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Hydraulic fracture diagnostics from Krauklis-wave resonance andtube-wave reflections
Chao Liang1, Ossian O’Reilly2, Eric M. Dunham3, and Dan Moos4
ABSTRACT
Fluid-filled fractures support guided waves known as Kraukliswaves. The resonance of Krauklis waves within fractures occursat specific frequencies; these frequencies, and the associated at-tenuation of the resonant modes, can be used to constrain the frac-ture geometry. We use numerical simulations of wave propagationalong fluid-filled fractures to quantify fracture resonance. Thesimulations involve solution of an approximation to the compress-ible Navier-Stokes equation for the viscous fluid in the fracturecoupled to the elastic-wave equation in the surrounding solid. Var-iable fracture aperture, narrow viscous boundary layers near thefracture walls, and additional attenuation from seismic radiation
are accounted for in the simulations. We then determine how tubewaves within a wellbore can be used to excite Krauklis waveswithin fractures that are hydraulically connected to the wellbore.The simulations provide the frequency-dependent hydraulicimpedance of the fracture, which can then be used in a fre-quency-domain tube-wave code to model tube-wave reflection/transmission from fractures from a source in the wellbore or atthe wellhead (e.g., water hammer from an abrupt shut-in). Tubewaves at the resonance frequencies of the fracture can be selec-tively amplified by proper tuning of the length of a sealed sectionof the wellbore containing the fracture. The overall methodologypresented here provides a framework for determining hydraulicfracture properties via interpretation of tube-wave data.
INTRODUCTION
Hydraulic fracturing is widely used to increase the permeabilityof oil and gas reservoirs. This transformative technology has en-abled production of vast shale oil and gas resources. Although en-gineers now have optimal control of well trajectories, the geometry(length, height, and aperture) of hydraulic fractures is still poorlyknown, which poses a challenge for optimizing well completion.Monitoring hydraulic fracture growth is desirable for many reasons.Tracking the length of fractures ensures that multiple subparallelfractures remain active, rather than shielding each other throughelastic interactions. It might also be used to prevent fractures innearby wells from intersecting one another. In addition, the abilityto measure fracture geometry would facilitate estimates of thestimulated volume and could be used to help guide the deliveryof proppant.
Due to their sensitivity to properties of the surrounding formationand fractures intersecting the wellbore, tube waves or water hammerpropagating along the well are widely used for formation evaluationand fracture diagnostics (Paillet, 1980; Paillet andWhite, 1982; Holz-hausen and Gooch, 1985a, 1985b; Holzhausen and Egan, 1986;Hornby et al., 1989; Tang and Cheng, 1989; Paige et al., 1992, 1995;Kostek et al., 1998a, 1998b; Patzek and De, 2000; Henry et al., 2002;Ziatdinov et al., 2006; Ionov, 2007; Wang et al., 2008; Derov et al.,2009; Mondal, 2010; Bakku et al., 2013; Carey et al., 2015; Livescuet al., 2016). Here, we are specifically concerned with low-frequencytube waves having wavelengths much greater than the wellboreradius. In this limit, tube waves propagate with minimal dispersionat a velocity slightly less than the fluid sound speed (Biot, 1952).Besides excitation from sources within the well or at the wellhead,tube waves can be generated by incident seismic waves from the sur-rounding medium (Beydoun et al., 1985; Schoenberg, 1986; Ionov
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] Baker Hughes, Palo Alto, California, USA; presently Stanford University, Department of Geophysics, Stanford, California, USA. E-mail:
[email protected] University, Department of Geophysics, Stanford, California, USA and Stanford University, Institute for Computational and Mathematical Engineer-
ing, Stanford, California, USA. E-mail: [email protected] Baker Hughes, Palo Alto, California, USA, presently . E-mail: [email protected].© 2017 Society of Exploration Geophysicists. All rights reserved.
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GEOPHYSICS, VOL. 82, NO. 3 (MAY-JUNE 2017); P. D171–D186, 12 FIGS., 1 TABLE.10.1190/GEO2016-0480.1
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and Maximov, 1996; Bakku et al., 2013) and by sources within hy-draulic fractures (Krauklis and Krauklis, 1998; Ionov, 2007; Derovet al., 2009). Furthermore, tube waves incident on a fracture in hy-draulic connection to the wellbore will reflect and transmit from thefracture, with frequency-dependent reflection/transmission coeffi-cients. As described in more detail below, these reflected and trans-mitted tube waves carry information about fracture geometry.A closely related concept is that of hydraulic impedance testing
(Holzhausen and Gooch, 1985a), initially developed to estimate thegeometry of axial fractures (fractures in the plane of the wellbore).This method, using very low frequency tube waves or water-ham-mer signals, was validated through laboratory experiments (Paigeet al., 1992) and applied to multiple field data sets (Holzhausenand Egan, 1986; Paige et al., 1995; Patzek and De, 2000; Mondal,2010; Carey et al., 2015). The method is typically presented usingthe well-known correspondence between hydraulics and electricalcircuits, with hydraulic impedance defined as the ratio of pressure tovolumetric flow rate (velocity times the cross-sectional area). Thefracture consists of a series connection of resistance (energy lossthrough viscous dissipation and seismic radiation), capacitance (en-ergy stored in elastic deformation of the solid and compression/expansion of the fluid), and inertance (kinetic energy of the fluid),which can be combined into a complex-valued hydraulic impedanceof the fracture. The hydraulic impedance is then related to the geom-etry of the fractures (Holzhausen and Gooch, 1985a; Paige et al.,1992; Carey et al., 2015). However, this method, at least as pre-sented in the literature thus far, assumes a quasi-static fractureresponse when relating (spatially uniform) pressure in the fractureto the fracture volume. This assumption is valid only for extremelylow frequencies, and it neglects valuable information at higherfrequencies in which there exists the possibility of a resonant frac-ture response associated with waves propagating within the fracture.The treatment of dissipation from fluid viscosity is similarly overlysimplistic.Thus, a more rigorous treatment of the fracture response is re-
quired, motivating a more nuanced description of the dynamicsof fluid flow within the fracture and of its coupling to the surround-ing elastic medium over a broader range of frequencies. The keyconcept here is a particular type of guided wave that propagatesalong fluid-filled cracks. These waves, known as crack waves orKrauklis waves, have been studied extensively in the context of theoil and gas industry, volcano seismology, and other fields (Krauklis,1962; Paillet and White, 1982; Chouet, 1986; Ferrazzini and Aki,1987; Korneev, 2008, 2010; Yamamoto and Kawakatsu, 2008;Dunham and Ogden, 2012; Lipovsky and Dunham, 2015; Nikitinet al., 2016). At the frequencies of interest here (approximately 1–1000 Hz), they are anomalously dispersed waves of opening andclosing that propagate along fractures at speeds approximately10–1000 m∕s. Experimental studies have confirmed the existenceand propagation characteristics of these waves (Tang and Cheng,1988; Nakagawa, 2013; Shih and Frehner, 2015; Nakagawa et al.,2016). Counterpropagating pairs of Krauklis waves form standingwaves (eigenmodes) of fluid-filled cracks, and the resonance fre-quencies and decay rates of these modes are sensitive to the fracturelength and aperture. Krauklis waves are thought to be responsiblefor the long period and very long period seismicity at active volca-noes (Aki et al., 1977; Chouet, 1988), and they have been suggestedas a possible explanation for harmonic seismic signals recorded dur-ing hydraulic fracturing treatments (Ferrazzini and Aki, 1987; Tary
et al., 2014). Most studies of Krauklis waves have been analytical orsemianalytical, with a major focus on deriving dispersion relationsfor waves in infinitely long fluid layers of uniform width. A notableexception is the work of Frehner and Schmalholz (2010) and Freh-ner (2013), in which finite elements in two dimensions were used tostudy Krauklis waves in finite-length fractures with variable aper-ture. Using a numerical approach provides a means to investigatethe reflection and scattering of Krauklis waves at fracture tips (Freh-ner and Schmalholz, 2010) and the excitation of Krauklis waves byincident seismic waves (Frehner, 2013). However, these studies didnot consider the interaction between the Krauklis and tube waves,which is a central focus of our study.Mathieu and Toksoz (1984) were the first to pose the mathemati-
cal problem of tube-wave reflection/transmission across a fracturein terms of the fracture impedance. In this work, we define the frac-ture hydraulic impedance in the frequency domain as
ZfðωÞ ¼pfðωÞqfðωÞ
; (1)
where pfðωÞ and qfðωÞ are the Fourier transforms of the pressureand volumetric flow rate into the fracture, both evaluated at the frac-ture mouth. The Fourier transform of some function gðtÞ is defined as
gðωÞ ¼Z
∞
−∞gðtÞeiωtdt: (2)
Mathieu and Toksoz (1984) treated the fracture as an infinite fluidlayer or a permeable porous layer with the fluid flow being describedby Darcy’s law. The fracture hydraulic impedance built on Darcy’slaw was then replaced with analytic solutions based on the dispersionrelation for acoustic waves in an inviscid (Hornby et al., 1989) orviscous (Tang and Cheng, 1989) fluid layer bounded by rigid fracturewalls. Later studies established that elasticity of the fracture wall rocksignificantly changes the reflection and transmission across a fracture(Tang, 1990; Kostek et al., 1998a, 1998b). The arrival time and am-plitude of reflected tube waves can be used to infer the location andeffective aperture of fractures (Medlin and Schmitt, 1994). However,most of these models assume an infinite fracture length, as is well-justified (Hornby et al., 1989) for the high frequencies (approximately1 kHz) that were the focus of these studies. By focusing instead onlower frequencies (< 100 Hz) and considering Krauklis waves re-flected from the fracture tip, Henry et al. (2002) and Henry (2005)argued that reflection/transmission of tube waves is affected by theresonance of the fracture, and this consequently provides sensitivityto fracture length.One approach to account for the finite extent of the fracture is to
use the dispersion relation for harmonic Krauklis waves to deter-mine the eigenmodes of a circular disk-shaped fracture of uniformaperture with a zero radial velocity boundary condition at the edgeof the disk (Hornby et al., 1989; Henry et al., 2002; Henry, 2005;Ziatdinov et al., 2006; Derov et al., 2009). This treatment fails toaccount for the decreased aperture near the fracture edge, whichdecreases the Krauklis-wave phase velocity and increases the vis-cous dissipation. It furthermore implicitly assumes perfect reflec-tion from the fracture edge, thereby neglecting attenuation fromseismic-wave radiation. Recent studies have established the impor-tance of this attenuation mechanism (Frehner and Schmalholz,2010; Frehner, 2013).
