SPE/DOE
`
`Society of
`Petroleum Engineers
`
`U.S. Department
`of Energy
`
`Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/85LPG/All-85LPG/SPE-13892-MS/3248495/spe-13892-ms.pdf/1 by Robert Durham on 20 September 2023
`
`SPE/DOE 13892
`
`Impedance of Hydraulic Fractures: Its Measurement and Use for
`Estimating Fracture Closure Pressure and Dimensions
`by G.R. Holzhausen, *Applied Geomechanics Inc., and R.P. Gooch, Stanford U.
`
`*SPE Member
`
`Copyright 1985, Society of Petroleum Engineers
`
`This paper was presented at the SPE/DOE 1985 Low Permeability Gas Reservoirs held in Denver, Colorado, May 19-22, 1985. The material is subject to
`correction by the author. Permission to copy is restricted to an abstract of not more than 300 words. Write SPE, P.O. Box 833836, Richardson, Texas
`75083-3836. Telex: 730989 SPE DAL.
`
`ABSTRACT
`
`The growth of a hydraulic fracture increases the
`period of free oscillations in a well. Simultaneously, the
`decay rate of free oscillations decreases. The properties
`of forced oscillations in a well also change during fracture
`growth. All of these effects result from the changing
`impedance of the hydraulic fracture that intersects the
`well. Fracture impedance can be determined directly by
`the ratio of downhole pressure and flow
`measuring
`oscillations, or determined
`indirectly
`from wellhead
`measurements using
`impedance
`transfer
`functions.
`Because impedance is a funCtion of fracture dimensions
`and the elasticity of the surrounding rock, impedance
`analysis offers a promising new approach for evaluating
`fracture geometry. Because oscillatory flow conditions
`occur continuously a hydraulic-fracturing treatment, data
`collection
`is simple and economical, adding
`to
`the
`attractiveness of this technique.
`
`INTRODUCTION
`
`This paper introduces impedance analysis as a tool
`for fracture diagnostics. Impedance analysis is based on
`the dynamics of wave propagation in a well and the effect
`the hydraulic fracture has on oscillatory pressures and
`flows.
`Impedance analysis is a logical extension of the two
`pressure analysis techniques currently used for evaluation
`transient
`of hydraulic fractures. The first, pressure
`analysis, is based on the solution of a diffusion equation
`the principle of
`derived
`from Darcy's
`law
`and
`conservation of ma.ss. 1
`In this method gradual pressure
`changes resulting from fluid flow through the pores of the
`fracture and
`formation are measured and used for
`estimating fracture size and permeability. The second
`pressure analysis method2
`is also derived from
`the
`principle of mass conservation and considers gradual
`pressure changes associated with the elasticity of an
`inflating fracture. Neither of these approaches considers
`
`the inertial component of fluid flow, an effect important
`in the study of wave propagation and reflection. Inertial
`forces are accounted for by invoking the principle of
`conservation of momentum. 3 This principle, along with
`that of conservation of mass, forms the basis of the study
`of oscillatory pressure and flow m wells and other
`conduits.
`This paper begins with a definition of impedance and
`then presents several field examples of oscillatory pressure
`the changing
`impedances of
`changes resulting from
`hydraulic
`fractures. Reasons
`for
`these changes are
`subsequently
`derived
`using
`impedance
`analysis
`techniques. 3•4 To
`illustrate
`the relationship between
`fracture impedance and fracture dimensions, we then
`construct a hydraulic model of a fracture intersecting the
`bottom of a well. The properties of the fracture are
`combined in two lumped parameters, a flow resistance
`and a capacitance, which determine the impedance at the
`well-fracture
`interface.
`These parameters
`can be
`expressed in terms of fracture dimensions and the elastic
`properties of the surrounding rock.
`It is not our purpose in this paper to provide a
`definitive recipe for measurement of fracture dimensions
`to
`based on
`impedance analysis. We hope
`instead
`illustrate the potential of the method and provide a
`framework for its further development.
