Copyright 1998, Society of Petroleum En~ineer~, Inc.

This paper was prepared for presentation at the 1998 SPE Annual Technical Con.ference and Exhibition held in New Orleans, Louisiana, 27.3o September 1998,

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ABSTRACT

The paper describes a 2-D model of acid fracturing which ismore rigorous than current acid fracturing simulators. Theequations are discretized and solved along fracture lengthand across the width (in x and y directions), as opposed to alumped formulation in the conventional model in which they-solution is replaced by a mass transfer coefficient Kg. Inthe 2-D model, correlations for Kg are not needed and themodel uses only directly measurable data, In addition, themodel can represent transient effects (Kg variable along fraclength and in time) combined with dispersion and thermaleffects.

Two models, one for lab experiments and one for field de-sign were developed. The lab model was validated againstresults of Roberts and Guin with excellent agreement andcompared with the 1-D approach on data typical for Phos-phoric dolomite. The results show large differences in regionsof high as well as low leak-off. Also, acid spending predicted

Sooiety of Petroleum Engineers

ASPE 48930

A New Two-Dimensional Model for Acid Fracturing Design

A. (Tony) Settari, Duke Engineering & Services, CalgaryR.B. Sullivan, Bass Enterprises Production Co., MidlandC.E. Hansen, Bass Enterprises Production Co., Denver*

● now with Enron Corp., Denver

by the 2-D model is higher even when the Kg values arecomparable, Correspondingly, the field design model pre-dicts larger spending and frac conductivity,

An example of the field design and post frac history matchfrom the Cottonwood Creek Unit, Washakie County, WY ispresented. It shows that. the new model is in better agree-ment with field data. Extensions to a 3-D modelling are alsodiscussed,

INTRODUCTION

Majority of acid fracturing modelling to date has beendone using the formulation which integrates the basic equa-tions across the fracture width (y-direction) and solve theproppant transport equations along the fracture length (d-irection). In the absence of the solution in the vertical (z-) direction, such formulation may be termed a 1-D model.This approach reduces the dimensionality of the problem(and computer work). On the other hand, it becomes neces-sary to introduce an “apparent mass transfer” coefficient Kg(first proposed by Roberts and Guinl), which relates the acidspending to average concentration across the width. This co-efficient then must be related to the measurable parameterswhich characterize the process, i.e., the effective diffusioncoefficient .De and the reaction rate constant k.

Recent examples of such 1-D models are found in the1iterature2–4 and their comparison is presented in Li et aI.5.The weak point in these models is the relation betweenli’g and flow properties and De. Usually, Kg is calculatedfrom dimensionless correlations which depend on the flow

39

2 A NEW TWO-DIMENSIONAL MODEL FOR ACID FRACTURING DESIGN SPE 48930

regime4’5. There are several problems with this approach:

1) Most of the data used to construct the correlations has

been interpreted using TeriH’s analytical solution (Nierode6)which assumes infinite reaction rate and a known value of dif-fusion (mixing) coefficient .De, Li et al.5 have shown the cor-relations for .De to be unreliable in the laminar flow regime.

2) The correlations are discontinuous at the boundary be-tween the laminar and transitional flow, and in addition theydepend on different sets of variables in the two regions. Asa result calculated Kg has a large discontinuity between thelaminar and transitional region when plotted as a function ofReynolds number NRe. While this discontinuity maybe real,an approach not relying on correlations would be preferable.

3) The correlations have been derived from data obtained

with zero or small leak-off. As shown previously, there mustbe a relation between .Kg and leak-off velocity v~ which is notimplicitly included in these correlations.

Thus, the motivation of the 2-D acid transport solution isto eliminate the need for Kg correlations by direct solutionof the equations across the u’idth. In this formulation, themass transfer then depends directly on the De and k and theflow field. Recently, Romero et al.’ presented a 3-D model,which is essentailly an extension of their 2-D x-z model tosolve across fracture width. Their 3-D extension chronologi-cally followed (and benefitted from) the 2-D model reportedhere (developed in 1994-96). The main focus of their workwas however on the vertical aspects of the problem. The com-parisons between models in Ref. 7 do not allow to isolate theeffect of the rigorous y-direction solution from the effects ofthe vertical flow comporrents. In contrast, this paper focuseson the more fundamental question: is it necessary to solvethe equations across fracture width, and if so, what conse-quence does it have on forecasting fracture conductivity andwell production. Once this question is answered, the effectof the vertical component of the solution, which is obviouslyalso very important, can be also better understood.

