STANFORD EXPLORATION PROJECT

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1 STANFORD EXPLORATION PROJECT James Berryman, Biondo Biondi, Stewart Levin, and Herbert Wang Report Number 98, September 1998

2 Copyright c 2001 by the Board o Trustees o the Leland Stanord Junior University Stanord, Caliornia Copying permitted or all internal purposes o the Sponsors o Stanord Exploration Project

3 Preace The electronic version o this report 1 makes the included programs and applications available to the reader. The markings [ER], [CR], and [NR], are promises by the author about the reproducibility o each igure result. ER CR NR denotes Easily Reproducible. The author claims that you can reproduce such a igure rom the programs, parameters, and data included in the electronic document. We assume you have a UNIX workstation with Fortran, C, X-Window system, and the sotware on our CD-ROM at your disposal. Beore the publication o the electronic document, someone other than the author tests the author s claim by destroying and rebuilding all ER igures. denotes Conditional Reproducibility. The author certiies that the commands are in place to reproduce the igure i certain resources are available. SEP sta have not attempted to veriy the author s certiication. To ind out what the required resources are, you can inspect a corresponding warning ile in the document s result directory. For example, you might need a large or proprietary data set. You may also need a super computer, or you might simply need a large amount o time (20 minutes or more) on a workstation. denotes Non-Reproducible. This class o igure is considered non-reproducible. Figures in this class are scans and artists drawings. Output o interactive processing are labeled NR. With the departure o Matthias Schwab or this report, the testing procedure has changed and might be not as thorough as beore. Readers suggestions are welcome. For more inormation on reproducing SEP s electronic documents, please visit < Sergey Fomel, Robert Clapp, and Jon Claerbout 1

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5 i SEP 98 TABLE OF CONTENTS 3-D Imaging Biondi B., 3-D Seismic Imaging th SEP Reunion Levin S. A., The History o the Mitchell Building Skylight Rock Physics Berryman J. G. and Wang H. F., Double porosity modeling in elastic wave propagation or reservoir characterization SEP article published or in press, SEP phone directory Research personnel SEP sponsors or

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7 Stanord Exploration Project, Report SEP 98, June 19, 2001, pages D Seismic Imaging Biondo Biondi biondo@sep.stanord.edu 1

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9 Stanord Exploration Project, Report SEP 98, June 19, 2001, pages The History o the Mitchell Building Skylight Stewart A. Levin1 ABSTRACT During recent earthquake reinorcement o the Ruth Wattis Mitchell Earth Sciences Building, unds were successully raised to install a skylight in the 4th loor. The skylight was completed in January o 1998 and dedicated during the Stanord Exploration Project s 25th year reunion on July 17th, Here is the story. 1 sep.stanord.edu 205

10 206 Levin SEP 98

11 SEP 98 Skylight 207 IN THE BEGINNING Ending Darkness Marie Prucha 2 Darkness. Deinition: Mitchell 4 th. 28 years o darkness. Unrelenting, unrelieved. Day or night, night or day, Always dark. Darkness. Students the prisoners, the victims. Working endlessly, not knowing sunlight, white pallor the rule. Darkness. Is there sun? Is there rain? Is it summer, spring, or all? Could be a blizzard. No way to tell. Darkness. Fluorescent bulbs. Plastic plants. Colorul screensavers. Not enough. The dark side will always be stronger. Darkness. But then a voice: This place is a dungeon in the sky. Seek reedom! Money incoming, Hope outpouring. Darkness still. Finally enough! Then Claerbout declared: Let there be light! And there was light. And there was much rejoicing. Light. Ater the Loma Prieta quake, Stanord began the process o reconstructing and reinorcing the buildings on campus, including reinorcement o the Mitchell Building. George Thompson astutely observed to Jon Claerbout that, while he was going to lose one o the windows in his third loor oice to a reinorcing pillar, the time was opportune to request that a skylight be installed in the 4 th loor as part o the construction activity. Jon Claerbout contacted his riends, colleagues, sponsors and students, both present and ormer, and raised the requisite unds to turn the idea into a reality. Facts: Over 60 donors contributed to the skylight und. Fund raising was completed in 5 months. Better than 90% o those solicited or donations contributed a response rate unprecedented in the history o Stanord University undraising. 2 Written or the skylight dedication ceremonies July 17 th, 1998.

12 208 Levin SEP 98 MORE THAN JUST A PLAQUE In the course o preparations or the SEP 25 th year reunion, Jon Claerbout opined that we should prepare a plaque or sign recognizing the donors who contributed to the skylight and who were going to be present, by and large, at the reunion. While a plaque itsel tends to be rather boring, the inside suraces o the skylight shat itsel provided our large 4 15 areas in which to place text and graphics. Having been inspired by a brass rubbing I made in St. Martin o the Fields while vacationing in London, I suggested a pictorial theme o old scientiic illustrations augmented by one or more quotations to complement the lengthy list o donors names. Having come up with a concept, the idea was then reined by other SEP ers to ocus on a primarily astronomical theme appropriate to a skylight and we began a search or such illustrations. To this end we went to the Stanord University Library rare book collections to begin the search. The Assistant Head o Special Collections, John Mustain, managed to ind quite a ew illustrated books or us to look through and, with some additional search through the stacks, we compiled a suiciency 3 o material. With concept and materials well in hand, we contacted a commercial sign-making irm in nearby Redwood City to turn our materials into a design and implement it in the skylight shat. Ater a series o and ax exchanges, the inal design was approved. This was installed during the week leading up to the reunion and was immediately covered over with a black plastic drop cloth to protect the design rom prying eyes beore the oicial dedication and unveiling. The inal design consisted o 4 panels, each divided into three areas horizontally. In each area either a graphic or a text display was place. These individual elements are discussed next. Flamsteed s sextant John Flamsteed, the irst Astronomer Royal o the Greenwich Observatory, was ortunately wealthy and well-connected enough to equip the Observatory with a ine set o the newest telescopic instruments. One o these instruments is the 7 oot sextant shown in the illustration rom Flamsteed s Historia Coelestis (1725) which was designed to measure inclinations to a precision o 10 seconds o arc. Hevelius sextant Johannes Hevelius was an accomplished astronomer who worked approximately hal a century beore Flamsteed. His 6 oot sextant shown in this illustration rom Machina Coelestis (Hevelius, 1969) was itted with an ordinary eyepiece rather than the telescopes on later instruments such as Flamsteed s. 3 Actually everyone wanted something rom Huygens, but his line drawings were pretty eeble.

13 SEP 98 Skylight 209 Figure 1: Flamsteed s Sextant stew1-lamsteed [NR] Figure 2: Hevelius Sextant stew1-hevelius [NR]

14 210 Levin SEP 98 Wright s starry spheres Thomas Wright o Durham, a navigation instrument maker, is credited (albeit somewhat erroneously) with explaining the luminescence o the Milky Way along a band o the sky as looking through a large layer o stars end on. Shown here is Plate XXVII rom An Original Theory or New Hypothesis o the Universe (1750) in which he actually attributes the eect to looking approximately tangential to a spherical shell o stars distributed uniormly about a central divine center o universal order. Appendix A gives the text that accompanied this and its preceding igure. Figure 3: Wright s starry spheres stew1-wright [NR] The Octagon Room o the Greenwich Observatory The Great Star Room or Octagon Room was part o John Flamsteed s House adjacent to the old Royal Observatory in Greenwich. Due to smog and light pollution, the Royal Observatory was moved to Herstmonceux Castle in Sussex in the late 1940 s and early 1950 s. The Royal Observatory will close permanently on October 31 st, Figure 4: The Octagon Room stew1-octagon [NR]

15 SEP 98 Skylight 211 Gauss diary This acsimile page rom Gauss diary ( ) dates rom the year beore he received his doctorate rom Göttingen. The terse nature o these entries makes translation diicult, but the highlighted note [84] is most probably related to the partial classiication o ternary quadratic orms that appears in his amous Disquisitiones Arithmetica (Gauss, 1801) where he also demonstrated the ruler-and-compass constructibility o the regular 17-gon. Figure 5: Facsimile page rom Gauss Diary... stew1-gauss-1 [NR] Figure 6:... and its typewritten transcription stew1-gauss-2 [NR]