`
`CONCEPT OF HYDRAULIC IMPEDANCE
`
`Imagine that a specialized tool is placed at the
`bottom of a well beside a low-permeability zone about to
`be fractured. This tool is able to precisely measure very
`small changes of both pressure and flow as injection rates
`the
`tool can measure
`are
`increased.
`In addition,
`oscillatory pressures and
`flows
`resulting
`from
`the
`reciprocating action of the pistons in
`the fracturing
`pumps. When injection begins, the pumps force fluid
`into the well, although flow into the formation is not
`possible since breakdown has not occurred. At the same
`time, the pressure begins to rise because the pumps are
`
`411
`
`IWS EXHIBIT 1057
`
`EX_1057_001
`
`

`

`2
`
`IMPEDANCE OF HYDRAULIC FRACTURES
`
`SPE/DOE 13892
`
`frequency w, frequency f
`(hertz) and period T (seconds)·
`are: f = wj21r and T = 21rjw = 1/ f .
`in our
`Another concept
`that will be valuable
`subsequent analyses is that of characteristic impedance
`Zc . The characteristic impedance can be considered as a
`hydraulic impedance that describes the proportionality
`between head and flow moving in one direction only. 3 In
`the phase difference
`an
`infinite frictionless conduit,
`between head and flow oscillations is either 0 or 1rjw,
`depending on whether the flow is moving in a positive or
`negative direction. The imaginary term in the expression
`for impedance (Eq. 1) vanishes and the characteristic
`impedance assumes a purely real value that can be shown
`to be3
`
`where a is the acoustic wavespeed in the conduit and A
`is the cross-sectional area of the conduit.
`
`(2)
`
`FREE AND FORCED OSCILLATIONS
`
`In the analysis of impedance in hydraulic systems, it
`is convenient to distinguish between free oscillations and
`forced oscillations. The latter is also referred to as
`steady-oscillatory behavior. In the forced oscillation of a
`fluid system, all oscillations are at the frequency of the
`forcing
`function.
`During
`a
`hydraulic
`fracturing
`treatment, forcing is provided by the reciprocating action
`of the pumps that inject fluid down the treatment well.
`The frequency of forcing is determined by the frequency
`In
`of the piston strokes and higher-order harmonics.
`contrast, free oscillations result from an initial, temporary
`excitation, such a.s the sudden removal of fluid from a
`pressurized well by valving, or the sudden opening of a
`hydraulic fracture at breakdown. Upon removal of the
`excitation, the oscillations attenuate as a result of natural
`physical damping in the system. The frequency of free
`oscillations is determined by the wavespeed of the fluid,
`the lengths of the system elements, and the physical
`properties of the system boundaries.
`Both free and forced oscillations occur throughout a
`typical hydraulic-fracturing treatment. Steady pumping
`results in a condition of forced oscillation, whereas free
`oscillations are caused by suddenly starting or stopping
`the pumping and by numerous other disturbances that
`naturally occur during pumping. The same theoretical
`is used . to evaluate both free and forced
`framework
`oscillations. In the former, the frequencies of interest are
`one or more of the natural frequencies of the system. In
`the latter, the frequencies of the forcing functions are
`used.
`
`FIELD OBSERVATIONS
`
`The configuration of surface and downhole pipe
`(tubing,
`casing,
`etc.)
`remains
`constant during a
`hydraulic-fracturing· treatment, whereas the geometry of
`the fracture changes continuously as it is being created.
`
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`
`compressing the fluid in the well. Our specialized tool
`would therefore measure large pressure oscillations (in
`addition to large static pressures) but would measure zero
`flow. The corresponding ratio of pressure to flow would
`be infinite.
`Continuing this thought experiment, we know that
`formation breakdown will occur when
`the downhole
`pressure becomes great enough to overcome both the rock
`strength and the minimum in situ compressive stress at
`If we were
`the treatment depth.
`to again measure
`pressure and flow after fracture growth has begun, we
`would expect static and oscillatory pressures to be less
`than before, while flow would be greater because fluid is
`now moving from the well into the fracture. The ratio of
`pressure to flow would thus· be less than in the pre(cid:173)
`fracturing case.
