GEOPHYSICS, VOL. 82, NO. 3 (MAY-JUNE 2017); P. D171–D186, 12 FIGS., 1 TABLE. 10.1190/GEO2016-0480.1

Hydraulic diagnostics from Krauklis- resonance and tube-wave reflections

Chao Liang1, Ossian O’Reilly2, Eric M. Dunham3, and Dan Moos4

ABSTRACT are accounted for in the simulations. We then determine how tube within a wellbore can be used to excite Krauklis waves Fluid-filled fractures support guided waves known as Krauklis within fractures that are hydraulically connected to the wellbore. waves. The resonance of Krauklis waves within fractures occurs The simulations provide the frequency-dependent hydraulic at specific frequencies; these frequencies, and the associated at- impedance of the fracture, which can then be used in a fre- tenuation of the resonant modes, can be used to constrain the frac- quency-domain tube-wave code to model tube-wave reflection/ ture geometry. We use numerical simulations of wave propagation transmission from fractures from a source in the wellbore or at along fluid-filled fractures to quantify fracture resonance. The the wellhead (e.g., water hammer from an abrupt shut-in). Tube simulations involve solution of an approximation to the compress- waves at the resonance frequencies of the fracture can be selec- ible Navier-Stokes equation for the viscous fluid in the fracture tively amplified by proper tuning of the length of a sealed section coupled to the elastic-wave equation in the surrounding solid. Var- of the wellbore containing the fracture. The overall methodology iable fracture aperture, narrow viscous boundary layers near the presented here provides a framework for determining hydraulic fracture walls, and additional attenuation from seismic radiation fracture properties via interpretation of tube-wave data.

INTRODUCTION Due to their sensitivity to properties of the surrounding formation and fractures intersecting the wellbore, tube waves or water hammer is widely used to increase the permeability propagating along the well are widely used for formation evaluation of oil and gas reservoirs. This transformative technology has en- and fracture diagnostics (Paillet, 1980; Paillet and White, 1982; Holz- abled production of vast shale oil and gas resources. Although en- hausen and Gooch, 1985a, 1985b; Holzhausen and Egan, 1986; gineers now have optimal control of well trajectories, the geometry Hornby et al., 1989; Tang and Cheng, 1989; Paige et al., 1992, 1995; (length, height, and aperture) of hydraulic fractures is still poorly Kostek et al., 1998a, 1998b; Patzek and De, 2000; Henry et al., 2002; known, which poses a challenge for optimizing well completion. Ziatdinov et al., 2006; Ionov, 2007; Wang et al., 2008; Derov et al., Monitoring hydraulic fracture growth is desirable for many reasons. 2009; Mondal, 2010; Bakku et al., 2013; Carey et al., 2015; Livescu Tracking the length of fractures ensures that multiple subparallel et al., 2016). Here, we are specifically concerned with low-frequency fractures remain active, rather than shielding each other through tube waves having wavelengths much greater than the wellbore elastic interactions. It might also be used to prevent fractures in radius. In this limit, tube waves propagate with minimal dispersion nearby wells from intersecting one another. In addition, the ability atavelocityslightlylessthanthefluidsoundspeed(Biot, 1952). to measure fracture geometry would facilitate estimates of the Besides excitation from sources within the well or at the wellhead, stimulated volume and could be used to help guide the delivery tube waves can be generated by incident seismic waves from the sur- of proppant. rounding medium (Beydoun et al., 1985; Schoenberg, 1986; Ionov

Downloaded 08/02/17 to 171.64.171.164. Redistribution subject SEG license or copyright; see Terms of Use at http://library.seg.org/ Manuscript received by the Editor 9 September 2016; published online 21 March 2017. 1Stanford University, Department of Geophysics, Stanford, California, USA. E-mail: [email protected]. 2Formerly Baker Hughes, Palo Alto, California, USA; presently Stanford University, Department of Geophysics, Stanford, California, USA. E-mail: [email protected]. 3Stanford University, Department of Geophysics, Stanford, California, USA and Stanford University, Institute for Computational and Mathematical Engineer- ing, Stanford, California, USA. E-mail: [email protected]. 4Formerly Baker Hughes, Palo Alto, California, USA, presently . E-mail: [email protected]. © 2017 Society of Exploration Geophysicists. All rights reserved.

D171 D172 Liang et al.

and Maximov, 1996; Bakku et al., 2013) and by sources within hy- et al., 2014). Most studies of Krauklis waves have been analytical or draulic fractures (Krauklis and Krauklis, 1998; Ionov, 2007; Derov semianalytical, with a major focus on deriving dispersion relations et al., 2009). Furthermore, tube waves incident on a fracture in hy- for waves in infinitely long fluid layers of uniform width. A notable draulic connection to the wellbore will reflect and transmit from the exception is the work of Frehner and Schmalholz (2010) and Freh- fracture, with frequency-dependent reflection/transmission coeffi- ner (2013), in which finite elements in two dimensions were used to cients. As described in more detail below, these reflected and trans- study Krauklis waves in finite-length fractures with variable aper- mitted tube waves carry information about fracture geometry. ture. Using a numerical approach provides a means to investigate A closely related concept is that of hydraulic impedance testing the reflection and scattering of Krauklis waves at fracture tips (Freh- (Holzhausen and Gooch, 1985a), initially developed to estimate the ner and Schmalholz, 2010) and the excitation of Krauklis waves by geometry of axial fractures (fractures in the plane of the wellbore). incident seismic waves (Frehner, 2013). However, these studies did This method, using very low frequency tube waves or water-ham- not consider the interaction between the Krauklis and tube waves, mer signals, was validated through laboratory experiments (Paige which is a central focus of our study. et al., 1992) and applied to multiple field data sets (Holzhausen Mathieu and Toksoz (1984) were the first to pose the mathemati- and Egan, 1986; Paige et al., 1995; Patzek and De, 2000; Mondal, cal problem of tube-wave reflection/transmission across a fracture 2010; Carey et al., 2015). The method is typically presented using in terms of the fracture impedance. In this work, we define the frac- the well-known correspondence between hydraulics and electrical ture hydraulic impedance in the frequency domain as circuits, with hydraulic impedance defined as the ratio of pressure to p^ ðωÞ volumetric flow rate (velocity times the cross-sectional area). The ðωÞ¼ f Zf ðωÞ ; (1) fracture consists of a series connection of resistance (energy loss q^ f through viscous dissipation and seismic radiation), capacitance (en- ðωÞ ðωÞ ergy stored in elastic deformation of the solid and compression/ where p^ f and q^ f are the Fourier transforms of the pressure expansion of the fluid), and inertance (kinetic energy of the fluid), and volumetric flow rate into the fracture, both evaluated at the frac- which can be combined into a complex-valued hydraulic impedance ture mouth. The Fourier transform of some function gðtÞ is defined as of the fracture. The hydraulic impedance is then related to the geom- Z ∞ etry of the fractures (Holzhausen and Gooch, 1985a; Paige et al., g^ðωÞ¼ gðtÞeiωtdt: (2) 1992; Carey et al., 2015). However, this method, at least as pre- −∞ sented in the literature thus far, assumes a quasi-static fracture response when relating (spatially uniform) pressure in the fracture Mathieu and Toksoz (1984) treated the fracture as an infinite fluid to the fracture volume. This assumption is valid only for extremely layer or a permeable porous layer with the fluid flow being described

low frequencies, and it neglects valuable information at higher by Darcy’s law. The fracture hydraulic impedance built on Darcy’s frequencies in which there exists the possibility of a resonant frac- law was then replaced with analytic solutions based on the dispersion ture response associated with waves propagating within the fracture. relation for acoustic waves in an inviscid (Hornby et al., 1989)or The treatment of dissipation from fluid viscosity is similarly overly viscous (Tang and Cheng, 1989) fluid layer bounded by rigid fracture simplistic. walls. Later studies established that elasticity of the fracture wall Thus, a more rigorous treatment of the fracture response is re- significantly changes the reflection and transmission across a fracture quired, motivating a more nuanced description of the dynamics (Tang, 1990; Kostek et al., 1998a, 1998b). The arrival time and am- of fluid flow within the fracture and of its coupling to the surround- plitude of reflected tube waves can be used to infer the location and ing elastic medium over a broader range of frequencies. The key effective aperture of fractures (Medlin and Schmitt, 1994). However, concept here is a particular type of guided wave that propagates most of these models assume an infinite fracture length, as is well- along fluid-filled cracks. These waves, known as crack waves or justified (Hornby et al., 1989) for the high frequencies (approximately Krauklis waves, have been studied extensively in the context of the 1kHz)thatwerethefocusofthesestudies.Byfocusinginsteadon oil and gas industry, volcano seismology, and other fields (Krauklis, lower frequencies (< 100 Hz) and considering Krauklis waves re- 1962; Paillet and White, 1982; Chouet, 1986; Ferrazzini and Aki, flected from the fracture tip, Henry et al. (2002) and Henry (2005) 1987; Korneev, 2008, 2010; Yamamoto and Kawakatsu, 2008; argued that reflection/transmission of tube waves is affected by the Dunham and Ogden, 2012; Lipovsky and Dunham, 2015; Nikitin resonance of the fracture, and this consequently provides sensitivity et al., 2016). At the frequencies of interest here (approximately 1– to fracture length. 1000 Hz), they are anomalously dispersed waves of opening and One approach to account for the finite extent of the fracture is to closing that propagate along fractures at speeds approximately use the dispersion relation for harmonic Krauklis waves to deter- 10–1000 m∕s. Experimental studies have confirmed the existence mine the eigenmodes of a circular disk-shaped fracture of uniform and propagation characteristics of these waves (Tang and Cheng, aperture with a zero radial velocity boundary condition at the edge 1988; Nakagawa, 2013; Shih and Frehner, 2015; Nakagawa et al., of the disk (Hornby et al., 1989; Henry et al., 2002; Henry, 2005; 2016). Counterpropagating pairs of Krauklis waves form standing Ziatdinov et al., 2006; Derov et al., 2009). This treatment fails to waves (eigenmodes) of fluid-filled cracks, and the resonance fre- account for the decreased aperture near the fracture edge, which

