Terapaper Modeling of AcidFracturing Tr-5034

8/13/2019 Terapaper Modeling of AcidFracturing Tr-5034

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Modeling of Acid Fracturing TreatmentsAntonln Settarl SPE, Simtech Consulting Services Ltd.

Summary. This paper gives the theory and numerical implementation of a comprehensive acid-fracturing model that solves for fracture geometry (2D or 3D), leakoff, heat transfer, and acid transport simultaneously. The acid-transport model integrates a number offeatures not accounted for in earlier design models: multiple fluids (acids, gels, and foams); acidizing controlled both by mass transferand rate of reaction; and leakoff, including the effects of matrix acidizing and wormholing. Examples of treatment design illustratethese features. Coupling with reservoir forecasting models gives the ability to optimize the job or compare it with a proppant treatment.IntroductionAcid fracturing currently is receiving renewed attention as a wellstimulation technique. Compared with advances in modeling toolsfor proppant fracturing, the models for acid-fracturing processeshave not changed much in the last decade. During a period of intense study in the 1970's, the basic understanding of the reactionkinetics and acid transport in the acid-fracturing process was established by laboratory and theoretical studies. I-3 The designmodels of acid-fracturing typically were decoupled from fracturegeometry models and use fixed fracture width and prescribedleakoff. 4-8 Also, it was recognized that quantitative prediction ofwormholing was a major obstacle to theoretical predictions of acidpenetration. t also was a principal reason for the discrepancy between predicted and observed acid-fracture lengths.Very little work on acid-fracturing design was reported in the1980's. The earlier models, with decoupled fracture-geometry calculations, were still being used. This was in sharp contrast to thegreat increase of sophistication in propped fracture design and madecomparisons of these two alternatives difficult. Only recently wasinterest in acid fracturing revived. Ben Naceur and Economides 9and Lo and Dean 10 presented improved models of acid transport,coupled with a specific 2D model of fracture geometry. Gdanskiand Lee II described a more general model, but their work lackedmathematical details necessary to other modelers. Theoretical andlaboratory work currently is being done to understand wormholebehavior l2-14 (which is the major factor responsible for largeleakoft) and to develop fluids with better reaction and leakoffcontrol. 15,16

This paper gives the theory of a comprehensive acid-fracturingmodel, the numerical implementation of the acid-transport andleakoff solution, and examples illustrating various features of themodel. The model includes a number of new features essential formodern design.1 Multiple fluids with different rheologies (reacting or not), including gel/acid sequences.2. Acidizing process controlled by both mass transfer and rateof reaction.3. Leakoff behavior, including a new method for calculating theincrease in leakoff caused by matrix acidizing and wormholing effects at the fracture wall.4. Coupling with heat transfer by means of the thermal dependency of reaction kinetics and the heat of the reaction.5. Implicit coupling with 2D or 3D fracture geometry and withreservoir forecasting models to optimize the job or compare it witha proppant treatment.

This formulation is considerably more general than the modeldescribed in Ref. 10, which assumes infinite reaction rate and treatsacid transport as steady-state. These features are illustrated by examples of acid transport under different physical conditions (limitedby mass transfer, reaction rate, effect of reaction order, etc.). Thecase study of a treatment with an alternating gel/acid sequence showsthe model's ability to assess the importance of various design parameters. n particular, the model can predict short penetration distances in agreement with field experience.Equations for Acid TransportFor simplicity, an element of a fracture with variable width butconstant height is considered (Fig. 1). In the following, it will beCopyright 1993 Society of Petroleum Engineers30

assumed that concentration, C, is defined as mass of acid per volumeof solution. f one wishes to use concentration C in terms of moles,C must be replaced by MwC, The equations describing the flowof acid in a channel of unit height are given below.The equation of continuity within a channel of variable width is

a(vz)/ax+O(vy)/iiy=aAlat-i, (1)whereA = elemental area dxdy. The term aAlat represents the changeof control volume caused by the change of fracture width, b withtime. The boundary conditions are

Vyly=bl2=Ve,where vf= leakoff velocity. By integrating this across y and z timing, we obtain the equation solved by the geometry module of afracture simulator (except for compressibility):

a(vAc)lax-2qe x)=aA clat-i, (2)where v x,t) = average volumetric velocity, Ac(x,t) = local crosssectional area, and qe(x,t)=leakoff rate on one fracture surface.f constant width vertically is assumed, Ac and qe are given by

c = Y3hf band qe=hffVe,where hf=total fracture height and hfe=leakoff height.

