J.H. ENG, D.B. BENNION, JoB. STRONGHycal Energy Research Laboratories Ltd.

Laboratory Tests to Evaluate FinesMobilization

An important parameter to be determined in the laboratof}' isthe critical interstitial velocity (actual velocity in the pore space)at which fines mobilization begins to occur.

The tests are commonly conducted on shon cylindrical coresamples (generally 2.54 to 3.81 cm in diameter by 4 to 8 cm inlength). The samples are generally clean and obtained fromextracted core material. The use of extracted core material can, insome instances, supply a "worst case" scenario. If a resef\'oir isoil-wet in the case of water injection, the migratory fines will gen-erally be encapsulated and stabilized by the immobile or lessmobile oil phase and fines mobilization may not be as severe asindicated by the use of clean, water-wet core. Also, the nonnaltoluene extraction cleaning process can damage clays by remov-ing the water of hydration causing, in general, greater susceptibili-ty to velocity-induced damage and other apparent impainnentmechanisms.

The use of preserved state core is technically preferable but hasdisadvantages with respect to both the cost and complexity of thetests involved. Preserved core implies the preservation of multiplephases within the core sample which can lead to complicationsassociated with relative penneability effects. Due to the high flowrates utilized in velocity sensitivity tests, the residual saturationsof oil, water, or gas can change due to an increase in capillarynumber within the system. Thus, while preserved samples candetect relatively severe damage due to fines mobilization, theaccuracy and analysis of the data for relatively subtle changes inpermeability is much more difficult.

Introduction

Description of Equipment Used in FinesMigration Tests

A schematic of the experimental apparatus used in finesmobilization tests appears as Figure 1. The core sample is encasedin tefIon shrink tubing and placed in a 2.54 or 3.81 cm 1.D.,0.635 cm heavy lead sleeve. The shrink tubing is used to ensurethat no fluid slippage occurs between the sleeve and the core andprevents direct fluid contact with the lead sleeve. The ductility ofthe lead sleeve allows a confiDing overburden pressure to betransferred to the core to simulate reservoir pressure. The core,mounted within the lead sleeve, is placed inside a 7.5 cm I.D. steelcore holder which is capable of simulating reservoir pressures ofup to 69 MPa. This pressure is applied by filling the annular spacebetween the lead sleeve and the core holder with light oil and thencompressing the oil with a hydraulic pump to obtain the desiredoverburden pressure.

0

The migration of small solid materials ("fines") within porousmedia has long been recognized as a source of potentially severepermeability impairment in formations containing oil and gas.Fines migration occurs when loosely attached particles are mobi-lized by fluid drag forces caused by the motion of fluid within thepore space. If a sufficient quantity of large enough fmes are mobi-lized, they can collect at pore throats within the medium and causepermeability impairment. Muecke(l) documents how a variety ofdifferent types of fmes such as clay minerals, quartz, amorphoussilica, feldspars, mica and carbonates can be susceptible to migra-tion. Gray and Rex(2) reported the migration of mica needles andkaolinite. They also found that migration could be induced bysalinity changes or abrupt reductions in the ratio of divalent tomonovalent ions present in the brines. Morgan(3) investigatedchanges in pH and salinity with respect to fmes migration.

Muecke<l) documented how fmes mobilization generally occursonly when the phase which wets the fines is in motion. For exam-ple, there are many cases of severe fines mobilization occurring inwater-wet reservoirs during waterfloods only after the waterbreaks through to the producing wells.

Other authors including Portet'4), Selby<S), Kwan(6), Prisholm(7),Gabriel(8), Borchardt(9), and Sengupta(IO) have documented phe-nomena which affect fines migration. Groesbeck and Collins(ll)describe a detailed phenomenological theory of entrainment anddeposition of fines in porous media.

October 1993, Volume 32, No.8 49

--

~~?J~::J_--4~-~ --

~I t

I ---.----

--~

-[~JE3=O:--=-_.~-

nGURE 1: FInes migration apparatus.

The core bolder ends each contain two ports. One of these portsis for fluid feed or production and the second is for pressure mea-surement. The portions of the core holder directly adjacent to theinjection and production ends of the core are equipped with radialdistribution plates to ensure evenly distributed fluid flow into andout of the core specimen.

