SPESociety of PetroIetn1 Engineers

SPE 20409

Theory of Lost. Circulation PressureN. Morita, Conoco Inc.; A.D. Black, Drilling Research Laboratory; and G-F. Fuh, Conoco Inc.

SPE Members

Copyright 1990, Society of Petroleum Engineers Inc.

This paper was prepared for presentation at the 65th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers held in New Orleans, LA, September 23-26, 1990.

This paper was selected for presentation by an SPE Program Committee following review of informatio~ contained in an abstract submitted by the author(s). Contents of the paper,as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The .ma!enal, ~s presen!ed~ does not necessanly ref~ect

any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE ~eetings are subjec,t to publication review by Edl~onal Co~mltteesof the Societyof Petroleum Engineers. Permission to copy is restricted to an abstract of not more than 300 words. illustrations may not be copied. The abstract should contain conspicuous acknowledgmentof where and by whom the paper is presented. Write Publications Manager, SPE, P.O. Box 833836, Richardson, TX 75083-3836. Telex, 730989 SPEDAL.

ABSTRACT

Conventional theories predicting boreholebreakdown pressure assume breakdown occurs whenthe tangential stress at the borehole exceeds thetensile strength of the formation. Fracturingtests conducted during the DEA-13 joint industryproject, however, showed that when drilling fluidwas used as an injection fluid, boreholebreakdown did not occur until the well pressuresignificantly exceeded the pressure whichresulted in a tangential stress equal to the rocktensile strength even with a large surface flaw.The test results have shown that all drillingmuds have a tendency to seal narrow naturalfractures or fractures created by high boreholepressure. The sealing effect of the mudstabilizes fractures and prevents fracturepropagation. This effect is one of the primaryfactors for controlling wellbore stability. Inthis paper, a theory of fracture initiation andfracture propagation around a borehole whosestability is enhanced by drilling fluidinteraction, has been developed and shown to beconsistent not only with all the DEA-13laboratory results, but also with various fieldevidence. The results show that lost circulationpressure is highly dependent on the Young'smodulus of the formation, wellbore size, and typeof the drilling fluid, although the conventionaltheories have ignored these facts.

INTRODUCTION

Since Anthony Lucas introduced drilling mud as acirculation fluid to remove drill cuttings andcool the drill bit, millions of wells have beensuccessfully drilled. Despite the inherent natureof formation strength heterogeneity, irregularityof boreholes and high borehole pressurefluctuations from surging, many wells have beendrilled with relatively few problems. Such high

References and illustrations at end of paper.

43

stability of boreholes cannot be explained by anyconventional theories (Ref.2 to 6) which estimateborehole stability .from only the stress statearound a borehole;

Two roles of drilling fluid for stabilizing theborehole are well known: (a) supporting theborehole wall with hydrostatic well columnpressure; and (b) laying down low-permeable caketo prevent pore pressure build-up and maintainhigh effective stress on the formation. Theseroles have been introduced in conventionaltheories. Any continuum theories developed inthe past calculate stress state or effectivestress state at the borehole surface and judgethe lost circulation pressure based upon whetherthe effective stress exceeds tensile rockstrength (zero tensile strength if existence ofcracks is assumed). Because conventionaltheories are based upon only stress concentrationaround a borehole, they cannot explain commonlyobserved field phenomena such as borehole sizeeffect, elastic modulus effect, and effect of mudtype upon lost circulation pressure.

Drilling fluids contain solids which form bridgesin the fracture aperture. Minute cracks areplugged by solids contained in drilling fluids.Although these functions of drilling fluids arewell known, they have never been introduced intotheory.

DEA-13 experiments (Ref.l) demonstrated that,before borehole breakdown occurs, a stablefracture develops. The fracture aperture issealed by drilling fluid solid bridging over thefracture inlet. The borehole breakdown thenoccurs when the drilling fluid begins to enterinto these fractures.

Another phenomenon found in these tests was thata narrow fracture tip zone exists which cannotallow drilling fluid invasion. Although theexistence of such a zone is well-known, it hasnever been empirically quantified. Sandia's

2 THEORY OF LOST CmCULATIDN PRESSURESPE 20409

The tests simulate the field conditions exceptfor the following items:

ppg and 16 ppgwere used. Water and oil were alsoused for pre-fracturing.

The difference between the laboratory conditionsand field conditions requires a new theory tointerpret the laboratory results before applyingthem to field problems.

A. The borehole diameter is 1.5 inches, which issignificantly smaller than real borehole sizes.B. The samples used for the tests were Bereasandstones. They represent intermediate strengthformations, but they don't represent soft ortight formations.C. Permeability of Berea sandstone is relativelyhigh (100 to 120 md).D: Down hole temperature was not simulated.

pressurefracturepsi, butinto the

the specificrevealed the

The laboratory results, withconditions indicated above,following facts:

A. The borehole breakdown and extension pressureswere abnormally high, as shown in Figs.2 and 3where WB, MOB and DOB denote the water, mineraloil and diesel oil base muds, respectively. Suchhigh pressures are not observed in the field.B. The borehole breakdown pressure varieddepending on confining pressure, but the threebase muds (water, diesel, and mineral oil basemuds) gave approximately the same breakdownpressures.C. The plots for low peak fracture extensionpressure versus fracture volume were similar forthe three base muds. Note that the peak pressuresare the high and low values of the fluctuatingfracture extension pressure as shown in Fig. 2.However, the water base mud had high peakfracture extension pressure which was 700 - 1,500psi higher than low peak pressure for a.Hl ­-1,800 psi and a

H2- -2,200 psi. The dieseI- and

mineral oil base muds had high peak fractureextension pressure, which is very close to thelow peak pressure as shown in Fig.2.D. Split rock samples showed that a fracture tipzone exists which was too narrow to allowdrilling fluid invasion. Consequently, thefracture tip is always several inches ahead ofthe drilling fluid front during boreholebreakdown and fracture extension (see Figs.4, 5,6 and 7).E. Fig.8 shows the fracture extensionand the status of the borehole. Theinitiation occurs between 5000 to 6000the drilling fluid cannot penetrate

The experiments typically consisted of fourinj ections: (a) the first inj ection to achievewellbore breakdown; (b) the second injection toreopen the fracture; (c) the third injection totest the healing effect of drilling fluids afterwaiting for one hour and (d) the final injectionto create a new fracture perpendicular to theoriginal crack by altering the directionalconfining pressures. The typical injection rateswas .5 cc/sec. For some tests, the borehole ispre-notched or pre-fractured with water or oil tosee the effect of the borehole flaws.

BRIEF DESCRIPTION OF THE DEA-13 EXPERIMENTALRESULTS

The most puzzling .result from the DEA-13experiments was that laboratory tests with1.5-inch borehole diameter gave abnormally largeborehole breakdown pressures (exceeding 9 kpsiwith 2 k confining pressure) even if theboreholes were prefractured hydraulically ormechanically (Note that the pre-cracked boreholehas no tensile strength). Conventionally, thedifference in borehole breakdown pressuresbetween laboratory versus field findings wasexplained by the existence of surface flaws forfield scale boreholes. Such a possibility waseliminated by introducing large surface flaws forthe laboratory scale tests. The above empiricalfacts led to treating borehole breakdown problemswith a new stable fracture extension theoryinstead of conventional continuum theory.

large scale fracture experiment showed such azone extends several feet in front of thefracturing fluid front with very low in-situstress. There are several hydraulic fracturetheories to explain the fracture tip zone, butall of them treat it with flow friction theory.In addition, the DEA-13 experiments showed thatthere exists a dehydrated mud zone behind theabove-mentioned fracture tip zone. This zone canalso seal fracture pressure although widening thefracture width can break the dehydrated mud plug.The size of fracture tip filled with dehydratedmud depends on spurt loss and filtrate loss,hence, depends on mud type.

The theory verified by the DEA-13 experimentsrevealed that lost circulation pressure dependson Young's modulus, wellbore size, mud solidbridging at the narrow fracture tip and degree ofdehydration, as well as the parameters which canbe derived from conventional theories such asstress, pore pressure, temperature, boreholeinclination, and pore pressure build-up.

The laboratory tests for DEA-13 fracturepropagation pressure measurements were conductedto simulate the real in-situ conditions. Sinceanother paper will be published by Alan D. Black(Ref .1) on the laboratory results, only a briefdesctription is given in this section.

The laboratory tests were conducted with cubicBerea sandstone samples approximately 30x30x30inches. A borehole 1.5 inches diameter wasdrilled at the center, had casings glued at thetop and the bottom one-third with packers to sealthe borehole pressure. The confining pressureswere kept at 2200 psi and 1800 psi in thehorizontal directions and 3000 psi in thevertical direction during the first threeinjection tests, but it was changed to 300 psiand 2500 psi in the horizontal directions in thefourth inj ection test for each sample. The porepressure was atmospheric pressure. The Bereasandstone had approximately 100 to 120 mdpermeability, .19 porosity, 520 psi.in**.5fracture toughness, and the approximate stressstrain curves are given in Fig.l. Water base,mineral oil base and diesel oil base muds with 10

44

SPE 20409 N. MORITA, A.D. BLACK AND G.F.FUH 3

There are six factors which disturb the stressstate around a borehole. They are:

This type of lost circulation problem can becontrolled only by the lost circulation materialswith large particles or cement.

