.

SPE 15069

A Complete Integrated Model for Design and RealTime Analysisof Hydraulic Fracturing Operationsby AR. Crockett and N.M. Okusu, Resources Engineering Systems Inc., and M.P. Cleary,Massachusetts Inst. of Technology

SPE Members

~opyright 1986, Society of Petroleum Engineers

rhis paper was prepared for presentation at the 5Sth Califomis Regional Meeting of the Society of Petroleum Engineers held m Oakland, CA, April 24,!9ss,

rhis pa~r was selected for presentation by an SPE Program Commmes following review of mlormation contained in an abstract submitted by thesuthor(a). Contents of the psper, as presented, have not bean reviewad by the Sosiefy of Petroleum Enginaare and are aubjact to correction by thaguthor(a). The material, aa presented, doas not necessarily reflect any poaifion of the $ociaty of Petroleum Engineers, its offiiere, or members. Paperapraaentad at SPE meetings are subject to publication review by Editorial Committees of the Socialy of Petroleum Enginaera. Permission to copy isrestricted to an abstract of not more than 200 words, Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of wherenndby whom the pa~r is presented. Write Publicatiina Manager, SPE, P.O. Sox 83SSSS,FUchardaon, TX 7S0S3-3S3S. Telex, 7309S9, SPEDAL.

Abstract IntroductionA new comprehensive model of hydraulic fracturing is pre-

sented which has been developed for the Gas Research Institute A substantial amount of effort has been invested, espe-

(GRI) mobile fracture monitoring and analysis facility. Thecially over the past decade, in the development of models fordesign and analysis of hydraulic fracturing. The resulting mod-

main purpose of the model is to simulate the hydraulic fractur-ing process in real-time, that is on+ite during the fracturing

els have varied in at least three major aspects: realism and

operation, but the model can also be used for pre-fracture de-generality of the assumptions made in formulating the mod-

sign and post-fracture analysis. Sensor data obtained duringela; complexity of the resulting computer codes and machine

the course of the job requirements; and flexibility of the input-output characterie-

such as wellhead pressure, flow rates, tics, especially in relation to real job conditions and operatorfrac-fluid viscosity, and proppant staging can be received .directly by the model se input, superseding the prefrac job

interfacing. Although some good progress haa been made by

design schedule, and making possible more accurate model es-many groups, there have not been any models which were sat-isfactory in all of the three important areas, and most models

timatea of current fracturing conditions and predictions of final hfracture geometry, as the job proceeds.

ave, at beet, been adequate in one aspect only.

The overall model haa four major components describing: Examples of previous work may be found in the consid-

flow of fluids and slurry in tubular goodserable literature which has evolved on this subject. The sim-

creation and propagation of the hydraulic fractureplest models, assuming 2D geometry with a constant specified

transport of proppant, deposition, and fracture closureheight, were those of Christianovich, Geertsma, de Klerk and

heat and fluid exchange between fracture and reservoir.Daneshy (CGD, Refs. 1, 2), having width dependent on length,

This fully-integrated, numerically robust model of the hy-and those of Perkine, Kern and Nordgren (PKN, Refs. 3, 4),having width dependent on height: these two models are really

draulic frauturing process takes directly into account as much valid only when length greatly exceeds height (or vita-versa ifof the essential physics as possible, given the computational the models are turned on end) so clearly neither are appro-limitations of the real-time application. Additional informa- priate for the typical field condition where the fracture is nottion, pertaining to very complex reservoir characteristics, can very @l contained.be indirectly supplied to the model through data-based resultsobtained prior to the job from other more comprehensive [e,g., Since both the height and length may grow substantially

3D, croes+wctional) fracture simulations. Having thereby em- during the job in contradiction to these 2D models an

bedded more elaborate fracture analysis into a simple lumped a priori specification of height is hardly acceptable, even if

model, accurate pre~lctions are achieved at execution speeds based on (usually tenuow) deductions from well 10SS. To re-

faster than real-time, allowing on-the-job analysis and even solve thm problem, Cleary (Ref. 5) developed a pseudo-three.

real-time history matching for unknown reservoir parameters. dlmeneional hydrafrac model (P3DH) of simultaneow length

Sample results are presented from model simulations of a and height growth, having width dependent mainly on the

fracture treatment performed in the Travis Peak formation oflesser of height or length. This model wee adopted by vari-

East Texas. Actual sensor data from the job was used as inputous groups in industry (e.g., Refis.6, 7) while others developed

and model predictions are compared with field measurements.specialii variationa on the same theme (e.g., R@. 8, 9). Al-though these P3DH-type models, if fully developed, do indeed

References and illustrations at end of paper.

.

9 A fY3MPl,F!TF! REAL-TIME MODEL OF HYDRAULIC FRACTURING SPE 15069m . . . . . . . . ---- .. ..

jerve to make first-order estimates of fracture geometry for 1. Frac-fluid/slurry flow in tubular ~oods and perfora-

.athe: uniform reservoir and pumping conditions, the cross- tions, including effects of cross-linking, foaming and proppantIeciional approach implicit in these models 10SSSits advantage concentration in gelled fluid flow rheology. This incorporates

or fractures which are not moderately to well contained and both laboratory and field data on pipe flow, the latter based

nore generally, they do not easily allow the representation of on simultaneous uphole and downhole pressure measurement

:omplex geometries and treatment schedules. wherever such data has been collected: a major purpose is

To provide a fracture height growth criterion valid for allto eliminate the need for downhole data collection, by beingconfidently able to calculate true downhole fracturing pressure

evels of confinement and a structure favorable to the eventual from measured uphole pressures.~corporation of general and complex reservoir characteristics,lore comprehensive 3D models have been developed, using Uphole and bottomhole pressures are related by dividingVOdistinct approaches (Refs. 10, 11). Although the former the slurry in the wellbore into batches or sections along thelef. 10) haa in many ways become a practical tool for analysis length of the wellbore (Fig. 1) and so!ving the flow equationsnd design (Ref. 12), providing comprehensive capabilities, the for each section. The flow equation for the jtb section islrrent generation of fully 3D models lacks the kind of com-putational efficiency and ease of operation required for many 1 dp 2fv2 p