16 212 Levin SEP 98 Copernicus manuscript These two handwritten Latin pages rom Copernicus amous De Revolutionibus ( ) are rom Book IV in which he analyzes lunar eclipses in terms o the relative precession o the moon due to the Earth s orbit around the sun. Appendix B give a translation o the text. Figure 7: Copernicus Manuscript stew1-copernicus-1 [NR] Figure 8: Copernicus Manuscript stew1-copernicus-2 [NR] Thomson s poem The quotation, ound during an Internet search, originates rom Thomson s The Castle o Indolence (1748) published less than hal a year beore his death. Some o the spelling and

17 SEP 98 Skylight 213 punctuation have been changed or the skylight. Appendix C gives the original text o that verse. Figure 9: Thomson s Poem stew1-thomson [NR] Fiat Lux In addition to its biblical translation Let there be light, Fiat Lux has a classical meaning o the dawning o new knowledge or enlightenment, which is how U.C. Berkeley (Boo! Hiss!) uses it in their motto. This short, classical quotation nicely balances the longer verbiage o the rest o the dedication. Figure 10: Fiat Lux. [NR] stew1-iat-lux Donor name lists The list o 62 names o donors was divided up into three alphabetized 4 groups according to space requirements, with the biggest monetary donors in the largest, most visible panel and so orth. All anonymous donors are acknowledged on the dedication panel. 4 At least Claerbout really belongs beore Chevron in SEP s opinion!

18 214 Levin SEP 98 DEDICATION On the evening o July 17 th, 1998, SEP reunion participants were herded rom the Hartley Conerence Center up to the 4 th loor o the Mitchell Building to witness the dedication. At the signal rom Dimitri Bevc that the last o the stragglers had arrived, Jon stood somewhat precariously on a chair to say a ew words (Appendix D) o welcome and explanation about the history o the skylight and the planned dedication events that evening. Marie Prucha took over and recited the poem she wrote or the occasion ater which people were careully herded onto the carpeted area under the still-masked skylight. The end o the poem signalled SEP s reenactment o... acta est lux resulting in the rapid removal o the black plastic cover and an unexpected, but delightul, shower o balloons on the spectators. Ater a minute or two o oohs and aahs, Dean Lynn Orr made a short impromptu(?) speech o recognition to Jon and his team s accomplishment in bringing light into the basement aerie to complement the intellectual lights that had graced the 4 th loor over the past 25 years. This was ollowed by a toast composed and read or the occasion by Biondo Biondi. AN INVITATION In conclusion, I invite everyone to come up and wander around the 4 th loor o the Mitchell Building and compare its lourescent-lit recessed corridors to the open, skylit expanse in ront o the elevator to really appreciate the improvement that the skylight has made. REFERENCES Copernicus, N., 1543,1974, Nicolaus Copernicus gesamtausgabe, de revolutionibus, aksimile des manuskriptes:, volume 1 Gerstenberg, Gebrüder, GmbH&Co., Rathausstr 18-20, D Hildesheim, Post 10055, D Hildesheim, Tele (05121) Fax Flamsteed, J., 1725, Historiae coelestis Britannicae:, volume 3 H. Meere, London. Gauss, C. F., ,1985, Mathematische tagebuch :, Ostwalds Klassiker der Exakten Wissenshaen, Bd. 256 Deutsch, Harri, GmbH, Grästr 47 u.51, D Frankurt, Tele (069) Fax Gauss, C. F., 1801, Disquisitiones arithmeticae: Gerh. Fleischer, Leipzig. Hevelius, J., 1969, Machinæ coelestis pars prior; Organographiam:, volume 1 Zentralantiquariat der Deutschen Demokratischen Republik, Leipzig. Thomson, J., 1748, The castle o indolence: An allegorical poem. Written in imitation o Spenser.: Andrew Millar, over against Catherine Street, in the Strand (London). Wallis, C. G., 1995, On the revolutions o heavenly spheres by Nicolaus Copernicus:, Great Minds Series Prometheus Books, 59 John Glenn Drive, Amherst, New York

19 SEP 98 Skylight 215 Wright, T., 1750, An original theory or new hypothesis o the universe: Grosvenor Street, London. H. Chapelle,

20 216 Levin SEP 98 APPENDIX A From page 64 o Thomas Wright s text: PLATE XXVI. Represents a Creation o a double Construction, where a superior Order o Bodies C, may be imagined to be circumscribed by the ormer one A, as posessing a more eminent Seat, and nearer the supream Presence, and consequently o a more perect Nature. Lastly, PLATE XXVII. Represents such a Section, and Segments o the same, as I hope will give you a perect Idea o what I mean by such a Theory. Fig. I. is a corresponding Section o the Part at A, in Fig. 2. whose versed Sine 5 is equal to hal the Thickness o the starry Vortice AC, or BA. Now I say, by supposing the Thickness o this Shell, I. you may imagine the middle Semi-Chord 6 AD, or AE, to be nearly 6 ; and consequently thus in a like regular Distribution o the Stars, there must o course be at least three Times as many to be seen in this Direction o the Sine, or Semi-chord AE, itsel, than in that o the semi-versed Sine 7 AC, or where near the Direction o the Radius o the Space G. Q.E.D. 5 minus the cosine o the angle 6 hal a chord 7 hal the versed sine

21 SEP 98 Skylight 217 APPENDIX B Translation rom Wallis (1995): But since segment EA is less than a semicircle, the centre o the epicycle will not be in it but in the remainder ABCE. Thereore let K be the centre, and let DMKL be drawn through both apsides 8, and let L be the highest apsis and M the lowest. Now, by Euclid, III, 36, it is clear that rect. AD, DE = rect. LD, DM. Now since LM, the diameter o the circle to which DM is added in a straight line is bisected at K, then rect. LD,DM+sq. KM=sq. DK. Thereore DK A L O N K E M [begin manuscript page] =1,148,556 where KL=100,000; and on that account, LK=8,706 where DKL=100,000 and LK is the radius o the epicycle. Having done that, draw KNO perpendicular to AD. Since KD, DE, and EA have their ratios to one another given in the parts whereo LK=100,000, and since NE= 1 / 2 AE=73,893: thereore, by addition, DEN=1,146,577. But in triangle DKN side DK is given, side ND is given, and angle N=90 ; on that account, at the centre, angle NKD= / 2 and arc MEO= / 2. Hence, arc LAO=180 arc NEO= / 2. Now arc OA= 1 / 2 arc AOE= / 2 ; and arc LA=arc LAO-arc OA=45 43, which is the distance or position o anomaly o the moon rom the highest apsis o the epicycle at the irst eclipse. But D 8 apsis (pl. apsides) point o closest or urthest approach o an orbit around a body.

22 218 Levin SEP 98 arc AB= Accordingly, by subtraction, arc LB=64 38, which is the anomaly at the second eclipse. And by addition arc LBC=146 14, where the third eclipse alls. Now it will also be clear that since angle DKN= / 2, where 4 rt. angles=360, angle KDN=90 angle DKN= / 2 ; and that is the additosubtraction which the anomaly adds at the irst eclipse. Now angle ADB = 7 42 ; thereore, by subtraction, angle LDB= / 2, which arc LB subtracts rom the regular movement o the moon at the second eclipse. And since angle BDC=1 21, and thereore, by subtraction, angle CDM=2 49, the subtractive additosubtraction caused by arc LBC at the third eclipse; thereore the mean position o the moon, i.e., o the centre K, at the irst eclipse was 9 53 o Scorpio, because its apparent position was at o Scorpio; and that was the number o degrees o the sun diametrically opposite in Taurus. And thus the mean movement o the moon at the second eclipse was at 29 1 / 2 o Aries; and in the third eclipse, at 17 4 o Virgo. Moreover, the regular distances o the moon rom the sun were or the irst eclipse, or the second, or the last. So Ptolemy. Following his example, let us now proceed to a third trinity o eclipses o the moon, which were painstakingly observed by us. The irst was in the year o Our Lord 1511, ater October 6th had passed. The moon began to be eclipsed 1 1 / 8 equal hours beore midnight, and was completely restored 2 1 / 3 hours ater midnight, and in this way the middle o the eclipse was at 7 / 12 hours ater midnight the morning ollowing being the Nones 9 o October, the 7 th. There was a total eclipse, while the sun was in o Libra but by regular movement at o Libra. We observed the second eclipse in the year o Our Lord 1522, in the month o September, ater the lapse o ive days. The eclipse was total, and began 2 / 5 equal hours beore midnight, but its midpoint occurred 1 1 / 3 hours ater midnight, which the 6 th day ollowed the 8 th day beore the Ides o September. The sun was in the 22 1 / 5 o Virgo but, according to its regular movement, in o Virgo. We observed the third in the year o Our Lord 1523, at the close o August 25. th It began 2 4 / 5 hours ater midnight, was a total eclipse, and the midtime was 4 5 / 12 hours ater the midnight prior to the 7 th day beore the Kalends o September. The sun was in o Virgo but according to its mean movement at 13 2 o Virgo. And here it is also maniest that the distance between the true positions o the sun and the moon rom the irst eclipse to the second was , but rom the second to the third it was Now the time rom the irst eclipse to the second was 10 equal years 337 days 3 / 4 hours according to apparent time, but by corrected equal time 9 Nones, Ides, Kalends Roman calendar designations (Beware the Ides o March!).