Field evidence shows that a long interval canoften be drilled with wellbore pressure above thepore pressure gradient. It suggests that the lostcirculation problems do not necessarily associatewith the highly fractured formations or vuggyzones. They can be induced at the zones with theabove conditions (a) to (c). A theory of lostcirculation pressure is developed for these threecases in this section.

As explained in a previous section, the mostimportant discovery for the DEA-13 fracturepropagation experiments was that the boreholebreakdown did not occur at the onset of afracture initiation at a borehole surface. Thedrilling fluid cannot penetrate into a fractureuntil the fracture aperture exceeds a criticalwidth. In this paper the borehole stabilityproblem is treated exactly by simulating thephenomena observed in the laboratory tests.

Eq.lPw <!:: Po

Figures in Tables 1 and 2 show the boundaryconditions. The well and part of the fracture (upto the fluid front) have pressure Pw and thepressure between the fluid front and the fracturefront is set to equal to the pore pressure.Inclination of borehole, temperature, and porepressure effect are also taken into account.Tables 1 and 2 also show base solutions of theborehole breakdown pressure and fracture aperturefor those fractures started from a smoothborehole and a borehole with a mud invaded crack,respectively. The approximate solutions wereobtained analytically assuming a fracture withouta borehole, then, the correction factor iscalculated with a finite element code to takeinto account the borehole effect. Tables 3 and 4show a set of equations to calculate the boreholebreakdown pressure and fracture apertureinclUding the temperature, pore pressure andborehole inclination. These terms are added tothe base solutions shown in Tables 1 and 2 byconsidering these stress disturbances as thereaction forces. The detailed description of thetheory is given in the Appendix.

fracture. A stable fracture grows until thefracture aperture becomes wide enough to allowthe drilling fluid entry at the peak pressure "A"in Fig.SA. After the borehole breakdown, thestable fracture propagates repeating the fractureinflation and propagation.F. Fracture reopening pressure were abnormallyhigh for those boreholes prefractured without mudfluid invasion (Fig.9). Fracture reopeningpressures were also high for those prefractureswhich had been previously invaded by drillingfluids. On the average, the 16 ppg water-base mudhad higher fracture reopening pressure than oilbase muds.The high fracture reopening pressures wereobserved instantly after fracture closure withouttime lapse, which means that the drilling fluidacts as a sealant without consolidation due todehydration. Fig.9 shows the degree of thesealing for various muds. A thin drilling fluidseals the fracture as shown in Fig.lO.D; whilethe muds forming a thick cake, seal the fractureas shown in Fig.lO.B or Fig.lO.C.G. The fracture width at the point where fracturepressure is sealed due to solid bridging (Wt inFig.S) varies between .01 and .01S inch, as shownin Fig.7.H. The fracture width at the point where fracturepressure is sealed due to dehydrated mud (Wm inFig.S) varies between .01 to .02S inch, dependingon the drilling base fluid and fracture size.High density water base mud has a large fracturewidth at the sealing point, while mineral- ordiesel-base mud has a small width.

3. THEORY OF LOST CIRCULATION PRESSURE

By introducing a stable fracture growth with highfracture extension pressure, all the laboratoryresults can be explained. Such a theory can alsoexplain the difference between field observedphenomena and the laboratory results.

The experiments conducted for this projectquantified two approximate fracture widths: onewhich absolutely does not allow drilling fluidentry (Wt in Fig.S) and one which increasesdepending upon the dehydration process (Wm inFig. S). The fracture pressure beyond these twosealing points (Wm for dehydrated muds or Wt withtrivial mud dehydration) is isolated from thepressure within the rest of the fracture. Becausethe fracturing pressure cannot reach beyond thesealing points, fracture extension becomessignificantly stable.

One of the important parameters affecting lostcirculation problems is the surface flaw. Foursurface conditions are observed in a borehole.(a) Flawless surface : the borehole surface issmoothly drilled without flaw.(b) Borehole with a closed fracture: the boreholeis intersected by a closed fracture.(c) Irregular borehole: the borehole surface hasirregularity due to a small open fracture, vuggsor drilling actions.(d) Highly fractured formations, vuggy zones orgranular zones.

The lost circulation for the case (d) is inducedif the wellbore pressure exceeds pore pressure:

A. Irregularity of borehole shapeB. Temperature coolingC. Inclined wellD. Pore pressure build-upE. Nonlinear rock deformationF. Shale swelling

Since shale swelling normally increases lostcirculation pressure by increasing the stressstate around a borehole, only the effect of (A)to (E) are discussed. Borehole shape may beelliptical, but the most irregular form ofborehole shape is a borehole with a notch (thenotch represents a natural fracture or a vuggypore). Hence, the irregularity of borehole shape

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4 THEORY. OF LOST CIRCULATION PRESSURE SPE 20409

is treated in this work as a fracture propagationfrom an existing fracture.

Fig.II shows the prediction of the theory on thetypical fracture extension pressure from aborehole. Note that if a wide fracture existsaround a borehole, the fracture extensionpressure should start from a certain fracturelength B. If the fracture toughness is large,there may initially be a small decline in thefracture extension pressure as shown in Fig.ll,although most sandstones do not have a largefracture toughness. As the fracture enlarges, thefracture extension pressure increases. Fractureextension pressure normally starts decliningafter a peak pressure if the fracture aperturebecomes wide enough to allow drilling fluid entry(Note: B-rw for no drilling fluid entry intofracture). Hence, for the normal case, thefracture extension pressure which allows drillingfluid to invade into a fracture coincides withborehole breakdown pressure. However, if stressdisturbance is significant, borehole breakdowndoes not occur due to further stable fractureextension even after drilling fluid entry. Asshown in Fig.ll, the peak fracture extensionpressure is lowered in accordance with themagnitude of stress disturbance around aborehole. The discussion of factors affecting thepeak fracture extension pressure (boreholebreakdown pressure) follows.

Factors Affecting Borehole Breakdown PressureWithout Artificial Stress Disturbance

A. In-situ Stress

Fig.12 shows the in-situ stress effect. It is aprimary factor affecting borehole breakdownpressure. The new fracture theory developed inthe appendix predicts significantly high boreholestability at high in-situ stress if the boreholesize is small.

If a crack with width, Wo, exists around aborehole, the pre-existing fracture width shouldbe subtracted when using Fig.13. For instance,suppose the width allowing drilling fluid entryinto fracture is .01 inch, the prefracture widthis .009 inch, and E/(1-v 2 ) -l.E6, then a curvefor WmE/(1-v2 )-(.01-.009)xl.E6-l.E3 in.psishould is used. The result shows that preexistingfracture width is an important factor forbreakdown pressure.

E. Nonlinear Deformation of Rock

There are three non-linear deformations oftenobserved for sedimentary rocks. They are Mode Inonlinearity (crack closure or Mode I microcrackgrowth), shear type plasticity, and plasticitydue to pore volume crushing. The Mode Inonlinearity has a similar effect as macrocrackgrowth, hence, it can be treated approximately asa part of a macrocrack growth problem. Plasticnonlinearity occurs if a rock is soft relative toin-situ stress. Fig.14 shows the effect ofplasticity for a relatively weak rock. Thetangential stress may become compressive due toplasticity if borehole pressure is too high. Forsome stress combinations, tangential stressremains compressive without opening a fracture.Such plastic effects increase borehole breakdownpressure.

Table 5 summarizes the well pressure inducingfluid invasion into the fracture. A failurelimit is imposed to plastic strain to those caseswhere tangential stress never becomes tension,assuming nonuniform deformation destabilizes theborehole if plastic strain exceeds a certainvalue. The table compares the nonlinear fracturetheory with the conventional linear elasticitytheory. The nonlinear theory consistentlypredicts a higher fluid invasion pressure intothe fracture.

B. Borehole SizeFactors Affecting BreakdownArtificial Stress Disturbance.

Pressure with

Fig.12 shows the borehole size effect. A slimhole may be several hundred psi stronger for lostcirculation pressure than a large hole. This sizeeffect predicted from the new theory matches thefield observation, while the conventionalcontinuum theory predicts a constant boreholebreakdown pressure regardless of the boreholesize.

C. Modulus Effect

Fig.13 shows the modulus effect. Note thatfracture width, Wm, which allows drilling fluidentry is reasonably constant (.01 to .015 inch),according to the laboratory results, before thewidth Wm grows with fracture extension. Hence,formation Young's modulus primarily affects thefracture initiation pressure. Note that theconventional linear continuum theory does notinclude Young's modulus in the equation of theborehole breakdown pressure.

D. Effect of Width of Pre-existing Fracture

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F. Irregular Borehole Shape

The most serious form of irregular boreholeshape is a wide crack around a borehole. Figs.lsand 16 show the effect of a wide crack uponbreakdown pressure. The borehole breakdownpressure is significantly reduced by wide cracks,but even with a relatively large crack, boreholebreakdown is still significantly higher than thein-situ stress if in-situ stress is high.

