Optimization of the Productivity Indexand the Fracture Geometry of a

Stimulated Well With Fracture Face andChoke Skins

Diego J. Romero, SPE,* and Peter P. Valkó, SPE, Texas A&M U., and Michael J. Economides, SPE, U. of Houston

Summary

For a given reservoir of known permeability and dimensions, theproppant mass injected to the pay determines a unique proppantnumber. Unique to each proppant number, there exists an optimumdimensionless fracture conductivity that exclusively determinesthe optimum fracture dimensions.1

Impairments affecting flow perpendicular to the fracture sur-face are accounted for as fracture-face-skin effect. On the otherhand, flow impairment caused by a reduction of the fracture con-ductivity near the wellbore is called choked fracture skin. Both effectshave a large influence on the productivity of a fractured well.

In this work, the performance of a fractured well is calculatedwith a direct boundary element method. This method provides thedimensionless productivity index, and the model allows for thepresence of each of the two different skin effects.

The fracture face skin was found to have a significant detri-mental effect on the dimensionless productivity index, even chang-ing the character of its dependence on the dimensionless fractureconductivity. The effect of the choke skin also was found to bepotentially detrimental but less complex to account for becauseit can be represented as an apparent reduction in the proppant number.

Introduction

The post-treatment performance of hydraulically fractured wellshas been a recurring theme in petroleum literature, covering thespectrum of understanding the physics of flow to the optimizationof design. Optimization itself has taken different comprehensiveeconomic hues, from just reducing execution costs to maximizingproduction or injection rates.

Irrespective of the ultimate criterion, the magnitude of reservoirpermeability has been central to fracture morphology. For a givenreservoir of known permeability and dimensions, the proppantmass injected into the pay determines a unique proppant number.Unique to each proppant number there exists an optimum dimen-sionless fracture conductivity1 that exclusively determines the op-timum fracture dimensions.

However, damaged hydraulic fracture performance deviates sub-stantially from that of undamaged fractures. This work is intended tocalculate and optimize the performance of hydraulically fracturedwells that are burdened by two types of flow impediments—fracture-face damage and damage at the connection between the fracture andthe well, referred to as a choke. Fracture-face damage can be actualdamage to the reservoir permeability from fracturing fluid and poly-mer leakoff, or it can be caused by the reduction in relative perme-ability because of a phase change. Choked fracture is mainly causedby proppant flowback or overdisplacement.

This work follows a considerable body of literature, postulatingthat the increase in the fractured well productivity (comparedto the unfractured state) depends on both reservoir and frac-ture characteristics.

In 1960, McGuire and Sikora2 studied the effect of verticalfractures on well productivity and showed how the productivitydepends on the fracture penetration and conductivity.

Prats et al.3 and Cinco-Ley and Samaniego4−6 are credited withthe introduction of dimensionless groups of variables to describethe performance of a fractured well. The concept of dimensionlessfracture conductivity has since been used as the dominant indicatorof relative improvement in fluid flow that is provided by the frac-ture compared to the alternative (i.e., no fracture).

Early in fractured well performance research, certain worksassumed an infinite-conductivity fracture. Prats et al.3 showed thatin the case of an infinite-conductivity fracture and relatively largedrainage area, the effective wellbore radius is equal to one-half thefracture half-length. In an infinite-conductivity fracture, the pres-sure drop is negligible with respect to that in the formation. Thissituation is achieved when the dimensionless fracture conductivityis greater than 300. Gringarten and Ramey7 first introduced themathematical solution for this kind of fracture in an infinite actingreservoir, and it has been used since in well test applications forwells intersecting large natural fractures.

Sawyer et al.8 presented a numerical simulation for the productionof wells intercepted by a finite-conductivity fracture. They showedthat the assumption of infinite fracture conductivity could lead toserious errors when calculating the fractured well performance.

In 1978, Cinco-Ley et al.5 demonstrated that the infinite-fracture-conductivity assumption is quite erroneous when the pres-sure drop along the fracture is considerable, which would be the caseif the dimensionless fracture conductivity were lower than 300.

The focus of much of this work was addressing well-testingtechniques.5 However, as early as 1962, Prats et al.3 showed thatan infinite-conductivity fracture, even if achievable, was not theone at which maximum well production would occur if the volumeof proppant is correctly accounted for as a constraint.

