54630_spe-paper

Copyright 1999, Society of Petroleum Engineers Inc.

This paper was prepared for presentation at the 1999 SPE Western Regional Meeting held inAnchorage, Alaska, 26–28 May 1999.

This paper was selected for presentation by an SPE Program Committee following review ofinformation contained in an abstract submitted by the author(s). Contents of the paper, aspresented, have not been reviewed by the Society of Petroleum Engineers and are subject tocorrection by the author(s). The material, as presented, does not necessarily reflect anyposition of the Society of Petroleum Engineers, its officers, or members. Papers presented atSPE meetings are subject to publication review by Editorial Committees of the Society ofPetroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paperfor commercial purposes without the written consent of the Society of Petroleum Engineers isprohibited. Permission to reproduce in print is restricted to an abstract of not more than 300words; illustrations may not be copied. The abstract must contain conspicuousacknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O.Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

AbstractThe dramatic effects of non-Darcy and multiphase flow inpropped hydraulic fractures have been documented by severalauthors.1,2,7,9-19 However, many engineers disregard theseeffects when designing fracture treatments under theassumption that they only apply to high rate wells. This papershows these effects are significant even in wells considered tobe low rate by current industry standards. Ignoring theseeffects will often lead to inaccurate production forecasts, sub-optimal fracture design, and selection of an inappropriateproppant type. These mistakes result in lost revenues whichcan exceed $2 million per fracture treatment for typical gasand oil well fracture treatments conducted in North America.

Proppant permeability values reported by the industry andused in most fracture design models are measured with asingle-phase fluid at extremely low velocities. The laboratoryrates stipulated in the American Petroleum Institute3 (API)Recommended Practice number 61 typically correspond tosuperficial fluid velocities ranging from 0.2 to 2.0 inches perminute. However, in real fractures, the actual fluid velocitymay exceed two feet per second, approximately 1000 timesgreater than in the laboratory measurements. Although theAPI warned that the lab-measured values would not berealistic under actual fracture conditions, the industry haslargely failed to incorporate correction factors into productionmodels to compensate for non-Darcy flow effects.

In addition to non-Darcy effects, the measured proppantpermeability values fail to incorporate the effects ofmultiphase flow. Most gas wells will produce some freeliquid (water or condensate), and almost all oil wells will beproduced below the bubblepoint of the oil, resulting in

substantial volumes of free gas in the fracture. This papershows that non-Darcy and multiphase flow effects frequentlyresult in an effective fracture conductivity 50% to 98% lowerthan the reference value obtained from the API testingprocedure. Incremental reductions for gel damage andproppant embedment often result in fracture permeabilities orconductivities (the product of fracture permeability andeffective fracture width) of 1 to 10% of the published values.This paper examines the mistakes that are likely to be madewhen ignoring these effects, and estimates the additionalcashflow which can be obtained by optimizing fracture designwith consideration of multiphase and non-Darcy flow.

IntroductionThe American Petroleum Institute has published standardtesting procedures 3 which involve flowing a single phaseliquid (water with 2% KCl) through a 7″ x 1½″ linearproppant pack cell at extremely low flowrates of 1 to 10ml/min. These rates correspond to a range of 4 to 40 bpd ofoil, or 13 to 130 mscfd of gas at 1000 psi pbh, produced fromtwo fracture wings of a 30 ft. high fracture. The superficialvelocity of the water during the test is on the order of 0.2 to2.0 inches per minute; while in actual fractures, the true fluidvelocity can be several feet per second. The API recognizedthat these tests are not representative of actual conditions andplaced the following “disclaimer” with the procedures:“CAUTION: The testing procedures in this publication arenot designed to provide absolute values of proppantconductivity under downhole reservoir conditions.”3 Theconductivity tests were designed for operational simplicity andselected to be in the laminar flow regime to improverepeatability. Unfortunately, proppant suppliers in theindustry have solely used these procedures to publishpermeability values for their products and many engineershave used these reference values in fracture design withoutmaking appropriate adjustments for the non-Darcy andmutiphase flow effects.

Erroneously using the published proppant permeabilityvalues without compensation for non-Darcy and multiphaseflow effects results in:

• Invalid fracture conductivity predictions – which may beoverestimated by more than an order of magnitude,

SPE 54630

Non-Darcy and Multiphase Flow in Propped Fractures:Case Studies Illustrate the Dramatic Effect on Well ProductivityMichael C. Vincent, SPE, Insight Consulting; C. Mark Pearson, SPE; and John Kullman, SPE, CARBO Ceramics, Inc.

2 M. C. VINCENT, C. M. PEARSON, J. KULLMAN SPE 54630

• Design of a sub-optimal stimulation treatment including ashorter effective producing fracture half-length,

• Significant overestimation of the post-stimulationproduction rates,

• Potential for selection of the incorrect proppant for thefracture treatment design, and

• Significant loss in production from the fractured wellcompared to what could have been achieved.

Non-Darcy Flow EffectsHenry Darcy initially developed his familiar correlation byobserving water draining through a sand column.4 His studiesshowed that at low velocities, the pressure drop throughporous media is proportional to the fluid velocity:

∆p/L = µ v / k ………………….…………….. (1)

where: ∆p/L = pressure drop per length of proppant packµ = fluid viscosityv = superficial fluid velocity (as if porosity were 100%)k = permeability of porous media

However, at the higher velocities typical in a proppedfracture, the pressure gradients become proportional to thesquare of the velocity, as represented by Forchheimer’sequation: 5

∆p/L = µ v / k + βρv2 ……………………….. (2)

where: β = coefficient of inertial resistanceρ = fluid density

Forchheimer’s equation states that the pressure gradient isthe sum of the viscous forces (µv/k) and the inertial forces(βρv2). At low velocities, where inertial forces are small,Forchheimer’s equation reduces to Darcy’s Law.