G. Thermal Cooling Effect

Because the thermal expansion coefficient isrelatively large, borehole cooling lowers lostcirculation pressure. The borehole breakdownpressure reduces with time after a formation isexposed to a cool drilling fluid. Figs. 17 and 18show the effect of thermal cooling. The effect ishigh for a tight rock, but since a tight rock hasa large fracture propagation resistance for asmall fracture, the temperature effect is offsetwhen a crack size is very small.

H. Effect of Borehole Inclination

SPE 20409 N. Morita, A.D. BLACK & G.F. FUH 5

Borehole inclination enhances the nonuniformstress field around a borehole, but thenonuniform stress field quickly diminishes withdistance. Hence stable fracture theory predictsless sensitivity to borehole angle. Fig.19 showsthe effect of borehole inclination. The wellinclination effect is non-trivial for a formationwith a low Young's modulus. However, once astable crack starts extending, borehole angleeffect is quickly reduced (Figs.20 and 21).

I. Pore Pressure Build-up Effect

Pore pressure build-up occurs if formationpermeability becomes small compared with drillingfluid cake permeability. Pore pressure build-upincreases total stress, which results inpreventing fracture propagation. However, if atight formation allows pressure build-up at thefracture tip, it offsets the higher in-situstress and reduces fracture propagation pressure.Fig.22 shows the pore pressure build-up effectfor a formation with a relatively smallpermeability (not extremely small though) .Fracture extension pressure is reduced by thehigher pore pressure when the fracture size issmall. However, the higher total stress offsetsthe reduced tip pressure, which results in higherfracture propagation pressure. It is emphasizedthat pressure build-up at the fracture tipsignificantly reduces borehole breakdown pressureand fracture extension pressure for animpermeable formation since the pressure at thefracture tip is always equal to borehole pressurewhen a formation has very low permeability.

Other Factors Affecting Breakdown Pressure

J. Dehydration Effect

Dehydrated mud plugs fracture tip with a widerfracture width, Wm, to prevent fracture pressurefrom transmitting to fracture tip. As fracturelength becomes larger, the width, Wm, increasesfrom .015 inch to a larger width. Fig.13 and 15show that a wider width gives a larger fractureextension pressure, especially for a smallfracture.

K. Fracture Reopening

After a casing shoe test, a fracture containingdehydrated drilling fluid exists at the casingshoe. After some pressure surges, a fracture maybe created with dehydrated mud within thefracture. The pressure to reopen such fracturesvaries depending on the sealing capability of thedehydrated mud as shown in Fig.9 (see the secondand third injection tests). The mud cakethickness, which can seal the fracture without anunstable fracture extension, varies dependingupon the mud type previously invaded into thefracture. DEA-13 experiments showed that formineral oil- or diesel-base muds, the mudthickness is close to the grain size or slightlyless as shown in Figs .10-A or 10-D. Hence, anadditional compressive load is required to seal afracture aperture. The water base mud has a thickcake as shown in Fig.lO-B or 10-C, resulting in a

47

higher borehole reopening pressure as shown inFig.9

CALCULATION PROCEDURE AND EXAMPLE CALCULATIONS

The equations of well pressure and fracture widthshown in Tables 3 and 4 (Table 3 for B-rw andTable 4 for B/rw.GT.l.33) givePw-A/rw-WmE/(1-v 2 ) relations as shown in Fig.23for various temperature, pore pressure andpre-crack sizes. If the critical fractureaperture Wm (or WmE/(1-v 2 )) with which a drillingfluid can invade into a fracture is given, theborehole breakdown pressure can be calculated.Since the fracture width is given as a functionof A/rw, a numerical iterative method is requiredto find A/rw from Wm if the calculation iscarried out with a computer program instead ofthe graphical method shown in Fig. 23. Inclinedwells and thermally cooled wells have asignificantly low tangential stress around aborehole. A very stable fracture grows without alost circulation. For these stable fracturegrowth problems, the equations for notchedborehole shown in Table 4 must be checked even ifthe borehole surface is initially smooth. The~aximum pressure found by varying B (the mud~nvaded crack radius) is the lost circulationpressure.

The equations predict the borehole breakdownpressure accurately if the correct input data aregiven. One problem of these equations is thatthey do not take into account the nonlinearbehavior of rock. Generally the nonlinearbehavior is negligible during the tensilefailure, however, as explained for the weak rockbehavior, the nonlinearity may affect thebore~ole b.reakdown significantly even during atens~le fa~lure if the porosity of rock exceeds30 percent.

For solid free drilling fluids, the stablefra:ture . growth is dominantly affected by theflu~d pressure distribution within the fracture.These equations give only a crude approximationfor these special problems. The following inputdata are required to calculate the lostcirculation pressure:

a. Well radius rw and well inclinationb. The radius B: rw plus the surface crack

lengthc. In-situ stress: vertical and horizontal

(negative sign for compression)d. Young's modulus, Poison's ratio,rock grain

compressibility: Note that the Young'smodulus to open a fracture is about half ofthe Young's modulus measured using acylindrical sample due to the nonlinearity toopen a fracture. Hence if the Young's modulusis measured by opening a fracture, themeasured value is directly used as an inputparameter; but if it is measured with acylindrical sample, the Young's modulus mustbe halved for input data.

e. Pore pressure at the borehole surface andpressure gradient: these values must becalculated at time t using the equation givenin the appendix.

6 THEORY OF LOST CIRCULATION PRESSURE SPE 20409

f. Temperature at the borehole surface andtemperature gradient: these values must becalculated at time t using the equation givenin appendix.

g. The critical fracture width Wm which allowsmud fluid invasion into the fracture tip: Wmis between .01 to .015 inch for standarddrilling fluid. Wm remains approximatelyconstant if the API filtrate loss is lessthan .5cc. However for high fluid loss mud Wmgrows with fracture length. For example Wm vsRf for the 16 ppg water base mud used in thiswork is given by Wm-.Ol"+.002xRf. For astable fracture growth problem (inclined wellor thermally cooled well), Wm must beincreased with the fracture length.

Table 6 shows the example calculations usingrealistic borehole data for a flawless borehole.Since the borehole is assumed to be flawless, theprediction matches the borehole breakdownpressure during a casing shoe test. Two boreholebreakdown pressures are calculated in the table.PBW denotes the pressure initiating drillingfluid invasion into a fracture and PB denotes thepressure inducing lost circulation. Note that aborehole breakdown induces a fracture, but it maystop propagating if it is stable.

Table 6 shows that the lost circulation pressurefor a flawless borehole normally exceeds thein-situ stress only by 100 to 500 psi if theeffective in-situ stress (in-situ stress minuspore pressure) is 500 psi since the wellboreconditions are usually hostile as shown in Table6. Note Wm is .01 to .015" for standard drillingfluids for flawless borehole surface, while Wm-WOmust be used instead of Wm for a cracked boreholewhere Wm is the critical fracture width whichallows the mud invasion and Wo is the crack widthexisting in a formation. The lost circulationpressure increases with the effective in-situstress, but it still ranges only 500 to 1000 psiover the in-situ stress for a flawless surfacewith uH--1000 psi and uV--2000 psi.

Lost circulation normally occurs at theborehole with cracks. Fig.20 shows that if theYoung's modulus is relatively high, the lostcirculation pressure is still 500 psi to 1000 psiabove the in-situ stress even if the boreholesurface is not smooth. Since there are variousfactors involved in the field lost circulationproblems, the detailed discussion of the fieldproblems will be published in another paper.

CONCLUSIONS

1. Two roles of drilling fluid for stabilizingborehole have been well known: (a) supportingborehole wall with hydrostatic well columnpressure; and (b) low permeable cakepreventing pore pressure build-up andmaintenance of a high effective stress on theformation. In addition to these well-knowneffects, this paper clarified that drillingfluid acts as a good sealant for minutecracks and narrow fracture tip. A theory andset of equations are developed to take intoaccount all of the above three effects.

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2. The conventional theories predict that a lostcirculation occurs when the tangential stressat the borehole surface exceeds the tensilestrength (normally assumed to be zero to takeinto account the surface flaw) of rock, whilethe new theory predicts a stable fracturegrows if the drilling muds contain solid andgel components and lost circulation occurswhen the fracture becomes unstable. Becausethe above assumption used by the conventionaltheory is contradictory to the laboratoryobservations, the conventional theory cannotexplain the various aspects of laboratory andfield test results.

3. The new theory predicts that the lostcirculation pressure depends on formationYoung's modulus, borehole size, existingcrack size, the width of the closed fractureand the type of drilling fluids, as well asall other parameters involved in theconventional theory. These effects wereobserved in the field and in the laboratorytests although the conventional theories haveignored them.