The productivity index of a fractured well, however, is oftenless than the one predicted, even when employing correct finite-conductivity fracture models. This is mostly caused by an extrapressure drop around and/or within the fracture that can be attrib-uted to damage to the formation immediately surrounding the frac-ture face or additional flow impediments in the fracture. Cinco-Leyand Samaniego6 proposed a pressure transient solution that con-sidered the fracture face skin. They assumed that the flow from theformation toward the fracture was linear, passing through two porousmedia in series. One medium is the undamaged formation, and theother is the damaged zone around the fracture. In the same work, theyalso studied the effects of flow impairments inside the fracture nearthe wellbore for what they termed the choke fracture skin.

Another effect that causes an additional pressure drop is thenon-Darcy flow within the fracture. Wattenbarger and Ramey9 andHolditch and Morse10 investigated how the fracture conductivity isaffected by the non-Darcy flow and found that the extra pressuredrop is proportional to the product of a turbulence factor andvelocity square. Methods to correct the dimensionless fracture con-ductivity, used in fracture design and well-test analysis, also have

*Currently with El Paso Production.

Copyright © 2003 Society of Petroleum Engineers

This paper (SPE 81908) was revised for publication from paper SPE 73758, first presentedat the 2002 SPE International Symposium and Exhibition on Formation Damage Control,Lafayette, Louisiana, 20−21 February. Original manuscript received for review 15 May2002. Revised manuscript received 9 October 2002. Paper peer approved 31 October2002.

57February 2003 SPE Production & Facilities

been developed by Guppy et al.11 and Gidley.12 The latter workshowed that fractured wells affected by non-Darcy flow within thefracture exhibit an apparent (reduced) fracture conductivity that isflow-rate dependent.

We note that the concepts of proppant number and optimumdimensionless fracture conductivity remain valid even for the caseof non-Darcy flow if the appropriate reduced proppant pack per-meability is substituted into the definitions of proppant numberand fracture conductivity. Because the reduction factor in equiva-lent permeability is flow-rate dependent, some iteration cyclesmight be needed during the optimization process.

In a much later work, Wang et al.13 demonstrated the produc-tion impairment of fractures in gas-condensate reservoirs causedby the formation of liquid condensate in the vicinity of the fractureface. They considered this effect, caused by relative permeabilityphenomena, as a type of fracture-face damage similar to the onedescribed for real damage by Cinco-Ley and Samaniego.6

With both analytical and numerical simulators, Azari et al.14

demonstrated the choking effect caused by low fracture conduc-tivity near the wellbore. Such a situation can arise if, for example,the proppant is overdisplaced at the end of a treatment by the flushor if the proppant settles significantly during fracture closure.

Until now, however, no rigorous method has been proposed todirectly calculate the pseudosteady-state performance of such a non-ideal fractured well without solving the model for all previous times(that is, in transient regime). The purpose of this work is to investigatethe effect of the flow impairments on the productivity index.

In the following sections, a solution methodology is suggestedto determine the inflow into a fully penetrating vertical fracturethat is intersected by a vertical well located in the center of asquare drainage area and is subject to the individual or combinedeffects of fracture face skin and choked fracture skin. With thesolution methodology, pseudosteady-state productivity indices arecalculated. In the presentation of results, we rely heavily on thepreviously introduced proppant-number concept. The approach’susefulness is illustrated in conjunction with fracture design opti-mization and fractured-well performance analysis.

A Direct Boundary Element Method To Calculatethe Fractured Well Productivity IndexInfluence Function. Ozkan15 suggested that the pseudosteady-state drawdown at any point in a reservoir (x,y) caused by a welllocated at (xw,yw) can be given in terms of an influence function, a.

p − p =�1�Bq

2�kha�xD, yD, xwD, ywD, xeD, yeD�. . . . . . . . . . . . . . . . (1)

Because the dimensionless productivity index, JD, is defined by

J =q

p − pwf

=2�kh

�1B�JD, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

the influence function can be used to calculate the dimensionlessproductivity index of a single vertical well as follows.