Pressure differential is monitored using a Validyne pressuretransducer. The b"ansdUCtt is mounted directly across d1e core andmeasures the pressure differential between the injection and pro-duction ends. The pressure transducer can have a variety of pres-sure ranges from 0-20 to 0-25 (XX) kPa depending on d)e expectedpermeability of the core material. The signal from the pressuretransducer is directly connected to a strip chart recorder whichprovides a continuous pressure profile of the test. A digital read-out also appears on a multi-channel Validyne terminal from whichd)e test operator takes readings as a backup. A highly accuratemanometer system can also be utilized as a check on d1e pressuretransducer measurement system.

A Ruska displacement pump is used to inject fluids into thecore. The pump is capable of injecting at rates from I to8200 cm3Jhr at pressures up to 69 MPa with an accuracy of iO.OIcmJ. The p11Dp is filled with distilled water which displaces var-soL This varsol then displaces the brine to be used in the test intothe core. This arrangement is used to avoid placing corrosivebrine solutions directly into d1e pump. EfDuent from the core iscollected in IS cm3 centrifuge tubes.

All injection fluids are filtered to 0.5 microns prior to use toremove any potentially plugging suspended particulates. An in-line 0.5 micron filter is also present directly before the core as abackup filtration system.

4. Displace water through the core at a low constant rate (i.e.20 cc/hr). Record initial absolute base permeability to brine.

S. Increase rate to 40 cc/hr, displace sufficient water to achievesteady state pressure differential. Then reduce rate to baselevel of 20 cc/hr and note any change in baseline penDeabili-ty caused by die elevated flow rate. (Rates given are exam-ples only and may vary according to sample penoeability.)

6. Repeat step 5 at elevated rates of 60. 100. 200.400, I<XX>.2:(MX). 4120, and 8200 cc/hr. Reduce rate back to base levelto 20 cc/hr after each elevated rate displacement and mea-sure permeability. This eliminates possible ~ perDle-ability measurements due to turbulence in die porous medi-um at high flow rates.

Figures 2 and 3 provide examples of results of a typical fmesmobilization test. Figure 2 illustrates the flow sequence profileutilized in a typical fines mobilization study. The "critical" inter-stitial velocity for die sample in Figure 3 is about 0.26 cmls. Thedamage profile represented in Figure 3 is an idealized case. Oftenthe damage profile is not linear as illustrated and may exhibitoscil1atory behavior due to the migration. entrainment and expul-sion of different size disbibutions of fines at increasing interstitialshear rates.

Interpretation of Test DataA problem with l~ory test data is that the critical velocity

data determined is for linear flow within a porous media.Although valuable to indicate whether ~ is velocity-induced

Procedure for Fines Mobilization TestsThe plug slated for the velocity sensitivity test is mounted

using the equipment described. Plug or full diameter core can beutilized. The following procedure is then employed:

I. Heat to reservoir temperattJre, apply net overburden pressureto simulate pore pressure.

2. Evacuate core for 24 hours to remove residual gas satura-tioo.

3. Satunte core at a low rate with filtered formation Wa&er orthe non-damaging test fluid of interest.

50 The Journal of Canadan Petrolam Tvd"8-..")kJgy

with fines migration in this high-velocity flow region.Since velocity patterns can vary significantly depending upon

values of the parameters affecting flow, it is desirable to be able tosimulate the flow in an efficient yet accurate manner so that flo"profiles can be easily calculated for specific configurations. Inview of this, the following assumptions have been incorporated inthe model developed. First of all, an incompressible, single phasesystem is considered. Although this assumption does introduce acertain degree of error into the results, die simulation results stillprovide a reasonable approximation for the purpose outlined.Secondly, the near wellbore region is assumed to be characterizedby a constant porosity throughout As far as permeabilities in thisregion are concerned, vertical permeabilities can vary significant-ly from horizontal pern1eabilities. The model, dlerefore, includesspecification of a horizontal penneability, which is constant in meradial plane, and a vertical permeability., which is constant in mevertical direction. A further assumption involves implementationof a simplified geometry in which multiple perforations are car-ried out at uniform vertical intervals in a radially symmetric man-ner as illustrated in Figure 4. The distance between perforationlayers is easily varied to match specific perforation patterns. Thenumber of perforations in a given layer is also adjustable to matchthe appropriate phase angle and shot density.