Downloaded 08/02/17 to 171.64.171.164. Redistribution subject SEG license or copyright; see Terms of Use at http://library.seg.org/ quencies and decay rates of these modes are sensitive to the fracture decreases the Krauklis-wave phase velocity and increases the vis- length and aperture. Krauklis waves are thought to be responsible cous dissipation. It furthermore implicitly assumes perfect reflec- for the long period and very long period seismicity at active volca- tion from the fracture edge, thereby neglecting attenuation from noes (Aki et al., 1977; Chouet, 1988), and they have been suggested seismic-wave radiation. Recent studies have established the impor- as a possible explanation for harmonic seismic signals recorded dur- tance of this attenuation mechanism (Frehner and Schmalholz, ing hydraulic fracturing treatments (Ferrazzini and Aki, 1987; Tary 2010; Frehner, 2013). Diagnostics with Krauklis and tube waves D173

In this work, we examine how Krauklis waves propagating in The fluid-filled fracture system is described by the elastic-wave fluid-filled fractures, as well as tube-wave interactions with fractures, equation, governing displacements of the solid, and the compress- can be used to infer fracture geometry. Figure 1 shows the system. ible Navier-Stokes equation for the viscous fluid in the fracture. In The overall problem is to determine the response of the coupled this work, we use a linearized, approximate version of the Navier- fracture-wellbore system to excitation at the wellhead, within the Stokes equation that retains only the minimum set of terms required wellbore, or at the fracture mouth. Other excitation mechanisms, to properly capture the low-frequency response of the fluid (Lipov- including sources in the fracture or incident seismic waves from sky and Dunham, 2015). By low frequency, we mean ωw0∕c0 ≪ 1, ω some external source in the solid (e.g., microseismic events or active where is the angular frequency, w0 is the crack aperture or width sources), are not considered here. The fracture and wellbore are (the two terms are used interchangeably hereafter), and c0 is the coupled in several ways, the most important of which is through 3 fluid sound speed. For w0 ∼ 1mmand c0 ∼ 10 m∕s, frequencies the direct fluid contact at the fracture mouth. Pressure changes at this must be smaller than approximately 1 MHz. This is hardly a restric- junction, for instance, due to tube waves propagating along the well- tion because fracture resonance frequencies are typically well less bore, will excite Krauklis waves in the fracture. Similarly, fluid can be than approximately 100 Hz. The derivation of the governing equa- exchanged between the wellbore and fracture. Other interactions, tions and details of the numerical treatment (using a provably stable, such as through elastic deformation of the solid surrounding the well- high-order-accurate finite-difference discretization) are discussed bore and fracture, are neglected in our modeling approach. by Lipovsky and Dunham (2015) and O. O’Reilly et al. (personal By making this approximation, and by further assuming that all communication, 2017). Here, we explain the geometry of our sim- perturbations are sufficiently small so as to justify linearization, the ulations and then briefly review the fluid governing equations response of the coupled fracture-wellbore system can be obtained in within the fracture to facilitate the later discussion of Krauklis two steps (Mathieu and Toksoz, 1984; Hornby et al., 1989; Kostek waves and the fracture transfer function. et al., 1998a; Henry, 2005). In the first step, we determine the re- Although a solution to the 3D problem is required to quantify sponse of the fracture, in isolation from the wellbore, to excitation at how the fracture transfer function depends on the fracture length, the fracture mouth. Specifically, we calculate the (frequency-depen- height, and width, in this preliminary study, we instead use a 2D dent) hydraulic impedance of the fracture, as defined in equation 1, plane strain model (effectively assuming an infinite fracture height). using high-resolution finite-difference simulations of the dynamic We anticipate that this 2D model will provide a reasonable de- response of a fluid-filled fracture embedded in a deformable elastic scription of axisymmetric fractures, such as the one illustrated in medium. This provides a rigorous treatment of viscous dissipation and seismic radiation along finite-length fractures with possibly Figure 1, although some differences should be expected from the complex geometries, including variable aperture. The numerical differing nature of plane waves and axisymmetric waves. However, solutions are compared with semianalytic solutions based on disper- the procedure for using the fracture transfer function, obtained from

sion relations for harmonic waves propagating along an infinitely numerical simulations, to solve the coupled wellbore-fracture prob- long fluid layer of uniform width. In the second step, we solve the lem, is completely general, as are the overall qualitative results con- tube-wave problem in the frequency domain, in which the fracture cerning matched resonance that are discussed in the context of tube- response is captured through the fracture hydraulic impedance, and wave interactions with fractures. Finally, when the fracture transfer then it is converted to the time domain by inverting the Fourier functions from 3D simulations are available, the coupling solution transforms. The fracture response, as embodied by the fracture hy- procedure can be used with no additional modifications. draulic impedance, features multiple resonance peaks associated with the eigenmodes of the fracture. The frequencies and attenua- Source input tion properties of these modes are sensitive to the fracture length at wellhead wellbore and aperture. These resonances furthermore make the tube-wave reflection/transmission coefficients dependent on frequency, with the maximum reflection at the resonance frequencies of the fracture. z Linear elastic solid We close by presenting synthetic pressure seismograms for tube Incident waves within the wellbore, along with a demonstration of the sen- tube wave tube wave sitivity of these signals to fracture geometry. Seismic Nonplanar radiation fracture geometry Krauklis wave Viscous dissipation FLUID-FILLED FRACTURES Fluid governing equations and numerical simulations of fluid-filled fractures Compressible The first step in the solution procedure outlined above is to de- Transmitted tube wave termine the hydraulic impedance of the fracture. In fact, we find it tube wave more convenient to work with a nondimensional quantity propor- Sealed bottom Downloaded 08/02/17 to 171.64.171.164. Redistribution subject SEG license or copyright; see Terms of Use at http://library.seg.org/ tional to the reciprocal of the fracture hydraulic impedance, which we refer to as the fracture transfer function FðωÞ. Although the frac- ture hydraulic impedance diverges in the low-frequency limit, the Figure 1. Tube waves, excited at the wellhead, are incident on a fluid-filled fracture intersecting the wellbore. Pressure changes and fracture transfer function goes to zero in a manner that captures the fluid-mass exchange between the wellbore and fracture excite Krau- quasi-static fracture response used in the original work on hydraulic klis waves within the fracture, leading to partial reflection of the impedance testing. tube waves and dissipation of energy. D174 Liang et al.