JD =1

a�xwD + rwD, ywD, xwD, ywD, xeD, yeD� + s, . . . . . . . . . . . . . . (3)

where rw�the wellbore radius and s�the skin factor.A novel application of Eq. 1 has been the generalization to

multiwell environment by Valko et al.16 and simultaneously byUmnuayponwiwat and Ozkan.17

p − p =�1�B

2�kh �i=1

nw

qia�xD, yD, xwD,i, ywD,i, xeD, yeD�, . . . . . . . . . . (4)

where nw�the number of wells (line sources). For a square drain-age area considered in this work, yeD�xeD and the influence func-tion depend only on the location of the source (denoted by wi insubsequent equations) and the observation point (denoted by oj).

Vertical Fracture Performance. If a fully penetrating verticalfracture intersects the wellbore, the well performance depends onthe following penetration ratio.

Ix =2xf

xe, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)

where xf�the fracture half-length and xe � the reservoir drainageextent (side length of a square), and the dimensionless fractureconductivity is defined by

CfD =kfw

kxf, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)

where kf � the proppant pack permeability, w�the propped frac-ture width, and k � the reservoir permeability.

In Ref. 1, the two expressions (Eqs. 5 and 6) are combinedthrough the proppant number

Nprop =4kf wxf

kxe2

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)

which, after multiplying and dividing by the reservoir thickness, h,leads to

Nprop =2kfV2w,prop

kVr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8)

In Eq. 8, V2w,prop � the volume of the two-wing propped fractureinside the pay, and Vr�the drained volume (pore plus matrix). Eq.8 is quite important because it shows that the proppant number isa constant quantity for a given mass of proppant injected into agiven drainage volume.1

Direct Boundary Element Method. Romero18 solved the prob-lem of calculating the pseudosteady-state dimensionless produc-tivity index of a vertically fractured well with flow impairment.The fracture is modeled as nw line sources located at wi (i: 1…nw)with corresponding (nonuniform) production rates (q1,…qnw). Eq.4 is applied at nw observation points in the fracture (at oi, locatedbetween wi−1 and wi). The pressure (drawdown) difference be-tween two observation points (o1 and o2) can be obtained from thecorresponding applications of Eq. 4, with

�p1−2 =�1�B

2�kh�q1�a�o1, w1� − a�o2, w1�� + …

+ qnw �a�o1, wnw� − a�o2, wnw��� . . . . . . . . . . . . . . . . . . . . (9)

Darcy’s law for the flow in the fracture results in

�pf,1−2 =2�1�B

kf hw�q2�xo2 − xo1� + . . . + qnw �xo2 − xo1�� . . . . . . (10)

Because Eqs. 9 and 10 describe the pressure drop between thesame two points, their right sides are equal. If the following di-mensionless variables are defined (along with the definition of thedimensionless fracture conductivity in Eq. 6),

qD =q B �

2� k h�p − pw�, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11)

xD = x � xe*, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12)

and Ix = xf � xe*, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13)

then we obtain the following equation in dimensionless form:

58 February 2003 SPE Production & Facilities

qD1 �a�o1, w1� − a�o2, w1�� +qD2 �a�o1, w2� − a�o2, w2�

−4�

CfDIx�xDo2 − xDo1�� + … + qDnw �a�o1, wnw�

− a�o2, wnw� −4�

CfDIx�xDo2 − xDo1�� = 0. . . . . . . (14)

Eq. 14 describes the pressure drop between observation points 1and 2. We can write nw−1 similar equations between o1 and re-examine observation points. (Note that in Eqs. 12 and 13, xe*=xe/2because of the symmetry of the problem.) The notation for Eq. 14is shown in Fig. 1.

As suggested in Ref. 18, the nw−th equation should be the directapplication of Eq. 4 at the wellbore with �pD�1. Once the systemis solved, the dimensionless productivity index is calculated fromthe following.

JD = 4�i=1

nw

qDi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (15)

(Note that a factor of four is needed in Eq. 15 because the obtainedindividual production rates add up to one-quarter of the total pro-duction from the well.) In the calculations, a maximum nw�512line sources were used, and the influence function was calculatedaccording to the method detailed in Ref. 16.

Results for Fractured Well Without Skin. In this section, resultsfrom the model are shown first for an undamaged fracture. Be−cause the proppant number, Nprop, directly reflects the amount ofproppant injected to the pay, it is used as the parameter on all figures.