FORMULATION OF THE 2-D ACID TRANS-PORT

For simplicity, we will consider an element of fracture withvariable width but constant height as shown on Fig. 1. Inthe following, it will be assumed that concentration C isdefined as mass of acid/volume of solution. If one wishes to

use concentration @ in terms of moles, C must be replace

by hlw~ everywhere. The equations describing the flow ofacid in a channel of unit height under laminar flow are:

Equation of continuity. The general equation within achannel of variable width is

d(v.)/az + a(vv)/ay = dA/8t – qinj (1)

where A represents the elemental area dxdy which representsthe change of control volume due to change of frac width bwith time (the actual form of this term depends on how thecoordinate system used accounts for the moving boundary;it is not important for the final form of the equations). The

boundary conditions are

~V[V=Y = vi, Y = b/2

where v~ is the leak-off velocity. By integrating this across yand z we get the equation solved by the geometry module ofa frac simulator (except for compressibility):

8( UAC)/dZ – 2q((~) = 8Ac/& – qi.j (2)

where ti(z, t) is the average volumetric velocity, AC(X, t) isthe local crossectional area and q~(z, t) is the leak-off rateon one surface of the fracture. Assuming constant widthvertically, A= and ql are given by

AC = Hfb ql = Hflvl

where Hj is the total and Hjl the leak-off height,

Equation of acid mass conservation. If we neglect diffu-sion in the x-direction, this is

where Vz and Vv are now 2-D components of velocity field:C is acid concentration and the last term is the injectionrate of acid. The effective diffusion coefficient De is definedbelow. Two boundary conditions apply:

i) At the wellbore [Z = O), C = Cinjii) At the fracture wall, y = b/2 = Y, the boundary

condition states that the total flux of acid to the surfaceequals to acid spent + acid leaking off with velocity V1, Thetotal mass transport to the wall is the sum of diffusion andconvection:

–q.zi = (–C’BVg + De~)[v=Y (4)

where De is the effective diffusion coefficient, which for lam-inar flow is equal to the molecular diffusion, but it increasesin presence of turbulence. The reacted amount is assumedto be expressed as:

r = k(l – +)(CB – Ceq)m= k’(CB – C,~)~ (5)

where m is the reaction order, k is the reaction rate constantand CB and Ceq are the boundary (wall) and equilibriumconcentrations. The Ceq takes into account the effect of re-verse reaction which may be important for weak acids. Theamount lost to formation is

qAl = v/c~ (6)

where Cl is the “loss concentration”. Also, by continuity theleak-off velocity vi and boundary velocity Vv must be equal,which gives the general boundary condition as:

‘(-CBW1 + ~ef)[g=y = k’(CB - C,,)m + v&’/ (7)

40

SPE48930 A. SETTARI, R.B. SULLIVAN and Cl?. HANSEN 3

The only unclear issue in this equation is the relationshipbetween CB and Cl. Note that this issue disappears if leakoffis zero. This will be discussed in detail later.

The above formulation is standard in acidizing literature.Because of possible future extensions to turbulent flow etc.,we have used substantially more general formulation givenby:

, i9c8 h’r~) + ;(hv~) =-:(C%) - ;(c%) +%(

ti’c——= at

!7injCinj (8)

where

A’r=Dez+a/lwzl+@tl~gl (9a)

Kv=Deg+ cr/llJg[+crtlt)zl (9b)

are the total dispersion coefficients which are composed ofthe molecular diffusion (in the isotropic case DeZ = Dev =D,), and hydrodynamic dispersion part dependent on flowvelocities. The al and crt are the logit udinal and transversedispersion coefficients which have been measured extensivelyin flows through porous media or atmospheric flows. Whilein practice the model is usually run with ~1 = crt = O,these terms may be useful later in explaining the differencebetween molecular diffusion D~ and effective diffusion Deor in modelling transport in turbulent flow,

The underlying assumptions (as in the model) are:

a) Flow is incompressible

b) The reaction has no effect on volumetric (i.e., reactionproducts have the same volume as reactants)

c) Concentration is uniform vertically (strictly speaking,this implies that the leak-off height Hfl and the reactiveheight ~~r are equal to hydraulic height Hj),

Boundary conditions at fracture wall

An attempt has been made to determine the leak-off concen-tration Cl by further analysis of the acid mass balance at thefracture wall. Detailed discussion of various limiting cases isfound in Appendix A. Unfortunately, the consideration of themass balance at the wall itself does not allow us to determineCl except that it has to satisfy CB > Cl > C.g. The lim-

iting case Cl = CB was used in our previous work4. In theabsence of better theory, a feature was provided to choose Clto be anywhere between these limits. This allows to assessthe effect of the uncertainty in the boundary condition asdiscussed later.

MODEL IMPLEMENTATION

The choice of a numerical method used to solve the acidtransport equations waa motivated by a requirement thatthe 2-D acid transport module should be driven by the so-lution from any conventional 2-D or 3-D fracture geometrysimulator, as well as be able to simulate simple laboratorygeometries. This dictated in particular the approach to theflow solution.

Solution of the flow field

Since conventional flow/geometry fracture models solve onlyfor an average velocity across fracture width, it was necessaryto avoid re-solving the flow equation (1). In order to obtainthe y-component of the velocity, an assumption of self-similarsteady state flow has been made, Assuming that at everyposition in the fracture the flow profile is fully developed,velocities can be calculated from the average flow rate ascomputed by the flow/geometry model and fluid properties.

