Summary. Introduction

Transcription

1 Optimal survey esign for marine borehole seismics Darrell Coles*, Schlumberger-Doll Research, Yi Yang, WesternGeco, Hugues Djikpesse, Michael Prange, Schlumberger-Doll Research, an Konstantin Osypov, WesternGeco Summary Optimal experimental esign is the theory an practice of maximizing the expecte information in ata in orer to minimize the post-analysis uncertainty in parameters constraine by those ata. In this article, a borehole seismic survey is optimize in the Gulf of Mexico with the D N -criterion, a nonlinear esign objective function amenable to efficient optimization. Several potential applications are propose, incluing: (1) ientifying the maximum offset for 3D vertical seismic profiling (VSP); (2) an annularspiral 3D VSP geometry to reuce acquisition time/cost; (3) a Z-geometry for presurvey ata borehole seismic acquisition; (4) optimizing source vessel placement for offset/azimuth checkshots; (5) ientifying optimal ata for near-real-time quality control an inversion; an (6) optimally controlling ata set ecimation to maximize the efficient an accurate analysis of large ata sets. Introuction Geological moel uncertainty is an important source of information in hyrocarbon exploration. Quantifie escriptions of uncertainty are particularly important because they can be use formally to ientify an characterize risks an to anticipate potential ifficulties throughout the reservoir life cycle, all of which being factors that vitally inform ecision-making. Failing to properly account for uncertainty is to isregar the value of information (Bratvol et al., 29) it brings to the ecision-making process (Osypov et al., 211). Consequently, it is also valuable to minimize this earth moel uncertainty because so oing minimizes the risk associate with potentially large investments. One way to minimize risk in this context is to apply statistical experimental esign (SED) the theory an practice of optimizing experiments to maximize the expecte information in ata observations. Nonlinear SED is a special branch eicate to nonlinear ata-moel relationships, in which the information content of ata varies with the earth moel, unlike in linear SED (Atkinson et al., 27, p. 248). It is important to aress nonlinearity because most ata-moel relationships (theoretical functions, hereafter) in hyrocarbon exploration are nonlinear an affect moel uncertainty in complicate ways (cf. Guest an Curtis, 211; Coles an Curtis, 211b; Winterfors an Curtis, 28). The information-epenence of ata on moels may be accommoate by representing the earth moel, the expecte ata arising from it, an measurement errors, with probability istributions. However, probabilistic approaches typically involve high-imensional integrals that are expensive to compute an thus limit the size of the experiments that can be optimize (Coles an Curtis, 211). The D N -criterion was introuce as a possible solution to the nonlinear problem (ibi). It is a nonlinear esign objective function that can be maximize using very efficient algorithms from linearize esign theory. This makes the D N -criterion capable of optimizing large-scale experiments, a feature common to only a few nonlinear esign methos (Coles an Prange, 211; Coles an Curtis, 211b; Coles, 211; Guest an Curtis, 211; Haber et al., 28). Here, we sketch a erivation of the D N - criterion which eluciates how it works while eliminating a restrictive assumption from the original erivation (Coles an Curtis, 211b). Theory A common goal in survey esign is to ientify an experiment that maximally iscriminates between two or more theoretical functions that are thought to moel some observe ata, an this is typically one by optimizing a hypothesis test (Feorov, 1972, p. 226). This iea may be moifie to hanle competing parameterizations of the same theoretical function as follows. Without loss of generality, one parameterization may be treate as true (null hypothesis) an another is treate as some alternative (alternative hypothesis). The object in optimizing a iscriminating survey is to maximize the os that the alternative hypothesis is rejecte, which ensures that the moel parameters most likely to explain an observe ata set are, in fact, the true ones. To evelop the iea, let ( m, ξ) = g( m, ξ) + ε( m, ξ ) (1) be a known mathematical moel, where is a vector of ata observations mae at observation points ξ, m is the vector of moel parameters, g is a known theoretical function relating an m, an ε is a vector of stochastic measurement errors. We assume that m has a known prior istribution, ρ ( m ), which characterizes the state of knowlege about m before any new ata are acquire, an likewise, ε has a known istribution. A iscriminating test commonly use in experimental esign is the log-likelihoo-ratio test (Feorov, 1972, p. 249), which expresses the os ratio of the null an alternative hypothesis. SEG Las Vegas 212 Annual Meeting Page 1

