Feasibility Study of Ultrasound for Oil WellStimulation on the Basis of Wave-Properties

ConsiderationsSergio Caicedo, PDVSA-Intevep

Summary

Ultrasound or a high-frequency (20 kHz to 100 kHz) pressurewave has been used in diagnosis and treatments in different areas,such as: medicine, dentistry, civil engineering, and many otherindustrial applications. In the oil industry, there are applications(i.e., pipeline inspections, fluid velocity measurements, etc.), butto the present, these applications in formation stimulation havebeen incipient, and only a few lab and field test experiences havebeen reported. Stimulation with ultrasound is not a commonoperation offered by oil service companies. To visualize the realpotential of ultrasound in oil well stimulation, it is necessary tounderstand the wave phenomenon, its properties, the parametersthat define its behavior, and its interaction with the propagationmedia. This basic knowledge and the understanding of thedifferent formation damage mechanisms are the keys to compre-hend the real potential and application window of the ultrasoundin oil well stimulation. This paper presents the theoretical basisof ultrasound and wave phenomena that must be consideredwhen considering stimulation with ultrasound. Finally, somesuggestions about the application window of this technologyare given.

Introduction

Ultrasound has been applied in many areas, such as diagnosis,quality control, inspections, cleaning, etc. Industrial cleaning isachieved by flaking out the particles with a mechanical action ofthe pressure waves (Fig. 1). Usually, the piece is submerged influids inside a container with walls that have ultrasonic sources.Clearly, there is a great difference with an application for oil wellstimulation, in which the source is running inside the hole, and thecleaning area is around the source.

Each application has a particular frequency and power asso-ciated according to the sample dimensions and the purpose. Forexample, the power and frequency used for control echography inpregnant mothers are different than ones used in muscular thera-peutic treatments. In the first case, it is enough to detect an echowith high resolution (higher frequencies). In the second case,energy is required to be transferred to the tissue, but high resolu-tion is not required (lower frequencies). It is clear that the purposeand the propagation media affect the ultrasound parameters, high-lighting the importance to understand which are the damagemechanisms in which ultrasound can be applied and vice versa.

The advantage of applying ultrasound comparing with conven-tional stimulation is that no invasion or external fluids arerequired. Ttherefore, fluid/rock interaction analysis is avoided,and the placement as well as the associated equipment and riskyoperation of handling high pressures at the wellhead is alsoavoided. Additionally, ultrasound allows underbalance treatmentswithout shutting in the well.

Ultrasound cleaning is not a common tool offered by servicecompanies in the field. Only field tests in China and Russia havebeen reported with more qualitative than quantitative information

making these tests inconclusive. Recent references about labexperiences and tool prototypes suggest the potential of this tech-nology. However, ultrasonic stimulation has little understandingof the phenomena taking place in the porous media, and how thewaves are interacting with the matrix and the trapped particles.The parameters for suitable cleaning with ultrasonic treatment arenot well defined, and how these parameters change while thewave is propagating in the porous media is also not clear.

Power requirements for stimulation and effective penetrationdepend on the elastic media (matrix), the radial geometry, andcompletion (i.e., either open, gravel packed, or case hole). Wavephenomena as reflection, transmission-refraction, diffraction, andinterference must be considered; otherwise, a successful applica-tion in Russia can be a failure in other places, because change inone or more parameters considerably affects the wave.

Previous Works

Most references in ultrasonic stimulation date from 1950 to1980 from the USSR Academy of Sciences. First, motivationwas the production increase reported after earthquakes in theUSSR (Babanov 1987) and the USA. After the earthquake, insome wells, not only production increased but also the casingpressure. However, the phenomena are so complex that in afew neighbor wells the production decreased. Because it was afield report, no reason was given to explain the change. Later,some theories were suggested to explain how seismic wavesaffected the saturated porous media (Kissan 1991, Mikhailovet al. 1975). It must be highlighted that the energy releasedand the area affected by an earthquake (with frequencies < 60Hz) can generate both microfractures and macrofractures, whichare hard to reproduce with a tool running into a well and withthe phenomena probably being different from those associatedwith ultrasound (with frequencies > 20 kHz).

