00057713

Fracture Width Logging While Drillingand Drilling Mud/Loss-Circulation-Material

Selection Guidelines in NaturallyFractured Reservoirs

Olivier Lietard,* SPE, Dowell; Tessa Unwin, Schlumberger Cambridge Research; and D.J. Guillot, SPE,and M.H. Hodder, SPE, Dowell

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SummaryIn this article we discuss drilling mud losses in naturally fracturreservoirs with fracture permeability larger than 50 odd md amuch larger than the matrix permeability.

Type curves are provided that describe mud loss volumetime and allow the determination of the hydraulic width of natufractures through the usual type curve matching technique oflog plots.

Real time logging while drilling of natural fracture width pemits the proper selection of loss-circulation material~LCM! withrespect to particle size distribution. Efficient~rapid! mud loss con-trol is made possible, increasing the chances of minimizing frture damage by the drilling mud, the cuttings and the LCMs theselves.

The total mud losses experienced when drilling a long horiztal section are shown to agree with reservoir properties~derivedby well testing! in particularly well-documented examples of thcentral North Sea.

The equations provided can therefore be utilized as wellreservoir engineers to properly evaluate the contribution of natfractures to the permeability and skin of any well, whether thfractures are clear or are invaded by drilling mud.

Introduction

The idea of monitoring massive mud losses in a naturally frtured reservoir to characterize formation rock characteristicsproposed about 30 years ago1 and is therefore nothing like a newone. Despite a sizeable list of studies since that date, most omethods derived so far are only qualitative, as evidenced bymost recent publications.2 An attempt of quantitative analysis halately been proposed for a Newtonian drilling mud based onassumption of a square root of time behavior of the mud loss ra3

In our opinion, the parallel made in that study with fluid leakoffhydraulic fracturing is not valid. Moreover, the assumption oNewtonian mud is extremely restrictive. Finally, such an appropromotes lots of confusion between normal invasion of natufractures~at an equivalent circulating density lower than the fomation rock fracturing, or parting, pressure!, which is the solescope of the present article, and forced invasion with concomipropagation of an artificial hydraulic fracture through a poromedium with no secondary porosity. It should be made extremclear at this point that the following study is focused on minvasion of natural fractures well below fracturing pressure.

*Now a consultant.

Copyright © 1999 Society of Petroleum Engineers

This paper (SPE 57713) was revised for publication from paper SPE 36832, first presentedat the 1996 SPE European Petroleum Conference held in Milan, Italy, 22–24 October.Original manuscript received for review 22 October 1996. Revised manuscript received 23April 1999. Paper peer approved 1 June 1999.

168 SPE Drill. & Completion14 ~3!, September 1999

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The increasing occurrence of oil field developments in fratured reservoirs in the North Sea4 has lately led to even morededicated efforts at understanding the impact of large mud loon formation damage and damage removal techniques.5

However, the need to come up with a detailed quantitatanalysis of the phenomenon has become urgent with the perstive of the startup of the Clair field west of the Shetlands. Thuge oil accumulation is the first and so far a unique examplenaturally heavily fractured sandstone reservoir in this part ofworld. Its successful development cannot be guaranteed unfracture damage by insoluble cuttings is minimized, if not pvented. Indeed, the large viscosity of the reservoir oil at downhconditions would not allow for economical production if skwere present. Methods to ensure immediate control of mud lomust be designed and implemented. They cannot be baseanything other than a precise, real time knowledge of the widththe fracture the drilling bit is running through.

In addition, the determination of the actual hydraulic widthnatural fractures when drilling wildcat and appraisal wells woumake extended well tests~EWTs! much less necessary. Formatiomicroimaging~FMI! ~and other related acronyms! indeed has onlyaccess to visual width and is merely reliable when extremely nwellbore values of fracture width and spacing are required. Ithe industry consensus that the data derived from such logtools must not be extrapolated at the scale of the reservoir. Hethe requirement for a technique of logging while drilling~LWD!of the hydraulic width of natural fractures to have a meaninginvestigation radius.

Reservoir Engineering of Horizontal Wells in NaturallyFractured ReservoirsMuskat6 and Jones7 have shown that the permeability (kf) and theporosity (f f) of a system of parallel fractures spread all overreservoir are given by

kf5w3/~24d!, ~1!

f f5w/d, ~2!

where~w! is the average width and~d ! the average spacing~i.e.,the average distance between two fractures!. For a 0.12 mm widthand a 1.2 m spacing, the permeability is 0.6310213 sq m ~about60 md! and the secondary porosity value is as small as 0.01%.permeability perpendicular to the fracture plane, however, is oequal to the matrix one (km).

When a well fully intercepts the fissure pattern~e.g., a verticalwell drilled all over the height of the payzone along the plane overtical fracture, or, more simply, a horizontal well!, it has alsobeen shown8 that a negative skin exists. This pseudoskin (Sf)stems from a larger apparent permeability at the vicinity ofwellbore, where full connectivity to the fracture~s! is present:

Sf5~p/2!~122r w /d!1 ln~2r w /d!, ~3!

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where (r w) is the wellbore radius.Gigeret al.9 give the productivity index (q/Dp) of a horizontal

well in a primary porosity oil reservoir with no permeability anisotropy ~applicable in the naturally fractured reservoirs of cosideration here, as long as the well does not run parallel tofracture plane!:

q/Dp5@2pkh/~mB!#@ ln~4r d /L !1~h/L !ln~h/2pr w!#21, ~4!

where~k! is the reservoir permeability,~h! the payzone height,~m!and ~B! the oil viscosity and formation volume factor, respetively, (r d) the drainage radius and~L! the well length, whenL,r d . Indeed, horizontal wells are of particular interest in narally fractured reservoirs since the fracture plane is vertical~drill-ing horizontally maximizes the chances of intercepting fractur!and there is no vertical permeability anisotropy. Lie´tard et al.10

have verified the definition of the skin (Ss) due to the presence odamage in horizontal wells and found it to be quite similar toexpression given by Renard and Dupuy:11

Ss5~h/L !~k/ks21!ln@$r s1~p21!r w%/~pr w!#, ~5!

where (ks) is the permeability of the damaged zone with rad(r s) around the wellbore.

Impact of Mud Losses on Well ProductivityBecausekf@km , most of the mud invasion damage in naturafractured reservoirs is in the fractures, and the invasion radiuvery large (r s@r w) due to the very small value of fracture poroity. Considering that the damage permeability (ks) is about equalto the matrix permeability (km) around the wellbore12 ~completeplugging of the fractures by mud damage!, and replacing~k! by(kf1km) in Eq. ~5!, it becomes

Ss'~h/L !~kf /km!ln@r s /~pr w!#, ~6!

and the productivity (Js) of the damaged horizontal well, compared to the undamaged one (J), is equal to

Js /J5pD /~pD1Ss!, ~7!

with (pD), the dimensionless pressure, equal to

pD5 ln~4r d /L !1~h/L !ln@h/~2pr w!#1Sf , ~8!

where Gigeret al.’s expression for primary porosity reservoirscomplemented by (Sf) for secondary porosity contributions. Ithe absence of loss-circulation-model~LCM! cake buildup insidethe fracture, the damage radius (r s) is5

r s5wDpD /~3ty!, ~9!

where (DpD) is the maximum drilling overpressure and (ty) isthe yield value of the drilling mud. Since the horizontal well itercepts (L/d) fractures, the total mud loss (V) experienced allalong the drilling period of the payzone is

V'pw~r s!2~L/d!5~8pL/3!~DpD /ty!2kf . ~10!

End of Job Analysis of Mud Losses and of ProductivityEvolution for the Well CaseThe completion of Machar well 18z in UKCS block 23/26adescribed in detail elsewhere.4,5 A total of 240 bbl of mud losseswere monitored during 11 events over 1,990 ft of drilling acrothe pay zone. The average fissure spacing~d ! is therefore about173 ft, and the fracture skin (Sf) is equal to23.92 according toEq. ~3! with r w50.36 ft with a ~81

2 in. drill bit!. The latter valueagrees with the post-stimulation well test interpretation4 whichalso evaluates a productivity index of 147 B/D/psi for~kh! about20,000 md ft. Sinceh5340 ft, m50.4 cp, andB51.45 rb/STB,we will conveniently usek560 md and derive (pD) as defined inEq. ~8! from Eq. ~4!: pD51.69. WithSf523.92 and the middle

Lietard et al.: Fracture Width Logging

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term of Eq.~8! equal to 1.02, ln(4rd /L)54.59 andr d546,800 ft~14.3 km!. The actual diapir radius is in the range of 6 km. Tdiscrepancy could stem from the heterogeneity of the reservActually, the fracturing is much more intense at the top of the sdome ~where well 20z, of shorter length, produces at 470 B/psi!, and this can explain why the drainage radius looks so lawhen assuming a constant 60 md permeability.

