Effect of fracture compliance on wave propagation within a fluid-filled fracture

$Page 1: Effect of fracture compliance on wave propagation within a fluid-filled fracture$
Effect of fracture compliance on wave propagation withina fluid-filled fracture

Seiji Nakagawa and Valeri A. KorneevEarth Sciences Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, MS74R120, Berkeley,California 94720

(Received 12 December 2013; revised 16 April 2014; accepted 25 April 2014)

Open and partially closed fractures can trap seismic waves. Waves propagating primarily within

fluid in a fracture are sometimes called Krauklis waves, which are strongly dispersive at low

frequencies. The behavior of Krauklis waves has previously been examined for an open, fluid-filled

channel (fracture), but the impact of finite fracture compliance resulting from contacting asperities

and porous fillings in the fracture (e.g., debris, proppants) has not been fully investigated. In this

paper, a dispersion equation is derived for Krauklis wave propagation in a fracture with finite

fracture compliance, using a modified linear-slip-interface model (seismic displacement-discontinuity

model). The resulting equation is formally identical to the dispersion equation for the symmetric

fracture interface wave, another type of guided wave along a fracture. The low-frequency solutions of

the newly derived dispersion equations are in good agreement with the exact solutions available for

an open fracture. The primary effect of finite fracture compliance on Krauklis wave propagation is to

increase wave velocity and attenuation at low frequencies. These effects can be used to monitor

changes in the mechanical properties of a fracture. [http://dx.doi.org/10.1121/1.4875333]

PACS number(s): 43.20.Mv, 43.20.Jr, 43.20.Bi [RKS] Pages: 3186–3197

I. INTRODUCTION

An open fracture containing fluid can carry highly

dispersive and slow seismic waves at low frequencies.1

These waves are sometimes called Krauklis waves, after

Pavel Krauklis’s pioneering theoretical work on the behavior

of the wave.2 Krauklis waves are a guided wave mode result-

ing from interactions between fluids in a fracture and its elas-

tic background. Similarly to borehole tube waves, most of

the wave’s energy is in the fluid motion parallel to the fluid–

solid interface. Many researchers have examined Krauklis

wave propagation along an open, fluid-filled fracture analyti-

cally and numerically, and have derived dispersion equations

(frequency equations) to explain the wave’s frequency-

dependent dispersive behavior.1,3,4 Recently, there has been

growing interest in the use of Krauklis waves for detection

and characterization of fractures in the field, particularly

hydraulically induced fractures. Korneev4 showed that it is

possible to induce the resonance of Krauklis waves within a

finite fracture, which can be related to the fracture length

and thickness. Dvorkin et al.5 also examined the resonance

of more general, proppant-filled fractures with varying

geometry, embedded within permeable and impermeable

backgrounds. However, because their analysis assumed a

rigid background (no deformation is allowed), the results are

not fully applicable to the behavior of Krauklis waves. Tary

and Van der Baan,6 observing the distinct resonances of seis-

mic waves during fluid injection in reservoir rock, argued

that these resonances may have been caused by low-

frequency Krauklis waves in the fractures.

Natural and induced subsurface fractures are likely

to have partial surface contacts and/or porous fillings (e.g.,

debris and proppants), which provide finite mechanical com-

pliance and reduced permeability to the fractures. To this

day, however, these effects have not been fully investigated.

Previously, dispersion equations for Krauklis waves were

derived for wave propagation within fluid contained in an

open gap.1–4 However, for investigating the effect of pore

pressure and diagenetic (chemical) changes in a fracture on

Krauklis wave propagation, both finite fracture compliance

and flow permeability of partially open and filled fractures

need to be considered. Pyrak-Nolte and Cook7 and Gu et al.8

examined the effect of fracture compliance on the dispersion

of Rayleigh surface waves coupled across a fracture. These

waves are called fracture interface waves (FIW), which can

have either symmetric or antisymmetric particule motions

across the fracture. Although one of these wave modes—the

symmetric FIW—can be related to Krauklis waves, fluid

flow within the fracture was not considered in the derivation

of its dispersion equation using the classical linear-slip-inter-

face model9 (LSIM).

Recently, the LSIM was extended for poroelastic

fractures, considering wave-induced fluid pressure and dis-

placement within a fracture.10,11 In the following, we use

this extended LSIM to derive dispersion equations for

Krauklis waves propagating along and within a flat, fluid-

filled fracture with finite mechanical compliance. For the

previously studied case of an open fracture containing a

mediating fluid layer, the results are compared to available

exact solutions.4,12 Subsequently, using the new equations,

we examine how fracture compliance affects the velocity

and attenuation of the Krauklis waves, together with other

parameters, including fracture thickness, fluid viscosity, and

fracture permeability.

II. THEORY

In this section, we first derive the extended LSIM for a

highly permeable fracture. Subsequently, using this model, dis-

persion equations for Krauklis waves and their low-frequency

3186 J. Acoust. Soc. Am. 135 (6), June 2014 0001-4966/2014/135(6)/3186/12/$30.00

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asymptotes are derived. A list of frequently used parameters is

given in Table I.

A. Extended linear-slip-interface model

An LSIM for a fracture can be derived by relating the

wave-induced stress and displacement on two parallel surfa-

ces bounding a compliant layer representing a fracture, under

the condition that the thickness of the layer is reduced to

zero. Previously, Nakagawa and Schoenberg11 used this

technique to derive an LSIM for poroelastic fractures.