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In this work, we examine how Krauklis waves propagating influid-filled fractures, as well as tube-wave interactions with fractures,can be used to infer fracture geometry. Figure 1 shows the system.The overall problem is to determine the response of the coupledfracture-wellbore system to excitation at the wellhead, within thewellbore, or at the fracture mouth. Other excitation mechanisms,including sources in the fracture or incident seismic waves fromsome external source in the solid (e.g., microseismic events or activesources), are not considered here. The fracture and wellbore arecoupled in several ways, the most important of which is throughthe direct fluid contact at the fracture mouth. Pressure changes at thisjunction, for instance, due to tube waves propagating along the well-bore, will excite Krauklis waves in the fracture. Similarly, fluid can beexchanged between the wellbore and fracture. Other interactions,such as through elastic deformation of the solid surrounding the well-bore and fracture, are neglected in our modeling approach.By making this approximation, and by further assuming that all
perturbations are sufficiently small so as to justify linearization, theresponse of the coupled fracture-wellbore system can be obtained intwo steps (Mathieu and Toksoz, 1984; Hornby et al., 1989; Kosteket al., 1998a; Henry, 2005). In the first step, we determine the re-sponse of the fracture, in isolation from the wellbore, to excitation atthe fracture mouth. Specifically, we calculate the (frequency-depen-dent) hydraulic impedance of the fracture, as defined in equation 1,using high-resolution finite-difference simulations of the dynamicresponse of a fluid-filled fracture embedded in a deformable elasticmedium. This provides a rigorous treatment of viscous dissipationand seismic radiation along finite-length fractures with possiblycomplex geometries, including variable aperture. The numericalsolutions are compared with semianalytic solutions based on disper-sion relations for harmonic waves propagating along an infinitelylong fluid layer of uniform width. In the second step, we solve thetube-wave problem in the frequency domain, in which the fractureresponse is captured through the fracture hydraulic impedance, andthen it is converted to the time domain by inverting the Fouriertransforms. The fracture response, as embodied by the fracture hy-draulic impedance, features multiple resonance peaks associatedwith the eigenmodes of the fracture. The frequencies and attenua-tion properties of these modes are sensitive to the fracture lengthand aperture. These resonances furthermore make the tube-wavereflection/transmission coefficients dependent on frequency, withthe maximum reflection at the resonance frequencies of the fracture.We close by presenting synthetic pressure seismograms for tubewaves within the wellbore, along with a demonstration of the sen-sitivity of these signals to fracture geometry.
FLUID-FILLED FRACTURES
Fluid governing equations and numerical simulationsof fluid-filled fractures
The first step in the solution procedure outlined above is to de-termine the hydraulic impedance of the fracture. In fact, we find itmore convenient to work with a nondimensional quantity propor-tional to the reciprocal of the fracture hydraulic impedance, whichwe refer to as the fracture transfer function FðωÞ. Although the frac-ture hydraulic impedance diverges in the low-frequency limit, thefracture transfer function goes to zero in a manner that captures thequasi-static fracture response used in the original work on hydraulicimpedance testing.
The fluid-filled fracture system is described by the elastic-waveequation, governing displacements of the solid, and the compress-ible Navier-Stokes equation for the viscous fluid in the fracture. Inthis work, we use a linearized, approximate version of the Navier-Stokes equation that retains only the minimum set of terms requiredto properly capture the low-frequency response of the fluid (Lipov-sky and Dunham, 2015). By low frequency, we mean ωw0∕c0 ≪ 1,where ω is the angular frequency, w0 is the crack aperture or width(the two terms are used interchangeably hereafter), and c0 is thefluid sound speed. For w0 ∼ 1 mm and c0 ∼ 103 m∕s, frequenciesmust be smaller than approximately 1 MHz. This is hardly a restric-tion because fracture resonance frequencies are typically well lessthan approximately 100 Hz. The derivation of the governing equa-tions and details of the numerical treatment (using a provably stable,high-order-accurate finite-difference discretization) are discussedby Lipovsky and Dunham (2015) and O. O’Reilly et al. (personalcommunication, 2017). Here, we explain the geometry of our sim-ulations and then briefly review the fluid governing equationswithin the fracture to facilitate the later discussion of Kraukliswaves and the fracture transfer function.Although a solution to the 3D problem is required to quantify
how the fracture transfer function depends on the fracture length,height, and width, in this preliminary study, we instead use a 2Dplane strain model (effectively assuming an infinite fracture height).We anticipate that this 2D model will provide a reasonable de-scription of axisymmetric fractures, such as the one illustrated inFigure 1, although some differences should be expected from thediffering nature of plane waves and axisymmetric waves. However,the procedure for using the fracture transfer function, obtained fromnumerical simulations, to solve the coupled wellbore-fracture prob-lem, is completely general, as are the overall qualitative results con-cerning matched resonance that are discussed in the context of tube-wave interactions with fractures. Finally, when the fracture transferfunctions from 3D simulations are available, the coupling solutionprocedure can be used with no additional modifications.
Seismic radiation
Viscous dissipation
Linear elastic solid Incident tube wavetube wave
Transmittedtube wave
Sealed bottom
wellbore
Nonplanarfracture geometry
Compressible
Source inputat wellhead
tube wave
Krauklis wave
z
Figure 1. Tube waves, excited at the wellhead, are incident on afluid-filled fracture intersecting the wellbore. Pressure changes andfluid-mass exchange between the wellbore and fracture excite Krau-klis waves within the fracture, leading to partial reflection of thetube waves and dissipation of energy.
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Returning to the 2D plane strain problem, let x be the distancealong the fracture and y be the distance perpendicular to the frac-ture; the origin is placed at the fracture mouth, where we will latercouple the fracture to a wellbore. We use an approximation similarin many respects to the widely used lubrication approximation forthin viscous layers (Batchelor, 2000), although we retain terms de-scribing fluid inertia and compressibility. In the low-frequency limit(ωw0∕c0 ≪ 1), the y-momentum balance establishes the uniformityof pressure across the width of the fracture (Ferrazzini and Aki,1987; Korneev, 2008; Lipovsky and Dunham, 2015). The linearizedx-momentum balance is
ρ∂v∂t
þ ∂p∂x
¼ μ∂2v∂y2
; (3)
where vðx; y; tÞ is the particle velocity in the x-direction, pðx; tÞ is thepressure, and ρ and μ are the fluid density and dynamic viscosity,respectively. We have retained only the viscous term correspondingto shear along planes parallel to y ¼ 0; scaling arguments show thatother viscous terms are negligible in comparison for this class ofproblems (Lipovsky and Dunham, 2015). The initial and perturbedfracture widths are defined as
w0ðxÞ ¼ wþ0 ðxÞ − wþ
0 ðxÞ; (4)
wðx; tÞ ¼ wþðx; tÞ − w−ðx; tÞ: (5)
Note that w0 refers to the full width, whereas some previous work(Lipovsky and Dunham, 2015) used this to refer to the half-width.Combining the linearized fluid mass balance with a linearized equa-tion of state, we obtain
1
K∂p∂t
þ 1
w0
∂w∂t
¼ −1
w0
∂ðuw0Þ∂x
; (6)
where
uðx; tÞ ¼ 1
w0ðxÞZ
wþ0ðxÞ
w−0ðxÞ
vðx; y; tÞdy (7)
is the x-velocity averaged over the fracture width and K is the fluidbulk modulus. The fluid sound speed is c0 ¼
ffiffiffiffiffiffiffiffiffiK∕ρ
p. Accumulation/
loss of fluid mass at some location in the fracture can be accommo-dated by either compressing/expanding the fluid (the first term on theleft side of equation 6) or opening/closing the fracture walls (the sec-ond term on the left side of equation 6). The latter is the dominantprocess at the low frequencies of interest in this study.Coupling between the fluid and solid requires balancing tractions
and enforcing continuity of normal and tangential particle velocityon the fracture walls (i.e., the kinematic and no-slip conditions). Atthe fracture mouth, pressure is prescribed; zero velocity is prescribedat the fracture tip. At the outer boundaries of the solid domain, ab-sorbing boundary conditions are used to suppress artificial reflec-tions. The computational domain in both directions is 12 times thelength of the fracture so as to fully capture the quasi-static displace-ments in the solid and to further minimize boundary reflections.Spatial variations in the fracture width are captured through w0ðxÞ.