4. Thermal cooling, hole inclination and porepressure build-up affect lost circulationpressure, but do not affect it as signifi­cantly as conventional theories predict.Thermal cooling lowers lost circulationpressure for formations with a high Young'smodulus. It takes several hours to losecirculation after drilling through suchformations since rock is a good thermalinsulator. Well angle lowers lostcirculation pressure for formations with lowYoung's modulus. Pore pressure build-uplowers lost circulation pressure forformations with low permeability.

5. Predicting lost circulation pressure requiresaccurate information on temperaturedistribution around a borehole, the Youngsmodulus, ~ore pressure, in-situ stress, wellangle and the existing crack dimension arounda well.

NOMENCLATURES

A well radius plus fracture lengthB well radius plus the fracture interval

invaded with drilling fluidc fluid compressibilityC specific heatc i ' cb - rock matrix and bulk compressibilitiesE,v - elastic moduliFl,F2,F3 - constants given in Table 1Gl,G2,G3 - constants given in Table 1Hl,H2,H3 H4 - constants given in Table 2K thermal diffusion coefficientKc - fracture toghnessKr, K

T- permeability, thermal diffusivity

P, Pw - pressure, well pressurePf pressure within a fracturePow pore pressure at the borehole surfaceq pore pressure gradient around a boreholeqT temperature gradient around a boreholer radial coordinater D r/rwrw well radiusRf fracture radius up to the pressure sealing

pointRm the length of dehydrated mud around the

SPE 20409 N. MORITA, A.D.BLACK & G.F.FUH 7

fracture tipRt length of the fracture tip without mud

invasionS,Sl defined in Table 2t timet

d_ dimensionless time defined in Appendix

T~T - temperature, temperature at wellboreW w_ fracture widthWl,W2,W3,W4,Wp - fracture widths corresponding to

the hydrostatic boundary stress,directional stress, borehole pressure,fracture surface pressure and porepressure, respectively.

Wm fracture width at the sealing point withmud cake

Wt fracture width at the point which does notallow the mud invasion

WO fracture width of a pre-existing fractureao defined in Table 3a thermal expansion coeficientfi,l - rw/A, well angle

U radial stressrun tangential stress

U axial stressu~l,uH2 - in-situ stresses(negative)uH,u

V- horizontal and vertical in-situ stress

~;~ :H2JiJ~~2B/Ap,~ - fluid density, viscosityif; porosity

REFERENCES

1. Black, A.D. "Investigation of Lost CirculationProblems with Oil Base Drilling Fluids" DRLreports: Phase I-May 1986.

2. Aanoy, B.S. et. a1. "Method forFracture-Gradient Prediction for Vertical andInclined Boreholes" SPE Transactions Vol.287,1989, pp 99-103.

3. Aanoy, B.S. "Modeling of the Stability ofHighly Inclined Boreholes in Anisotropic RockFormations", SPEDE Sept.88, pp 259-268

4. Bradlly, W.B. "Failure of Inclined Boreholes",J. Energy Resources Technol., Trans.,ASME(Dec.1979) vol.10l,pp 232-239

5. Giin-Fa Fuh "Use of Borehole StabilityAnalysis for Successful Drilling of High AngleHoles" SPE 17235, 1988.

6. Yew, C.H., and Li, T. "Fracturing of aDeviated Well" SPE Transactions Vol. 285, 1988,pp 429-437.

ACKNOWLEDGEMENTS

The authors thank all the DEA-13 participantsfor inputting valuable knowledge and ideas intothe fracture initiation and propagationexperiments using the water, diesel oil andmineral oil base drilling fluids.

APPENDIX NEW THEORY TO PREDICT LOST CIRCULATIONPRESSURE

surface. Two phenomena should proceed before afracture starts allowing drilling fluid entry.They are growth of micro-cracks existing in rocksand stable macro-crack growth. To break down aborehole, borehole pressure should exceedpressures to extend both existing micro-cracksand a stable macro-crack.

A linear theory gives Eq. a-1.1 (Table 1) toextend a micro-crack existing around a borehole.If fracture length and toughness are uniform,Equation a-l.l is independent of wellbore radiusand agrees with fracture initiation pressurederived from linear continuum mechanics.However, they are actually nonuniform and alarger borehole has a higher probability ofcontaining larger cracks with a weaker fracturetoughness.

As a fracture becomes a macro-fracture, therelations between fracture extension pressureversus relative crack size (A/r ), and fracturewidth versus relative crack si~e are given byEqs.a-l.l to a-l.6. The closed form of theequations are derived assuming no borehole, then,the correction factors Fl to F3 are calculated totake into account the borehole effect using afinite element code.

A.2 Borehole Breakdown Pressure from ExistingCracks

Cracks with various sizes and widths exist informations. When a borehole is drilled throughthem, some cracks are wide enough to allowdrilling fluid to penetrate, while other cracksare narrow enough to prevent drilling fluidentry. Since the stability of a boreholedistinctly differs depending on the area on whichborehole pressure acts, the effects of the twotypes of pre-existing cracks on wellborestability are discussed independently in thefollowing section.

Narrow Pre-existing Cracks

Cracks created by thermal or tectonic forces havea width narrower than grain size. They arenarrow enough not to allow drilling fluidinvasion. However since they do not have anyfracture toughness, drilling fluid can penetrateinto a fracture with a smaller borehole pressurethan for a borehole without any preexistingfracture. Hence, the P - A/r - W relationgiven in Table 1 should b~ modifi:d.

The crack remains open under in-situ stress dueto grain mismatching between upper and lowerfracture surfaces. The fracture width becomeslarge if borehole pressure is increased. Let Abe the length where the fracture width isaffected by borehole pressure change. Then therelations between P versus A/r and W versusA/rw are given by w w

A.l Borehole Breakdown Pressure A - r w

A sudden borehole pressure decline occurs ifdrilling fluid starts entering into a fractureand fluid pressure starts acting on the fracture

49

Pw

I 2JA/1r, fi - rw/A

[- uH1

I{1r/2 + fijl-fi2 - ArcsinfilFl

8 THEORY OF LOST CIRCULATION PRESSURE SPE 20409

Wide Cracks

A.3 Fracture Reopening Pressure

For an inclined well, a stable fracturepropagates after fracture initiation. Thefracture growth is stable until fracture aperturebecomes wide enough to allow drilling fluidentry. For a stable fracture growth around aborehole, the assumption of nondirectiona1horizontal in-situ stress (G

H1G

H2)

significantly simplifies the problems.

A. Fracture initiated at the minimum tangentialstress direction propagates as an in-planefracture.B. Numerical experiments showed that out-of-p1anestresses did not affect stress intensity factorand fracture width if fracture initiated at theminimum tangential direction and fracture surfaceremained perpendicular to minimum stress.

base muds and all the 10 ppg muds could not seala fracture gap which had been partially inducedby grain mismatching at fracture surface andembeddment of hard components contained indrilling solids.

The above two facts simplify the calculations ofaperture and stress intensity factor for a stablefracture propagating from a borehole since theydepend upon O"n1y in-plane, in-situ stressperpendicular to a borehole axis. Let G

H1and

GH2 be in-plane, in-situ stress given by

Equations B-3 and B-4 give the most pessimisticfracture reopening pressures for drilling fluidswith and without sealing capability,respectively. However, if taking the averagevalue of the fracture reopening pressures, theDRL experiments showed that for muds with perfectsealing capability, fracture reopening did notoccur until fracture aperture becameapproximately .004 inch. For fluids which actedas leaky sealants, fracture reopening did occurat a pressure less than the minimum in-situstress but the leak stopped with the fractureinflation. The fracture inflation occured untilthe fracture pressure became high enough torestart fracture propagation.

A.4 Theory of Lost Circulation for an InclinedHole

GH1 - GH

, GH2

- cos2~ GH

+ sin2~ GV

(B-S)

Then all the equations (Equations in Tables 1 to4) developed for a vertical well can be applied.However, because the two principal in-situstresses perpendicular to a borehole may besignificantly different for a highly inclinedwell, the borehole breakdown may not occur at thepressure when drilling fluid starts penetratinginto a fracture. A stable fracture may growwithin a highly stress concentrated area around awell before it becomes unstable.

Since thermal expansion coefficient of rock isrelatively large, temperature affects boreholebreakdown pressure, fracture reopening pressure,and fracture extension pressure. However the

A.S Temperature Effect Upon Borehole BreakdownPressure. Fracture Reopening Pressure. andFracture Extension Pressure

(B-4)Pw

The DRL experiments showed that the dehydratedmud embedded between fracture surfaces should besufficiently thick and strong to act as a perfectsealant. The test samples which were split apartshowed that the 16 ppg water-base mud and LVT oilbase mud formed a mud cake thick enough to coverthe rough fracture surface. Some mud cakesqueezed out into the borehole from the fractureindicated that it can seal fractures even iflarger particles are embedded between surfacesand widen the preexisting fracture aperture. Onthe other hand, the 16 ppg mineral and diesel oil

For a drilling fluid with perfect sealingcapabil i ty :

For a drilling fluid without perfect sealingcapability:

Cracks created by a dissolution process arenormally vuggy and wide. Drilling fluid canenter these cracks if a borehole intersects them.A borehole becomes unstable if such wide cracksintersect it since a wider pressure boundary caneasily induce cracks from existing wide cracks.Approximate P - A/B - W correlations for a crackcreated at a ;ide crack tip are given by the Eqs.a-2.1 to a-2.8 in Table 2. The closed form ofequations is first derived assuming no borehole,then, the· correction factors H1 to H4 arecalculated to take into account the boreholeeffect using a finite element code.