Most recent works (i.e., 1990) are more quantitative and mea-sure permeabilities. Cherskiy et al. (1994) determined the perme-ability of saturated cores under an acoustic field. The permeabilityincreased when ultrasound was applied and returned to its originalvalue after treatment.

The lab tests of Roberts et al. (2000) and Venkitaraman et al.(1995) measured the effects of ultrasound on damage removal forthe following mechanisms: fine migration, mud cake, organicdeposit, and paraffin. No physical explanation about the interac-tion wave-fine particles is given, but requirements for power andacoustic intensity are reported to remove such damages as well asthe frequencies and type of treatment (i.e., pulsed or continuos).These papers are important, because not only magnitudes arereported but also experimental assembly and procedure. Themost relevant quantities reported are the minimum intensity of20 W/m2 for fine removal, 400 W/m2 for paraffin, and a pene-tration depth of 15 cm. However, Venkitaraman only addressedremoval in the core and did not consider wave propagationphenomena in cylindrical geometry.

The effects of cylindrical geometry in the near wellbore arereported by Van der Bas et al. (2004). The experiments aredivided in linear and cylindrical geometry. With the linear geo-metry, Van der Bas obtained more equivalent results than Venki-taraman, while the experiments with cylindrical geometry wereonly to test screen cleanup. No wave phenomena were reported byVan der Bas.

Copyright ã 2009 Society of Petroleum Engineers

This paper (SPE 107083) was accepted for presentation at the 2007 SPE Latin Americanand Caribbean Petroleum Engineering Conference, Buenos Aires, 15–18 April, and revisedfor publication. Original manuscript received for review 17 January 2007. Revised manu-script received for review 10 April 2008. Manuscript peer approved 7 May 2008.

February 2009 SPE Production & Operations 81

Theroretical Background

To determine the ultrasound application window for well stim-ulation, it is required to understand the damage mechanismsand comprehend the definition of an ultrasound as well as itswave phenomena. As a general principle, well stimulationis applied to improve well productivity, usually in tight zones(i.e., low permeability), in viscous fluids, and damagedwells—with the last ones probably being the most commoncase. When dealing with undamage low-permeable zones, theonly way to increase productivity is creating a long fracture orpreferential channels in which the fluids can flow easily intothe wellbore. Clearly ultrasound cannot create channels norincrease permeability; therefore, it is not recommended forthis case.

In wells with high viscous fluids (i.e., heavy oil and/or emulsi-fied oil) ultrasound can be applied for heating or breaking theemulsion, but other heating techniques and demulsifiers are moreefficient in this task.

In damage mechanisms (i.e., wettability changes, fluid inva-sion, etc.) in which there are not particles to be removed, ultra-sound has a low potential of being a stimulation technique.However, it could be applied together with other stimulation pro-cedures to enhance the effects, but this is not the goal of thepresent work.

In damage caused by plugging of the pore throat by particles(i.e., solid invasion during drilling, fine migration, or inorganicprecipitation) there is a high potential for applying ultrasound,because mechanical removal of particles can be performed bypressure waves. But, keep in mind that the penetration and energyintensity have to be high enough for that purpose. Again, there areother stimulation technologies for this case that compete or can beused together with ultrasound.

It is clearly necessary to calculate the intensity of theultrasound for a given penetration; for that purpose, a basicunderstanding of wave properties of ultrasound is required,which is the main objective of this research. The ultrasoundis defined as a pressure wave created by a source or generator(usually a piezoelectric device) with a frequency higher than20 kHz that propagates in a continuous media with elasticproperties.