Results of previous studies reflect the need to evaluate flo"" inthree dimensions. With the assumptions incorporated into themodel, the potential equation describing flow in a steady-statesystem in cylindrical co-ordinates is given by the following equa-tion:

a( aP ) 1 a ( aP ) a ( Pi )- IT - +-- T.- +r- T.- -0r r r r~ ~ - -

The pressures defined in Equation (1) are related to velocities inthe manner described by Fon:hbeimer's equation:

-~_~_PPV2

ax k

For low velocity flows, the secoIKI tenn of Equation (2) becomesnegligible and the equation reduces to the well-known Darcyequation. Inclusion of the velocity-squared term, however.becomes necessary as velocities ~ causing nubulence andinterstitial effects in the interstices of the porous media. Sincehighest velocities appear at the flow-convergent region surround-ing the perforation, inclusion of this turbulence term may signifi-cantly affect the magnitude of the velocities calculated. In order toincorporate Equation (2) into the solution of the potential equa-tion, the transtnissibilities defined in Equation (I) are representedby the following manipulation of Equation (2):

damage. it provides no direct insight into what a safe injection orproduction rate would be in the field.

Actual flow into or out of the reservoir is rarely simple (as inthe case of an openhole completion) where a simple radial flowmodel can be used to evaluate the maximum sandface velocitybased strictly on the hole diameter, porosity of the formation, andthe net pay. More commonly, wells are cased and then perforated.This IaUlts in a very complex flow geomeuy governed by:

a) perforation type, size and lengthb) perforation density (number of shots/metre)c) veI1ica1 VemlS horizontal permeability ratiod) porositye) net payf) reservoir heterogeneityg) number of open and effective perforationsh) localized damage adjacent to the perforations

In order to obtain reliable estimates of the maximum velocitieswhich will occur in the near perforation region. where potentialdamage will be most severe, detailed modelling of flow in thisarea is necessary. Formulation of such a flow model will thenallow direct application of the laboratory test data back to fieldinjection or production conditions.

T- kp+~ 3)

raroozabadi and Katz{16) studied high-velocity flow through sever-al sandstone cores and developed the following correlation for thevelocity coefficient. [i.

~ - L80SxlO-5tL»l .(4)

10 onier to develop an efficient, accurate model the numericalmethod utilized to solve the potential equation is critical. Fmitedifference techniques have long been employed in solving the par-tial differential equations in black-oil as weD as compositionalsimulators. The drawback of fmite difference approximations.however, is the introduction oftnlncation errors. the magnitude ofwhich are a function of the discretization intervals selected. Ofcourse these emxs may be minimized by increasing the total num-ber of gridblocks. In addition, solution of partial differential equa-tions by means of finite difference techniques yields pressure~ts which are aVetaged over the gridbl~ defined For mod-

Simulation Model DevelopmentNumerous authors have conducted numerical simulations to

describe the steady-state flow characteristics in the near weUboreregion associated with perforated completions. The earliest studieswere carried out utilizing finite difference techniques to solve dtepartial differential equation derived from material balance consid-erations of flow within a porous media. Harris(12) first applied thistechnique in an attempt to define the productivity enhancementachieved by utilization of perforated completions as opposed toopenhole completions. Hong<13) also used this approach to calcu-late productivity ratios for various completion configurations. Ascomputing capabilities have increased. more rigorous finite ele-ment models have been developed to define dJe flow patterns sur-rounding a perforation. More recently, Locke<14) and Tariq<15) havedeveloped three-dimensional fmite element models to calculatedie productivity ratios associated widt more complex geometricalperforation configurations. namely. helical perforation patterns. Inmany of these studies. nomographs have been developed to assistin the calculation of productivity ratios for various perforationconfigurations and to assist in ascertaining the most productive

complction tbign.In none of dtese studies. however. have velocity profiles been

p-ovided to alert engineers of dJe potential problems associated

Odober 1993. V~ 32. No. 8

weUbore radius, the assumption is made of perfect radial flow. Asfar as the actual perforation is concerned. the following equationis incorporated into the model.