f we neglect diffusion in the x direction,a a a ( ac ) ac- - C v x ) - - C vy - De-= iCj , (3)ax ay iiy iiy at

where Vx and vy=2D components of the velocity field, C=acidconcentration, and iCj=acid injection rate. The effective diffusioncoefficient, De is defined below. Two boundary conditions apply:(1) at the wellbore (x=O), C=C j and (2) at the fracture wally=b/2), the boundary condition states that the total flux of acidto the surface equals the acid spent plus acid leaking off with velocity ve The total mass transport to the wall is the sum of diffusion and convection:-qA= -CBVy+De aC)1 , (4)ay y=bl2

where De = effective diffusion coefficient, which for laminar flowis equal to the molecular diffusion but increases in the presenceof turbulence. The reacted amount is assumed to be

r=k l-q,) CB-ceq)m, ............................ (5)where m=reaction order, k=reaction-rate constant, and CB andCeq = boundary (wall) and equilibrium concentrations, respectiveIy Ceq takes into account the effect of reverse reaction, which maybe important for weak acids. 8 The amount lost to the formation is

qAe=veCf ....................................... (6)where Ce= loss concentration . Also, by continuity, the leakoffvelocity, ve, must equal the boundary velocity, vY which gives thegeneral boundary condition as

- -CBVe+De aC)1 =k 1-q,) CB-ceq)m+vf Ce (7)ay y=bl2SPE Production Facilities, February 1993


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c IC vi yx

I

h I O ~ i ~ 7 _

Fig. 1 Physlcal system for acid transport model.The only unclear issue in this equation is the relationship between

CB and Ce. Note that this issue disappears if leakoff is zero. Mostauthors5-7,lO assume thatCB=Ct . (8)

However, analysis of a control volume at the fracture wall, assuming that all reaction occurs on the surface, suggests that the acidconcentration would have a discontinuity at the surface so thatCt7,000):NSh =0.026(NRe )O.8(Nsc ) II . 15a)In transitional flow 1 ,800


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2.50.00.00 0.25 0.50

CUM.ACID CLBIFT2) 6 ~0.0 '0 . : ; ; ; ; ; ; , = = ~ + - - ~ ~ 3 f - - F 1 - L - T R A + 4 - T 1 - 0 - N - r t > , - M - E - , S-Q-+:-TC-M-IN- 71 - - - -+ - - - - 1

Fig. 3-Measured and simulated leakoff with pad and acidfluids-Laboratory Experiment 2.wood number is fairly constant, but Roberts and Guin'sl work indicates strong influence of free convection (see their Fig. 6). Unfortunately, such results were not correlated with the Reynoldsnumber. Lo and Dean 10 recently presented a correlation for thelaminar flow region. They define the Nusselt number as

NNu = -(bIC)(aClaY) y=bI2 (16)and assume that CB=0. Their definition can be generalized for thecase of CB 0, which makes it equivalent to the Sherwoodnumber:

NNu = - _ b ac I =bKglDe =NSh12. . . . . . . . . . (17)C-CB) ay y=bl2

Ref. 10 then correlates NNu with the Peclet number, NPe =vf bl2De by

NNu =4.10+ 1.26Npe a 2 N ~ e . . . (lSa)if NPe < 10 and

NNu =2Npe (lSb)if NPe ~ 2 0 where the constant a2 (quoted as 0.04 in Ref. 10) mustbe a2 =0.02675 to maintain continuity. Eqs. 15 and IS are usedin the model by default. However, they predict a large (an orderof magnitude) discontinuity in Kg at the boundary between laminar and transitional flbw, which makes them far from satisfactory.Also, the correlating dimensionless groups change at this point.Because of the difficulties of a priori prediction of mixing, thebest method of determining the mass-transport coefficients is bysimulating the experiments for acid flow between parallel reactivesurfaces and matching the measured acid concentration profile. Forthis purpose, a separate model was developed. This model simulates results of a laboratory experiment with a fracture of constantwidth and with prescribed leakoff. The matched values of Kg andk can then be used for field predictions. The model also can be usedfor field-size simulation with a fracture of constant width andprescribed leakoff, which is comparable to earlier design models.eakoff and Wormhollng