Consider plane wave propagation within the Cartesian coor-

dinate system ðx1; x2; x3Þ shown in Fig. 1. The governing

equations of poroelastic wave propagation13 can be written

using the following coupled matrix equation:11

@

@x3

_u1

s33

�pf

s13

_u3

_w3

266666664

377777775¼ �ix

0 QXY

QYX 0

� �_u1

s33

�pf

s13

_u3

_w3

266666664

377777775; (1)

QXY �1=G n 0

n q qf

0 qf ~q

264

375;

QYX �

�4Gn2 1� G

HD

� ��

q2f � q~q

~qn 1� 2G

HD

� �n �

qf

~qþ a

2G

HD

� �

n 1� 2G

HD

� �1

HD� a

HD

n �qf

~qþ a

2G

HD

� �� a

HD

a2

HDþ 1

M� n2

~q

2666666664

3777777775:

ui and wi (i¼ 1,2,3) are the solid frame displacement and the

fluid displacement relative to the solid frame [i.e., the Darcy

flux wi ¼ /ðufi � uiÞ; where / is the porosity and uf

i is the

fluid displacement], respectively. sij (i,j¼ 1,2,3) are the total

stress and pf is the fluid pressure. Note that these displace-

ments and stresses are locally averaged quantities at the pore

scale. G, HD, a, M are the frame shear modulus, drained uni-

axial frame modulus (HD � KD þ 4G=3; where KD is the

drained bulk modulus), Biot-Wills effective stress coeffi-

cient, and the Biot’s storage modulus, respectively. q is the

bulk density, qf is the fluid density, and the parameter ~q� igf =xkðxÞ is defined via fluid viscosity gf and the

frequency-dependent permeability k(x).14 The over-dot “_”

indicates temporal partial differentiation @=@t. Assuming

plane waves with the same time t and frequency x depend-

ence expixðnx1 � tÞ, both time and the 1-direction spatial

differentiations can be substituted via @=@t$ �ix and

@=@x1 $ ixn, where n is the 1-direction wave slowness.

Also note that we consider only compressional and shear

waves with particle motions within the 1-3 plane, and do not

consider shear waves with particle motions parallel to the

fracture, which are independent of the other waves.

Therefore, @=@x2 ! 0.

Assuming that the material properties do not vary across

the fracture, Eq. (1) is integrated over the fracture thickness

h to obtain a set of boundary (or jump) conditions,

sþ31 � s�31 ¼ðh=2

�h=2

@s31

@x3

dx3 ¼ h 4GðxnÞ2 1þ 2

3

G

HD

� �þ x2

q2f

~q� q

!" #�u1

�ixnh 1� 2G

HD

� ��s33 � ixnh �

qf

~qþ a

2G

HD

� �ð��pf Þ; (2)

sþ33 � s�33 ¼ðh=2

�h=2

@s33

@x3

dx3 ¼ �ixnh@s31

@x1

� x2h q�u3 þ qf �w3ð Þ; (3)

�pþf � ð�p�f Þ ¼ðh=2

�h=2

@ð�pf Þ@x3

dx3 ¼ �x2h qf �u3 þ ~q �w3

� �; (4)

uþ1 � u�1 ¼ðh=2

�h=2

@u1

@x3

dx3 ¼h

G�s31 � ixnh�u3; (5)

J. Acoust. Soc. Am., Vol. 135, No. 6, June 2014 S. Nakagawa and V. A. Korneev: Wave propagation in a fluid-filled fracture 3187


uþ3 � u�3 ¼ðh=2

�h=2

@u3

@x3

dx3 ¼ ��D

1� �D

� �h@

@x1

�u1

þ h

HD�s33 � að��pf Þ�

; (6)

wþ3 �w�3 ¼ðh=2

�h=2

@w3

@x3

dx3¼� �qf

~qþa

2G

HD

� �ixnh�u1

�ah

HD�s33�að��pf Þ�

þ h

Mð��pf Þ�

hn2

~qð��pf Þ:

(7)

Note that the superscripts “þ” and “�” indicate quantities at

x3¼þh/2 and �h/2, respectively, and the overline “�” indi-

cates the average across the fracture. The linear-slip-inter-

face approach simplifies the above equations by taking the

thin-fracture-thickness limit O(h)!0. In this process, non-

vanishing fracture compliances are defined and maintained,

which are

gT � h=G (specific shear fracture compliance),

gD � h=HD (specific normal drained fracture compliance),

gM � h=M (specific storage fracture compliance).

For modeling a fracture with large fluid-flow permeabil-

ity, we must also define a parameter that does not vanish in

the thin-fracture limit. This is

~j � kðxÞh (fracture transmissivity).

Note that previously,11 for a low-permeability fracture,

another characteristic fracture-permeability parameter called

membrane permeability was defined by j � kðxÞ=h, which

allows discontinuous fluid pressure across a fracture. In this

paper, however, we consider only a high-permeability frac-

ture within which fluid pressure does not vary in the

fracture-normal direction.

By taking the thin-fracture limit and using these defini-

tions, the boundary conditions become

sþ31 � s�31 ¼ ð�ixÞqf ~jðxÞ

gf

ixnð��pf Þ þ x2qf �u1

h i¼ �qf hx2 �w1; (8)

sþ33 � s�33 ¼ 0; (9)

pþf � p�f ¼ 0; (10)

uþ1 � u�1 ¼ gT�s31; (11)

uþ3 � u�3 ¼ gND�s33 � að�pf Þ�

; (12)

wþ3 � w�3 ¼ �agND�s33 � að��pf Þ�

þ gMð��pf Þ

þ n~jðxÞgf

ixnð��pf Þ þ qf x2�u1

h i: (13)

Note that the following dynamic Darcy’s law is used:

~jðxÞgf

@ð��pf Þ@x1

� qf

@2�u1

@t2

� �¼ h

@ �w1

@t; (14)

FIG. 1. Cartesian coordinate system considered in the theory. Plane waves

are assumed to propagate along the 1-3 plane. Shear waves with particle

motions in the 2-direction are decoupled from other waves and are not

considered.

