receivers and the point source described above. The position o the source is identical to one o the receivers. Numerical results Numerical simulations were run or 10s physical time. Fourier transorms o the recorded solid velocity time signals were calculated or all three receivers (Figure 4). They all show a very distinct peak at the requency o the external orce at 0.3Hz, as expected. In addition a second peak appears in all spectra. It coincides with the eigenrequency o the pore luid oscillations ω 0. The excited pore luid oscillations are transerred to the elastic porous matrix. This transer is strong enough that the corresponding requency, the eigenrequency o the oscillations is carried on top o the externally applied requency Ω. Both signals can be measured in the solid velocity signal at any point o the model. The trough o the spectra at receiver R 1 (blue line in Figure 4) is an artiact o the ast Fourier transorm (FFT) and has no physical meaning. Figure 4: Fourier transorms o solid velocity time signals or three receivers R 1 - R 3 (Figure 3). Positions o receivers are measured rom top o model. Dashed line: Frequency o external orce (0.3Hz); Solid line: Eigenrequency o pore luid oscillations (3Hz) The pore luid oscillations are initiated by the incident monochromatic wave generated by the external orce. While the wave itsel is monochromatic, the wave ront consists o all requencies. Thereore the pore luid starts to oscillate with its eigenrequency. Ater the wave ront has passed, the only externally applied requency is Ω. Its amplitude stays constant over time. All other requencies decay in amplitude over time (Figure 5). Figure 5: Spectra o solid velocity measured at receiver R 1 (Figure 3) ater dierent amount o time. Longest time signal is 10s, shortest is 3.5s. Black vertical line: Frequency o external orce; Red vertical line: Eigenrequency o pore luid oscillations; Green vertical line: Frequency with minimum power rom 1 to 3Hz = Hz Figure 6: Spectral amplitude decay over time at the eigenrequency o pore luid oscillations (3Hz) or dierent porous layer thicknesses. This amplitude is normalized with the amplitude at the requency o the external source (0.3Hz), which stays constant over time. The solid velocity to calculate this is measured outside the porous layer. EAGE 69 th Conerence & Exhibition London, UK, 11-14 June 007
The pore luid continues to oscillate with its eigenrequency and constantly transers energy to the elastic porous rock. Although this requency also decays in amplitude over time it is relatively higher in the spectrum (Figure 5). A second set o numerical simulations were perormed. The homogeneous model setup was expanded with two purely elastic layers on top and at the bottom o the model. The external point source was applied below the porous layer and an additional receiver was added on top, both 7m away rom the interace between porous and purely elastic layer. Several runs with a dierent thickness o the porous layer were perormed. The solid velocity at the newly added receiver was recorded and the Fourier spectra calculated. As in the homogeneous model (Figure 5) the amplitude at the eigenrequency o the pore luid oscillation is elevated and it decays with time. Interestingly, this decay over time is dierent or dierent thicknesses o the porous layer (Figure 6). A thick porous layer creates higher amplitudes which decay linearly with time in a log-log-diagram. A thin porous layer creates smaller amplitudes, but the decay is slower until the amplitude asymptotically reaches the values or thicker layers. A saturation o this eect occurs at around 70m thickness o the porous layer. Discussion and conclusions Oscillations o entrapped luid droplets in a porous elastic rock can be transerred to the solid. This transer is strong enough that the additional requency, the eigenrequency o the pore luid oscillations is visible in the Fourier spectrum o the solid velocity. Modiications o seismic background noise in the low requency range were observed in nature above hydrocarbon bearing structures (Figure, right gray bar; Dangel et al. 003). The physical explanation or these modiications is the subject o discussion (Gra et al. 007). Among other phenomena, such as seismic attenuation, relections or scattering, oscillations o pore luids have to be considered as one possible explanation. First results suggest that thickness inormation o hydrocarbon bearing structures can be extracted rom seismic background noise measurements. Reerences Aki, K. and Richards, P.G. [1980] Quantitative Seismology: Theory and Methods. W.H. Freeman and Co., San Francisco Beresnev, I.A. and Johnson, P.A. [1994] Elastic-wave stimulation o oil production: A review o methods and results. Geophysics 59, 1000-1017. Beresnev, I.A. [006] Theory o vibratory mobilization on nonwetting luids entrapped in pore constrictions. Geophysics 71, N47-N56. Dangel, S., Schaepman, M.E., Stoll, E.P., Carniel, R., Barzandji, O., Rode, E.-D. and Singer, J.M. [003] Phenomenology o tremor-like signals observed over hydrocarbon reservoirs. Journal o Volcanology and Geothermal Research 18, 135-158. Dvorkin, J. Mavko, G. and Nur, A. [1990] The oscillations o a viscous compressible luid in an arbitrarilyshaped pore. Mechanics and Materials 9, 165-179. Gra, R., Schmalholz, S.M., Podladchikov, Y. and Saenger, E. [007] Passive low requency spectral analysis: Exploring a new ield in geophysics. World Oil January 007, 47-5 Graham, D.R. and Higdon, J.J.L. [000a] Oscillatory low o droplets in capillary tubes. Part 1. Straight tubes. J. Fluid Mechanics 45, 31-53. Graham, D.R. and Higdon, J.J.L. [000b] Oscillatory low o droplets in capillary tubes. Part. Constricted tubes. J. Fluid Mechanics 45, 55-77. Hilpert, M., Jirka, G.H. and Plate, E.J. [000] Capillarity-induced resonance o oil blobs in capillary tubes and porous media. Geophysics 65(3), 874-883. Holzner, R., Eschle, P., Frehner, M., Schmalholz, S. and Podladchikov, Y. [006] Hydrocarbon microtremors interpreted as oscillations driven by oceanic background waves. 68 th EAGE Conerence and Exhibition, Vienna, Austria. Ionescu, D.-C. and Igel, H. [003] Transparent boundary conditions or wave propagation on unbounded s. ICCS, Melbourne and St. Petersburg 003, LNCS 659, 807-816, Sloot, P.M.A. et al. (Eds), Springer Verlag, Berlin Heidelberg, ISBN: 3-540-40196-. Virieux, J. [1986] P-SV wave propagation in heterogeneous media: Velocity-stress inite-dierence method. Geophysics 51(4), 899-901. EAGE 69 th Conerence & Exhibition London, UK, 11-14 June 007