Returning to the 2D plane strain problem, let x be the distance wðx; tÞ¼wþðx; tÞ − w−ðx; tÞ: (5) along the fracture and y be the distance perpendicular to the frac- ture; the origin is placed at the fracture mouth, where we will later Note that w0 refers to the full width, whereas some previous work couple the fracture to a wellbore. We use an approximation similar (Lipovsky and Dunham, 2015) used this to refer to the half-width. in many respects to the widely used lubrication approximation for Combining the linearized fluid mass balance with a linearized equa- thin viscous layers (Batchelor, 2000), although we retain terms de- tion of state, we obtain scribing fluid inertia and compressibility. In the low-frequency limit ω ∕ ≪ 1 ∂p 1 ∂w 1 ∂ðuw Þ ( w0 c0 1), the y-momentum balance establishes the uniformity þ ¼ − 0 ∂ ∂ ∂ ; (6) of pressure across the width of the fracture (Ferrazzini and Aki, K t w0 t w0 x 1987; Korneev, 2008; Lipovsky and Dunham, 2015). The linearized x-momentum balance is where 2 ∂v ∂p ∂ v Z þ w ðxÞ ρ þ ¼ μ ; (3) 1 0 ∂t ∂x ∂y2 uðx; tÞ¼ vðx; y; tÞdy (7) ð Þ − w0 x w ðxÞ where vðx; y; tÞ is the particle velocity in the x-direction, pðx; tÞ is the 0 pressure, and ρ and μ are the fluid density and dynamic viscosity, is the x-velocity averaged over the fracture widthpffiffiffiffiffiffiffiffiffi and K is the fluid respectively. We have retained only the viscous term corresponding bulk modulus. The fluid sound speed is c ¼ K∕ρ. Accumulation/ ¼ 0 to shear along planes parallel to y 0; scaling arguments show that loss of fluid mass at some location in the fracture can be accommo- other viscous terms are negligible in comparison for this class of dated by either compressing/expanding the fluid (the first term on the problems (Lipovsky and Dunham, 2015). The initial and perturbed left side of equation 6) or opening/closing the fracture walls (the sec- fracture widths are defined as ond term on the left side of equation 6). The latter is the dominant ð Þ¼ þð Þ − þð Þ process at the low frequencies of interest in this study. w0 x w0 x w0 x ; (4) Coupling between the fluid and solid requires balancing tractions and enforcing continuity of normal and tangential particle velocity a) time = 39.59 ms 0.6 on the fracture walls (i.e., the kinematic and no-slip conditions). At the fracture mouth, pressure is prescribed; zero velocity is prescribed 0.45 seismic radiation at the fracture tip. At the outer boundaries of the solid domain, ab- 0.3 sorbing boundary conditions are used to suppress artificial reflec- fracture mouth fracture tip 0.15 tions. The computational domain in both directions is 12 times the length of the fracture so as to fully capture the quasi-static displace- 0 ments in the solid and to further minimize boundary reflections. –0.15 ð Þ Spatial variations in the fracture width are captured through w0 x . –0.3 A uniformly pressurized fracturepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in an infinite medium has an ellip- 10 m ð Þ ∝ 2 − 2 tical opening profile, w0 x L x , and a singularity at

vertical solid velocity (mm/s) –0.45 ð Þ the fracture tips. Here, we use a related solution for w0 x in which 4 mm –0.6 (vertically exaggerated) closure at the tip occurs in a more gradual manner so as to remove horizontal fluid velocity (cm/s) that singularity. In particular, we use expressions for w0ðxÞ from a -8 -6 -4 -2 02468 cohesive zone model (Chen and Knopoff, 1986), appropriately modi- 0.6 fied from antiplane shear to plane strain by replacement of the shear time = 48.57 ms b) modulus G with G ¼ G∕ð1 − νÞ,whereν is Poisson’s ratio. The dispersive Krauklis waves 0.45 cohesive zone region is approximately 25% of the fracture length. 0.3 In addition, we blunt the fracture tip so that it has a finite width (usu- 0.15 ally a small fraction of the maximum width at the fracture mouth, unless otherwise indicated). 10 m 0 Figure 2 shows snapshots of the numerical simulation of Krauklis –0.15 waves propagating along a fluid-filled fracture. For this example 4 mm –0.3 (vertically exaggerated) only, we add to w0ðxÞ a band-limited self-similar roughness, horizontal fluid velocity (cm/s)

vertical solid velocity (mm/s) –0.45 as in Dunham et al. (2011) with an amplitude-to-wavelength ratio of −2 –8 –6 –4 –2 02468 –0.6 10 , similar to what is observed for natural fracture surfaces and faults (Power and Tullis, 1991; Candela et al., 2012). Material prop- Figure 2. Snapshots from simulation of Krauklis and elastic waves erties used in this and other simulations are given in Table 1. The excited by an imposed pressure chirp at the fracture mouth, for a 10 m fracture opens in regions of converging fluid flow, and it contracts long fracture with a 4 mm width at the fracture mouth. The back- ground shows the solid response (to scale), and the inset shows the where the flow diverges. Krauklis waves with different wavelengths

Downloaded 08/02/17 to 171.64.171.164. Redistribution subject SEG license or copyright; see Terms of Use at http://library.seg.org/ fluid response (vertically exaggerated). Colors in the solid show the separate as they propagate along the fracture due to dispersion. particle velocity in the direction normal to the fracture walls; discon- Krauklis waves arise from the combined effects of fluid inertia and tinuities across the fracture indicate opening/closing motions charac- the restoring force from fracture wall elasticity. Because the elastic teristic of Krauklis waves. Colors in the fluid show the particle velocity; note the narrow viscous boundary layers near the nonplanar wall is more compliant at longer wavelengths, all else being equal, fracture walls. The multimedia version of Figure 2 can be accessed long-wavelength waves experience smaller restoring forces and through this link: s1.mp4. hence propagate more slowly. Elastic waves in the solid are excited Diagnostics with Krauklis and tube waves D175

at the fracture mouth and along the fracture due to inertia of the ω Kc f ¼ el ¼ 0 ; (10) solid during Krauklis-wave reflection from the fracture tip and frac- el π π 2 G w0 ture mouth. Note that elastic waves propagate nearly an order of magnitude faster than Krauklis waves; this is evident in Figure 2a. the elastic wall deformation becomes appreciable. This additional As seen in the Figure 2 insets, viscous effects are primarily re- compliance leads to reduced phase velocity, given approximately as stricted to narrow boundary layers near the fracture walls and there (Krauklis, 1962) can even be flow reversals and nonmonotonic velocity profiles. The   Gw ω 1∕3 decreased width near the fracture tip decelerates the Krauklis waves ≈ 0 c ρ : (11) and enhances the viscous dissipation in the near-tip region. After 2 being reflected multiple times, pairs of counterpropagating Krauklis waves form standing waves along the fracture, which set the fluid- Lower frequency waves propagate slower than higher frequency filled fracture into resonance. Shorter period modes are damped out waves because the elastic walls are more compliant at longer wave- first, by the viscous dissipation and seismic radiation, leaving long- lengths. Viscous dissipation is confined to thin boundary layers period modes with wavelengths of the order of the fracture length. around the fracture walls. The frequency and decay rate, or attenuation, of these resonant At even lower frequencies, below modes are captured by the location and width of spectral peaks ω 2μ in the fracture transfer function, as we demonstrate shortly. f ¼ vis ¼ ; (12) vis π πρ 2 2 w0 Krauklis waves Before discussing the fracture transfer function, we review some Table 1. Material properties. key properties of Krauklis waves. This is most easily done in the context of an infinitely long fracture or fluid layer (with the constant initial width w0) between identical elastic half-spaces. Seeking Solid (rock) iðkx−ωtÞ 3 e solutions to the governing equations, and neglecting inertia Density ρs (kg∕m ) 2489 of the solid (which is negligible for Krauklis waves at the low P-wave speed cp (m∕s) 4367 frequencies of interest to us), leads to the dispersion relation (Lip- S-wave speed c (m∕s) 2646 ovsky and Dunham, 2015) s    Fluid (water) ξ 2 ρ ∕ 3 tan c0k 2K Density 0 (kg m ) 1000

ð ωÞ¼ − þ þ ¼ DK k; ξ 1 ω 1 0; (8) ∕ G kw0 Sound wave speed c0 (m s) 1500 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Viscosity μ (Pa:s) 0.001 ξ ¼ − 2ω∕ ν ν ¼ μ∕ρ where iw0 4 , is the fluid kinematic viscosity, k is the wavenumber, and ω is the angular fre-

quency. Using the full linearized Navier-Stokes 3 a) c) 10 equation for the fluid and retaining inertia in 103 10 mm the solid give rises to a more complex dispersion 3 mm Cutoff Limit 2 1 mm 10 2 relation (Ferrazzini and Aki, 1987; Korneev, 10 0.5 mm 10 mm 2008) that has fundamental and higher mode sol- 1 10 1 mm fvis utions. In contrast, our approximate fluid model 3 mm 0.5 mm 1 fel 10 only captures the fundamental mode. However, 100 Phase velocity (m/s)

the higher modes exist only above specific cutoff Phase velocity (m/s) –1 frequencies that are well outside the frequency 10 100 10–3 10–210–1 100 101 102 103 104 105 106 range of interest. Here, we first discuss solutions 1 2 5 10 20 50 200 1000 3000 b) Frequency (Hz) d) Wavelength (m) to equation 8 for real ω and complex k. The spatial 2 10 0 10 Q

attenuation of the modes is quantified by Q –1 1 10 0.5 mm 10 1 mm 1 Im k 3 mm ¼ ; (9) 10–2 10 mm 100 2Q Re k 10–3 10–1 0.5 mm f 1 mm where Q is the quality factor, approximately the vis 3 mm 10–4 10–2 number of spatial oscillations required for an ap- fel 10 mm

Spatial Attenuation 1/2 –5 –3

10 Temporal attenuation 1/2 10 preciable decay of the amplitude. Plots of the phase 10–3 10–210–1 100 101 102 103 104 105 106 1 2 5 10 20 50 200 1000 3000 velocity and attenuation are presented in Figure 3. Frequency (Hz) Wavelength (m)