23 SEP 98 Skylight 219 4/ 5 hours. From the second to the third there were 354 days 3 hours 5 minutes; but according to equal time 3 hours 9 minutes. During the irst interval the mean movement o the sun and the moon measured as one not counting the complete circles amounted to , and the movement o anomaly to , subtracting approximately 5 rom the regular movement; in the second interval [end manuscript page] the mean movement o the sun and moon was ; and the movement o anomaly was , adding 2 59 to the mean movement.

24 220 Levin SEP 98 APPENDIX C The original text in Canto II, Verse III o James Thomson s The Castle o Indolence reads: I care not, Fortune, what you me deny : You cannot rob me o ree Nature s Grace ; You cannot shut the Windows o the Sky, Through which Aurora shews her brightening Face ; You cannot bar my constant Feet to trace The Woods and Lawns, by living Stream, at Eve : Let Health my Nerves and iner Fibres brace, And I their Toys to the great Children leave ; O Fancy, Reason, Virtue, nought can me bereave.

25 SEP 98 Skylight 221 APPENDIX D Jon Claerbout s dedication remarks: Most o you know me as the most aggressive proponent o this skylight. I you were slow to make your pledge, you REALLY know me as an aggressive advocate. I d like to introduce you to a ew other people who also conspired to get us this skylight. George Thompson spotted the architects Dudley Kenworthy knew how to make good things happen Julie Hardin CAN-DO attitude, transmit ideas to architects. I asked each o the SEP PhD graduates to contribute. 90The Stanord development oice was amazed. Our rate o return is unprecedented. Not only SEP PhD graduates contributed, but aculty, Biondo Biondi, Jerry Harris, Simon Klemperer, and Lynn Orr and about hal o the students living up here contributed. Even our administrative secretaries contributed. And all your contributions were generous contributions. 17 o you contributed $1000 or more. 35 o you contributed $500 or more. 52 o you contributed $200 or more. Many current students gave a week s pay. O course you know the reason why. This place was a dungeon in the sky, and you all knew it. Most o you recall walking down the stairwells, not knowing whether you would emerge into sun or rain, light or dark. Years ago we ALWAYS went outdoors or lunch, even when it rained. With El Nino this year, we had many January lunches right here where we stand. It was delightul, as you will soon see. In years gone by, I elt the need to apologize to our new students and visitors or the quality o our quarters. I elt cheated when the Geologists and Petroleum Engineers aced us out o the new Cecil Green building. But no more. Soon you will see that we now have an excellent home. So I thank you again, or your generous contributions and, on behal o many uture generations, I express to you our heartelt thanks. Marie Prucha will recite a poem she has prepared or this occasion. Then we will proceed with the unveiling.

26 222 Levin SEP 98 Thanks to the many donors, both named and anonymous, whose generous contributions made this skylight possible. Fiat Lux Genesis 1:3 Dedicated this Seventeenth day o July in the year One Thousand Nine Hundred and Ninety Eight.

27 Stanord Exploration Project, Report SEP 98, June 19, 2001, pages Double porosity modeling in elastic wave propagation or reservoir characterization James G. Berryman 1 and Herbert F. Wang 2 ABSTRACT To account or large-volume low-permeability storage porosity and low-volume highpermeability racture/crack porosity in oil and gas reservoirs, phenomenological equations or elastic wave propagation in a double porosity medium have been ormulated and the coeicients in these linear equations identiied. The generalization rom single porosity to double porosity modeling increases the number o independent inertial coeicients rom three to six, the number o independent drag coeicients rom three to six, and the number o independent stress-strain coeicients rom three to six or an isotropic applied stress and assumed isotropy o the medium. The analysis leading to physical interpretations o the inertial and drag coeicients is relatively straightorward, whereas that or the stress-strain coeicients is more tedious. In a quasistatic analysis o stress-strain, the physical interpretations are based upon considerations o extremes in both spatial and temporal scales. The limit o very short times is the one most relevant or wave propagation, and in this case both matrix porosity and ractures are expected to behave in an undrained ashion, although our analysis makes no assumptions in this regard. For the very long times more relevant or reservoir drawdown, the double porosity medium behaves as an equivalent single porosity medium. At the macroscopic spatial level, the pertinent parameters (such as the total compressibility) may be determined by appropriate ield tests. At the mesoscopic scale pertinent parameters o the rock matrix can be determined directly through laboratory measurements on core, and the compressiblity can be measured or a single racture. We show explicitly how to generalize the quasistatic results to incorporate wave propagation eects and how eects that are usually attributed to squirt low under partially saturated conditions can be explained alternatively in terms o the double-porosity model. The result is thereore a theory that generalizes, but is completely consistent with, Biot s theory o poroelasticity and is valid or analysis o elastic wave data rom highly ractured reservoirs. 1 berryman@sep.stanord.edu 2 Department o Geology and Geophysics, University o Wisconsin, Madison, WI wang@geology.wisc.edu 223

28 224 Berryman & Wang SEP 98 INTRODUCTION It is well-known in the phenomenology o earth materials that rocks are generally heterogeneous, porous, and oten ractured or cracked. In situ, rock pores and cracks/ractures can contain oil, gas, or water. These luids are all o great practical interest to us. Distinguishing these luids by their seismic signatures is a key issue in seismic exploration and reservoir monitoring. Understanding their low characteristics is typically the responsibility o the reservoir engineer. Traditional approaches to seismic exploration have oten made use o Biot s theory o poroelasticity [Biot, 1941; 1956; 1962; Gassmann, 1951]. This theory has always been limited by an explicit assumption that the porosity itsel is homogeneous. Although this assumption is known to be adequate or acoustic studies o many rock core samples in a laboratory setting, it is probably not a very good assumption or applications to realistic heterogeneous reservoirs. One approach to dealing with the heterogeneity is to construct a model that is locally homogeneous, i.e., a sort o inite element approach in which each block o the model satisies Biot-Gassmann equations. This approach may be adequate in some applications, and is certainly amenable to study with large computers. However, such models avoid the question o how we are to deal with heterogeneity on the local scale, i.e., much smaller than the size o blocks typically used in the codes. Although it is clear that porosity in the earth can and does come in virtually all shapes and sizes, it is also clear that two types o porosity are most important: (1) Matrix porosity that occupies a inite and substantial raction o the volume o the reservoir. This porosity is oten called the storage porosity, because this is the volume that stores the luids o interest to us. (2) Fracture or crack porosity that may occupy very little volume, but nevertheless has two very important eects on the reservoir properties. The irst eect is that ractures/cracks drastically weaken the rock elastically, and at very low eective stress levels introduce nonlinear behavior since very small changes in stress can lead to large changes in the racture/crack apertures (and at the same time change the racture strength or uture changes). The second eect is that the ractures/cracks oten introduce a high permeability pathway or the luid to escape rom the reservoir. This eect is obviously key to reservoir analysis and the economics o luid withdrawal. It is thereore not surprising that there have been many attempts to incorporate ractures into rock models, and especially models that try to account or partial saturation eects and the possibility that luid moves (or squirts) during the passage o seismic waves [Budiansky and O Connell, 1975; O Connell and Budiansky, 1977; Mavko and Nur, 1979; Mavko and Jizba, 1991; Dvorkin and Nur, 1993]. Previous attempts to incorporate have generally been rather ad hoc in their approach to the introduction o the ractures into Biot s theory, i Biot s theory is used at all. The present authors have recently started an eort to make a rigorous extension o Biot s poroelasticity to include ractures/cracks by making a generalization to double-porosity/dual-permeability modeling [Berryman and Wang, 1995]. The previously published work concentrated on the luid low aspects o this problem in order to deal with the interactions between luid withdrawal and the elastic response (closure) o ractures during reservoir drawdown.

