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Azimuthal anisotropy in microseismic monitoring: Part 1 – Theory Vladimir Grechka (Marathon Oil Company) Sergey Yaskevich (Institute of Petroleum Geology and Geophysics SB RAS) Summary Modern downhole microseismic surveys employ geometries in which ray trajectories generated by locatable events provide full azimuthal and polar coverage, making it possible to estimate in-situ seismic anisotropy. Traveltimes and particle motions of the direct P- and shear-waves acquired in such geometries can constrain stiffness tensors of triclinic media. While obtaining all 21 stiffness coefficients of a homogeneous triclinic space simultaneously with locating pertinent microseismic events from records collected in a single vertical well appears relatively straightforward, the same methodology does not necessarily apply to layered formations because combination of their vertical heterogeneity and azimuthal anisotropy might invalidate the commonly adopted approximation of the event azimuths by those of the P-wave polarization vectors. When the event azimuths cannot be derived from the particle motions, traveltimes observed in two or more wells are required to locate the events and build layered triclinic or higher-symmetry azimuthally anisotropic velocity models. The multiwell event-location methods are expected to perform better than their single-well counterparts because they rely solely on triangulation and eliminate the usually pronounced azimuthal event-location uncertainties that stem from noises adversely affecting the results of hodogram analysis.

Introduction Exploitation of unconventional oil and gas reservoirs necessitates their hydraulic fracturing as a means of enhancing natural permeability of tight formations to levels that make the production of hydrocarbons economically meaningful. Hydraulic stimulations are often monitored by three-component geophones placed in one or several boreholes adjacent to the hydraulically treated wells. These geophones record weak earthquakes, termed the microseismic events, that help assess the geometries of hydraulic fractures and evaluate the treatment efficiency. Primary information for such an assessment comes from the locations of microseismic events associated with a particular stage of treatment. To locate events accurately, a good velocity model is needed. Here we follow the ideas of Grechka et al. (2011), who demonstrated that such velocity models can be built simultaneously with locating the microseismicity. When full-aperture geometries are available, quantifying triclinic anisotropy becomes feasible. We begin our theoretical study with inversion of 21 stiffnesses characterizing homogeneous triclinic media, the inversion that utilizes traveltime and polarization information supplied by microseismic events only (no perforation shots). Extension of our inversion to layered triclinic formations reveals

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that the presence of vertical heterogeneity and rather moderate azimuthal anisotropy might render the Pwave polarization azimuths unusable for representing the azimuths of microseismic events. If this happens, multi-well microseismic surveys could be a viable option for locating events simultaneously with building layered azimuthally anisotropic velocity models of the subsurface.

Notation and statement of inverse problem Similarly to Grechka and Duchkov (2011) and Grechka et al. (2011), our data are the traveltimes tQ of the direct P-, S1 -, and S2 -waves (Q = P, S1 , and S2 ) excited by Np perforation shots, whose spatial locations ξp ≡ {ξp,1 , ξp,2 , ξp,3 } (p = 1, . . . , Np ) are assumed to be known, and by Ne microseismic events, whose locations ξe ≡ {ξe,1 , ξe,2 , ξe,3 } (e = 1, . . . , Ne ) are sought. Since we aim at extracting information from the microseismic events themselves, perforation shots might be absent, making Np = 0 and array ξp empty. The traveltimes tQ are recorded by Ng geophones placed at xg ≡ {xg,1 , xg,2 , xg,3 } (g = 1, . . . , Ng ) in a single well or in several wells. Although in field applications, the size of the data vector d ≡ tQ (xg ) + noise (1) is usually smaller than 3 Ng (Np + Ne ) because the signal-to-noise ratio of either P, or S1 , or S2 mode, or any their combination might be too low at some geophones for the traveltimes of those modes to be pickable, in our theoretical study, we assume traveltimes tQ to be available at all xg . If a downhole microseismic survey is acquired with geophones located in a single vertical well, additional data are required for estimating ξe . They are usually obtained from the hodogram analysis (e.g., Montalbetti and Kanasewich, 1970; Rutledge and Phillips, 2003) of the P-waves, whose polarization vectors U P,e yield the azimuths βU ≡ arctan(UP,e,2 /UP,e,1 ) of the particle motion conventionally used to approximate the event azimuths βe : βe ≈ βU .