Downloaded 08/02/17 to 171.64.171.164. Redistribution subject SEG license or copyright; see Terms of Use at http://library.seg.org/ At high frequency (but still sufficiently low frequency as to justify the ωw0∕c0 ≪ 1 approxi- Figure 3. (a) Phase velocity and (b) spatial attenuation of Krauklis waves for a real fre- mation), the phase velocity approaches the fluid quency. (c) Phase velocity and (d) temporal attenuation for a real wavelength. Dispersion and attenuation curves are plotted for four fracture widths: 0.5, 1, 3, and 10 mm, as labeled. sound speed because the fracture walls are effec- The dashed black and red lines in (a and b) mark the characteristic frequencies f and f tively rigid relative to the compressibility of the vis el defined in the text. Dashed lines in (c and d) mark the cutoff wavelength λc,beyondwhich fluid. At frequencies below waves cease to propagate. D176 Liang et al.

viscous effects are felt across the entire width of the fluid layer and Fracture transfer function the velocity v approaches the well-known parabolic Poiseuille flow profile. Viscous dissipation is sufficiently severe as to damp out Now, we turn our attention to finite-length fractures. Our objec- tive is to quantify the response of the fracture to forcing at the frac- waves over only a few cycles of oscillation; in addition, phase ture mouth, where the fracture connects to the wellbore. This is velocity decreases below that given in equation 11. done through the dimensionless fracture transfer function FðωÞ that An important consequence of dissipation, from the viscous ef- relates pressure pð0;tÞ and width-averaged velocity uð0;tÞ at the fects and from elastic-wave radiation, is that Krauklis-wave reso- fracture mouth: nance will only occur in sufficiently short fractures. We can gain deeper insight by plotting the phase velocity and attenuation of ðωÞ ð ωÞ¼F ð ωÞ Krauklis waves for real wavelength λ and complex frequency in u^ 0; ρ p^ 0; ; (15) c0 Figure 3c and 3d, which correspond to standing waves formed by pairs of counterpropagating Krauklis waves. The temporal attenu- where u^ð0; ωÞ and p^ ð0; ωÞ are the Fourier transform of width-aver- ation is quantified by aged velocity and pressure, respectively. The fracture transfer func- tion is related to the more commonly used fracture hydraulic ðωÞ 1 Im ω impedance Zf defined in equation 1 by ¼ ; (13) 2Q Re ω ρc ∕A FðωÞ¼ 0 f ; (16) ZfðωÞ for temporal quality factor Q. When the wavelength exceeds a cut- off wavelength (Lipovsky and Dunham, 2015) where Af is the cross-sectional area of the fracture mouth. We have nondimensionalized FðωÞ using the fluid acoustic impedance ρc0,   such that FðωÞ¼1 for an infinitely long layer of inviscid, com- 60μ2 −1∕3 λ ¼ 2π ; (14) pressible fluid between parallel, rigid walls. c ρ 5 G w0 To calculate the fracture transfer function, numerical simulations such as that in Figure 2 are performed by imposing the pressure at the fracture mouth pð0;tÞ and measuring the resulting width-aver- the phase velocity drops to zero and temporal attenuation diverges. aged velocity at the fracture mouth uð0;tÞ. Figure 4 illustrates the This expression, corresponding to the dashed lines in Figure 3c and sensitivity of uð0;tÞ to the fracture geometry, given the same chirp 3d, provides a crude estimate of the maximum length of fractures input pð0;tÞ. that can exhibit resonant oscillations. However, it is essential to ac- Then, pð0;tÞ and uð0;tÞ are Fourier transformed and the fracture count for additional dissipation that occurs from the decreased width transfer function is calculated using equation 15. An example is near the fracture tip, and this is best done numerically. We therefore shown in Figure 5. Figure 6 compares the amplitude of the transfer quantify the detectability limits of the fractures, using our 2D plane- function jFj for fractures of different lengths and widths. The trans- strain simulations, in a later section. fer functions exhibit multiple spectral peaks, corresponding to the resonant modes of the fracture (with a constant pressure boundary a) 1 condition at the fracture mouth). These peaks are finite because of dissipation from viscosity and seismic radiation. Longer and nar- 0.5 rower fractures have lower resonance frequencies, which is due to higher compliance and slower Krauklis-wave phase velocities. 0 We next examine the asymptotic behavior of the fracture transfer

Pressure (kPa) Pressure function at low frequencies, which facilitates comparison with –0.5 quasi-static fracture models that have been widely used in the b) 10 mm literature on hydraulic impedance testing and fracture diagnostics us- 4 mm ing water-hammer signals (Holzhausen and Gooch, 1985a, 1985b; 2 mm Holzhausen and Egan, 1986; Paige et al., 1995; Mondal, 2010; Carey ω ≪ ðωÞ∕ ðωÞ 1.6 mm et al., 2015). By low frequency, we mean c L,wherec 14 mm/s 0.8 mm is the Krauklis-wave phase velocity and L is the fracture length. Us-

Velocity (mm/s) ing equation 11 for cðωÞ, the low-frequency condition is ω ≪ 0 0.5 1.0 1.5 2.0 3 1∕2 ðG w0L ∕2ρÞ or ω∕2π ≪ 15 Hz for L ∼ 1mand w0 ∼ 1mm. Time (s) The low-frequency condition results in effectively uniform pressure Figure 4. Response of fractures to (a) broadband pressure chirp at across the length of the crack (at least when viscous pressure losses the fracture mouth, quantified through (b) width-averaged velocity can be neglected) and the fracture response can be described by a at the fracture mouth. The response, shown for five fractures having much simpler model. The global mass balance for the fracture, within the same length (L ¼ 10 m) and different widths (the value given the context of our linearization, is

Downloaded 08/02/17 to 171.64.171.164. Redistribution subject SEG license or copyright; see Terms of Use at http://library.seg.org/ above each curve), is characterized by several resonance frequen- Z cies. The resonance frequency decreases as the width decreases, as L ð Þ ≈ d ð Þ anticipated from the dependence of the Krauklis-wave phase veloc- w0u 0;t w x; t dx; (17) ity on the width (equation 11 and Figure 3a). Higher frequencies dt 0 decay quickly, leaving only the fundamental resonant mode. Wider fractures experience less viscous dissipation (Figure 3b), and hence where we have assumed that the fluid is effectively incompressible at they oscillate longer. these low frequencies, such that inflow of fluid (the left side) is bal- Diagnostics with Krauklis and tube waves D177

anced by changes in width (the right side). Given that the fracture is lations accurately capture the quasi-static response. However, this very thin and approximately planar, the change in the opening is then asymptotic behavior only applies in the extreme low-frequency limit related to the pressure change using the well-known solution for a (far below the first resonance frequency) and deviates substantially plane strain mode I crack in an infinite medium (Lawn, 1993). The from the actual response at higher frequencies. The traditional hy- result is draulic impedance testing method (Holzhausen and Gooch, 1985a; Holzhausen and Egan, 1986) thus leaves the vast spectrum at higher πL2 dp frequencies unexplored. uð0;tÞ ≈ : (18) 4w0G dt We also derive an approximate analytical solution to the transfer function based on the dispersion relation (rather than using simula- It follows, upon Fourier transforming equation 18 and using equa- tions). This dispersion-based approach has been used in several stud- tion 15,that ies (Hornby et al., 1989; Kostek et al., 1998a; Henry, 2005; Derov et al., 2009), and it is illustrative to compare it with the more rigorous − π 2ρ ðωÞ ≈ i L c0 ω ω → simulation-based solution. The solution, derived in Appendix A,as- F as 0: (19) sumes uniform width and imposes a zero-velocity boundary condi- 4G w0 tion at the fracture tip. The resulting fracture transfer function is This asymptotic behavior is essentially the same as that quantified by −ikðωÞc tan½kðωÞL Holzhausen and Gooch (1985a) as the fracture capacitance, although FðωÞ¼ 0 ; (20) our result is for the 2D plane strain case to permit comparison with ω our simulations. ðωÞ The low-frequency asymptotes (equation 19) are plotted as dashed where k is the complex wavenumber obtained by solving the ω lines in Figures 5a (inset) and 6, verifying that our numerical simu- dispersion equation 8, assuming real angular frequency . Our numerical simulations permit an exploration of how the de- creased width near the fracture tip alters the fracture response, relative a) to the more idealized models based on dispersion (equation 20)that 200 Quasi-static 200 ) ω approxmation assume tabular (uniform-width) fractures. Figure 7 compares transfer

ω Re F( ) 100 (

F functions for 1 m long fractures with the same width (2 mm) at the 0 100 fracture mouth but different profiles near the tip. These range from a –100 Frequency (Hz) 03612 45