The carbonate rock itself has limited permeability, sokm

!kf , and Eq. ~1! gives w5420mm and Eq. ~10! (DpD /ty)5364,420~240 bbl total loss!. By transferring both values in Eq~9! we find that (r s), the damage radius, is 51 m~about 167 feet!.The damage skin from Eq.~6! becomesSs'53.7/km with (km) inmd. A well test conducted before stimulation~with mud losses inplace! gave 2.9 B/D/psi.4 Hence from Eq.~7! we determine thatthe actual skin value was184 ~recall thatpD51.69!. The testinterpretation gave177, the discrepancy coming from slightldifferent assumptions of~kh! and the productivity index. By com-paring the skin value itself with the outcome just above from E~6!, the matrix permeability is estimated to be about 0.64 md. Tis in line with what was reported4 ~,1 md! and justifies the as-sumptionkm!kf made above.

All the equations provided match very well with ‘‘postmortem’’ data derived from pre- and post-stimulation well teand end of job drilling records. In addition, the average fisswidth and loss volume per fissure~420 mm, about 1/60 in. and21.8 bbl, respectively! do agree with Dyke’s description of a situation where ‘‘mud will block fracture at a distance from wellbowith detectable loss’’~Table 3 of Ref. 2!.

Nevertheless, these matches should not hide the fact thareal time LWD of the natural fracture width is made feasiblethe above equations. Making use of the average estimates defrom Machar 18z experience to tailor the particle size of the LCto be used when drilling Machar 20z two years later would habeen a waste of time. The above methodology is merely a tooreservoir engineers, not for drilling and stimulation people. Tlatter require detailed, local information rather than values avaged at reservoir scale.

Transient Equations of Radial Mud Loss Invasion froma Borehole into a Fracture PlaneThe local pressure drop (dp/dr) for a Bingham fluid of plasticviscosity (mp) and yield value (ty) in laminar flow in a slot ofwidth ~w! is13

dp/dr 512mpvm /w213ty /w, ~11!

with the local velocity (vm) in radial flow conditions~Vm is themud loss volume! being

vm5qm /~2prw !5~dVm /dt !/~2prw !. ~12!

Assuming a constant drilling overpressure (DpD) during the localmud loss~constant equivalent circulating density!,

DpD5E @6mp~dVm /dt !/~pw3r !13ty /w#dr ,

the integration being made from (r w) to (r i), with Vm5pw(r i2

2r w2 ), (r i) being the depth of mud invasion at time~t!. The inte-

gration over the mud bank is straightforward and leads to

DpD5~12mp /w2!r i ln@r i /r w#~dr /dt !1~3ty /w!@r i2r w#.~13!

Dimensionless radius (r D) and time (tD) are defined for easiehandling and integration of Eq.~13!, which becomes

dtD /dr D54r D ln~r D!/@12a~r D21!#, ~14!

tD5bt,

SPE Drill. & Completion, Vol. 14, No. 3, September 1999 169

VASION FACTOR „a…, PLUS ABSOLUTE/RELATIVE

a50err.%0 (numer.) 0 (anal.)

2.54545 2.54518 0.010911.7757 11.7750 0.005629.3625 29.3614 0.003856.4735 56.4719 0.002994.0088 94.0067 0.0023

142.702 142.699 0.0019203.172 203.169 0.0016275.954 275.950 0.0014460.297 460.291 0.0014

698.962 698.953 0.00131,349.62 1,349.60 0.00162,245.30 2,245.27 0.00153,399.64 3,399.60 0.00145,223.22 5,223.16 0.0013

10,205.6 10,205.4 0.001625,880.8 25,880.3 0.002149,691.9 49,690.9 0.0020

82,105.8 82,104.4 0.0017202,984 202,980 0.0021383,874 383,866 0.0019627,694 627,684 0.0017936,696 936,682 0.0015

1,757,300 1,757,270 0.00162,857,350 2,857,300 0.00155,930,160 5,930,060 0.0017

10,210,000 10,210,000 0.0016

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TABLE 1– DIMENSIONLESS TIMES AT VARIOUS DIMENSIONLESS RADII FOR SEVEN VALUES OF THE DIMENSIONLESS FINITE INCOMPARISONS OF NUMERICAL/ANALYTICAL VALUES AT a50

R

a

0.04 0.02 0.01 0.004 0.002 0.001

2 2.61714 2.58074 2.56296 2.55243 2.54894 2.547193 12.4669 12.1103 11.9404 11.8410 11.8082 11.79194 32.0482 30.6401 29.9863 29.6085 29.4850 29.42365 63.6235 59.8122 58.0898 57.1079 56.7888 56.63066 109.461 101.083 97.4040 95.3351 94.6668 94.33667 171.965 155.821 148.942 145.128 143.903 143.3008 253.775 225.356 213.628 207.216 205.172 204.1669 357.864 311.016 292.325 282.254 279.065 277.500

11 647.139 536.132 495.014 473.520 466.805 463.526

13 1,075.25 842.486 763.306 723.212 710.859 704.85517 2,587.35 1,750.17 1,521.09 1,412.84 1,380.43 1,364.8321 5,923.10 3,155.63 2,614.86 2,378.43 2,309.75 2,277.0225 17,879.0 5,226.12 4,096.56 3,644.61 3,517.42 3,457.4230 ¯ 9,137.25 6,581.92 5,685.40 5,443.48 5,330.8440 ¯ 25,562.0 14,180.5 11,461.2 10,793.1 10,490.260 ¯ ¯ 45,835.8 31,055.1 28,206.5 26,988.280 ¯ ¯ 127,049 64,084.0 55,872.2 52,581.2

100 ¯ ¯ 650,417 114,623 95,356.5 88,180.9150 ¯ ¯ ¯ 361,607 257,198 226,524200 ¯ ¯ ¯ 991,545 536,397 445,963250 ¯ ¯ ¯ 6,124,400 981,082 760,719300 ¯ ¯ ¯ ¯ 1,670,250 1,187,030400 ¯ ¯ ¯ ¯ 4,547,730 2,455,470500 ¯ ¯ ¯ ¯ 31,893,700 4,465,300700 ¯ ¯ ¯ ¯ ¯ 12,420,900900 ¯ ¯ ¯ ¯ ¯ 36,390,000

Fig. 1–Type curves for mud losses in natural fractures. By increasing values of the invasion plateau „i.e., from left to right …:r D max5111/a526, 51, 101, 251, 501, and 1,001. The bottom curve is a Newtonian case „a50….

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with b5(w/r w)2@DpD /(3mp)#, r D5r i /r w and a5(3rw /w)3(ty /DpD).

Newtonian Case.Eq. ~14! has an analytical solution when thmud is Newtonian~i.e., ty50!:

tD5r D2 @ ln~r D

2 !21#11, ~15!

which is the early time invasion behavior whatever the actyield value of the mud. As is shown later, the complete invasfor any Bingham mud is grossly approximated by Eq.~15! up tor D5(111/a), at which distance the invasion can be said to sat once. Equation~15! tells us that a Newtonian mud would nostop invading natural fissures since (r D) in this case does not tentowards any limit when (tD) goes to infinity.

General Case.To our knowledge, Eq.~14! has no analytical so-lution for positive yield values. A numerical integration schemhas been devised so as to generate tables of (tD) vs. (r D) valuesfor any value of~a!. This is easily taken care of by any spreasheet software. The only small problem is the startup atr D51.Due to the presence of ln(rD) in the derivative, it becomes necesary to estimate@ t (n1I)D# as a function of@r (n1I)D# rather than@r (n)D#:

t ~n1I!D5t ~n!D14r ~n1I!D ln~r ~n1I!D!Dr D /@12a~r ~n1I!D21!#.~16!

As a consequence, all (tD) values for very small (r D) are overes-timated by up to 100% of@ t (1)D# as shown by the comparison onumerical and analytical values for (a)50. This overestimationdecreases below 1% in less than 125 steps whatever (Dr D),0.1. To minimize deviations from analytical estimates, tscheme has been run with an initial (Dr D) of 1024 in the range1,r D,9, and then progressively increased.Table 1 lists somekey values of (r D) and (tD) for seven different~a! factors~cor-responding to final invasions ranging from 26 to 1,001 wellboradius!. Also provided are the relative errors made about the Netonian base line when comparing the scheme estimate witha)


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50 to the analytical solution of Eq.~15!. It is assumed that similarerrors affect numerical solutions for positive~a! values.

Eq. ~14! illustrates dr D /dtD˜0 when r D˜(111/a)5r D max. In other words, a Bingham mud, merely because ofpositive yield value, has a finite invasion (r max) equal to

r max5r w1wDpD /~3ty!'wDpD /~3ty!. ~17!

This means that bridging of cuttings or of LCM particles, or mthickening by leakoff through fracture walls, is not necessarilyexplanation for limited mud invasion.

Type Curves for Mud Loss in Natural FracturesMethodology. To make the above equations useful to the drilliengineer and actually allow a real time LWD estimate of fractuwidth, the following considerations are made.