The increase of leakoff from the combined effect of matrix acidizing and wormholing is believed to be a primary reason for the generally short penetrations of acid fractures. The increase of leakoffin laboratory cores has been measured by a number of investigators, 15,16,19 and an example of actual data is presented below. Inthe field situation, the increase measured in the laboratory will affect primarily the invaded Cl ) region. Therefore, one would expect field leakoff to increase to the same degree in gas reservoirs32

.: r. . ,' . . ,Vlm

o ,f o o, . . C------ - . - - - - - - - c

Fig. 4-Representation of wormhole leakoff in the model.and to a somewhat lesser degree in oil reservoirs. Nierode* andKing** found that the increased leakoff was necessary for agreement between designed and measured fracture lengths. Huckabee,20 however, reported that field testing indicated no increasein the overall leakoff coefficient in an oil reservoir.The wormholing effect is particularly difficult to predict analytically. Hung'sl3 method allows calculation of single wormholegrowth and simulation of the competition of several wormholes,but it makes simple assumptions about the pressure field. The calculations also depend on the knowledge of statistical distributionof pore sizes, which requires laboratory work. Daccord et at 14used fractal concepts to characterize the wormhole pattern. Whiletheoretical models of wormholing may become predictive in thefuture, the laboratory leakoff measurement currently is the meansof quantifying measured leakoff for modeling. The method hereis proposed to express the increase in reacting fluid leakoff compared with leakoff of an inert fluid. Then acid leakoff is simulatedwith the general model of Ref. 21 and increased in this ratio. Methodimplementation is as follows.1. Generate a theoretical prediction ofIeakoff velocity, vf, vs.time from the leakoff model used in the fracturing simulator, withrealistic data for the acid fluid but without acidizing effects. Sucha forecast can be obtained by running a laboratory leakoff test withinert fluid properties and matching it with the model developed spe-cifically to simulate leakoff data.2. In the laboratory, measure the leakoff velocity, v )A' for theacid. From the comparison of (vf)A measured in the laboratorywith the theoretical inert fluid leakoff ve for the same fluid, determine the ratioRAf = ve)A IVe=f(MA) ............................. 19)

where MA = some measure of the cumulative amount of acid contacting the fracture wall.3. When the function RAe has been established, calculate theleakoff velocity in the fracture-geometry model:(ve)A(x,t)=RAe ,)ve(x,t), ........................ (20)

where ve is computed the usual way (without the effect ofacidizing) .In the model, RAe can be correlated with one of two quantitiesthat are internally computed as part of the solution: RAf = f(MAf )where MAi is the cumulative mass of acid loss to the formation perunit area; or RAf=f(MAt ), where MAt is the cumulative mass ofacid contacting the wall per unit area (Le., acid spent and leaked off).Note that, although the method of representing the data throughRAf was motivated by an increase in leakoff, it also can be usedto model a decrease that may result from pore collapse or solidprecipitation with some acids. The function RAf can be determinedby matching two laboratory measurements. An example is shown

'Personal communication with D.E. Nierode, Exxon Production Research Co.,Houston (1988).Personal communication with G.E. King, Amoco Production Co., Tulsa (1987).

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TABLE 1-THERMAL REACTION DATA FOR COMPARISON WITHLEE AND ROBERTS18 DATA (LIMESTONE AND DOLOMITE FORMATIONS)

Acid typeMass-transfer coefficient, KgReaction constant, em/secReaction orderEquilibrium concentration,Dissolving power, kg/kg acidDensity of soluble rock material, kg/m 3Activation energy, kJ/kgmolHeat of reaction, kJ/kgReference temperature for reaction data, CKey to add loss velocity to Kg