4+-=9T; =exp(~)~l+~?al-time applications, especially those which involve repeated v dz2 p dz2 Pr(1)

Kecutions of the model, at rates faster than real-time (e,g.,m the determination of unknown reservoir characteristics by where p is the pressure along the wellbore, and f the frictionistory-matching, or for the prediction of final fracture config- factor characterizing the viscous drag (Ref. 13). The kineticrat ions corresponding to alternate treatment schedules). energy or convective term udu/dz2 haa not been neglected

The foregoing 3D, P3DH and 2D efforts have substan- and, in order to model compressible flows ,whlch occur dur-

ally enhanced abilities to model many aspects of hydrafrac; ing foam treatments, a closure relation is assumed in which

owever, for different reasons, none has produced an adequate the slurry density p is written in terms of pressure, the slurry

~odel for realtime analysis of fracture operations. Thus, the bulk modulus B, and the slurry density p, determined for some

urpose of the model described in this paper is not only to be reference pressure (usually either atmospheric or bottomhole).

sed for comprehensive hydrafrsc design, but also for real-time This expression for p is linearized as shown because p is suffi-

nalysis of the treatment. To achi~ve this, the models had to ciently less than 13; the resulting linearized form of Eqn, 1 and

atisfy the three major conditions ~iet!orih at the outset: they its solution isre realistic and general in the I,hysics and geometry whichhey represent; they run effectively on all levels of computer

dp~-C1p=CO; Pj= Pj-1(1 + cl~B) + cO~B, (Z4

ystem, from mainframe to portable; and they operate on thectual data generated by the job itself, running in real-time as co = (Pr9 zf(Pj-1 uj-1)2/Pr~)/c2, (2b]ceded and providing an instantaneous view of the job status tohe operator including pre-frac design and post-frac analy- Cl = (Prg + zf(Pj-lvj-1)2/Pr~) /~c2t (2C]is. By offering all of these essential capabilities, the rnodellingf hydrafrac finally takes on a greater and more meaningful and

C2 = 1 (~j-lvj-1)2/prB (2d:ractical significance.where pj - 1 and pj are the pressures at the top and bottom o

%h batch respectively, and b?B is the batch length, SinCthe Jthe fluid is assumed to be compressible, the batch length is un

Overall Description of Model Capabilitiesknown and must be determined from an integral expression fo]the mass of the batch MB, given values for the j -1 variables

The overall modelling capability consists of four fully in-

/

entegrated modules, each representing a major aspect of the hy- MB = pAwdzz (38draulic fracturing process. In order to be run during the frac- 0

ture treatment at rates faster than real-.me and :.ing aC- yieldingtual sensor data as input, these modules weie .. . kematical lyformulated to accept a wide range of input values, and were (Pj-lcl + co)t~ + (~+ pj_,/B)tB B-=0. (3befficiently implemented on the computer to minimize execu- 2B prAw

tion time. None of thw was accomplished, however, at theexpense of accuracy or completeness; on the contrary, we have The frictim factor is ba deification of Clapp

taken considerz- ; care to incorporate as much of the easen- equation (Ref. 13)tial physics as possible dkectly in the modules an indir ect 2,75 4.-.method of data-baaing results provides a means to account for f;lfa =o.45- -

(n-.2)/2)n q(ReCfC (4a

whatever remains. The physics which have been incorporateedn

into each of these modules is described in the following four in which jc ia the Clapps friction factor and Rec is the Re}sect ions: nolds number for power law fluids, The friction factor f i

related b fc by

999.-.

.

RPF. 1 MIRO A R flRf)f!KRTT- N.M. (3K1JS1J (!z M.P. CT,TCAR.Y 3a - *V . . . . . . -A ----- -. - , ------- ----- - ..- .-. - -----

1

f

E = ac-bclOgRe R,, = ;;:_:m (4b)vhere ac and be are coefficients found to be necessary in ordero match laboratory and field pressure data for pipe flow, The:ffect of proppant on viscosity is described by

In all practical hydrafrac operations, the pumping-rateis specified and the pressure is a consequence of the specifiedflow-rate. Hence, the most important equation determiningpressure and crack-opening is the theological law for flow ofthe fluid; although this can be quite complex, for instance in-volving viscoelastic memory effects for cross-linked polymers(e.g., Ref. 5), we will employ here the more conventional power-Iaw model,

n which p: is the apparent viscosity for gel and s is the volu- A 2n+1 ~pFnetric sand concentration. The effect of foam quality on vis-

()tii m= 7i4 ~

~Xi ;i=l,2 (7a)

osity is accounted for in a similar fashion; however, the precise PForm of this effect has not yet been finalised,

!. Fracture creation. extension. shutin and closure:rhis component is a completely general integrated 3D model ofLydraulic fracturing, incorporating explicitly to first crder all~fthe essential physics, including the rock mechanics of defor-mation and fracture, the coupled fluid flow within the fracture,md the fluid and heat exchange with the surrounding reser-foir. Any required degree of accuracy may be obtained bylmbedding, in functions or data-bases, the results of compre-hensive (e, g,, fully 3D or cross%ectional) fracture simulationsnto the integration parameters or gamma (-Y)factors of thenodel, thus taking into account the effects of complex reser-{oircharacteristics (i. c,, multiple strata, inhomogeneities, etc.)m the spatial variation of fracture attributes (i. e., crack open-ng, excess pressure, fracture height, etc.). Default values for~amma factors have also been determined, which are either:onstants when variation in the factors does not greatly af-kct fracture propagation, or functions of relevant parameterswhen the parameter variation dominantly affects the fracture

in which @is the effective channel-flow viscosity

P = 2K(4 +2/n)* ; r = K~n (7b)

phrased in terms of the consistency index K and the flowbehavior index n which appear in the relationship betweenshear-stress r and shear strain-rate ~. Both K and n canbe measured with a conventional viscometer. In previous ef-forts (Ref. 5, 15), the lateral/vertical mass flow rates per unitheight/length, W1and W2,respectively, have been written as

Wl = W/2 L2 (8a)

where the mass flow rate into the fracture wing is divided bythe fracture height, and