(2)

When the quality of approximation 2 is satisfactory, the number of unknown coordinates of ξe reduces from three to two: ξe ≡ {ζe,1 cos βe , ζe,1 sin βe , ζe,2 } ≈ {ζe,1 cos βU , ζe,1 sin βU , ζe,2 } , (e = 1, . . . , Ne ) ,

(3)

where ζe,1 and ζe,2 are the radial distance of a microseismic event from a vertical observation well and the

DOI http://dx.doi.org/10.1190/segam2013-0018.1 Page 1987

Main Menu Azimuthal anisotropy in microseismic monitoring: Part 1 – Theory Table 1: Tsvankin’s (1997) parameters of layers used in our modeling. The quantities α(1) and α(2) denote the dip of the crystallographic symmetry plane [x1 , x2 ] and the azimuth of the crystallographic plane [x1 , x3 ] measured from east to north. Layer number

VP0 (km/s)

VS0 (km/s)

ϵ(1)

ϵ(2)

δ (1)

δ (2)

δ (3)

γ (1)

γ (2)

α(1) (◦ )

α(2) (◦ )

1

3.5

2.0

0.20

-0.05

0.00

-0.20

-0.05

2

2.5

1.5

0.10

-0.15

0.05

-0.15

0.00

0.20

0.05

20

10

0.10

-0.05

20

3

3.0

1.8

0.20

-0.05

0.25

0.05

0.20

40

0.15

0.00

20

70

(a)

event depth, respectively. Alternatively, when approximation 2 is unacceptable, ξe cannot be replaced with ζ e , requiring records of microseismic events in more than one well in order to locate the events.

400

300

200

In addition to ξe or ζ e , the vector of unknowns [ ] m ≡ c [l] , {ξe or ζ e }, d [l] , τn (4) North (m)

100

of our problem includes the density-normalized stiffness tensors c [l] of triclinic layers (l = 1, . . . , Nl ), the depths of horizontal interfaces d [l] (l = 1, . . . , Nl−1 ) separating those layers, and the origin times τn (n = 1, . . . , Np + Ne ) of the perforations shots and events.

0

−100

−200

Our goal is to find the conditions under which vector m (equation 4) is uniquely invertible from data d (equation 1).

−300

−400

−400

−300

−200

−100

Homogeneous media

0 East (m)

100

200

300

400

100

200

300

400

(b)

We begin our investigation with the event-location problem in a homogeneous triclinic space in the absence of perforation shots (Np = 0). Figure 1 displays our geometry in which linear patterns of events oriented north-south (Figure 1a) are meant to correspond to hydraulic fractures extended from the treatment wells drilled in the east-west direction. The rays between the events and receivers in our geometry create a dense, full-aperture directional coverage. We choose the parameters of layer 2 in Table 1 for our modeling. Although its symmetry is orthorhombic rather than triclinic, nonzero dip α(1) of the crystallographic symmetry plane [x1 , x2 ] and nonzero azimuth α(2) of the crystallographic plane [x1 , x3 ] yield nonzero components cIJ (I, J = 1, . . . , 6) of tensor c and make c appear as if its symmetry were triclinic. As we aim at estimating all 21 stiffness elements, we will not be using our knowledge of the symmetry of c. The maximum difference between the azimuths of rays and the P-wave polarization vectors in our model is 3.2◦ , its mean value is 1.3◦ . Such small differences, well in line with the findings of Crampin et al. (1982), ensure the applicability of approximation 2 and allow us to search for vector m (equation 4) that depends on two-component event coordinates ζ e . Frech´ et-derivative matrix and its condition number The size of vector of unknowns m for our problem is 441, including 21 stiffness elements, 2 × 140 = 280 components of event-location vectors ζ e , and 140 origin times of the events. The question of primary importance is whether unambiguous estimation of m

0

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Depth (m)

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600

−400

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0 East (m)

Fig. 1: (a) Plan and (b) depth views of positions of microseismic events (dots) and receivers (triangles) used for simultaneous inversion of event locations and elastic stiffness tensor.

(equation 4) from d (equation 1) is possible without relying on the shear-wave splitting, which may or may not be observed in field data (e.g., Gajewski et al., 2009; Grechka et al., 2011). We answer this question by examining the singular value decomposition (SVD) of the Frechét-derivative matrix (Grechka and Duchkov, 2011) F ≡

∂tQ ∂m

(5)

computed at the true model m. Figure 2, showing the singular values s of F computed for the direct P- and S1 -waves in the geometry displayed in Figure 1, is our main result that proves the feasibility of building triclinic velocity models simultaneously with locating microseismic events. Logarithm of the

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Main Menu Azimuthal anisotropy in microseismic monitoring: Part 1 – Theory (a)

0 400

−0.4

300

−0.6

200

−0.8 100

−1

North (m)

Log 10 of normalized singular value

−0.2

−1.2 −1.4

0

−100

−1.6 −200

−1.8 −300

−2 −2.2 0

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50

100

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200 250 300 Parameter combination

350

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450

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0 East (m)

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(b)

Fig. 2: Singular values of matrix 5 for geometry shown in Figure 1. The SVD is performed for the P-, S1 -waves.