`As the fracture continues to grow, we would expect
`the relative values of downhole pressure and downhole
`flow to continue to change. Because fracture growth is
`accompanied by an increase in the cross-sectional area of
`the fracture where it intersects the wellbore, the ease
`with which fluid can flow into or out of the fracture
`should increase. At the same time, the pressure gradient
`required to maintain that flow should decrease. Since
`fracture growth is accompanied by an increase in the
`fluid stored
`in
`the fracture,
`the quantity of fluid
`contained in a single flow oscillation should become a
`smaller and smaller fraction of total fracture volume.
`The fracture thus behaves as a large capacitor becoming
`more and more effective at holding downhole pressure
`constant as its size increases. In the limiting case, a very
`large fracture would behave as a constant pressure
`boundary, although it is questionable whether this case is
`ever attained
`in practice. We
`therefore expect
`the
`downhole pressure oscillations associated with oscillatory
`flow to diminish as the fracture grows.
`As anticipated
`from
`the above discussion, our
`specialized pressure-and-flow-measurement
`tool should
`detect a decreasing ratio of oscillatory pressure
`to
`oscillatory flow as the fracture grows. If we knew how to
`analyze the pressure-to-flow ratio, we could use it as a
`means of interpreting fracture dimensions.
`In hydraulics, the ratio of oscillatory pressure (or
`hydraulic head) to oscillatory flow is called the hydraulic
`impedance, Z. 3 The impedance is a complex number
`defined by the amplitude, frequency, and phase of the
`pressure and flow oscillations at a point. It is a function
`of the physical properties of the piping system and fluid.
`Impedance can be written in terms of oscillatory head H
`and flow Q as
`z
`
`.
`H
`= -e•w¢>
`Qeiwt
`Q
`where w is the circular frequency in radians per second, t
`is time in seconds, <P is the phase difference between the
`i = V-i The
`head
`and
`flow
`oscillations,
`and
`relationship between head H
`and pressure P
`is
`P = pgH where p is fluid mass density and g
`is
`gravitational acceleration. Useful relationships between
`
`He iw(t +¢>)
`
`(1)
`
`412
`
`IWS EXHIBIT 1057
`
`EX_1057_002
`
`

`

`SPE/DOE 13892
`
`G.R. Holzhausen & R.P. Gooch
`
`3
`
`Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/85LPG/All-85LPG/SPE-13892-MS/3248495/spe-13892-ms.pdf/1 by Robert Durham on 20 September 2023
`
`Changes in oscillatory behavior observed under conditions
`of uniform excitation and constant fluid properties should
`therefore be related to the changing geometry of the one
`variable in the system:
`the hydraulic fracture. Three
`recent experiments provide examples of the effects of
`fracture growth on wellbore pressure oscillations.
`the first experiment (Figure 1), a
`In
`transient
`condition was
`initiated by rapidly removing a small
`volume of fluid ( < 10 liters) by abruptly valving at the
`wellhead. The well was cased to a total depth of 1589 m.
`There were 18 casing perforations between 1448 and 1395
`m in the production zone whose permeability was several
`microdarcies. A packer was set in the 16-cm I.D. casing
`at a depth of 1296 m. Tubing from the wellhead to the
`packer had an I.D. of 6.2 em. The viscosity of fluid in
`the well was approximately 80 cp 5. Figure 1a shows free
`oscillations measured at the wellhead prior to fracturing.
`The period of these oscillations was 2L /a, where L is
`well depth and a is sonic wavespeed in the fluid in the
`well, about 1400 mfsec. Figure 1b
`illustrates free
`oscillations at the wellhead recorded a few minutes after
`the completion of the 320,000-liter fracturing treatment.
`The period of these oscillations was approximately double
`the pre-fracturing case. Doubling of the period of free
`oscillations has also been reported by Anderson and
`Stahl6 who performed tests on three wells in the 1960s.