TABLE I. Frequently used symbols.

Symbol Meaning Unit

ui solid displacement vector (i¼ 1, 2, 3) (m)

ufi fluid displacement vector (i¼ 1, 2, 3) (m)

wi Darcy flow flux vector (i¼ 1, 2, 3) (m)

sij total stress tensor (i, j¼ 1, 2, 3) (Pa)

pf fluid pressure (Pa)

x circular frequency (rad/s)

n fracture-parallel slowness (s/m)

nS S-wave slowness (s/m)

nP P-wave slowness (s/m)

nA acoustic wave slowness (s/m)

nS3 3-direction S-wave slowness (s/m)

nP3 3-direction P-wave slowness (s/m)

h fracture thickness (m)

G shear modulus (Pa)

HD drained P-wave modulus (Pa)

KD drained bulk modulus (Pa)

M storage modulus (Pa)

�D drained Poisson ratio

a Biot-Willis effective stress coefficient

q total density (kg/m3)

qf fluid density (kg/m3)

~q effective fluid density (kg/m3)

/ porosity

gf fluid viscosity (Pa s)

k(x) frequency-dependent permeability (m2)

k0 static permeability (m2)

gT specific shear fracture compliance (m/Pa)

gD specific drained normal fracture compliance (m/Pa)

gM specific storage fracture compliance (m/Pa)

g�M specific effective storage fracture compliance (m/Pa)

g�N specific effective normal fracture compliance (m/Pa)

~j fracture transmissivity (m3)– average across a fracture (overline)(B) background medium quantity (superscript)

3188 J. Acoust. Soc. Am., Vol. 135, No. 6, June 2014 S. Nakagawa and V. A. Korneev: Wave propagation in a fluid-filled fracture


which results directly from the linear momentum balance for

the fluid pressure in the 1-direction:13

@ð��pf Þ@x1

¼ qf

@2�u1

@t2þ ~q

@2 �w1

@t2: (15)

Because of the low-frequency approximation of LSIM, we

further neglect O(x2)-terms in the above equations. As a

result, the poroelastic linear-slip-interface model for a per-

meable fracture is written as

sþ31 ¼ s�31 ¼ �s31; (16)

sþ33 ¼ s�33 ¼ �s33; (17)

pþf ¼ p�f ¼ �pf ; (18)

uþ1 � u�1 ¼ gT�s31; (19)

uþ3 � u�3 ¼ gD �s33 � að��pf Þ�

; (20)

wþ3 � w�3 ¼ �agD �s33 � að��pf Þ�

þ gMð��pf Þ

þ ixn2~jðxÞ=gf � ð��pf Þ: (21)

Equations (16)–(18) state the continuity of shear and normal

stresses and fluid pressure across the fracture. Equation (19)

states the proportionality between the shear displacement

jump (slip) across the fracture and the shear stress, via the

specific shear fracture compliance gT. Similarly, Eq. (20)

states a linear relationship between the normal displacement

jump and the effective normal stress. The sixth and last

equation [Eq. (21)], the fluid flux discontinuity across a frac-

ture, can be viewed as a linearized mass (volume) conserva-

tion rule. The three terms in the right-hand side of this

equation consist of fluid flux due to fracture closure

�agD �s33 � að��pf Þ�

¼�a uþ3 � u�3� �

[via Eq. (20)],

pressure-induced volume changes of the fluid and solid

within the fracture gMð��pf Þ, and the fracture-parallel trans-

port of fluid ixn2~jðxÞ=gf � ð��pf Þ. The last two terms can be

combined to define an effective fracture storage compliance

parameter g�Mðx; nÞ � gM þ ixn2~jðxÞ=gf , which implies

that the wave-induced fluid flow in the fracture alters the

apparent compliance of the fluid.

B. Dispersion equations

For the current derivation, we assume that the back-

ground medium is impermeable. Therefore, there is no fluid

flux in the 3-direction (i.e., wþ3 � w�3 ¼ 0). A more general

case involving a permeable, poroelastic background medium

is considered in the Appendix. This relationship can be used

in Eq. (21) to express the fluid pressure �pf using the total

normal stress �s33 on the fracture:

��pf ¼agD

a2gD þ g�M�s33: (22)

The coefficient of �s33 in Eq. (22) can be viewed as an effec-

tive, uniaxial Skempton coefficient. This equation is used to

eliminate the pressure in the displacement discontinuity

equation, yielding

uþ3 � u�3 ¼ gD 1� a2gD

a2gD þ g�M

" #�s33; (23)

or �s33 ¼1

gD

þ a2

g�M

" #ðuþ3 � u�3 Þ: (24)

From this result, the linear-slip boundary conditions for a

permeable fracture with an impermeable background can be

reduced to

sþ31 ¼ s�31 ¼ �s31; (25)

sþ33 ¼ s�33 ¼ �s33; (26)

uþ1 � u�1 ¼ gT�s31; (27)

uþ3 � u�3 ¼ g�Nðx; nÞ � �s33; (28)

where a frequency-dependent, effective normal fracture

compliance g�N is defined via

1

g�Nðx; nÞ� 1

gD

þ a2

g�M¼ 1

gD

þ a2

gM þ ixn2~jðxÞ=gf

: (29)