) and Im 0 ω

( a) 1

F ω Im F( ) 0

w 0 –100 / Re y –1 0 1020304050 0 0.2 0.4 0.6 0.8 1 x/L Frequency (Hz) b) c)150 b) 250 150 | | L = 1 m; w = 4.7 mm L = 3.6 m; w = 2.1 mm F F 100 100 ) 200 ω ( F 150 50 50 Amplitude | Amplitude | 0 0 100 0 50 100 150 200 0 50 100 150 200 150 150

50 | L = 2.2 m; w = 4.7 mm | L = 3.6 m; w = 5.9 mm Amplitude of F F 100 100 0 0 1020304050 Frequency (Hz) 50 50

c) π/2 Amplitude | Amplitude | 0 0 0 50 100 150 200 0 50 100 150 200 π 150 150

) /4 | L = 6.5 m; w = 4.7 mm | L = 3.6 m; w = 12.3 mm ω F F (

F 100 100 0 50 50

−π/4 Amplitude | Amplitude | Phase of 0 0 0 50 100 150 200 0 50 100 150 200 −π/2 Frequency (Hz) Frequency (Hz) 0 1020304050 Frequency (Hz) Figure 6. Fracture transfer functions for different fracture lengths

Downloaded 08/02/17 to 171.64.171.164. Redistribution subject SEG license or copyright; see Terms of Use at http://library.seg.org/ and widths. (a) Normalized fracture geometry with the fracture tip Figure 5. Fracture transfer function FðωÞ for a 10 m long fracture of width equal to 1∕50 of the width at the fracture mouth. (b) Fracture 5 mm width at the fracture mouth and the profile shown in Figure 6a. transfer functions for 4.7 mm wide fractures with varied lengths. (a) Real and imaginary parts of FðωÞ. Inset shows low-frequency Longer fractures have lower resonance frequencies. (c) Fracture response, which is compared with the quasi-static approximation transfer functions for 3.6 m long fractures with varied widths. Wider (equation 19) used in hydraulic impedance testing. Also shown are fractures have higher resonance frequencies. In panels (b and c), the the (b) amplitude and (c) phase of FðωÞ. dashed lines plot the quasi-static limit (equation 19). D178 Liang et al.

tabular fracture to ones in which the width at the tip is decreased to Figure 8a shows a graphical method for estimating fracture only 1∕50 of the maximum width. As the fracture tip width is re- geometry from measurements of the frequency and temporal quality duced, the resonance frequencies and the fracture transfer function factor Q of the fundamental mode. This is similar to Figures 6–8 in amplitude at resonance peaks are shifted to lower values. This is Lipovsky and Dunham (2015), but here the fundamental mode due to higher viscous dissipation and slower Krauklis-wave phase properties are determined from transfer functions from numerical velocity for narrower fractures. The differences are most pronounced simulations using the tapered fracture shape shown in Figure 6a. ¼ ∕Δ Δ at higher frequencies and for higher resonant modes. This finding Specifically, Q f0 f for fundamental frequency f0 and f, the highlights the importance of accounting for the near-tip geometry full width at half of the maximum amplitude in a plot of jFj (Fig- when using high-frequency data to determine fracture geometry. ure 6b). We also provide, in Figure 8b and 8c, a comparison between The tabular or flat fracture model (equation 20) can lead to substantial our numerical results (for a tapered width) and predictions based on errors; in this example, there is approximately 20% error in the fun- the dispersion relation (for a uniform width). Note that the dispersion damental frequency and 100% error in the amplitude of the funda- solutions are calculated for zero velocity at the fracture tip and con- mental mode spectral peak. stant pressure at the fracture mouth, whereas Lipovsky and Dunham (2015) assume zero velocity at both tips. Viscous dissipation is under- Fracture geometry inference and detectability limits estimated using the dispersion-based approach, which ignores the narrow region near the fracture tip. There are also differences in As evidenced in Figures 4–7, the fracture response, as embodied the resonance frequency; the finite-length fractures in our numerical by the fracture transfer function, is sensitive to fracture geometry simulations are stiffer, and hence they resonate at higher frequencies, (length and width). We now consider the inverse problem, that than those suggested by the dispersion-based method. is, determining the fracture geometry from properties of the reso- We finish this section by noting that quantitative results in Fig- nant modes of the fracture. Lipovsky and Dunham (2015), building ure 8 are specific to the 2D plane strain problem. We anticipate that on work by Korneev (2008) and Tary et al. (2014), show how mea- an axisymmetric (penny-shaped) fracture would have a similar re- surements of the resonance frequencies and attenuation or decay rates sponse, but we caution that these results are unlikely to apply to 3D associated with resonant modes could be used to uniquely determine fractures when the length greatly exceeds the height. That case, the length and width. Their work uses dispersion relations derived for which is of great relevance to the oil and gas industry, warrants harmonic waves in an infinitely long fluid layer, and we have seen, in study using 3D numerical simulations. Figure 7, some notable discrepancies as compared with our numerical simulations of finite-length fractures. We thus revisit this problem, TUBE-WAVE INTERACTION WITH FLUID-FILLED focusing in particular on the detectability limits. FRACTURES

As the fracture length grows, the resonance wavelengths increase and eventually reach a cutoff limit, beyond which the temporal at- Having focused on the fracture response in the previous sections, tenuation diverges (Figure 3d). This provides the most optimistic we now return to the overall problem of determining the response of estimate of fracture detectability using resonance; noise in real mea- the coupled wellbore-fracture system. We present a simple model surements will further limit the detectability. for low-frequency tube waves, and we derive reflection/transmis- sion coefficients that quantify tube-wave interaction with a fracture a) intersecting the wellbore. We then examine the system response to 1 excitation at the wellhead or at one end of a sealed interval, noting 0 the possibility of a matched resonance between tube waves in the

(mm) −1 y 0 0.2 0.4 0.6 0.8 1 wellbore and Krauklis waves in the fracture. x (m) b) 400 Tube-wave governing equations and wellbore-fracture Flat fracture coupling 1/5 Dispersion-based z 300 Let be the distance along the well. The wellbore, with a constant 1/10 approximation 2 radius a and cross-sectional area AT ¼ πa , is intersected by a frac- | Quasi-static F 1/20 ture at z ¼ 0 with aperture w at the fracture mouth. Low-frequency approximation 0 1/50 tube waves are governed by the linearized momentum and mass 200 balance equations (and the latter is combined with linearized con- stitutive laws for a compressible fluid and deformable elastic solid Amplitude | surrounding the wellbore): 100 ρ ∂q ∂p þ ¼ 0; (21) A ∂t ∂z 0 T 0 100 200 300 400 500 600

Downloaded 08/02/17 to 171.64.171.164. Redistribution subject SEG license or copyright; see Terms of Use at http://library.seg.org/ Frequency (Hz) ∂ ∂ AT p þ q ¼ − ð Þδð Þ Figure 7. (a) Geometry of a 1 m long fracture with the tip width ∂ ∂ Afu t z ; (22) varied from 1, 1∕5, 1∕20,to1∕50 of the width at the fracture mouth M t z (2 mm). (b) Amplitude of the fracture transfer function with varied fracture tip width compared with the dispersion-based approxima- where qðz; tÞ is the volumetric flow rate along the wellbore, pðz; tÞ tion solution (equation 20) and the quasi-static limit (equation 19). is the pressure, uðtÞ is the velocity into the fracture at the fracture Diagnostics with Krauklis and tube waves D179

mouth, ρ is the fluid density (assumed to be the same as in the frac- respectively. Satisfying the governing equations 21 and 22 and the ture), M is a modulus that is typically close to the fluid bulk modu- fracture junction conditions (equations 24 and 25) yields lus (Biot, 1952), and Af is the fracture mouth area. For a fracture rðωÞ∕2 1 intersecting the wellbore perpendicularly, as we assume in the exam- RðωÞ¼− ¼ − ; (27) ¼ π þ ðωÞ∕ þ ∕ ðωÞ ples presented below, Af 2 aw0. This can be generalized to ac- 1 r 2 1 2 r count for fractures intersecting the wellbore at other angles (Hornby et al., 1989; Derov et al., 2009). These equations describepffiffiffiffiffiffiffiffiffiffi nondisper- ∕ ðωÞ ¼ ∕ρ 1 2 r sive wave propagation at the tube-wave speed cT M ,which TðωÞ¼ ¼ ; (28) we solve in the frequency domain using Fourier transforms. How- 1 þ rðωÞ∕2 1 þ 2∕rðωÞ ever, they could also be solved by a time-domain finite-difference method (Liang et al., 2015; Karlstrom and Dunham, 2016)orby a) the method of characteristics (Mondal, 2010; Carey et al., 2015), both 5 100 HZ

of which permit spatial variation of properties. In all examples shown 37 HZ below, we set the wellbore diameter to 2a ¼ 0.1 m and assume M ¼ 2 20 15 13.9 HZ ¼ 0.1 HZ K (and hence c c ) for simplicity, although expressions are given = 0.3 HZ T 0 5.2 HZ 1 10 f 0 for the general case. The model can, of course, be generalized to ac- 1.9 HZ 5 count for permeable walls (Tang, 1990), irregularly shaped boreholes 0.5 0.7 HZ (Tezuka et al., 1997), spatially variable properties (Chen et al., 1996; 2