A drilling fluid which acts as a perfect sealantafter dehydration does not leak into apre-existing fracture if we11bore pressure isless than the well pressure giving zero hoopstress or the in-situ stress, whichever islarger. On the other hand, a drilling fluidwhich ac ts as a leaky sealant (1. e., a sealantwhich can gradually build up pressure in afracture) leaks into a fracture at a pressureless than the minimum in-situ stress but the leakstops if the we11bore pressure is less than thein-situ stress. Hence the following equationsgive the most pessimistic fluid loss pressureinto a preexisting fracture previously invaded bya drilling fluid.

W _ W 4(1-v 2) [A_ G1+A( ) G2o + E nuH1 GH1 -GH2

2R P+ -Yff..J!. (G3+1n(A/r » 1 (B-2)

where F1 to F3 and G1 t~ G3 are givenWin Table 1.The above equations show that the zero fracturetoughness and pre-existing fracture widthsignificantly reduce the we11bore breakdownpressure.

50

SPE 20409 N. MORITA, A.D.BLACK & G.F.FUH 9

effect is localized within several feet around aborehole since rock is a good thermal insulator.

The solution of Equation B-6 with boundarycondition (B-7) is given by (conduction of heatin solids, Carslaw and Jaeger, p. 336)

Suppose a wellbore surface cools down due todrilling fluid circulation and is kept at aconstant temperature Tw' or

rD

> eTJqT

(B-ll)surface is

--Lu(J - - I-v aTw (B-12)

to the continuum theory, boreholepressure should be reduced byif the well temperature declines by

The tcingential pressure at boreholegiven by

Eau(J - 4(1-v) [(qT-2Tw)-(2Tw+qT)rD2+2qT InrD]

for rD

< eTJqT

EaT2(1-:) [1 + (r/rw) 2] (B-13)

AccordingbreakdownEaTJ(l-v)T .After a few hours, the temperature within theradial distance crossing fracture tip approachesthe well temperature T for a small fracture.The equivalent surface w load in Figure 24 thusapproaches

(B-6)

(B-7)

for 0 < t < co

for r < r < cow-

fl + 1 il _ L aTar2 r ar ~at

~ - K/pC .

Tlr

_r

- Tw

w

Tlt_O - 0

where

Borehole cooling occurs due to fluid circulation.It is significant at the bottom of the well, butit is generally trivial at the casing shoe.Hence, it affects the borehole breakdown pressureif lost circulation occurs around a boreholebottom. The thermal diffusion equation is givenby:

T T - qT In r/rwfor r < r eTJqT (B-9)

w w

T 0 for r ~ r eTJqTw

where

qT - qT (~t/r~)

Several approximate equations have been suggestedin the petroleum engineering literatures for thefunction qT (Investigation radius or zone ofinfluence) .

The continuum mechanics gives the followingstress distribution when temperature distributionT(r) is present around a borehole.

Since it is tedious to integrate Equation (B-7),it is approximated with a simpler form.The temperature distribution with a logarithmicscale show a constant slope for T vs (r/rw) up toa certain radius. The constant slope indicatesthat temperature distribution can be expressed bya logarithmic approximation up to a certaindistance, or

the stressinduced byinduced bysmall andIt acts as

(B-IS)EaT__w_2(1-v)

Equations B-13 and B-14 show thatintensity factor and fracture widthtemperature are analogous to thosein-situ stress if a crack istemperature sufficiently spreads out.if the in-situ stress is lowered by

Pore pressure build-up normally occurs ifformation permeability is similar or smaller thanmud cake permeability (1.E-3 to 1.E-S md).

The tangential stress around a wellbore inducedby uniform horizontal in-situ stress is given by

A.6 Pore Pressure Build-Up Effect Upon BoreholeBreakdown Pressure. Fracture ReopeningPressure. and Fracture Extension Pressure

according to Equation B-ll.

The value a is relatively constant for sandstonesand is approximately .8 x E-S in/in/OF. Enormally ranges between 105 to 107 psi forsandstones. Hence EaT. .I[2(1-v)] ranges 4.7 psito 470 psi for every "to°F temperature change.Such reduction of borehole breakdown pressure issignificant for tight and low in-situ stressformations. However excessive temperature dropmay not seriously reduce lost circulationpressure if a stable fracture reaches beyond thetemperature cooling zone. Fracture extension maystop due to its high fracture propagationresistance caused by the high pseudo-fracturetoughness.

JoCur)Yo(ur )-Yo(ur)Jo(ur )dw w-yJ o 2(r u) + Yo 2(r u) u

w w (B-8)

--M!r D

U r 2 f rDT drDr I-v D1

--M! r 2r

D --L aTU - f rDT drD(J I-v D

1I-v

U--L aT (B-IO)

z I-v

where the plane strain is assumed.

Substituting Equation B-9 into Equation B-IO

gives

2Tw f -K u 2T -T + e T

w 1r 0

51

lO THEORY OF LOST CIRCULATION PRESSURE SPE 20409

Let's consider the most simple pore pressurebuild-up problem to check its effect. Assumingone phase flows with constant viscosity,porosity, and permeability, the diffusivityequation becomes

~Qfk at

r(B-l6)

Since borehole pressure fluctuates, we assume anaverage pore pressure at the interface betweendrilling fluid cake and rock surface.

plr=rw

= Pow

for 0 < t < co

pi - 0t-Ofor r

w::; r < co (B-17)

The solution of Equation B-16 with the boundarycondition (B-17) is given by Equation B-8 byreplacing T by P . Hence the pore pressurefield is icfenticalOWto the temperature field ifnondimensional time k t/¢~cr2 is used instead ofkTt/r 2. The differebce, h~wever, is that porepress~re build-up affects wider regions thantemperature field since k t/¢~cr 2 can besignificantly larger than kTtfr 2 be~ause of therelatively small pore fluid comIrressibility.

The solution of Equation B-16 with the boundarycondition B-17 can be approximated by

P Pow - q(tn ) In(r/rw) for r < r w Exp(Po~q)

P 0 for r > r Exp(P /q) (B-18)w ow'

where

q(tn)-[2(lnt

n+·23l8)-1-1.154(lnt

n+·23l8)-2) Pow

tn

= k t/¢~cr2r w

for a relatively large tn. The fracturepropagation through a formation with lowpermeability is complex since pore pressuredistribution is disturbed by filtrate through thefracture surface. Such problems cannot be solvedwithout a complex numerical model.

However some extreme cases can be solved bysimplifying the problem, which provides insightinto the effect of pore pressure build-up. Iffracture speed is very high, the fracture tippressure, Pf' becomes significantly smaller dueto the sudden tip volume expansion. It increasesfracture propagation pressure. However, thereduction of lost circulation pressure occurs ifthe fracture tip pressure P

fbecomes sufficiently

high because of slow fracture speed or largespurt loss. With sufficient low fracture speed,the fracture tip pressure ranges:

Pore Pressure at tip < Pf

< Pw

Tables 3 and 4 show equations for boreholebreakdown pressure and fracture extensionpressure with both temperature and pore pressureeffects.

52

SPE 20409

Borehol e Breakdown Pressure From An Open Crack

Tabl e 2 Breakdown Pressure from An Open Crack

(3) For a large A/rw:(A/rw·GT.3), use Fl-I. F2-1 and F3-1.

Equation of Fracture Aperture: W- WI+W2+W3+W4 (a-2.4)

I. For hydrostatic boundary stressWI - 4(I-v2)A HI/E x

. [~+ (2/,,)(pjIf - Arcsinp) In((I+~)/~)]X''Hl(a-2.5)

2. For directional stressW2 - 4(l-v2)A H2/E x ["Hl-"H2][ (2/,,)p' .tr=7J2 ;r-;j2/~2]

(a.2.6)3. For borehol e pressure

W3 - 8(I-v2)A H3/("E)xPw pjr-p ln «I+~/~)

4. For fractue surface presure (a-2.7)W4 - 8(I-v2)A~"E)xPwx

[~ Arcsin(~)j!-;i"2/~ -~ln(~)-Arcsin(p)ln «I+JI=;i2)/~)](a-2.8)

where HI-I, H2-I, H3-1. H4-1 are good approximationsfor IP-:I.5rw' For BsI.5ry, they are gi ven by a function ofA/B and B/r • For examp e HI to H4 are gi venin the fOll~ing table for B/rw-I.333. ['O'h

zO"h

I~ ffi ~ ~ ~ 1I 1.1 1.049 1.116 1.035 1.079 w2 1.3 1.047 1.076 0.998 1.073 /3 1.5 1.030 1.057 0.983 1.062 Pw P4 2.0 1.020 0.978 0.950 1.025 LJ a5 2.5 1.013 0.945 0.950 1.006 I6 3.0 1.011 0.928 0.950 0.9957 4.0 1.010 0.914 0.950 0.998 SA'8 6.0 1.009 0.892 0.950 0.9939 8.0 1.005 0.868 0.950 0.983----·---,