In the static limit x!0 and in the zero-fluid-mobility limit

~jðxÞ=gf!0, this effective compliance becomes the

undrained specific-normal-fracture compliance gU ¼ h=HU

¼ h=ðHD þ a2MÞ, via

g�M ! gM;1

g�N! 1

gD

þ a2

gM

¼ 1

gU

:

The derived reduced boundary conditions in Eqs. (25)–(28)

are formally identical to the original linear-slip-interface

conditions used by Gu et al.8 for studying fracture interface

waves (Rayleigh interface waves)—except that the effective

specific-normal-fracture compliance g�N is used in place of

the simple specific-normal-fracture compliance (gD or gU).

Therefore, using their results, the frequency equations of

waves guided by a fluid-filled fracture are

Rðn2Þ þ i2nS

xqðBÞgT

� �nS3n

3S ¼ 0; (30)

Rðn2Þ þ i2nS

xqðBÞg�N

� �nP3n

3S ¼ 0; (31)

Rðn2Þ � ð2n2 � n2SÞ

2 þ 4n2nP3nS3 (Rayleigh equation).

In Eqs. (30) and (31), qðBÞ is the density of the back-

ground medium and nP3 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2

P � n2q

and nS3 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2

S � n2q

are the 3-direction P- and S-wave slownesses. nP and nS are

the P- and S-wave slownesses of the background medium,

respectively.

Equation (30) is the dispersion equation for antisymmet-

ric fracture interface waves. Because these waves do not

depend on the properties of the fluid in the fracture for low



frequencies, we will not discuss their behavior further in this

paper. Equation (31) is for the symmetric fracture interface

waves, but its specific fracture compliance (or stiffness) is

given as an effective specific-fracture compliance, which

depends upon both wave frequency and slowness, as well as

fluid viscosity and fracture permeability.

C. Frequency-dependent permeability models and thezero-viscosity limit

For a fracture containing porous fillings (debris, prop-

pant) [Fig. 2(a)], we can define the frequency-dependent

fracture transmissivity ~j½¼ kðxÞh� using the Johnson et al.permeability model14 as

~jðxÞ ¼ kðxÞh ¼ k0h

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� i

4

nJ

xxJ

r� i

xxJ

!; (32)

where k0 is the static permeability, nJ is a finite parameter

determined by the pore geometry (for example, a value of 8

is recommended for common sandstones), and xJ is the

viscous-boundary characteristic frequency given by xJ

� gf=qf Fk0 ¼ gf /=qf a1k0; where F is the electrical forma-

tion factor and a1 is the high-frequency limit pore-space tor-

tuosity, both of which approach unity for an open fracture.

In contrast, for an open fracture or a fracture with sparse

asperity contacts [Figs. 2(b) and 2(c)], the fracture thickness

has a different influence on permeability. In this case, Biot’s

dynamic, oscillating fluid-flow model for a fluid channel

between rigid parallel walls15 can be used to specify the

frequency-dependent fracture transmissivity:

~jðxÞ ¼ kðxÞh ¼ h3

4h21� tanhh

h

� �; h � h

2

ffiffiffiffiffiffiffiffiqf x

igf

s:

(33)

Note that h is a dimensionless parameter. In the static limit

(x!0), this reduces to the cubic law16 ~jð0Þ ¼ k0h ¼ h3=12.

For the special case of vanishing fluid viscosity (gf!0), both

permeability models result in

~jðxÞ !igf h

xqf F;

where F¼ 1 for an open channel (fracture).

D. Low-frequency asymptotes

Low-frequency asymptotes of Eq. (31) can be obtained

by considering the very small phase velocity of Krauklis

waves (examples are provided in Sec. III). Assuming jn2Pj

and jn2Sj � jn2j, Eq. (31) can be reduced to

xqðBÞ

n2S

1� n2P

n2S

!nþ 1

g�N¼ 0: (34)

Using Eq. (29), Eq. (34) can be written in the form of a

third-order polynomial in n as

rbn3 þ bgD

n2 þ rgMnþ gM

gD

þ a2 ¼ 0; (35)

where

r � xqðBÞ

n2S

1� n2P

n2S

!;

b � ix~jðxÞ=gf :

Equation (35) can be solved explicitly, for example using the

Cardano formulas. However, the resulting solution formula

is very complex and not presented here.

For the special case of an open fracture (gap), gD!1,

a! 1, and the storage coefficient M! n2A=qf ; where nA is

the acoustic wave slowness and qf the fluid density. If the

fluid within the fracture is stiff (e.g., water), jn2Aj � jn2j.

Using these assumptions, Eq. (35) can be reduced further to

rbn3 þ 1 ¼ 0; (36)

yielding an explicit expression for the phase velocity:

1

n¼ h �i

x2GðBÞ

12gf

1� n2P

n2S

!" #1=3

; (37)

which is identical to the asymptote for a thin fracture, previ-

ously obtained by Korneev.4 [Note that GðBÞ � qðBÞ=n2S and

~jðxÞ � h3=12 for small frequencies.]

For a rigid background (nS ! 0 and r!1), Eq. (35)

becomes

bn2 þ gM ¼ 0; (38)

yielding an asymptote

1

n¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixkðxÞM

igf

s; (39)

which is the low-frequency phase velocity of the Biot’s slow

P wave13 for a porous medium with a rigid frame.