`The second experiment was conducted in a 330-m
`deep test well at Mounds, Oklahoma that was cased
`(0.126-m I.D.) to a depth of 311 m. Below this depth was
`the Skinner sandstone
`an open-hole completion
`in
`(porosity ~ 20%, permeability ~ 20 millidarcies). All
`injections were down the casing with no tubing in the
`well. The two pressure records· shown in Figure 2 were
`made with water in the well after the sandstone had been
`hydraulically fractured. The well was shut in at these
`times and the pressure was declining as a result of leakoff
`into the formation. The first oscillations were recorded
`when the wellbore pressure was about 0.4 MPa above the
`statically determined fracture closure pressure. The
`oscillations continued for several cycles before damping
`out (Figure 2a).
`In contrast, they damped out almost
`immediately after excitation at fracture closure pressure
`(Figure 2b ). Subsequently, free oscillations were initiated
`above fracture closure pressure after different volumes of
`water had been injected into the already-created fracture.
`In every case, greater volumes were characterized by
`reduced rates of attenuation of the oscillations (Figure 3).
`Plots of peak-to-peak amplitude versus time (Figure 4)
`clearly illustrate this effect.
`The third experiment was conducted in the same
`Mounds, Oklahoma test well with tubing run to the
`bottom of the casing and an open annulus. Pressure
`oscillations were measured at the top of the annulus, on
`the tubing at the wellhead, and on the treatment line
`near the two pump trucks used to pump the fracturing
`fluid (Figure 5). Pressure records made during proppant
`injection indicated that the ratio of oscillatory annulus
`pressure
`to oscillatory wellhead pressure declined as
`pumping progressed (Figure 6a and 6b ). This observation
`
`413
`
`the expected effect of a fracture
`is consistent with
`behaving as a large capacitor if we make the assumption
`that
`annulus
`pressure
`oscillations
`are
`directly
`proportional
`to pressure oscillations at
`the fracture
`orifice. W ~ also observed that the relative phase of
`pressure oscillations on the annulus fell further behind
`the phase of oscillations at the wellhead and pump trucks
`as proppant injection progressed (Figure 7).
`
`INTERPRETATION OF PRESSURE
`AND FLOW OSCILLATIONS
`
`In this section, a hydraulic well-fracture model is
`developed and used to derive expressions for fracture
`impedance and
`the
`frequency
`and decay of
`free
`oscillations. We subsequently show how
`impedances
`derived from pressure and flow measurements can be used
`to evaluate fracture closure and dimensions.
`
`Impedance Analysis
`We can illustrate the effect of fracture growth on
`wellbore pressure oscillations using a simple model of a
`fracture intersecting the bottom of a well. In this model,
`the physical properties of the fracture are lumped into
`two parameters: the flow resistance R 1 at the fracture(cid:173)
`well interface and the hydraulic capacitance, or storage of
`the fracture, c1 (Figure 8). The R 1 and c1 elements
`are combined in series to reflect the fact that flow into
`the fracture must first overcome a resistance before
`fracture capacitance can be increased.
`A change of hydraulic head A.H1 in a fracture gives
`rise to a change of fracture volume A v1 . We define the
`ratio of volume change to head change as the capacitance
`the fracture. Sneddon 7 derived the relationship
`of
`between internal pressure and opening of an oblate(cid:173)
`ellipsoidal (penny-shaped) fracture in an infinite elastic
`medium. Following his results, we can write fracture
`capacitance as
`
`(3)
`
`where h is fracture radius, v Poisson's ratio and p the
`shear modulus of the medium. Because capacitance is
`proportional
`to
`the cube of the fracture radius,
`it
`increases rapidly as the fracture grows. When the radius
`exceeds a few meters, flow oscillations of a few liters per
`second should produce pressure oscillations at the well(cid:173)
`fracture interface of no more than a few hundredths of a
`megapascal {i.e., several psi) (Figure 9). The larger the
`fracture, the more effective it is at maintaining itself at
`relatively constant pressure during periods of pressure
`and flow oscillations in the wellbore.