For Newtonian laminar flow, the well known solution is:

‘?JZ(?J)= vz(o)(l – (?J/Y)2) (lo)

where Y = b/2 is the fracture half-width. The average flowvelocity ~Z is defined by

JY

Dzb = 2 vzdy = Qo

(11)

where Q is the flow rate through the fracture at that position,

For Non-Newtonian power law fluid, an analytical solutionto the flow between parallel plates leads to the following gen-eralization of Eqn. (10) (see Ast arit a and Maruccis):

Vs(y) = Vz(o)(l – (y/Y) (l+~/@) (12)

where n is the power law exponent (fluid consistency index).Note that Eqn. (12) assumes zero slip velocity at the wall;more general equation could be also used.

In both forms, we have vu = O everywhere as a resultof the assumption of fully developed laminar flow, and zeroleak-off. For flow with leak-off, WV will be nonzero, but onecan compute the flow distribution on a finite difference gridexplicitly as follows:

●

●

●

●

At the inlet to the fracture, the injection rate Qi =qiwj is known and the block flow rates (Qz)l,j =

(~x)~,jAyj can be computed based on the velocitY dis-tribution, Eqn. (12).

For the outflow from the first row of blocks ( at i+l/2for i=l), the total flow rate is

(13)

where AQa,-c is the accumulation of the fluid in the gridblock (1/2 element ). Then the flows (Qz)2,j can beconstructed from Q1 and will be self-similar to (Qz)l ,j.The y-flow components (QY )1 ,j can be now computedby continuity (assuming steady state) starting from thecenterline (j= 1) explicitly.

Repeat the procedure for the next row of blocks (i=2)etc. until the end of the system is reached

Note that the above procedure is numerical (in effect an ex-plicit soluton for the flow components) and therefore alwaysmass conservative. For a channel of a constant width, theresulting velocity components a ree with the analytical ve-

flocity profiles derived by Terilll (see also Appendix of Ref.

41

4 A NEW TWO-DIMENSIONAL MODEL FOR ACID FRACTURING DESIGN SPE 48930

8) which are also self-similar. For the field design model,both the grid and the velocity field must be recomputed ev-ery time step due to variation in fracture width and leakoff.

Solution of the concentration equation

The concentration equation is solved by standard finite dif-ference discretization. The convective terms are discretizedusing a l-point or 2-point upstream weighting formula. Thediffusion terms are discretized using conventional 5-pointscheme. Time discretization by backward difference, CrankNicholsen and 2nd order Runge Kutta are available in thecode. Option for explicit time discretization has also beencoded.

Although several options for reducing numerical disper-sion are available internally in the code, most of the testingand applications have been carried out using the l-point up-stream space and backward time discretization. This combi-nation offers the most robust model (unconditionally stable,guaranteed not to overshoot or oscillate). However, it alsohas the largest numerical dispersion.

Grid construction

Proper gridding is critical for obtaining accurate answersto the reaction problem. The grid used is point-centeredg,which gives grid points directly at the fracture wall and atthe centerline. In order to reduce the numerical errors andcomputation time, methods for an automatic irregular gridconstruction were developed as follows:

a) In the x-direction, a regular grid with Nz blocks is setup for the laboratory model, based on the length of the sys-tem L.. For the field model, the x-gridding is determineddynamically by the fracture flow/geometry model.

b) In the y-direction, half of the system is solved withy = Oat the centerline. The y-grid is irregular. The first methodused was derived from the velocity profile such that constantchange of velocity occured between two grids. For IVY grid

blocks, the Ay are then given by

Ayj = yj – yj-~, yj = (1 – ~/NJnlil+njY (14)

However, the solutions with this grid converged slowly (asfVv increased) as some cases. By numerical experimentationit was found that the accuracy is controlled mostly by thesmallest grid size at the wall. A new grid algorithm has beentherefore added, which is based on a log distribution of .Ay.User needs to enter only the smallest grid size Ayl at thewall. A ratios of Y/AYl between 100,000 and 1,000,000 with.NV = 20-40 gave good, consistent accuracy to all problemstested. Note that the grid sizes at the wall become extremelysmall (10–7 cm or less).

Handling of a general reaction order

The reaction term results in a transmissibility term in thematrix equation of the form:

Tri(c;~l – Ceg)m (15)

where the “reaction transmissibility” is

(?B,~ is the concentration at the boundary at location i andm is the reaction order (m < 1). Since the grid is pointcentered, we have cB,~ = ci,~u. If m # 1, the above wouldintroduce nonlinearity in the system of equations. For thisreason, the term is linearized as follows:

Similar method was used in the 1-D code4.