A uniformly pressurized fracture in an infinite medium has an ellip-tical opening profile, w0ðxÞ ∝
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 − x2
p, and a stress singularity at
the fracture tips. Here, we use a related solution for w0ðxÞ in whichclosure at the tip occurs in a more gradual manner so as to removethat singularity. In particular, we use expressions for w0ðxÞ from acohesive zone model (Chen and Knopoff, 1986), appropriately modi-fied from antiplane shear to plane strain by replacement of the shearmodulus G with G� ¼ G∕ð1 − νÞ, where ν is Poisson’s ratio. Thecohesive zone region is approximately 25% of the fracture length.In addition, we blunt the fracture tip so that it has a finite width (usu-ally a small fraction of the maximum width at the fracture mouth,unless otherwise indicated).Figure 2 shows snapshots of the numerical simulation of Krauklis
waves propagating along a fluid-filled fracture. For this exampleonly, we add to w0ðxÞ a band-limited self-similar fractal roughness,as in Dunham et al. (2011) with an amplitude-to-wavelength ratio of10−2, similar to what is observed for natural fracture surfaces andfaults (Power and Tullis, 1991; Candela et al., 2012). Material prop-erties used in this and other simulations are given in Table 1. Thefracture opens in regions of converging fluid flow, and it contractswhere the flow diverges. Krauklis waves with different wavelengthsseparate as they propagate along the fracture due to dispersion.Krauklis waves arise from the combined effects of fluid inertia andthe restoring force from fracture wall elasticity. Because the elasticwall is more compliant at longer wavelengths, all else being equal,long-wavelength waves experience smaller restoring forces andhence propagate more slowly. Elastic waves in the solid are excited
time = 39.59 ms
time = 48.57 ms
horizontal fluid velocity (cm/s)
0 2 4 6 8-2-4-6-8
vert
ical
sol
id v
eloc
ity (
mm
/s)
0
0.15
0.45
0.3
0.6
–0.15
–0.3
–0.45
–0.6
10 m
4 m
m
10 m
4 m
m
(vertically exaggerated)
(vertically exaggerated)
seismic radiation
dispersive Krauklis waves
fracture mouth fracture tip
vert
ical
sol
id v
eloc
ity (
mm
/s)
0
0.15
0.45
0.3
0.6
–0.15
–0.3
–0.45
–0.6
horizontal fluid velocity (cm/s)
0 2 4 6 8–2–4–6–8
a)
b)
Figure 2. Snapshots from simulation of Krauklis and elastic wavesexcited by an imposed pressure chirp at the fracture mouth, for a 10 mlong fracture with a 4 mm width at the fracture mouth. The back-ground shows the solid response (to scale), and the inset shows thefluid response (vertically exaggerated). Colors in the solid show theparticle velocity in the direction normal to the fracture walls; discon-tinuities across the fracture indicate opening/closing motions charac-teristic of Krauklis waves. Colors in the fluid show the particlevelocity; note the narrow viscous boundary layers near the nonplanarfracture walls. The multimedia version of Figure 2 can be accessedthrough this link: s1.mp4.
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at the fracture mouth and along the fracture due to inertia of thesolid during Krauklis-wave reflection from the fracture tip and frac-ture mouth. Note that elastic waves propagate nearly an order ofmagnitude faster than Krauklis waves; this is evident in Figure 2a.As seen in the Figure 2 insets, viscous effects are primarily re-stricted to narrow boundary layers near the fracture walls and therecan even be flow reversals and nonmonotonic velocity profiles. Thedecreased width near the fracture tip decelerates the Krauklis wavesand enhances the viscous dissipation in the near-tip region. Afterbeing reflected multiple times, pairs of counterpropagating Kraukliswaves form standing waves along the fracture, which set the fluid-filled fracture into resonance. Shorter period modes are damped outfirst, by the viscous dissipation and seismic radiation, leaving long-period modes with wavelengths of the order of the fracture length.The frequency and decay rate, or attenuation, of these resonantmodes are captured by the location and width of spectral peaksin the fracture transfer function, as we demonstrate shortly.
Krauklis waves
Before discussing the fracture transfer function, we review somekey properties of Krauklis waves. This is most easily done in thecontext of an infinitely long fracture or fluid layer (with the constantinitial width w0) between identical elastic half-spaces. Seekingeiðkx−ωtÞ solutions to the governing equations, and neglecting inertiaof the solid (which is negligible for Krauklis waves at the lowfrequencies of interest to us), leads to the dispersion relation (Lip-ovsky and Dunham, 2015)
DKðk;ωÞ ¼�tan ξ
ξ− 1
��c0kω
�2
þ 1þ 2KG�kw0
¼ 0; (8)
where ξ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi−iw2
0ω∕4νp
, ν ¼ μ∕ρ is the fluid kinematic viscosity, kis the wavenumber, and ω is the angular fre-quency. Using the full linearized Navier-Stokesequation for the fluid and retaining inertia inthe solid give rises to a more complex dispersionrelation (Ferrazzini and Aki, 1987; Korneev,2008) that has fundamental and higher mode sol-utions. In contrast, our approximate fluid modelonly captures the fundamental mode. However,the higher modes exist only above specific cutofffrequencies that are well outside the frequencyrange of interest. Here, we first discuss solutionsto equation 8 for realω and complex k. The spatialattenuation of the modes is quantified by
1
2Q¼ Im k
Re k; (9)
where Q is the quality factor, approximately thenumber of spatial oscillations required for an ap-preciable decay of the amplitude. Plots of the phasevelocity and attenuation are presented in Figure 3.At high frequency (but still sufficiently low
frequency as to justify the ωw0∕c0 ≪ 1 approxi-mation), the phase velocity approaches the fluidsound speed because the fracture walls are effec-tively rigid relative to the compressibility of thefluid. At frequencies below
fel ¼ωel
2π¼ Kc0
πG�w0
; (10)
the elastic wall deformation becomes appreciable. This additionalcompliance leads to reduced phase velocity, given approximately as(Krauklis, 1962)
c ≈�G�w0ω
2ρ
�1∕3
: (11)
Lower frequency waves propagate slower than higher frequencywaves because the elastic walls are more compliant at longer wave-lengths. Viscous dissipation is confined to thin boundary layersaround the fracture walls.At even lower frequencies, below
fvis ¼ωvis
2π¼ 2μ
πρw20
; (12)
Table 1. Material properties.
Solid (rock)
Density ρs (kg∕m3) 2489
P-wave speed cp (m∕s) 4367
S-wave speed cs (m∕s) 2646
Fluid (water)
Density ρ0 (kg∕m3) 1000
Sound wave speed c0 (m∕s) 1500
Viscosity μ (Pa:s) 0.001
Wavelength (m)
Tem
pora
l atte
nuat
ion
1/2Q
10–3
10–2
10–1
100
101
102
0.5 mm
1 mm
3 mm
10 mm
Wavelength (m)
Pha
se v
eloc
ity (
m/s
)
100
101
102
103
1 2 5 10 20 50 200 1000 3000
0.5 mm
1 mm3 mm
10 mm Cutoff Limit
1 2 5 10 20 50 200 1000 3000
Frequency (Hz)10–3 10–210–1 100 101 102 103 104 105 106
Spa
tial A
ttenu
atio
n 1/
2Q
10–5
10–4
10–3
10–2
10–1
100
0.5 mm1 mm3 mm10 mm
fvis
fel
Pha
se v
eloc
ity (
m/s
)
10–1
100
101
102
103
0.5
mm
1 m
m
3 m
m10 m
m
Frequency (Hz)10–3 10–210–1 100 101 102 103 104 105 106
fvis
fel
a)
b)
c)
d)
Figure 3. (a) Phase velocity and (b) spatial attenuation of Krauklis waves for a real fre-quency. (c) Phase velocity and (d) temporal attenuation for a real wavelength. Dispersionand attenuation curves are plotted for four fracture widths: 0.5, 1, 3, and 10mm, as labeled.The dashed black and red lines in (a and b) mark the characteristic frequencies fvis and feldefined in the text. Dashed lines in (c and d) mark the cutoff wavelength λc, beyond whichwaves cease to propagate.