.!A2l- [T ~..-'.. '-,

T, aP "dP+--+T -co

r ae "az .(5)

erate gradients this is not a serious discrepancy, but again theinaccuracy associated with this approximation is a function of the

grid specification.Finite element techniques have been increasingly utilized in

applications for the solution of this numerical method's fonnula-tion. Increased accuracy can be achieved by varying tile sius ofthe finite elements and/or the specification of higher order polyno-mial approximations of partial differential equations; particularly,the solution of momentum and continuity equations. This methodis based upon the use of functional approximations to the solutionover discrete "elements". The attractiveness of this approach liesin its ability to conform to irregular shapes or regions involvingsteep gradients. Due to the nature of this numerical method's for-mulation, increased accuracy can be achieved by varying tile sizesof the finite elements and/or the specification of higher orderpolynomial approximations. While implementation of fmite ele-ment methods hold great promise, tile vast majority of reservoirsimulators still employ finite difference techniques due to tile sub-stantial effort involved in the design of the fmite element mesh for

specific applications.Methods of Weighted Residuals unite the finite element

approach described with another technique known as Collocation.These methods, as described in Villadsen and Micbe1sen<l7) alsoemploy polynomial approximations to the solution of partial dif-ferential equations with formulations such that errors at nodeschosen are zero so long as tile functions are continuous and thedegree of tile polynomial approximations used is sufficiently high.Using families of orthogonal polynomials, these nodal points areselected as tile roots of tile orthogonal polynomial or collocationpoints. The manner in which this method is fonnulated facilitatesadaptation of the grid which enables study for minimi7~tion of thenumber of grids while still maintaining accuracy. Due to theincreased ~uracy obtainable with this method, as compared withfinite difference approximations, the orthogonal collocation tech-nique has been applied here in solving the ~-dimensiona1 flowproblem describing flow in the vicinity of tile perforation. Despitetile accuracy and versatility of collocation methods, tile resultantmatrix which must be solved is a Spule, non-uniformly-baodedmatrix. Hence, in order to maintain a diagonal matrix, orthogonalcollocation is implemented in both the angular and vertical direc-tions and fmite difference is applied in tile radial ~on. In thismanner, the ~tions in which maximum gradients are expectedto occur are modeUed utilizing the most accurate approximations.

Stipulation of the simplified perforation g~metry dictates thattile region to be simulated need cover only one-qUBrter of the flowarea. This assumption assumes a homogeneous reservoir systemwith an undamaged, open perforation. This three-dimensionalregion is illustrated in Figure S. Because of tile symmetry of theflow patterns for the simplified g~metry, each of tile vertical andangular boundaries are specified by Neumann boundary condi-tions. Radially, the presence of the wellbore also dictates a noflow boundary. At the outenDOSt radius, specified as 60 times tile

Derivation of this equation is based upon the following twoassumpIions. FIrStly, the perforation is approximated by a squarecross-sectional tube whose cross-sectional area is equivalent tod1at of a cylinder of diameler dp. and secondly, die perforation isrepresented as line ~ located at the inrersecrion between thelower horizontal plane and the inner radial plane as shown inFigure 5. Saucier<'8) pedormed laboratory tests on pressure pro-files in fonnarion-filled and gravel-packed perforations, revealinga large range in permeabilities in perforations as a function of dienature of the porous material in dJem. In onIer (0 allow for Ibis.die penneability in the perforation must be specified as an inputparameter. In addition, a crushed zone at die end of the perfora-tion can be ~ted for by reduction of the radial ttaosmissibili-ty at that point. Angular transmissibility can also be altered toaccount for permeability impairment due to crushing along theperforation walls. The model has further been SUUctured to requireboth a pressure at the outer boundary and a bottomhole pressureas input to the simulator. Producrion or injection rates, dependingupon the pressure values assigned, can then be calculated from diepressure gradients calculated by the model.

Iniriarion of the simulation is made by use of Darcy's equationfor calculation of transmissibilities. The resultant maIrix is thensolved using an iterative method (0 arrive at the nodal pressurevalues. From these values, velocities are calculated for each of dienodes. For high velocity flows, ~-!!!OOariOD for tulbuIent flowis made by iterative correction of the transmissibility terms inaccordance with Equations (3) and (4).