FIG. 2. Permeable gouge filled fracture and open and partially closed frac-

tures. The behavior of these fractures is modeled via LSIM using two per-

meability models with different frequency dependency.



III. EXAMPLES AND DISCUSSION

In this section, we will provide several examples of the

velocity and attenuation of Krauklis waves, predicted by the

newly derived dispersion equations.

A. Comparison to an exact solution

The dispersion equation for symmetric FIWs and

Krauklis waves [Eq. (31)] was derived by taking the thin

fracture thickness limit O(h)!0 and by assuming small

FIG. 3. (Color online) Comparison of LISM and exact solutions for a fluid-filled open channel (fracture). The same fracture with low-velocity fluid

(VA¼ 200 m/s) (a),(b) and high-velocity (VA¼ 1482 m/s) (c),(d) are examined. Note that the upper panels of the velocity plots (a),(c) are in linear scale, while

the lower panels and the attenuation plots are in log scale. The computed velocities and attenuations show good agreement at low frequencies. The velocity of

the lower-frequency mode increases with frequency while the attenuation decreases. The higher-frequency mode above the cutoff frequency exhibits slight

decreases in velocity, with only very small attenuation. For the case with lower acoustic velocity of the fluid (a),(b), the exact solution shows multiple branches

in the velocity and attenuation (rapid changes occur corresponding to the velocity changes and its cutoffs), induced by multiple reflections of the waves within

the fracture. Shear and Rayleigh velocities of the background medium (VS and VR, respectively) and the acoustic velocity of the fluid within the fracture (VA)

are also indicated. (a) Phase velocity (VA¼ 200 m/s). (b) Attenuation (VA¼ 200 m/s). (c) Phase velocity (VA¼ 1482 m/s). (d) Attenuation (VA¼ 1482 m/s).



wave frequencies [O(x2)!0]. However, because real frac-

tures have finite thickness, and wave measurements are

made at finite frequencies, the accuracy of the approximation

using LSIM needs to be examined.

The validity of the newly derived dispersion equation is

examined by comparing its solutions to the solutions from

the exact dispersion equation for Krauklis waves, within an

open parallel channel (fracture) containing viscous fluid.3,4

This exact equation was derived by solving a linearized

Navier-Stokes equation for wave propagation within the

fluid channel, which provides both P- and S-wave slow-

nesses in the fluid. Solutions of both exact equations and the

LSIM equation are computed numerically from a grid-search

method.4 The particular models considered here assume a

background medium with density of 2650 kg/m3, a P-wave

velocity of 4758 m/s, an S-wave velocity of 2747 m/s (no

attenuation is assumed in the fluid), and a flat, infinite frac-

ture (channel) containing fluid with density 1000 kg/m3.

FIG. 4. (Color online) Comparisons of guided-wave velocity and attenuation computed from the exact solution (red broken curves) and the LSIM approxima-

tion (black solid curves). The effect of the fracture thickness for a constant fluid viscosity of 1 cP (a),(b) and the effect of the fluid viscosity for a constant frac-

ture thickness of 1 cm (c),(d) are examined. For the range of parameters examined here, both solutions are in excellent agreement for frequencies below 1 kHz.

Generally, velocities are in better agreement than attenuation, and the disagreement increases for large fracture thicknesses and fluid viscosity. (a) Phase veloc-

ity (varying fracture thickness). (b) Attenuation (varying fracture thickness). (c) Phase velocity (varying viscosity). (d) Attenuation (varying viscosity).



In the first example, we assume fracture thickness of

1 cm and fluid viscosity 1 cP, and compare the results for

both low and high wave velocities in the fluid (VA¼ 200 m/s

and 1482 m/s) [Figs. 3(a) and 3(b), respectively]. The solu-

tions include both the Krauklis wave (the dispersive mode

without a cutoff frequency) and higher-velocity fracture

interface waves (for LSIM solutions), and multiple modes of

low-velocity guided waves (for the exact solutions).

Generally, LSIM solutions agree well with the exact solu-

tions for low frequencies. Also note that the velocity of the

fracture-interface wave predicted by LSIM corresponds to

the low-dispersion part of the multiple guided wave modes

for the exact solution. The predicted attenuation for the frac-

ture interface wave and corresponding multiple guided wave

modes of the exact solutions may not appear to agree well,

but the actual overall attenuation is both very small

(1/Q � 1/1000). In Figs. 3(a) and 3(c), the velocities of the

guided waves appear to approach the Rayleigh wave velocity

of the background and the acoustic velocity of the fluid.

However, strictly speaking, they should approach the

pseudo-Rayleigh wave velocity17 (leaky Rayleigh wave

propagating along a fluid�solid interface) and the Scholte

wave velocity18 (nonleaky wave propagating along a fluid-

solid interface), respectively.

Next, we compare the LISM and exact solutions for a

range of fracture thicknesses and fluid viscosities, focusing

only on the Krauklis waves. The following two sets of exam-

ples assume either constant fluid viscosity (1 cP) and a range

of fracture thickness (10 lm–10 cm), or constant fracture

thickness (1 mm) and varying fluid viscosity (0.001–1000

cP). The P-wave velocity of the fluid is 1482 m/s. Both phase

velocity and attenuation are presented in Figs. 4(a) and 4(b)

(for the varying fracture thickness case) and in Figs. 4(c) and

4(d) (for the varying viscosity case). Note that the phase ve-

locity is computed by V¼Re[1/n] and the attenuation by

1=Q ¼ �Im½1=n2�=Re½1=n2�, where n is the slowness of the

guided wave. Generally, for the tested parameters in these

examples, low-frequency (�1 kHz) results using the LSIM

approximation agree very well with the exact solutions.