(8b)

configuration (i. e., confinement, frictional drag, etc.). Havingrelegated the complex numerical analysis to the determination where W2equals the time rate of change of the vertical cross-

~f functions or data-bases for the integration parameters, it section of the fracture plus the mass loss rate per unit length.

is possible to perform accurate, faster-than-real-time analy- 7x and 72s are the vertical crocxwwctionand fluid loss shape

ses during the fracture treatment, as well as routine pre- and f~tors respectively. Specifying WI as the mass flow rate input

post-frac analyses on PC-based systems. W, however, reduces the order of the governing set of equationsby one; this causes discontinuous jumps in estimates oi fracture

One of the primary equations is overall mass conservation, height and width when the flowrate W varies with time. Weuse another closure condition similar to Eqn. (8b) for both

W 2WL = F V. A (2L2) LI (5) lateral and vertical directions

in which W and 2W& are injected and lost fluid masses , pfthe fracture slurry density, while Ll, 2L2 and A are the length,total height and width of one fracture wing. ~Vis the numericalfactor, which describes the volumetric shape of the fracture.

The width A determined by the pressure in the fractureand the results of numerical modelling (e.g,, as described ex-tensively in Ref. 10) may be phrased as an integrated crack-opening relation

in which pF is the fracture pressure, ac is the closure stressand E is the crack-opening modulus, e.g., E/4(1 - U2) for anisotropic homogeneous rock structure. Of course, the actualopening depends in a complex way on the geometry (not justthe shortest length scale 4) and on the rock structural variation:these characteristics are embedded in the coefficient VI.

Wi = PF ~is A hi + 7i6 wL/Lkv i=l,2; k=2,1 (8c)

which yields the same solution to withh a constant as Eqns. (8ab) for steady pumping, while allowing smooth transitions inheight and width through surges.

The factor 7i4 is intended to account for the differenc~in the flow from that between parallel plates; in additionEqn, (7a) must be embedded in a spatially-dhtributed modeto numerically determine the pressure gra&ent (which is a WC.tor -F in the full 3D simulations, (e.g., Re!%. 10, 12) thaisupport the present model). However, the results of these simulationa can be recorded as coefflciente qi2 and simply used uthe form

~pf=aZi

-7i2u/Li, i = 1, 2 (9:

where the pressure gradient coefficient 7i2 reflects all of th~complexities associated with stratification, tluid rheology, Wlctional drag and earth streaa gradenta.

I---

. .

----- ..- .-.

Equations 5-9 can be combined to create two governing These solutions repraent the two extremes bounding therst order differential equations in the length and height Li, aspect ratios of fractmes created during moat treatments; in-s follows: termediate levels of confinement occur smoothly as 7M -+ O.

d (L?) + BiL~ = Ai (lea) Note that the excess preaeure u of the uncontained and com-E plete!y contained fractures diffem substantially u decreases

here by t-lis and is ~wiant to flowrate for the former; whereaslV = (3n + 6)/n, (lOb) it incre~~ by tlis and depends on flowrate for the latter.

[[

L Interpretation of this pressure renpome has become a popu-A _ ~ ~i2~i~E WZWL Li +2Li o lar method of determining the extent of fracture confinement(Ioc) (e.g., Ref. 14). Cleary has previously derived a number of spe-i .71 -Y&D 2pF7u ~ ~ cial analytical solutions (Ref. 15) using the closure conditions

nd expressed in Eqne. (8a, b) and has compared hia solutions withthose of earlier models (e.g., Refk. 1-4) whkh, incidently, are

Bi = N_ ~w2~w~) , i = 1, 2; k = 2, 1. (lOd) special cases of the present model. Clearys solutions corre-spond very closely to Eqna. (11, 12) even though he altemEqn. 8a to account for radial flow patterns in a circular frac-

Once the length and height are determined from Eqns. 10, ture.le other parametem, such as the width and excess pressure,}Iloweasily from bad substitution into Eqns, 5-9. In actual- What has been described so far ieh spatially integrated

y, there are three differential equations governing length and statement of the governing equatione to be solved, but equally

oth upper and lower height extensions as shown in Fig. 1; our important is the issue oi values for the gamma factors. Previ-

lodel possesses this general feature, but, for simplicity, only ous approaches (e. g,, Refs, 6, 7) which developed forma for the

~e symmetric fracture formulation is presented. vertical pressure gradient coefficient, 722, were baaed on croes-

C1early, the fluid 10SS2WL plays a dominant role in thesectional models of the khid described in Refi. 6, 16; themforms have now been developed into the following structure

dutiona just derived and fluid 10SSwill be discussed in the w~l~ include alterations not only to 722, but also to 712, ~1,ext section. However, it is interesting to derive some analyt- and Vis::al results which can be obtained in the special case where a

71 = S, &S9qf, (13a)Vewtonian fluid is assumed (n = 1) and fluid 10SSis neglected.rhe solution for a circular fracture (no confinement) is found

~i2 = S.i Sdi Spi Spi %)9 i = 1* 29 (13b),0 be

= [:-:[+13], *d qis = Li7~8* = 12(ha) (13C)

A= [[i%w2[a] ~

The shape factors (S) appearing in Eqns. 13 denote the(llb) following influences on fracture evolution: 1) confining stress

variation S,; 2) modulus stratification Sd ; 3) spatial variationmd of frac fluid viscosity SP ; 4) frictional drag induced by prop-

E 4 T1T13 P lt_~[

0=-=1

pant SP ; S) stratified fluid 10SSSL ; d 6) certain geometric(llC

) effects (e.g., crack-opening dependence on geometry) S~. 7:,71 9 qlzqlq ~

uhereaa the solution for a constant height fracture (complete 7?w ad V$ denote b~e VZLIUSSof the g~a f~tors when

confinement) becomesnone of these effects is significant (in which case all of the S-factore equal unity). If any additional physical influences need

1= [[:-lwi]i~ (12a)

to be incorporated, representative shape factom can simply beadded to the strings of factom in Eqne. 13.