in Fig. 2, where a laboratory run on a low-permeability gas reservoir core with the pad fluid was followed by a displacement withacid. The measured and simulated leakoffvolumes from these tworuns are shown. The simulated match was obtained with the function RAi shown. After the match, a second laboratory test on adifferent core was simulated and matched with the same functionRAi . This result, shown in Fig. 3, supports the validity of the concept expressed by Eq. 19. For field application, however, reducing the effect in liquid-filled reservoirs may be appropriate if theleakoff can be calibrated from good minifracture data. Further improvement of this method is possible by applying the correctiononly to the C1 region in the field situation.Relation Between Leakoff and Mass Transfer. When the modelwas tested with increased leakoff, it was found that, for certain lowvalues of mass transfer,' the computed acid concentrations in thefracture would increase above the input concentrations, which isphysically impossible. This problem (also reported in Ref. 10) iscaused by ignoring the relationship between Kg and Vi and by heterogeneity of leakoff (wormholing).In the homogeneous-Ieakoff case (i.e., no wormholing), the phenomenon can be explained as follows. As discussed, the concentration in the fracture will remain constant if the total amount ofacid leaving the fracture face is equal to the leakoff rate times thein-situ concentration, qAi=ViC. If we now have a reaction that ismass-transport-limited, CB will be close to zero (or Ceq), and thetotal amount of acid leaving is according to Eq. 14, viCB+Kg(C-CB), which potentially can be smaller than ViC. However,the mass-transfer coefficient is physically related to boundary-layerthickness, and the boundary layer will diminish with increasingleakoff. Therefore, Kg must increase with leakoff. To the best ofmy knowledge, this dependency has not been measured and reportedin the literature; however, this atgUment can be used to give alimiting condition on the value of Kg:

V i C B + K g ( C - C B ) ~ C V i (21)The worst case is if CB=O; therefore, the above condition will always be satisfied if

K g ~ V f (22)Because most of the reaction data are obtained for low leakoff

rates, one could argue that the leakoff velocity should be addedto the measured data-i.e., one should use an effective coefficient,Kge=Kg+vi 23)In the model, an option is provided either to increase Kg automatically to always satisfy Eq. 22 or to add leakoff velocity to Kgaccording to Eq. 23.In the presence of wormholing, one must also consider the heterogeneity of leakoff. Once a wormhole develops, the leakoff intothe wormhole is at a concentration close to C nstead of CB Also,the leakoff rate is nonuniform because most of the leakoff rate isthrough the wormhole (Fig. 4). Therefore, for calculation of theloss term, a weighted average of the two concentrations should betaken, which will restore physically correct mass balance. Then

SPE Production Facilities, February 1993

Limestone FormationH ICorrelation4.129x10- 40.441o1.3727100.23000 x 10 5

0.10910x 10 437.770o

DolomiteFormationH ICorrelation1.252x 10- 30.669o1.3727100.40480 x 10 5

0.10910x 10 493.330oKg can remain at a value consistent with the much lower matrixleakoff rate.The effect can be quantified by the use of the leakoff increasefunction, RAe, defined by Eq. 19. We assume that part of the increase results from increased matrix leakoff and part from flow intothe wormhole. We can calculate the two components of leakoff.The total leakoff velocity is

(vi)A =RAeve, (24)where Vi is the inert fluid leakoff. Therefore, the total increase is

Ai -l~ v i = RAi -1)ve = . v i k 25)RAeThe wormhole leakoff velocity, V l w and the matrix velocity, vimare

Ai -lvew=Fw .ve=Fw Ve)A 26a)RAiand Vim =(Ve)A -Viw (26b)

We can now reformulate the boundary condition for total fluxof acid to the fracture surface (Eq. 14 if we assume that the leakoffto the wormholes is with the concentration Cand the cross-sectionalarea of the wormholes is small so that the decrease in the reactivearea ofthe matrix is negligible. Then Eq. 14 can be generalized to

QAt=2[QimCB+qiwC+hjrKg(C-CB)] (27)Implementation of Eq. 27 requires only additional input of thewormhole fraction, w If w is high, use of this equation eliminates problems with nonphysical concentrations. To guarantee consistency under any conditions, we must apply the argument givenabove to the matrix component. The conditions of Eqs. 22 and 23change to

g ~ v e m 28a)and Kge =Kg+vim 28b)Thermal eatures of the odelThe fracturing simulator that hosts the acid-transport model canbe run in an isothermal or thermal mode. Correspondingly, the formulation of the acid model includes several features that are coupled with heat transfer. These features become active if thefracturing simulator is run in the thermal mode.Mass Transport. The internally generated mass-transport coefficient, Kg, is a function of temperature. This is reflected in the calculation method of Eqs. 15 or 18 because De and the fluid PVTproperties and viscosity that enter the dimensionless groups are functions of temperature. An alternative is to use an Arrhenius typeof equation, as is customary for the reaction rate.The reaction-rate constant, k, is assumed to be a function of temperature according to the Arrhenius equation:

k=ko exp(-EaIRT), (29)33


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r---------------------------------------,