B. Impact of fracture compliance

For the same fracture as in the previous section, with

fluid viscosity 1 cP and fracture width 1 mm, the specific

drained normal fracture compliance now varies from 10�9 to

10�14 m/Pa. Phase velocities and attenuation computed from

Eq. (31) are shown in Fig. 5.

Reducing fracture compliance (increasing fracture stiff-

ness) results in increases in wave velocity and attenuation,

especially at low frequencies. As a result, the phase velocity

dispersion can be greatly reduced for low-compliance frac-

tures. Also, the changes in velocity can be extremely large,

exhibiting more than a one-order-of-magnitude change. Note

that for gD¼ 0, the wave becomes the oscillating fluid flow

within a fluid channel between rigid walls examined by

Biot.15 Wave velocity and attenuation in a fracture with

finite fracture compliance vary between this Biot limit (rigid

fracture limit) and an open fracture limit.

C. Impact of fracture permeability models

Because an open fracture and a fracture containing

porous permeable filling (e.g., gouge, proppant in a hydrau-

lic fracture) have different frequency-dependent behavior,

velocity dispersion, and attenuation of the guided waves also

FIG. 5. (Color online) Comparison of Krauklis wave phase velocity and attenuation for a range of drained specific normal fracture compliance gD. The direct

effect of decreasing compliance is to increase both velocity and attenuation for low frequencies, and the velocity and attenuation vary between the high and

low fracture compliance limits. For the example shown here, the behavior of the velocity does not change for compliances larger than 10�9 m/Pa and smaller

than 10�14 m/Pa. Also, the changes in the attenuation is small (compared to the velocity), except for very small frequencies below 1 Hz. (a) Phase velocity.

(b) Attenuation.



are different, even if the static permeability (transmissivity)

of the fractures is the same.

We consider the two different fracture models in

Sec. II C [Eqs. (31) and (32)] with identical static permeabil-

ity, but consisting of (a) a porous permeable layer and (b) an

open channel. For the permeable layer model, we assume a

fracture (gap) thickness of h¼ 1 mm filled with D250 lm

size sand grains with porosity /¼ 0.5. Using the Kozeny-

Carman model, assuming the grain sphericity¼ 1, the static

transmissivity (k0h) of a fracture containing the layer is

given by

k0h ¼ D2

180

/3

ð1� /Þ2h ¼ 1:74 10�13 m3:

FIG. 6. (Color online) Comparison of Krauklis wave phase velocity and attenuation for two fracture permeability models with the same static permeabil-

ity. Biot’s open parallel wall model (“Open fracture,” shown in a broken line) and Johnson et al. general model for a porous medium (“Porous

fracture,” shown in a solid line) are considered. The two models shown here exhibit different permeability at frequencies above 1–10 Hz (a).

Corresponding to this, both velocity (b) and attenuation (c) predicted by the porous fracture models are higher than the open fracture model. This

behavior is the same for different fracture compliances (gD¼ 10�14 m/Pa and1). Note that the velocities and attenuations for different fracture compli-

ance values converge at the high-frequency limit, but not for the different permeability models. (a) Permeability models. (b) Phase velocity. (c)

Attenuation.



The thickness heq of an open fracture with the same static

transmissivity is computed using the cubic law k0h ¼ h3eq=12,

resulting in heq¼ 128 lm.

Figure 6(a) compares the frequency-dependent trans-

missivity of the two fractures, computed using the Johnson

et al. model14 (for a porous layer) and the Biot model15 (for

an open fracture). Fluid and solid material properties are the

same as in previous sections above. The two permeability

models agree for low frequencies (<10 Hz for this exam-

ple), but the transmissivity of the porous fracture model is

higher than the open fracture model for high frequencies.

(Note that the fracture thickness is wider for the porous frac-

ture model, so that the same static permeability can be

obtained.)

Velocity dispersion and attenuation of Krauklis waves

are computed for these models [Figs. 6(b) and 6(c)], for

both infinite normal fracture compliance (gD¼1) and fi-

nite fracture compliance (gD¼ 10�11 m/Pa). For high fre-

quencies, the two models diverge where the permeability

of the fractures is different, but the effect of the different

fracture compliance is small. In contrast, the effect of

fracture compliance is more prominent for lower

frequencies.

IV. CONCLUSIONS

In this paper, frequency (dispersion) equations for

guided waves along a flat, fluid-filled fracture are presented.

The equations are derived using a modified linear-slip-inter-

face model for a poroelastic fracture, which allows us to

examine the effect of finite mechanical compliance of a frac-

ture. The derived equations are formally identical to the frac-

ture interface waves (Rayleigh interface waves), but with

their fracture-compliance terms replaced by frequency and

slowness-dependent, effective specific-fracture compliance.

Of the multiple modes of guided waves predicted by the

equations, the lowest-frequency mode with symmetric parti-

cle motions is for the fluid-guided waves (i.e., Krauklis

waves).

The significance of the new dispersion equation is that it

predicts the velocity and attenuation of the Krauklis waves

within a fracture that contains proppant and gouge materials,

and/or is subjected to confining stress, resulting in finite frac-

ture compliance. Compared to a fracture with infinite com-

pliance (open gap), a fracture with finite compliance exhibits

reduced velocity dispersion at low frequencies, increased

phase velocity, and increased attenuation. The new equation

can also model the effect of different frequency-dependent

permeability behavior for an open (or partially open) fracture

and a gouge-filled fracture.