When the drilling bit hits a natural fracture of width~w!, theloss volume (Vm) is given by

Vm5pwrw2 @r D

2 21#, ~18!

with (r D) being defined at (tD), and keeping in mind that theactual and dimensionless times are related byt5btD . Hence plot-ting

log10 X5 log10@Vm /~pr w2 !#

as a function of

log10 Y5 log10@ tDpD /~3mpr w2 !#

allows conventional superposition with a type curve of log10(r D2

21) vs. log10(tD). Fig. 1 depicts this particular type curve. Notthat for convenience during the type curve plotting process@andbecause (tD)’s are generated from@r D# ’s, not the opposite# times@~Y! or (tD)# are along the~y! axis whereas invasions@~X! or(r D

2 21)] are along the~x! axis; the natural tendency would cafor the opposite.Table 2 lists the particular values of log10(r D

2

21) and log10(tD) which have been derived through the numecal scheme and used to generate the type curves.


S FINITE INVASION FACTOR „a…

a

0.01 0.004 0.002 0.001 0 anal.

2.3305 2.3173 2.3130 2.3108 2.30872.4618 2.4467 2.4417 2.4393 2.43692.5921 2.5748 2.5692 2.5664 2.56362.7215 2.7017 2.6953 2.6922 2.68902.8502 2.8276 2.8203 2.8167 2.81322.9784 2.9525 2.9443 2.9402 2.9362

3.1063 3.0767 3.0673 3.0628 3.05823.2340 3.202 3.1895 3.1844 3.17923.3618 3.3230 3.3110 3.3051 3.29933.4899 3.4455 3.4318 3.4251 3.4186

3.6186 3.5675 3.5520 3.5445 3.5371

3.7482 3.6893 3.6717 3.6632 3.65493.8790 3.8110 3.7910 3.7814 3.7720

a

0.004 0.002 0.001 0 anal.

6.50896.67796.9715

¯ 5.9975 5.8862 5.8023¯ 6.1413 6.0078 5.9121¯ 6.2940 6.1311 6.0217¯ 6.4601 6.2566 6.1311¯ 6.6492 6.3848 6.2404¯ 6.8891 6.5165 6.3494¯ 7.0618¯ 7.1616¯ 7.3278¯ 7.4106¯ 7.5782¯ ¯ 6.6527 6.4582¯ ¯ 6.7952 6.5669¯ ¯ 6.9464 6.6753¯ ¯ 7.1111 6.7837¯ ¯ 7.2986 6.8918¯ ¯ 7.5361 6.9999¯ ¯ 7.7064¯ ¯ 8.2073 7.1077¯ ¯ 8.2679¯ ¯ 8.3699¯ ¯ ¯ 7.6451¯ ¯ ¯ 8.1795

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TABLE 2– DIMENSIONLESS TIMES AT VARIOUS DIMENSIONLESS INVASIONS FOR SEVEN VALUES OF THE DIMENSIONLES„TYPE CURVES INPUT DATA IN DECIMAL LOGS …

log10

(R221)

a log10

(R221)0.04 0.02 0.01 0.004 0.002 0.001 0 anal. 0.04 0.02

0.0 20.4081 20.4106 20.4118 20.4125 20.4128 20.4129 20.4131 1.8 2.4054 2.35380.1 20.2291 20.2319 20.2336 20.2345 20.2348 20.2349 20.2351 1.9 2.5492 2.48860.2 20.0531 20.0568 20.0586 20.0597 20.0600 20.0602 20.0605 2.0 2.6945 2.62300.3 0.1196 0.1152 0.1131 0.1118 0.1113 0.1111 0.1108 2.1 2.8419 2.75720.4 0.2892 0.2839 0.2813 0.2797 0.2792 0.2790 0.2787 2.2 2.9927 2.89150.5 0.4556 0.4493 0.4462 0.4443 0.4437 0.4434 0.4430 2.3 3.1483 3.0263

0.6 0.6188 0.6113 0.6076 0.6054 0.6046 0.6043 0.6039 2.4 3.3111 3.16190.7 0.7791 0.7701 0.7657 0.7631 0.7622 0.7618 0.7613 2.5 3.4853 3.29890.8 0.9364 0.9258 0.9206 0.9175 0.9165 0.9160 0.9154 2.6 3.6786 3.43790.9 1.0910 1.0785 1.0724 1.0687 1.0675 1.0670 1.0663 2.7 3.9107 3.57961.0 1.2431 1.2284 1.2213 1.2170 1.2156 1.2149 1.2142 2.75 4.06031.1 1.3931 1.3758 1.3674 1.3625 1.3609 1.3601 1.3593 2.8 4.2807 3.7253

1.2 1.5410 1.5207 1.5110 1.5053 1.5034 1.5025 1.5016 2.81 4.35131.3 1.6873 1.6636 1.6523 1.6457 1.6436 1.6425 1.6414 2.82 4.45431.4 1.8322 1.8045 1.7915 1.7839 1.7814 1.7801 1.7789 2.829 4.75681.5 1.9761 1.9438 1.9287 1.9199 1.9170 1.9156 1.9142 2.8292 4.82861.6 2.1193 2.0816 2.0641 2.0541 2.0508 2.0491 2.0475 2.9 ¯ 3.87651.7 2.2622 2.2182 2.1980 2.1865 2.1827 2.1808 2.1789 3.0 ¯ 4.0356

log10

(R221)

a log10

(R221)0.02 0.01 0.004 0.002 0.001 0 anal.

3.1 4.2071 4.0116 3.9327 3.9099 3.8990 3.8885 4.773.2 4.3995 4.1464 4.0545 4.0286 4.0163 4.0044 4.793.3 4.6361 4.2844 4.1767 4.1471 4.1332 4.1197 4.7993.35 4.7958 4.83.4 5.0697 4.4266 4.2992 4.2654 4.2497 4.2346 4.93.41 5.2066 5.03.414 5.3553 5.13.4149 5.5356 5.23.5 ¯ 4.5746 4.4225 4.3838 4.3659 4.3489 5.33.6 ¯ 4.7310 4.5467 4.5023 4.4820 4.4628 5.353.7 ¯ 4.9002 4.6720 4.6209 4.5979 4.5763 5.373.8 ¯ 5.0912 4.7989 4.7399 4.7138 4.6894 5.393.9 ¯ 5.3289 4.9278 4.8593 4.8296 4.8021 5.3953.95 ¯ 5.4933 5.3994.0 ¯ 5.8138 5.0593 4.9794 4.9454 4.9144 5.44.0043 ¯ 5.8918 5.54.0080 ¯ 6.0609 5.64.0085 ¯ 6.1922 5.74.1 ¯ ¯ 5.1942 5.1003 5.0614 5.0264 5.84.2 ¯ ¯ 5.3336 5.2222 5.1776 5.1381 5.94.3 ¯ ¯ 5.4791 5.3454 5.2942 5.2495 5.954.4 ¯ ¯ 5.6334 5.4702 5.4111 5.3606 6.04.5 ¯ ¯ 5.8011 5.5972 5.5286 5.4714 6.00054.6 ¯ ¯ 5.9920 5.7268 5.6469 5.5819 6.00084.7 ¯ ¯ 6.2339 5.8599 5.7660 5.6922 6.54.75 ¯ ¯ 6.4080 7.0

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When superposing field data onto type curves, both~x! and~y!shifts give direct access to~w!. This is because

log10~X!5 log10~w!1 log10~r D2 21!

and

log10~Y!5 log10~1/w2!1 log10~ tD!522 log10~w!1 log10~ tD!.

Sincew!1 m, actual times~Y! shift by 22 log10(w), i.e., posi-tively ~compared to the type curve!, whereas actual invasions~X!shift negatively by log10(w).

In addition, field data line up at late times with one particuinvasion plateau and therefore provide more redundant infortion about~w! through Eq.~17!. The proposed type curve therefore actually represents a triple match over the fracture hydrawidth.

Causes of Poor Match.We should distinguish failures originating from the limitation of the mathematical model and thoselated to physical phenomena.

First, it must be emphasized that the possible case of turbuflow is not considered, and that the laminar case has been derately simplified for easier solving. Unwin14 has indeed shownthat the above equations are linearized forms of a full analytsolution containing a cubic power of the pressure drop whcannot be disregarded at large rates. In other words, this wmerely extends to radial flow conditions the approach usedyears ago by Hillet al.15 to describe linear flow pressure droalong propagating fractures in hydraulic fracturing.

Second, departures from the proposed type curve either sifracturing conditions~the frac gradient is exceeded!, bridging bycuttings or loss control materials, and/or excessive leakoff ofinvading mud into the primary porosity. Another cause is theture of the pattern of fractures, which can be conjugated at theof anticlines ~there are two fracture azimuths at an angle tgenerally is 60°!.