The rock or porous media can be considered an elastic media ifthe external perturbation or deformation is not so strong to breakor fracture the rock. A deformation is called “elastic” when theparticles that conform the continuos media return to the initialposition once the deforming agent is retired. The stress analysis,together with Hooke’s law, allows finding the relationshipbetween normal and shear stresses (sij), elasticity properties ofthe media given by the elasticity tensor (Cijkl) and normal andshear deformations (ejk).

sij ¼ cijklekl: (1)

For isotropic media,

sxx ¼ ðaþ 2mÞexx þ aeyy þ aezz; (2)

syz ¼ 2meyz; (3)

where either sxx represents normal or pressure “p” stresses andsxy represents shear stresses and a, m are the Lame constants. Forfluids m = 0, because fluids do not return to their original shapeunder static shear stresses. Lame’s constants, bulk module, Pois-son’s ratio, and Young’s modulus are equivalent parameters.

When including dynamics (SF = ma), the Newton’s equationapplied into an infinitesimal volume of the media in which exter-nal forces are given by Hooke’s law then, two equations rise(Auld 1973):

ðaþ 2mÞr2p ¼ r@2p

@t2; (4)

mr2ðr � uÞ ¼ r@2

@t2ðr � uÞ: (5)

Both equations have the form:

@2f ðx; tÞ@x2

¼ 1

V2

@2f ðx; tÞ@t2

; (6)

that is called wave equation, because it admits the solution

f ðx; tÞ ¼ f ðx� VtÞ; (7)

meaning that the solution is a function f that moves forward and/orbackwards in the X axis with velocity V. The first big differencebetween Eq. 4 and Eq. 5 is that Eq. 4 shows that the particles ofmedia deform and return to their position parallel to the propaga-tion axe, and Eq. 5 shows that one particles moves perpendicularto the axe of propagation. The first equation particles are calledP waves, and the second equation particles are called shear waves.Now, it is clear that pressure p(x,t) is a p wave.

The second great difference is that shear waves do not propa-gate in fluids, because m = 0. The third difference is the velocities.

Vp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiaþ 2m

r

sVS ¼

ffiffiffimr

r: (8)

By looking at the numerator, it is clear that p waves are fasterthan S waves. Another interesting observation is that wave propa-gation velocity is a media property and not affected by the sourceor perturbation.

Basic properties of ultrasound can be understood if weconsider a harmonic solution associated with a source as follows:

pðx; tÞ ¼ po cosðkx � wtÞ; (9)

where po is the amplitude; k, the wave number; and w, the angularfrequency. By physical definition of period (T) and wavelength (l), the following conditions hold: p(x,t) = p(x,t+T) andp(x,t) = p(x+l,t). These conditions together with the harmonicsolution is conducive to the following two relationships:

kl ¼ 2p; (10)

wT ¼ 2p: (11)

By introducing this solution into the wave equation, we can geta third basic relationship as follows:

k2 ¼ w2

V2: (12)

Using Eq. 10 and Eq. 11, the following called dispersion rela-tionship is obtained:

l ¼ V

f: (13). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Fig. 1—Piece before and after industrial ultrasound cleaning.

82 February 2009 SPE Production & Operations

Because velocity V is a property of the media, then wavelengthand frequency are related in inverse proportion. Table 1 showsan example of wavelength for different frequencies assuming avelocity of propagation of 3000 m/s, which is the order of magni-tude expected in rocks.

It should be highlighted that ultrasound wavelength is compa-rable with those penetration values reported by Venkitaraman(1995) and Van der Bas et al. (2004). Clearly, if frequency isincreased for better cleaning the wavelength is shorter, which isnot good for stimulation purposes. The wave equation does notinclude attenuation (i.e., friction) effects; when considering fric-tion, another undesirable behavior occurs—the higher the fre-quency the higher the attenuation. Propagation effects are alsoimportant, with the most important factors being interphases andgeometry. In cased-hole wells, for ultrasound waves propagatingfrom the wellbore to the formation, the most important interphasesto be considered are the following: the liquid-steel, steel-cement,and cement-rock. In openhole wells there is just one interphaseliquid-rock. In each interphase, when the wave arrives perpendic-ular to the surface (the most convenient case when the tool is infront of the pay zone), there is one reflected and one transmittedwave. Finally interference and diffraction of waves also must beconsidered, because to apply waves for cleaning is completelydifferent than fluid placement with coil tubing. All these consid-erations have to be taken into account for source power require-ments to have enough power inside the formation where thedamage is present.