0

100

condition number of F max s , min s

200

(6)

which is equal to 2.21 (Figure 2), indicates that the problem at hand is better posed than those analyzed in Grechka and Duchkov (2011) and Grechka et al. (2011) for their narrow-angle microseismic geometries. Replacing the pair of direct waves {P, S1 } with {P, S2 } yields κ = 2.37, again implying a well-posed inverse problem. Interestingly, using both shearwaves S1 and S2 in the inversion improves its properties only slightly, resulting in κ = 2.06.

Depth (m)

κ ≡ log10

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500

600

−400

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0 East (m)

Fig. 3: (a) Plan and (b) depth views of event locations (open circles) obtained from 50 realizations of noisecontaminated data.

main source of the event-location errors in our geometry.

Inversion Having demonstrated the feasibility of the inversion, we now estimate 441 unknowns discussed in the previous section from traveltimes of the direct P-, S1 waves by minimizing the least-squares objective function ]2 ∑ [ comp F (m) ≡ tQ (xg , m) − tdata , (7) Q (xg )

Despite significant azimuthal errors in the event locations, all 21 stiffness coefficients are tightly constrained by the data. The error in the obtained stiffness tensor, defined as ∆c ≡

e, g, Q

in which the computed βU and tdata are contaminated Q with Gaussian noise that has the zero mean and the standard deviations of 5◦ and 1 ms, respectively. The event locations for 50 realizations of noise are displayed in Figure 3. The plan view exhibits approximately circular patterns related to errors in the event azimuths (open circles in Figure 3a). Naturally, azimuthal errors cover greater distances as events move away from the observer (triangle in Figure 3a), yielding the mean horizontal event-location error of 19.5 m. The behavior of depth errors in event locations is different. While clouds in Figure 3b tend to point to the middle of the receiver array, they grow little with the distance of events from the recording well and stay close to the horizontal, making comp true mean|ξe,3 − ξe,3 | to be equal to 2.4 m. The difference in magnitudes between the horizontal error (19.5 m) and the depth error (2.4 m) clearly identifies uncertainties in the hodogram analysis as the

||ccomp − ctrue || × 100% , ||ctrue ||

(8)

is equal to ∆c = 1.9%.

Layered media The success of inversion discussed in the previous section suggests the possibility of parameter estimation in realistic models containing triclinic layers. While such an extension appears quite plausible because the growth of the model space due to the presence of several layers and interfaces can be mitigated by including information from perforation shots, something we have avoided so far, a combined influence of vertical heterogeneity and azimuthal anisotropy in our model (Table 1) causes the differences between azimuths βU and βe to exceed 20◦ and renders the resulting event locations too inaccurate for practical applications. For this reason, we examine the event-location problem in a dual-well geometry. Figure 4 depicts its layout, in which the horizontal treatment well located at north = −150 m contains four perforation shots and

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before, influences the estimates. The obtained event locations (open circles in Figure 4) reveal superior accuracy of our dual-well recording geometry compared to the single-well one. Perhaps the most significant improvement is the absence of azimuthal swings in the horizontal plane (compare Figures 3a and 4a). As a consequence of eliminating the source of the largest uncertainty, the event locations obtained from noise-contaminated data overlay their true locations except for a few outliers, implying the virtual absence of difference comp in errors of different components of vectors ξe,i : comp true mean|ξe,i − ξe,i | ≈ 1 m (i = 1, 2, 3). Those small errors suggest that uncertainties in event locations obtained from field data recorded in similar geometries are expected to be dominated by the bandlimited nature of data or unaccounted lateral velocity variations rather than by the absence of tight constraints provided by the traveltimes. There is little surprise that accurate event locations entail the high-quality estimates of the stiffnesses. Their errors defined by equation 8 are ∆c[l] = [2.3%, 1.0%, 3.2%]. The parameters of layer 2 are estimated more accurately than parameters of layers 1 and 3 because both the perforation shots and the horizontal array of receives are located in the second layer. The depths of the interfaces (horizontal lines in Figure 4b) have been estimated too, both with the standard deviations of approximately 1 m.

