Estimating flow properties of porous media with a model for dynamic diffusion

Transcription

1 Estiating low properties o porous edia with a odel or dynaic diusion Chuntang Xu*, Jerry M. Harris, and Youli Quan, Geophysics Departent, Stanord University Suary We present an approach or estiating eective copressibility and pereability ro Dierential acoustic resonance spectroscopy (DARS laboratory easureents. The eective copressibility o luid-saturated porous ediu, located in a haronic pressure ield, is a unction o the loading requency and the luid in the pore space, as well as the pereability and porosity o the ediu. The process is describable by a diusion process that relates eective copressibility and pereability. DARS is used to easure the eective copressibility. Then the diusion odel is used to estiate the pereability. This ethod is tested with DARS lab data. Introduction Harris et al. (5 presented a ethod or estiating acoustic attenuation with DARS. The key coponent o DARS consists o a resonant luid-illed cavity. At resonance, the standing wave inside the cavity provides a spatially varying but haronic pressure ield along the ajor axis o the cavity. The introduction o a porous saple o rock (Figure 1 perturbs the resonant odes o the cavity in a way that allows estiation o an eective attenuation and eective copressibility. I the saple is placed at a pressure anti-node, according to Morse and Ingard (1968, and Harris et al. (5 the perturbed requency o the resonator can be expressed as ω s ω = ω V s Φ δκ. (1 λ V c In equation (1, ω s and ω are the resonance requencies o the cavity with and without the saple, respectively; Φ is the acoustic pressure inside the cavity; λ is a constant coeicient; V s is the volue o the saple and V c is the volue o the cavity. The paraeter δκ is deined by δκ = κ κ / κ, where κ and κ s are the copressibilities ( s (reciprocal o bulk odulus o the luid and the saple, ( 1, in which ρ and c respectively; κ is deined by κ = ρ c are the density and acoustic velocity o the luid, respectively, and κ s is given by κ s = ρ s V p V ( s, where v p and v s are the P-wave and S-wave velocities o the saple. Rearranging (1, we get an expression or copressibility: κ s = (1+ Aξκ, ( where ωs ω Vc ξ =, λ A =. ω Vs Φ In equation (, ξ is the easured requency perturbation, κ is the copressibility o the luid inside the cavity, and the coeicient A can be obtained ro calibrations with a reerence saple. The calibration actor A is considered not to change or the other unknown saples. The copressibility or bulk odulus o an unknown saple can be calculated ro equation ( with the perturbation it produces in the acoustic resonator. Figure 1: The key coponent o DARS is the resonator. In the easureent, the unknown saple is placed at the center o the cavity, where the acoutic pressure is axiu at the irst ode. We used DARS to estiate the copressibilities o both nonporous and porous aterials and copared the results with that derived by ultrasound easureents. We ound that the copressibilities obtained by two dierent ethods are coparable or nonporous aterials (Figure but not always or porous saples. Figure 3 shows the coparison result o eight rock saples. The data points o the saples with extreely low pereability, such as the coal, chalk and the granite, are along the 45-degree line, which indicates that the copressibilities obtained by the two dierent techniques are equivalent or the three aterials. The data or saples with edian and high pereability, such as two Berea sandstones and the Boise sandstone, plot away the 45-degree line. The saples with higher pereability illustrate larger deviation ro the 45- degree line. Porosity does not show this eect. For instance, the Chalk has high porosity; but plots close to the 45-degree line. Because the ultrasound easureents are on dry rocks, we thereore estiated the equivalent wet rae copressibility by using Gassann luid substitution procedure, and then copared the corrected results with that derived by the perturbation observations. However, the data points deviated even arther away ro the 45-degree line. This phenoenon indicates that the copressibility derived by DARS easureents is apparently not the copressibility usually quantiied by other techniques, e.g., ultrasound ethod. This observation otivated us to investigate the echanis o the luid and solid atrix interaction in the DARS easureents. SEG/New Orleans 6 Annual Meeting Downloaded 1 Dec 11 to Redistribution subject to SEG license or copyright; see Ters o Use at