To quantify the effects mentioned in the previous paragraph,start the analysis with reflected and transmitted waves. For normalincidence, the three waves involved are the incident—thereflected and the transmitted—which can be expressed by thefollowing:

piðtÞ ¼ poi cosðk1x� wtÞprðtÞ ¼ por cosð�k1x� wtÞptðtÞ ¼ pot cosðk1x� wtÞ

: (14)

The root medium square (RMS) average power associated to awave-traveling trough, a cross section area (A) is given by (Auld1973) as follows:

PowerRMS ¼ ðaþ 2mÞðaþ 2m=3Þ

1

2

p2orV

A: (15)

Therefore, the intensity, which is defined as the power per crosssection area, is given by the following:

I ¼ PowerRMS

A¼ ðaþ 2mÞ

ðaþ 2m=3Þ1

2

p2orV

: (16)

Venkitaraman et al. (1995) reported some intensity values (W/m2) required at lab test that needed cleaning with ultrasound, butthey are useless with out estimated values inside the formation.Keeping this in mind, the Snell’s law gives the amplitude ratiosbetween incident, reflected, and transmitted waves as follows(Fowler 1990):

Ramp ¼ poipor

¼ r2v2 � r1v1r2v2 þ r1v1

Tamp ¼ poipot

¼ 2r1v1r2v2 þ r1v1

: (17)

The energy ratios (Fowler 1990) are as follows:

Rpow ¼ PotiPotr

¼ ðr2v2 � r1v1Þ2ðr2v2 þ r1v1Þ2

Tpow ¼ PotiPotT

¼ 4r1v1r2v2ðr2v2 þ r1v1Þ2

: : : : : : : : : : : : : : : : : : : : : : : : : ð18Þ

As can be appreciated, these ratios depend on the acousticimpedance (rn) of both media at the interphase. Table 2 showsreference values calculated from Eq. 17 and Eq. 18 evaluated withdensity and sound velocities for water steel, cement, and rock,which represents open and casedhole wells.

As can be noticed in casedhole wells, only approximately 9%of the source energy is transmitted to the reservoir face and 47%for openhole wells. An additional correction is needed, becauseintensity falls as penetration increases. To study this attenuation, itis necessary to express the wave equation in radial coordinates asfollows:

1

r

@

@rðr @p

@rÞ ¼ 1

V2

@2p

@t2: (19)

The exact solution is the Bessel function Jo, which canbe approximated, for radius greater than wellbore radius, by thefollowing:

pðr; tÞ � po cosðkr � wtÞffiffiffiffiffiffiffipkr

p : (20)

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February 2009 SPE Production & Operations 83

Then, pressure decays as 1/r0.5. But, keeping in mind thatintensity is related to the square of pressure, then the intensity ata given distance r (from the center of the wellbore) is related tothe intensity Io at the sand face by the following:

IðrÞ ¼ Iorwr: (21)

When introducing friction proportional to the velocity of theparticles (different than velocity of wave propagation) to the sys-tem, the wave equation is transformed to the following:

1

r

@

@rðr @p

@rÞ ¼ 1

V2

@2p

@t2þ m

@p

@t: (22)

Then, the pressure and the intensity changes to the following:

pðr; tÞ � Po cosðkr � wtÞffiffiffiffiffiffiffipkr

p e�bðwÞkr; (23)

IðrÞ ¼ Iorwre�2bðwÞkðr�rwÞ: (24)

Then, there is attenuation not only for radial geometry butalso because of the frequency; as was previously explained,the higher the frequency the higher the attenuation. The b(w)must be measured at the lab with the core saturated withreal fluids, because it is a property of the saturated media(Bourbie 1987).