Two figures are presented here for an undamaged fracture. Fig.2 is for an Nprop of less than 0.1, while Fig. 3 is for an Nprop ofgreater than 0.1. In both figures, the dimensionless productivityindex, JD, is plotted vs. the dimensionless fracture conductivity,CfD, at constant values of Nprop. The limiting penetration ratio (seeEq. 13) equal to 1 is also plotted on the figures with a dashed line.Note that the maximum possible value for JD is 6/�, which cor-responds to fully linear flow.

As seen in Figs. 2 and 3, there is an optimal dimensionlessfracture conductivity for a given proppant number, Nprop, that rep-resents the optimum relation between the two functions of the

fracture—its ability to collect fluid from the reservoir and to con-duct the fluid into the well. For low and moderate proppant num-bers (Nprop�<0.1), this relation occurs at a dimensionless fractureconductivity that is equal to 1.6. We note that in formations ofmedium and high permeability, realistic proppant numbers arealways in the low to moderate range (i.e., less than 0.1).

Note from Fig. 3 that when the propped volume increases orwhen the permeability contrast is very large, the optimal dimen-sionless productivity index occurs at larger dimensionless fractureconductivity values. Also note that for values of Nprop equal to 10or more, the maximum dimensionless productivity index isachieved when the reservoir is penetrated from “wall to wall” (i.e.,when the penetration ratio, Ix, equals 1). It is questionable, how-ever, that such large proppant numbers can be realized in practice.In any case, the optimum fracture geometry is given by:

xf = �kfV2w,prop�2

CfD,optkh �1�2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (16)

and w = �CfD,optkV2w,prop�2

kfh�1�2

, . . . . . . . . . . . . . . . . . . . . . . . . . . . (17)

where CfD,opt can be directly read from Figs. 2 or 3.19

The optimization of hydraulic fracture geometry presented pre-viously does not include the effect of a damage zone around thefracture face and/or within the fracture. Because both effects havea large influence on the productivity of a fractured well, theyshould be considered during the optimization process.

Fracture-Face Skin EffectFracture-face damage implies permeability reduction normal to thefracture face and includes flow impairments caused by severalfactors. A filter cake may be formed on the inside fracture face thatis difficult to eliminate, even with proper breaking practices. Thereis always a zone around the fracture that is invaded by someportion of the polymer contained in the fracturing fluid. The filtratecomponent of the fracturing fluid penetrating the formation causessome permeability impairment in a larger zone.

Cinco-Ley and Samaniego6 described the fracture-face-skin ef-fect, sff, in terms of damage penetration and damaged permeability.

sff =�ws

2xf� k

ks− 1�, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18)

where the variables are shown in Fig. 4.

Fig. 1—Variables of the direct boundary element method for the productivity index calculation.

59February 2003 SPE Production & Facilities

The previous skin factor can be used to calculate an approxi-mate dimensionless productivity index according to

JD,A =1

1

JD|s=0+ sff

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (19)

where JD|s=0 � the dimensionless productivity index of the frac-tured well with zero fracture face skin.

Eq. 18 is valid only for the case of uniform influx and damagealong the fracture face. For rigorous calculations, we need to in-corporate a skin factor distribution.

sff�x� =�

2xf�� k

ks− 1�ws�

@x

, . . . . . . . . . . . . . . . . . . . . . . . . . . . (20)

and the pressure drop caused by the varying fracture face skin willbecome

�psff =��B xf

� k hq̃�x�sff�x�, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (21)

where q̃ (x)�the influx normal to the fracture face per unit area.To take the distributed skin into account, Eq. 14 should be adjustedto include the additional pressure drop given by Eq. 21.

Fig. 3—Fractured well performance for high proppant numbers.

Fig. 2—Fractured well performance for low and medium proppant numbers.

60 February 2003 SPE Production & Facilities

qD1�a�o1, w1� − a�o2, w1� + �4nw�sff,w1�

+ qD2�a�o1, w2� − a�o2, w2� −4�

CfDIx�xDo2 − xDo1��

+ … + qDnw �a�o1, wnw� − a �o2,wnw�

−4�

CfDIx�xDo2 − xDo1�� = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . (22)

In other words, the diagonal elements of the coefficient matrixshould be increased by the appropriate (local) skin factor.