29 SEP 98 Elastic waves or double porosity 225 It is the purpose o the present work to point out that a similar analysis applies to the wave propagation problem. We expect it will be possible to incorporate all o the important physical eects in a very natural way into this double-porosity extension o poroelasticity or seismic wave propagation. The price we pay or this rigor is that we must solve coupled equations o motion locally. Within traditional poroelasticity, there are two types o equations that are coupled. These are the equations or the elastic behavior o the solid rock and the equations or elastic and luid low behavior o the pore luid. In the double-porosity extension o poroelasticity, we will have not two types o equations but three. The equations or the elastic behavior o the solid rock will be unchanged except or a new coupling term, while there will be two types o pore-luid equations (even i there is only one luid present) depending on the environment o the luid. Pore luid in the matrix (storage) porosity will have one set o equations with coupling to racture luid and solid; while luid in the ractures/cracks will have another set o equations with coupling to pore luid and solid. Although solution o these equations is no doubt more diicult than or simple acoustics/elasticity, it is probably not signiicantly more diicult than traditional single-porosity poroelasticity. We are not going to solve these equations in the present paper. We will instead derive them and then show that the various coeicients in these equations can be readily identiied with measurable quantities. EQUATIONS OF MOTION The seismic equations o motion or a double-porosity medium have been derived recently by Tuncay and Corapcioglu [1996] using a volume averaging approach. (These authors also provide a thorough review o the prior literature on this topic.) We will present instead a quick derivation based on ideas similar to those o Biot s original papers [Biot, 1956; 1962], wherein a Lagrangian ormulation is presented and the phenomenological equations derived. Physically what we need is quite simple just equations embodying the concepts o orce = mass acceleration and dissipation due to viscous loss mechanisms. The orces are determined by taking a derivative o an energy storage unctional. The appropriate energies are discussed at length later in this paper, so or our purposes in this section it will suice to assume that the constitutive laws relating stress and strain are known, and so the pertinent orces are the divergence o the solid stress ield τ i j, j and the gradients o the two luid pressures p (1),i and p (2),i or the matrix and racture luids, respectively. (In this notation, i, j index the three Cartesian coordinates x 1, x 2, x 3 and a comma preceding a subscript indicates a derivative with respect to the speciied coordinate direction.) Then, the only work we need to do to establish the equations o motion or dynamical double-porosity systems concerns the inertial terms arising rom the kinetic energy o the system, and drag terms arising rom dissipation due to relative motion o the constituents. Generalizing Biot s approach [Biot, 1956] to the ormulation o the kinetic energy terms, we ind that, or a system with two luids, the kinetic energy T is determined by 2T = ρ 11 u u + ρ 22 U (1) U (1) + ρ 33 U (2) U (2) +2ρ 12 u U (1) + 2ρ 13 u U (2) + 2ρ 23 U (1) U (2), (1)

30 226 Berryman & Wang SEP 98 where u is the displacement o the solid, U (k) is the displacement o the kth luid which occupies volume raction φ (k), and the various coeicients ρ 11, ρ 12, etc., are inertial mass coeicients that take into account the act that the relative low o luid through the pores is not uniorm, and that oscillations o solid mass in the presence o luid leads to induced mass eects. Clariying the precise meaning o these displacements is beyond our current scope, but recent publications help with these interpretations [Pride and Berryman, 1998]. Dissipation plays a crucial role in the motion o the luids and so cannot be neglected in this context. The appropriate dissipation unctional will take the orm 2D = b 12 ( u U (1) ) ( u U (1) ) + b 13 ( u U (2) ) ( u U (2) ) +b 23 ( U (1) U (2) ) ( U (1) U (2) ). (2) This ormula assumes that all dissipation is caused by motion o the luids either relative to the solid, or relative to each other. (Other potential sources o attenuation, especially or partially saturated porous media [Miksis, 1988], should also be considered, but will not be discussed here.) We expect the coeicient b 23 will generally be small and probably be negligible, whenever the double-porosity model is appropriate or the system under study. Lagrange s equations then show easily that ( ) T + D = τ i j, j, t u i u i or i = 1,2,3, (3) and that ( T t U (k) i ) + D U (k) = p (k) i, or i = 1,2,3;k = 1,2. (4) i These equations now account properly or inertia and elastic energy, strain, and stress, as well as or the speciied types o dissipation mechanisms, and are in complete agreement with those developed by Tuncay and Corapcioglu [1996] using a dierent approach. In (4), the parts o the equation not involving the kinetic energy can be shown to be equivalent to a two-luid Darcy s law in this context, so b 12 and b 13 are related to Darcy s constants or two single phase low and b 23 is the small coupling coeicient. Explicit relations between the b s and the appropriate permeabilities (see Eqs. (53) and (54) o Berryman and Wang [1995]) are not diicult to establish (see the next section). The harder part o the analysis concerns the constitutive equations required or the right hand side o (3). Ater the section on inertia and drag, the remainder o the paper will necessarily be devoted to addressing some o these issues concerning stress-strain relations. In summary, equations (3) and (4) can be combined into ρ 11 ρ 12 ρ 13 ü i b 12 + b 13 b 12 b 13 u i ρ 12 ρ 22 ρ 23 Ü (1) i + b 12 b 12 + b 23 b 23 ρ 13 ρ 23 ρ 33 Ü (2) b i 13 b 23 b 13 + b 23 i U (1) i U (2) τ i j, j = p (1),i, (5) p (2),i

31 SEP 98 Elastic waves or double porosity 227 showing the coupling between the solid and both types o luid components. In the next section we show how to identiy the inertial and drag coeicients with physically measureable quantities. INERTIAL AND DRAG COEFFICIENTS Inertial coeicients It is easy to understand that the inertial coeicients appearing in the kinetic energy T must depend on the densities o solid and luid constituents ρ s and ρ, and also on the volume ractions φ (1) and φ (2) o the matrix and racture porosities. The total porosity is given by φ = φ (1) + φ (2) and the volume raction occupied by the solid material is thereore 1 φ. For a single porosity material, there are only three inertial coeicients and the kinetic energy can be written as 2T = ( u ( )( ) ρ11 ρ U) 12, (6) ρ 12 ρ 22 u U where U is the velocity o the only luid present. Then, it is easy to see that, i u = U, the total inertia ρ ρ 12 + ρ 22 must equal the total inertia present in the system (1 φ)ρ s +φρ. Furthermore, Biot [1956] has shown that ρ 11 + ρ 12 = (1 φ)ρ s, and that ρ 22 + ρ 12 = φρ. These three equations are not linearly independent and thereore do not determine the three coeicients. So we make the additional assumption that ρ 22 = αφρ, where α was termed the structure actor by Biot [1956] but has more recently been termed the tortuosity [Brown, 1980; Johnson et al., 1982]. Berryman [1980] has shown that ( ) 1 α = 1 +r φ 1, (7) ollows rom interpreting the coeicient ρ 11 as resulting rom the solid density plus the induced mass due to the oscillation o the solid in the surrounding luid. Then, ρ 11 = (1 φ)(ρ s +rρ ), where r is a actor dependent on microgeometry that is expected to lie in the range 0 r 1, with r = 1 2 or spherical grains. For example, i φ = 0.2 and r = 0.5, equation (7) implies α = 3.0, which is a typical value or tortuosity o sandstones. For double porosity, the kinetic energy may be written as ρ 11 ρ 12 ρ 13 2T = ( u U (1) U (2) ) ρ 12 ρ 22 ρ 23 ρ 13 ρ 23 ρ 33 u U (1) U (2). (8) We now consider some limiting cases. First, suppose that all the solid and luid material moves in unison. Then, we again have the result ρ 11 + ρ 22 + ρ ρ ρ ρ 23 must equal the total inertia o the system (1 φ)ρ s + φρ. Next, i we suppose that the two luids can be