2 Denoting the true moel an its corresponing ata by m an g m ε, respectively, an enoting an alternative moel ( ) + m 1, the log-likelihoo-ratio in question is L ( ) ln ln g m + ε m Λ= L g( m) + ε m 1, (2) where L is the ata likelihoo function (epenence on ξ is suppresse for ease of notation). Maximizing Λ with respect to ξ maximizes the os that the true moel, m, will be accepte an the alternative moel, m 1, rejecte. The log-likelihoo ratio in expression 2 is efine for a single m an m 1 an may be generalize for all probable moel parameterizations by averaging it over m, m1 ~ ρ ( m ), which gives L ( ) π ln π (, 1) ln g m + ε m E Λ= m m mm1, (3) L g( m) + ε m1 where is the expectation operator over the joint Eπ istribution of m an m 1, π (, ) = ρ( ) ρ( ) m m m m. 1 1 When ε is Gaussian with zero mean an covariance Σ ε, it is easy to show that equation 3 simplifies to E πln Λ= tr Σε ΣΣε, (4) where tr is the trace operator an Σ ( ) ( ) ( ) ( ) T 1 E π g m g m g m g m. (5) Maximizing the average log-likelihoo ratio in expression 4 shoul therefore maximize the os that the true parameterization is accepte over all probable alternatives. The trace in expression 4, however, oes not penalize zero eigenvalues. All eigenvalues must be nonzero for the atamoel system to be well etermine. Recalling that the trace equals the sum of eigenvalues, we replace it with the sum of log eigenvalues, which is given by the log of the eterminant. This sum goes to negative infinity for any ξ that causes Σε ΣΣε to be singular, which is equivalent to having an unetermine system. Thus, we arrive at the D N -criterion, Σ Φ= ln et Σε ΣΣε = ln Σ. (6) This erivation avois the assumption that ( ) ( ) ε g m g m is 1 multivariate Gaussian (multinormal), a strongly limiting assumption from the original erivation (Coles an Curtis, 211b). It is easy to show that V iag Σ is the iagonal variance matrix an R is the correlation matrix of the theoretical ata. Therefore, as a rule, Φ must generally increase with the variance, an ecrease with the correlation ( R ecreases as correlation increases), of the theoretical ata. This means that it is easier to iscriminate Σ V R, where ( ) between moel parameterizations if the preicte ata vary greatly from parameterization to parameterization, accounting for ata noise (this is the likelihoo objective). It also means that it is easier to iscriminate between parameterizations if their preicte ata are expecte to be uncorrelate (this is the egrees of freeom or objective). Several optimization algorithms exist that may be use to efficiently maximize Φ, incluing the sequential Construction, Exchange, an Decimation Algorithms (cf. Coles an Curtis, 211a). These algorithms are greey in that a solution is optimize through a sequence of locally optimal upates in the hope that the result is close to the global optimum. Coles an Curtis (211a) showe evience that some greey techniques may approach quite close to global optima. 3D VSP optimization We emonstrate D N -optimization for 3D VSP at a real fiel locate on the Walker Rige in the Gulf of Mexico. Several exploratory wells have been rille in a site of interest an have shown signs of hyrocarbons. A reservoir has been conjecture beneath a salt trap at this site an is the impetus for this emonstration. An existing borehole was chosen which transects the salt trap an passes through the inferre reservoir beneath. This borehole was equippe with a virtual string of forty equispace geophones encompassing the reservoir interval (Yi Yang, personal communication). Receiver spacing was 22.9 m, an the top receiver was place 953 m below sea surface. A virtual carpet of caniate shots was place at the sea surface, centere on the wellhea, with 91.4-m shot spacing in each carinal irection. From this caniate set, an optimal subset of shots was to be selecte. The carpet covere ~325 km 2 an comprise 4 caniate shots (Figure 1). Figure 1 A borehole (white line) is centere in the target region an a virtual string of 4 geophones (black ots) is place over the inferre reservoir interval. Blue-white-re inicates P-wave velocities for one of the 5 moels. The caniate shot carpet is exemplifie by orange ots. The horizontal surface (blue-green-re) epicts the salt trap horizon. Earth moel is from Osypov et al., 211. The earth moel was previously characterize by a structural uncertainty workflow (Osypov et al., 21; Osypov et al., SEG Las Vegas 212 Annual Meeting Page 2