.

Simulation ResultsSimulation results are obtained for a 0.20 m diameter injection

well with four perforations per layer. Distance between successivelayen of perforations is 0.5 m with each perforation extending outradially into the fonnation a distalM:e of 0.30 In. Penneabilities inthe horizontal direction have been set at 100 roD, while in the ver-tical direction, a permeability of 50 mD is specified.Compensation fcx the crushed ~ at the radial eI)d of the peifo-ration is accounted for by a lOO-fold reduction in the penneabilityat this point. For simplicity the walls of the perforation areassumed to be nOD-damaged in dIese examples. In most cases azone of impaired penneability is present along die wall of the pe1'-foration (which can be taken into consideration in the input pea--meability specifications fcx the model if de$ired). Flow Ibrougb

The Journal of Can~Petraeum Technology

die perforation itself is dictated by a 10 D penneability, simuJat-ing a relatively poorly conductive packed perforation. (These areusumptions utilized sttictly for this example case.)

For simulation of a Water injection well, the fluid is assigned adensity of 1100 kg/m3 and a viscosity of 1 mPa.s (1 cP). All pres-IW'CS in the model are oonnalized and a simulation is carried outfor an injection pressure of 1.0 and a normalized pressure speci-fied at the outer boundary of 0.1, which llanslate into pressures of20 MPa and 15.5 MPa. respectively. Solution of the resultantpressure profiles enables calculation of the injection rates basedupon gradients at the outer radial boundary. Back calculation inIbis ~ yields an injection rate of 0.630 m3/day of water intodie single perforation. This is furtbec verified by integration of thevelocity gradients along the length of the perforation.

Solution of the matrix describing flow within the wellboreregion is obtained by iterative correction of an initial guess of thepressure profile. The pressures calculated for each of the grid-points then allow calculation of the velocities, which are functionsof the pressure gradients. Figure 6 illustrates the velocity profilescalculated by the model assuming Darcy flow for the horizontalplane i~~iAtP.ly adjKent to the perforation. The lengths of thearrows displayed are proportional to die magnitude of the veloci-ties emanating from the initial points of each of the vectors in thedirections sOOwn. It should be DOted that these velocity vectors donot include vertical components which also vary with distancefrom die perforation. Since penneabilities are largest in the hori-zontal direction. however, the horizontal component of the veloci-ties with the greatest magninJdes are illUStrated in this diagram.The results of the simulation reveal that the largest velocities areencountered at the perforation-wellbore interface and decreasemonotonically with distance from this point. These results areintuitively expected. A ~ observation is the decrease in mag-

nitude of the velocity vectors along the length of the perforation.This ~uction is related both to the penneabilit)' along the lengthof the perforation and to the presence of the crushed zone at theradial boundary of the perforation. Due to these tWo factors, pres-sures decrease rapidly along the length of the perforation whichresult in lower angular and vertical pressure gradients and. hence.lower vekJCities in each of these directions.

After calculation of the velocity profiles generated using theDarcy flow assumption, compensation for non-Darcy flow isachieved by correction of the transmissibility tenDS in accordancewith Equation (3) followed by iterative solution of the revisedmatrix. While this does not significantly alter velocity profilesthroughout the region. a small effect is noticeable. Comparison ofvelocity magnitudes for gridpoints, situated along a radial axis onthe same horizontal plane as the perforation and deviating only4,50 from the radial axis of the perforation. is provided inFigure 7, Although the injection rate into the formation is smallfor this particular case, allowance for non-Darcy phenomena doesindicate an effect which may be much more significant for higherinjection rates. As far as actual velocities are concerned. bothcurves indicate that velocities may be sufficient to cause finesmigration near the weUbore dependent, of course. upon the actualvalue of the critical velocity which can be measured in laboratoryfines migration tests. Of interest in the simulation results are notonly the points at which fmes may begin to be mobilized. but alsothe directions in which fines may be displaced.