Conceptually, Darcy’s Law states that the pressure drop inporous media is controlled by friction. At low velocities, theviscous drag of the molecule of fluid upon the proppantcontrols the pressure drop. Forchheimer’s equation recognizesthat additional energy loss is incurred by the repeatedacceleration and deceleration of the fluids. A molecule of oilor gas will travel ~50,000 proppant grain diameters within 100ft. of fracture length. At each encounter with a proppant grain,the molecule must change direction to travel around the grain.The molecule will undergo substantial acceleration to travelthrough the narrow pore throat and decelerate into the largerpore body. Forchheimer’s equation recognizes that the energylosses due to this repeated acceleration and deceleration of themolecule are highly significant at realistic fluid velocities.

Using the above equations, the pressure drop per foot offracture length can be calculated as a function of velocity inthe fracture or production rate through a given cross-sectionalarea of fracture. Figure 1 shows the pressure drop through abi-wing fracture with a 50 ft. frac height when propped with20/40 Jordan sand at 2 lb/ft2. It is apparent that Darcy’s Lawis not valid for calculation of the pressure drop down thefracture at even moderate rates below 500 mscfd. Figure 1

was developed for a bottom hole pressure (pbh) of 2000 psi.At lower pbh, the gas will expand and subject the fluid toincreasingly higher velocities, resulting in even greater inertialpressure loss. Examination of Figure 1 reveals that theinertial pressure drops are much more significant than theviscous losses, suggesting that beta is more important toultimate productivity than the laminar conductivities reportedby proppant manufacturers using the API test procedure.

Interestingly, Forchheimer’s equation for pressure drop inporous media is similar in form to the equations used tocalculate the pressure drop in pipes. At low (laminar) flowrates, viscous forces dictate the pressure drop, with ∆p/L beingproportional to the fluid velocity. But at higher (turbulent)flow rates, the pressure drop becomes proportional to thesquare of the velocity. Because of this similarity, the non-Darcy flow effects are often referred to as “turbulent” effects.This is not entirely accurate. A turbulent flow regimetypically begins to occur in pipe flow at Reynolds numbersabove 2300.6 However, in porous media, the non-Darcyeffects become significant even if the overall velocity is withinthe laminar flow regime and has been observed with Reynoldsnumbers as low as one.7 The nature of flow in porous mediacauses a very complex flow regime even at low velocities.

The non-Darcy inertial forces are primarily due to theacceleration and deceleration of the fluid as it travels throughthe tortuous flow path of the porous media. The oil, water orgas in the fracture must continually change direction,accelerating through pore throats and decelerating in the largerpore spaces. The force necessary to effect those accelerationscan also be derived from Newton’s Laws of Motion, in whichthe kinetic energy is equal to ½ the mass multiplied by thesquare of the velocity.8 Therefore, the form of Forchheimer’sexperimentally derived relationship is intuitively reasonable.

It is important to recognize that both Darcy’s andForchheimer’s equations are based on the superficial velocityof the fluid, which effectively assumes 100% porosity in thefracture. The same fluid velocity is assumed for both adamaged fracture containing poorly sorted, severely crushed,highly angular proppant and a theoretically ideal proppedfracture containing perfectly sphericial proppant of uniformsize. In reality, the pressure loss in the fracture is clearlyrelated to changes in the true fluid velocity, which is afunction of the proppant characteristics. Forchheimer’sequation incorporates the differences in proppantcharacteristics via the beta factor (β).

Beta Factor. Alternatively referred to as the “inertial flowcoefficient,” the beta factor is a proportionality coefficient thatis determined by laboratory measurements. Beta is essentiallya measure of the tortuosity of the flow path.9 Beta isdetermined by flowing more realistic velocities through theAPI conductivity cell and solving Equation 2 for beta to matchthe observed pressure drops.

Many factors influence the inertial flow effects for singlephase flow including: 9-12

SPE 54630 NON-DARCY AND MULTIPHASE FLOW IN PROPPED FRACTURES:CASE STUDIES ILLUSTRATE THE DRAMATIC EFFECT ON WELL PRODUCTIVITY 3

• Initial proppant permeability,• Porosity of the proppant,• Curvature of streamlines (proppant angularity),• Relative aperture of pore throat to pore space,• Proppant size distribution,• Heterogeneities, and• Surface roughness.

Figure 1 indicates that the pressure drop due to friction isminimal compared to inertial losses for most gas wells withmoderate production rates. A well producing 500 mscfd ofdry gas through two 50 ft. high fracture wings at a pbh of 2000psi will have a total pressure drop approximately five timesgreater than Darcy’s Law would predict. To minimizepressure drop through the fracture, it is necessary to minimizethe βρv2 term from Equation 2 . Figure 2 shows that the betafactors for common proppants can vary by a factor of six.Since the pressure drop in the fracture is typically dominatedby non-Darcy effects, the beta factor can be more importantthan reference permeability when selecting a proppant.

Unfortunately, beta factors are not typically measured andreported by the industry. Selecting a proppant by referencepermeability alone can be misleading. It is possible forproppants with similar reference permeabilities to havedrastically different effective permeabilities when subjected tothe realistic fluid velocities in typical fractures.