32 228 Berryman & Wang SEP 98 made to move in unison, but independently o the solid, then we can take U = U (1) = U (2), and telescope the expression or the kinetic energy to 2T = ( u ( U) ρ 11 (ρ 12 + ρ 13 ) (ρ 12 + ρ 13 ) (ρ ρ 23 + ρ 33 ))( u U ). (9) We can now relate the matrix elements in (9) directly to the barred matrix elements appearing in (6), which then gives us three equations or our six unknowns. Again these equations are not linearly independent, so we still need our more equations. Next we consider the possibility that the racture luid can oscillate independently o the solid and the matrix luid, and urthermore that the matrix luid velocity is locked to that o the solid so that u = U (1). For this case, the kinetic energy telescopes in a dierent way to ( )( 2T = ( u U (2) (ρ11 + 2ρ ) 12 + ρ 22 ) (ρ 13 + ρ 23 ) u ). (ρ 13 + ρ 23 ) ρ 33 U (2) (10) This equation is also o the orm (6), but we must be careul to account properly or the parts o the system included in the matrix elements. Now we treat the solid and matrix luid as a single unit, so ρ ρ 12 + ρ 22 = (1 φ)ρ s + φ (1) ρ + (α (2) 1)φ (2) ρ, (11) and where α (2) is the tortuosity o racture porosity alone. ρ 13 + ρ 23 = (α (2) 1)φ (2) ρ, (12) ρ 33 = α (2) φ (2) ρ, (13) Finally, we consider the possibility that the matrix luid can oscillate independently o the solid and the racture luid, and urthermore that the racture luid velocity is locked to that o the solid so that u = U (2). The kinetic energy telescopes in a very similar way to the previous case with the result ( )( 2T = ( u U (1) (ρ11 + 2ρ ) 13 + ρ 33 ) (ρ 12 + ρ 23 ) u ). (ρ 12 + ρ 23 ) ρ 22 U (1) (14) We imagine that this thought experiment amounts to analyzing the matrix material alone without ractures being present. The equations resulting rom this identiication are completely analogous to those in (11)-(13), so we will not show them explicitly here. We now have nine equations in the six unknowns and six o these are linearly independent, so the system can be solved. The result o this analysis is that the o-diagonal terms are given by 2ρ 12 /ρ = (α (2) 1)φ (2) (α (1) 1)φ (1) (α 1)φ, (15)

33 SEP 98 Elastic waves or double porosity 229 2ρ 13 /ρ = (α (1) 1)φ (1) (α (2) 1)φ (2) (α 1)φ, (16) and 2ρ 23 /ρ = αφ α (1) φ (1) α (2) φ (2). (17) The diagonal terms are given by ρ 11 = (1 φ)ρ s + (α 1)φρ, (18) and ρ 33 is given by (13). ρ 22 = α (1) φ (1) ρ, (19) Estimates o the three tortuosities α, α (1), and α (2) may be obtained using (7), or direct measurements may be made using electrical methods as advocated by Brown [1980] and Johnson et al. [1982]. Drag coeicients The drag coeicients may be determined by irst noting that the equations presented here reduce to those o Berryman and Wang [1995] in the low requency limit by merely neglecting the inertial terms. What is required to make the direct identiication o the coeicients is a pair o coupled equations or the two increments o luid content ζ (1) and ζ (2). These quantities are related to the displacements by ζ (1) = φ (1) (U (1) u) and ζ (2) = φ (2) (U (2) u). The pertinent equations rom Berryman and Wang [1995] are ( ζ η (1) ) ( k (11) k ζ (2) = (12) )( p (1) ) k (21) k (22),ii p (2), (20),ii where η is the shear viscosity o the luid, and the ks are permeabilities including possible cross-coupling terms. We can extract the terms we need rom (5), and then take the divergence to obtain ( b12 + b 23 b 23 b 23 b 13 + b 23 )( ( U (1) ) u) ( U (2) = u) ( p (1) ),ii p (2). (21),ii Comparing these two sets o equations and solving or the b coeicients, we ind b 12 = ηφ(1) (k (22) k (21) ) k (11) k (22) k (12), (22) k (21) and b 23 = b 12 = ηφ(2) (k (11) k (12) ) k (11) k (22) k (12), (23) k (21) ηφ (1) k (21) k (11) k (22) k (12) k (21) = ηφ (2) k (12) k (11) k (22) k (12). (24) k (21)

34 230 Berryman & Wang SEP 98 For many applications it will be adequate to assume that the cross-coupling vanishes. In this situation, b 23 = 0, and b 12 = ηφ(1) k (11) (25) b 13 = ηφ(2), (26) k (22) which also provides a simple interpretation o these coeicients in terms o the porosities and diagonal permeabilities. This completes the identiication o the inertial and drag coeicients introduced in the previous section. STRESS-STRAIN FOR SINGLE POROSITY In the absence o driving orces that can maintain pressure dierentials over long time periods, double porosity models must reduce to single porosity models in the long time limit when the matrix pore pressure and crack pore pressure become equal. It is thereore necessary to remind ourselves o the basic results or single porosity models in poroelasticity. One important role these results play is to provide constraints or the long time behavior in the problems o interest. A second signiicant use o these results (see Berryman and Wang [1995]) arises when we make laboratory measurements on core samples having properties characteristic o the matrix material. Then the results presented in this section apply speciically to the matrix stinesses, porosity, etc. For isotropic materials and hydrostatic pressure variations, the two independent variables in linear mechanics o porous media are the conining (external) pressure p c and the luid (pore) pressure p. The dierential pressure p d p c p is oten used to eliminate the conining pressure. The equations o the undamental dilatations are then or the total volume V, δv V = δp d K + δp K s (27) δv φ V φ = δp d K p + δp K φ (28) or the pore volume V φ = φv, and δv V = δp K (29) or the luid volume V. Equation (27) serves to deine the various constants o the porous solid, such as the drained rame bulk modulus K and the unjacketed bulk modulus K s or the

35 SEP 98 Elastic waves or double porosity 231 composite rame. Equation (28) deines the jacketed pore modulus K p and the unjacketed pore modulus K φ. Similarly, (29) deines the bulk modulus K o the pore luid. Treating δp c and δp as the independent variables in our poroelastic theory, we deine the dependent variables δe δv/v and δζ (δv φ δv )/V, both o which are positive on expansion, and which are respectively the total volume dilatation and the increment o luid content. Then, it ollows directly rom the deinitions and rom (27), (28), and (29) that ( ) ( )( ) δe 1/K 1/Ks 1/K δpc =. (30) δζ φ/k p φ(1/k p + 1/K 1/K φ ) δp Now we consider two well-known thought experiments: the drained test and the undrained test [Gassmann, 1951; Biot and Willis, 1957; Geertsma, 1957]. (For a single porosity system, these two experiments are sometimes considered equivalent to the slow loading and ast loading limits respectively. However, these terms are relative since, or example, the ast loading equivalent to undrained limit is still assumed to be slow enough that the average luid and conining pressures are assumed to have reached equilibrium.) The drained test assumes that the porous material is surrounded by an impermeable jacket and the luid is allowed to escape through a tube that penetrates the jacket. Then, in a long duration experiment, the luid pressure remains in equilibrium with the external luid pressure (e.g., atmospheric) and so δp = 0 and hence δp c = δp d ; so the changes o total volume and pore volume are given exactly by the drained constants 1/K and 1/K p as deined in (27) and (28). In contrast, the undrained test assumes that the jacketed sample has no passages to the outside world, so pore pressure responds only to in conining pressure changes. With no means o escape, the increment o luid content cannot change, so δζ = 0. Then, the second equation in (30) shows that 0 = φ/k p (δp c δp /B), (31) where Skempton s pore-pressure buildup coeicient B [Skempton, 1954] is deined by B δp δp (32) c δζ =0 and is thereore given by B = K p (1/K 1/K φ ). (33) It ollows immediately rom this deinition that the undrained modulus K u is determined by (also see Carroll [1980]) K u = K 1 αb, (34) where we introduced the combination o moduli known as the Biot-Willis parameter α = 1 K/K s. This result was apparently irst obtained by Gassmann [1951] or the case o microhomogeneous porous media (i.e., K s = K φ = K m, the bulk modulus o the single mineral