`
`The fracture resistance R 1
`is the proportionality
`constant relating a change of flow into or out of the
`fracture to a corresponding change of hydraulic head:
`
`( 4)
`
`IWS EXHIBIT 1057
`
`EX_1057_003
`
`

`

`IMPEDANCE OF HYDRAULIC FRACTURES
`SPE /DOE 13~92
`4
`---- --------------------------------------------~-------------------------------------------------,
`We wish to analyze the impedance in our model well
`several different source frequencies
`to give downhole
`(Figure 8) and its relationship to fracture growth, as
`impedance as a function of frequency.
`expressed by the R 1 and c1 parameters. For forced(cid:173)
`We now turn our attention to the case of free(cid:173)
`the
`oscillation conditions, ·a reciprocating pump at
`oscillation testing. Our goal is to derive an equation that
`wellhead generates sinusoidal flow at a given frequency.
`expresses
`the
`fracture
`impedance
`in
`terms of
`the
`For free-oscillation conditions, excitation is by a sudden
`frequency and rate of decay of the free oscillations. It is
`temporary flow change. The tubing or casing leading
`a well known result from steady-state Laplace analysis
`the wellhead to the fracture
`is modeled as a
`from
`that the character of free oscillations is determined by
`length L . The characteristic
`frictionless pipe of
`the singularities of
`the
`impedance with . respect
`to
`impedance Zc of the well is given by equation (2) and the
`frequency. 4 It can be shown that the wellhead impedance
`propagation constant "/is given by 4
`is
`iw
`"1= (cid:173)
`a
`Pressure transducers are connected to the top and bottom
`qf the well to measure oscillatory pressure behavior. In
`addition, we assume that there is a sensor at the bottom
`of the well that measures flow oscillations into and out of
`the fracture. The bottom of the well is characterized by
`a lumped impedance Z 1 that is a function of the fracture
`constants R 1 and c1 :
`z1 = R1 + -:----0
`
`where r I
`
`(5)
`
`1
`
`1
`
`fW
`
`I
`
`(6)
`
`(8)
`
`(9)
`
`Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/85LPG/All-85LPG/SPE-13892-MS/3248495/spe-13892-ms.pdf/1 by Robert Durham on 20 September 2023
`
`1 + r I e 2"~L
`z tD = zc . 1
`r
`2"~L
`-
`I e
`is the downhole reflection coefficient given by
`z1 -z
`r I = zl +;c
`Replacing i w by s and using the definition of "/, it can be
`that 27L =sTd where 8 =a+i w
`shown
`is complex
`frequency and Td =2L /a is the two-way travel time up
`and down the wellbore. The poles (singularities) of Eq.
`(8) occur at values of 8 for which the denominator goes
`to zero, i.e.,
`
`fracture
`the
`test,
`forced-oscillation
`a
`During
`impedance can be determined in the following manner: A
`sinusoidal fluid flow of known frequency w and known
`magnitude is set up at the wellhead. Once steady state is
`reached,
`the downhole pressure sensor will show a
`sinusoidal oscillation of the same frequency as
`the
`wellhead source but of a different magnitude. Similarly,
`the downhole flow oscillation will be at
`the same
`frequency but different magnitude. The magnitudes and
`phases of the downhole pressure and flow oscillations are
`recorded. Since flow and pressure have been measured,
`the magnitude of
`the
`fracture
`impedance can be
`determined by dividing these two quantities. Once this
`has been done,
`the frequency of the flow source is
`changed and the whole process repeated. In this manner,
`the magnitude of
`the
`fracture
`impedance can be
`determined as a function of frequency.
`In many fracturing jobs, downhole pressure and flow
`measurements
`are
`not
`available
`but wellhead
`measurements are.
`In these situations, the down.hole
`(fracture) impedance must be determined by applying an
`impedance transformation to the wellhead impedance. If
`t~e magnitude and phases of wellhead pressure and flow
`have been measured,
`the complex-valued wellhead
`impedance
`is easily determined by vector division.