Wang et al., 2008),andfriction(Livescu et al., 2016). The source 0.2 Overdamped region term on the right side of the mass balance equation 22 describes Average half width (mm) Q = 0.5 the mass exchange between the wellbore and the fracture at the frac- 0.1 ture mouth. Because the fracture aperture is much smaller than the 12 5102050100 tube-wave wavelength, the source term is approximately a delta func- Fracture length L (m) tion at the fracture location. Integrating equations 21 and 22 across the junction at the well- b) 5 bore-fracture intersection, we obtain the following jump conditions: 100 20

þ − 10 qð0 ;tÞ − qð0 ;tÞ¼−AfuðtÞ; (23) 2 20 1 10

0.5 Q = 0.5 pð0þ;tÞ − pð0−;tÞ¼0: (24) Q = 0.5 0.2 The pressure is continuous across the junction, whereas the volu- Numerical solution metric flow rate (and fluid particle velocity through the wellbore) Average half width (mm) 0.1 Solution based on dispersion experiences a jump that accounts for mass exchange with the frac- 12 5102050100 ture. The volumetric flow rate into the fracture AfuðtÞ is related to the pressure at the fracture mouth pð0;tÞ through the fracture transfer Fracture length L (m) function using equation 15 (or, equivalently, the fracture hydraulic c) impedance ZfðωÞ). Although we focus on one intersecting fracture 5 1 HZ in this study, extension to multiple fractures is straightforward by 0.1 HZ 100 HZ 10 HZ adding multiple jump conditions similar to equations 23 and 24. 2

By Fourier transforming equations 23 and 24 and using equation 15, 0.1 HZ

100 HZ we obtain 1 1 HZ 10 HZ A FðωÞ p^ ð0;ωÞ 0.5 q^ð0þ;ωÞ−q^ð0−;ωÞ¼− f p^ ð0;ωÞ¼− : (25) ρ ðωÞ Q=0.5 c0 Zf 0.2 Numerical solution Average half width (mm) 0.1 Solution based on dispersion Tube-wave reflection/transmission coefficients 12 5102050100 We now derive the reflection and transmission coefficients of Fracture length L (m) tube waves incident on the fracture. Assuming an infinitely long Figure 8. (a) Graphical method to determine the fracture length and well, we seek a Fourier-domain solution of the form: width from frequency (red curves) and temporal quality factor Q

Downloaded 08/02/17 to 171.64.171.164. Redistribution subject SEG license or copyright; see Terms of Use at http://library.seg.org/  − (black curves) of the fundamental resonant mode (i.e., from the frac- eikz þ Re ikz;z< 0; ture transfer function from numerical simulations using the tapered p^ ðz; ωÞ¼ (26) Teikz;z>0; fracture geometry; see Figure 6). In the overdamped region (Q < 0.5), viscous dissipation prevents resonant oscillations and the geom- etry cannot be determined with this method. (b) Quality factor and ¼ ω∕ where k cT is the wavenumber, the incident wave has unit am- (c) frequency, comparing numerical simulation results with solutions plitude, and R and T are the reflection and transmission coefficients, based on the dispersion relation (equation 8). D180 Liang et al.

in which Response of the wellbore-fracture system We now consider a finite-length section of the well intersected by ρ ∕ a single fracture, with boundary conditions prescribed at both ends ðωÞ¼ ZT ¼ cT AT ðωÞ r ðωÞ ρ ∕ F (29) of the well section. These ends might coincide with the wellhead Zf c0 Af and well bottom or the two ends of a sealed interval. Let h1 and h2 be the lengths of the two sections above and below the fracture, is the ratio of the hydraulic impedance of tube waves in the well- respectively, with the fracture at z ¼ 0 as before. We seek the pres- sure and velocity within the wellbore, given some excitation at the bore, ZT ≡ ρcT ∕AT , to the fracture hydraulic impedance ZfðωÞ. The factor of two in the expressions for RðωÞ and TðωÞ arises because a end of the upper well section. pair of tube waves propagate away from the fracture. Equations 21 and 22 are supplemented with boundary conditions ¼ − Figure 9 explores the relation between the fracture transfer func- at the top and bottom of the wellbore. At the top (z h1), we set tion and the reflection/transmission coefficients. The maximum re- the volumetric flow rate into the well equal to a prescribed injection flection approximately coincides with the resonance peaks in the rate QðtÞ, not to be confused with the quality factor discussed earlier: transfer functions, corresponding to the eigenmodes of the fracture ð− Þ¼ ð Þ with a constant pressure boundary condition at the fracture mouth. q h1;t Q t : (30) At these frequencies, the hydraulic impedance of the fracture is In Appendix B, we show that this boundary condition is mathemati- much smaller than that of tube waves (i.e., rðωÞ∕2 ≫ 1), and only cally equivalent to the case of a sealed end (i.e., qð−h ;tÞ¼0)witha small pressure changes at the fracture mouth are required to induce 1 monopole source placed within the wellbore just below the end. At a large flow into or out of the fracture. The reflection coefficient the bottom (z ¼ h ), we assume a partially reflecting condition of the goes to −1 in this limit, so that waves at these specific frequencies 2 form are reflected as if from a constant pressure boundary. We also note that reducing the fracture width decreases reflection because the pðh ;tÞ − Z qðh ;tÞ¼R ½pðh ;tÞþZ qðh ;tÞ; (31) fracture mouth area decreases relative to the wellbore cross-sec- 2 T 2 b 2 T 2 tional area AT and because the fracture resonance peak amplitude in which Rb is the well bottom reflection coefficient (satisfying decreases due to increased viscous dissipation in narrower fractures jRbj ≤ 1). This boundary condition can be equivalently written in (captured in F or Zf). terms of the hydraulic impedance Zb at the bottom of the well: 1 þ R pðh ;tÞ¼Z qðh ;tÞ;Z ¼ b Z : a) w = 1 mm w = 5 mm 2 b 2 b T (32) 0 0 − 250 1 Rb

200 We assume constant, real R (and hence Z ), although it is possible

| b b F to use a frequency-dependent, complex-valued Rb if desired. For 150 Rb ¼ 0, there is no reflection from the bottom, Rb ¼ 1 corresponds ð Þ¼ ¼ − ð Þ¼ 100 to q h2;t 0,andRb 1 corresponds to p h2;t 0. The solution to the stated problem is Amplitude | 50  ð Þþ ð Þ − ð ωÞ¼ a1 sin kz a2 cos kz ; h1

w = 1 mm, |R| w = 5 mm, |R| b) 0 0  w = 1 mm, |T | w = 5 mm, |T| 0 0 −ia cosðkzÞþia sinðkzÞ; −h < z < 0; 1 ð ωÞ¼ 1 2 1 ZT q^ z; ikz − −ikz b1e b2e ; 0 < z < h2; | T 0.8 (34) |, |

R 0.6 where k ¼ ω∕cT and the coefficients a1, a2, b1,andb2 are deter- 0.4 mined by the top boundary condition (equation 30), bottom boundary condition (equation 31), and fracture junction conditions (equa-

Amplitude | 0.2 tions 24 and 25):

0 0 1020304050 iZ Q^ ðωÞ a ¼ T ; (35) Frequency (Hz) 1 DðωÞ Downloaded 08/02/17 to 171.64.171.164. Redistribution subject SEG license or copyright; see Terms of Use at http://library.seg.org/ Figure 9. (a) Amplitude of the fracture transfer functions jFj for two 10 m long fractures with different widths (1 and 5 mm). (b) Am- plitude of the tube-wave reflection and transmission coefficients, jRj and jTj, across the fracture (equations 27 and 28) for the well- ^ ðωÞ ¼ ZT Q bore diameter 2a ¼ 0.1 m. Maximum reflection occurs at the res- a2 ; (36) onance frequencies of the fracture. ½rðωÞþΛðωÞDðωÞ Diagnostics with Krauklis and tube waves D181