Multiphase Flow EffectsSeveral researchers have attempted to quantify the effect ofmultiphase flow in fractures with laboratory work or byanalyzing production data.13-18 While these authors reportdiffering results regarding the absolute value of theconductivity loss, all conclude that the effects are substantialand should not be ignored. Although he recommendsmeasuring multiphase effects for specific cases, Pennyestimates that the effective conductivity is typically reducedby a factor of three for each 4 barrels of liquid produced permmscfd.13

Evans reported that beta factors increase by tenfold withvery small mobile liquid saturations.14 Milton-Tayler reported“order of magnitude” reductions in effective conductivity dueto multiphase flow.15 Schubarth reviewed actual productionresults from 550 gas wells and found a strong correlationbetween condensate yield (multiphase flow) and reducedproductivity.16 In the literature, as in this paper, the effect ofmultiphase and non-Darcy flow can be reported in severalways. If the pressure drop across the proppant pack increasesby ten-fold, this is synonymous with reporting a ten-foldreduction in effective conductivity or a ten-fold increase inbeta. Converting the pressure drop to an effectiveconductivity is one technique to account for multiphase andnon-Darcy effects in a production model utilizing Darcy’slaw.19

Incremental pressure drop due to multiphase flow isbelieved to be the result of three primary causes: saturationchanges, relative permeability effects and phase interactions

within complex flow regimes. For oil wells produced belowbubble point, the evolution of solution gas to free gas will alsodrastically increase the fluid velocities in the fracture.

It is relatively simple to calculate the saturation changesand relative permeability effects due to multiphase flow in thefracture. However, the complex interactions resulting frommultiphase flow are exceedingly difficult to model. Due tomarked viscosity differences, the gas phase travels atsubstantially higher velocity than the liquid phase in thefracture. However, due to the tortuous flow path inside theproppant pack, the two phases often remain in intimatecontact. Liquid droplets may be picked up (entrained) by thegas and substantial energy is consumed as the liquid isaccelerated to many times its superficial velocity. When thedroplet collides with a proppant grain, it undergoes substantialdeceleration only to be accelerated again by the gas phase.The extreme inefficiencies in flow path and inertial losses areresponsible for the surprisingly large pressure drops observedin the laboratory with addition of a few volume percent ofliquid to a gas system.

A common mistake with oil wells is failure to recognizethat the pressure in the fracture is typically far below bubblepoint. Although the average reservoir pressure can often bemaintained at or above bubble point, the bottom hole pressure(pbh) of the producing well is typically reduced to maximizedrawdown and production rates from the well. It is importantto realize that an oil well with a gas-oil ratio (R) of 560 scf/bblproduces 99% gas by volume and only 1% liquid whenevaluated at atmospheric pressure conditions. The bottomhole pressure may be 150 psi for wells with submersiblepumps and may yield 90% gas and 10% liquid by volume. Atsome distance away from the wellbore, the pressure in thefracture may be 1500 psi; in which case, the fracture may befilled with 50% liquid and 50% gas. Unless the productionmodel is linked to a compositional simulator, it is unlikely thatit rigorously handles the evolution of solution gas in the bodyof the fracture. Failing to consider that 50% to 90% of thefracture can be filled with free gas has obvious implicationsregarding the effective capacity of the fracture to produce oil.

Experimental measurements of multiphase and non-Darcyflow effects have been documented by Penny.13 As shown inFigure 3, the proppant conductivity measured at extremelylow flowrates according to API specifications can beexceedingly optimistic. Even with a fairly modest well (1mmscfd from a bi-wing 50 ft. fracture height), the non-Darcyflow effects reduce the effective conductivity of the proppantpack by approximately 70%. Addition of a mere 10 bwpd tothe 1 mmscfd causes an incremental reduction ofapproximately 25% of the reference conductivity. Although10 bbls of water in 1 mmscfd appears to be a negligiblevolumetric percentage (0.006%) when evaluated atatmospheric pressure, the volumetric percentage is ~1% whenevaluated at 3000 psi. Note, however, that adding a mere 1%liquid to the system increased the pressure drop by 550%!Saturation changes and relative permeability changes alonecannot account for this dramatic effect. It is believed that the


primary factor is the increased complexity of the flow regimedue to phase interactions. Additional damage to the proppantpack due to gel damage, embedment, long-term degradation,fines migration and other factors can result in effectiveconductivities that are 99% lower than suggested by the APItesting procedures.

Production ModelsFew production models accurately handle non-Darcy flow.Although sophisticated 3-phase models can predict the effectsof saturation changes and relative permeability, it appears thatno mechanistic model currently available can accuratelypredict the extreme pressure drop due to phase interactionswithin the intimately mixed flow regimes typical in fractures.

Production models which assume infinite fractureconductivity are often unsuitable for these analyses. Theassumption of infinite fracture conductivity providesreasonable results when the dimensionless fractureconductivity (CfD) is greater than 500.20 In concept, infiniteconductivity models assume that the pressure drop in thefracture is negligible compared to the pressure drop in theformation. For a constant fracture length, Holditch has shownthat to optimize economic return, the CfD should be in therange of 10 to 30.21 Elbel has shown that fracture designs witha CfD of three or less cannot be improved significantly byincreasing fracture length with the same conductivity. If theCfD is greater than 30, increasing length will be morebeneficial than increasing conductivity.22 The case studies inthis paper indicate that the pressure drop in the fracture isfrequently significant and fracture conductivities obtained inpractice will typically be far lower than optimal.

Production and economic forecasting in this paper wasprepared with Stim-Lab’s SLFrac production model. This toolwas chosen because it has the highest quality permeability andbeta factor data in the industry and the production model isavailable and used by the 40+ companies who are members ofthe Stim-Lab proppant consortium. All the major servicecompanies are members of the consortium and may use theprogram to evaluate potential fracture designs. Theproduction model is based on Agarwal type curves, so itincludes implicit assumptions regarding an infinite reservoirwith no pressure support.20 However, if incremental benefitsof multiple cases are compared, the inaccuracies of thesimplified type curve solution will essentially cancel out. Atthis time, the SLFrac program has the ability to estimatemultiphase effects in gas wells by correlating trends inmeasured laboratory data, but does not accurately predict theconductivity reduction for oil wells with pbh below bubblepoint.