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viscous effects are felt across the entire width of the fluid layer andthe velocity v approaches the well-known parabolic Poiseuille flowprofile. Viscous dissipation is sufficiently severe as to damp outwaves over only a few cycles of oscillation; in addition, phasevelocity decreases below that given in equation 11.An important consequence of dissipation, from the viscous ef-
fects and from elastic-wave radiation, is that Krauklis-wave reso-nance will only occur in sufficiently short fractures. We can gaindeeper insight by plotting the phase velocity and attenuation ofKrauklis waves for real wavelength λ and complex frequency inFigure 3c and 3d, which correspond to standing waves formed bypairs of counterpropagating Krauklis waves. The temporal attenu-ation is quantified by
1
2Q¼ Imω
Reω; (13)
for temporal quality factor Q. When the wavelength exceeds a cut-off wavelength (Lipovsky and Dunham, 2015)
λc ¼ 2π
�60μ2
G�ρw50
�−1∕3; (14)
the phase velocity drops to zero and temporal attenuation diverges.This expression, corresponding to the dashed lines in Figure 3c and3d, provides a crude estimate of the maximum length of fracturesthat can exhibit resonant oscillations. However, it is essential to ac-count for additional dissipation that occurs from the decreased widthnear the fracture tip, and this is best done numerically. We thereforequantify the detectability limits of the fractures, using our 2D plane-strain simulations, in a later section.
Fracture transfer function
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-ture mouth, where the fracture connects to the wellbore. This isdone through the dimensionless fracture transfer function FðωÞ thatrelates pressure pð0; tÞ and width-averaged velocity uð0; tÞ at thefracture mouth:
uð0;ωÞ ¼ FðωÞρc0
pð0;ωÞ; (15)
where uð0;ωÞ and pð0;ωÞ are the Fourier transform of width-aver-aged velocity and pressure, respectively. The fracture transfer func-tion is related to the more commonly used fracture hydraulicimpedance ZfðωÞ defined in equation 1 by
FðωÞ ¼ ρc0∕Af
ZfðωÞ; (16)
where Af is the cross-sectional area of the fracture mouth. We havenondimensionalized FðωÞ using the fluid acoustic impedance ρc0,such that FðωÞ ¼ 1 for an infinitely long layer of inviscid, com-pressible fluid between parallel, rigid walls.To calculate the fracture transfer function, numerical simulations
such as that in Figure 2 are performed by imposing the pressure atthe fracture mouth pð0; tÞ and measuring the resulting width-aver-aged velocity at the fracture mouth uð0; tÞ. Figure 4 illustrates thesensitivity of uð0; tÞ to the fracture geometry, given the same chirpinput pð0; tÞ.Then, pð0; tÞ and uð0; tÞ are Fourier transformed and the fracture
transfer function is calculated using equation 15. An example isshown in Figure 5. Figure 6 compares the amplitude of the transferfunction jFj for fractures of different lengths and widths. The trans-fer functions exhibit multiple spectral peaks, corresponding to theresonant modes of the fracture (with a constant pressure boundarycondition at the fracture mouth). These peaks are finite because ofdissipation from viscosity and seismic radiation. Longer and nar-rower fractures have lower resonance frequencies, which is due tohigher compliance and slower Krauklis-wave phase velocities.We next examine the asymptotic behavior of the fracture transfer
function at low frequencies, which facilitates comparison withquasi-static fracture models that have been widely used in theliterature on hydraulic impedance testing and fracture diagnostics us-ing water-hammer signals (Holzhausen and Gooch, 1985a, 1985b;Holzhausen and Egan, 1986; Paige et al., 1995; Mondal, 2010; Careyet al., 2015). By low frequency, we mean ω ≪ cðωÞ∕L, where cðωÞis the Krauklis-wave phase velocity and L is the fracture length. Us-ing equation 11 for cðωÞ, the low-frequency condition is ω ≪ðG�w0L3∕2ρÞ1∕2 or ω∕2π ≪ 15 Hz for L ∼ 1 m and w0 ∼ 1 mm.The low-frequency condition results in effectively uniform pressureacross the length of the crack (at least when viscous pressure lossescan be neglected) and the fracture response can be described by amuch simpler model. The global mass balance for the fracture, withinthe context of our linearization, is
w0uð0; tÞ ≈ddt
ZL
0
wðx; tÞdx; (17)
where we have assumed that the fluid is effectively incompressible atthese low frequencies, such that inflow of fluid (the left side) is bal-
0
0.5
1
1.6 mm
10 mm
14 mm/s
Pre
ssur
e (k
Pa)
Time (s)
Vel
ocity
(m
m/s
)
0 0.5 1.0 1.5 2.0
4 mm
2 mm
0.8 mm
–0.5
a)
b)
Figure 4. Response of fractures to (a) broadband pressure chirp atthe fracture mouth, quantified through (b) width-averaged velocityat the fracture mouth. The response, shown for five fractures havingthe same length (L ¼ 10 m) and different widths (the value givenabove each curve), is characterized by several resonance frequen-cies. The resonance frequency decreases as the width decreases, asanticipated from the dependence of the Krauklis-wave phase veloc-ity on the width (equation 11 and Figure 3a). Higher frequenciesdecay quickly, leaving only the fundamental resonant mode. Widerfractures experience less viscous dissipation (Figure 3b), and hencethey oscillate longer.
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anced by changes in width (the right side). Given that the fracture isvery thin and approximately planar, the change in the opening is thenrelated to the pressure change using the well-known solution for aplane strain mode I crack in an infinite medium (Lawn, 1993). Theresult is
uð0; tÞ ≈ πL2
4w0G�dpdt
: (18)
It follows, upon Fourier transforming equation 18 and using equa-tion 15, that
FðωÞ ≈ −iπL2ρc04G�w0
ω as ω → 0: (19)
This asymptotic behavior is essentially the same as that quantified byHolzhausen and Gooch (1985a) as the fracture capacitance, althoughour result is for the 2D plane strain case to permit comparison withour simulations.The low-frequency asymptotes (equation 19) are plotted as dashed
lines in Figures 5a (inset) and 6, verifying that our numerical simu-
lations accurately capture the quasi-static response. However, thisasymptotic behavior only applies in the extreme low-frequency limit(far below the first resonance frequency) and deviates substantiallyfrom the actual response at higher frequencies. The traditional hy-draulic impedance testing method (Holzhausen and Gooch, 1985a;Holzhausen and Egan, 1986) thus leaves the vast spectrum at higherfrequencies unexplored.We also derive an approximate analytical solution to the transfer
function based on the dispersion relation (rather than using simula-tions). This dispersion-based approach has been used in several stud-ies (Hornby et al., 1989; Kostek et al., 1998a; Henry, 2005; Derovet al., 2009), and it is illustrative to compare it with the more rigoroussimulation-based solution. The solution, derived in Appendix A, as-sumes uniform width and imposes a zero-velocity boundary condi-tion at the fracture tip. The resulting fracture transfer function is
FðωÞ ¼ −ikðωÞc0 tan½kðωÞL�ω
; (20)
where kðωÞ is the complex wavenumber obtained by solving thedispersion equation 8, assuming real angular frequency ω.Our numerical simulations permit an exploration of how the de-
creased width near the fracture tip alters the fracture response, relativeto the more idealized models based on dispersion (equation 20) thatassume tabular (uniform-width) fractures. Figure 7 compares transferfunctions for 1 m long fractures with the same width (2 mm) at thefracture mouth but different profiles near the tip. These range from a
−
0 10 20 30 40 50
–100
0
100
200
–100
0
100
200
0 3 61 2 4 5
Frequency (Hz)
Re F(ω)
Im F(ω)
Quasi-staticapproxmation
a)
0 10 20 30 40 500
50
100
150
200
250b)
0 10 20 30 40 50
π/2c)
π/4
0
/4
− /2
Re
F(ω
) an
d Im
F(ω
)A
mpl
itude
of F
(ω)
Pha
se o
f F(ω
)
Frequency (Hz)
Frequency (Hz)
Frequency (Hz)
π
π
Figure 5. Fracture transfer function FðωÞ for a 10 m long fracture of5 mm width at the fracture mouth and the profile shown in Figure 6a.(a) Real and imaginary parts of FðωÞ. Inset shows low-frequencyresponse, which is compared with the quasi-static approximation(equation 19) used in hydraulic impedance testing. Also shown arethe (b) amplitude and (c) phase of FðωÞ.