Another factor demanding attention is the permeability of theperforation itself. The properties of the perforation pecking mater-ial will strongly affect penneability which. in rum. will affect boththe injectivity and the nature of the velocity profiles. In order toinvestigate this. a separate run is carried out for the same condi-tions. but with a very low penneability in the perforation of onlyI D (this simulates a badly infilled gravel packed perforation).Simulation results indicate that this reduction causes the overallinjection rate to decrease to 0.376 m3/day. The comparison ofvelocity magnitudes (along the same radial axis described inFigure 6) for d1is case, in contrast to those for die base case, areshown in Figure 8. As anticipated. the reduction in perforationpermeability causes greater flow velocity out of the perforation atthe perforation-weUbore interface and decreased flow velocityfwther along its length.

Reservoir geology and perforation patterns will also affect flow- both in terms of injectivity and velocity profiles SUn'OUnding theperforation. The magnitude of the effects caused by variations ineither reservoir geology or the perforation pattern. however, aredifficult to quantify. For example, another run carried out for diecase of a 10 mD vertical permeability (k.Ik. = 10 as comparedwith k/k. = 2 for the base case) indicates a ~ injection rate of0.596 m'/day, a reduction of only 5%. As far as velocities are con-cenled, a comparison of velocity magnitudes is presented inFigure 9. In comparison with the magnitude of the velocitiesobserved in die base case. the redIK:rlon in vertical permeability

53Ocki)er 1993. Vdwne 32. No. 8

results in lower velocities near the wellbore, but higher velocitiesradially outward. While this is an expected consequence of a ver-tical flow restriction which results in higher pressures (i.e. lowerpressure gradients) near the well bore and, consequently, highervelocities at greater radial distances, the magnitude of the differ-ences is difficult, if not impossible, to ascertain without the use ofa simulation.

ConclusionsFines migration is a source of potentially severe penneability

impainnent Laboratory tests have been developed and equipmenthas been constructed to detennine critical velocities at which finesbegin to mobilize. Laboratory procedures facilitate the calculationof these critical velocities which represent linear flows withinporous media. In order to apply this information to an actual reser-voir, a three-dimensional simulator has been developed to calcu-late the velocity profiles surrounding a perforation. Although notdiscussed in this paper, development of this model enables modi-fication of reservoir geological properties (i.e. penneabilities) andperforation patterns, including the number of perforations perlayer, the penneability of the perforation, and the length anddiameter of the perforation. In addition. allowance is made fornon-Darcy flow, a phenomenon which may become significant inthe high velocity flow region surrounding a perforation. Due tothe complex nature of three-dimensional flow, this type of a simu-lation is necessary to quantify velocity profiles and thus apply theresults of laboratory experimentation.

AcknowledgementsThe assistance and technical expertise of staff at Hycal Energy

Research Laboratories Ltd. are greatly appreciated.

R.A., A Laboratory StUdy of Pemleability Damage to Cold Lake TarSands Cores;; Journal 01 Conadian Petroleum TechnolOg;\.. Vol. 18-No. I. February, 1989.. pp. 56 - 62.

7. PRISHOLM, S., KIELSON, B.L. and HASLUND, 0.. FinesMigration, Blocking. and Clay Swelling of Potential GeothermalSandstone Reservoirs: Denmark, Society of Petroleum EngineersFonn Evaluation, June 1987.

8. GABRIEL, G.A. and INAMDAR, G.R., An ExperimentalInvestigation of Fines Migration in Porous Media: Society ofPetroleum Engineers Paper 12168, presented at the 1983 Societ). ofPetroleum Engineers Annual Technical Conference and Exhibition.San Francisco, CA, October 5-8, 1983.

9. BORCHARDT, J.K.. ROLL, D.L. and RAYNE, L.M.. Use of aMineral Fines Stabilizer in Well Completions; Society of PetroleumEngineers Paper 12757, presented at the 1984 Society of PetroleumEngineers California Regional Meeting, Long Beach, CA, April II.13,1984.

10. SENGUPTA,S.K. et al., Effect of Row ~and Rheology on ShearStrength of Migrating Formation Fines Due to Row of PseudoplasticRuids; Society of Petroleum Engineers Paper 10669, presented atthe 1982 Society of Petroleum Engineers Fom1ation Damage ControlSymposium, Lafayette. Los Angeles, CA, March 24-25, 1982.