36 232 Berryman & Wang SEP 98 present) and by Brown and Korringa [1975] and Rice [1975] or general porous media with multiple minerals as constituents. Finally, we condense the general relations rom (30) together with the reciprocity relations [Brown and Korringa, 1975] into symmetric orm as ( ) δe = 1 ( )( ) 1 α δpc. (35) δζ K α α/b δp The storage compressibility, which is a central concept in describing poroelastic aquier behavior in hydrogeology, is related inversely to one deined in Biot s original 1941 paper by S δζ = α δp BK. (36) δpc =0 This storage compressibility is the change in increment o luid content per unit change in the luid pressure, deined or a condition o no change in external pressure. It has also been called the three-dimensional storage compressibility by Kümpel [1991]. We may equivalently eliminate the Biot-Willis parameter α and write (35) in terms o the undrained modulus so that ( ) δe = 1 ( )( ) 1 (1 K/K u )/B δpc δζ K (1 K/K u )/B (1 K/K u )/B 2. (37) δp Equation (37) has the advantage that all the parameters have very well deined physical interpretations, and are also easily generalized or a double porosity model. Finally, note that (35) shows that K p = φk/α, which we generally reer to as the reciprocity relation. The total strain energy unctional (including shear) or this problem may be written in the orm 2E = δτ i j δe i j + δp δζ, (38) where δe i j is the change in the average strain with δe ii δe being the dilatation, δτ i j being the change in the average stress tensor or the saturated porous medium with 1 3 δτ ii = δp c. It ollows that and δp c = E (δe) δp = (39) E (δζ ), (40) both o which are also consistent with Betti s reciprocal theorem [Love, 1927] since the matrices in (35) and (37) are symmetric. The shear modulus µ is related to the bulk modulus and Poisson s ratio by µ = 3(1 2ν)K/2(1 + ν). Then, it ollows that the stress equilibrium equation is τ i j, j = (K u µ)e,i + µu i, j j BK u ζ,i = 0 (41)

37 SEP 98 Elastic waves or double porosity 233 and Darcy s law takes the orm k η p,ii = ζ, (42) where η is the single-luid shear viscosity. STRESS-STRAIN FOR DOUBLE POROSITY p c p c p c p c p (1) p (2) Figure 1: The double porosity model eatures a porous rock matrix intersected by ractures. Three types o macroscopic pressure are pertinent in such a model: external conining pressure p c, internal pressure o the matrix pore luid p (1), and internal pressure o the racture pore luid p (2). A single porosity medium is one in which either matrix or racture porosity are present, but not both. jim-pic [NR] We now assume two distinct phases at the macroscopic level: a porous matrix phase with the eective properties K (1), µ (1), K (1) m, φ (1) occupying volume raction V (1) /V = v (1) o the total volume and a macroscopic crack or joint phase occupying the remaining raction o the volume V (2) /V = v (2) = 1 v (1). The key eature distinguishing the two phases and thereore requiring this analysis is the very high luid permeability k (22) o the crack or joint phase and the relatively lower permeability k (11) o the matrix phase. We could also introduce a third independent permeability k (12) = k (21) or luid low at the interace between the matrix

38 234 Berryman & Wang SEP 98 and crack phases, but or simplicity we assume here that this third permeability is essentially the same as that o the matrix phase, so k (12) = k (11). We have three distinct pressures: conining pressure δp c, matrix-luid pressure δp (1), and joint-luid pressure δp (2). Treating δp c,δp (1), and δp (2) as the independent variables in our double porosity theory, we deine the dependent variables δe δv/v (as beore), δζ (1) = (δv (1) (1) φ δv )/V, and δζ (2) = (δv (2) (2) φ δv )/V, which are respectively the total volume dilatation, the increment o luid content in the matrix phase, and the increment o luid content in the joints. We assume that the luid in the matrix is the same kind o luid as that in the cracks or joints, but that the two luid regions may be in dierent states o average stress and thereore need to be distinguished by their respective superscripts. Linear relations among strain, luid content, and pressure then take the general orm δe a 11 a 12 a 13 δp c δζ (1) = a 21 a 22 a 23 δp (1) δζ (2). (43) a 31 a 32 a 33 δp (2) By analogy with (35) and (37), it is easy to see that a 12 = a 21 and a 13 = a 31. The symmetry o the new o-diagonal coeicients may be demonstrated by using Betti s reciprocal theorem in the orm 0 (δe δζ (1) δζ (2) ) δ p (1) 0 = ( δe δζ (1) δζ (2) cr ) 0 0 δp (2), (44) where unbarred quantities reer to one experiment and barred to another experiment to show that δζ (1) δ p (1) = a 23 δp (2) δ p (1) = a 32 δ p (1) δp (2) = δζ (2) δp (2). (45) Hence, a 23 = a 32. Similar arguments have oten been used to establish the symmetry o the other o-diagonal components. Thus, we have established that the matrix in (43) is completely symmetric, so we need to determine only six independent coeicients. To do so, we consider a series o thought experiments, including tests in both the short time and long time limits. The key idea here is that at long times the two pore pressures must come to equilibrium (p (1) = p (2) = p as t ) as long as the cross permeability k (12) is inite. However, at very short times, we may assume that the process o pressure equilibration has not yet begun, or equivalently that k (12) = 0 at t = 0. We nevertheless assume that the pressure in each o the two components have individually equilibrated on the average, even at short times. Undrained joints, undrained matrix, short time There are several dierent, but equally valid, choices o time scale on which to deine Skemptonlike coeicients or the matrix/racture system under consideration. Elsworth and Bai [1992] use a deinition based on the idea that or very short time both luid systems will independently

39 SEP 98 Elastic waves or double porosity 235 act undrained ater the addition o a sudden change o conining pressure. This idea implies that δζ (1) = 0 = δζ (2) which, when substituted into (43), gives Deining δe = a 11 δp c + a 12 δp (1) + a 13 δp (2) 0 = a 12 δp c + a 22 δp (1) + a 23 δp (2) B (1) E B δp(1) δp c δζ (1) =δζ (2) =0 0 = a 13 δp c + a 23 δp (1) + a 33 δp (2). (46) and B (2) E B δp(2) δp c δζ (1) =δζ (2) =0 we can solve (46) or the two Skempton s coeicients and ind the results and B (1) E B = a 23a 13 a 12 a 33 a 22 a 33 a 2 23 B (2) The eective undrained modulus is ound to be given by 1 K u E B (47) (48) E B = a 23a 12 a 13 a 22 a 22 a 33 a23 2. (49) δe δp c δζ (1) =δζ (2) =0 = a 11 + a 12 B (1) E B + a 13B (2) E B. (50) These deinitions will be compared to others. Drained joints, undrained matrix, intermediate time Now consider a sudden change o conining pressure on a jacketed sample, but this time with tubes inserted in the joint (racture) porosity so δp (2) = 0, while δζ (1) = 0. We will call this the drained joint, undrained matrix limit. The resulting equations are showing that the pore-pressure buildup in the matrix is B[u (1) ] δp(1) δp c δe = a 11 δp c a 12 p (1) δζ (2) = a 31 δp c a 32 δp (1), (51) δζ (1) =δp (2) =0 = a 21 a 22. (52) Similarly, the eective undrained modulus or the matrix phase is ound rom (51) to be determined by 1 K [u (1) ] δe = a 11 + a 12 B[u (1) ]. (53) δp c δζ (1) =δp (2) =0 Notice that i a 23 = 0 then (48) and (52) are the same.