Mud loss due to excessive downhole pressure~particularly dur-ing surges! and to the initiation of hydraulic fracturing intoprimary porosity porous medium obeys different flow equatioThere is no possible confusion with the process of normal insion described in this article. The evolution of the drilling pressbefore the mud loss event is extremely different. Hydraulic frturing is the consequence of a rapid, often instantaneous preincrease, whereas normal invasion is the mere consequencecollision between the well path and a natural fracture at conspressure conditions.

Bridging should occur when the fracture width is less thantimes the average diameter of the particles of the invading mrial ~this particular ratio is the default value for tip screenoutmost proppant fracturing simulators, but drilling people tendmake use of any ratio between 3 and 7!. A cake is built up thatdramatically decreases the local conductivity of the fracture.the same time the mud loses most of its solids in the cake formprocess, which causes the mud filtrate, which has nearly Newian behavior, to flow past the cake into the clean, deeper pathe fracture. It is not the purpose of this study to derive the cresponding modeling equations, however it is well known tbridging lessens losses and should therefore lead to plateau vat reduced (r D

2 21) values~compared to type curve!—the thirdmatch on~w! no longer holds~at this point the calculated widthwould be much thinner than what the early time fit would indcate!.

Excessive leakoff through the walls of the natural fracture ithe primary porosity of the reservoir rock of sizeable (km) can beanother source of mismatch. In this case the overall effect isadditional loss of fluid and the actual invasion data should t


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towards increased (r D2 21) values—the third match on~w! is de-

stroyed as well~the fracture looks thicker than expected from tearly time match!. In the extreme cases where leakoff is so intenthat both mud dehydration and bridging are promoted, theresult reverts back to bridging consequences. Hence there shexist intermediate cases where a good late time third match on~w!is obtained by mere coincidence. It must be emphasized that mnaturally fractured carbonate reservoirs have reduced matrixmeability and do not experience significant leakoff. The propotechnique should therefore work very well in these instances~e.g.,the work of Machar, Banff, and Kyle in the North Sea5!. In thecase of Clair’s sandstone, the fracture and the matrix’s permeaity are very similar~about 100 md each!, and the method shouldnot lead to very accurate results unless a mud with high thixotrand/or yield [email protected]., mixed metal hydroxide MMH mud# isused. Xanthan-based formate muds with extremely large elontional viscosity16 should also control leakoff with good efficiencyeven in the case of large matrix permeability.

When the well trajectory runs through a pattern of conjugafractures, most of the equations@Eqs.~1! to ~10!# must be adaptedand all further equations are only valid until the mud flow hitsintersect between the invaded fracture and the nearest fractuthe other azimuth. Later on, the flow divides itself into three coponents. The reduction of the mud rate at this stage leadsdecrease of the shear rate and therefore to an apparent increathe mud viscosity. There is more of an open section to flohowever the fluid looks more viscous: whether this leads toacceleration or a slowdown of the mud loss is not answered inarticle. Further studies are required in order to derive all the eqtions adapted to the case of a reservoir with a pattern of cogated natural fractures.

Validation of the Mud Loss Type Curve by Field Data. Due tothe unavailability of the method outlined in this article, mud loreports have so far been limited to indications of static andnamic ~i.e., while drilling! loss rates as well as to total daily losvolumes. (Vm) charts are almost never produced~except for inRef. 3!, which precludes any attempt at validating the propostype curve by field data. We hope the present study triggers inest so that drilling reports become more detailed in the future

Mud Loss Rates: Machar 20z Well Example

Machar 20z is the twin of well 18z discussed earlier. It was copleted and stimulated the same way as well 18z, however righthe top of the diapir is a zone of intense fracturing. Pre- apost-stimulation productivity indices are even more impressibeating Machar 18z and its 1994Petroleum Engineer Journarecord breaking entry. Indeed, productivity soared from 1.02470 B/D/psi after a mud and silt remover acid treatment was pformed August 5, 1995 by the same stimulation vessel thatpumped the Machar 18z job in December 1993.4 A comparisonwith well 18z productivity evolution—2.9 to 147 B/D/psi—already signals the larger reservoir permeability at the top ofdome: a larger number of wider natural fractures over a shopayzone length~1,270 ft! means increased damage buildup aremoval potentials at the same time. The drilling reports confithe trend, totaling 4,033 barrels of mud losses through thelimestone target.Table 3 is a summary of loss events17 after the 958 casing was set too deep—past the Paleocene sandstone—while drilling the final 81

2 in. hole section. Because losses at tvery top of the payzone~2,189 bbl! occurred during long, unex-pected trips into the 1214 hole, the following calculations do notake them into account and only consider 2,844 barrels of losand eight events over 1,130 ft.


174 Lietard et a

TABLE 3– DRILLING REPORT FOR MACHAR 23/26a-20 LOSSESWHILE DRILLING THE 8 1

2 SECTION

Losses were encountered at the following depths:2581 m Slight Losses—2bbl/hr No action taken2707 m 12 bbl/hr—drilling Pump 50 bbl LCM pill, lost 20 bbl

9 bbl/hr—static Before stopping2737 m 50 bbl/hr—static Pump 50 bbl LCM pill2745 m 28bbl/hr—static Pump 50 bbl LCM pill2765 m 50 bbl/hr—static Pump 50 bbl LCM pill

The mud weight was reduced from 1.38 to 1.35 s.g. to help lessen the loss rates when drillinginto a fracture. On all occasions, up to this point, when LCM was pumped the losses were cured.2798 m 140 bbl/hr—static Pump 62 bbl LCM pill-pulled one

Stand circulated above reducing to 5bbl/hr and the ECD allowed to do agentle squeeze on the LCM

2802 m 50 bbl/hr—static Pump 30 bbl LCM at 2550 m2809 m 35 bbl/hr—static Run to bottom, pump 50 bbl LCM,

POOH to 2550 m, circulate, no losses

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Treating the data in the same way as was done for Macherleads to the following estimates: spacingd5141.5 ft, Sf523.62, andpD52.04 ~with r d520,000 ft, about 6 km!. The res-ervoir ~or fissure! permeability (kf) is 231 md, therefore the average width ~w! is 620 mm ~slightly less than 1/40 in.! and(DpD /ty)5828,320. The invasion radius is 170.6 m~about 560ft! and the actual productivity folds of increase~461! are matchedwhen in Eq.~7! the skin takes the enormous value of1938. Eq.~5!, however, accepts the latter together with (kf ) and (r s) pro-viding the matrix permeability (km) is 0.45 md. This is quite a bismaller than in the 18z case~0.64 md! but can be the result of amore intense cementation of the Tor ‘‘chalk’’ closer to the sdome~a larger exposure to saturated brine!.

With this preliminary check completed, it is now possiblemove confidently to the study of transient mud loss data. Tparameters in Eq.~14! can be estimated as

tD5bt,

with b'8.795(ty /mp),

a50.0006436, and, therefore,r Dmax5(111/a)'1,555.

With a plateau value log10(r D2 21) in the type curve equal to

about 6.38, mud losses can be grossly approximated by the Ntonian curve of Eq.~15! until log10(tD) is about 7.5~see Fig. 1!, atwhich time the loss stops.

The drilling report indicates that the mud density decreabetween the fifth and the sixth losses ‘‘to lessen the loss rate wdrilling into a fracture.’’ This of course helps since the stapressure is decreased by 90 psi. In the process, the (ty /mp) ratioalso evolved from 21/35 to 18/26 in field units~lb/100 sq ft andcp!, which also helped under the dynamical conditions sinceannular friction is decreased as well. Overall the equivalent cirlating density~ECD! might be somewhat quite improved; however severe surges seem to have ruined these efforts@withDpD /ty5828,320 and an average yield value of 19.5 lb/100 sqthe average (DpD) is 1,120 psi, twice the excess pressure nmally provided by the ECD#. Decreasing both the plastic viscosiand the yield value, under these conditions of overpressure, conly lead to larger losses. This actually was the case after thewas reconditioned~Table 3!. Indeed, looking at the expressionfor the ~a! and ~b! parameters, one can see that the rheologproperties of the mud work opposite to the drilling overpressu

To estimate the average mud loss rate for our well exampleuse an averagety /mp50.64 in field units, corresponding to 30reciprocal seconds. This leads tob52,695~the same unit!. Hence

l.: Fracture Width Logging

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2 21) at that time is about 5.7~Table 2~b!!, andEq. ~18! becomes

Vm~1 hour!5pwrw2 3105.695511.582 m3.

This corresponds to about 73 bbl/hr and is well in line with moof the reported loss rates~Table 3!, the mean of which is 45.5bbl/hr. This most probably translates the poor efficiency ofLCM pills, as emphasized by one of the end of job conclusioand recommendations that says, ‘‘...the LCM formulation amechanism in which it works should be revisited.’’17

Mud Loss Curve for the Field CasesAverage mud loss curves have been roughly generated forMachar wells, with a numerical scheme (Dr D) increment equal to1. For Machar 18z, the~b! factor has been evaluated by the muparameters of Machar 20z. Curves are provided in a log/log~Fig. 2! which can be used for type curve matching exercises~XandY log10 coordinates!.