The shape factom occuring in Eqns. 13 are determinedfrom detailed (3D and croes+ectional) numerical simulations

= [:-;(&)2+

(e.g., Refa. 12, 17) and comparison with laboratory and field(12b) data (e.g., Ret%. 12, 18). The information derived from these

studka can be either dwectly incorporated through a numer-

md ical data-base, or more Preferably, condemsad into ftmctimalrepreaentations of the S-fwtore whkh depend on the essential

u=%~*:($2#

parameters of the physics involved. Cleary origiidly devel.

(12c) oped this latter approach, whkh was implemented by Settariand Ckary (Ref. 6), and further studkd in the context oiP3DH and croes-eectiond models by Settari (Ref. 16) andNarendran (Ref. 17). The determination of comprehenaiwand accurate functional forma for the shape factors ia at thheart of 0U2 model development, and many detdb have not

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-. ----

~een finalised; however, to consolidate ideas, several illustra- 3) The effect of temperature-dependent viscosity variationive examples based on cross+ectional analyzes are described along the frmture b expr~sed ss~elow:

1) If the confining stress is greater in the adjacent strata Spi =

+ kwlexp(-~ia f)p{ew]

,han in the reservoir by an amount Auc, vertical height growth p(eR)s impeded. Such a step stress contrast affects the pressureyadient 7i2 and the crack opening 71 coefficients when Lz > in which both the characteristic distance from the wellbore LejH,; the corresponding S-factors take on the following forms: (e.g., as in Ref. 20) over which the temperature varies from

wellbore e~ to reservoir 9R temperature, and the exponentialcoefficient ~Pi, determine the scale of the thermal dependence.

S~l = ~ [~- AoC[l -e~p(~.i(l - ~))]], (14a) It is also true of viscosity variation that it can only impede,not stop fracture growth; however, the breakdown of the frac-

[[

sin-(~,z~)

1]

fluid viscosity, from heat-up and other causes, aids confinementS82= : u- AuC 1-

sin-l (0,2) (14b) when wting in conduction with stress contrasts because a leSS

viscous fluid produces a smaller excess pressure which in turnand decreazes the equilibrium height,

[ 1

l-t (1-(LLE)2)* + 4) The geometric shape factor Sg, represented bys, = S*2+ 2 Au=;~~,$b!n . (14C)

)1- (1-(/3, ~)2 $ % = 1 + [~ -1] exp(-~c(% 1)),

S. and S.i have been constructed so that the excess pressure{

L1/Lz, Lz < LIL = L2/Ll, L1

(lg)u in the governing equations (Eqns. 5-9) is replaced with the < L2

proper values for confined fracture growth: note that in theIirzt limiting case where the fracture height just reaches the changes the crack opening coefficient between limiting values

confining barrier (L2 = l..,) the three S-factors are all unity appropriate for a well contained VS. a circular fracture, de-

(homogeneous reservoir conditions); for the second case where pending on the aspect ratio of the fracture.

the fracture has grown far out of zone (L2 >> H,) all of the 9. Refactors approximately equal (0

servoir Simulation: This component represents,- Au,)/u, In the latter case, as accurately as needed for practical purposes, the fluid flow

the characteristics of the adjacent strata predominate and thefracture again grows ae if were in a homogeneous medium,

and heat transfer withb the rock in the strata surroundingthe fracture. It thus allows calculation of fluid Ioas and f&-

except the conlining stress UChaa been replaced with UC+AuC. ture efficiency; heat-up of the the-fluid and, consequently,The coefficients ~,~ and /3, are determined from more de- ita changing thermorheolog~ poro-induced backstrezsea; and

tailed analyses; however, several constraints on them can be eventually, production of oil and gas through the propped frac-identified. First, @,l and Pa2 must be chosen so that S82 < Sal, ture.in order that the height growth maybe condned. Second, when The main featurea of the fiuid-leas model are shown inAu. >0 there exists an equilibrium height, beyond which Fig. 2 in which the fracture surroundings are characterizedno further growth occurs at the particular value of G for thl s by four major zones indicated previously (e.g., Refs. 19, 20):case, and especially for highly elongated well contained frac-tures, @,l = B, = 1.

a filter-cake zone 6C, a leaked-off fluid zone 6L, a thermallycooled zone 6h if the thermal front exceeds the leak-off front,

2) Modulus stratification also affects both qi2 and -yl: and a distributed reservoir zone. The penetration depths outinto the reservoir of the thermal and the leak-off fluid fronts

d = : + [1 -:1 xd-@d *)are small relative to the scale of the fhc-pressure diffusion so

(1d ) that the incompressible flow through these zones a~acent tothe fracture can be described by straightforward expressions oi

and DArtys law:Sdi = 1 + [S~ -1] exp(-~di ~). (14e) ~L_2~pF pc=2kL p&pL=2kRp&-pJ

PC & ~~ (15a)PR 6h 6L

Since there is no equilibrium height, the modulus contras t relating the fracture pressure pp to the interface preesure PIcan only impede, not stop fkacture growth; consequently, in th e The viscosity of the fluid occupying each of the zones is mitablllimit of a very large fracture the adjacent strata characterizti Cs averaged to take into account its thermal sensitivityagain predominate arid the fracture grows as if it seen only 1the modulus of the adjacent strata in thw knit both S~l an d /*=C6

~a (0) dzs (lsb;

sdz must have returned to a value of unity and $j ta E/EA inorder to have replaced E with ~A everywhere in the govemin g

where a may stand for the filtercake, the leak~ff, or the ther

equations. The exponential coefiiciente fid and ~di control th emal zone (C, L, and h subscript respectively).

rate of thu replacement and are determined from the rnoredetailed &nalyaes.

Zza

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6 A CX3MPLETE REAL-TIME MODEL OF HYDRAULIC FRACTURING SPE 15069

.

----- .-- -.