Calculation of 1-D lumped parameters

Throughout th~ solution, one can calculate the averageconcentration C(z) and the local mass transfer coefficient.Kg(z). These can be then compared with the 1-D solutionsand wit h correlations.

The average concentration is defined by

/

}T

CQ=2 C(y)v.(y)dyo

which in finite difference form is

= ~Ci,j(Qx)i,j/Qi (17)j=l

where Qi is the total rate through crossection i. Because thevelocity profile is self-similar, the inlet flow rates and Qinjcan be used for all crossections (as long as n is not allowedto vary along fracture length).

The mass transfer coefficient Kg is defined by

Kg(c – CB) = –l?-(m/q/)ly=Y

In finite difference form, at location i

Average value of lfg for the whole system can be then com-puted by averaging over A~i. However, as discussed later,lfg varies strongly in the entrance region and then stabilizes,making the stabilized value a more appropriate choice.

Alternatively, one can express the results in terms of theequivalent Sherwood or Nusselt numbers given by NNU =NSh /2 = bKg /D,. We chose to make the comparisons interms of .Kg because it is an input variable for the simulator.

42

STE 4893tI A.SETTARI, R.B. SULLIVAN and C.E. HANSEN 5

MODEL VERIFICATION

The laboratory model was verified against numerical solu-tions of Roberts and Guin1° (abbreviated as R&G). Theseauthors solved the simplified version of steady-state acidtransport. In order to match them, the model was run withconstant fracture width and Dez = O, Dev = De, CY1=at = Ounder variety of conditions until a steady state wasreached.

The R&G solutions were presented in the form of plots ofPeclet number JVpe vs dimensionless distance LD for con-stants values of dimensionless concentration C/C;. Eachplot is for a given value of dimensionless reaction parameterP and reaction order m. The definition of the variables is:

NPe = VIY/De LD = vIx/ijY = ‘2vl/Qi.j

P = (1 – q5)kYC~-1/D. = k’YC~-l/D. (19)

The common data used for this comparison was as follows:

Y = b/2 = 0.025 ft, ~j~ = 1 ft, Cin3 = 1., V/ = 20 ft/day.flow index n = 1.

-De was varied from 2 ft/day to .25 ft/day which givesNpe in the range of .25 to 2. Reaction parameter k’=10000ft2/day was used to model infinite reaction. For the case offinite reaction, k’ was recalculated to give a constant P. Forexample, for P = 2, the required values of k’ for the casesrun are in Table 1.

The length of system and total simulation time was variedaccording to P and Npe to fully utilize the grid. This re-sulted in Lz = 1 to 4 ft and total simulated time from 1 to 6minutes. Grid refinement experiments established grid sizenecessary for sufficient accuracy for different cases (case withlarge reaction rate and therefore largest concentration gra-dients haa the largest numerical errors). Results presentedhere were obtained with N. = 41 and NY = 21.

Results oft he comparison

Comparisons were run for all cases reported by R&G. Notethat the penetration in all cases is limited to LD = 1, atwhich point the convective x-velocity becomes zero due toleak-off as indicated by Eqn. (13). R&G solution does notappear to have been carried out close to this limit. As anexample, the comparison for the case of finite reaction with-F’ = 2 aud reaction order m = 1 is shown in Fig. 2 (R&Gshown by lines; our results by dots and broken lines) .Theagreement is excellent, considering the differences in the nu-merical techniques used (R&G used explicit discretization ofthe steady-state equations and analytical velocity profiles).

Similar quality results have been obtained for reaction or-der m = 0.4 (note that R&G denote reaction rate by n).A series of runs were made to determine also the effect ofNon-Newtonian flow (in the laminar region), using the basedata with P = 2 and m = 1. The comparison of results forn = 1.0, 0.6 and 0.3 showed that the effect is only moderateand the acid penetration distance decreases with decreasingn.

Finally, the simulations showed that, in all cases, Kg asdefined by Eqn. (18) was variable along the channel length.Typical variation starts high in the entrance and then rapidlystabilizes to a constant value about half of the entrance value.For further comparisons with the 1-D design model, the sta-bilized value (as opposed to a mean value) was used through-out this work.

MODELING OF LAB EXPERIMENTS

Comparisons of 1-D and 2-D solutions was done for hypothet-ical laboratory experiments on Phosphoric Dolomite. Reac-tion data and actual lab experiment matching was reported

511. Two sets of base data were used andin previous work ‘the results were examined in a range of D. and leak-off ve-locity V1. The same sample dimensions were used for bothexperiments: channel of a length L= 0.5878 ft, height Hji= 0.1354 ft and slot width b = 0.025 ft.

Case 1. The example simulates flow of 22 % HCL acid inEdwards limestone at 190 deg F. The reaction and other datais in Table 2 (based on Fig. 11 of Ref. 5; note that reactiondata measured at 150 deg F was extrapolated to referencevalue of 300 deg F). 1-D and 2-D results were compared interms of the acid spent (lUfA.P), A’g coefficient (taken at theend of the system if variable along the length) and acid lost(ikfA~O,S).