~a! and ~b! in the Machar 18z case are equal to, respective0.00215 (r Dmax

5468) and 548 reciprocal seconds. It has toemphasized that the ultimate radius of invasion is related onlthe yield value of the mud, whereas the loss rate is a functionits plastic viscosity. When the mud rheology is unknown,~a! canstill be derived from well test data~in the Machar 18z case!, but~b! cannot be evaluated.

Additional Outcomes of the Proposed ModelPressure Surge Effects While Drilling.Drill pipe trips into thehole are widely known to provoke pressure surges as large aspsi.18 These hammering effects cause additional mud invaswith the final surge radius (r DS) being related to the surge duration (DtS) through

bS~DtS!'~r DS21/aD!2@2 ln~aDr DS!21#, ~19!

with (bS) being the surge time factor, with (DpD) replaced by(DpS) in Eq. ~14!, and (aD) the finite invasion factor corresponding to the normal drilling overpressure. Eq.~19! shows that acomplete description of the final mud invasion would requirecomplete recording of the bottomhole pressure all along the ding through the naturally fractured payzone. Such monitorwould also tell whether the above model is applicable: surwould have to be controlled so as to not exceed the frac press

SPE Drill. & Completion, Vol. 14, No. 3, September 1999

Fig. 2–Mud loss curves for field cases „log/log plot …. Machar 18z: triangles „left side …; Machar 20z: squares „right side ….

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Damage Removal.There are two ways of regaining full communication between the clean~deep! part of the fissure system anthe wellbore.

The first one consists of pushing the mud bank far away frthe well deep into the rock. This is accomplished with a sequeof fluids5 aimed at achieving a positive mobility ratio in the firstep, and turbulence and acid attack later. The treatment fluiddesigned to not present any residual viscosity when the weflowed back. This technique is implemented by pumping the trement above the parting pressure. In that respect it is very efficwhen cakes have been built up deep in the natural fractures. Tcakes can be dislodged by fracture openings and pushed awathe mud filtrate.

The second technique, which has very recently been succfully implemented for the first time in the North Sea, consistsapplying a sudden pressure drawdown in the well, making usa formation surge tool. For obvious reasons, it cannot be appin wells prone to collapse and/or sand production. Providing tthe drawdown is large enough to overcome the mud yield valuall the natural fractures, the mud bank is put into motion. For tbank to be backproduced in full, a minimum dynamic drawdoshould be maintained, one that corresponds to the much smpressure drop caused by the plastic viscosity of the mud. Applythe required initial~static! drawdown is easy and only necessitatemptying the wellbore~by nitrogen displacement with coiled tubing, for instance! above a plug with a remotely operated valvMaintaining the minimum dynamical drawdown might provmore difficult when the bank produced fills up the wellbore. Onthis has prevented introduction of the technique until recenFurther studies are planned that will consider the integrationEq. ~12! with boundary conditions applicable to the problem cosidered; however they will not provide any information about tefficiency of the instantaneous drawdown technique in the pence of mud cakes. It is possible that, under the pressure cotions used, the cakes get pinched by ‘‘collapsing’’ fractures, qua disastrous scenario indeed. For this reason only, avoidinguse of LCM’s as much as possible is often recommended, andwill be discussed in the next section.


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A design methodology for both damage removal methodsbadly needed. Its development will require further treatment ofmodeling equations proposed in this article. The statistical disbution of fracture widths, in particular, has to be considered siit certainly has a great influence on the efficiency of mud remoin terms of zone coverage. The averaging approach used inarticle is no longer applicable.

Mud/LCM Selection Guidelines and CementingConditions in Naturally Fractured ReservoirsDrilling Mud. Mud invasion and losses are minimized when bothe plastic viscosity and the yield value of the mud are increasA large yield value, however, is definitely the best way to contlosses, and MMH muds are therefore preferable~over silicatemuds which have a large plastic viscosity and generate a lofriction pressure!.

Loss Control Materials. The influence of depositing cakes oLCM in natural fractures has not been studied here. It is howeobvious that LCMs, when properly selected, can dramaticallycrease mud losses. There is unfortunately a price to pay for tpermanent damage of the most prolific part of the reservoir roi.e., the natural fractures. Whatever is good at stopping any inflalso impairs production. Acid soluble materials have therefbeen recommended, however they do not ensure compcleanup.

• The amount of acid required and its pumping rate are relato the length of the payzone; in horizontal wells the designsfor thousands of barrels and the horsepower is often in exces5,000 hhp, a costly treatment indeed.

• Diversion of the acid from fracture to fracture is often prolematic, particularly in the most common open hole completio

• Acid attack is greatly impaired when the LCM is oil weeither by inclusion into an oil-based mud, or by contact with fomation oil; various water-wetting additives have been tried wvery limited success.


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• In carbonate rocks~the main core of naturally fissured resevoirs!, most of the acid reacts with the fracture walls first, leavithe LCM untouched, particularly when the cake has been deited deep into the rock.

In order to minimize LCM damage, extreme care must bevoted to selecting the proper particle size. A prerequisite of couis an accurate estimate of the width of the natural fractures, whis the main subject of this article. The LCM should be fine enouto be squeezed into the fracture, so that the cake builds itseand is protected from permanent wear at the wellbore. ButLCMs should at the same time be coarse enough to preventinvasion. The deeper the LCM can get into the natural fractbefore building up a cake, the larger the amount of materialwill be deposited and the more intense the damage problem. Wa compromise needed, we think that the best LCM is one wcoarser particles with a diameter about 40% of the width offissure being considered. It must, in addition, present a verycial particle size distribution obeying Fuller’s19 empirical rule of‘‘granulometric filling’’ that ensures minimum permeability of thcake. Since such a distribution is not straightforward~grindingmust be carefully monitored! and is very sensitive to dispersionflocculation conditions, there are only a couple of LCMs wacceptable properties at the surface. Modifications of theses perties due to additional grinding through the drill bit and/or intemixing with cuttings complicate the selection.

For all these reasons of poor control of the downhole LCparticle size distribution and of its adaptation to the natural frture width, it is proposed that LCMs be avoided as much as psible. This is particularly recommended in the case of LCMs ttend to agglomerate into a compact, zero-porosity type cemenmaterial~e.g., rock salt!. This type is often preferred for its ultimate efficiency in curing loses, however it can permanently soff natural fractures as well. In some ways, making use of ttype of LCM is like allowing cement slurries to invade and bcome set in fissures. A much better, option~over LCM use! isliving with the mud losses, and minimizing them as muchpossible. This is accomplished through (ty) increase and (DPD)decrease. It must be understood that full communication betwthe wellbore and the pattern of natural fractures is much measily re-established when flowing back a mud bank than wattempting to dislodge a jammed LCM cake. The mud bank stliquid ~dehydration does not occur due to the very low matpermeability! whereas the LCM cake is packed and solid.

Cementing.As mentioned before, there would be nothing worthan cementing the natural fissures of a secondary porosity revoir. Huge values of the skin would result, and would be impsible to alleviate unless a short hydraulic fracturing treatm~‘‘skin frac’’ ! were performed.

Cement slurry invasion, however, is easily prevented, provithat the cementing engineer is made aware of the value ofmaximum equivalent circulating density while drilling, andgiven orders not to exceed this pressure at any time duringplacement of the cement slurry along the part of the annulusposite the payzone. The mud bank then protects the fissuredtem from any cement invasion. Perforating clusters across thezones recovers full hydraulic communication between the natfractures and the wellbore, despite the presence of liquid mbeyond the tips of the perforations. This is evidenced by theperience of Machar wells 18z and 20z.

Conclusions• The hydraulic width of a natural fracture intercepted by a dring bit can be estimated by the analysis of the subsequentloss history. Type curves are provided for the superposition ofactual loss data. An easy derivation of the fracture width is

176 Lietard et al.: Fracture Width Logging

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tained from the shifts of both log/log axes. An additional late-timmatch with the maximum invasion radius assists the analysithe case of a Bingham mud.

The proposed method requires two~in and out! mud flowmeters, continuous monitoring of the equivalent circulating dsity at the drill bit, and good knowledge of the rheological proerties of the drilling mud at downhole conditions.

• This new logging while drilling technique should help drillinengineers in rapidly selecting the least damaging loss controlterial immediately after occurrence of the loss. A careful matchthe LCM particle size to a rigorous estimate of the width of tnatural fractures is now possible. Preferably, LCM-free, laryield value mud systems should be chosen, despite the largepected mud loss, as a compromise for the objectives of producengineers.

• The method also represents a valuable tool for reservoirgineers in their search for information on the contribution ofpattern of natural fractures to the total permeability of a reservrock.