Z Q^ ðωÞ of the well around the fracture is sealed at both ends and a source ¼ T – b1 2ikh ; (37) is placed at one end. The solution given in equations 33 40 still ð1 þ Rbe 2 Þ½rðωÞþΛðωÞDðωÞ applies, but we select a smaller length h1. There is now complex interference between multiply reflected waves; whether this inter- ference is constructive or destructive depends on the well section lengths h and h and the fracture reflection/transmission coeffi- R e2ikh2 Z Q^ ðωÞ 1 2 ¼ b T cients (and hence frequency). As we demonstrate, the signals in b2 2ikh ; (38) ð1 þ Rbe 2 Þ½rðωÞþΛðωÞDðωÞ

where a) 4 ð Þ i sin kh1 DðωÞ¼cosðkh Þ − ; (39) 2 1 rðωÞþΛðωÞ

0 Velocity (mm/s) 1 − R e2ikh2 –2 ΛðωÞ¼ b ; (40) 0 0.005 0.01 0.015 0.02 0.025 0.03 þ 2ikh2 1 Rbe Time (s) b) and rðωÞ is the hydraulic impedance ratio defined as before (equa- 0.01 tion 29). Solutions in the time domain are obtained by inverting the Fourier transform. 0.005 Excitation at the wellhead Here, we use the solution derived above to demonstrate how frac-

ture growth might be monitored using tube waves or water-hammer Spectral amplitude (mm) 0 0 200 400 600 800 1000 signals generated by excitation at the wellhead. The wellbore has a Frequency (Hz) total length of 3 km, a diameter of 2a ¼ 0.1 m, and a partially sealed bottom with reflection coefficient Rb ¼ 0.8. A single fracture Figure 10. Chirp used for examples shown in Figures 11 and 12:

is placed 2 km from the wellhead and 1 km from the well bottom (a) time series and (b) spectrum. ¼ ¼ (i.e., h1 2kmand h2 1km). At the well- head, we prescribe a broadband chirp (up to ap- a) b) c) proximately 500 Hz) in velocity, as shown in Source input w0 = 2 mm L = 3 m Figure 10. As mentioned earlier and detailed at wellhead L = 6 m –2.0 1 kPa L = 10 m in Appendix B, this is equivalent to placing a mo- Sensor TP 4 kPa

nopole source a short distance below the sealed (km) –1.5 z wellhead. Pressure perturbations could be gener- Pressure Sensor BT h –1.0 ated by abruptly shutting in the well, as is done in 1 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 Time (s) hydraulic impedance testing (Holzhausen and Sensor TP –0.5 d) Egan, 1986; Carey et al., 2015). However, an en- L = 10 m w = 0.4 mm 0 0 w = 1.0 mm gineered source would provide better control over Sensor BT 1 kPa 0 w = 5.0 mm h Sensor TP 0 the source spectrum. To account for dissipation 2 0.5 Distance from fracture,

during tube-wave propagation along the wellbore, Pressure 1.0 we add a small imaginarypffiffiffiffiffiffiffiffiffiffi part to the tube-wave Sensor BT − ¼ð − 3 Þ ∕ρ 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 speed: cT 1 10 i M . Figure 11 shows Sealed bottom 012345 a schematic of this system and the synthetic bore- Time (s) Time (s) hole record section. The interplay between tube Figure 11. Response of the wellbore-fracture system to chirp excitation (Figure 10)at waves in the wellbore and dispersive Krauklis the wellhead. (a) Schematic of the system, with one fracture 2 km below the wellhead waves within the fracture is evident. and 1 km above the well bottom. Sensor TP is 500 m above the fracture, and sensor BT is 250 m below the fracture. (b) Record section of pressure along the wellbore (every 250 m) with a fracture that is 10 m long and 2 mm wide at the fracture mouth. Multiple Matched resonance reflections from the fracture, well bottom, and wellhead are observed. Dashed boxes mark the time window (1.5–2.2 s) examined in panels (c and d) for sensors TP and The example shown in Figure 11 illustrates the BT. (c) Comparison of pressure at sensors TP and BT for fractures with the same width

Downloaded 08/02/17 to 171.64.171.164. Redistribution subject SEG license or copyright; see Terms of Use at http://library.seg.org/ response when the entire long wellbore is hy- but different lengths. (d) Same as panel (c) but for fractures with the same length but draulically connected to the fracture. Distinct re- different widths. In panels (c and d), sensor TP shows waves reflected from the fracture; flections can be seen, and interference between first arriving are direct tube-wave reflections from the fracture mouth, followed by tube waves generated by Krauklis waves in the fracture that have reflected from the fracture different reflections is confined to short time in- tip. Sensor BT shows transmitted waves, which have similar arrivals. Longer fractures tervals. We next examine the more complex re- show a more dispersed set of Krauklis-wave arrivals. Reflections are smaller from nar- sponse that arises when a much smaller section rower fractures. D182 Liang et al.

the upper well section that contains the source can be selectively (q ¼ 0) and constant pressure condition (p ¼ 0) at the fracture ðω ∕ Þ¼ amplified when a resonance frequency of the upper well section satisfy cos h1 cT 0. For example, the lowest resonance fre- ¼ ω∕ π ¼ ∕ is tuned to one of the resonance frequencies of the fracture. We refer quency is f 2 cT 4h1. Matched resonance occurs when to this phenomenon as matched resonance. one of these frequencies matches a fracture resonance frequency. Reflection of waves from the fracture is most pronounced at To justify this more rigorously, and explain some possible com- frequencies that permit maximum exchange of fluid between the plications, we note that the tube-wave eigenfunctions associated wellbore and fracture. These frequencies correspond approximately with this resonant wellbore response are sinðωz∕cT Þ, corresponding to the resonance frequencies of the fracture with a constant pressure to the first term on the right side of equation 33. The amplitude of boundary condition at the fracture mouth (i.e., corresponding to the this term is given in equation 35, which is largest when the denom- peaks of the fracture transfer function; see Figure 9). The resonance inator DðωÞ, given in equation 39, is smallest. The hydraulic imped- frequencies of the upper wellbore section with a sealed top end ance ratio is quite large (ideally, jrðωÞj ≫ 1) around the fracture resonance frequencies. A further requirement is that jΛðωÞj be suf- j ðωÞj a) 0.5 ficiently small compared with r , at least around the targeted (both: h = 100 m) R = 0.9 1 fracture resonance frequency. When these conditions are satisfied, b ðωÞ ≈ ð Þ ¼ ω∕ ðωÞ¼ D cos kh1 with k cT . Setting D 0 then yields the ðω ∕ Þ¼ 0 wellbore section resonance condition cos h1 cT 0 stated above. Of course, this resonance condition is only an approxima- R = 0 (nonreflecting well bottom) pressure (kPa) b tion, especially when jrðωÞj does not greatly exceed unity or when –0.5 0246810 jΛðωÞj is not small. In these cases, the matched resonance condition time (s) ω b) can be more precisely determined by finding the frequencies that ðωÞ ΛðωÞ 0.4 1 minimize the exact D in equation 39. However, can be |R|, fracture reflection coefficient made arbitrarily small by sealing the bottom end of the lower well ¼ ΛðωÞ¼− ðω ∕ Þ section (Rb 1 and hence i tan h2 cT ) and decreas- 0.2 0.5 ing h2, such that ωh2∕cT ≪ 1. R = 0.9 b Figure 12a and 12b illustrates matched resonance. The frequency Rb = 0 of the fundamental mode of the fracture (the 10 m long, 5 mm wide 0 0 ≈ spectral amplitude of pressure (kPa s) 0 1020304050 example shown in Figures 5 and 9)isf 3.8 Hz, so the upper well- ¼ ≈ ∕ frequency (Hz) bore section is chosen to be h1 100 m cT 4f long to satisfy the c) 0.5 matched resonance condition. The system is excited by a broadband h = 200 m 1 (both: Rb = 0.9)

chirp at the upper end of this wellbore section. The 3.8 Hz reso- 0 nance frequency clearly dominates the system response. The re- sponse is shown for two bottom boundary conditions. The first is

h1 = 50 m the relatively simple case of a well with a nonreflecting bottom boun- pressure (kPa) –0.5 ¼ Λ ¼ 0246810 dary (Rb 0, for which 1 regardless of h2). This eliminates the time (s) possibility of resonance within the lower wellbore section and pre- vents bottom reflections from then transmitting through the fracture d) 0.4 1 to return to the upper well section. The second case has h ¼ 10 m h = 200 m |R|, fracture reflection coefficient 2 1 and R ¼ 0.9, corresponding to a partially sealed end a short distance h = 50 m b 0.2 1 0.5 below the fracture. Although the initial response in the second case contains a complex set of high-frequency reverberations associated 0 0 with reflections within the lower well section, these are eventually

amplitude (kPa s) 01020304050 damped out to leave only the prominent 3.8 Hz resonance. This il- frequency (Hz) lustrates a rather remarkable insensitivity to characteristics of the Figure 12. Matched resonance: The length of the upper wellbore lower well section because similar results (not shown) were found section containing the source is chosen so that the resonance fre- even for a wide range of lower well section lengths h2. quency of tube waves in this wellbore section matches the fundamen- We next demonstrate how changing the length of the upper well tal mode resonance frequency of the fracture (3.8 Hz). (a) Pressure section h alters the resonant response of the system. Figure 12c and response at a receiver in the center of the upper wellbore section (at 1 ¼ − ∕ 12d shows the responses for values of h1 that are larger and smaller z h1 2) to the chirp excitation shown in Figure 10. The 3.8 Hz resonance is selectively amplified, regardless of the length of the than the matched resonance length. The fundamental mode reso- lower wellbore section and the bottom boundary condition: Compare nance peak shifts to a lower frequency when h1 increases and to the case with a nonreflecting well bottom (red, Rb ¼ 0)topartially ¼ ¼ a higher frequency when h1 decreases. Many resonant modes are sealed (blue, Rb 0.9 and h2 10 m). (b) Fourier amplitude spec- trum of pressure time series shown in (a), with prominent peak at the excited for the large h1 case, leading to a rather complex pressure 3.8 Hz matched resonance frequency. Although additional spectral response. The situation is simpler for the small h1 case, but as h1 peaks appear for the sealed bottom case, the matched resonance peak continues to decrease, the amplitude of the fundamental mode peak