3 211) which escribe knowlege of the anisotropic moel parameters by a multinormal istribution base on previous inversion of wie-azimuth (WAZ) surface seismic ata. From this multinormal istribution, five hunre VTI earth moels were ranomly sample, creating an ensemble characterization of the current state of moel uncertainty. The 5 moels were 3D meshes of the elastic properties V p, ε, an δ. The moel cubes were also centere on the wellhea, with areal extent ientical to the caniate shot carpet an extening 15 km in epth from the sea surface. The moel ensemble was use as prior information by the D N -optimizer. Because D N -optimization operates in ata space, it was necessary to compute the P-wave traveltimes for all combinations of shots, receivers, an moels. In all, 8 million traveltimes = 4 (receivers) x 4 (shots) x 5 (moels) were compute. Figure 3a shows in map view the spatial correlation of the traveltime at one caniate shot location with respect to the entire shot carpet. Notice the preferential alignment of correlation along a SW-NE irection, which is probably a manifestation of moel anisotropy. Furthermore, notice the strong an wiesprea istribution of correlation, which inicates that neighboring shots in a fairly large area will provie little complementary information relative to a selecte shot. This oes not mean that nearby shots shoul be ignore, only that they reinforce the information provie by the selecte shot. Figure 3b shows spatial correlations for a (a) Before looking at the optimal borehole survey results, let us examine the expecte ata variability an correlation over the ensemble of 5 anisotropic earth moels. The D N -criterion shoul select shot locations where there is large expecte ata variability an low ata correlation. (b) Figure 2: Traveltime stanar eviations over 5 moel samples at four receivers, which are numbere from the top of the string. Wellhea is at center (white ot). Figure 2 shows in map view the traveltime variability of the caniate shot carpet for a selection of geophones. The color at any position signifies the variability of the traveltime (over the ensemble of 5 moels) for a caniate shot locate there with respect to the note receiver. Each receiver exhibits a crue annulus of high variability that increases with geophone epth. Notice the raial fingers an ring-like features that appear, isappear, or persist at ifferent receiver epths. These are probably ue to moel structures an anisotropy. Most notably, the greatest traveltime variability occurs in the northwest quarant, associate with upper an mile receivers, an in the southwest quarant, for eeper receivers. Figure 3 Spatial correlation of traveltimes over 5 moel samples for a shot (black ot) (a) in the northwest an a shot (b) WNW of the first. Wellhea is at center (white ot). shot positione roughly 1 km WNW of the shot in Figure 3a. Notice the reuction in correlation; the shallow receivers show little or no spatial correlation at all, an while the eeper receivers exhibit a larger area of correlation, the correlation magnitue is less than in Figure 3a. SEG Las Vegas 212 Annual Meeting Page 3

4 The D N -criterion woul, therefore, favor the shot in Figure 3b over that in Figure 3a because it has low spatial correlation an high traveltime variability (Figure 2). Figure 4 shows a D N -optimal survey of 15 shots. The optimization was carrie out with respect to the complete borehole string because the whole string woul be use to observe any one shot in a real situation. Thus, 6 = 4 (receivers) x 15 (shots) shot-receiver pairs were actually optimize. The optimize shots roughly occupy an annulus between 6 an 9 km raial istance from the wellhea, which was alreay anticipate in our iscussion of Figure 2. This suggests that shots closer than 6 km offset are relatively uninformative. One explanation for this is that the earth moel parameters sensitive to shots within this region have alreay been constraine by the WAZ surface seismic ata. The most informative regions are now associate with long offsets an non-vertical incience. The clustering of optimal shots in Figure 4, particularly in the northwest quarant, was also anticipate in Figure 2 an Figure 3, where coincient regions of high expecte ata variability an low ata correlation were seen. that the most informative ata are collecte while minimizing acquisition costs. In Figure 4 we can also ientify a minimum raius for spiral 3D VSP. Shots closer than 6 km to the wellhea are uninformative an can be isregare for tomography. We may use this fact to prescribe an annular-spiral 3D VSP that covers only the annular region between 6 an 9 km from the wellhea. By foregoing offsets less than 6 km, the acquisition time can be reuce by more than 4% compare with a complete spiral from the wellhea to 9 km. D N -optimization coul also be use to esign a presurvey acquisition geometry in which the expecte measurement noise woul be optimally characterize to ensure maximum ata quality uring actual ata acquisition. In this example, the geometry coul be a Z-survey, a walkaway-azimuthal VSP in which the top bar of the Z woul be an azimuthal segment through the hot region in the northwest, the iagonal part woul be a walkaway traversing the wellhea, an the bottom bar (which might be reverse in this case) woul be a secon azimuthal segment through the hot