There are three interesting cases for varying fracture face skinfactor—when the skin decreases linearly toward the tip, when it in-creases linearly, and when it is constant. The first case may reflectdamage caused by fracturing fluid leakoff, whereas the second mayreflect uneven fluid cleanup following the fracture treatment.

The mean value of the skin is used as a first approximation toevaluate the effect of damage on well performance.

sff =1

xf

0

xf

sff �x�dx. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (23)

Comparing the three cases is of particular interest when the meanvalue of the skin factor calculated from Eq. 23 is the same. In ourcalculations, Case A corresponds to a linearly decreasing fractureface skin from the well toward the fracture tip. Similarly, Case Bcorresponds to a linearly increasing fracture face skin. Finally, weassume a constant fracture face skin along the fracture in Case C.

In all three cases, however, the mean value of the skin is kept equalto unity.

The solid lines in Fig. 5 denote the zero-skin calculations. Thefirst corresponds to the base case (i.e., Nprop�0.1 with no fracture-face skin effect). For comparison purposes, the no-skin curve forNprop�0.01 is also included.

It can be seen from Fig. 5 that the fracture face skin has a largeinfluence on the dimensionless productivity index. In addition, thedamage distribution along the fracture greatly affects the perfor-mance. In Case A, which is the most likely to happen, a significantreduction in the productivity index is observed. It is also noted thatfor Case A, the location of the optimum CfD with respect to thezero-skin location (i.e., 1.6) is to the left but is to the right for CaseB. The optimum width and length can still be calculated from Eqs.16 and 17, but the resulting dimensions will be different than thoseobtained from CfD�1.6.

The largest reduction in performance happens in Case C (whenthe damage is uniformly distributed along the fracture). For in-stance, if the skin factor is one unit, its overall effect is equivalentto an order of magnitude reduction in the proppant number. Thatis, a uniformly distributed fracture face skin equal to 1 is roughlyequivalent to placing only 10% of the original proppant volumeand avoiding any damage.

Choked-Fracture-Skin EffectChoked-fracture skin effect refers to the presence of a damagedzone of the fracture that is near the well and has a conductivityreduction. The conductivity reduction can be caused by an over-displacement of proppant at the end of a fracture treatment job, by

Fig. 4—Fracture face damage variables.

Fig. 5—The effect of fracture-face skin distribution on well productivity. Case A shows decreasing damage toward the tip; Case B,increasing damage toward the tip; and Case C, constant damage along the fracture. For all three cases, Nprop = 0.1 and the averageskin=1. The line labeled Eq. 19 denotes the approximate calculation with only the average value of skin, in this case sff=1.

61February 2003 SPE Production & Facilities

settling of the proppant during fracture closure or by fines migra-tion and accumulation at the wellbore during production. A chokedfracture with a significant flow impediment at the vicinity of thewellbore is shown in Fig. 6, in which w�the unaltered fracturewidth, kf�the unaltered fracture permeability, and wck�the al-tered fracture width in the near-well region of the fracture.

Equivalent flow impediment can be caused by a reduced per-meability (kf,ck) zone in the fracture, even if the width is unaltered.

The extra pressure drop in the fracture is

�psck =�1�Bq

2�khsck, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (24)

where sck is given by

sck =�xck

xf� w

wck− 1� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (25)

or sck =�xck

xf� kf

kf,ck− 1�, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (26)

depending on whether the damage is expressed as a reduced width(wck) or a reduced permeability (kf,ck).

Because the damage is located inside the fracture, it will onlyaffect the pressure drop caused by flow through the fracture.Therefore, Eq. 14 should be replaced by

qD1 �a�o1, w1� − a�o2, w1�� + qD2 �a�o1, w2�

− a�o2, w2� −4�

CfDIx�xDo2 − xDo1� − 4sck�

+ … + qDnw �a�o1, wnw� − a�o2, wnw�

−4�

CfDIx�xDo2 − xDo1� − 4sck� = 0 . . . . . . . . . . . . (27)

To analyze the effect of the choked fracture skin on the fractured-well performance, two different values for choke skin (sck � 0.5and 1) were studied. As in the previous parametric studies, theproppant number was equal to 0.1. Fig. 7 illustrates the results.The solid line represents Nprop�0.1 without skin (base case). Forcomparison purposes, the Nprop�0.01 line is also shown (as an-other solid line) without skin.