10 10. 1.003 0.849 0.950 0.975

S- r(Arcsin(B/A)-Arcsin(p», r-2jA/". p-rw!A. SI-rp;r::B2 F3(a-2.1)

(I)A-rw

Pw - [Kc+(-3"HI+"H2)XI.1215 .;;p::r;;J1I[1.1215 j"CA-rw)+S]

(a-2.~)

(2) 1.01rw < A < 10rw

Pw - [Kc - "H1 r(,,/2 + pjr-p-Arcsin(p»Fl (a-2.3)-("Hl-"1I2) rp'Jr-iJ'I F2]/[S+SI]

Fl, F2 ana F3 are given in Table I

Borehole Breakdown Pressure

(I) A - rw : Pw - -3"HI+"H2 + KC/[1.1215~]

(a-I. I)(2) 1.01r ~ A < lOr

Pw - YK - "H1 rY,,/2 + pJr-P-Arcsin(p»FI-("Hl-"H2) rp':n:7f2 F2]/[rp;r::B2 F3] (a-I.2)

(3) For a large A/r ,use Fl-I, F2-1 and F3-1.For a small A/r:, use the following table forestimating Fl,F~ and F3.

Equation of Fracture Aperture : W- WI+W2+W3 (a-1.3)

1. For hydrostatic boundary stress: WI - 4(I-v2)A GI/E x"Hl(a-I.4)

2. For directional stress: W2 - 4(I-v2)A 62/E x ["1I1-"H2](a-I.5)

3. For borehole pressure: W3-B(I-v2)rw!("E)x[G3 +In(A/rw)]Pw(a-1.6)

(a) A - r w :

GI-2. 916( I-rw/A), G2-1.458(I-rw/A) .63-2.29(A/rw)-ln(A/rw)

(b) 1.0lrw < A < 10rw : GI,G2,G3 given the following table.

(c) A larger than lOrw : GI-1; G2-0; and G3-.39 ~hz

I A/Rw Fl F2 F3 GI 62 G3 0"hI 1.01 1.228 1.236 1.229 12 1.02 1.227 1.232 1.224 w3 1.04 1.212 1.222 1.2144 1.06 1.196 1.210 1.2005 1.08 1.183 1.201 1.1876 1.10 1.171 1.192 1.1757 1.151.1461.1731.1528 1.201.1251.157 1.133 .385 .160 .1629 1.25 1.109 1.145 1.117

10 1.30 1.095 1.134 1.104 .495 .189 .20611 1.40 1.075 1.117 1.08412 1.50 1.060 1.105 1.069 .638 .207 .25813 1.60 1.049 1.095 1.05914 1.80 1.035 1.082 1.04415 2.00 1.026 1.072 1.034 .814 .197 .32416 2.50 1.015 1.063 1.02417 3.00 1.009 1.053 1.015 .926 .147 .36018 4.00 1.004 1.042 1.013 .964 .114 .37219 6.00 1.002 1.034 1.010 .991 .077 .38320 8.00 1.001 1.025 1.008 1.0 .058 .38521 10. 1.000 1.018 1.005 1.0 .047 .385

Table I The Basic Solution for The Borehole Breakdown Pressure

(a-3.9)

Table 3.b Equation of Fracture Aperture with Pore Pressureand Temperature

(Note: Applicable only for q< pow!ln{A/rw)' qT< Tw/ln(A/rw)

Equation of Fracture ApertureW- WI+W2+W3+WpT (a-3.5)

I. For hydrostatic boundary stressWI - 4(I-v2)A 61/E x["Hl+ Pf +E(QqT-2QTwfuoq - 2Qlow)f(4(I-v»]

(a-3.6)2. For directional stress

W2 - 4(l-v2)A G2/E x ["Hl-"H2] (a-3.7)

3. For borehole pressureW3 - 8(I-v2)rw!("E)x[G3 +In(A/rw)][Pw-Pf -E(QqTfuoq)/(2(I-v))]

(a-3.8)

WpT -2(I+v)(QqTfuoq)A[2/" JI (s/(s2-p2)**.5){(,,/2)ln(s)-. p

Arcsln(p/s) 1n(s)+.5("1n(,,/2) -,,-,,'/72) ­Arcsin(pfs) 1n(Arcsin(p/s) )+Arcsin(pfs) +(Arcsin(p/s»'/18) ds -In(p)(.;r::ar +(2/,,)pln(p)- (2/,,)jIf Arcsin(p»]

w

(a-3.2)

(a-3.1)

UHl • uH'

"H2 - cos2-y "H + sin2

-y "V

Borehole Breakdown Pressure

(I)A-rw

Pw- -Pf -3"HI+"H2

+E(QT fu Pow)/(I-v)+Kc/[1.1215~w]w 0 (a-3.3)

(2) 1.0lrw < A < 10rw

Pw - Pf +E(QoqfuqT)/[2(I-v)]+

[Kc - ("Hl+ Pf +E(QqT-2QTwfuoq - 2Qlow)/(4(I-v»X

r(,,/2 + pjIf-Arcsin(p»Fl -("Hl-"H2) rp'jl-p2 F2 ­

E(QqTfuoq)/(4( I-v) )xr (2Arcs i n(p) 1n(p) -,,1 n(p)+2Arcsi n(p) ­

2Arcsin(p)1n(Arcsin(p) )+(Arcsin(p» '/9

+(,,In(,,/2)-''-''''/72)}]/[rpJ!=P2 F3] (a-3.4)

(3) For a large A/r ,use Fl-I, F2-1 and F3-1.For a small A/r:, use Table I for estimating Fl,F2 and F3.

Pore Pressure Oistribution:P - Pow - q(tD) In(r/rw) for r < rw Exp(Pow!q)

P - 0 for r > rw ExP(Pow!q)

Temperature Distribution:T - Tw - qT In(r/rw) for r < rw Exp(Tw!qT)

T - 0 for r ~ rw Exp(Tw!qT)

Q 0- (l-c/cb)(I-2v)/E. r - 2jA/". p - r w/A

Table 3.a Breakdown Pressure with Pore Pressure and Temperature(Note: Applicable only for q< pow/ln(A/rw)' qT< Tw!ln(A/rw) )

53

SPE 20409

W3 - 8~I-v~)A H3/(trElxll'w-Pf-E(aqT+aoq)/(Z(1-v»]Xf! of! 1n «l+~)/q) (a-4.8)

Table 4.b Equation of Fracture Aperture at 8with Fracture Borehole and Temperature

(Note: Applicable only for q( Pawlln(A/rw)' qT< Twlln(A/rw) )

where HI to H4 are calculated as shown in Table Z.

W4 - 8(l-v')A H4/("E)x[Pw-Pf ]X[q Arcsin(q)~ /q -q1n(q)-Arcsin(f!)ln «l+;r-;;o)/q)]

(a-4.9)WpT - Z(I+vHaqT+aoq)A[q Arccos(q)-~ - 1n(Z)Jr-;i2 +

(ZMf!(l-ln(f!})ln«I+~)/q) -(Jr-iiT - (ZMf!ln«I+~/q»ln(f!)] (a-4.10)

4. For fractue surface presure

EDuati on of Fracture Aperture

W- Wl+WZ+W3+WpT (a-4.5)

1. For hydrostatic boundary stress

WI - 4~I-v~)A Hl/E x[ -q + (Z/trHf!.[r-7i'F - Arcsinf!) 1n«I+~/q)]x[UH1+ Pf +E(aqT-ZaTw+aoq -ZaoPow)/(4(I-v))]

(a-4.6)Z. For di rec:ti ona1 stress .

Wz - 4(l-v')A HZ/E x [uHl-uHZll(Z/tr)f!' ;r-p .;r=;;o/q'](a-4.7)

3. For borehole pressure

(a-4.1)

Pore Pressure Distribution:P - Pow - q(tD) 1n(r/rw) for r < rw Exp(Pawlq)

P _ 0 for r> rw Exp(Pawlq)

Temperature Distribution:T - Tw - qT 1n(r/rw) for r < rw ExP(TwlqT)

T - 0 for r ~ rw Exp(TwlqT)

a - U:.!=i/cbHI-Zv)/E. _r _OZ.[1:[i<. f! - rwlA. Sl-rf!Jl-f!' F3

S- r (Arcsin(B/A)-Arcsin(f!».uHl-uH' uHZ-cos'., uH+ sin'., Uy

Borehole Breakdown Pressure From ·~"""fAn Open Crack ~t I(I)A-rw ~

Pw - -Pf+[Kc+(-3uHl+uHZ+E(aTw+aoPow)/(l-v»x .