Nomenclature

B 5 formation volume factor, dimensionless, rm3/stm3

dp 5 pressure differential, m/Lt2, Padr 5 radius differential, L, m

dr D 5 dimensionless radius differentialdt 5 time differential, t, s

dtD 5 dimensionless time differentialdVm 5 volume differential, L3, m3

h 5 payzone height, L, mJs 5 damaged productivity index, L4t/m, stm3/s/PaJ 5 undamaged productivity index, L4t/m, stm3/s/Pak 5 total reservoir permeability, L2, m2

kf 5 permeability of natural fractures, L2, m2

km 5 permeability of matrix rock, L2, m2

ks 5 permeability of the damaged zone, L2, m2

L 5 horizontal well producing length, L, mpD 5 dimensionless pressure

q 5 producing rate, L3/t, m3/sqm 5 mud loss rate at time~t!, L3/t, m3/s

r 5 radius, L, mr d 5 drainage radius, L, mr D 5 dimensionless radius

r D max 5 maximum dimensionless invasion radiusr DS 5 dimensionless surge invasion radius

r i 5 invasion radius at time~t!, L, mr max 5 maximum invasion radius, L, m

r (n)D 5 dimensionless radius step, numerical schemer s 5 damage radius, L, mr w 5 wellbore radius, L, mSf 5 natural fracture pseudoskip, dimensionlessSs 5 damage skin, dimensionless

t 5 time, t, stD 5 dimensionless time

t (n)D 5 dimensionless time step, numerical schemevm 5 local mud velocity, L/t, m/sV 5 total mud loss through payzone, L3, m3

Vm 5 mud loss volume at time~t!, L3, m3

w 5 average hydraulic width of natural fractures, L, mX 5 superposition ‘‘volume,’’ L, mY 5 superposition ‘‘time,’’ L22, m22

a 5 dimensionless finite invasion factoraD 5 dimensionless finite invasion factor~drilling!

b 5 time factor, t21, s21

bs 5 surge time factor, t21, s21

d 5 average spacing of natural fractures, L, m

SPE Drill. & Completion, Vol. 14, No. 3, September 1999

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Dp 5 production drawdown, m/Lt2, PaDpS 5 surge overbalance, m/Lt2, PaDpD 5 overbalance while drilling, m/Lt2, PaDr D 5 incremental dimensionless radius, numerical scheDtS 5 pressure surge duration, t, sf f 5 porosity of natural fractures, dimensionlessm 5 reservoir oil viscosity, m/Lt, Pa s

mp 5 plastic viscosity of the mud, m/Lt, Pa sty 5 yield value of the mud, m/Lt2, Pa

AcknowledgmentsThe authors thank Schlumberger for permission to publishinitial version of this article. The authors’ gratitude went at thtime to Mike Martin for his corrections and comments, and pticularly to Ken Nolte for his assistance concerning surge effeand to Anthony Pearson, Isaac Newton Institute, UniversityCambridge, for checking the mathematics and attempting to cup with an analytical solution of the general differential equatio

References1. Drummond, J.M.: ‘‘An Appraisal of Fracture Porosity,’’Bull. Can.

Pet. Geol.~1964! 12, 226.2. Dyke, C.G., Wu, B., and Milton-Taylor, D.: ‘‘Advances in Chara

terising Natural Fracture Permeability From Mud Log Data,’’SPEFE~September 1995! 160.

3. Sanfilippo, F., Brignoli, M., Santarelli, F., and Bezzola, C.: ‘‘Charaterization of Conductive Fractures While Drilling.’’ paper SPE 381presented at the 1997 SPE European Formation Damage ConferThe Hague, The Netherlands 2–3 June.

4. Gilchrist, J.M., Stephen, A.D., and Lie´tard, O.M.N.: ‘‘Use of High-Angle, Acid-Fractured Wells on the Machar Field Developmenpaper SPE 28917 presented at the 1994 European Petroleum Coence. London, 25–27 October.

5. Lietard, O., Bellarby, J., and Holcomb, D.: ‘‘Design, Execution aEvaluation of Acid Treatments of Naturally Fractured Carbonate,Reservoirs of the North Sea,’’SPEPF~May 1998! 133.

6. Muskat, M.: Physical Principles of Oil Production, McGraw–HillBook Co., New York~1949! 246.

7. Jones, Jr., F.A.: ‘‘A Laboratory Study of the Effects of ConfininPressure on Fracture Flow and Storage Capacity in CarboRocks,’’ JPT ~January 1975! 21.

8. Lietard, O.: ‘‘Permeabilities and Skins in Naturally Fractured Resvoirs: An Overview and an Update for Wells at Any Deviation,paper SPE 54725 presented at the 1999 SPE European FormDamage Conference, The Hague. The Netherlands, 31 May–1 J

9. Giger, F., Reiss, L.H., and Jourdan, A.: ‘‘The Reservoir EngineerAspects of Horizontal Drilling,’’ paper SPE 13024 presented at1984 SPE Annual Technical Conference and Exhibition, Hous16–19 September.

10. Lietard, O., Ayoub, J., and Pearson, A.: ‘‘Hydraulic FracturingHorizontal Wells: An Update of Design and Execution Guidelinespaper SPE 37122 presented at the 1996 International ConferencHorizontal Well Technology, Calgary, Canada, 18–20 November

11. Renard, G. and Dupuy, J.G.: ‘‘Formation Damage EffectsHorizontal-Well Flow Efficiency’’JPT ~July 1991! 786.


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ctsofmen.

-

c-7nce,

,’’nfer-

dil

gate

r-’’ationne.

ingheon,

f,’’e on

on

12. Lietard, O. and Daccord, G.: ‘‘Acid Wormholing in Carbonate Reervoirs: Validation of Experimental Growth Laws Through Field DaInterpretation,’’ paper presented at the 195th National Meeting ofAmerican Chemical Society, Toronto, 5–11 June 1988.

13. Guillot, D.: ‘‘Digest of Rheological Equations,’’Well Cementing,N.E.B. ~ed.!, Schlumberger Educational Services, SMP-7031~1990!App. 1, A-7.

14. Unwin, T.: ‘‘Effects of Mud Rheology on Massive Mud Losses inFracture.’’ Schlumberger Cambridge Research Scientific Note SSN/1995/015/FLM/C, 24 May 1995.

15. Hill, O.F., Ward, A.J., and Clement, C.C.: ‘‘Austin Chalk FracturinDesign Using a Cross-Linked Natural Polymer as a DivertiAgent,’’ JPT ~December 1978! 1795.

16. Hands, N.et al.: ‘‘Optimizing Inflow Performance of Long Multi-Lateral Offshore Well in Low-Permeability, Gas-Bearing SandstoK14-FB-102 Case Study,’’ paper SPE 50394 presented at the 1International Conference on Horizontal Well Technology, CalgaCanada, 1–4 November.

17. Nielsen, K. and Chambers, B.: BP Exploration Drilling Fluids Sumary for Turnkey Additional Production~TAP! Machar DevelopmentWell 23/26a-20, 3 October 1995.

18. Wagner, R.R., Halal, A.S., and Goodman, M.A.: ‘‘Surge Field TeHighlight Dynamic Fluid Response,’’ paper SPE/IADC 25771 prsented at the 1993 SPE/IADC Drilling Conference, Amsterda23–25 February.

19. Joisel, A.: ‘‘Composition des Be´tons Hydrauliques,’’Ann. Inst. Tech.Bat. Travaux Publics~October 1952! 58.

SI Metric Conversion Factorsbbl 3 1.589 8 E201 5 m3

bpm 3 2.649 667 E203 5 m3 s21

cp 3 1.0* E203 5 Pa sft 3 3.048* E201 5 m

gal 3 3.785 2 E203 5 m3

in. 3 2.54* E202 5 mlb/100 ft2 3 0.478 803 E 005 Pa

md 3 9.869 233 E216 5 m2

mm 3 1.0* E206 5 mpsi 3 6.894 757 E103 5 Pa

*Conversion factors are exact. SPEDC

Olivier Lietard is an independent petroleum consultant. Hepreviously worked for Dowell as a stimulation specialist. He hasworked in a variety of positions including marketing, sales, andtechnical field assignments in well stimulation and productiontechnology. Lietard holds a PhD degree in physicochemistry.Tess Unwin was a research scientist with Schlumberger Cam-bridge Research. Dominique Guillot is a cementing specialistwith Dowell. He has worked in well stimulation and construc-tion. Guillot is a member of an Annual Meeting technical com-mittee. Mike Hodder works for Dowell in New Orleans. He hasworked in research, technical services, field engineering, qual-ity, health, safety, and environment (QHSE), and marketing.Hodder holds a degree in natural sciences from Cambridge U.