It can be observed from Fig. 7 that the choke skin reduces theproductivity index of the well in a rather straightforward manner.For a proppant number of 0.1 and choke skin of 1, the dimension-less productivity index is equivalent to a fracture without damagebut with a reduced proppant number of approximately 0.01. How-ever, the location of the optimum dimensionless fracture conduc-tivity (1.6 for the given proppant number) is not altered by thepresence of the choke-fracture skin. We notice that the approxi-mate formula (Eq. 19) works satisfactorily for choke skin.

If we compare the effect of fracture face skin and that of chokeskin, we see that the latter is less complex. The plausible expla-nation is that the choke skin causes an additional pressure dropright at the vicinity of the wellbore without changing the shape ofthe influx distribution along the lateral direction, x. On the other

hand, the fracture face skin causes a relative redistribution of theinflux of produced fluids along the lateral direction, and nonuni-form damage amplifies this effect.

Application ExamplePlace 240,000 lbm of proppant (pack porosity�0.35, specificgravity�2.65, and equivalent permeability�60,000 md) into a65-ft-thick formation of 1.5-md effective permeability. Assumethat 50% of the proppant goes to pay because of some heightgrowth of the fracture to the adjacent shales. The drainage radius,re, is 2,100 ft; the well radius, rw, is 0.328 ft; and the skin factorbefore fracturing, spre, is 5.

Problem. Determine the maximum possible “folds of increase”and the optimum propped length and width.

Solution: The volume of proppant reaching the pay is 50% ofthe 240,000-lbm proppant volume: V2w,prop�1,116 ft3. The prop-pant number is

Nprop =2 � �60 � 103 md � 1,116 ft�

�1.5 md � �2,1002 ft2� � � � 65 ft�. . . . . . . . . . . . (28)

The maximum achievable dimensionless productivity index (seeFig. 2) is

JD,max p = 0.466. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (29)

According to the definition of folds of increase in the productivityindex (with respect to the originally damaged well), it can beobtained as

Jpost

Jpre=

JD,max

1

ln0.474re

rw+ spre

=0.466

1

ln0.472 � 2,100

0.328+ 5

= 6.1. . . . . . (30)

As seen in Fig. 2, the optimum is realized with CfD�1.6, and,hence, the optimum fracture dimensions are

xf = �0.5�1,116 ft3� �60,000 md�

1.6 � �65 ft��1.5 md��1�2

= 463 ft, . . . . . . . . . . . (31)

and w =0.5�1,116 ft3�

�65 ft��463 ft�= 0.0185 ft = 0.222 in. . . . . . . . . . . . . . . (32)

Problem. Determine the actual folds of increase if 10,000 lbm ofproppant has inadvertently been flowed back. Assume the prop-pant comes from the part of the fracture that is in the pay near thewellbore where a two-grain width (0.06 in.) is stabilized during thefracture-healing process.

Solution: As indicated previously, according to our assump-tions, optimum fracture geometry is created, but the proppantflowback then produces a choke with the following width-reduction ratio.

w

wck=

0.222

0.06= 3.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (33)

The length ratio corresponding to our assumptions is

Fig. 6—Notation for choked fracture.

62 February 2003 SPE Production & Facilities

xck

xf=

10,000

120,000

0.222

�0.222 − 0.06�= 0.114. . . . . . . . . . . . . . . . . . . . (34)

From Eq. 25, the choke skin is calculated as

sck =�xck

xf� w

wck− 1� = � � 0.114 � �3.70 − 1� = 0.97 . . . (35)

From Fig. 7, we see that for a proppant number of approximately0.1, the choke skin, sck,�1, reducing the JD,max from 0.466 to0.32. Another way to obtain the same result is to use a form similarto Eq. 19.

JD,ck =1

1

JD|s=0+ sck

=1

1

0.466+ 0.97

= 0.32. . . . . . . . . . . . . . . . . (36)

Hence, we can predict that the actual folds of increase decreasefrom 6.1 to 4.8 because of proppant flowback.