0 50 100 150 200
Am
plitu
de |F
|
0
50
100
150L = 3.6 m; w = 2.1 mm
0 50 100 150 200
Am
plitu
de |F
|
0
50
100
150L = 3.6 m; w = 5.9 mm
Frequency (Hz)0 50 100 150 200
Am
plitu
de |F
|
0
50
100
150L = 3.6 m; w = 12.3 mm
0 50 100 150 200
Am
plitu
de |F
|
0
50
100
150L = 1 m; w = 4.7 mm
0 50 100 150 200
Am
plitu
de |F
|
0
50
100
150L = 2.2 m; w = 4.7 mm
Frequency (Hz)0 50 100 150 200
Am
plitu
de |F
|
0
50
100
150L = 6.5 m; w = 4.7 mm
0 0.2 0.4 0.6 0.8 1
y/w
0
–1
0
1
x/L
a)
b) c)
Figure 6. Fracture transfer functions for different fracture lengthsand widths. (a) Normalized fracture geometry with the fracture tipwidth equal to 1∕50 of the width at the fracture mouth. (b) Fracturetransfer functions for 4.7 mm wide fractures with varied lengths.Longer fractures have lower resonance frequencies. (c) Fracturetransfer functions for 3.6 m long fractures with varied widths. Widerfractures have higher resonance frequencies. In panels (b and c), thedashed lines plot the quasi-static limit (equation 19).
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tabular fracture to ones in which the width at the tip is decreased toonly 1∕50 of the maximum width. As the fracture tip width is re-duced, the resonance frequencies and the fracture transfer functionamplitude at resonance peaks are shifted to lower values. This isdue to higher viscous dissipation and slower Krauklis-wave phasevelocity for narrower fractures. The differences are most pronouncedat higher frequencies and for higher resonant modes. This findinghighlights the importance of accounting for the near-tip geometrywhen using high-frequency data to determine fracture geometry.The tabular or flat fracture model (equation 20) can lead to substantialerrors; in this example, there is approximately 20% error in the fun-damental frequency and 100% error in the amplitude of the funda-mental mode spectral peak.
Fracture geometry inference and detectability limits
As evidenced in Figures 4–7, the fracture response, as embodiedby the fracture transfer function, is sensitive to fracture geometry(length and width). We now consider the inverse problem, thatis, determining the fracture geometry from properties of the reso-nant modes of the fracture. Lipovsky and Dunham (2015), buildingon work by Korneev (2008) and Tary et al. (2014), show how mea-surements of the resonance frequencies and attenuation or decay ratesassociated with resonant modes could be used to uniquely determinethe length and width. Their work uses dispersion relations derived forharmonic waves in an infinitely long fluid layer, and we have seen, inFigure 7, some notable discrepancies as compared with our numericalsimulations of finite-length fractures. We thus revisit this problem,focusing in particular on the detectability limits.As the fracture length grows, the resonance wavelengths increase
and eventually reach a cutoff limit, beyond which the temporal at-tenuation diverges (Figure 3d). This provides the most optimisticestimate of fracture detectability using resonance; noise in real mea-surements will further limit the detectability.
Figure 8a shows a graphical method for estimating fracturegeometry from measurements of the frequency and temporal qualityfactor Q of the fundamental mode. This is similar to Figures 6–8 inLipovsky and Dunham (2015), but here the fundamental modeproperties are determined from transfer functions from numericalsimulations using the tapered fracture shape shown in Figure 6a.Specifically, Q ¼ f0∕Δf for fundamental frequency f0 and Δf, thefull width at half of the maximum amplitude in a plot of jFj (Fig-ure 6b). We also provide, in Figure 8b and 8c, a comparison betweenour numerical results (for a tapered width) and predictions based onthe dispersion relation (for a uniform width). Note that the dispersionsolutions are calculated for zero velocity at the fracture tip and con-stant pressure at the fracture mouth, whereas Lipovsky and Dunham(2015) assume zero velocity at both tips. Viscous dissipation is under-estimated using the dispersion-based approach, which ignores thenarrow region near the fracture tip. There are also differences inthe resonance frequency; the finite-length fractures in our numericalsimulations are stiffer, and hence they resonate at higher frequencies,than those suggested by the dispersion-based method.We finish this section by noting that quantitative results in Fig-
ure 8 are specific to the 2D plane strain problem. We anticipate thatan axisymmetric (penny-shaped) fracture would have a similar re-sponse, but we caution that these results are unlikely to apply to 3Dfractures when the length greatly exceeds the height. That case,which is of great relevance to the oil and gas industry, warrantsstudy using 3D numerical simulations.
TUBE-WAVE INTERACTIONWITH FLUID-FILLEDFRACTURES
Having focused on the fracture response in the previous sections,we now return to the overall problem of determining the response ofthe coupled wellbore-fracture system. We present a simple modelfor low-frequency tube waves, and we derive reflection/transmis-sion coefficients that quantify tube-wave interaction with a fractureintersecting the wellbore. We then examine the system response toexcitation at the wellhead or at one end of a sealed interval, notingthe possibility of a matched resonance between tube waves in thewellbore and Krauklis waves in the fracture.
Tube-wave governing equations and wellbore-fracturecoupling
Let z be the distance along the well. The wellbore, with a constantradius a and cross-sectional area AT ¼ πa2, is intersected by a frac-ture at z ¼ 0 with aperture w0 at the fracture mouth. Low-frequencytube waves are governed by the linearized momentum and massbalance equations (and the latter is combined with linearized con-stitutive laws for a compressible fluid and deformable elastic solidsurrounding the wellbore):
ρ
AT
∂q∂t
þ ∂p∂z
¼ 0; (21)
AT
M∂p∂t
þ ∂q∂z
¼ −AfuðtÞδðzÞ; (22)
where qðz; tÞ is the volumetric flow rate along the wellbore, pðz; tÞis the pressure, uðtÞ is the velocity into the fracture at the fracture
0 100 200 300 400 500 6000
100
200
300
400
Frequency (Hz)
Am
plitu
de |F
|
Flat fracture
1/5
1/10
1/20
1/50
Dispersion-based approximation
Quasi-static approximation
0 0.2 0.4 0.6 0.8 1−1
01
x (m)
y (m
m)
a)
b)
Figure 7. (a) Geometry of a 1 m long fracture with the tip widthvaried from 1, 1∕5, 1∕20, to 1∕50 of the width at the fracture mouth(2 mm). (b) Amplitude of the fracture transfer function with variedfracture tip width compared with the dispersion-based approxima-tion solution (equation 20) and the quasi-static limit (equation 19).