The comparison as a function of D, is on Fig. 3. In this, aswell as other figures, the solid lines are the 1-D solution anddashed lines with dots the 2-D solution. The 1-D solution(using the model of Ref. 4) predicts that the Kg and acidspent is independent of De and only a function of leak-offin a wide range. The 2-D solution approximates this in afairly long range, but starts to deviate sooner. This meansphysically that if De < 10-5 cm2/s, acid transport to thewall by diffusion is negligible compared to convection (Ieak-

off velocity v1). If .De > 10-5, both mechanisms interact,and the 1-D solution is quite inaccurate. This prompts oneto look at the sensitivity to V(. Fig. 4 shows the comparisonfor the same data as a function of V[. The 1-D correlationpredicts that A’g ~ v~ in the entire range, while the 2-Dsolution starts to deviate at both small and large V1.

It appears that for this data the results are fairly similarexcept for the smalled acid lost &fA~O~~(which is howeververy small compared to M.4~p). This agreement is explainedby the high efficiency of acid spending coupled with leak-offdominated transport.

Case 2. This data represents emulsified acid in Phosphoricat 80 deg F with reaction data referenced at 150 deg F (Table3). Two other variations of this data were also simulated:

“plain” acid with De = 2.13x10-5 cm2/sec and “gelled” acid

with De = 7.9x10–6 cm2/sec as opposed to the emulsified

acid value of 2.80x10–8 cm2/sec. The sensitivity to leak-offfor the emulsified, gelled and plain fluids is shown in Figs. 5,6 and 7 (l-D solution by solid lines, 2-D solution by dashedlines with dots). The 1-D solution is now deviating at bothsmall and large leak-off values in terms of kfASp, while the

43

6 A NEW TWO-DIMENSIONAL MODEL FOR ACID FRACTURING DESIGN SPE 48930

Kg’s always converge together at large leak-off where Kg ~V1. Again, the marked common difference between the 1-D and 2-D solutions is the smaller amount of lost acid inthe 2-D solution. The boundary concentrations CB are alsocorresponding y much smaller.

Effect of CB assumption

As discussed above, the formulation of the kinetics does notyield unique determination of the leak-off concentration C’/.A question naturally arises as to whether the differences seenare affected by this uncertainty.

A feature was added in the model such that the leak-offconcentration CZ can be assumed to be anywhere betweenCB and C’eg: Cl = CBF1 + C.q(l – Fl), where F1 is aninput papameter. A series of runs with the Case 2 data for

SXE using leak-off velocity of 0.01 ft/min and different valuesof F1 were then made. The results are in Table 4. The totalacid spent and leaked-off is the same for any choice of F/;only the split is affected. The boundary concentration CBincreases towards the 1-D value as F1 decreases, but the acidleaked off is at the same time decreasing. Therefore the 1-Dsolution is not approached for any F1.

SCALE-UP TO FIELD CONDITIONS

The above lab comparisons clearly indicate that the lumped1-D model predicts, in most circumstances, less efficient acidspending compared to more rigorous 2-D solution. In orderto see if this also applies on field scale, a number of typicaltreatments were modeled using the lab model with dimen-sions scaled up to field conditions. The constant fracturewidth and leak-off make the interpretation easier comparedto the model coupled with fracture simulator.

The 2-D model predicted approximately the same acidizedlengths but larger acidized widths and conductivity com-pared to the lumped model, in agreement with the lab scalesimulations.

FIELD EXAMPLE

The following field example demonstrates that the 2-D modelis also in better agreement with actual data. The exam-ple presented is from the Cottonwood Creek Unit, WashakieCounty, Wyoming. This well was originally fracture stimu-lated during April, 1986 using 13,000 gals of 40 lb/regal ofcross-linked gelled water and 20,000 gaIs of 28% geIled acidin four stages. The treatment was pumped at an averageinjection rate of 12 BPM. All fluids contained 25% C02 byvolume.

To model the fracture treatment and resulting production,

a four layer model was constructed with the parameters givenin Table 5. The initial production period of 1420 days wasmodeled with a 200 ft fracture having an average conduc-tivity of 800 md-ft. The well was then re-fractured duringFebruary, 1990 using 65,000 gals of 20 lb linear gelled wa-ter, 80,000 gals of 28% emulsified acid (70% of 28% HC1,30% diesel), and 12,300 gals of 15% HCI in multi-stages withrates varying between 15-30 BPM.

The re-fracture treatment was pumped down 3 1/2 inchtubing with no packer. Hence, the annular pressure is usedas a reflection of bottomhole pressure to obtain net fractur-ing pressure. Figure 8 is a net pressure match of the actualtreatment used to estimate the created and etched geometry.All rock and fluid data was used without alteration exceptfor fracture toughness and containment which had to be in-creased. Also, the leak-off viscosity of the emulsified acid,which is difficult to estimate, was adjusted to 12 cp in orderto match the fall-off. The 1-D and 2-D model matches usedidentical data; the differences in the acid transport(and re-sulting fracture width) generated noticeable difference in thefall-off pressure.