ConclusionsThe performance of a fractured well is primarily determined by theproppant number (i.e., by the volume contrast of proppant placedinto the pay and by the permeability contrast of proppant pack toformation). For every proppant number, there is a unique maxi-mum productivity index that is realized only at the optimum di-mensionless fracture conductivity. In turn, the optimum dimen-sionless fracture conductivity determines the unique width andlength to provide optimum performance.

In previous works,1 we obtained the productivity index with thedirect boundary element method. In this work, the direct boundaryelement method was extended to calculate the effect of fracture faceskin with various damage distributions and choked fracture skin.

It was found that nonuniform fracture face skin significantlydecreases the productivity of the fractured well and also shifts thelocation of the optimum dimensionless fracture conductivity.Therefore, not only the maximum achievable productivity indexbut also the optimum fracture geometry will differ from the zero-skin case. A uniform damage distribution has the most detrimentaleffect on productivity but leaves the location of the optimum di-mensionless fracture conductivity intact.

The effect of choked fracture skin is less complex to accountfor: it is essentially equivalent to an apparent reduction of the prop-pant number and does not affect the optimum fracture geometry.

As illustrated by the example calculations, using the dimension-less productivity index and proppant number facilitates understandingfractured-well performance and makes the analysis transparent.

Nomenclaturea � influence functionB � formation volume factor, resbbl/STB

CfD � dimensionless fracture conductivityh � pay thickness, ftIx � dimensionless penetration ratioJ � productivity index, STB/D/psi

JD � dimensionless productivity indexk � formation permeability, mdkf � proppant pack permeability, md

nw � number of line sources (“wells”)Nprop � proppant number

o � observationp � pressure, psip � average pressure of drainage, psi

pw � wellbore flowing pressure, psiq � flow rate, STB/D

rw � wellbore radius, fts � skin factor

V2w,prop � propped volume in pay (2 wings)Vr � drained volume, ft3

w � fracture width, ftwck � choked width in one wing, ft.ws � fracture face skin zone extent, ft

x � coordinate, ftxck � choke length in one wing, ftxe � side length of drainage area, ftx*

e � half-side length of drainage area, xe/2, ftxf � fracture half length, ft

xw � coordinate of well, fty � coordinate, ft

ye � side length of drainage area, ftyw � coordinate of well, ft�1 � conversion factor (for field units 887.22)�p � drawdown, psi� � viscosity, cp

Fig. 7—The effect of choke skin on fractured well performance.

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SubscriptsA � approximate

ck � choked fractureD � dimensionless variablee � drainagef � fracture

ff � fracture facei � index of source

max � maximumo � observation

opt � optimumpost � post-treatmentpre � pretreatment

s � skinx � x directiony � y directionw � well

References1. Economides, M.J., Oligney, R., and Valko, P.P.: Unified Fracture De-

sign, ORSA Press, Alvin, Texas (2002).2. McGuire, W.J. and Sikora V.J.: “The Effect of Vertical Fractures on

Well Productivity,” Trans., AIME (1960) 401.3. Prats, M., Hazebroek, P., and Strickler, W.R.: “Effect of Vertical Frac-

tures on Reservoir Behavior—Compressible-Fluid Case,” SPEJ (June1962) 87.

4. Cinco-Ley, H., and Samaniego, V.F.: “Effect of Wellbore Storage andDamage on the Transient Pressure Behavior of Vertically FracturedWells,” paper SPE 6752 presented at the 1977 SPE Annual Fall Meet-ing, Denver, Colorado, 9−12 October.

5. Cinco-Ley, H. and Samaniego, V.F.: “Transient Pressure Analysis forFractured Wells,” JPT (September 1981) 1749.

6. Cinco-Ley, H., and Samaniego, V.F., and Dominguez, N.: “TransientPressure Behavior for a Well With a Finite-Conductivity Vertical Frac-ture,” SPEJ (August 1978) 253.

7. Gringarten, A.C. and Ramey, H.J. Jr.: “An Approximate Infinite Con-ductivity Solution for a Partially Penetrating Line Source Well,” SPEJ(April 1975) 325.

8. Sawyer, W.K., Locke, C.D., and Overbey, W.K. Jr.: “Simulation of aFinite-Capacity Vertical Fracture in a Gas Reservoir,” paper SPE 4593presented at the 1971 SPE Annual Fall Meeting, Las Vegas, Nevada, 30September−3 October.