If an engineer does not have access to a program such asSLFrac, it may be possible to compensate manually for theseeffects. Some finite-conductivity models will allow the userto modify the reference proppant conductivity tables or input adamage factor to compensate for multiphase and non-Darcyeffects.

Results - Case Studies:Following are four specific case studies which have beenhighly simplified for this paper. For actual analyses, theauthors recommend comparing a wider range of proppanttypes, sizes and performing sensitivities on uncertain valuessuch as fracture width, oil/gas prices and formationpermeability to develop a better understanding of the effect ofeach uncertainty upon fracture design. These wells were notselected because of particularly innovative frac designs, butrather to depict four very different wells with modest flowrateswhich illustrate the effects of non-Darcy and multiphase flowin a variety of flowing conditions.

Assumptions for Case Studies.The Appendix contains a complete listing of reservoircharacteristics and product prices used for all analyses. Tosimplify the paper, the results for only three proppants areshown in Cases 1 through 4: natural frac sand, a premiumresin-coated sand (RCS) and light weight ceramic (LWC).The three-dimensional plots for Case 5 include both standardand premium RCS, and also include heavy weight ceramics.These analyses do not consider the problem of proppantflowback. In unconsolidated formations, it is often necessaryto use curable resins to maintain proppant pack integrity. Theanalyses presented in this paper assume that consolidation ofthe pack is not required.

Case 1: Deep Gas well.Case 1 is a mid-continent gas well in the Anadarko Basin,Oklahoma, at 12,500’ total depth, with 25’ of pay and 0.1 mdperm. The service company’s predicted fracture geometry is555’ half-length, 50’ frac height, with an average proppantconcentration of 1.5 lb/ft2. This geometry equates to 83,250lbs. proppant. Production history-matching on adjacent wellsindicated that a gel damage factor of 50% provided reasonableforecasts.

As shown in Table 1, the initial stress on the proppant forthis deep well was calculated to be 9279 psi. At this stress, thestandard industry practice is to place ceramic proppant. Thethird column of Table 1 shows the predicted producingcharacteristics of the well when non-Darcy and multiphaseeffects are ignored. Disregarding multiphase and non-Darcyflow, simplistic production models would indicate that 1.5lb/ft2 LWC would provide a CfD of 12.9, yielding a fractureapproaching infinite conductivity relative to the formation.Expected well rates with LWC are 3.8 mmscfd, whichgenerates superior economic returns compared to natural fracsand or resin coated sand (RCS).

Columns four and five show the effects of including non-Darcy and multiphase flow. Although LWC is still predictedto be the most economic proppant, the effective CfD is 0.16and the predicted well rate is reduced to 1.6 mmscfd.Although disregarding the non-Darcy effects allowed selectionof the correct proppant, it is clear that 1.5 lb/ft2 of LWC doesnot provide an infinite conductivity fracture, and thatincreasing fracture width or placing higher permeabilityproppants should be considered. Disregarding these flow


effects results in the loss of 2 mmscfd of potential productionrate and lost revenue of $2,000,000 within the first three yearsof production.

Case 2: Shallow Gas well.The preceding example showed that an optimized fracturedesign will be flawed unless multiphase and non-Darcy effectsare considered. However, the proppant choice for Case 1 didnot change. Based on experience and “rules of thumb”, thetraditional wisdom is to use premium proppants with deep gaswells. But what happens to this analysis with a shallow well?

As detailed in the Appendix, Case 2 models the same 25’of pay at 3000’ depth. Reservoir pressure has been reduced to1800 psi, and fracture geometry has been maintained with 1.5lb/ft2 average concentration. For this well, a pbh of 800 psi isassumed, which generates an effective stress on the proppantof merely 1780 psi. Traditional wisdom specifies that naturalsand would suffice in this shallow, low stress well.

Table 2 shows the predictions for the well with eachassumption. As expected, with laminar flow assumptions, the20/40 sand provides ample conductivity to handle thepredicted 700 mscfd, with a CfD of 32. Any investment topurchase premium proppant is wasted, as it results in less than5 mscfd incremental rate. However, with multiphase and non-Darcy flow effects, the sand fracture is shown to beconductivity limited, with a CfD of 2.4 and total productionrate would be constrained to under 500 mscfd. Here theceramic proppant is shown to produce an incremental 59mscfd (a 13% increase in initial productivity), which pays outthe incremental investment within one year and produces a88% return on the investment within 3 years. While theeconomics are not outstanding, many companies wouldconsider an investment with expected annual returnsexceeding 20%.

These results are counter to current industry designguidelines. Ceramic proppant is the most economic proppantfor a shallow well with 1780 psi stress and producing only 500mscfd? These results indicate that the pressure drop down thefracture is significant when propped with 1.5 lb/ft2 naturalsand and subjected to multiphase flow. It is desirable toincrease the conductivity in some manner. In these analyses,selecting a higher permeability proppant was evaluated. Analternative remedy may be to increase fracture width ifoperational constraints allow. It is interesting to note that theselection of proppant had little to do with proppant strength orcrush resistance, but was due to the lower beta factor with theless angular, high porosity ceramic proppant.

While it may be obvious that deep wells merit higherstrength proppant, it is more challenging to convince thatpremium proppants should be considered at very low stresses.Case Study 2 suggests that proppant selection should not bemade by strength, but rather by effective conductivity requiredto handle the expected flowrates.

Case 3: Moderate Depth Oil Well.The third case study is a moderate stress oil well from theKuparuk River Field, Alaska. Total well depth is 6000 feetand expected pbh is 800 psi. With a normal frac gradient of 0.7psi/ft, the effective closure stress is calculated to beapproximately 3400 psi. As shown in Table 3, the formationproductivity is good, resulting in oil rates often exceeding 200bopd for any of the 20/40 proppants studied. Several SPEpapers have been written showing the advantage of increasingfracture width and proppant permeability in moderatepermeability reservoirs.23-25 A typical fracture geometryassumed for the Kuparuk case study was 180’ half-length, 55’frac height and 5 lb/ft2 proppant concentration equating toapproximately 100,000 lbs. of proppant per stimulation.