(b)

Conclusions The findings of our study can be summarized as follows.

Fig. 4: (a) Plan and (b) depth views of positions of microseismic events (solid dots), receivers (triangles), and perforation shots (stars) used for simultaneous inversion of event locations and layered anisotropic model. Horizontal lines at depths of 250 m and 350 m in (b) are the model interfaces; the layer parameters are listed in Table 1. Open circles indicate the event locations obtained from 50 realizations of noise-contaminated traveltime data. Most of those locations overlap with the true ones (solid dots).

another treatment well at north = 150 m is turned into the second observer. This time, we avoid using the polarization information and invert the P- and S1 -wave traveltimes measured in both wells for 629 pertinent model parameters: 3 × 21 = 63 stiffness coefficients, 2 interface depths, 3 × 140 = 420 components of event-location vectors ξe , and 144 origin times τn of the events and perforation shots. The SVD analysis, yielding logarithm of the condition number κ = 3.28, indicates the feasibility of such an inversion. Although the above value of κ is greater than those cited earlier, perhaps because events are being located in 3D space rather than in 2D vertical planes specified by azimuths βU , errors in βU are no longer relevant for the inversion and only traveltime noise, which has the same standard deviation of 1 ms as

(1) Full-aperture microseismic data recorded in single-well geometries can be inverted for the stiffnesses of a homogeneous triclinic medium simultaneously with locating the events. Measurements of the shear-wave splitting, although certainly helpful, are not absolutely necessary because either pair of direct arrivals {P, S1 } or {P, S2 } suffices to constrain the inversion. (2) Since trajectories of rays propagating in layered azimuthally anisotropic media deviate from vertical planes, event azimuths derived from the P-wave particle motions might be significantly biased. According to our modeling, azimuthal anisotropy does not have to be strong to render the P-wave polarization azimuths unusable as the azimuths of microseismic events. (3) Microseismic surveys recorded in multiple wells might be the only, albeit relatively expensive, option for obtaining accurate event locations in heterogeneous azimuthally anisotropic media. If multi-well acquisition is chosen, it should be expected to lead to superior event-location precision compared to that achievable in single-well monitoring. (4) Building layered triclinic models for processing of downhole microseismic is feasible. Our companion paper (Grechka and Yaskevich, 2013) describes application of the presented methodology to field data.

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http://dx.doi.org/10.1190/segam2013-0018.1 EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2013 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web.

REFERENCES Crampin, S., R. A. Stephen, and R. McGonigle , 1982, The polarization of P-waves in anisotropic media: Geophysical Journal of the Royal Astronomical Society, 68, 477–485, http://dx.doi.org/10.1111/j.1365-246X.1982.tb04910.x. Gajewski, D., K. Sommer, C. Vanelle, and R. Patzig, 2009, Influence of models on seismic event localization: Geophysics, 74, no. 5, WB55–WB61, http://dx.doi.org/10.1190/1.3183941. Grechka, V., and A. A. Duchkov, 2011, Narrow-angle representations of the phase and group velocities and their applications in anisotropic velocity-model building for microseismic monitoring: Geophysics, 76, no. 6, WC127–WC140, http://dx.doi.org/10.1190/geo2010-0408.1. Grechka, V., P. Singh, and I. Das, 2011, Estimation of effective anisotropy simultaneously with locations of microseismic events: Geophysics, 76, no. 6, WC143–WC153, http://dx.doi.org/10.1190/geo20100409.1. Grechka, V., and S. Yaskevich, 2013, Azimuthal anisotropy in microseismic monitoring: Part 2 study: 83rd Annual International Meeting, SEG, Expanded Abstracts, this issue.

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Montalbetti, J. R., and E. R. Kanasewich, 1970, Enhancement of teleseismic body phases with a polarization filter: Geophysical Journal of the Royal Astronomical Society, 21, 119–129, http://dx.doi.org/10.1111/j.1365-246X.1970.tb01771.x. Rutledge, J. T., and W. S. Phillips, 2003, Hydraulic stimulation of natural fractures as revealed by induced microearthquakes, Carthage Cotton Valley gas field, east Texas: Geophysics, 68, 441–452, http://dx.doi.org/10.1190/1.1567214. Tsvankin, I., 1997, Anisotropic parameters and P-wave velocity for orthorhombic media: Geophysics, 62, 1292–1309, http://dx.doi.org/10. 1190/1.1444231.