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mouth, ρ is the fluid density (assumed to be the same as in the frac-ture), M is a modulus that is typically close to the fluid bulk modu-lus (Biot, 1952), and Af is the fracture mouth area. For a fractureintersecting the wellbore perpendicularly, as we assume in the exam-ples presented below, Af ¼ 2πaw0. This can be generalized to ac-count for fractures intersecting the wellbore at other angles (Hornbyet al., 1989; Derov et al., 2009). These equations describe nondisper-sive wave propagation at the tube-wave speed cT ¼ ffiffiffiffiffiffiffiffiffiffi
M∕ρp
, whichwe solve in the frequency domain using Fourier transforms. How-ever, they could also be solved by a time-domain finite-differencemethod (Liang et al., 2015; Karlstrom and Dunham, 2016) or bythe method of characteristics (Mondal, 2010; Carey et al., 2015), bothof which permit spatial variation of properties. In all examples shownbelow, we set the wellbore diameter to 2a ¼ 0.1 m and assumeM ¼K (and hence cT ¼ c0) for simplicity, although expressions are givenfor the general case. The model can, of course, be generalized to ac-count for permeable walls (Tang, 1990), irregularly shaped boreholes(Tezuka et al., 1997), spatially variable properties (Chen et al., 1996;Wang et al., 2008), and friction (Livescu et al., 2016). The sourceterm on the right side of the mass balance equation 22 describesthe mass exchange between the wellbore and the fracture at the frac-ture mouth. Because the fracture aperture is much smaller than thetube-wave wavelength, the source term is approximately a delta func-tion at the fracture location.Integrating equations 21 and 22 across the junction at the well-
bore-fracture intersection, we obtain the following jump conditions:
qð0þ; tÞ − qð0−; tÞ ¼ −AfuðtÞ; (23)
pð0þ; tÞ − pð0−; tÞ ¼ 0: (24)
The pressure is continuous across the junction, whereas the volu-metric flow rate (and fluid particle velocity through the wellbore)experiences a jump that accounts for mass exchange with the frac-ture. The volumetric flow rate into the fracture AfuðtÞ is related tothe pressure at the fracture mouth pð0; tÞ through the fracture transferfunction using equation 15 (or, equivalently, the fracture hydraulicimpedance ZfðωÞ). Although we focus on one intersecting fracturein this study, extension to multiple fractures is straightforward byadding multiple jump conditions similar to equations 23 and 24.By Fourier transforming equations 23 and 24 and using equation 15,we obtain
qð0þ;ωÞ− qð0−;ωÞ¼−AfFðωÞρc0
pð0;ωÞ¼−pð0;ωÞZfðωÞ
: (25)
Tube-wave reflection/transmission coefficients
We now derive the reflection and transmission coefficients oftube waves incident on the fracture. Assuming an infinitely longwell, we seek a Fourier-domain solution of the form:
pðz;ωÞ ¼�eikz þ Re−ikz; z < 0;Teikz; z > 0;
(26)
where k ¼ ω∕cT is the wavenumber, the incident wave has unit am-plitude, and R and T are the reflection and transmission coefficients,
respectively. Satisfying the governing equations 21 and 22 and thefracture junction conditions (equations 24 and 25) yields
RðωÞ ¼ −rðωÞ∕2
1þ rðωÞ∕2 ¼ −1
1þ 2∕rðωÞ ; (27)
TðωÞ ¼ 1
1þ rðωÞ∕2 ¼ 2∕rðωÞ1þ 2∕rðωÞ ; (28)
Ave
rage
hal
f wid
th (
mm
)
0.1
0.2
0.5
1
2
5
Fracture length L (m)1 2 5 10 20 50 100
Fracture length L (m)1 2 5 10 20 50 100
Fracture length L (m)1 2 5 10 20 50 100
0.1
0.2
0.5
1
2
5
0.1
0.2
0.5
1
2
5A
vera
ge h
alf w
idth
(m
m)
Ave
rage
hal
f wid
th (
mm
)
Q = 0.5
2
5
1015
20
0.1
HZ
f 0=
0.3
HZ
0.7
HZ
1.9
HZ
5.2
HZ
13.9
HZ37
HZ
100
HZ
Overdamped region
Q = 0.5
10
20
Q = 0.5
1020
100
Numerical solution
Solution based on dispersion
Numerical solution
Solution based on dispersion
Q=0.5
0.1
HZ
1 H
Z
10
HZ
100
HZ
0.1
HZ
1 H
Z
10 H
Z
100
HZc)
b)
a)
Figure 8. (a) Graphical method to determine the fracture length andwidth from frequency (red curves) and temporal quality factor Q(black curves) of the fundamental resonant mode (i.e., from the frac-ture transfer function from numerical simulations using the taperedfracture 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(c) frequency, comparing numerical simulation results with solutionsbased on the dispersion relation (equation 8).
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in which
rðωÞ ¼ ZT
ZfðωÞ¼ ρcT∕AT
ρc0∕AfFðωÞ (29)
is the ratio of the hydraulic impedance of tube waves in the well-bore, ZT ≡ ρcT∕AT , to the fracture hydraulic impedance ZfðωÞ. Thefactor of two in the expressions for RðωÞ and TðωÞ arises because apair of tube waves propagate away from the fracture.Figure 9 explores the relation between the fracture transfer func-
tion and the reflection/transmission coefficients. The maximum re-flection approximately coincides with the resonance peaks in thetransfer functions, corresponding to the eigenmodes of the fracturewith a constant pressure boundary condition at the fracture mouth.At these frequencies, the hydraulic impedance of the fracture ismuch smaller than that of tube waves (i.e., rðωÞ∕2 ≫ 1), and onlysmall pressure changes at the fracture mouth are required to inducea large flow into or out of the fracture. The reflection coefficientgoes to −1 in this limit, so that waves at these specific frequenciesare reflected as if from a constant pressure boundary. We also notethat reducing the fracture width decreases reflection because thefracture mouth area decreases relative to the wellbore cross-sec-tional area AT and because the fracture resonance peak amplitudedecreases due to increased viscous dissipation in narrower fractures(captured in F or Zf).
Response of the wellbore-fracture system
We now consider a finite-length section of the well intersected bya single fracture, with boundary conditions prescribed at both endsof the well section. These ends might coincide with the wellheadand well bottom or the two ends of a sealed interval. Let h1 andh2 be the lengths of the two sections above and below the fracture,respectively, with the fracture at z ¼ 0 as before. We seek the pres-sure and velocity within the wellbore, given some excitation at theend of the upper well section.Equations 21 and 22 are supplemented with boundary conditions
at the top and bottom of the wellbore. At the top (z ¼ −h1), we setthe volumetric flow rate into the well equal to a prescribed injectionrateQðtÞ, not to be confused with the quality factor discussed earlier:
qð−h1; tÞ ¼ QðtÞ: (30)
In Appendix B, we show that this boundary condition is mathemati-cally equivalent to the case of a sealed end (i.e., qð−h1; tÞ ¼ 0) with amonopole source placed within the wellbore just below the end. Atthe bottom (z ¼ h2), we assume a partially reflecting condition of theform
pðh2; tÞ − ZTqðh2; tÞ ¼ Rb½pðh2; tÞ þ ZTqðh2; tÞ�; (31)
in which Rb is the well bottom reflection coefficient (satisfyingjRbj ≤ 1). This boundary condition can be equivalently written interms of the hydraulic impedance Zb at the bottom of the well:
pðh2; tÞ ¼ Zbqðh2; tÞ; Zb ¼1þ Rb
1 − RbZT: (32)
We assume constant, real Rb (and hence Zb), although it is possibleto use a frequency-dependent, complex-valued Rb if desired. ForRb ¼ 0, there is no reflection from the bottom, Rb ¼ 1 correspondsto qðh2; tÞ ¼ 0, and Rb ¼ −1 corresponds to pðh2; tÞ ¼ 0.The solution to the stated problem is
pðz;ωÞ¼�a1 sinðkzÞþa2 cosðkzÞ; −h1<z<0;b1eikzþb2e−ikz; 0<z<h2;
(33)
ZTqðz;ωÞ ¼�−ia1 cosðkzÞ þ ia2 sinðkzÞ; −h1 < z < 0;
b1eikz − b2e−ikz; 0 < z < h2;
(34)
where k ¼ ω∕cT and the coefficients a1, a2, b1, and b2 are deter-mined by the top boundary condition (equation 30), bottom boundarycondition (equation 31), and fracture junction conditions (equa-tions 24 and 25):
a1 ¼iZTQðωÞDðωÞ ; (35)
a2 ¼ZTQðωÞ
½rðωÞ þ ΛðωÞ�DðωÞ ; (36)
Frequency (Hz)0 10 20 30 40 50
Am
plitu
de |F
|
0
50
100
150
200
250
Frequency (Hz)0 10 20 30 40 50
Am
plitu
de |R
|, |T
|
0
0.2
0.4
0.6
0.8
1
w0 = 1 mm w0 = 5 mm
w0 = 1 mm, |R |w0 = 1 mm, |T |
w0 = 5 mm, |R |
w0 = 5 mm, |T |
a)
b)
Figure 9. (a) Amplitude of the fracture transfer functions jFj fortwo 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-bore diameter 2a ¼ 0.1 m. Maximum reflection occurs at the res-onance frequencies of the fracture.
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b1 ¼ZTQðωÞ
ð1þ Rbe2ikh2Þ½rðωÞ þ ΛðωÞ�DðωÞ ; (37)
b2 ¼Rbe2ikh2ZTQðωÞ
ð1þ Rbe2ikh2Þ½rðωÞ þ ΛðωÞ�DðωÞ ; (38)
where
DðωÞ ¼ cosðkh1Þ −i sinðkh1Þ
rðωÞ þ ΛðωÞ ; (39)
ΛðωÞ ¼ 1 − Rbe2ikh2
1þ Rbe2ikh2; (40)
and rðωÞ is the hydraulic impedance ratio defined as before (equa-tion 29). Solutions in the time domain are obtained by inverting theFourier transform.
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-hammersignals generated by excitation at the wellhead. The wellbore has atotal length of 3 km, a diameter of 2a ¼ 0.1 m, and a partiallysealed bottom with reflection coefficient Rb ¼ 0.8. A single fractureis placed 2 km from the wellhead and 1 km from the well bottom(i.e., h1 ¼ 2 km and h2 ¼ 1 km). At the well-head, we prescribe a broadband chirp (up to ap-proximately 500 Hz) in velocity, as shown inFigure 10. As mentioned earlier and detailedin Appendix B, this is equivalent to placing a mo-nopole source a short distance below the sealedwellhead. Pressure perturbations could be gener-ated by abruptly shutting in the well, as is done inhydraulic impedance testing (Holzhausen andEgan, 1986; Carey et al., 2015). However, an en-gineered source would provide better control overthe source spectrum. To account for dissipationduring tube-wave propagation along the wellbore,we add a small imaginary part to the tube-wavespeed: cT ¼ ð1 − 10−3iÞ ffiffiffiffiffiffiffiffiffiffi
M∕ρp
. Figure 11 showsa schematic of this system and the synthetic bore-hole record section. The interplay between tubewaves in the wellbore and dispersive Kraukliswaves within the fracture is evident.