Beyond the thermal and leak-off zones, DArty flow and Since the scale of thermal penetration into the reservoir

DmpressibiliQ must both be taken into account; the Greens relative to the fracture length is also small, heat transfer in the

Influence) function solution to the resulting diffusion equa- reservoir is assumed to be ID and normal to the fracture sur-

Ion which governs pore pressure is used to relate the interface face. Further assuming that the reservoir possesses thermally

ressure PI to the loss rate q~: homogeneous properties, a Greens function solution can be

/

t written for the governing thermal transport equation, analo-

PI PR = ~ qL (r) 7~ (~, T) d?, gous to that for fluid loss (Eqn. 15c)

1

+ t(15C)Yf(h ~) = * [4n :_ ~) eff-t3(Z3, t) = A

/~k ~ hL (~) ~~ (2s, O; t, r)-

Combining Eqns. (15a, c) produces the overall governingI%[r&[ 1

eff e (Z, T)] ~h (z3, Z; t, T) dX d?,quation relating fluid loss to fracture pressure:

/

tPF PR - ~ 9~(~)7j(t, r)dr =

[

fiR ~hF( - 6L)H

7h(~3, z; ~, ~) =

1

[-rexp[~~~:l (a)+ [@ce)+&cc],L ~ W) except there is a second integral representing the effect of ther-

mal convection which is absent in the case of fiuid-diffusion.

vhere If=lfor 6~-6rj>Oand H=0 for6h-6LS0. The The effect of fracture area creation is taken into account just

eak-off penetration can be found from the expression as in fluid loss:

s ~= (~n).t ~L (k) = & (tN]6L=A

/

(lb)~d ~ qL (~) d~. (15e)

Yote that since the depth of the frac-pressure diffusion into theThe average fracture temperature is calculated simply from

an expression of energy balance for the fracture:eservoir is typically small relative to the fracture length during;he treatment time, it is sufficient to aesume ID loss normal to;he fracture, provided that the creation of new fracture area is [$ (P C) FVF (eF -

eR)] = (Pc)wd(% - eR)

Lakeninto account; this is accomplished by dividing the volume + AFhL - (@) UILAF(eF - eR) (17C)!bssrate which occured at time level tm by the current fracturemea Ap(tN): where each term respectively takes into account: 1) thermal

qL (tn)= -qL(~J

storage in the fracture; 2) convection flow from the wellbore,(lsf) 3) conduction from the reservoir rock; and 4) convection into

the reservoir.md using this expression in the loss rate integral of Eqn, 15d.Previous comparisons with fully distributed calculations of loss The loss of fluid from the fracture into the surrounding

(Ref. 20) have shown this technique to be sufficiently accurate reservoir causes a swelling or strain in the rock which inFormost practical applications. turn increases the confining stress on the fracture. A ?ormal

If the pressure difference pp pff is assumed constant andexpression for this back-stressn or poro-induced stress UB is

the thermal effects are ignored (lf = O), the resulting special OB =Ug+u:+ug (18a)solution to Eqns. (15d, e) for the fluid loss rate qL has squareroot time behavior, in which the three regions of Figure 2 contribute to the overall

qL = KL/& (16a) stress (Ref. 20). Each of these stresses takes a different form,represented by the following expressions:

where KL is the overall fluid loss coefficient. This !OSScoeffi-cient is related to filtercake, kink-off, and reservoir zone prop- Ug = 2J#CCi.?dL/%t, (18b)erties, in addition to the preaure dMerence PF - pff, by thefollowing quadratic expression in KL: 0; = 2fi~(1 - #C,)~(pF -pi)/?rt(l - V), (MC)

$[&(-@c)+ @c]K~bn &(PI - PR)[l ep(-t-2) -t $erfc (~)] (18d)u: =

+pR@iZ~R z KL(pF-pR)=O.

where ~ = 2 @/t is the dimeaionless diffwion time; 4 is the(16b) lesser of the fracture height or length; q = (1 - ~/&)(l -

2u)/2(1 - v) is the poroelaetic induced stress factor; and b =

The easy solution of this quadratic is provided and discusse d 0.56 is a numerical factor (Ref. 20). Thermally-induced strasws

further in Ref. 18. do not greatly affect hydraulic fracture growth, except in steamand water fioodhg, because the time scale of the thermalconvective-&ffuaion in reservoir rock is much larger thau thefracture growth time.

---

4. Prormant transmort and rdacement: A rigorousmalysis of proppant transport would require the knowledge}f detailed flow patterns in the fracture and the spatial dis-tribution of leak-off, as in Ref. 12; however, such a schemes at present too computationally demanding for many real-,ime applications. Therefore, in keeping with the approach ofhis model, the essential characteristics of proppant transport:e.g., mass conservation, concentration distribution, differen-tial settling, bank formation, screen-out, and closure) are in-:luded explicitly, and gamma coefficients provide a means toIata-bsae the more accurate modelling results.

In order to calculate the penetration into the fracture of:ach proppant concentration band (Fig. 3) corresponding to;he treatment staging, the slurry volumes in the fracture VP-~fter leak-off must be calculated:

tvP. = /[ Q- Qbn]~r (19)

tpm

where tpm is the time when the rnth proppant stage enteredthe fracture, and Q~~ is the volume leak~ff rate through thehcture area adjacent to VP-. Once VP- is obtained, the ex-tension into the fracture of each concentration band Lpm isFoundfrom mess conservation yielding

v(723 - %) (%)3 -723 (%)2 + % ~ = o (Zoa)

where VF is the fracture volume. The height extension of eachband lfPm is then related to LP~ by a ratio of fracture heightto length

Hp. =LZ

7P LP. ~ (20b)

h which :iw default value of 7P is 1. The proppant volume ssa whole ISthen settled downward in accordance with relationsfor the unhindered and hindered settling velocities (e.g., as inRef. 21):

(h+l)d (Pp-Pf)~ *V8 =

108 n [ 72 K I (21a)and

Vef:h = l(J1.s2(l-f.)

(21b)

where V*and v~bare respectively the unhindered and hinderedsettling velocities, and fv is the fractional volume of suspendedproppant.

Sample Implementation of the Models

To illustrate the capabilities of the overall integrated mo-tlel, we analyse a fracture treatment performed in the TravisPeak formation of Eeet Texas on which the models were imple-mented in real-time, using the Gas Research Institute (GRI)fracture treatment monitoring and analysis facility developedby Resources Engineering Systems (RES) and described inRef. 22. The well was one of many monitored and analysedunder the auspices of the GRI Tight Gas Sand Program, in con-junction with a number of ~,ther GRI contractors (e.g., Ref. 23).