Comparison of 1-D and 2-D acid solutions

The difference in the acidized fracture width is shown in Fig.9. In a model which does not solve in the vertical direction,fracture width is an approximation and must be averagedover some height. In the model, one can specify two sce-narios. If the surrounding layers are reactive, 11~, = Hjand etched width is also an average over 17~ .If the surround-ing layers are non-reactive, IIjr = leak-off (reservoir) heightHjl and width is averaged over HjI. In our case, the firstassumption was used, but because the surrounding layers arenot as reactive as the reservoir rock, the computed widthswere adjusted by using only 1/2 of fracture heigh~ outsidethe reservoir.

The comparison of conductivity is in Fig. 10. This conduc-tivity was calculated from Nierode-Kruk correlation13 usingembedment strength of 60,000 psia and assumed (constant)closure stress of 4400 psia, Variable closure stress was usedin the reservoir simulation described below.

In summary, the 2-D solution predicts significantly moreetching, confirming the results of lab simulations.

Comparison with reservoir performance

Following the re-fracture treatment using the emulsifiedacid system, the well was produced for an additional 2418days (6.62 years) prior to conducting a 28-day bottomholepressure buildup test. This production period, as well as thepressure buildup test, was history matched using a 3-phasereservoir model with a fracture represent ation14. The matchresulted in 700 ft fracture half-length with an average widthof 0.1 inch. Conductivity was calculated in the model againusing Nierode Kruk correlation, but with variable closurestress according to current pressure. The width is in goodagreement with the fracture width profile from the 2-D frac-ture model (Fig. 9) but considerably higher than from the1-D model.

Production was modeled with the well in the center of a 45acre drainage area. Hence, the 700 ft fracture extends to thereservoir boundary. The geologic depositional model consistof amalgamated tidal flats. We believe the drainage area ofthis well primarily represents a “sweet spot” in one of thesetidal flats. The result of the overall production match is inFig. 11 and of the buildup at the end of the simulated period

44

SPE 409’30 A.SETTARI, R.B. SULLIVAN and C.E. HANSEN 7

in Fig. 12. The same quality match has been obtained byusing the width distribution from the 2-D solution of Fig. 9.

DISCUSSION

The above example demonstrates the much closer agree-ment of the 2-D model with well performance. Similar re-sults were obtained with other wells in the Cottonwood CreekUnit. All of the fracture simulations performed using the 1-Dacid fracturing model required a conductivity multiplier beapplied to the Nierode-Kruk correlation relating the amountof rock consumed to a resulting conductivity. Typically, thismultiplier was required to be in the range of 10-20 in orderto get a realistic value for fracture conductivity. However,

we found that the 2-D model presented in this paper in mostcases predicted greater acid spending compared to the 1-Dmodel. Consequently, the more rigorous solution has allowedus to remove the conductivity multiplier and merely us theNierode-Kruk correlation as originally presented.

Romero et al.7 presented examples in which the 3-D modelgave both larger and smaller conductivity compared to a 1-D model. However, in their work the differences are due toa combined effect of the vertical aspects (z-direction) andreaction kinetics aspects (y-direction) and the effect of they-solution cannot be separated. In addition, the method ofaveraging the width in the z direction is not clear. On theother hand, the comparisons presented in this work capturesolely the effect of the y-direction solution.

The petroleum industry has generally been reluctant toperform large acid fracturing treatments due to the com-plexity and uncertainties associated with treatment design.This includes both the acidizing physics and the correspond-ing prediction of post fracture performance. Due to theseinherent design problems and potentially large capital costs,acid fracturing treatments have generally been very smallsuch as to minimize the amount of capital exposed. Thework performed in this paper as well as Refs. 5 and 11 ad-dressed these unknowns. We found that when the properphysics are accounted for, acid fracturing treatments can bemodeled more accurately, As a result, acid fracturing, and inparticular the use of emulsified acids, is a viable competitorfor proppant fracturing in calcarious formations.

CONCLUSIONS

1. A model of 2-D acid transport which solves the equa-tions across the fracture width has been developed. The for-mulation uses self-similar velocity assumption, allowing touse directly velocity components from a conventional 2-D or3-D fracture model.

2. The solution is in good agreement with the work ofRoberts and GuinlO.

3. In lab experiments, the 2-D model predicts larger acidspending, lower boundary concentration, and less acid leak-off compared to the conventional model.

4. In field applications, the 2-D model predicts largeramout of etching and larger acidized widths, in agreementwith the trends observed on the lab scale. This translated to

larger fracture conductivities compared to the standard 1-D(or 2-D x-z) modeling method.