40 236 Berryman & Wang SEP 98 Drained matrix, undrained joints, intermediate time Next consider another sudden change o conining pressure on a jacketed sample, but this time the tubes are inserted in the matrix porosity so δp (1) = 0, while δζ (2) = 0. We will call this the drained matrix, undrained joint limit. The equations are showing that the pore-pressure buildup in the cracks is B[u (2) ] δp(2) δp c δe = a 11 δp c a 13 p (2) δζ (1) = a 21 δp c a 23 δp (2) 0 = a 31 δp c a 33 δp (2), (54) δζ (2) =δp (1) =0 = a 31 a 33. (55) Similarly, the eective undrained modulus or the joint phase is ound 1 K [u (2) ] δe = a 11 + a 13 B[u (2) ]. (56) δp c δζ (2) =δp (1) =0 We may properly view Eqs. (52), (53), (55), and (56) as deining relations among these parameters. Notice that i a 23 = 0 then (49) and (55) are the same. Drained test, long time The long duration drained (or jacketed ) test or a double porosity system should reduce to the same results as in the single porosity limit. The conditions on the pore pressures are δp (1) = δp (2) = 0, and the total volume obeys δe = a 11 δp c. It ollows thereore that a 11 1 K, (57) where K is the overall drained bulk modulus o the system including the ractures. Undrained test, long time The long duration undrained test or a double porosity system should also produce the same physical results as a single porosity system (assuming only that it makes sense at some appropriate larger scale to view the medium as homogeneous). The basic equations are δp (1) = δp (2) = δp, δζ δζ (1) + δζ (2) = 0, (58)

41 SEP 98 Elastic waves or double porosity 237 assuming the total mass o luid is conined. Then, it ollows that δe = a 11 δp c (a 12 + a 13 )δp, 0 = (a 21 + a 31 )δp c (a 22 + a 23 + a 32 + a 33 )δp, (59) showing that the overall pore-pressure buildup coeicient is given by B p p (60) c δζ =0 Similarly, the undrained bulk modulus is ound to be given by 1 δe K u δp = a 11 + (a 12 + a 13 )B. (61) c δζ =0 Fluid injection test, long time The conditions on the pore pressures or the long duration, single porosity limit or the threedimensional storage compressibility S are δp (1) = δp (2) = δp, while the conining pressure remains constant. It ollows thereore that S ζ p = a 22 + a 23 + a 32 + a 33. (62) δpc =0 This completes the main analysis o the elastic coeicients or double porosity. Further details may be ound in Berryman and Wang [1995].

42 238 Berryman & Wang SEP 98 TABLE 1. Material Properties Barre Bedord Parameter Granite Limestone K (GPa) 13.5 a 23.0 a K s (GPa) 54.5 a 66.0 a α 0.75 a 0.65 a K (1) (GPa) 21.5 a 27.0 a ν (1) K s (1) (GPa) 55.5 a 66.0 a α (1) 0.61 a 0.59 a K (GPa) φ (1) a B (1) S (1) (GPa 1 ) v (2) a a From Coyner [1984] EXAMPLES Table 1 presents data or Barre granite and Bedord limestone taken rom laboratory measurements by Coyner [1984]. Coyner s experiments included a series o tests on several types o laboratory scale rock samples at dierent conining pressures. The values quoted or K and K s are those or a moderate conining pressure o 10 MPa (values at lower conining pressures were also measured but we avoid using these values because the rocks generally exhibit nonlinear behavior in that region o the parameter space), while the values quoted or K (1) are at 25 MPa, which is close to the value beyond which the constants cease depending on pressure and thereore or which we assume all the cracks were closed. Thus, based on the idea that the pressure behavior is associated with two kinds o porosity in the laboratory samples a crack porosity, which is being closed between 10 and 25 MPa, and a and K (1) s residual matrix porosity above 25 MPa, we assume the available data are K, K s, K (1), K s (1), K, φ (1), and v (2). We ind that these data are suicient to compute all the coeicients. In Table 2, we ind in both types o rock that the coeicient a 23 is positive and small about an order o magnitude smaller than the other matrix elements. Another unusual eature o the results computed using these laboratory data is the occurrence o values larger than unity or B and B[u (1) ] in Barre granite; similar results were observed by Berryman and Wang [1995] or Chelmsord granite, and in subsequent work or Westerly granite. Also note that α (2) or both rocks is close to unity or these calculations. In this example, seven measurements (together with Poisson s ratio) are suicient to determine completely the mechanical behavior o the double-porosity model. Having a direct measurement o K eliminates the necessity o assuming a 23 = 0, which we have ound is sometimes necessary when dealing with ield data [Berryman and Wang, 1995]. TABLE 2. Double porosity parameters computed rom material properties in Table 1.

43 SEP 98 Elastic waves or double porosity 239 Barre Bedord Parameter Formula Granite Limestone a 11 (GPa 1 ) 1/K a 12 (GPa 1 ) α (1) K s (1) /K (1) K s a 13 (GPa 1 ) α/k a a 22 (GPa 1 ) v (1) α (1) /B (1) K (1) a 23 (GPa 1 ) v (1) α (1) /K (1) a a 33 (GPa 1 ) v (2) /K + v (1) /K (1) (1 2α)/K + 2a a 33 (GPa 1 ) a 33 v (2) /K α (2) (a 33 a 12 a 13 a 23 )/(a 11 a 23 a 13 a 12 ) B (a 12 + a 13 )/(a a 23 + a 33 ) B (1) (a 12 + a 23 )/a B[u (1) ] a 12 /a B (1) E B (a 23 a 13 a 12 a 33 )/(a 22 a 33 a23 2 ) B (2) (a 33 a 12 a 13 a 23 )/[a 33 (a 12 + a 23 ) a 23 (a 13 a 33 )] B[u (2) ] a 13 /a B (2) E B (a 23 a 12 a 13 a 22 )/(a 22 a 33 a23 2 ) K u (GPa) [a 11 (a 12 + a 13 ) 2 /(a a 23 + a 33 )] K (2) (GPa) v (2) (a 12 + a 13 )/(a 11 a 23 a 13 a 12 ) S (GPa 1 ) α/bk S (2) (GPa 1 ) α (2) /B (2) K (2) CONCLUSIONS We now have a complete set o equations or the double-porosity/dual-permeability model. The next step in the analysis will be to solve these equations and ind their wave propagation properties. This work is in progress and will be reported at a later date. REFERENCES Berryman, J. G., 1980, Conirmation o Biot s theory: Appl. Phys. Lett., 37, Berryman, J. G., and Wang, H. F., 1995, The elastic coeicients o double-porosity models or luid transport in jointed rock: J. Geophys. Res., 100, Biot, M. A., 1941, General theory o three dimensional consolidation: J. Appl. Phys., 12, Biot, M. A., 1956, Theory o propagation o elastic waves in a luid-saturated porous solid. I. Low-requency range: J. Acoust. Soc. Am., 28, Biot, M. A., 1962, Mechanics o deormation and acoustic propagation in porous media: J. Appl. Phys., 33,

44 240 Berryman & Wang SEP 98 Biot, M. A., and Willis, D. G., 1957, The elastic coeicients o the theory o consolidation: J. App. Mech., 24, Brown, R. J. S., 1980, Connection between ormation actor or electrical-resistivity and luidsolid coupling actor in Biot equations or acoustic-waves in luid-illed porous media: Geophysics, 45, Brown, R. J. S., and Korringa, J., 1975, On the dependence o the elastic properties o a porous rock on the compressibility o a pore luid: Geophysics, 40, Budiansky, B., and O Connell, R. J., 1976, Elastic moduli o a cracked solid: Int. J. Solids Struct., 12, Carroll, M. M., 1980, Mechanical response o luid-saturated porous materials: in Theoretical and Applied Mechanics, F. P. J. Rimrott and B. Tabarrok (eds.), Proceedings o the 15th International Congress o Theoretical and Applied Mechanics, Toronto, August 17 23, 1980, North-Holland, Amsterdam, pp Coyner, K. B., 1984, Eects o Stress, Pore Pressure, and Pore Fluids on Bulk Strain, Velocity, and Permeability o Rocks, Ph. D. Thesis, Massachusetts Institute o Technology. Dvorkin, J., and Nur, A., 1993, Dynamic poroelasticity: A uniied model with the squirt and the Biot mechanisms: Geophysics, 58, Gassmann, F., 1951, Über die elastizität poröser medien: Veirteljahrsschrit der Naturorschenden Gesellschat in Zürich, 96, Geertsma, J., 1957, The eect o luid pressure decline on volumetric changes o porous rocks: Trans. AIME, 210, Johnson, D. L., Plona, T. J., Scala, C., Pasierb, F., and Kojima, H., 1982, Tortuosity and acoustic slow waves: Phys. Rev. Lett., 49, Kümpel, H.-J., 1991, Poroelasticity: parameters reviewed: Geophys. J. Int., 105, Love, A. E. H., 1927, A Treatise on the Mathematical Theory o Elasticity, Dover, New York, pp Mavko, G., and Jizba, D., 1991, Estimating grain-scale luid eects on velocity dispersion in rocks: Geophysics, 56, Mavko, G., and Nur, A., 1979, Wave attenuation in partially saturated rocks: Geophysics, 44, Miksis, M. J., 1988, Eects o contact line movement on the dissipation o waves in partially saturated rocks: J. Geophys. Res., 93, O Connell, R. J., and Budiansky, B., 1974, Seismic velocities in dry and saturated cracked solids: J. Geophys. Res., 79,