The flow, rheology and proppant data obtained, and usedor real-time input to the models, are shown in Figure 4(a).rhe resulting model calculations of uphole pressure are shownn Figures 4(b, c), where comparisons are made with the actualneasured pressures.

Unknown reservoir end friction parametem were obtained)y history-matching, a technique that involves comparing the)bserved pressures and model predictions in order to minimize;he error between them. When history-matchhg is performedm the early response to pad injection, the parameter values>btained may then be used during the rest of the job to en-mre the beat model predictions possible. Use of our wellboretransport model may be seen to give an excellent fit to upholepressure in Figl re 4(b). Although the form and coefficients ofthe pipe friction component of the model have been chosen tobest fit an average of all industry lab and field data availableto us (e.g., as in Ref. 24), fine tuning of these coefficients byhistory-matching has improved the accuracy within 5 % overthe entire duration of the fracture treatment.

Even more intereating is the compsmson of excess pres-LiUreS (PF - crc) in Figure 4(c), remembering that the extentof fracture containment may be estimated from the pressurerise or fall-off (e.g., as in Eqne. (llc, 12c)) es long se otherinfluences on fracture pressure can be identified and separatedout. After the initiation phase, which is still under study withthe distributed 3D model (Ref. 12), we see that the reservoirparametem associated with no containment produce an excel-lent fit during the early part of the job, including the period ofthe great rise in pressure which is due to the first appearanceof 50 lb. cross-linked gel in the fracture. After thw early phaseof no containment, the fracture pressure continues rtilng thebest fit with model predictions during thw second phase seemsto indicate that the fracture has now encountered subatantiafconfining barriers; however, thb pressure rise could also havebeen caused by a greater flow resistance, due to proppant stag-ing, than we had incorporated into the proppant shape factomSPi of our model. Late in the job, when the heaviest proppantloading was reached, the pressure response turned sharply up-wards. A screenaut was clearly developing, as no amount ofcontainment in the models could have produced by itself suchan increase in pressure,

Although we have been able to identify mechanisms suchss sliding resistance, bank buildup, and screen-out as poten-tially dominant contributors to fracture (excess) pressure accounting for much of the pressure rise in Fig. 4(c) de-scriptions of these mechanisms have not yet been finalised inour models. Until enough reliable field data haa been matched,the confinement inferred from the pressure response will remainuncertain for many treatments. A Nolte-Smith type of analysis(e.g., Refs. 14, 25) would also require detailed incorporation ofsuch modelling and mechanistic interpretatiom, if proper ex-planations are to be made: although such analyses are nowbeing conducted as a major focus of treatment monitoring ve-hicles in the field (e.g., R&. 24, 26), it has become clear thatmuch more detailed modelling will be required in order to dw-tinguish between the effect on observed pressure of fracturecontainment and the other equally important influences.

7

A COMPLETE REAL-TIME MODEL OF HYDRAULIC FRACTURING

.

SPE 1506$

Given that this is the case, the range of possible fracture Nomenclature,mmetries corresponding to different levels of confinement, as AF area of fracture winghewn in Figure 4(d), is often as close as one can come to A area of wellbore crossaectionscertaining fracture geometry. Designing or analyzing frac-ure treatments based on this quality of information consists of B bulk modulus of wellbore slurry

bounding the fracture within the limits of acceptable best and cc volumetic fraction of filtercake polymer in frac-rorst case estimates. If the fracture pressure can be history- fluidnatched with certainty, having eliminated ambiguous or un- c~ pore-fluid diffusivity of the reservoir,nown influences on the pressure, it is then possible to narrowhe interval of probable fracture geometries to a single best

CW,CF, CL specific heats of the wellbore slurry, fracture

stimate of final fracture dimensions. The dominant reservoirslurry, leak-off fluid

parameters determined by such historymatching on fracture D hydraulic diameter of the wellbore

jressure are the confining stress contrmt, the fluid leak-off co- E Youngs modulus for the reservoirfficients and the reservoir modulus. The extent of proppant E crackapening modulus for the reservoir,ransport is indicated in Figure 4(e), corresponding to the lim- E/4(1 U2)ts of no containment and high containment. With these es- erf error fimctionimates of the final propped length and width of the fracture,alculations of oil or gas production can be made (e.g., Refs. 20, erfc complementary error function

;3); eventually, our program may be employed to make these &J, EL, EA crack-opening moduli for upper, lower and ad-)redict ions at any stage during the job, allowing decisions to jacent strata (when symmetric)

hange schedules or stop pumping based on real-time predic- f friction factor for wellbore flowions of well productivity.

f. Clapps friction factor for power law fluids

9 gravitational constant

& half height to modulus barrier

ConclusionshL conduction heat exchange rateHa half height to stress barrier

A comprehensive integrated model of hydraulic fracturin g;and wellbore transport has been developed and incorporated

spatial coordinate index

into a system for real-time monitoring and analysis of fracture K consistency index, viscosity for linear fluid

treatment operations. The model is baaed on a full 3D simula - K, Ka bulk moduli of the overall porous matrix andtion of hydrafrac creation, but it has been formulated so as to solid constituent, respectively, under drainedallow rapid execution, using data-based coefficients previousl y conditionsdetermined by other more elaborate numerical analyses, and &C,&L,k~by comparison with field observations. This allows both eazy

effective permeability of the filtercake, leak-offand reservoir zonea

identification of major features and isolation of possible error sduring design and evaluation, but especially it permits mult1-

Kk conductivity of reservoir rock-fluid matrix

ple executions of the models, faster than real-time, thus mak - L overall fluid loss coefficient

ing possible the on-ite determination of reservoir and frictio n L1 length of one fracture wingparameters, by history-matching on observed pressures as th e L2, L2U, L2L half-height (when symmetric), upper, lowerjob proceeds. This real-time feature of using flow and rheolog y heights of the fracturefield data as input, to calculate actual pressures and determin eLHparameters from the job response, itself sets the model apar t. Pm! Pm wing length and half height of rra:h proppant