5. The higher conductivities calculated by the new modelare in good agreement with the field data from the Cotton-wood Creek Unit, while the 1-D model requires large upwardadjustment of fracture conductivity. Therefore, it is believedthat the 2-D method provides more accurate (and different)results compared to the 1-D method. The same differenceswould be expected when comparing a 2-D x-z model with a

3-D model.

6. Based on the new modeling tool and our field experi-ence, acid fracturing is a more viable alternative to proppantfracturing for stimulation of calcarious formations, comparedto current thinking.

NOMENCLATURE

A = elemental areaAC = local crossectional area, L2

b = fracture width, L

C = concentration, m/VCi~j = injection concentration, m/V

II, = effective diffusion coefficient, L2/tE. = activation energy, E/mF1 = weighting factor for leak-off concentration

R~ = fracture full heightHfl = fracture leak-off heightHf. = fracture reactive height

HTzn = specific heat of reaction, E/mk = reaction rate constant, L/t

k’ = reaction rate parameter, = k(l – @), L/t

K=,Y = dispersion coefficients, L2/tLD = dimensionless x-distance (Eqn. 19)

ikfA,P = total mass of acid spent, mkfA~O.$ = total mass of acid leaked off, m

Npe = Peclet number, Np, = vIY/DeNz,v = number of grid blocks in x and y direction

P = dimensionless reaction parameter (Eqn. 19)

qinj = inj~ction rate for one wingQ = finite difference flow rate

t = timeT = temperature

T,.j = reference temperaturevz,v = flow velocity components

V1= leak-off velocityu = average velocity across fracture widthx = coordinate along fracture length, Ly = coordinate across fracture width, L

Y = channel half-width, Y = b/2, Lz = vertical coordinate, L

SubscriptsA = acid.0 = dimensionless

1 = leak-off (loss)1? = boundary (fracture wall)eq = equilibrium

Superscripts

45

a A ‘NEW TWO-DIMENSIONAL MODEL FOR ACID FRACTURING DESIGN SPE 48930

n, n + ~ = beginning and end of a time stepJ = average value of j

ACKNOWLEDGEMENTThe authors wish to express their gratitude to the man-

agement of Bass Enterprises Production Company for theirsupport in the development of the new two dimensional acidfracturing model and their permission to publish this paper.

REFERENCES

1. Roberts, L.D. and J .A. Guin: “A New Method for Pre-dicting Acid Penetration Distance”, SPEJ, August 1975, pp.277-285.

2. Lo, K.K. and Dean, R. H.: “Modeling of Acid Fracturing”,

SPE Prod. Eng., May 1989, PP.194-200.

3. Ben Naceur, K. and Economies, M. J.: “Design andEvaluation of Acid Fracturing Treatments”, SPE 18978, 1989Joint Rocky Mountain / Low Permeability Reservoir Sym-posium, Denver, March 1989

4. Settari, A.: “Modelling of Acid Fracturing Treatments”,SPE PE, Feb. 1993, PP.30-38.

5. Li, Y., Sullivan, R. B., de Rozieres, J., Gaz, G.L. andHinkel, J. J.: “An Overview of Current Acid Fracturing Tech-nology With Recent Implications for Emulsified Acids”, SPE26581, 68th Annual Technical Meeting of SPE, Houston, Tx,,Ott 3-6, 1993, pp.709-724.

6. Nierode, D.: Private communication, Exxon ProductionResearch, cca 1990.

7. Romero, J., Gu, H. and Guirajani, S. N.: “Three-Dimensional Transport in Acid Fracturing Treatments:Theoretical Development and Consequences for Hydrocar-bon Production”, SPE 39956, SPE Rocky Mountain Re-gional/Low Permeability Reservoirs Symposium, Denver,Colo., 5-8 April 1998.

8. Astarita, G. and Marucci, G.: lVon-iVewtonian .FluidMechanics, McGraw Hill, London, 1974.

9. Aziz, K. and Settari, A.: Petroleum reservoir simula-tion, Applied Science Publ., New York & London, 1979,476

P.10. Roberts, L.D. and Guin, J. A.: “The Effect of SurfaceKinetics in Fracture Acidizing”, SPEJ, Aug. 1974, pp.38.5-395.

11. de Rozieres, J., Chang, F.F. and Sullivan, R.B.: “Mea-suring Diffusion Coefficients in Acid Fracturing Fluids andTheir Application to Gelled and Emulsified Acids”, SPE28552, SPE 69th Annual Tech. Conf. and Exhib., New Or-leans, La, 25-28 Sept. 1994, pp.373-385.

12. Terill, R. M,: “Heat Transfer in Laminar Flow and MassTransfer”, Int. J. of Heat and Mass Transfer, Vol. 8, 1965,pp. 1491-1497.

13. Nierode, D.E. and Kruk, K. F.: “An Evaluation of AcidFluid Loss Additives, Retarded Acids, and Acidized FractureConductivity”, Paper SPE 4549, 48tIi Annual Meeting, LasVegas, Sept. 30- Oct. 3, 1973.