The predicted behavior of the Krauklis waves for a

thin fracture indicates that these waves can be very attenu-

ative at low frequencies. This implies that the use of the

waves in the field may be limited to near the seismic

energy source, depending upon the fracture thickness,

fracture compliance, and the fluid viscosity. For example,

for a 1-mm thick, open fracture saturated with water, the

seismic quality factor is on the order of 5–20 for a fre-

quency range of 10–100 Hz. Therefore, particularly for a

high-compliance fracture, the slow velocity of the wave

limits the wave propagation distance to a few tens of

meters.

Finally, the modified linear-slip-interface model

derived in this paper can improve the efficiency of nu-

merical wave propagation simulations within a medium

containing fluid-filled fractures. Because of the small ve-

locity of Krauklis waves at low frequencies, explicit

modeling of the fractures requires the use of dense, vari-

able grids within and around a fracture, which can be

computationally very demanding even for two-

dimensional models.19 The use of LSIM for modeling a

fracture, as originally proposed by Coates and

Schoenberg,20 potentially eliminates the need for a dense

numerical grid inside a fracture, if several issues resulting

from the more complex structure of the new boundary

conditions can be resolved—such as their implicit

dependence on the wave slowness and frequency, which

was not the case for the classical LSIM.

ACKNOWLEDGMENTS

This research was supported by the Office of Science,

Office of Basic Energy Sciences, Division of Chemical

Sciences of the U.S. Department of Energy, and by the

Research Partnership to Secure Energy for America

(RPSEA) through the Ultra-Deepwater and Unconventional

Natural Gas and Other Petroleum Resources Research and

Development Program, as authorized by the U.S. Energy

Policy Act (EPAct) of 2005, supported by the Assistant

Secretary for Fossil Energy, Office of Natural Gas and

Petroleum Technology, through the National Energy

Technology Laboratory, of the U.S. Department of Energy

under Contract No. DE-AC02-05CH11231.

APPENDIX: KRAUKLIS WAVE AND FRACTUREINTERFACE WAVE DISPERSION EQUATION FOR APLANE FRACTURE WITH FINITE COMPLIANCE,EMBEDDED WITHIN A POROUS, PERMEABLEBACKGROUND MEDIUM

When the background of a fracture is porous and perme-

able, a simple, compact dispersion equation for the Krauklis

waves, as we derived for an impermeable background, is dif-

ficult to obtain. In the following, using the Nakagawa and

Schoenberg’s approach,11 we will derive the dispersion

equation in the form of matrix equation, without providing

numerical examples of the solutions.

First, we express velocity and stress vectors on the frac-

ture in the following form:

b6X �

_u1

s33

�pf

264

375 ¼ ð�ixÞXa6eixðnx1�tÞ; (A1)

b6Y �

s13

_u3

_w3

264

375 ¼ 6ð�ixÞYa6eixðnx1�tÞ; (A2)



Xðx; nÞ �n=nPf n=nPs nS3=nS

�nPf HBU þ fPf C

B� �

þ 2n2GB=nPf �nPs HBU þ fPsC

B� �

þ 2n2GB=nPs 2nnS3GB=nS

�nPf CB þ fPf MB

� ��nPs CB þ fPsM

B� �

0

2664

3775;

Yðx; nÞ ��2nnPf 3GB=nPf �2nnPs3GB=nPs �ðn2

S � 2n2ÞGB=nS

nPf 3=nPf nPs3=nPs �n=nS

fPf nPf 3=nPf fPsnPs3=nPs �fSn=nS

2664

3775;

a6 � a6S a6

Pf a6Ps

h iT:

The signs in the superscript “6” indicate either positive or

negative side of the fracture along the x3 axis, and equiva-

lently, up or down-going waves radiating away from the

fracture. nS, nPf, and nPs are the S wave, fast P wave, and

slow P-wave slownesses in the background medium, respec-

tively. nS3, nPf 3, and nPs3 are the corresponding 3-direction

slownesses. GB, HBU, CB, and MB are the shear modulus, uni-

axial (P-wave) strain modulus, Biot’s coupling and storage

moduli for the background, respectively. The coefficients fS,fPf, and fPs and are the complex-valued ratios of the relative

fluid displacement to the solid frame displacement.13

Finally, the coefficient vectors a6 contain displacement

amplitude of the three wave modes in the up(þ) and

down(�)-going directions.

Arranging the displacement (or velocity) and stress (and

pressure) components in this way facilitates subsequent analyses

of the plane waves.21 From Eqs. (16)–(21) obtained in Sec. II A,

the modified LSIM can be expressed in the following form:

_uþ1 � _u�1sþ33 � s�33

�pþf � ð�p�f Þsþ13 � s�13

_uþ3 � _u�3_wþ3 � _w�3

26666666664

37777777775¼ �ixh

0 QXY

QYX 0

" #0

�s33

��pf

�s13

0

0

26666666664

37777777775;

QXY �1

h

gT 0 0

0 0 0

0 0 0

264

375;

QYX �1

h

0 0 0

0 gD �agD

0 �agD a2gD þ g�M

264

375:

Noticing the sparse structure of the matrix, the above equa-

tions can be written in the following form:

bþX � b�XbþY � b�Y

" #¼ � ixh

2

0 QXY

QYX 0

" #bþX þ b�XbþY þ b�Y

" #:

By writing the equations via wave amplitude coefficients a6,

Xðaþ � a�Þ ¼ ð�ixh=2ÞQXYYðaþ � a�Þ;