Matched resonance
The example shown in Figure 11 illustrates theresponse when the entire long wellbore is hy-draulically connected to the fracture. Distinct re-flections can be seen, and interference betweendifferent reflections is confined to short time in-tervals. We next examine the more complex re-sponse that arises when a much smaller section
of the well around the fracture is sealed at both ends and a sourceis placed at one end. The solution given in equations 33–40 stillapplies, but we select a smaller length h1. There is now complexinterference between multiply reflected waves; whether this inter-ference is constructive or destructive depends on the well sectionlengths h1 and h2 and the fracture reflection/transmission coeffi-cients (and hence frequency). As we demonstrate, the signals in
0 0.005 0.01 0.015 0.02 0.025 0.03
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–2
0
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4
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)0 200 400 600 800 1000
Frequency (Hz)
0
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am
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Figure 10. Chirp used for examples shown in Figures 11 and 12:(a) time series and (b) spectrum.
Sealed bottom
Source inputat wellhead
Sensor TP
Sensor BT
h1
Pre
ssur
eP
ress
ure
a) b) c)
d)
4 kPa
Time (s)0 1 2 3 4 5
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e fr
om fr
actu
re, z
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)
0
–0.5
–1.0
–1.5
–2.0
0.5
1.0
w0 = 2 mm
Sensor TP
Sensor BT
L = 3 mL = 6 mL = 10 m
Time (s)1.5 1.6 1.7 1.8 1.9 2 2.1 2.2
Sensor TP
Sensor BT
w0 = 0.4 mmw0 = 1.0 mmw0 = 5.0 mm
L = 10 m
Time (s)1.5 1.6 1.7 1.8 1.9 2 2.1 2.2
h2
1 kPa
1 kPa
Figure 11. Response of the wellbore-fracture system to chirp excitation (Figure 10) atthe wellhead. (a) Schematic of the system, with one fracture 2 km below the wellheadand 1 km above the well bottom. Sensor TP is 500 m above the fracture, and sensor BT is250 m below the fracture. (b) Record section of pressure along the wellbore (every250 m) with a fracture that is 10 m long and 2 mm wide at the fracture mouth. Multiplereflections from the fracture, well bottom, and wellhead are observed. Dashed boxesmark the time window (1.5–2.2 s) examined in panels (c and d) for sensors TP andBT. (c) Comparison of pressure at sensors TP and BT for fractures with the same widthbut different lengths. (d) Same as panel (c) but for fractures with the same length butdifferent widths. In panels (c and d), sensor TP shows waves reflected from the fracture;first arriving are direct tube-wave reflections from the fracture mouth, followed by tubewaves generated by Krauklis waves in the fracture that have reflected from the fracturetip. Sensor BT shows transmitted waves, which have similar arrivals. Longer fracturesshow a more dispersed set of Krauklis-wave arrivals. Reflections are smaller from nar-rower fractures.
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the upper well section that contains the source can be selectivelyamplified when a resonance frequency of the upper well sectionis tuned to one of the resonance frequencies of the fracture. We referto this phenomenon as matched resonance.Reflection of waves from the fracture is most pronounced at
frequencies that permit maximum exchange of fluid between thewellbore and fracture. These frequencies correspond approximatelyto the resonance frequencies of the fracture with a constant pressureboundary condition at the fracture mouth (i.e., corresponding to thepeaks of the fracture transfer function; see Figure 9). The resonancefrequencies of the upper wellbore section with a sealed top end
(q ¼ 0) and constant pressure condition (p ¼ 0) at the fracturesatisfy cosðωh1∕cTÞ ¼ 0. For example, the lowest resonance fre-quency is f ¼ ω∕2π ¼ cT∕4h1. Matched resonance occurs whenone of these frequencies matches a fracture resonance frequency.To justify this more rigorously, and explain some possible com-
plications, we note that the tube-wave eigenfunctions associatedwith this resonant wellbore response are sinðωz∕cTÞ, correspondingto the first term on the right side of equation 33. The amplitude ofthis term is given in equation 35, which is largest when the denom-inatorDðωÞ, given in equation 39, is smallest. The hydraulic imped-ance ratio is quite large (ideally, jrðωÞj ≫ 1) around the fractureresonance frequencies. A further requirement is that jΛðωÞj be suf-ficiently small compared with jrðωÞj, at least around the targetedfracture resonance frequency. When these conditions are satisfied,DðωÞ ≈ cosðkh1Þ with k ¼ ω∕cT . Setting DðωÞ ¼ 0 then yields thewellbore section resonance condition cosðωh1∕cTÞ ¼ 0 statedabove. Of course, this resonance condition is only an approxima-tion, especially when jrðωÞj does not greatly exceed unity or whenjΛðωÞj is not small. In these cases, the matched resonance conditioncan be more precisely determined by finding the frequencies ω thatminimize the exact DðωÞ in equation 39. However, ΛðωÞ can bemade arbitrarily small by sealing the bottom end of the lower wellsection (Rb ¼ 1 and hence ΛðωÞ ¼ −i tanðωh2∕cTÞ) and decreas-ing h2, such that ωh2∕cT ≪ 1.Figure 12a and 12b illustrates matched resonance. The frequency
of the fundamental mode of the fracture (the 10 m long, 5 mm wideexample shown in Figures 5 and 9) is f ≈ 3.8 Hz, so the upper well-bore section is chosen to be h1 ¼ 100 m ≈cT∕4f long to satisfy thematched resonance condition. The system is excited by a broadbandchirp at the upper end of this wellbore section. The 3.8 Hz reso-nance frequency clearly dominates the system response. The re-sponse is shown for two bottom boundary conditions. The first isthe relatively simple case of a well with a nonreflecting bottom boun-dary (Rb ¼ 0, for which Λ ¼ 1 regardless of h2). This eliminates thepossibility of resonance within the lower wellbore section and pre-vents bottom reflections from then transmitting through the fractureto return to the upper well section. The second case has h2 ¼ 10 m
and Rb ¼ 0.9, corresponding to a partially sealed end a short distancebelow the fracture. Although the initial response in the second casecontains a complex set of high-frequency reverberations associatedwith reflections within the lower well section, these are eventuallydamped out to leave only the prominent 3.8 Hz resonance. This il-lustrates a rather remarkable insensitivity to characteristics of thelower well section because similar results (not shown) were foundeven for a wide range of lower well section lengths h2.We next demonstrate how changing the length of the upper well
section h1 alters the resonant response of the system. Figure 12c and12d shows the responses for values of h1 that are larger and smallerthan the matched resonance length. The fundamental mode reso-nance peak shifts to a lower frequency when h1 increases and toa higher frequency when h1 decreases. Many resonant modes areexcited for the large h1 case, leading to a rather complex pressureresponse. The situation is simpler for the small h1 case, but as h1continues to decrease, the amplitude of the fundamental mode peakcontinues to decrease and might be difficult to observe in noisy data.
Possible measuring techniques
Although it is easy to implement different excitation sources andreceiver configurations in our theoretical calculations, it is not trivial
0 2 4 6 8 10time (s)
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0 10 20 30 40 50frequency (Hz)
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h1= 50 m
h1= 200 m
h1= 50 m
h1= 200 m |R|, fracture reflection coefficient
(both: Rb
= 0.9)
Figure 12. Matched resonance: The length of the upper wellboresection containing the source is chosen so that the resonance fre-quency of tube waves in this wellbore section matches the fundamen-tal mode resonance frequency of the fracture (3.8 Hz). (a) Pressureresponse at a receiver in the center of the upper wellbore section (atz ¼ −h1∕2) to the chirp excitation shown in Figure 10. The 3.8 Hzresonance is selectively amplified, regardless of the length of thelower wellbore section and the bottom boundary condition: Comparethe case with a nonreflecting well bottom (red, Rb ¼ 0) to partiallysealed (blue, Rb ¼ 0.9 and h2 ¼ 10 m). (b) Fourier amplitude spec-trum of pressure time series shown in (a), with prominent peak at the3.8 Hz matched resonance frequency. Although additional spectralpeaks appear for the sealed bottom case, the matched resonance peakis nearly identical to the nonreflecting bottom case. Also, shown isthe reflection coefficient of tube waves from the fracture, which haspeaks at the fracture resonance frequencies. (c) Same as panel (a) butfor h1 ¼ 200 and 50 m (with Rb ¼ 0.9 and h2 ¼ 10 m). Thesystem does not satisfy the matched resonance condition and thespectrum, shown in panel (d), is more complicated and lacks the pro-nounced peak at the fundamental resonance mode.