Fig. 2 shows amplitude and intensity (both normalized to thevalue at the sand face) as a function of penetration for a frequencyof 20 kHz, a velocity of 3500 m/s for a frictionless and a frictionmedia with a b = 0.02.

The graph in Fig. 2 shows why the wavelength is such animportant reference value, because after 5 or 10 wavelengths. theintensity dropped considerably, and also showed the effectivepenetration of ultrasound is a maximum of 3 ft. Now using Ven-kitaraman’s et al. (1995) intensity requirements, together withtransmission coefficients and attenuation expressions, it is possi-ble to estimate the power requirements at the source.

The last and more complex phenomena are interference anddiffraction. These effects arise from the interaction of waves withobjects with sizes comparable to the wavelength. Consideringcasedhole wells, then perforations diameter and geometrical dis-tribution can create an interference pattern equivalent to a double-slit interference pattern obtained with surface waves in a pool withtwo single sources (Fig. 3).

The superposition of waves creates points in which there isdestructive interference; in other words, the signal is null at anytime (in Fig. 3, they appear like light radial lines) independentlyof the power of the source, meaning there is no cleaning at thesepoints. Interference is a consequence of phase shift caused by thedifference in path length; when the difference is an odd number ofthe wavelength, then there is a phase shift of 180�, which isequivalent to a negative signal.

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Fig. 2—Ultrasound attenuation for 20 kHz and 50 kHz.

Fig. 3—Interference pattern by two sink sources. Fig. 4—Interference and difraction of a double slit.

84 February 2009 SPE Production & Operations

Diffraction is more related to the size of the object; when thesize of the object is comparable to the size of the wavelength, thenthe wave is affected or distortioned. Both effects can occur simul-taneously. Analyze the double-slit interference pattern witha monochromatic and coherent light (LASER). The intensity isgiven by the following (Caicedo and Mora 2004):

IRMS ¼ E2o cos

2

�pdzlL

�sen2ðpbzlL ÞðpbzlL Þ2

; (25)

where “d” is the distance between slits, and “b” is the width ofeach slit. The cosine expression is related to diffraction (i.e., theratio between wavelength and size of the object), and the sineexpression is related to interference (i.e., the ratio between thedistance of two local sources and the wavelength). Fig. 4 showsthe function and the corresponding real picture.

For ultrasound values assuming perforations of ½ inches ofdiameter 20 kHz and 1500 m/s up to 3500 m/s, the ratio b/l,ranging from 0.08 to 0.2 suggests that difraction should be con-sidered. However, Razi’s et al. (1995) lab test for fluid velocitymeasurements in the perforations with ultrasound indicates thatfor some frequencies, range distortion is not a problem.

How do you deal with diffraction and interference? Knowingthe perforations diameter (b), distortion can be handled by chang-ing the wavelength making the ratio b/l large (>10) throughfrequency selection. On the other hand, intereference alwaysoccurs, meaning that for a fixed frequency there are points withzero intensity (no matter the power of the source). However, thelocation of the points change with frequency, therefore, by doing asweep in frecuency, all the points can be clean with ultrasound.It should be pointed out that a frequency range selected to solvethe difraction and interference problems is not necessarly the bestfor the cleaning job.

Potential Application Window

Once all wave phenomena have been disscused and some refer-ence numbers calculated, it is possible to define the applicationwindow for ultrasound as a stimulation technique. First of all, thefollowing facts have to be considered (independently of the ultra-sound application):

• Cleaning requires a minimun frequency and intensity.• Penetration is approximately 2 or 3 ft.• The higher the frequency, the lower the wavelenght and the

attenuation.• There are power transmition losses in each media interphase.• There are wave phenomena related with wavelength and the

size of an object that affect ultrasound intensity patterndistribution; placement is not as easy as injecting a fluid.