Of course, the model can also be used for pre-frac design andstage penetration in the fracture

post-frac analysis, running on personal computers, minicom - Le,, Le, lateral, vertical thermal penetration scales im

puters and mainframes, the fracture

Since all of the essential physics of the hydrafrac proce Ss1 the lesser of the fracture wing length or half-

height; denotes generic length dominating craclappear explicitly in the model, routine improvements in its ac- openingcuracy are simply made by updating the gamma coefficients;thus future advances in modelling and experience gained from .t!B length of the j:h slurry section in the wellbor~

field applications can be continually incorporated without any MB mass of the jth slurry section in the wellborerestructuring of the model. Current and future efforts are fO-n flow behavior indexcused on improving and verifying default forms of the gamm acoefficients by comparison with fully 3D fracture simulation s, P

pressure

accurately incorporating all of the physics (e, g., wall frictio n, PFv pR fracture preesure, reservoir pore pressure

screenaut) affecting the fracture pressure response, and opti- PC, PLY PI pore pressures at the fihercake, Ieak*ff, thermizing procedures for determination of reservoir and fricti on mal interfacesparameters by real-time history-matching. Q slurry volume flowrate into one wing of th~

fracture

.-z--

.

>W I KnRo A R fiRfW!Kl?TT. N.M. OKUSU & M.P. CLEARY {LJ J.uvvcz . . . . . . ---- ----- -

, _.. _.. .

!Lm volume loss rate through fracture area adjacent P effective channel flow viscosityto mth proppant stage P. apparent viscosity of the slurry

L volume loss rate per unit area of one fracture PC, pL> #R viscosities of the filtercake, 1eak4f, and reser-face voir fluids

t radius of a circular fracture v Poissons ratio

tc= Reynolds number for a power law fluid d porosity

d,Sdi shape factors for modulus contr~t p, pF wellbore, fracture slurry densities

9 shape factor for fracture geometry Pp, Pf proppant, frac-fluid densities

inl inverse sine function Pr slurry density at a reference pressure

La*. shape factors for fluid loss Pw, PL wellbore slurry, leak-off fluid densities

Jpa~, shape factors for proppant 9 temperature,)#j shape factors for stress contrast @w, @F, @R wellbore, fracture, and reservoir temperatures,pi shape factors for thermal heatup u excess pressure

time uB induced stress due to thermal and poro-elastic

Pm time of entry of the mth proppant stage into effectsthe fracture u:, Ug, Ug poro-induced stress for the filtercake, leak-off

{F fracture volume and reservoir zones

/Pm fracture volume up to the m:h proppant stage u. confining (or closure) stress in reservoir

/ slurry velocity down the wellbore AOCU,AOCL stress contrasts for upper, lower strata

l,, vah unhindered and hindered settling velocities Au. stress contrast for adjacent strata (when sym-

*, w mass flow rate and total pumped mass into one metric)fracture wing r shear stress

ifiL , 2WL mass loss rate and total mass lost out of onefracture wiruz Acknowledgements

01, tiz lateral/vertical mass flow rates per unitheight/length in the fracture

The work presented in this paper has been performed as part

cl, Z2, z~of the GRI Tight Gas Sands Project, a comprehensive effort to

spatial coordlnatea in the lateral, vertical, and imp we many aspects of hydraulic fracturing operations. Wenormal directions with respect to fracture are grateful to H, D, Vo and R. M. Willis for their assistance in

dc S-factor coefficient for fracture geometry implementing the model on the computer and to Marie Dok-

~d, ~di S-factor coefficients for modulus contrast oupil for helping to prepare this manuscript.

flev@.i S-factor coefficients for stress contrast

0Hi S-factor coefficients for thermal penetration Figures

1 engineering shear strain-rate 1. Hydraulic fracture and wellbore with reservoir surroundYf Greens function for pore-fluid diffusion ings, showing principal variables used in modelling.

Yh Greens function for thermal conduction 2. Zones of leak-off and thermal penetration in the reservoil7P gamma factor for proppant shown as ID normal to the fracture surface.

7. gamma factor for fracture volume 3. Bands of proppant in the fracture corresponding to th71 gamma factor for crack-opening fracture treatment staging.

712, 722 gamma factors for lateral, vertical stress gra-dient

4(a). Flow, rheology, and proppant data; time in minutes.

?13, 723 gamma factors for lateral, vertical cross~ectio n 4(b). Uphole pressure data showing comparison of model w

area actual pressurea; time in minutes.

714, 724 gamma factors for lateral, vertical channel flow 4(c). Downhole (excess) pressure data showing comparison f

%S, 726 gamma factom for fluid 10ss model vo.. actual pressur=, time in minutes.

A width of fracture at wellbore 4(d). Model estimates for final hydraulic fracture length, heigh6C, lfL, dh filtercake, leak-off and thermal penetration and width showing dependence on confining stress co]

deptha trast, the principal containment parameter.

n poro-elastic induced stress factor 4(e). Model eotimates of proppant extent, variation with tinq = (1 - *)(1 -2u)/2(1 -v) and confinement; time in minutes.

n thermal diffusivity of the reservoir

~L thermal diffusivity of the leak-off fluidZzv

.

A COMP1.F!TR REAL-TIME MODEL OF HYDRAULIC FRACTURING SPE 15069

.

. . ------ . -. -. -.-..- .-- -

References

. Geertsma, J. and F. de Klerk, A Rapid Method of Predict- 14, Nolte, K. G. and M. B. Smith, Interpretation of Fracturinging Width and Extent of Hydraulically-Induced Fracturesn, Pressures, J, of J%t. Tech., Sept. 1981, pp. 1767-1775.J. Rt. Tech., Dec. 1969, pp. 1571-1581. 15. Cleary, M. P., Theoretical and Laboratory Simulation of

. Danezhy, A. A., On the Design of Vertical Hydraulic Frac- Underground Fracturing Operations, Firat Annual Reporttures, J. Rt. Tech., Jan. 1973, pp. 83-97. of the MIT UFRAC Project, MIT Resource Extraction Lab-

. Perkins, T. K. and L. R. Kern, Widths of Hydraulic Frac- oratory Report, Aug. 1981,

tureen, J. Rt. Tech., Sept. 1961, pp. 937-949. 16. Settari, A., Quantitative Analysis of Factors Influencing