`Zw
`Transformation
`from wellhead
`impedance
`to
`impedance z1
`downhole
`(fracture)
`is accomplished
`through the transformation4:
`[ Zw -Zc ) e -21L
`Zw +Zc
`
`1+
`zl - Z·
`c
`
`(7)
`
`[ Zw-Z,
`Zw +Zc
`The impedance transformation can be carried out at
`
`1-
`
`) e -21L
`
`414
`
`-·T~ -
`-
`e
`
`r
`I
`
`(10)
`
`To study the effect of fracture resistance on the rate
`of decay of free oscillations, we assume a purely real
`impedance of z1 =R 1 and a
`frictionless well of
`characteristic impedance Zc . The downhole reflection
`coefficient is then real and given by
`R1 -z
`ri=RI+;c
`
`(ll)
`
`In terms of this reflection coefficient, there are multiple
`values of s for which equality (10) is met. These are
`w = imaginary[s] =
`
`(12)
`
`and
`
`a = real[8] =
`
`-In I r I I
`
`(13)
`
`where n is odd for r I <O and n is even for r 1 >O. The
`imaginary part of 8 determines the frequency of the free
`oscillations while the real part of 8 determines the rate of
`decay. More
`specifically,
`the
`fundamental natural
`frequency is
`
`(14)
`
`a /4L
`a /2L
`
`{
`
`r 1 <O
`r I >O
`I = wj2rr =
`The system oscillates at odd harmonics for fracture
`resistances below the characteristic impedance of the
`wellbore (r 1 <O) and at even harmonics for fracture
`resistances above the characteristic impedance (r 1 >O).
`This behavior was observed in our field tests described
`earlier (Figures 1 and 3). The time constant of the free(cid:173)
`oscillation decay is
`
`IWS EXHIBIT 1057
`
`EX_1057_004
`
`

`

`Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/85LPG/All-85LPG/SPE-13892-MS/3248495/spe-13892-ms.pdf/1 by Robert Durham on 20 September 2023
`
`SPE/DOE 13892
`
`G.R. Holzhausen & R.P. Gooch
`
`5
`
`i
`
`(15)
`
`1
`I a I
`Notice that ; approaches infinity (zero decay) as the
`magnitude of the reflection coefficient approaches one.
`That is, the rate of decay becomes extremely slow as the
`fracture impedance approaches either zero (no fracture)
`or infinity (completely open fracture). Also notice that
`the
`time constant T goes
`to zero as
`the reflection
`coefficient goes to zero. That is, free oscillation will not
`occur when the fracture is open to a point where the
`fracture impedance and the characteristic impedance of
`the wellbore are equal. This explains the effect observed
`in Fig. 2b. The theoretical' effect of a purely resistive
`fracture impedances on free oscillations in a frictionless
`well is shown in Figure 10.
`
`To study the effect of fracture capacitance on the
`frequency of free oscillation, assume a purely capacitive
`the form z1 = 1/ c1 8 The
`fracture
`impedance of
`downhole reflection coefficient of Eq. (9) then becomes
`_ 1-Zc c1 8
`-
`l+Zc Cl 8
`
`f
`
`I
`
`(
`
`)
`16
`
`the
`For a frictionless well and capacitive fracture,
`singularities of Eq. (8) must occur at purely imaginary
`values of 8. Thus we can assume that 8 =i w and find
`the values of w for which equality (10) is met. This turns
`out to give an equation of the form
`
`(17)
`Equation (17) was evaluated for several values of c1 to
`generate the plot of natural frequency versus fracture
`the
`capacitance shown
`in Figure 11. A value for
`characteristic well impedance was taken from the Mounds
`well: Zc =11,250 sec/m2. Notice that as c1 varies from
`0 to oo, the natural frequency of the free oscillations
`shifts from a /2L (an even harmonic) down to a / 4L (an
`odd harmonic). Most of the shift is accomplished when
`fracture capacitances are between about 10-6 and 10-4
`m2.
`
`Fracture Closure and Dimensions
`the amplitude and decay of
`As shown above,
`the wellbore
`are
`strongly
`pressure oscillations
`in
`dependent on the resistive characteristics of the hydraulic
`If fracture capacitance is large or excitation
`fracture.