Discussion of Fracture Width LoggingWhile Drilling and Drilling Mud/

Loss-Circulation-Material SelectionGuidelines in Naturally

Fractured ReservoirsS.J. Sawaryn, SPE, BP Amoco

The discussion of drilling mud losses in naturally fractured reser-voirs by Liétard et. al.1 (SPE 57713, SPEDC, September 1999,page 168) provides valuable insight to a common drilling problem.The authors derived a differential equation describing the transientradial mud-loss invasion from a borehole into a fracture plane.They used numerical methods to solve the equation, having con-cluded that an analytical solution was not possible for muds withpositive yield values. The results were used to produce the typecurves for the mud losses. Further analysis shows that an analyticalsolution exists and confirms the efficiency of the numerical algo-rithm employed by the authors. Unlike the numerical method, theanalytical solution exhibits no problem at startup. A comparisonbetween the analytical solution and the numerical results is pre-sented in Table 1. It is hoped that the simplification afforded bythe analytical solution provides greater encouragement to thosewanting to try the technique.

Analytical Solution for Positive Yield ValuesLiétard et al.1 derived the differential Eq. 1 for the transient radialmud-loss invasion from a borehole into a fracture plane. The di-mensionless invasion radius rD and time tD are defined as rD�ri/rw and tD��t, where ��(w/rw)2[�p�/(3�p)]. The maximum in-vasion radius, rDmax, was shown to be (1 + 1/�), where ��(3rw/w)× (�y/�p�).

dtD

drD=

4rD ln�rD�

1 − ��rD − 1�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Introduce the change of variable u � (1 + 1/�)/rD with the re-striction that � � 0, and the following occurs.

dtD

du=

−4

��1 +1

�� ln ��1 +1

�� 1

u�u2 �u − 1�

. . . . . . . . . . . . . . . . . . . . . . (2)

Now substitute rDmax � (1 + 1/�) to simplify the manipulation andexpress the integrand in partial fractions.

= −4rDmax

� � 1

u − 1−

1

u−

1

u2� ln �rDmax

u �. . . . . . . . . . . . . . . . . . (3)

= −4rDmax ln�rDmax�

� � 1

u − 1−

1

u−

1

u2�+

4rDmax

� � ln�u�

u�1 − 1�u�−

ln�u�

u−

ln�u�

u2 �. . . . . . . . . . . . . . . . . . . . (4)

The expression u−1(1 − 1/u)−1 in Eq. 4 is equivalent to (u − 1)−1).By inspection, 0 < 1/rDmax � 1/u � 1, and the fact (1 − 1/u)−1 canbe regarded as the limit of the sum of the geometric series �u−n,as n tends to infinity. Expanding this factor gives

�dtD = −4rDmax ln�rDmax�

�� 1

u − 1−

1

u−

1

u2� du

+4rDmax

��

n=0

ln�u�

u n+1−

ln�u�

u−

ln�u�

u2 � du. . . . . . . . . . . . (5)

All expressions involving the variable u in Eq. 5 are now in stan-dard forms and can be integrated directly.2 The terms in u−1 andu−2 in the series cancel, and, hence, the integration of the series isstarted with the term n equal to 2.

tD = −4rDmax ln�rDmax�

� �ln�u − 1� − ln�u� +1

u�+

4rDmax

� �n=2

�−ln�u�

nun−

1

n2un� + A. . . . . . . . . . . . . . . . . . . . . (6)

Collecting terms and substituting rDmax/rD in Eq. 6 for u gives

tD = −4rDmax ln�rDmax�

� �ln �1 −rD

rDmax� +

rD

rDmax�

−4rDmax

� �n=2

1

n � rD

rDmax�n �ln �rDmax

rD� +

1

n� + A. . . . . . . . . . (7)

The boundary condition rD = 1 at tD determines the integrationconstant A.

Copyright © 2002 Society of Petroleum Engineers

247December 2002 SPE Drilling & Completion

www.spe.org/oil

A =4rDmax ln �rDmax�

� �ln �1 −1

rDmax� +

1

rDmax�

+4rDmax

� �n=2

1

n � 1

rDmax�n �ln�rDmax� +

1

n�. . . . . . . . . . . . . . . . (8)

Combining Eqs. 7 and 8 gives the final expression, Eq. 9,for dimensionless time as a function of the dimensionlessinvasion radius.

tD =4rDmax ln�rDmax�

� �ln � rDmax − 1

rDmax − rD� +

rD − 1

rDmax�

+4rDmax

� �n=2

1

n � 1

rDmax�n ��ln�rDmax� +

1

n�− rD

n �ln�rDmax

rD� +

1

n��. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

NomenclatureA � integration constant, dimensionless

�p� � overbalance while drilling, m/Lt2, Pan � index, dimensionless

rD � dimensionless radiusrDmax � maximum dimensionless invasion radius

ri � invasion radius at time (t), L, m

rw � wellbore radius, L, mt � time, t, s

tD � dimensionless timeu � substituted variable, dimensionlessw � average hydraulic width of natural fractures, L, m� � dimensionless finite invasion factor� � time factor, t–1, s–1

�p � plastic viscosity of the mud, m/Lt, Pa·s�y � yield value of the mud, m/Lt2, Pa

References1. Liétard, O. et al.: “Fracture Width Logging While Drilling and Drilling

Mud/Loss-Circulation-Material Selection Guidelines in Naturally Frac-tured Reservoirs,” SPEDC (September 1999) 168.

2. Abramowitz and Stegun: Handbook of Mathematical Functions, Do-ver, New York City (1972) Chap. 4, 69.

SI Metric Conversion Factorscp × 1.0* E–03 � Pa·sft × 3.048* E–01 � min × 2.54* E–02 � m

lb/hundred ft2 × 0.478803 E–00 � Papsi × 6.894757 E+03 � Pa

*Conversion factor is exact.

(SPE 75283)

Author’s Reply to Discussion of FractureWidth Logging While Drilling and DrillingMud/Loss-Circulation-Material Selection

Guidelines in Naturally Fractured ReservoirsOlivier Lietard, SPE, Consultant, and Dominique Guillot, SPE, Schlumberger

We are extremely honored by the congratulations of SteveSawaryn. We appreciate his interest in our work and his efforts atexpanding it. Particularly encouraging is the fact that these com-ments come from a person involved in day-to-day operations, witha much better understanding of what is valuable to the field thananybody else in any organization of the oil industry.

There is not much to comment about concerning the math-ematical developments of our colleague. Mathematics are not ar-guable. His general solution to our differential equation is validand elegant as well. Thank you, Steve, for a job well done.

With this solution now available, we are sure that the wish youmade in the last sentence of your introduction will eventually befulfilled. Just as well testing moved from type curves to computers,your analytical breakthrough allows for an automatic treatment ofmud-loss data. This is a significant help for field engineers. We

share your hopes that our technique will become mere routine inthe near future, as soon as some other courageous people developand offer the necessary software.

The limitations of our approach must, however, be kept inmind. We are quite concerned by the deviations to our law thatshould occur in reservoirs with conjugated fractures and/or withfair primary porosity (allowing for leakoff and mud dehydration).Another problem might originate from slippage at the fracturewalls (particularly in dolomites). There is still a lot of work to do,and we invite other creative thinkers like Steve Sawaryn to tackleit. We will always be grateful to see our paper used as a startingpoint for further developments. This is called progress!

[email protected]@clamart.oilfield.slb.com(SPE 75284)

248 December 2002 SPE Drilling & Completion

A =4rDmax ln �rDmax�

� �ln �1 −1

rDmax� +

1

rDmax�

+4rDmax

� �n=2

1

n � 1

rDmax�n �ln�rDmax� +

1

n�. . . . . . . . . . . . . . . . (8)

Combining Eqs. 7 and 8 gives the final expression, Eq. 9,for dimensionless time as a function of the dimensionlessinvasion radius.

tD =4rDmax ln�rDmax�

� �ln � rDmax − 1

rDmax − rD� +

rD − 1

rDmax�

+4rDmax

� �n=2

1

n � 1

rDmax�n ��ln�rDmax� +

1

n�− rD

n �ln�rDmax

rD� +

1

n��. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

NomenclatureA � integration constant, dimensionless

�p� � overbalance while drilling, m/Lt2, Pan � index, dimensionless

rD � dimensionless radiusrDmax � maximum dimensionless invasion radius

ri � invasion radius at time (t), L, m

rw � wellbore radius, L, mt � time, t, s

tD � dimensionless timeu � substituted variable, dimensionlessw � average hydraulic width of natural fractures, L, m� � dimensionless finite invasion factor� � time factor, t–1, s–1

�p � plastic viscosity of the mud, m/Lt, Pa·s�y � yield value of the mud, m/Lt2, Pa



2. Abramowitz and Stegun: Handbook of Mathematical Functions, Do-ver, New York City (1972) Chap. 4, 69.

SI Metric Conversion Factorscp × 1.0* E–03 � Pa·sft × 3.048* E–01 � min × 2.54* E–02 � m

lb/hundred ft2 × 0.478803 E–00 � Papsi × 6.894757 E+03 � Pa


(SPE 75283)

Author’s Reply to Discussion of FractureWidth Logging While Drilling and DrillingMud/Loss-Circulation-Material Selection

Guidelines in Naturally Fractured ReservoirsOlivier Lietard, SPE, Consultant, and Dominique Guillot, SPE, Schlumberger

We are extremely honored by the congratulations of SteveSawaryn. We appreciate his interest in our work and his efforts atexpanding it. Particularly encouraging is the fact that these com-ments come from a person involved in day-to-day operations, witha much better understanding of what is valuable to the field thananybody else in any organization of the oil industry.