9. Wattenbarger, R.A. and Ramey, H.J. Jr.: “Well Test Interpretation ofVertically Fractured Gas Wells,” JPT (March 1969) 246.

10. Holditch, S.A. and Morse, R.A.: “The Effects of Non-Darcy Flow onthe Behavior of Hydraulically Fractured Wells,” JPT (October 1976)1169.

11. Guppy, K.H. et al.: “Non-Darcy Flow in Wells with Finite-Conductivity Vertical Fractures,” SPEJ (October 1982) 681.

12. Gidley, J.L.: “A Method for Correcting Dimensionless Fracture Con-ductivity for Non-Darcy Flow Effects,” SPEPE (November 1991) 391.

13. Wang, X. et al.: “Production Impairment and Purpose-Built Design ofHydraulic Fractures in Gas-Condensate Reservoirs,” paper SPE 64749presented at the 2000 SPE International Oil and Gas Conference andExhibition, Beijing, 7−10 November.

14. Azari, M. et al.: “Performance Prediction for Finite-Conductivity Ver-tical Fractures,” paper SPE 22659 presented at the 1991 SPE AnnualTechnical Conference and Exhibition, Dallas, 6−9 October.

15. Ozkan, E.: “Performance of Horizontal Wells,” PhD dissertation, U. ofTulsa, Tulsa (1988).

16. Valko, P.P., Doublet L.E., and Blasingame, T.A.: “Development andApplication of the Multiwell Productivity Index (MPI),” SPEJ (March2000) 21.

17. Umnuayponwiwat, S. and Ozkan, E.: “Evaluation of Inflow Perfor-mance of Multiple Horizontal Wells in Closed Systems,” J. of EnergyResources Technology (2000) 122, 8.

18. Romero, D.J.: “Direct Boundary Method to Calculate Pseudosteady-State Productivity Index of a Fractured Well with Fracture Face Skinand Choked Skin,” Masters thesis, Texas A&M U., College Station,Texas (2001).

19. Spreadsheet, FracPI, http://pumpjack.tamu.edu/∼valko.

SI Metric Conversion Factorsbbl × 1.58987 E−01 � m3

cp × 1.0* E–03 � Pa�sft × 3.048* E–01 � m

in. × 2.54* E+00 � cmmd × 9.869223 E–04 � �m2

psi × 6.894757 E+00 � kPa

*Conversion factor is exact.

Diego J. Romero is currently employed by El Paso ProductionCo. in Houston. Romero holds a BS degree in petroleum engi-neering from Foundation U. of America, Santafé de Bogotá,Colombia, and an MS degree in petroleum engineering fromTexas A&M U. Peter P. Valkó is an associate professor at theHarold Vance Dept. of Petroleum Engineering at Texas A&M U.He previously taught in Austria and Hungary and also workedwith the Hungarian oil company MOL. Valko holds BS and PhDdegrees in chemical engineering and an MS degree in tech-nical mathematics from Veszprem U. (Hungary) and from theInst. of Catalysis, Novosibirsk (Russia). Valkó is currently servingon the SPE Journal Editorial Review Board and has served onthe SPE Forum Steering Committee. Michael J. Economides is aprofessor of chemical engineering at the U. of Houston. Previ-ously, he was the Noble Professor of Petroleum Engineering atTexas A&M U. and served as a Chief Scientist of the GlobalPetroleum Research Inst. Before joining Texas A&M U., he wasthe Director of the Inst. of Drilling and Production at the LeobenMining U., Austria. From 1984 to 1989, he worked with Schlum-berger companies. Economides holds BS, MS, and PhD de-grees from Kansas State and Stanford U. He has beenawarded the following honors: Doctor Honoris Causa, Petro-leum and Gas U., Ploeisti, Romania; Russian Academy of Natu-ral Sciences, inducted as Foreign Member; Society of Petro-leum Engineers, 1997 Production Engineering Award; DoctorHonoris Causa and Honorary Professor, The Gubkin RussianState Academy of Oil and Gas, Moscow; Distinguished Mem-ber, Society of Petroleum Engineers; and the Outstanding Fac-ulty Award (U. of Alaska, School of Mineral Industry).

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