Downloaded 08/02/17 to 171.64.171.164. Redistribution subject SEG license or copyright; see Terms of Use at http://library.seg.org/ is nearly identical to the nonreflecting bottom case. Also, shown is continues to decrease and might be difficult to observe in noisy data. the reflection coefficient of tube waves from the fracture, which has peaks at the fracture resonance frequencies. (c) Same as panel (a) but ¼ ¼ ¼ Possible measuring techniques for h1 200 and 50 m (with Rb 0.9 and h2 10 m). The system does not satisfy the matched resonance condition and the spectrum, shown in panel (d), is more complicated and lacks the pro- Although it is easy to implement different excitation sources and nounced peak at the fundamental resonance mode. receiver configurations in our theoretical calculations, it is not trivial Diagnostics with Krauklis and tube waves D183

to make downhole pressure measurements in practice. In general, and measure fracture eigenmodes, which can then be used to infer pressure is measured at the pump or in the pipe connecting the the fracture geometry. pump to the wellhead during hydraulic fracturing treatments. In- stalling downhole pressure transducers or hydrophones with com- ACKNOWLEDGMENTS plex pumping and treatment equipment in place during hydraulic fracturing would be challenging. In addition, downhole sensors This work was supported by a gift from Baker Hughes to the can occupy a significant volume of the borehole and create complex Stanford Energy and Environment Affiliates Program and seed tool effects on measured tube waves (Pardo et al., 2013). Quantify- funding from the Stanford Natural Gas Initiative. O. O’Reilly was ing such effects is beyond the scope of this study. A possible alter- partially supported by the Chevron fellowship in the Department of native approach would be to install geophones or pressure gauges in Geophysics at Stanford University. nearby monitoring wells. The model presented in this study would need to be extended to predict the signals in the monitoring wells, APPENDIX A and it is uncertain if these signals would have as much useful in- formation regarding the fracture geometry. Another technology that DISPERSION-BASED FRACTURE TRANSFER reduces the influence of the tool on tube-wave propagation is fiber- FUNCTION optic distributed acoustic sensing (DAS) (MacPhail et al., 2012; Molenaar et al., 2012; Boone et al., 2015). DAS cables can run In this appendix, we use the Krauklis-wave dispersion relation to through grooves across swellable packers along the casing in the derive an approximate expression for the fracture transfer function. sealing elements in uncemented packer and sleeve completion, and Let x be the distance into the fracture measured from the fracture ¼ they can be buried in the cement in cemented plug-and-perf com- mouth at x 0. The fracture has length L and uniform aperture w0. pletions (Boone et al., 2015). This technology not only avoids any Solutions in the frequency domain are written as a superposition of interaction with tube waves, but it can also provide continuous mea- plane waves propagating in the þx and −x directions: surements along the full length of the wellbore. p^ ðx; ωÞ¼Aeikx þ Be−ikx; (A-1)

CONCLUSION ρ ð ωÞ¼ ikx − −ikx We have investigated the interaction of tube waves with fractures. cu^ x; Ae Be ; (A-2) Although this problem has received much attention in the literature, ¼ ðωÞ ¼ ω∕ ðωÞ previous work has been restricted to relatively idealized analytical where k k and c k are the wavenumber and phase

or semianalytical models of the fracture response. In contrast, we velocity determined by the Krauklis-wave dispersion relation (equa- ω advocate the use of numerical simulations to more accurately de- tion 8, solved for real and possibly complex k), and A and B are scribe the fracture. These simulations account for variable fracture the coefficients determined by boundary conditions at the ends of width, narrow viscous boundary layers adjacent to the fracture walls, the fracture. At the fracture tip, the fluid velocity is set to zero: and dissipation from viscosity and from seismic radiation. The sim- ρ ð ωÞ¼ ikL − −ikL ¼ ulations feature Krauklis waves propagating along the fracture; coun- cu^ L; Ae Be 0: (A-3) terpropagating pairs of Krauklis waves form the eigenmodes of the The fracture transfer function, defined in equation 15, is obtained by fracture. As many authors note, the frequency and attenuation of combining equations A-1–A-3: these modes can be used to constrain fracture geometry. Although this initial study uses a 2D plane strain fracture model, − ðωÞ ½ ðωÞ ðωÞ¼ ik c0 tan k L the overall methodology can be applied when 3D models of the F ω : (A-4) fracture are available. Such models would permit investigation of fractures that have bounded height, a commonly arising situation in the industry, and one for which the 2D model presented here is not Peaks in FðωÞ correspond to the resonance frequencies of the well-justified. fracture with a constant pressure condition at the fracture mouth. We then showed how to distill the simulation results into a single, Therefore, besides equation A-3, we have complex-valued function that quantifies the fracture response, spe- cifically its hydraulic impedance (or the normalized reciprocal of p^ ð0; ωÞ¼A þ B ¼ 0: (A-5) the impedance, the fracture transfer function). The fracture transfer function can then be used to determine how tube waves within a Combining equations A-3 and A-5, we obtain the resonance con- wellbore reflect and transmit from fractures intersecting the well, dition and to solve for the response of the wellbore-fracture system to excitation at the wellhead or within the wellbore. cos kL ¼ 0; (A-6) The coupled wellbore-fracture system has a particularly complex ¼ð − ∕ Þπ ¼ Downloaded 08/02/17 to 171.64.171.164. Redistribution subject SEG license or copyright; see Terms of Use at http://library.seg.org/ response, potentially involving resonance within wellbore sections or knL n 1 2 for positive integers n 1; 2; :::. This can adjacent to the fracture or within the fracture itself. We found that it be solved, assuming a real k and a complex ω, for the resonance is possible to selectively amplify tube waves at the eigenfrequencies frequencies and (temporal) decay rates of the eigenmodes. of the fracture by properly choosing the length of the well section We next derive the resonance frequencies for the case of negli- containing the source and intersecting the fracture. This phenome- gible dissipation (i.e., for an inviscid fluid). In this case, k and ω are non, which we call matched resonance, could prove useful to excite real, and we have (Krauklis, 1962) D184 Liang et al.

ðωÞ¼ð ω∕ ρÞ1∕3 ¼ð π ∕ρÞ1∕3 c G w0 2 G w0 f : (A-7) Constants a and b are determined by substituting equations B-5 and B-6 into equations B-7 and B-8: It follows that the resonance frequencies are ω∕ ¼ ðωÞ is cT qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ZT m^ e ; (B-9) ¼ ðπ ∕ρÞ½ð − ∕ Þ∕ 3 fn G w0 n 1 2 2L : (A-8) ω∕ ¼ ðωÞ ð ω∕ Þ is cT This expression provides a reasonably accurate prediction of the b ZT m^ cos s cT e : (B-10) resonance frequencies for the examples shown in this work, pro- vided that w in equation A-8 is interpreted as average width. 0 Substituting equations B-9 and B-10 into equations B-5 and B-6, we obtain the solution below the source (z>s):

APPENDIX B iωz∕cT p^ ðz; ωÞ¼ZT m^ ðωÞ cosðsω∕cT Þe ; (B-11) EQUIVALENCE OF MONOPOLE SOURCE AND FLOW RATE BOUNDARY CONDITION ω ∕ q^ðz; ωÞ¼m^ ðωÞ cosðsω∕c Þei z cT : (B-12) In this appendix, we establish the equivalence, for sufficiently T low frequencies, between a monopole source placed just below the sealed end of a wellbore section and a prescribed flow rate boundary Similarly, we obtain the solution to the tube-wave problem with ¼ condition at that end. For convenience, we take z 0 to coincide no internal source but with the top boundary condition being with the end. The tube-wave equations with a monopole source qð0;tÞ¼QðtÞ: mðtÞ at z ¼ s are ð ωÞ¼ ^ ðωÞ iωz∕c ρ ∂q ∂p p^ z; ZT Q e ; (B-13) þ ¼ 0; (B-1) AT ∂t ∂z q^ðz; ωÞ¼Q^ ðωÞeiωz∕c: (B-14) A ∂p ∂q T þ ¼ mðtÞδðz − sÞ: (B-2) M ∂t ∂z

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