There is not much to comment about concerning the math-ematical developments of our colleague. Mathematics are not ar-guable. His general solution to our differential equation is validand elegant as well. Thank you, Steve, for a job well done.

With this solution now available, we are sure that the wish youmade in the last sentence of your introduction will eventually befulfilled. Just as well testing moved from type curves to computers,your analytical breakthrough allows for an automatic treatment ofmud-loss data. This is a significant help for field engineers. We

share your hopes that our technique will become mere routine inthe near future, as soon as some other courageous people developand offer the necessary software.

The limitations of our approach must, however, be kept inmind. We are quite concerned by the deviations to our law thatshould occur in reservoirs with conjugated fractures and/or withfair primary porosity (allowing for leakoff and mud dehydration).Another problem might originate from slippage at the fracturewalls (particularly in dolomites). There is still a lot of work to do,and we invite other creative thinkers like Steve Sawaryn to tackleit. We will always be grateful to see our paper used as a startingpoint for further developments. This is called progress!

[email protected]@clamart.oilfield.slb.com(SPE 75284)


Further Discussion of Fracture WidthLogging While Drilling and Drilling

Mud/Loss-Circulation-Material SelectionGuidelines in Naturally

Fractured ReservoirsFaruk Civan, SPE, and Maurice L. Rasmussen, U. of Oklahoma

The formulation of drilling mud losses into naturally fracturedformations by Liétard et al.1 and its analytical solution bySawaryn2 are instrumental for estimating the transient-state radialmud invasion in the near-wellbore region. In the following, a sim-pler analytical solution is presented that yields the same results asthose obtained by Sawaryn2 but is more practical for routinecalculations.

Analytical SolutionConsider the following ordinary differential equation, derived byLiétard et al.,1 expressing the dimensionless mud invasion radius,rD, as a function of dimensionless time tD.

dtD

drD=

4rD ln rD

1 − �D �rD − 1�, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

in which �D � a dimensionless mud invasion factor. The maxi-mum dimensionless mud invasion radius is given by1,2

rDmax = 1 +1��D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

Substituting Eq. 2 and integrating the right of Eq. 1 by parts withrespect to rD yield

tD = C + 4rDmax �rDmax − 1�

�− ln rD � rD

rDmax+ ln �1 −

rD

rDmax��

+rD

rDmax+ � 1

rDln �1 −

rD

rDmax� drD

� , . . . . . . . . . . . . . . (3)

in which C � an integration constant. When the series expansiongiven by

− ln�1 − z� = �n=1

� zn

n, − 1 � z � 1 . . . . . . . . . . . . . . . . . . . . . . . . (4)

is applied, in which z � a variable, the integral term in Eq. 3 canbe evaluated as

� 1

rDln �1 −

rD

rDmax�drD = − �

n=1

� 1

n2 � rD

rDmax�n

. . . . . . . . . . . . . . (5)

The initial condition is given by

rD = 1, tD = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)

Hence, substituting Eq. 5 and applying the initial condition, Eq. 6,in Eq. 3 yield the following analytical solution

tD = 4rDmax �rDmax − 1�

�− ln rD � rD

rDmax+ ln �1 −

rD

rDmax��

+ �n=2

� 1

n2 �� 1

rDmax�n

− � rD

rDmax�n�� . . . . . . . . . . . . . . . . (7)

The expression given by Eq. 7 is much simpler than the solutiongiven by Sawaryn.2 Also, the sign in front of the (rD−1)/rDmax termin Sawaryn’s2 solution should be minus.

Table 1 shows a comparison of the analytical and numericalsolutions for �D�0.04 along with the number of terms requiredfor evaluation of the series in Eq. 7 and Sawaryn’s2 solution for thesame analytical results. As the value of the dimensionless radiusapproaches the maximum dimensionless mud invasion radius,Sawaryn’s2 solution requires increasingly more terms than theCopyright © 2002 Society of Petroleum Engineers

249December 2002 SPE Drilling & Completion

present analytical solution for comparable results. While reproduc-ing exactly the same analytical results given by Sawaryn,2 Eq. 7here is more practical for routine calculations.

NomenclatureC � an integration constantn � index for series sum

nmax � maximum number of terms needed in seriesrD � dimensionless radius

rDmax � maximum dimensionless radius of mud invasiontD � dimensionless timez � variable

�D � dimensionless mud invasion factor



2. Sawaryn, S.J.: “Discussion of Fracture Width Logging While Drillingand Drilling Mud/Loss-Circulation-Material Selection Guidelines inNaturally Fractured Reservoirs,” SPEDC (December 2001) 268.

SI Metric Conversion Factorft × 3.048* E−01 � m


(SPE 81590)

Authors’ Reply to Further Discussion of“Fracture Width Logging While Drilling

and Drilling Mud/Loss-Circulation-MaterialSelection Guidelines in Naturally

Fractured Reservoirs”

Once again, we are extremely pleased at this contribution to thedevelopment of our work (SPE 57713). We hope that Mr. Sawarynalso appreciates the efforts of MM. Civan and Rasmussen at im-proving the numerical treatment of the equations that make thebasis our analysis of mud losses in naturally fractured reservoirs.Our thanks go to the three of them.

We would, however, like to reiterate our concerns about thelimits of our approach on the physical phenomena involved and tokeep encouraging interested colleagues to challenge it at least intwo directions.

• The description of mud losses in reservoirs with conjugatedfractures, when the mud invasion is deeper than the spacing of thefractures, which, without doubt, is the most frequent case.

• The effect of slippage at the wall for a mud flowing infractures—particularly in dolomites—at low macroscopicrates.

Although we are not involved in this area of research any longer,we look forward to reading studies done along these lines in thenear future.

Olivier Lietard, SPE, Consultant, and Dominique Guillot,SPE, Schlumberger

The simplification of the derivation and the final expression ofthe analytical solution presented by Civan and Rasmussen im-proves both the clarity and convergence of the resulting equationas well as its ease of implementation. I congratulate the authors ontheir work, and am pleased that further progress has been made inthis area. I am also grateful to them for pointing out the typo-graphical error in my earlier discussion paper.

Steve Sawaryn, SPE, BP plc(SPE 81591)


present analytical solution for comparable results. While reproduc-ing exactly the same analytical results given by Sawaryn,2 Eq. 7here is more practical for routine calculations.

NomenclatureC � an integration constantn � index for series sum

nmax � maximum number of terms needed in seriesrD � dimensionless radius

rDmax � maximum dimensionless radius of mud invasiontD � dimensionless timez � variable

�D � dimensionless mud invasion factor



2. Sawaryn, S.J.: “Discussion of Fracture Width Logging While Drillingand Drilling Mud/Loss-Circulation-Material Selection Guidelines inNaturally Fractured Reservoirs,” SPEDC (December 2001) 268.

SI Metric Conversion Factorft × 3.048* E−01 � m


(SPE 81590)

Authors’ Reply to Further Discussion of“Fracture Width Logging While Drilling

and Drilling Mud/Loss-Circulation-MaterialSelection Guidelines in Naturally

Fractured Reservoirs”

Once again, we are extremely pleased at this contribution to thedevelopment of our work (SPE 57713). We hope that Mr. Sawarynalso appreciates the efforts of MM. Civan and Rasmussen at im-proving the numerical treatment of the equations that make thebasis our analysis of mud losses in naturally fractured reservoirs.Our thanks go to the three of them.

We would, however, like to reiterate our concerns about thelimits of our approach on the physical phenomena involved and tokeep encouraging interested colleagues to challenge it at least intwo directions.

• The description of mud losses in reservoirs with conjugatedfractures, when the mud invasion is deeper than the spacing of thefractures, which, without doubt, is the most frequent case.

• The effect of slippage at the wall for a mud flowing infractures—particularly in dolomites—at low macroscopicrates.

Although we are not involved in this area of research any longer,we look forward to reading studies done along these lines in thenear future.

Olivier Lietard, SPE, Consultant, and Dominique Guillot,SPE, Schlumberger

The simplification of the derivation and the final expression ofthe analytical solution presented by Civan and Rasmussen im-proves both the clarity and convergence of the resulting equationas well as its ease of implementation. I congratulate the authors ontheir work, and am pleased that further progress has been made inthis area. I am also grateful to them for pointing out the typo-graphical error in my earlier discussion paper.

Steve Sawaryn, SPE, BP plc(SPE 81591)


www.spe.org/oil

00057713

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