II. GRUESBECK, C. and COLLINS, R.E., Entrainment and Depositionof Fine Particles in Porous Media; Society of Petroleum EngineersJournal, December 1982, pp. 846-56.

12. HARRIS, M.H., The Effect of Perforating on Well Producti\it).:Journal 01 Petroleum Technology, April 1966, pp. 518-18; Trans.AIME,237.

13. HONG, K.C., Producti\ity of Perforated Completions in FormationsWith or Wid\out Damage; Journal 01 Petroleum Technolog)', AugUSt1975, pp. 1017-38; Trans., AIME, 259.

14. LOCKE, S., An Ad\"3Dced Method for Predicting the Producti\it).Ratio of a Perforated Well; Journal 01 Petroleum Technology,December 1981, pp. 1381-88.

15. TARIQ, S.M., Evaluation of Row Characteristics of PerforationsIncluding Nonlinear Effects With the Finite-Element Method;Society of Petroleum Engineers, May 1987, pp. 104-112.

16. FlROOZABADI, A. and KATZ, D.L, An Analysis of High-velocityGas Row Through Porous Media; Journal of Petroleum TechnolOg)',February 1979, pp. 211-16.

17. VILLADSEN, J. and MICHELSEN, M.L., Solution of DifferentialEquation Models by Polynomial Approximation; Prentice-Hall Inc..New Jersey, 1978.

18. SAUCIER, RJ., Considerations in Gravel Pack Design; Society ofPetroleum Engineers. February, 1974, pp. 205-212.1

NOMENCLATUREd, = Diameter of Perforation (m)k = Permeability (D)r = Radial Distance (m)v = Velocity (m/s)z = Vertical Distance (m)P = Pressure (kPa)T = Transmissibility (D/Pa.s)

Authors' Biographies

John Eng is currently working on his Ph.D. at Brigham YoungUniversity in Utah. John was employed by Hycal EnergyResearch Laboratories Ltd. as a Senior Research Engineer worlc-ing in the area of numerical modelling and multi-phase flow inporous media. John received B.Sc. and M.Sco degrees from theUniversity of Calgary in Chemical Engineering and is a memberof APEGGA.

Greek Symbols~ = Non-Darcy Coefficient (mID)JL = Fluid Viscosity (Pa.s)9 = Angular Distance (rad)

Subscriptsr = Radiale = Angularz = Verticala =area1

Brant Bennion is currently President ofHycal Energy Research Laboratories Ltd.of Calgary, Alberta. He is involved inresearch in multi-phase flow in porousmedia, formation damage studies, miSCloleand thermal E.O.R. Brant received a B.Sc.ftom the University ofCaigary in 1984 andis a member of APEGGA and a Boardmember of the Calgary section of ThePetroleum Society of CIM.

Jeff Strong is currently a Senior ResearchEngineer at Hycal Energy ResearchLaboratories Ltd. Jeff's interests includefluid phase behavior and formation dam-age. Jeff received a B.Sc. degree from theUniversity of Calgary and is a member ofAPEGGA and The Petroleum Society ofCIM.

REFERENCES1. MUECKE. T. W., Formation Fines and Factors Controlling Their

Movement in Porous Media; Journal of Petroleum Technology.February, 1979, pp. 144-50.

2. GRAY, D.H. and REx. R. W., Formation Damage in SandstoneCaused by Clay ~oo and Migratioo; Clay and Clay Minerals,p.26,l966.

3. MORGAN, J.T. and GORDON, D.T., Influence of Pore Ge(!metryon Water-Oil Relative Permeabilities; Journal of PetroleumTechIIOlogy, October, 1970. pp. 1199.1208.

4. PORTER, K.B., An Overview of Formation Damage; Journal ofPetroleum Technology. VoL 4J, No.8. August. 1989.

5. SELBY, R.J., and FAROUQ-ALI, S.M., Mechanics of SandProduction And The Flow of Fines in Porous Media; Journal ofCanadian Petroleum Technology, Vol. 27, No.3, May, J988, pp. 55 -63.

6. KW AN, M. Y., CULLEN, M.P., JAMIESON, P.R. Md FOR"nER,

54 The Journal of Canadian Petroleum Technology