51 110

UMBSTONB6DOLOMITBx NO BUT 01' u e nON

Fig. 5 Temperature profiles with and without heat ofreaction limestone and dolomite formations.

where Ea = activation energy constant. The value of k entered inthe input data is assumed to be k, at the reference temperature, T,.Then the value at any other temperature can be computed ask(T) =k,exp[ Ea ( - ~ ] . . (30)R T, T

Heat of Reaction. The heat generated from the reaction can significantly change the temperature and therefore the acid spendingalong the fracture. 9,18 The heat energy generated is related to themass of acid reacted byQ,=k(1-cP)(c-cB)mHrxn =rHrxn (31)

where Q = ate of heat generation per unit area and Hrxn =heat ofreaction (energy/mass of acid reacted). In the model, the amountof acid reacted in each gridblock during timestep n Wp is computed and stored. During the solution of the heat balance (whichfollows the solution of acid transport), the heat generated in thefracture gridblock is W,H xn and the accumulation side of theenergy equation is modified to include this term, which is essentially a heat source in the energy equation.The feature was tested in the model with Lee and Roberts'18data. They solved two example problems assuming a limestone ordolomite formation, with other common data about the treatmentshown in Table 1. Unfortunately, the reaction data are incomplete,and their model appears to be decoupled from fracture geometry(constant fracture width). Consequently, we cannot expect to u p l i ~cate their results. Using the interpretation of the reaction data inTable 1, we obtained the temperature distribution in the fractureshown in Fig. 5. The results are similar to Fig. 2 of Ref. 18, butthe temperature peaks are closer to the wellbore.Numerical mplementationThe above theory of acid transport and leakoff was implementedin three separate codes: stand-alone leakoff (LOSS), stand-aloneacid-transport model (ACID), and a fully integrated fracturing simulator (FRACANAL). The two auxiliary programs are importantparts of the system because they can be used to characterize thedata necessary to make field predictions, as shown above for fluidloss behavior.Acid-Balance Equation. The acid-balance equation is discretizedwith the same basic technique as for propp ant transport. In fact,the two problems are very similar because the equations for proppant concentration in suspension and acid concentration are analogous, and it is possible to maintain parallel structure of the twomodules. The major difference is that there is no equivalent of settlement in acid transport.34

TABLE 2 ACID TRANSPORT DATAFOR LO AND DEAN'S EXAMPLE 10Pumping Schedule

Treatment volume, STBPumping rate, bbl/minAcid concentration, o lReaction Data

Acid typeType of formationDiffusion coefficient, De cm 2 /secReaction constant, cm/secReaction orderEquilibrium concentration, G lDissolving power, Ibm/lbm acidDensity of soluble rock material, Ibm/ft 3Isothermal Run

Reference temperature for reactiondata, OF

2002015HCIlimestone1 x 10- 40.11.0o1.37169.1853

60.000Time-Independent Fracture-Fluid Rheology

Newtonian viscosity, cp 100Constant Fracture-Geometry Data

Fracture height, ftDepth to fracture center, ftFracture width, ftFluid density, Ibmlft3Average leakoff velocity, ftlminWall porosity

1003,936o62.40.00100.1500

Fracture grid in x direction: 35 blocks, Ax= 10 ftThe problem is solved numerically as hyperbolic because maintaining the fronts between different fluids with no dispersion is desirable. This requires logic to track the position of a front withina gridblock and to subdivide the gridblock with the front into twoelements for the solution of mass balance. The details of this procedure are given in Ref. 22 and apply directly to the acid model.The mandatory additional term is the rate of decrease of acid massin a block caused by spending (reaction) and by loss, expressedby Eq. 14 or 27. These equations contain two unknowns, CandCBi for each gridblock, but the boundary concentration can beeliminated. For example, the implicit discretization of Eq. 14 is(qAt)t+ 1= 2[qe;Cr 1 +Ai,Kg(C-CB)t+1 32)The reaction boundary condition, Eq. 13, is discretized asKg(C-CB)r+ 1 =k(l-cP)[(CB)r+l -ceq)m . 33)If m= I, CB can be expressed as

K k(CB)r+ 1=---g-Cr+l +-----Ceq=FbCr+l +Fe, (34)Kg+k Kg+kwhere k =k(l-cP). After elimination from Eq. 32, the final expression is

(qAt)r 1= 2[qfiFb+Ai,Kg(l-Fb)Cr 1+2(qi -Ai,Kg)Fe(35)This term can be incorporated into the finite-difference equationsas an implicit sink/source term.