2 Estiating low properties o porous edia with a odel or dynaic diusion κ DARS (GPa Aluinu Delrin Lucite PVC Telon κ ultrasound (GPa -1 Figure : Coparison o the copressibility estiated by DARS and by ultrasound ethod or ive solid plastic aterials. The aluinu is used as reerence saple or calibration. κ - DARS (GPa κ - ultrasound (GPa -1 Berea Boise Chalk Granite Coal Sandstone Figure 3: Coparison o the copressibilities evaluated by perturbation ethod and ultrasound easureent or eight rock saples. The ultrasound results are or dry rocks. Diusion Theory Consider a luid-saturated porous ediu that is subjected to a sall aplitude oscillatory pressure gradient; the pressure luctuation will cause icro luid low through the saple to release the dierential pressure. This phenoenon can be described by dynaic diusion. In a cylindrical coordinator syste, the pressure diusion equation in a hoogeneous porous ediu with circular syetric geoetry has the ollowing or: 1 p p 1 p r + =, (3 r z D t with D = k (φηβ. Here, φ and k are porosity and pereability o the porous saple, respectively; p is the acoustic pressure in the luid, and η is the viscosity o the luid; β is the copressibility actor involving both the luid and the solid atrix siultaneously. Detailed derivation o the diusion equation is presented, e.g., in Barenblatt et al. (199. I we ignore the low in the axial direction, then we can sipliy the expression as 1 p 1 p r =. (4 r D t Furtherore, i acoustic pressure is a sinusoidal unction o iωt tie, i.e., p( r, t = P( e, we can rewrite equation (4 as P 1 P iω + P =. (5 r D The solution o this equation is / D / Dr PK P( =. (6 K Here, K is the zero order odiied Bessel unction o the second kind. P is the aplitude o pressure luctuation at saple surace r = r, and ω is the resonant requency o the acoustic syste. Eective copressibility (EC o porous aterials in a periodic pressure environent can be expressed by the ratio o the net voluetric strain o the aterial to the corresponding stress on the saple. The net volue change o the saple consists o contributions ro the solid atrix and the pore luid. Thereore, the EC o the porous saple can be written as ( V + V 1 κ e =, (7 V P t where V and V are the net volue change o the rock atrix and the pore luid inside the pore space, respectively; V t is the bulk volue o the saple. According to the deinition o copressibility, the net volue change o the rae, V can be written as V = κ V ( 1 φ P, where φ is the saple porosity and t κ is the copressibility o the rae. The net volue change o the pore luid is equal to the aount o luid lows into and out the pore space driven by the changing pressure. Because o the spatial distribution o the pressure inside the rock, the luid volue change can be written as V κ P( dv, where dv = φπrdr. = Substituting the expression o V, V and P( into equation (7, we get the EC expression, φκ r K / D κ e = κ (1 φ + rdr. (8 r K / Dr It can be seen ro equation (8 that the EC o a luidsaturated porous aterial is a unction o the requency o the pressure ield, the property o the pore luid, and the porosity and pereability o the saple. SEG/New Orleans 6 Annual Meeting Downloaded 1 Dec 11 to Redistribution subject to SEG license or copyright; see Ters o Use at 183

3 Estiating low properties o porous edia with a odel or dynaic diusion Experient The key coponent o DARS is an open-ended cylindrical cavity, which is iersed in a luid-illed tank. In the easureent, the saple is placed at the center o the cavity, where the acoustic pressure is highest (or the irst resonant ode. As equation (1 shows, at the pressure antinode, the saller copressibility o the saples shits the requency higher than the epty cavity resonance response. A typical resonance response o the syste with and without the saple is shown in Figure 4. cavity. Figure 5 also shows that the Boise and the Chalk exhibit siilar perturbation eect, which indicates that they have equivalent copressibility even though the two saples have draatically dierent acoustic and low properties (Table 1. Figure 5: Perturbation responses o six dierent rock types. Figure 4: Power spectru o the acoustic syste. ω is the resonant requency without the saple. Under the intererence o a sall saple, the resonant requency shits to ω s. The easureent results discussed in this paper involved eight rock saples. Their acoustic and low properties are listed in Table 1. All the saples were ully saturated with the sae luid contained in the cavity. Furtherore, we considered only the irst resonance ode. v p (k/sec v s (k/sec ρ (g/cc φ (% k (Darcy Berea Boise Chalk Coal Granite Sandstone Table 1: Acoustic and low properties o eight rock saples. The density, P- and S-wave velocities are easured in the dry state. Results The perturbation responses o several o the saples are shown in Figure 5. As Figure 5 illustrates, the resonance requency o the syste always increases under the intererence o each saple; however, the agnitude o the increent is dierent. According to the perturbation theory, equation (1, this phenoenon states that the copressibilities o the relevant saples are dierent ro each other but all are larger than that o the luid inside the We calculated the copressibility o the eight rocks by the perturbation data using equation (, and also estiated the corresponding theoretical values by the EC odel, equation (8. The results derived by two dierent ethods are copared in Figure 6 and Table. As Figure 6 shows, the cross plot o the eight rocks is along a 45-degree line. The correlation o the two observations is.998 and standard deviation o the data points ro the 45-degree line is.4. These eatures veriy that the interaction o the luid and the solid skeleton in DARS easureents on pereable saples is a dynaic diusion process, thus we can use the EC concept to interpret the DARS observations. As equation (8 shows, the EC o porous aterials contains the inoration o the loading requency, the properties o the luid stored inside the pore space, and ost iportantly, the pereability and porosity o the edia. Thereore, DARS provides the potential to investigate the low properties o porous edia. The luctuation o the data points around the 45-degree line ay attribute to: easureent errors in DARS and ultrasound easureents; the studied aterials ight be heterogeneous; the low in the axial direction should also be considered in the derivation o the eective copressibilities. We inally investigated the easibility to estiate pereability o porous aterials through the cobination o DARS observations and the diusion odel. and the two sandstone saples were used with the assuption that the pereability was the only unknown paraeter. The procedure was as ollows: we irst calculated the copressibility o the three saples by using DARS observations. Then we searched the optial pereability by orcing the theoretically calculated EC to atch with that estiated by DASR. The easured SEG/New Orleans 6 Annual Meeting Downloaded 1 Dec 11 to Redistribution subject to SEG license or copyright; see Ters o Use at