. Nordgren, R. P., Propagation of a Vertical Hydraulic Frac- Vertical and Lateral Fracture Growth, SPE/DOE Paper

ture, J. Sot. M. Eng,, Aug. 1982, pp. 306-314. No. 13862, presented at the SPE/DOE Low Permeability

. Cleary, M. P., Comprehensive Design Formulae for Hy-Gaa Reservoir Symposium, Denver, May 19&i,

draulic Fracturing, SPE Paper No. 9259, presented at the 17 Narendr~t M V*Analysis of the Growth and Interaction

55th SPE Annual Fall Technical Conference, Dallas, Sept. of Multiple Plane Hydraulic Fractures, Ph.D. thesis, MIT,

1980 (see also SPE Paper No. 9260, Analysis of Mecha- Dept. of Mech. Eng., February 1986.

nisrna and Procedures for Producing Favorable Shapes of 18. Crockett, A. R., Willis, R. M. and M. P. Cleary, lrnprov-Hydraulic Fractures ). ing Hydraulic Fracture Predictions by Real-Time Hietory-

1. Settari, A. and M. P, Cleary, Three-Dimensional Simula- Matching on Observed Pressures, SPE Paper No. 15264,

tion of Hydraulic Fracturing, J. of Rt. Tech., July 1984, presented at the SPE Unconventional G= Recovery Sym.

pp. 1177-1190 (see also SPE Paper No. 10505, Develop- posium in Louisville, KY, May 1986.

ment and Testing of a Pseudo-Three-Dimensional Model of 19. Settari, A., A General Model of Fluid Loss in HydraulicHydraulic Fracture Geometry, presented at the SPE Sym- Fracturingn, SPE/DOE Paper No. 11625, presented atposium on Numerical Simulation, New Orleans, February the SPE/DOE Joint Symposium on Low Permeabiliw Gas1982). Reservoirs, Denver, March 1983.

~. Meyer, B. R., Frac Model in 3D - Parts 1-4, Oil and Gas 20. Crockett, A. R., Vo, H., and M, P. Cleary, Studies of FluidJournal,, June 17 (p, 87), July 1 (p. 62), July 22 (p. 83) and Flow, Heat Transfer and Induced Stresses In and AroundJUIY 29 (p. 132), 1985. Underground Fractures, Report No. REL-84-1, MIT Re-

J. Palmer, I. D., and C. T. Luis, A Model of the Hydraulic source Extraction Laboratory, March 1984.

Fracturing Processes for Elongated Vertical Fractures and 21. Danezhy, A. A., Numerical Solution of Sand Transport inComparison of Results with Other Models, SPE/DOE Pa- Hydraulic Fracturing, J. ht. Tech., Jan, 1978, pp. 132-per No, 13864, presented at the SPE/DOE Low Permeabil- 140.ity Gas Reservoir Symposium, Denver, May 1985. 22. Cleary, M, P., Buharali, A. M., Crockett, A. R., and I. A

). Advani, S. H., Finite Element Model Simulations Asso- Salehi, Computerized Field System for Real-Time Moniciated with Hydraulic Fracturing, SPE/DOE Paper No. toring and Analysis of Hydraulic Fracturing Operations8941, presented at the SPE/DOE Unconventional Gae Re- SPE Paper No. 14087, presented at the 2nd Internationalcovery Symposium, Pittsburgh, May 1980, SPE Meeting, Beijing, March 1986.

1. Cleary, M. P., Kavvadaa, M., and K. Y. Lam, Develop- 23. Holditch, S. A., Robinson, B. M., and W. S. Whiteheadment of a Fully Three-Dimensional Simulator for Analy- Pre-Fracture and Pozt-Fracture Formation Evaluationssis and Design of Hydraulic Fracturing, SPE Paper No. Necessary to Characterize the 3D Shape of Hydraulic ~11631, presented at the Symposium of Low Permeability Fracture, SPE Paper No. 14086, presented at the 2n~Gas Reservoirs, Denver, March 1983. (See also Report International SPE Meeting, Beijing, March 1986.No. REL-82-12, MIT Resource Extraction Laboratory $ 24. Hannah, R. R., Barrington, L. J., and L. C. Lance, ThDec. 1982.) Real-Time Calculations of Accurate Bottomhole Fractur

1. Abou-Sayed, A. S., Sinha, K. P., and R. J. Cifton, Evalu - ing Pressure fro: LSurface Measurements, Using Measure~ation of Hydraulic Fractures Using a 3-D Simulator: Part 1 Pressured as a Base, SPE Paper No. 12062, presented a Technical Approach, SPE/DOE/GRI Paper No, 12877, 58th SPE Annual Fall Technical Conference, San Francisccand Abou-Sayed, A. S., Clifton, R. J., Dougherty, R. L., an d SerA. 1983.R. H. Morales, Evaluation of Hydraulic Fractures Using a3-D Simulator: Part 2 Caae Studies, SPE/DOE/GRIPaper No. 12878, presented at the SPE/DOE/GRI Uncon-ventional Gas Recovery Symposium, PM.sburgh, 1984.

2. Lam, K. Y., Barr, D. T. and M. P, Cleary, A CompleteThree-Dimensional Simulator for Analysis and Design ofHydraulic Fracturing, SPE Paper No. 15266, presentedat the SPE Unconventional Gas Recovery Sympozium inLouisville, KY, May 1986.

3. Govier, G. W. and K. Aziz, !l%enow of Cornplez Mixturesin Hpss, N.Y. Van Nodrand RAnhold Co., 1972.

25. Nolte, K. G., Determination of Proppant and Fluid Schedules from Fracturing Pressure Decline, SPE Paper Nc13278, preeented at the 59th Annual Fall Teciudcal Conference, Houston, Sept. 1984.

26. Cooper, M. P., Nelson, S. G. and M. D. Schoppe-, Cornparison of Methods for Determining In-Situ Leak-Off RatBaaed on Analysis with an On-Site Computer, SPE PapeNo. 13223, presented at the 59th SPE Annual Fall TechniciConference, Houston, Sept. 1984.

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