1.1Z15 ~JI[I.1Z15 ~+S](a-4.3)

(Z) 1.01rw <A < 10rw

Pw - Pf + [SIXE(aoq+aqT)/(Z(l-v»+

KC - (uH

1+ Pf +E(aqT-ZaTw+aoq -ZaoPow)/(4(I-v»)X

r{,,/Z + f!JI=P-Arcsin(f!»Fl -(uH1-uHZ) rf!'Jl-ij'F FZ ­

E(aqT+aoq)/(4( I-v) )xr (ZArcsin(f!)1n(f!) -,,1 n(f!)+ZArcsin(f!) ­

ZArcsin(f!)l n(Arcsin(f!) )+(Arcsin(f!» '/9

+(,,1 n(,,/Z)-,,-,,'/7Z» ]/[S+SI] (a-4.4)

Table 5 Well bore Pressure Inducing Fl uid Invasion Into Fracture

(5.b) Rock type:Intermediately weak (Unconfined compressive strength- lkpsi)

Table 4.a Breakdown Pressure from An Open Crack Pore(Note: App1 icab1e only for q< Powl1n(A/rw)' qT< Twlln(A/rw)

(3) For a 1arge A/r • use Fl-l. FZ-l and F3-1.For a small A/r:. use Table 1 for estimating Fl.FZ and F3.

PBW- Borehole pressure initiating fluid invasion intb fracturePB - Actual borehole breakdown pressure inducing a lost circulation

Data:r -6 inches. uniform pore pressuretllerm1 expantion coeff. (a) - .8E-5(/degree F) .KT-.01l6cm'/sec5.5 housrs after formation is exposed to low temperaturefracture toughness - 1. psi*in**.5, v -.1

uH Uy Well Temp E wm-.Ol" wm-.OZ" wm-.03"Angle (E6*psi) PBW PB PBW PB PBW PB

(psi) (psi) (deg.)(deg.F) (psi )(psi) (psi )(psi) (psi) (psi)

-500 -1000 O. O. .Z 1074 1074 1142 1142 1211 1211-500 -1000 O. O. 1.0 1345 1345 1655 1655 1946 1946-500 -1000 O. O. 5.0 2478 Z478 3663 3663 47294729-500 -1000 O. -50. .Z 989 989 1060 1060 1132 1132-500 -1000 O. -50. 1. 988 988 1371 1371 1733 1733-500 -1000 O. -50. 5. 2010 2010 3663 3663 4729 4729-500 -1000 O. -100. .2 904 904 979 979 1054 1054-500 -1000 O. -100. 1. 662 885 1118 1118 1540 1540-500 -1000 O. -100. 5. 1513 1607 3663 3663 4729 4729-500 -1000 45. O. .Z 829 829 904 904 970 970-500 -1000 45. O. 1.0 1096 1096 1376 1405 1630 1677-500 -1000 45. O. 5.0 2100 Z148 3139 3148 4090 4090-500 -1000 45. -50. .2 743 743 820 820 888 888-500 -1000 45. -50. 1. 707 894 1036 1229 1343 1510-500 -1000 45. -50. 5. 1329 1644 3139 3139 4090 4090-500 -1000 45. -100. .2 658 658 736 756 806 848-500 -1000 45. -100. 1. 331 879 717 1099 1079 1342-500 -1000 45. -100. 5. 724 1601 3139 3139 4090 4090-500 -2000 90. -50. .2 496 638 578 749 649 841-500 -2000 90. -50. 1. 456 874 766 1188 1042 1471-500 -2000 90. -50. 5. 739 1635 2618 2618 3591 3725-500 -2000 O. -50. .2 989 989 1060 1060 1132 1132-500 -2000 O. -50. 1. 988 988 1371 1371 1733 1733-500 -ZOOO O. -50. 5. 2010 2010 3663 3663 4729 4729-500 -2000 45. -50. .2 248 630 333 718 410 799-500 -2000 45. -50. 1. 214 868 522 1147 785 1433-500 -2000 45. -50. 5. 249 1625 1996 2558 3193 3691-500 -2000 90. -50. .2 -495 615 -410 689 -324 754-500 -2000 90. -50. 1. -507 851 -181 1072 95 1323-500 -2000 90. -50. 5. -817 1596 613 2501 1953 3591

-1000 -2000 O. -50. .2 1983 1983 206Z 2062 2132 2132-1000 -2000 O. -50. 1. 1953 1953 2327 2327 2680 2680-1000 -2000 O. -50. 5. 2467 2467 4500 4500 6178 6178-1000 -2000 45. -50. .2 1493 1493 1571 1571 1652 1652-1000 -2000 45. -50. 1. 1458 1569 1800 1978 2113 2324-1000 -2000 45. -50. 5. 15Z0 2405 3369 3588 5039 5039-1000 -2000 90. -50. .2 1002 1176 1077 1298 1161 1425-1000 -2000 90. -50. 1. 976 1491 1314 1893 1620 2241-1000 -2000 90. -50. 5. 815 2390 2455 3513 3975 4712

Table 6 Example Calculations of Borehole Breakdown Pressure andLost Circulation Pressure

54

Z8Z0

1650

ZOOO

1000500

1900

-500

1000

1000 psi 1060 psi

500 630

1000psi 1600psi

750 lZ30

500 840

angle

oo

45o

90

(UH'Uy) (-.5kpsi .-lkpsi) (-.5kpsi .-Zkpsi) (-lkpsi .-Zkpsi)Linear Nonlinear Linear Nonl inear Linear Non1 inearcontinuum fracture continuum fracture continuum fracture

Well angle·0

(5.a) Rock type: Weak - (unconfined compressive strength-150psi)

(uH'Uy) (-.5kpsl.-lIcpsl)Linear Nonl inearcontinuum fracture

Wello

oo

90

SPE 204Qq0-0.008 -0.004 0.000 0.000 0.008 0.012 0.011 $;;! q 0.0 60.0 100.0 160~b~ ~ • :t '<t~-.

~ q

~ ~ l:>!

q q$2 $2

~ ~ ~~~ 0

i ..::>r.n

I~0 ffiq 0l:! a:'" ,.;

11.

~...J Oq

~~C! J:'<t:;) W<C .. a:

~q~...

~ 0..~ ~

0.0 60.0 100.0 160.0INJECTION VOWME(CC)

C! C!0 0

-0.008 -0.004 0.000 0.004 0.008 0.012 0.018 Fig.2- Fracture initiation and extensionSTRAINIIN/INI pressures from a borehole (uHl--1800 psi;

Fig.l- Reconstructed stress strain results uH --2200 psi; uV--3000 psi; mineral oil basesimulating the Berea sample used for the DEA-13 mu3; Berea sandstone; rw-.7S")fracture experiments (compression loading path).

Fig.3- Breakdown pressure (uHl--1800 psi;uH2--2200 psi for Injection 1; uHl--300 psi;uH2--2S00 psi for Injection 2; Berea sandstone;rw-.7S").

'"o

o...

\

\\

\

~. I'.: I

DEHYDRATED MUD ~::.

I .,..1:':<>/::<" .:K:,· /. :- = : ", 4 ....... /'

' •• : ..' : " : eo '.::

.':-,.

.:.:..: :':'--. :: '... ,'.

--~

e~:lll~ ,8)l 0'. •000"

0'0

FRACTURE FRONT{--- ---

r /'~:: f;tj·:::·;:;:~1:·r :/.,

r ::: FLUID FRONT

..YI

I II

\

\

I\

\

"\

Fig.4- Side view of mud cake formed in afracture.

0 0 .0 to 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.'6N N

I(Jza:FwoOC'"

~IE0",~o

~

27

26

SoLlll FREE

PRe~fRACTURE

9OOpSl;llOQpsl

XU."

MOO

10 PPM

INJeCTION 4

\.

INJECTION 1

INJECTION !I

, .X 7 • '"+u KI5

fRACTURE FRONT

----.....--R;, ~----~ -4IIII

-'I

."x,

M08 ooa16 PPSI

D ••

·+S.•

'''.000

12,000

""Q

10,000wor

~ 0000

w0:g

4000

.000

OIL BASE HUD

Fig. S- Cross sectional view of mud cake formedaround fracture tip.

M W W U U M M W M M ~

R.UID FRONT RADIUS(INCH)

Fig.6- Sealed fracture tip length vs fracturefluid front radius (calculated from Fig.2).

55

Fig.7- Fracture width at sealing point vsfracture size (calculated from Fig.2).

2.0 3.0 4.0 5,0 6.0 7.0

AUlD FRONT RADIUS(lNCH)

~u(8) WEDGED CRACK

(D) PARALLEL CRACK WITHOUT

PERFECT SEAL

)-<A) PARALLEL CRACK

(C) PARAllEL CRACK WITH

SQUEEZED-OUT•. CAKE

)-rFig.lO- Typical crack configuration after mudinvasion.

'"l:!o

ooqo

~o

10.~oo

9.08.0

8.0

7.06050403.02.0~ 0.0 to00!z~~O

Cll'l;;i!;o~o(J) oot(~

503:5W O

a:

too«~lEo

§00.0 to

A"'--'

FIG. -A RIGHT BEFORE BOREHOLEBREAte:"DOWN

FIG. -B fRACTURE EXTENSIOIt WITH

FLUID HON-PENETRATING ZONE

FIG. ·-D AFT£R SUDDEN MOVEf1EHT OfFLUID fRQNT

UJ

""::>VIVI

~0-

ZoVI:z:UJl­X...,UJ

'"::>I­u<'"u...