4 Estiating low properties o porous edia with a odel or dynaic diusion pereability o the three saples by gas-injection are 37, 185 and 4 Darcy respectively. The corresponding estiated pereability o each o the saples are 394, 177 and 43 Darcy separately. The little isatching between the easured pereability and that estiated by the EC odel ay attribute to errors within DARS observations and the gas-injection easureent o pereability. In estiating the EC o the studied saples by equation (8, the involved porosity and skeleton copressibility are obtained by other techniques. We anticipate quantiying theses paraeters solely by DARS easureents. The ethodology will be addressed in a uture study. κ - dynaic diusion odel (GPa Berea Boise Chalk Granite Coal Sandstone..4 κ - DARS (GPa -1 Figure 6: Cross plot the copressibility estiated by DARS and by the dynaic diusion odel or eight rock saples. κ-ultrasound κ-dars κ-diusion (GPa -1 (GPa -1 (GPa Berea Boise Chalk Granite Coal Sandstone Table : Cross plot the copressibility evaluated by perturbation observations and the analytical results ro the dynaic diusion odel or the eight rock saples. Conclusion The interaction o the luid and the solid skeleton in DARS easureents on pereable saples is a dynaic diusion process. The copressibility estiated by the perturbation easureents is a unction o pereability and porosity, requency o the pressure variation, and the properties o the luid inside the pore space. k predicted (Darcy Sandstone k - Gas injection (Darcy Figure 7: Cross plot o the pereability obtained by gas-injection easureent and estiated by the dynaic diusion odel or three rock saples. The experiental results o eight rock saples show that pereability has considerable eect on the EC o porous edia. For saples with very low pereability (<1D, the copressibility derived ro the perturbation observations are coparable to that derived by ultrasound ethod. Contrarily, or those aterials with relatively high pereability, the copressibilities observed by the perturbation easureents are uch lower than that obtained by ultrasound ethod. We derived an eective copressibility odel based upon a dynaic diusion process, and copared the analytical results or eight rocks with that derived by the perturbation observed. The results agree well, which indicates that we can utilize the dynaic diusion odel to interpret the perturbation easureents, and estiate the low properties o porous edia. We use three rock saples to validate the easibility to estiate pereability o porous aterials through DARS observations. The estiated pereability o the three rocks is coparable with that easured by gas-injection ethod. Reerences Barenblatt, G.I., Entov, V.M., and Ryzhik, V.M., 199, Theory o Fluid Flows through Natural Rocks: Dordrecht, Kluwer Acadeic Publishers. Harris, J. H., Quan, Y. L., Xu, C. T., 5, Dierential Acoustic Resonance Spectroscopy: An experiental ethod or estiating acoustic attenuation in porous edia, SEG expanded abstract, Johnson, D. J., 199, Probing porous edia with superluid acoustics, J. phys., SA449-SA455. Morse, P. M. & Ingard, K. U., Theoretical acoustics. Mcgraw-Hill Book Copany. New York. SEG/New Orleans 6 Annual Meeting Downloaded 1 Dec 11 to Redistribution subject to SEG license or copyright; see Ters o Use at

5 EDITED REFERENCES Note: This reerence list is a copy-edited version o the reerence list subitted by the author. Reerence lists or the 6 SEG Technical Progra Expanded Abstracts have been copy edited so that reerences provided with the online etadata or each paper will achieve a high degree o linking to cited sources that appear on the Web. REFERENCES Barenblatt, G. I., V. M. Entov, and V. M. Ryzhik, 199, Theory o luid lows through natural rocks: Dordrecht: Kluwer Acadeic Publishers. Harris, J. H., Y. L. Quan, and C. T. Xu, 5, Dierential acoustic resonance spectroscopy: An experiental ethod or estiating acoustic attenuation in porous edia: 75th Annual International Meeting, SEG, Expanded Abstracts, Johnson, D. J., 199, Probing porous edia with superluid acoustics: Journal o Physics, SA449 SA455. Morse, P. M., and K. U. Ingard, 1968, Theoretical acoustics: McGraw-Hill Book Co. SEG/New Orleans 6 Annual Meeting Downloaded 1 Dec 11 to Redistribution subject to SEG license or copyright; see Ters o Use at