Yðaþ þ a�Þ ¼ ð�ixh=2ÞQYXXðaþ þ a�Þ:

Or,

Xþ ðixh=2ÞQXYY�

ðaþ � a�Þ ¼ 0;

Yþ ðixh=2ÞQYXX�

ðaþ þ a�Þ ¼ 0:

Note that we can define new independent coefficients

aasym � aþ � a� and asym � aþ þ a�. These are for anti-

symmetric and symmetric components of displacement

(velocity) and stress across the fracture through Eqs. (A1)

and (A2). Therefore, by requiring the determinant of the

coefficient matrices to vanish, we obtain two sets of inde-

pendent dispersion equations for guided waves with antisym-

metric and symmetric particle motions.��Xþ ixh

2QXYY

�� ¼ 0 ðAntisymmetric guided wavesÞ;

(A3)��Yþ ixh

2QYXX

�� ¼ 0 ðSymmetric guided wavesÞ: (A4)

Also note that the dispersion equation for the symmetric

waves Eq. (A4) involves only the normal fracture compli-

ance, while the antisymmetric wave Eq. (A3) involves only

the shear fracture compliance.

1V. Ferrazzini and K. Aki, “Slow waves trapped in a fluid-filled infinite

crack: Implication for volcanic tremor,” J. Geophys. Res. 92, 9215–9223,

doi:10.1029/JB092iB09p09215 (1987).2P. V. Krauklis, “About some low frequency oscillations of a liquid layer

in elastic medium,” PMM 26(6), 1111–1115 (1962) (in Russian).3A. S. Ashour, “Propagation of guided waves in a fluid layer bounded by

two viscoelastic transversely isotropic solids,” J. Appl. Geophys. 44,

327–336 (2000).4V. A. Korneev, “Slow waves in fractures filled with viscous fluid,”

Geophysics 73(1), 1–7 (2008).5J. Dvorkin, G. Mavko, and A. Nur, “The dynamics of viscous compressi-

ble fluid in a fracture,” Geophysics 57, 720–726 (1992).6J.-B. Tary and M. Van der Baan, “Potential use of resonance frequencies

in microseismic interpretation,” Leading Edge 31(11), 1338–1346 (2012).



7L. J. Pyrak-Nolte and N. G. W. Cook, “Elastic interface waves along

a fracture,” Geophys. Res. Lett. 14(11), 1107–1110,

doi:10.1029/GL014i011p01107 (1987).8B. Gu, K. T. Nihei, L. R. Myer, and L. J. Pyrak-Nolte, “Fracture interface

waves,” J. Geophys. Res., [Solid Earth], 101(B1), 827–835,

doi:10.1029/95JB02846 (2012).9M. A. Schoenberg, “Elastic wave behavior across linear slip interfaces,”

J. Acoust. Soc. Am. 68, 1516–1521 (1980).10S. Nakagawa and L. R. Myer, “Fracture permeability and seismic wave

scattering—Poroelastic linear-slip interface model for heterogeneous

fractures,” in Extended Abstract, 79th Annual Meeting of the Society ofExploration Geophysicists, Houston (2009), pp. 3461–3465.

11S. Nakagawa and M. A. Schoenberg, “Poroelastic modeling of seismic bound-

ary conditions across a fracture,” J. Acoust. Soc. Am. 122(2), 831–847 (2007).12V. A. Korneev, “Low-frequency fluid waves in fractures and pipes,”

Geophysics, 75, N97–N107 (2010).13See, for example, S. R. Pride, “Relationships between seismic and hydro-

logical properties,” in Hydrogeophysics, edited by Y. Rubin and S.

Hubbard (Kluwer Academic, New York, 2003), pp. 1–31.14D. L. Johnson, J. Koplik, and R. Dashen, “Theory of dynamic permeability

and tortuosity in fluid-saturated porous media,” J. Fluid Mech. 176,

379–402 (1987).

15M. A. Biot, “Theory of propagation of elastic waves in a fluid-saturated

porous solid. II. High frequency range,” J. Acoust. Soc. Am. 28, 179–191

(1956).16See, for example, J. S. Witherspoon, Y. Wang, K. Iwai, and J. E. Gale,

“Validity of cubic law for fluid-flow in a deformable rock fracture,”

Water Resour. Res. 16, 1016–1024, doi:10.1029/WR016i006p01016

(1980).17See, for example, W. L. Roever, T. F. Vining, and E. Strick, “Propagation

of elastic wave motion from an impulsive source along a fluid/solid inter-

face. I. experimental pressure response. II. Theoretical pressure response.

III. The pseudo-Rayleigh wave,” Philos. Trans. R. Soc. London, A 251,

455–523 (1959).18See, for example, E. Strick and A. S. Ginzbarg, “Stoneley-wave

velocities for a fluid–solid interface,” Bull. Seismol. Soc. Am. 46,

281–292 (1956).19M. Frehner and S. Schmalholz, “Finite-element simulations of Stoneley

guided-wave reflection and scattering at the tips of fluid-filled fractures,”

Geophysics 75, T23–T36 (2010).20R. T. Coates and M. A. Schoenberg, “Finite-difference modeling of faults

and fractures,” Geophysics 60, 1514–1526 (1995).21M. A. Schoenberg and J. S. Protazio, “ ‘Zoeppritz’ rationalized and gener-

alized to anisotropy,” J. Seismic Explor. 1(2), 125–144 (1992).



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Effect of fracture compliance on wave propagation within a fluid-filled fracture