Table 3 shows that using a laminar production model, anengineer may recognize that the well was conductivity limitedwith natural sand and could justify the incremental expense ofceramic proppant. At these low closure pressures, ceramicproppant is selected not for strength but for improvedpermeability.

Although use of a laminar flow model allowed the correctproppant to be selected, it seriously over-estimates productionrates that can be produced through the fracture. While aDarcy model will predict 700 bopd to be produced through thefracture with modest pressure drops, inclusion of non-Darcyeffects indicates that 300 bopd is more representative of actualflow capacity even when propped with 5 lb/ft2 20/40 sand orRCS. As has been proven from production data, increased useof larger, higher conductivity ceramic proppants and tip screenout techniques were very beneficial in development of thisfield.23,24

The SLFrac model does not have a compositionalsimulator and, therefore, cannot accurately predict how muchfree gas will be present in the fracture. However, consideringonly non-Darcy flow effects of single-phase oil, it can beshown that the ceramic proppant can generate a 8500% returnon incremental investment over three years. If necessary, PVTdata could be examined to estimate the volumetric percent ofsolution gas that would be present in the fracture as free gas.However, the analyses of non-Darcy flow effects alone haveindicated that the fracture is conductivity-limited and thepresence of gas would only exaggerate the decision to use aLWC proppant and increase fracture width where possible.

Case 4: Lower Rate, Low Stress Oil Well.The fourth case study is a low stress oil well in LakeMaracaibo, Venezuela. Total well depth is 6500 feet,expected pbh is 350 psi and, with a relatively low frac gradientof 0.54 psi/ft, the effective closure stress is calculated to beless than 3200 psi. As shown in Table 4, the productivity ismodest, resulting in oil rates of ~200 bopd per frac zone. Thewells are typically propped with 4 lb/ft2 20/40 proppants.Which proppant type will be most economic?

Table 4 shows that using a laminar flow productionmodel, an engineer would recognize that the well wasconductivity-limited with natural sand, but could not justify


the incremental expense of premium ceramic proppant. ADarcy model suggests that ceramic proppant will yield anadditional 13 bopd, which takes more than a year to payout theincremental investment. Although the ceramic will ultimatelygenerate a higher cash flow, the incremental oil rate would behard to document and justify. However, when non-Darcy floweffects are considered, the expected production rates drop tobelow 100 bopd for the natural sand frac compared to 266bopd for 20/40 lightweight ceramic. Placing natural sand asrecommended by the Darcy production model will result inselection of the incorrect proppant type, sacrificing over 160bopd and over $1,000,000 of additional cash flow during thefirst three years of production.

The SLFrac model cannot be used to predict the effects offree gas present in the fracture. However, considering onlynon-Darcy flow effects, it can be shown that the 20/40 ceramicproppant will generate a 2600% return on incrementalinvestment over three years. Although the data are notincluded here, evaluation of larger mesh ceramic proppantsshow additional production benefit for this field. Again, theseresults suggest that traditional “rules of thumb” for proppantselection do not agree with more rigorous engineeringanalyses and economic optimization.

Case 5: Three Dimensional PlotsTo further illustrate how proppant selection is affected by

non-Darcy and multiphase assumptions, Figure 4 wasdeveloped. The preceding analyses demonstrate that proppantselection should not be made solely on strength. It ishypothesized that the fluid velocity in the fracture might bestcorrelate to proppant selection. However, to generate a moreuser-friendly correlation, three-dimensional graphs were madethat relate proppant selection to closure stress and formationpermeability for a specific fracture geometry.

Again, these plots are highly simplified and include thefollowing assumptions:

Gas well with 50’ pay; 200,000 lbs of 20/40 proppant(500’ half-length, 100’ frac height, 2 lb/ft2) 250°F,Proppant prices = 40% discount from book price, Gasprice = $2/mscf.

The vertical axes display the incremental cash flow (afterproppant cost) relative to sand after three years of production.Therefore, sand is shown as a horizontal plane at anincremental cash flow of zero, and RCS and ceramic aredisplayed as intersecting surfaces relative to this plane.

Figure 4a shows the projected cash flow benefits for eachproppant when multiphase and non-Darcy effects are ignored.This plot illustrates that at low closure pressure and lowformation perm, natural sand is the most economic proppant.Curves for RCS and ceramic result in a negative cash flowrelative to sand at these conditions. However, with the scaleof the plot to $35 Million, the incremental cost of the premiumproppants (up to ~$100,000) is hard to discern.

As fracture closure pressure and/or permeability increase,the benefit of premium proppants is seen to increase. It is alsonoted that the $35mm benefit to ceramic at 1 md and 9000 psistress is overstated by the Darcy model. Ignoring multiphase

and non-Darcy effects allows prediction of unrealistically highgas rates exceeding 30 MMSCFD through a fracturecontaining 2 lb/ft2 20/40 proppants.

Figure 4b shows the forecast for an identical fracturewhen non-Darcy effects are included in the calculation.Notice that the scale of the vertical axis is reduced to amaximum benefit ~$10mm, which reflects more reasonableincremental gas rates through the fracture.

Figure 4c shows the impact of including both non-Darcyand multiphase flow effects in the fracture. Notice that therelative area where natural sand is the preferred proppant isreduced due to the additional pressure drops of these floweffects.

These plots are not meant to be used as precise “tables” todetermine the most economic proppant for a given well asthey are rigorous only for the fracture geometry and productprices shown. Instead, they are presented to illustrate thefallacy of existing “rules of thumb” and to illustrate thedramatic impact of non-Darcy and multiphase flow onproppant selection and/or fracture design.