For n I, the elimination cannot be carried out. However, thefollowing linearization (by Ben Naceur*) solves the problem. Findan equivalent value of k so that the reaction rate would be the samewith m=l:k;(CB-Ceq)=k (CB-ceq)m . (36)I f CB in Eq. 36 is evaluated explicitly at the beginning of thetimestep, the effective value can be computed and the eliminationcarried out as before.

Personal communication with K Ben Naceur. Dowell Schlumberger, Nigeria (1989).

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t.1.OWiDmoAVG.CONC. 0.WALLCONC.

i 7 '.tTi , ,Z Q5 0.esSu De emll.eeZ

, '.110. 1

25 225 250Fig. 6-Acid-transport solution-comparison with Ref. 10(simple reaction data from Table 2).

Integration of Acid Transport Into the Fracturing Simulator.The acid transport described here was integrated into a general simulator that includes options for 2D and 3D fracture geometry, 23,24generalized leakoff treatment,25 and a number of other sophisticated features. 26-28 In the overall solution structure, the acid transport is solved sequentially after the geometry and leakoff solutionhave been obtained and before the heat-transfer solution has beenupdated. The acid module is essentially interchangeable with theproppant-transport module and can be integrated into the structureas follows. .1. During the fracture-geometry/leakoff solution, the elastic fracture width, b is increased by the acidized width, b:/. This is doneonly to calculate flow in the fracture (transmissibilities and flowvelocities) and not for the fracture-opening equations. Also, the acidized width is not added to the mass-balance equations (to fractureelemental volumes); this corresponds to an assumption that the products of reaction occupy the same volume as the dissolved rock.2. The computed leakoff velocities are increased by the factor

R f determined by table look-up from the input data; on the basisof the amount of acid leaked off at the beginning of the timestep.The leakoff treatment of foams is different and accounts for theseparation of the liquid and vapor components in porous media.3. After the geometry/leakoff solution is completed, new acidflow velocities are computed (with a no-slip assumption) and theacid transport is solved.4. After the new acid solution is obtained, the increase in acidized volumes of all blocks is determined and the cumulative acidmasses lost and reacted for each gridblock are updated. The acidized volumes and widths bA are computed on a 2D grid withhorizontal and vertical definition and are used for coupling withreservoir simulators as described in Ref. 26.5. Temperature in the fracture is updated by solving the timestepof the energy-balance equation, including the heat of reaction computed from the acid-transport solution.At the end of the simulation, the geometry of the acidized fracture [i.e., bA x,z)] is transferred to the reservoir simulator, and itsconductivity is calculated as a function of the original acidized widthand the local effective stress by Nierode and Kruk's 19 method orby tables obtained from laboratory measurements.Validation of AcidTransport SolutionTo illustrate the various aspects of the physics, a number of caseswere simulated with the stand-alone ACID model and Lo andDean'slO data. Table 2 shows the common data. Because themodel of Ref. 10 assumes an infinite rate of reaction and CB= 0,a high value of reaction constant, k=O.1 cm/sec, n=l and Ceq =0are used here for comparison. Also, a constant value ofleakoffve locity, vf =0.001 ft/min, was used. Fig. 6 shows the result withSPE Production & Facilities, February 1993

1 ...------------------- 0 .11OWlnmOAVG.CONC.XWALLCONC. 0..

'.tT0."