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to make downhole pressure measurements in practice. In general,pressure is measured at the pump or in the pipe connecting thepump to the wellhead during hydraulic fracturing treatments. In-stalling downhole pressure transducers or hydrophones with com-plex pumping and treatment equipment in place during hydraulicfracturing would be challenging. In addition, downhole sensorscan occupy a significant volume of the borehole and create complextool effects on measured tube waves (Pardo et al., 2013). Quantify-ing such effects is beyond the scope of this study. A possible alter-native approach would be to install geophones or pressure gauges innearby monitoring wells. The model presented in this study wouldneed to be extended to predict the signals in the monitoring wells,and it is uncertain if these signals would have as much useful in-formation regarding the fracture geometry. Another technology thatreduces the influence of the tool on tube-wave propagation is fiber-optic distributed acoustic sensing (DAS) (MacPhail et al., 2012;Molenaar et al., 2012; Boone et al., 2015). DAS cables can runthrough grooves across swellable packers along the casing in thesealing elements in uncemented packer and sleeve completion, andthey can be buried in the cement in cemented plug-and-perf com-pletions (Boone et al., 2015). This technology not only avoids anyinteraction with tube waves, but it can also provide continuous mea-surements along the full length of the wellbore.
CONCLUSION
We have investigated the interaction of tube waves with fractures.Although this problem has received much attention in the literature,previous work has been restricted to relatively idealized analyticalor semianalytical models of the fracture response. In contrast, weadvocate the use of numerical simulations to more accurately de-scribe the fracture. These simulations account for variable fracturewidth, narrow viscous boundary layers adjacent to the fracture walls,and dissipation from viscosity and from seismic radiation. The sim-ulations feature Krauklis waves propagating along the fracture; coun-terpropagating pairs of Krauklis waves form the eigenmodes of thefracture. As many authors note, the frequency and attenuation ofthese modes can be used to constrain fracture geometry.Although this initial study uses a 2D plane strain fracture model,
the overall methodology can be applied when 3D models of thefracture are available. Such models would permit investigation offractures that have bounded height, a commonly arising situation inthe industry, and one for which the 2D model presented here is notwell-justified.We then showed how to distill the simulation results into a single,
complex-valued function that quantifies the fracture response, spe-cifically its hydraulic impedance (or the normalized reciprocal ofthe impedance, the fracture transfer function). The fracture transferfunction can then be used to determine how tube waves within awellbore reflect and transmit from fractures intersecting the well,and to solve for the response of the wellbore-fracture system toexcitation at the wellhead or within the wellbore.The coupled wellbore-fracture system has a particularly complex
response, potentially involving resonance within wellbore sectionsadjacent to the fracture or within the fracture itself. We found that itis possible to selectively amplify tube waves at the eigenfrequenciesof the fracture by properly choosing the length of the well sectioncontaining the source and intersecting the fracture. This phenome-non, which we call matched resonance, could prove useful to excite
and measure fracture eigenmodes, which can then be used to inferthe fracture geometry.
ACKNOWLEDGMENTS
This work was supported by a gift from Baker Hughes to theStanford Energy and Environment Affiliates Program and seedfunding from the Stanford Natural Gas Initiative. O. O’Reilly waspartially supported by the Chevron fellowship in the Department ofGeophysics at Stanford University.
APPENDIX A
DISPERSION-BASED FRACTURE TRANSFERFUNCTION
In this appendix, we use the Krauklis-wave dispersion relation toderive an approximate expression for the fracture transfer function.Let x be the distance into the fracture measured from the fracturemouth at x ¼ 0. The fracture has length L and uniform aperture w0.Solutions in the frequency domain are written as a superposition ofplane waves propagating in the þx and −x directions:
pðx;ωÞ ¼ Aeikx þ Be−ikx; (A-1)
ρcuðx;ωÞ ¼ Aeikx − Be−ikx; (A-2)
where k ¼ kðωÞ and c ¼ ω∕kðωÞ are the wavenumber and phasevelocity determined by the Krauklis-wave dispersion relation (equa-tion 8, solved for real ω and possibly complex k), and A and B arethe coefficients determined by boundary conditions at the ends ofthe fracture. At the fracture tip, the fluid velocity is set to zero:
ρcuðL;ωÞ ¼ AeikL − Be−ikL ¼ 0: (A-3)
The fracture transfer function, defined in equation 15, is obtained bycombining equations A-1–A-3:
FðωÞ ¼ −ikðωÞc0 tan½kðωÞL�ω
: (A-4)
Peaks in FðωÞ correspond to the resonance frequencies of thefracture with a constant pressure condition at the fracture mouth.Therefore, besides equation A-3, we have
pð0;ωÞ ¼ Aþ B ¼ 0: (A-5)
Combining equations A-3 and A-5, we obtain the resonance con-dition
cos kL ¼ 0; (A-6)
or knL ¼ ðn − 1∕2Þπ for positive integers n ¼ 1; 2; : : : . This canbe solved, assuming a real k and a complex ω, for the resonancefrequencies and (temporal) decay rates of the eigenmodes.We next derive the resonance frequencies for the case of negli-
gible dissipation (i.e., for an inviscid fluid). In this case, k and ω arereal, and we have (Krauklis, 1962)
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cðωÞ ¼ ðG�w0ω∕2ρÞ1∕3 ¼ ðG�w0πf∕ρÞ1∕3: (A-7)
It follows that the resonance frequencies are
fn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðπG�w0∕ρÞ½ðn − 1∕2Þ∕2L�3
q: (A-8)
This expression provides a reasonably accurate prediction of theresonance frequencies for the examples shown in this work, pro-vided that w0 in equation A-8 is interpreted as average width.
APPENDIX B
EQUIVALENCE OF MONOPOLE SOURCE ANDFLOW RATE BOUNDARY CONDITION
In this appendix, we establish the equivalence, for sufficientlylow frequencies, between a monopole source placed just below thesealed end of a wellbore section and a prescribed flow rate boundarycondition at that end. For convenience, we take z ¼ 0 to coincidewith the end. The tube-wave equations with a monopole sourcemðtÞ at z ¼ s are
ρ
AT
∂q∂t
þ ∂p∂z
¼ 0; (B-1)
AT
M∂p∂t
þ ∂q∂z
¼ mðtÞδðz − sÞ: (B-2)
For the sealed end, qð0; tÞ ¼ 0, whereas the prescribed flow rateboundary condition is
qð0; tÞ ¼ QðtÞ: (B-3)
We now show equivalence of these two problems, in the sense thatQðtÞ ≈mðtÞ, for frequencies satisfying
sω∕cT ≪ 1: (B-4)
This is done by requiring that the outgoing waves below the source(z > s) are identical for the two problems.The solution to equations B-1 and B-2 in a semi-infinite wellbore
with a sealed end at z ¼ 0 is
pðz;ωÞ ¼�a cosðωz∕cTÞ; 0 < z < s;beiωðz−sÞ∕cT ; z > s;
(B-5)
ZTqðz;ωÞ ¼�ia sinðωz∕cTÞ; 0 < z < sbeiωðz−sÞ∕cT ; z > s;
(B-6)
where a and b are the constants to be determined.Fourier transforming equations B-1 and B-2 and integrating
across the source yields the jump conditions across the source:
pðsþ;ωÞ − pðs−;ωÞ ¼ 0; (B-7)
qðsþ;ωÞ − qðs−;ωÞ ¼ mðωÞ: (B-8)
Constants a and b are determined by substituting equations B-5and B-6 into equations B-7 and B-8:
a ¼ ZTmðωÞeisω∕cT ; (B-9)
b ¼ ZTmðωÞ cosðsω∕cTÞeisω∕cT : (B-10)
Substituting equations B-9 and B-10 into equations B-5 and B-6,we obtain the solution below the source (z > s):
pðz;ωÞ ¼ ZTmðωÞ cosðsω∕cTÞeiωz∕cT ; (B-11)
qðz;ωÞ ¼ mðωÞ cosðsω∕cTÞeiωz∕cT : (B-12)
Similarly, we obtain the solution to the tube-wave problem withno internal source but with the top boundary condition beingqð0; tÞ ¼ QðtÞ:
pðz;ωÞ ¼ ZTQðωÞeiωz∕c; (B-13)
qðz;ωÞ ¼ QðωÞeiωz∕c: (B-14)
We now determine the condition for which the wavefield belowthe source (z > s) is identical between the two problems. Specifi-cally, we require the equivalence of equations B-11 and B-13 andsimilarly for equations B-12 and B-14. The necessary condition is
QðωÞ ¼ mðωÞ cosðsω∕cTÞ: (B-15)
Moreover, for sources placed just below the sealed end, or equiva-lently at sufficiently low frequencies, sω∕cT ≪ 1 and cosðsω∕cTÞ≈1. Thus, QðωÞ ≈ mðωÞ or QðtÞ ≈mðtÞ, as claimed.
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