`frequency is high (Eq. 6), the selection of a purely real
`(resistive)
`fracture
`impedance can be
`justified. By
`to
`the
`characteristic
`equating
`fracture
`resistance
`impedance of a frictionless conduit, we can then define a
`procedure
`for
`deriving
`fracture
`size
`estimates.
`Characteristic impedance of a frictionless fracture is now
`given by:
`
`a
`gA
`where A is the area of the fracture where it intersects the
`well and a is the wavespeed in the fracture. Fracture
`wavespeed can be derived from the following expression3
`
`(18)
`
`415
`
`a =
`
`[
`
`1/2
`
`]
`
`(19)
`
`K jp
`1 + (K /A )(~A /~P)
`where ~A/ ~p is the area change corresponding to a
`fluid pressure change ~p . In a stiff conduit, such as a
`cased well, the denominator is very close to 1 and the
`wavespeed is close to K / p, that of a perfectly rigid pipe.
`For water, this limiting wavespeed is about 1485 m/sec.
`W avespeeds in the two test wells discussed previously
`were measured at about 1400 m/sec.
`For a very compliant conduit such as a hydraulic
`fracture, the denominator in Eq. (19) is large with respect
`to unity and the wavespeed in the conduit is very slow.
`In this case Eq. ( 19) simplifies to
`[(~P A )/(p~A ) ] 112
`(20)
`a =
`We can estimate wavespeeds in hydraulic fractures and
`their relationship to fracture dimensions by considering
`the expansion of a penny-shaped fracture resulting from
`changes of internal pressure. The change of area of a
`cross-section drawn through the center of such a fracture
`is7
`
`(21)
`
`Substituting (21) into (20) and using the formula for the
`area of an ellipse, A = rrbh , the following expression for
`wavespeed results:
`
`a
`
`1
`2
`1
`
`]
`
`rrb Jl
`[
`.2ph(l-v)
`
`(22)
`
`where b is half-width of the ellipse. Evaluation of this
`equation reveals that wavespeeds in the fracture can be
`extremely slow with respect to those in the well (Figure
`12).
`
`We can now define the fracture impedance in terms
`of fracture dimensions and elastic properties. Assuming
`that the fracture has an elliptical cross section where it
`intersects the well and that it is hi-winged so that it
`intersects the well on two sides, we can use Eqs. ( 18) and
`(22) to write
`
`,---- I 8g 2p:(l-v) I [ b~3 ~
`
`Rl
`
`23
`
`(
`
`)
`
`where R 1 is the characteristic impedance of a frictionless
`penny-shaped fracture. The corresponding expression for
`the impedance of an infinitely long (two-dimensional)
`fracture can be derived using the formula for opening of
`such a fracture under internal pressure8 and is
`
`In Figure 13 we have plotted curves of impedance
`versus fracture half-width for penny-shaped fractures of
`several different radii (solid lines). One curve (dashed
`line) is the impedance of an infinitely long fracture with a
`half-height of 6.10 m. Figure 13 illustrates a way in
`which changes of fracture impedance may help track
`fracture growth. Assume that the fracture begins to grow
`
`IWS EXHIBIT 1057
`
`EX_1057_005
`
`

`

`6
`
`IMPEDANCE OF HYDRAULIC FRACTURES
`
`SPE/DOE 13892
`
`Downloaded from http://onepetro.org/SPERMPTC/proceedings-pdf/85LPG/All-85LPG/SPE-13892-MS/3248495/spe-13892-ms.pdf/1 by Robert Durham on 20 September 2023
`
`DISCUSSION
`
`The interpretation of fracture impedance in the
`preceding pages is based on results from five test wells,
`two of which were tested by the authors. The other
`three wells were tested in the 1960's as part of a different
`study6, and only sketchy information dealing with their
`free-oscillatory behavior is available. There is no doubt
`that the equations of oscillatory flow and impedance
`analysis can be applied to wells as they can to other
`piping systems. The most difficult question to answer
`concerns the relationship between fracture impedance and
`fracture dimensions. Certain conclusions are obvious on
`the basis of the existing field data and the analysis
`presented in this paper. For example, it seems clear that
`the high compliance of an internally pressurized fracture
`gives rise to the high capacitance and low impedance that
`result in the doubling of the period of free oscillations
`observed after breakdown. Because
`the compliance
`changes as the fracture grows, we expect it to produce
`changes in oscillatory pressure behavior regardless of the
`details of the hydraulic model chosen for the fracture.