Figure 5 shows 3-d plots in the same format for an oilwell. Input values are shown in the Appendix. The range ofpermeabilities have been increased by one order of magnitudeto more accurately depict the characteristics of economic oilreservoirs. Interestingly, the shape of the gas and oil cases aresimilar when non-Darcy effects are ignored. However, withnon-Darcy flow, the benefit to increased permeability is morepronounced in oil wells at moderate stresses. A plot withmultiphase flow has been omitted, as the SLFrac programdoes not accurately handle the evolution of solution gas fromthe liquid phase below bubble point.

Interpretation of ResultsIgnoring multiphase and non-Darcy effects can lead toincorrect decisions regarding the required fracture width andproppant selection.

It is common that the CfD is below 10. Infiniteconductivity fracture models are frequently invalid. Economicpotential of the fracture design is not optimized.

Proppant selection should not merely be a function ofstress, well depth, or other simplistic rules of thumb.Economic optimization will lead to selection of a preferredfracture geometry and proppant based on stress, formationdeliverability and other factors.

Although highly simplified, these analyses illustrate thatfracture design will frequently be flawed if multiphase andnon-Darcy effects are ignored.

Alternative Strategy for Frac Design/Prop SelectionIn lieu of applying “rules of thumb” to design fracturetreatments, the following alternative is proposed:

• Estimate reservoir and fracture parameters, bracketuncertain values.

• Use a production model which includes calculation ofnon-Darcy and multiphase effects.


• With available production data from offset wells,perform sensitivities to history match effectivefracture length, width and gel damage.

• Determine what conductivity is needed and the mosteconomic way to achieve this.

• Optimize economics (cash flow, Net Present Value,payback period, ROI, etc.) by adjusting fracturegeometry and proppant selection within rangesdictated by operational constraints.

How to handle uncertainties:• Try to bracket unknown values . If the expected

proppant concentration is 2 lb/ft2., it will be instructionalto perform sensitivities at 1 and 3 lb/ft2. concentrations todetermine if that estimate changes your decision. If thedirectional answer remains the same (i.e., if increased fraclength is beneficial in both cases), then the precise valuefor concentration appears not to change the decision.

• Potential of multiple fractures . If multiple fractures areexpected, then fracture conductivity predicted bytraditional planar models will be optimistic.

• Cycling of pressure/stress on proppant. Permeabilitydata gathered according to API specifications are forsingle-cycle crush tests and conductivity tests performedfor ~50 hours per stress level in a conductivity cell. Itmay be appropriate to reduce these values further ifmultiple pressure cycles or long-term degradation areexpected.

• Additional stress due to induced fracture width or netpressure generated during a screen out. Substantialstress can be applied to the formation face during thefracturing treatment. After the fluids leak off, theproppant must be competent to support this additionalstress. The fracture gradient typically used in the industryis measured at a fracture width of zero. If the fracture isnot allowed to close, that additional stress must besupported by the proppant. These additional stresses maybe calculated from rock properties or can be estimatedfrom observing the net pressure during the fracture.26 Fora fully-packed fracture, many engineers consider that thefracture is unable to relax and the full net pressure appliedduring the treatment is added to the effective fractureclosure stress.

• Varying pressure conditions. Proppant crushing shouldbe evaluated at the lowest pbh that may occur during theproductive life of the well. In a gas reservoir, this willtypically occur late in the life of the field as the reservoirpressure has dropped resulting in an increase in theeffective stress on the proppant. If a cleanout isperformed following the frac job, that pbh may be lowerthan typical production pressures and expose the proppantto substantially greater effective stress.

• Non-uniform proppant distribution . Conductivitytesting is performed with uniform proppant packs. Inactual fractures, crush should be expected to be greater

due to imperfect proppant distribution concentrating thestress on fewer proppant grains.

Conclusions1. Non-Darcy and multiphase flow effects are substantial

and should be considered in fracture design.2. The actual pressure drop down the fracture may be 100x

greater than predicted using Darcy’s Law with referenceproppant permeabilities measured under API testconditons.

3. If a production model is incapable of modeling thecomplex phase interactions involved with multiphase andnon-Darcy flow, these effects need to be approximated byreducing the effective proppant permeability.

4. Multiphase flow effects can be substantial. Even a fewpercent (by volume) of a second phase can increase thepressure drop by up to an order of magnitude. Thesechanges are much more dramatic than intuition wouldpredict. It is believed that the pressure loss is due torepeated acceleration of the denser phase.

5. Failure to consider these factors can lead to inadequatefracture conductivity, which may account for shorteffective fracture lengths.

6. Applying “rules of thumb” for frac design appears tofrequently give the wrong answer. Proppant selectionshould be made by optimizing economic returns, not as afunction of depth or stress alone. The most economicproppant is highly dependent upon formationpermeability. Even if the most economic proppant isdetermined for a reservoir with low formationpermeability (e.g. 0.1 md), it will frequently not be themost economic proppant for a similar depth formationwith a higher permeability. Rules of thumb that relateproppant selection to a single variable, such as well depthor proppant stress, are inherently flawed.

7. A simple production model is available which can handlemultiphase and non-Darcy flow in gas wells. The modelis easy to use and can give a “10-minute” analysis thatdisplays the dramatic result of these flow effects. It is notnecessary to rely on inaccurate rules of thumb.