Keeping in mind that ultrasound is a mechanical phenomenomthat consists on a presure wave with frequency higher than20 kHz, it should be applyed for damages that can be removedmechanically. The phenomena and reference numbers disscussedin this paper together with previous studies suggest a highpotencial of ultrasound for the following applications in wellstimulation:

• Damage removing in the near (less than 3 ft) of the wellborecaused by fines migration, as well as organic and inorganic pre-cipitation

• Gravelpack and screens cleaning• Perforations cleaningUltrasound can also be used with other techniques to enhance

the previous cases.As a reference of the power requirements of ultrasound, con-

sider Venkitaraman et al. (1995) intensity values between 3.7 to60 W/m2 and 2 ft of penetration. Table 3 shows the powerrequirements for casehole and openhole wells.

Table 3 shows that cased-hole wells require six times morepower than openhole wells; because the power is given per foot, italso gives an idea of the treatment costs depending on the layerthickness. Because the hypotetical tool has a given size, it alsogives the required power and how long the stimulation lasts know-ing the radiation time recommended by Venkitaraman et al.(1995). These computations only considered p waves, becausethe liquid in the wellbore allows only p waves. Besides, normalincidence has the highest power transmition coeficient. It shouldbe noticed that angular incidence (some feet above and under theemissor) of p waves generates both p and s waves.

Conclusions

Wave theory and previous works show the feasibility to use ultra-sound as a mechanical stimulation technique with penetrationslower than 3 ft for damage associated with mechanical pluggingof the pore throats, screens, and gravelpacks.

Any ultrasound application or study for well stimulation mustconsider the following facts: (1) cleaning requires a minimumfrequency and intensity; (2) the higher the frequency, the lowerthe wavelength; (3) and the higher the attenuation, penetration isabout 2 or 3 ft; there is power transmission losses in each mediainterphase.

There are wave interference and diffraction phenomena thataffect the treatment making the “placement” not so easy as it iswith invading fluids. The phenomena always take place and can-not be ignored, but understanding the parameters involved in thesephenomena, a frequency range selection can be done to reduce theundesirable effects.

Research in ultrasound, as mechanical stimulation, should con-tinue because it is an attractive technique and it can be performedwhile the well is producing avoiding placement of invading fluidsand deferred production.

Nomenclature

f = frequency, Hz

I = intensity, W/m2

k = wave number, 1/ft [1/m]

p = pressure, psi [N/m2]

r = radius, ft [m]

rw = wellbore radius, ft [m]

R = reflection coefficient, dimensionless

t = time, s

. . . . . . . . . . . . . . . . . . . .

February 2009 SPE Production & Operations 85

T = period, s

Ttran = transmission coefficient, dimensionless

Vp = P wave’s velocity, ft/s [m/s]

Vs = S wave’s velocity, ft/s [m/s]

w = angular frequency, rad/s

a = Lame’s Constant for normal stress, psi [N/m2]

b = attenuation constant, dimmensionless

l = wavelength, ft [m]

m = Lame’s Constant for shear stress, psi [N/m2]

r = density, lbm/ ft3 [kg/m3]

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age. SPEDC 15 (1): 19–24. SPE-62046-PA. DOI: 10.2118/62046-PA.

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Batenburg, D., Birchak, B., and Yoo, K. 2004. Radial Near Wellbore

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Sergio Caicedo has been working as artificial lift specialist for16 years at PDVSA-Intevep, where he has researched anddeveloped simulation software for gas and pumping lift meth-ods. Caicedo also has been and instructor and consultant forPDVSA E&P field engineers. He has worked in interdisciplinaryprojects related with production, such as water conformance,sand control, intelligent completions, uncertainty effects onproduction, thermal process, and reservoir simulation. He hasauthored and co-authored over 15 technical papers andarticles. Caicedo holds a BS degree in physics and computerengineering from the Universidad Simon Bolıvar, Venezuela,and an MSc degree in petroleum engineering from theUniversity of Texas at Austin.

86 February 2009 SPE Production & Operations