45 SEP 98 Elastic waves or double porosity 241 O Connell, R. J., and Budiansky, B., 1977, Viscoelastic porperties o luid-saturated cracked solids: J. Geophys. Res., 82, Pride, S. R., and Berryman, J. G., 1998, Connecting theory to experiment in poroelasticity: J. Mech. Phys. Solids, 46, Rice, J. R., 1975, On the stability o dilatant hardening or saturated rock masses: J. Geophys. Res., 80, Rice, J. R., and Cleary, M. P., 1976, Some basic stress diusion solutions or luid-saturated elastic porous media with compressible constituents: Rev. Geophys. Space Phys., 14, Skempton, A. W., 1954, The pore-pressure coeicients A and B: Geotechnique, 4, Tuncay, K., and Corapcioglu, M. Y., 1996, Wave propagation inractured porous media: Transp. Porous Media, 23,

46 SEP SEP ARTICLES PUBLISHED OR IN PRESS Alkhaliah, T., Biondi, B., and Fomel, S., 1998, Time-domain processing in arbitrarily inhomogeneous media: 68th Ann. Internat. Meeting, Soc. Expl. Geophys., Expanded Abstracts, Alkhaliah, T., 1998, An acoustic wave equation or anisotropic media: 68th Ann. Internat. Meeting, Soc. Expl. Geophys., Expanded Abstracts, Bachrach, R., and Rickett, J., 1998, Seismic detection o viscous contaminent using shallow seismic relection: Seismic detection o viscous contaminent using shallow seismic relection:, Env. Eng. Geophys. Soc., Symposium on the Application o Geophysics to Environmental and Engineering Problems. Biondi, B., Fomel, S., and Chemingui, N., 1998, Azimuth moveout or 3-D prestack imaging: Geophysics, 63, no. 2, Biondi, B., 1998a, Azimuth moveout vs. dip moveout in inhomogeneous media: 68th Ann. Internat. Meeting, Soc. Expl. Geophys., Expanded Abstracts, Biondi, B., 1998b, Cost-eective prestack depth imaging o marine data: Ann. Internat. Meeting, Oshore Technology Conerence. Biondi, B., 1998c, Robust relection tomography in the time domain: 68th Ann. Internat. Meeting, Soc. Expl. Geophys., Expanded Abstracts, Claerbout, J., 1998a, Multidimensional recursive ilters via a helix: Geophysics, 63, Claerbout, J., 1998b, Multidimensional recursive ilters via a helix with application to velocity estimation and 3-D migration: 68th Ann. Internat. Meeting, Soc. Expl. Geophys., Expanded Abstracts, Clapp, R., Biondi, B., Fomel, S., and Claerbout, J., 1998, Regularizing velocity estimation using geologic dip inormation: 68th Ann. Internat. Meeting, Soc. Expl. Geophys., Expanded Abstracts, Crawley, S., 1998, Shot interpolation or Radon multiple suppression: 68th Ann. Internat. Meeting, Soc. Expl. Geophys., Expanded Abstracts, Ecker, C., Dvorkin, J., and Nur, A., 1998, Estimating the amount o ree gas and hydrate rom surace seismic: 68th Ann. Internat. Meeting, Soc. Expl. Geophys., Expanded Abstracts, Malcotti, H., and Biondi, B., 1998, Accurate linear interpolation in the extended split-step migration: 9th Ann. Internat. Mtg., Sociedad Venezolana de Ing. Geoisicos, Expanded Abstracts, submitted.

47 278 SEP 98 Rickett, J., and Lumley, D., 1998, A cross-equalization processing low or o-the-shel 4- D seismic data: 68th Ann. Internat. Meeting, Soc. Expl. Geophys., Expanded Abstracts, Rickett, J., Claerbout, J., and Fomel, S., 1998, Implicit 3-D depth migration by waveield extrapolation with helical boundary conditions: 68th Ann. Internat. Meeting, Soc. Expl. Geophys., Expanded Abstracts, Sava, P., and Fomel, S., 1998, Huygens waveront tracing: A robust alternative to ray tracing: 68th Ann. Internat. Meeting, Soc. Expl. Geophys., Expanded Abstracts, Sun, Y., and Alkhaliah, T., 1998, Building the 3-D integral DMO operator in the slant-stack domain: 68th Ann. Internat. Meeting, Soc. Expl. Geophys., Expanded Abstracts, Sun, Y., and Fomel, S., 1998, Fast-marching eikonal solver in the tetragonal coordinates: 68th Ann. Internat. Meeting, Soc. Expl. Geophys., Expanded Abstracts,

48 SEP SEP PHONE DIRECTORY Name Phone Login Name Alkhaliah, Tariq (650) tariq Berryman, James berryman Biondi, Biondo biondo Brown, Morgan morgan Chemingui, Nizar nizar Claerbout, Jon jon Clapp, Robert bob Crawley, Sean sean Fomel, Sergey sergey Lau, Diane diane Malcotti, Hermes hermes Prucha, Marie marie Rickett, James james Sava, Paul paul Sun, Yalei yalei SEP ax number: (650) Our Internet address is sep.stanord.edu ; i.e., send Jon electronic mail with the address jon@sep.stanord.edu. WORLD-WIDE WEB SERVER INFORMATION The login name or the sponsors area on our world-wide web server ( is sepsponsor. We have set the password to be wb@sp#95. SUPERTIER CODE PASSWD ON SEP-CDS The Connection Machine codes developed during the SuperTier activities have been encrypted on SEP-CDs. The passwords are: SEP-CD-2: TnX4BkS SEP-CD-6: AvS+Cm5 SEP-CD-3: Cm54SeP SEP-CD-7: Cm5&MiG SEP-CD-5: Up&Rng

49 280 SEP 98

50 SEP Research Personnel James G. Berryman received a B.S. degree in physics rom Kansas University (Lawrence) in 1969 and a Ph.D. degree in physics rom the University o Wisconsin (Madison) in He subsequently worked on seismic prospecting at Conoco. His later research concentrated on sound waves in rocks at AT&T Bell Laboratories ( ) and in Earth Sciences at Lawrence Livermore National Laboratory (1981- ). He is currently a physicist in the Earth and Environmental Sciences Directorate at LLNL. Continuing research interests include seismic and electrical methods o geophysical imaging and waves in porous media containing luids. He is a member o APS, AGU, SEG, and EEGS. Biondo L. Biondi graduated rom Politecnico di Milano in 1984 and received an M.S. (1988) and a Ph.D. (1990) in geophysics rom Stanord. SEG Outstanding Paper award During 1987 he worked as a Research Geophysicist or TOTAL, Compagnie Francaise des Petroles in Paris. Ater his Ph.D. at Stanord Biondo worked or three years with Thinking Machines Co. on the applications o massively parallel computers to seismic processing. Ater leaving Thinking Machines Biondo started 3DGeo Development, a sotware and service company devoted to high-end seismic imaging. Biondo is now Associate Proessor (Research) o Geophysics and leads SEP eorts in 3-D imaging. He is a member o SEG and EAGE.