Nomenclature∆p/L = pressure drop per length of proppant pack

µ = fluid viscosityv = superficial fluid velocity (if porosity were 100%)k = permeability of porous mediaβ = coefficient of inertial resistanceρ = fluid density

(µv / k) = viscous term(βρv2) = inertial term

pbh = bottom hole pressureR = gas-oil ratio (producing)KCl = potassium chlorideg = gram


References1. Cooke, C.E., “Conductivity of Fracture Proppants in Multiple

Layers,” paper SPE 4117, JPT, Sep 1973.2. Martins, J.P., et al.: “The Effects of Non-Darcy Flow in Propped

Fractures,” paper SPE 20709 presented at the 1990 AnnualTechnical Converence and Exhibition, Sep 23-26.

3. American Petroleum Institute.: “Recommended Practices forEvaluating Short Term Proppant Pack Conductivity,” API RP61, Oct. 1989.

4. Frick, Thomas C., et al.: Petroleum Production Handbook,Millet the Printer, Dallas (1962) Vol 2, p 23-11.

5. Forchheimer, P.: “Wasserbewegung durch Boden,” ZVDI(1901) 45, 1781.

6. Welty, J.R., et al.: Fundamentals of Momentum, Heat, and MassTransfer, John Wiley & Sons, New York (1984) p 164.

7. Stim-Lab Proppant Consortia Notes, 1990, p 12-v.8. Tipler, P.A.: Physics, Worth Publishers, New York (1982) 165.9. Geertsma, J.:”Estimating the Coefficient of Inertial Resistance

in Fluid Flow through Porous Media,” paper SPE 4706, SPEJ,Oct 1974.

10. Pursell, et al .: “Lab Investigation of Inertial Flow in HighStrength Proppants,” paper SPE 18319 presented at the 1988Annual Technical Conference and Exhibition, Oct. 2-5.

11. Milton-Tayler, David: “Non-Darcy Gas Flow: From LaboratoryData to Field Prediction,” paper SPE 26146 presented at the1993 Gas Technology Symposium, June 28-30.

12. Stim-Lab Proppant Consortium 1994-1998 Reports.13. Penny, G.S., and L Jin: “The Development of Laboratory

Correlations Showing the Impact of Multiphase Flow, Fluid, andProppant Selection Upon Gas Well Productivity,” paper SPE30494 presented at the 1995 Technical Conference andExhibition, Oct. 22-25.

14. Evans, E.V and R.D. Evans: “Influence of an Immobile orMobile Saturation on Non-Darcy Compressible Flow of RealGases in Propped Fractures,” paper SPE 15066, JPT, Oct. 1988.

15. Milton-Tayler, David: “Realistic Fracture Conductivities ofPropped Hydraulic Fractures,” paper SPE 26602 presented at1993 Annual Technical Conference and Exhibition, Oct. 3-6.

16. Schubarth, S.K., et al: “Moxa Arch Frontier DevelopmentSuccess Through Increased Fracture Conductivity – Part 2,”paper SPE 30717 presented at the 1995 Annual TechnicalConference, Oct 22-25.

17. Jin, L, and G.S. Penny: “A Study on Two Phase, Non-Darcy GasFlow through Proppant Packs”, paper SPE 49248 selected forpresentation at 1998 Annual Technical Conference andExhibition, Sep. 27-30.

18. Holditch, S.A. and R.A. Morse: “The Effects of Non-DarcyFlow on the Behavior of Hydraulically Fractured Gas Wells,”paper SPE 5586, JPT, Oct 1976.

19. Gidley, J.L.: “A Method for Correcting Dimensionless FractureConductivity for Non-Darcy Flow Effects,” paper SPE 20710,SPEPE, Nov 1991.

20. Agarwal, R.G. et al,: “Evaluation and Performance Prediction ofLow-Permeability Gas Wells Stimulated by Massive HydraulicFracturing,” paper SPE 6838, JPT, Mar. 1979.

21. Holditch, S.A.: “Criteria of Proppant Agent selection,” preparedfor the Norton Company, 1979.

22. Elbel, J.L.: “Considerations in Fracture Design”, in Economidesand Nolte’s Reservoir Stimulation, Prentice Hall, New Jersey(1989) p 9-1.

23. Pearson, C.M. et al,:”Optimal Fracture Stimulation of aModerate Permeability Reservoir, Kuparuk River Unit,” paperSPE 20707 presented at the 1990 Technical Conference, Sept23-26.

24. Pospisil, G. et al, “Results of a Large-Scale RefractureStimulation Program, Kuparuk River Unit, Alaska,” paper SPE24857.

25. Martins, J.P. et al, “Tip Screenout Fracturing Applied to theRavenspurn South Gas Field Development,” paper SPE 19766.

26. Schubarth, S.K. et al, “Understanding Proppant Closure Stress,”paper SPE 37489 presented at the 1997 Production OperationsSymposium, March 9-11.

Fig. 1 - Comparison of Viscous and Inertial Flow Effects

0

1 0

2 0

3 0

4 0

5 0

0 1 2 3 4 5

W e l l F l o w r a t e ( M M S C F D )

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V i s c o u s F o r c e s ( D a r c y ' s L a w )

Reference Conductivity measured at this rate

Conditions:

50' Frac Height

2#/sq ft ->0.15" after embedment2000 psi BHFP, 200 F

20/40 Jordan Sand2000 psi closure stress

Reference Perm = 243 Darcy55% gel damage


Fig. 2 - Beta Factor Comparison, Common Proppants

0.000

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Data: SLFrac, July ‘98, Gas Well, 6000 psi stress, 200F, 3e6 Modulus, 50% gel damage, 2#/sq ft

Fig. 3 - Impact of Multiphase Non-Darcy Flow20/40 Proppants at 2 lb/sq ft, 6500 psi and 225 F

Adapted from SPE 30494

4 6 7

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Reference Conductivity Conditions: Velocity equivalent to 65 MSCFD dry gas from bi-wing, 50’ fracture height at 3000 psi bottom hole pres.