`is dominated by
`Because
`fracture
`impedance
`wavespeed,
`impedance changes should be detected in
`both open-hole
`and perforated-casing
`completions.
`that
`impedance
`changes
`are
`Figure 1
`illustrates
`detectable through perforations, although more field data
`from perforated wells is needed. The interpretation of
`impedance may differ in the open-hole and perforated
`cases because of differences in orifice size.
`In analyzing the relationship between impedance and
`fracture
`dimensions, we made
`two
`important
`assumptions. The first was the assumption of frictionless
`flow conditions. This assumption greatly simplifies the
`mathematics but may lead to unacceptable errors in
`interpretation. Procedures for the treatment of friction
`are available and should be invoked to examine the
`sensitivity of the analysis to frictional effects. The second
`assumption was that the equation for static opening of a
`fracture could be used to define wavespeed, capacitance,
`and impedance under oscillatory flow conditions. This
`assumption may
`be
`justifiable
`for
`low-frequency
`oscillations at the fundamental period of the well (about
`one second for every 300 m of well depth), but it is less
`certain for high-frequency excitations such as forced
`oscillations during pumping. The deformation of a
`fracture at these higher frequencies and its effect on
`wavespeed and impedance requires further investigation.
`
`CONCLUSIONS
`
`Impedance analysis is a promtsmg new tool for
`hydraulic-fracture diagnostics. Growth of a hydraulic
`fracture results in a continuous change of downhole
`impedance at the well-fracture interface. This change
`can be used to evaluate fracture dimensions. The first
`to measure downhole
`step in such an evaluation is
`
`as an expanding penny, maintaining a constant aspect
`ratio ( b /h) until it meets a barrier to vertical growth.
`When the barrier is met, height growth is arrested and
`the fracture begins to lengthen. As it lengthens, fracture
`width and aspect ratio increase. Path A-B-E (Figure 13)
`shows the rapid drop of impedance that occurs during
`uniform radial expansion. When the growth barrier is
`met, say at h =6.10 m, continued
`lengthening and
`widening of the fracture will result in a rate of impedance
`fall-off (path B-C-D) that is reduced with respect to the
`previous rate of fall-off. In contrast, if the fracture were
`to continue to grow radially without confinement, the
`impedance
`fall-off would
`follow
`the path B-E.
`the
`impedance during a
`Continuous monitoring of
`fracturing
`treatment should
`therefore reveal whether
`fracture growth was dominated by radial expansion or
`whether vertical containment was established and
`maintained.
`This analysis also helps explain why well and
`fracture impedances may be matched at the fracture
`closure pressure (Figure 2b ). Figure 13 indicates that the
`fracture impedance rises abruptly when
`the fracture
`width drops below about one millimeter. As the width
`the
`drops below a few hundredths of a millimeter,
`fracture impedance rises rapidly toward the characteristic
`impedance of the well. The characteristic impedance of
`the Mounds, Oklahoma test well is fairly typical at
`11,250 secjm2, assuming frictionless conditions. Thus, by
`the time the fracture and well impedances match, the
`fracture has become so narrow that it can be considered
`closed for all practical purposes.
`We can use equations (23} and (24} to obtain a range
`of fracture radii (heights) and widths that are compatible
`with a particular value of impedance. In Figure 14, we
`have shown curves of fracture radius versus width
`corresponding to a fracture impedance of 108 secjm 2• By
`themselves,
`these curves place broad constraints on
`fracture size. However, if a mean pressure in the fracture
`can be established, e.g. by subtracting fracture closure
`pressure from ISIP, it may be possible to place much
`closer limits on fracture dimensions.
`The half-width b of a penny-shaped fracture in an
`infinite elastic body is8