B~Rw

/I' ..........- DRILLING FLUID ENTRY BV INTO FRACTURE

I- 1----'" No STRESS DISTURBANCE ~ I

PORE PRESSURE ----f \ Rw 'I.....:..-BUlD-UP \..J I

Ip;--/

TEMPERATURECOOLING

SOLID FREE FLUID(NO PERMEABILITY ISASSUMED)

Fig.8- Typical sequence of fracture extension forwater base muds. A

Fig .11- Fracture extension pressure for typicalwell conditions.

.014.i5o 14 Q 18 (1.Ole"

1!l 0 1 : i: (.J3ec>

-- -- - - - - z ~~(.l~ccf - - - - - - - -- - - -.~2-o~O<l~06c~~~;;~;;

o .. (O.ec)3()"H,'+O"'H)-1t 0 u : ~~( 6Occ)07 .19°21'

G 5 <'37c" 0 <d~ce

o5000

3000

2000

3 (1.Olccl

t:l PRE-fRACTURED HITH]OLlD-fREEFLUID

PRE-fRACTURED WITH DRILL"ING FLUID

SECOND hWECTION

THIRD INJEeTi ON

fiRE-fRACTURE

j, HoTCH(.2S- DEPTH)

• 3

011 ..

.10

..o 3 (3.37cC>

o •

.. 11 a.7ed0 10

. .

16pPG !'IUD lOpPG !'IUD1000

9. 10.

rw andKc-600

6. 7. 8.Rw(INCHES)

s.3.2.1.

Fig .12- Borehole breakdown pressure vsin-situ stress (WmE/(1-v 2 )-25000 in*psi;psi*in**.5) .

IIOB DOBI1IlB DOB Ce~O

Fig.9- Fracture reopening pressure forpre-fractured borehole (Number withinparenthesis: in-situ filtrate rate converted tothe API 30 minutes filtrate rate; uHl--1800 psi;uH2--2200 psi; rw-.75").

56

SPE 20409

goo INCLINATION

O'tl--·St::ps'

\, lO:--I .•",_:VERTICAL HOLE

~:~PRESSURE TO OPEN STAIU fRACTURE

X PRES$~E TO .....LOM rHtllllNG FLUUIflilRY INTO FRACTURE

\\

" \\ \

\\\.,--~~. r--, \j

·500PSI

.,,,,

-300~

~ir·2:10

J -100

o5000

2000

3000

(.s~r6000

1000 LINEAR CoHTlIIUIoltIt. EU,ST1CITY<Ctc."'150psl)

~ ·100 W25-0~-5OOL-~--~-""">-I-000-~~""":"""'-15...JOO-Bo~ •..j,"f-"-'-'-"-"-",-'--<2001JpSI

8 9Rw(INCH£S}

10

Fig.14- Nonlinear effect upon the tangentialstress at borehole top surface (intermediatelyweak rock: unconfined rock strength-1000 psi).

Fig.l3- Borehole breakdown pressure vs rv andWmE/(l-1I2) (in-situ stress--500 psi; Kc-600psi*in**.5) .

Pw(PSI)

5000

4000

3000

-S-

3000

2000

01I'.151JOpSI

-lOOOPsl

\ -5OOpSI

'----- ---.l-oo-.-S,--------~'5xloS

•25xlO

------.125x105 ; :::::::::::::::~;;;;;~;;;;;:;;;;;;;;;;;;;;;::~

.0625x10

1000

10 20 50 60

Fig.15- Breakdown pressure for a borehole withwide cracks (In-situ stress--500 psi; Kc-600psi*in**.5; rw-6").

10 20 30 40BCINCH)

50 60 Fig.l6- Breakdown pressure for a borehole withwide cracks (WmE/(l-v2)-25000 in*psi; Kc-600psi*in**.5; rw-6").

og~

67.5 100~

g

f

~i000g

E-SE6 PSIE-2E6 PSIE-.5E6 PSI

50 DEGREE CDDLlNG(E-3E6pSI)

.\\ .

~ 0.0 12.5 25.0 37.5 50.0 62.5 75.0gT~~..L~-""""~-"""'_~-'-'''''''''~~L~'-:.L~_::L:''''''~+

~ :::;::;;;;;;;, .. :.::..:.""'''''"."''".,,,, ,.................................. 0o .:::::: ':. -:. .. : ..",....... .. .______________ ci

5!f':-'"""~..,.~~..,.~~~~~~~~~~........:~:.::::::::;=:..~_._J.~M u ~ ~ ~ ~ ~ ~ ~

FlUID FRONT POsmON(lN.)

Fig.18- Temperature effect on fracture extensionpressure for various moduli (50 degree F cooling;uH--1000 psi;~-.01l6 cm**2/s; time - 2.3 days;Kc-O; 0-.8E-5 per degree F; v-.l; lIm-.Ol";rw-6").

ooo~

o

~

oolS

o

-~

60.§

re50.0

on fractureF cooling;0-.8E-5 per

40.030.020.0

No TEMPERATURE

10.0

\~ 0.0

re

----~__ - --AFTER 2.30AYSc----. . . ~DAY_S__

08" HORIZONTAL IN-SITU STRESS 0

m g-i0-:-.0-:----l0r.~0--~2To-.o---3...,Or.0----4rO.~0--~50..,--.O---6+0.:

flUID FRONT POsmON(lNJ

Fig.17- Transient temperature effectextension pressure (100 degreeuH--1000 psi;K.r-.01l6 cm**2/s;Kc-0;degree F; v-.l; lIm-.Ol"; rw-6").

57

SPE 20409

og"''"

90.Qloo..,

fracturethan 1)strength

75.0

75.0

······························0(-500.1500.600.1.)" g

~

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

60.0

60.0

45.0

o-....:.,:::---=~ -====~_-+o

:il

30.015.0

15.0

Conventional Theory

0.0

~ 0.0oo..,

~ ..

l750 ~tfrg ~

o , .... :

~ N 1-----·.!ifi,-.'~====S~(-;500;;'-~I;OO;of,,;oO;';1~'3;33~)=~~~~~~~=len # (-500.-1000.0,1.333)

ffi~ 0a:: 0 gno te (-SOO.-ISOO.600.1.333) (-500,-1500.0.1.33) ~

Z~q 0

g8t===""",=:::--===~L...I:.~~~~~li....!~~~~L__-tg~~{ ~

i1ia::qceo

:il

30.0 45.0

WELL ANGLE{OEGREE)

Fig.20- Vell inclination effect uponextension pressure (i.e.,B/rw greater(low in-situ stress; rw-6"; intermediaterock: YmE/(1-1I 2 )=2500 in*psi).

ooi

···· L~

fracturevarious

O'H-- 2500

75.0

------__.L~

C!--- J 52

30.0 45.0 00.0Will ANGLE(DEGREEl

75,000 ./

15.00.0

og

'b C!iO_.0~...........=15k·OT=:;:~::3=0:.0:::::~~_45L·0~~~_0.L0._0~~~7.l5_.0~~~9-t0'~'b.- $::! S!'-

W",ElCl-,] )-100,000",. PSI

Fig.19- Yell inclination effect uponinitiation pressure (i.e.,B/rw-l) forelastic modulus (high in-situ stress:psi, O'v--5000 psi; rw-6"; Kc-O).

C!oog

ogl'l

ogo..

100~og

uponpsi;

87.575.0

E" 3.£6 PSIE- 2.E6 PSI

E- .5E6pSI

50.037.525.012.5~ 0.0

g

~i~~:=::::~~§§~§§~~~~~::~~~l~M ~ ~ ~ ~ ~ ~ ~ ~

FLUID FRONT POSmON(IN,)

Fig. 22 - Pore pressure build-up effectfracture extension pressure (Pow-1000Td-100; rvs6"; Kc-O; O'H--1000 psi; ao-.7).

g0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 80.0 1OO~

l" l'l0

"l'~'0

0 0g g~C! 0

!!>8 00

(/)0 52(/)-

~:; °v·-lOOOpsi \ C!0

0 5:z'"0~:;

Frolll Top (1-0,15,30.45.60,7$ and 90 degree 00

~~ ~XWO 0'0 0

Uo g<l:~

fE o :;0g 0..~

00g

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 80.0 100.0

FLUID FRONT POSmON{lN,)

Fig.21- Fracture extension pressure vs drillingfluid front position for an inclined well(E-.2E6 psi;II-.1; Ym-.02"; Kc-O; rw-6").

, /(r)'

(A) BOREHOLE WITH A FRA':TUR-t:: dNDER TEMPERATURE FIELD

load forfracture

(B) EQUIVALENT FRACTURE SURFACE LOAD

Fig.24- Equivalent fracture surfacecalculating fracture toughness andwidth.

a.

1..loS

.5l1'lO5'

*

Fig.23- Pw-A/rw-W relation (rw-6").

3000

7000

58