Non-Darcy Flow Conditions: Velocity equivalent to 1 MMSCFD dry gas from bi-wing 50’ fracture height at 3000 psi bottom hole pressure.Multiphase Flow Conditions: Equivalent of 10 bwpd added to 1 MMSCFD of dry gas at same conditions.


Table 1. Deep Gas Well, Proppant Stress = 9279 psi

Disregarding Includes non-Darcy Including bothAnalytical Predictions: Proppant Multiphase and Effects, non-Darcy and

Non-Darcy Effects Disregards multiphase Flow Multiphase flow*Initial Production Rate 20/40 Sand 2781 423 354

(x1000 scfd) 20/40 RCS 3614 820 47020/40 LWC 3763 1756 1578

Effective Conductivity 20/40 Sand 78 1.5 0.4(md-ft) 20/40 RCS 387 4.2 2.1

20/40 LWC 717 15.4 8.8Dimensionless Fracture 20/40 Sand 1.4 0.03 0.01

Conductivity (Fcd) 20/40 RCS 7.0 0.08 0.0420/40 LWC 12.9 0.28 0.16

Cash Flow after 3 years 20/40 Sand 3773 717 602of production 20/40 RCS 4457 1352 776

($000, net of proppant cost) 20/40 LWC 4662 2718 2507Most Economic Proppant

for assumed frac geometry 20/40 LWC 20/40 LWC 20/40 LWCTime required to payout

incremental investment for 20/40 RCS < 2 months < 2 months < 2 monthspremium proppant 20/40 LWC < 2 months < 2 months < 2 months

% return on incremental investment after three years 20/40 RCS 3600% 3342% 916%

20/40 LWC 3419% 7696% 7327%

Table 2. Shallow Gas Well, Proppant Stress = 1780 psi

Disregarding Includes non-Darcy Including bothAnalytical Predictions: Multiphase and Effects, non-Darcy and

Non-Darcy Effects Disregards multiphase Flow Multiphase flow*Initial Production Rate 20/40 Sand 711 614 461

(x1000 scfd) 20/40 RCS 711 502 42820/40 LWC 716 635 520

Effective Conductivity 20/40 Sand 1769 282 132(md-ft) 20/40 RCS 1886 180 86

20/40 LWC 2633 387 198Dimensionless Fracture 20/40 Sand 32 5.1 2.4

Conductivity (Fcd) 20/40 RCS 34 3.2 1.620/40 LWC 47 7.0 3.6

Cash Flow after 3 years 20/40 Sand 815 722 599of production 20/40 RCS 797 615 550

($000, net of proppant cost) 20/40 LWC 794 711 622Most Economic Proppant

for assumed frac geometry 20/40 Sand 20/40 Sand 20/40 LWCTime required to payout

incremental investment for 20/40 RCS never never neverpremium proppant 20/40 LWC never ~ 4 years 1 year

% return on incremental investment after three years 20/40 RCS -95% -563% -258%

20/40 LWC -81% -42% 88%


Table 3. Moderate Depth Oil Well, 5 lb/sq ft 20/40 Proppants, Proppant Stress = 3400 psi

Disregarding Includes non-DarcyAnalytical Predictions: Multiphase and Effects,

Non-Darcy Effects Disregards multiphase FlowInitial Production Rate 20/40 Sand 724 280

(bopd) 20/40 RCS 734 29320/40 LWC 747 654

Effective Conductivity 20/40 Sand 5903 484(md-ft) 20/40 RCS 7863 515

20/40 LWC 16336 1677Dimensionless Fracture 20/40 Sand 0.66 0.05

Conductivity (Fcd) 20/40 RCS 0.87 0.0620/40 LWC 1.82 0.19

Cash Flow after 3 years 20/40 Sand 5607 2482of production 20/40 RCS 5635 2556

($000, net of proppant cost) 20/40 LWC 5704 5124Most Economic Proppant

for assumed frac geometry 20/40 LWC 20/40 LWCTime required to payout

incremental investment for 20/40 RCS 1.25 years 6 monthspremium proppant 20/40 LWC 5 months < 1 month

% return on incremental investment after three years 20/40 RCS 122% 322%

20/40 LWC 313% 8523%

Table 4. Lower Rate Oil Well, 4 lb/sq ft 20/40 Proppants, Proppant Stress = 3160 psi

Disregarding Includes non-DarcyAnalytical Predictions: Multiphase and Effects,

Non-Darcy Effects Disregards multiphase FlowInitial Production Rate 20/40 Sand 280 97

(bopd) 20/40 RCS 283 10420/40 LWC 293 266

Effective Conductivity 20/40 Sand 3193 261(md-ft) 20/40 RCS 4089 300

20/40 LWC 8392 931Dimensionless Fracture 20/40 Sand 0.53 0.04

Conductivity (Fcd) 20/40 RCS 0.68 0.0520/40 LWC 1.40 0.16

Cash Flow after 3 years 20/40 Sand 2043 728of production 20/40 RCS 2031 753

($000, net of proppant cost) 20/40 LWC 2067 1813Most Economic Proppant

for assumed frac geometry 20/40 LWC 20/40 LWCTime required to payout

incremental investment for 20/40 RCS never 1.6 yearspremium proppant 20/40 LWC 1.25 years < 1 month

% return on incremental investment after three years 20/40 RCS -40% 83%

20/40 LWC 59% 2646%


Fig. 4 – Incremental Cash Flow due to Proppant Selection – Gas Well4a) Disregards Multiphase and Non-Darcy Effects4b) Disregards Multiphase Flow, Includes Non-Darcy Effects4c) Includes Multiphase and Non-Darcy Effects


Fig. 5 – Incremental Cash Flow due to Proppant Selection – Oil Well5a) Disregards Multiphase and Non-Darcy Effects5b) Includes Multiphase and Non-Darcy Effects

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