Qo.os55o ..o.J'.12O.tl

Fig. 7-Effect of increased leakoff, with n=0.5, Ceq =0.2and k =0.01 cm/sec.internal generation of mass-transfer coefficients. Because Ref. 10used a constant value of De=O OOOI cm2/sec, the correspondingsolution with this value is also shown in comparison with Lo andDean's results. The solution here is very close to the 2D solutionof Ref. 10, which validates the method used to calculate mass transfer from De The same agreement was obtained for the casewithout leakoff.To test other features (not included in the model of Ref. 10), thesame basic set of data was used with modifications. The use of amore reasonable value for the reaction constant, k=O.OI cm/sec,results in higher boundary concentration and therefore smaller acidized width and demonstrates that the formulation of the model canhandle acidizing controlled by mass transfer, reaction rate, or acombination of both. The use of nonzero Ceq results in unspentacid and much smaller acidized width. This may be important fortreatments with weak acids or low concentrations. Having a reaction order lower than 1 increases the rate of reaction. This makesthe result for k=O OI and n=0.5100k more like the infinite-reactingsystem of Fig. 6. The feature is important because the measurements often show low values of n.The above parameters, although changing the fracture widths,do not have a large effect on penetration. The final, more realisticexample shows the effect of increased leakoff. The reaction datawere k=O.OI, n=0.5 and Ceq=0.2, and the laboratory leakoffincrease function shown in Fig. 2 was used. The result is a significant decrease in fracture penetration (Fig. 7). In higher-permeabilityreservoirs, the increase in leakoff can cause the fracture to stopgrowing or even to collapse, as shown later.The above cases show how the ACID model can be used to determine the sensitivity of acid penetration to various parameters.However, its most important use is to obtain the best reaction datafor the field design by matching acid flow experiments on slabbedcores. This procedure, together with matching the leakoff data described previously, determines the key parameters for simulationof the performance of the fracture job.Example of Fracture DesignAs an example, consider treating a formation with properties givenin Table 3. Two treatments will be compared here: (1) a job consisting of a gelled pad followed by straight 28 %acid and (2) a treatment consisting of gel/acid sequence. In both treatments, the totalamount of acid used is the same (1,200 bbl), as shown in Table4 Alternating the gel and acid has been claimed to reduce acidloss29; here the effect is quantified through use of the model. Thedifference in the leakoff of straight acid as opposed to alternatinggel and acid was investigated first on a laboratory scale. Fig. 8 showsthe hypothetical laboratory runs simulated with the LOSS model.In both runs, the first pad was assumed to have a spurt loss and

35


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TABLE 3-RESERVOIR DATA FOR THE FIELD EXAMPLERock-Mechanics

Vertical stress, pSiaHorizontal stress, psiaYoung's modulus, psi aPoisson's ratioBiol's constantCritical stress-intensity factor, Kc psialJitReservoir Properties

Average pressure, psiaIn-situ horizontal permeability, mdIn-situ porosityReservoir fluid viscosity, cpTotal in-situ compressibility, pSia- 1k to reservoir fluid at Swck to filtrate in invaded zoneIn-situ water saturation, SwcResidual reservoir fluid saturation to filtrateThickness, ftDepth to center of pay, ft

Thermal PropertiesInitial reservoir temperature, ofDensity of fluid-saturated rock, Ibm/ft3

13,0006,14110,100,0000.25000.60001,368.00

2,680.001.20000.08000.020600.0003300.18300.50000.25000.1000130.00011,877.00

Heat capacity of fluid-saturated rock, Btullbm-oFThermal conductivity of saturated rock, Btu/ft-D_oFDrained coefficient of thermal

167.0130.000.2150.00expansion, 1/oF OAOOx10-

a wall-building coefficient of Cm =0.002 ft/min l->. As acid comesin contact with the fracture wall, the filter cake previously deposited is assumed to be effectively destroyed, as shown experimentallyin Ref. 15. Subsequent exposure to gel will build a new filter cake,but without spurt loss. Also, because the gel will enter the wormholes preferentially, the decrease in filtrate viscosity is smaller andthus the effect of C1 is larger. This was modeled by assuming a90% viscosity screenout factor for the pad but only 50% for subsequent gel stages (see Ref. 21 for the formulation of viscosityscreenout).Only the initial pad leakoff follows the classic theoretical curve.During the acid stage, leakoff accelerates; in the gel stage leakoffis retarded. As a result, the loss with the alternating fluids is aboutone-half that with straight acid. Similar improvement would be expected in the field case, as Fig. 9 shows.

1/ACIDONLY ,/

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Terapaper Modeling of AcidFracturing Tr-5034

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