region in the south. Such a geometry woul ensure that the most informative ata are acquire, ensuring goo coverage for the characterization of noise an ata quality. Another use for D N -optimization woul be to prouce realtime information maps for steering an acquisition vessel or to place it in the vicinity where maximally-informative shots are expecte to occur. Such an approach coul be use, for example, to ientify far-offset checkshot positions to optimally constrain look-ahea moels in real-time rilling. D N -optimization coul also be use for rapi post-acquisition quality control. The most informative shots occur in the aforesai annulus, an they ientify the best ata to be inverte to rener a maximally-informative quick-look at the moel, which woul be emerging as ata are acquire. Figure 4: D N -optimal borehole seismic survey of 15 shots (black ots). Colore backgroun is the D N -value (equation 6) of a prospective 151 st shot (warm colors have greater D N -value), which conveys the relative information content of various regions of the caniate shot carpet, given the current experiment. Wellhea is at center (white ot). Discussion an conclusion Base on the result in Figure 4, we can ientify a maximum raius for a spiral 3D VSP, which woul be aroun 9 km from the wellhea. The ability to systematically recommen a maximum raius for spiral VSP is useful because it ensures One last iea woul be to use D N -optimization to ecimate an existing ata set that is too large to be practically analyze in toto or that nees to be rapily analyze for ecision-making purposes. The iea woul be to systematically fin a suitably small subset of the ata that maximally constrains the earth moel or any targete subset thereof. This has broa potential application, incluing for tomography, least-squares migration, full-waveform inversion, an reverse-time migration for local seismic imaging, to name a few. Apart from the several notable contributions above, D N - optimization can hanle inustrial-scale nonlinear esign problems. Though a moest boy of work exists on nonlinear esign, much remains to be one to make it practical for large problems. The ability to probabilistically optimize experiments for the nonlinear case is essential because posterior moel istributions are complicate by nonlinearity an D N -optimization accomplishes this while still being computationally feasible for real-worl problems. SEG Las Vegas 212 Annual Meeting Page 4

5 EDITED REFERENCES Note: This reference list is a copy-eite version of the reference list submitte by the author. Reference lists for the 212 SEG Technical Program Expane Abstracts have been copy eite so that references provie with the online metaata for each paper will achieve a high egree of linking to cite sources that appear on the Web. REFERENCES Atkinson, A. C., A. N. Donev, an R. D. Tobias, 27, Optimum experimental esigns, with SAS: Oxfor University Press. Bratvol, R., J. E. Bickel, an H. P. Lohne, 29, Value of information in the oil an gas inustry: past, present, an future: SPE Reservoir Evaluation & Engineering, 12, Coles, D., 211, Generalizing the DN-criterion for nonlinear survey esign: 81st Annual International Meeting, SEG, Expane Abstracts, Coles, D., an A. Curtis, 211a, A free lunch in linearize experimental esign?: Computers & Geoscience, 37, Coles, D., an A. Curtis, 211b, Efficient nonli near Bayesian survey esign by D N -optimization: Geophysics, 76, no. 2, Q1 Q8. Coles, D., an M. Prange, 212, Towar efficient computation of the expecte relative entropy in nonlinear experimental esign: Inverse Problem (in press). Feorov, V.V., 1972, Theory of optimal experiments: Acaemic Press. Guest, T., an A. Curtis, 29, Iteratively constructive sequential esign of experiments an surveys with nonlinear parameter-ata relationships: Journal of Geophysical Research, 114, B437. Guest, T., an A. Curtis, 211, On stanar an optimal esigns of inustrial-scale 2-D seismic surveys: Geophysical Journal International, 186, Guest, T., an A. Curtis, 21, Optimal trace selection for AVA processing of shale -san reservoirs: Geophysics, 75, no. 4, C37 C47. Haber, E., L. Horesh, an L. Tenorio, 28, Numerical methos for experimental esign of large -scale linear ill-pose inverse problems: Inverse Problems, 24, Osypov, K, D. Nicho ls, Y. Yang, F. Qiao, M. O'Bria n, an O. Zraveva, 21, Q uantifying structural uncertainty in anisotropic epth imaging Gulf of Mexico case stuy: 72n Conference & Exhibition, EAGE, Extene Abstracts, Osypov, K, D. Nichols, M. Woowar, O. Zraveva, F. Qiao, E. Yarman, Y. Yang, Y. Liu, an N. Ivanova, 211, From quantifying seismic uncertainty to assessing E & P risks an the value of information: 81st Annual International Meeting, SEG, Expane Abstracts, Winterfors, E., an A. Curtis, 28, Numerical etection an reuction of non -uniqueness in nonlinear inverse problems: Inverse Problems, 24, SEG Las Vegas 212 Annual Meeting Page 5