14. Settari, A., Puchyr, P.J. and Bachman, R.C. : “PartiallyDe-Coupled Modelling of Hydraulic Fracturing Processes”,SPE PE, February 1990, pp,37-44.

APPENDIX A

Determination of the leak-off concentration Cl

Equation (7) can be written as a mass balance for a controlvolume V at the wall as

QD + Q. - Q1 - Q, = Vaclat (A -1)

where QD, Qc, Q/, Qr are the diffusion, convection, leak-offand reaction rates. We can now consider different specialcases:

a) No reaction, no diffusion. This gives

Q. -Q, = v~clat (A -2)

In order that this boundary condition can exist in steadystate, we must have Qc = Q1 and if V1 = VB this requiresc1 = CB.

b) No reaction, convection and diffusion. We have

Q.+ QD - Q[ = V8C/8 (A - 3)t

In steady state C’ approaches c~nj and QD becomes zero.This requires again Cl = CB. Only the rate of the approachto steady state is different.

c) No flow, no diffusion. This would result in

-Qv = V8C/&

which is a non-physical situation - some mass transport isrequired.

d) No bulk flow, diffusion. This case is described by

QD -Q, = v8C/8t (A -4)

If initially C = const, this equation indicates that a neg-ative OC’/~t will be created and C’ will decline with a. ratecontrolled by De.

e) Reaction and convection, no diffusion. Here we have

Q. -Q, = Vaclat (A -5)

By continuity w’e must have vi = vB = v and the above

reduces to

A(CB – C1)V – Q, = V8C/8t

where A is the element area. If we now assume that Cl =CB, this reduces to –Qr = V8C/i3t which would indicatethat 8C/& = –Qr /V, i.e., concentration C will declineto Ceq . Note that the decline is independent of convectionwhich does not seem correct.

46

sm 48930 A.SETTARI, R.B. SULLIVANand C.E. HANSEN

If we assume more generally that C,q < Cl < CB, steadystate can be reached at a value of CB at which

A(CB – Cl)v = Q, = Ak’(CB – C,q)m

The upper limit of Cl = CB gives CB = C,q as alreadyderived above. However, the lower limit leads to a conditionv/k’ = (CB – C.g)m-l which has physical solution only inthe limit for large v. Therefore in general all one can say isthat Cl > Ceq.

A slightly different approach would be to consider thesteady state directly, which leads to the same equation.Thus, the mass balance on the fracture wall (which is alreadyexpressed by the boundary condition (7)) does not allow oneto determine the leak-off concentration Cl.

47

TABLE 1- ACID TRANSPORT DATA FORCOMPARISON WITH ROBERTS & GUIN

Np, 0.25 0.5 1.0 1.5 2.0

De(i?/d) 2.0 1.0 0.5 0.333 0.25

k’ (t?/d) 160. 80. 40. 26.66 20.

TABLE 2- DATA FOR LAB SIMULATIONS - CASE 1

Treatment data:

Volume pumped (solution) 1000 ml

Rate 100 mllminConcentration (volume) 22 ‘Y,

Temperature 190 deg F

Reaction data:

Reference temperature

Diffusion coefficient DeReaction rate constant k’Equilibrium cone. Ce~Reaction order m

Dissolving power ~100

Activation energy E.Heat of reaction Hm

Density of soluble material

300 deg F7.6x1 0-7 cm21s

0.050421 C122/S

o.

0.411.37 lbflb acid

12016 Btullbmole2269 Btu/lb

172 lbicufl

TABLE 2- DATA FOR LAB SIMULATIONS - CASE 2

Treatment data:

Temperature 80 deg FOther data same as Case 1

Reaction data:

Reference temperatureDiffusion coef%cient D.Reaction rate constant k’

Equilibrium cone. Ce~Reaction order m

Dissolving power ~looActivation energy E.Heat of reaction Hm

Density of soluble material

150 cieg F2.8x1 0-8 cm21s

0.0020958 CdS

o.

0.499

1.37 lb/lb acid15290 Btu/lbmole

2269 Btu/lb

172 lb/tuft

48

TABLE 4- EFFECT OF LEAK-OFF CONCENTRATIONASSUMPTION (FACTOR F,) - CASE 2

1-D .508 E-2 .0322 .2011 18.00 .2333

2-D Fl=l.O .504 E-2 .167 .06240 5.60 ,22942-D F1=0.5 .504 E-2 .188 .04130 7.16 .22932-D FI=O.O1 .504 E-2 .225 .00446 10.31 .2294

TABLE 5- FIELD EXAMPLE: RESERVOIR DATA

SaturationsLayer Net Pav Porosity Permeability Oil Water ~

(f-t) (%) (red) (%) (acres)1 47 7 0.2 90 10 452 10 8 0.3 90 10 453 3 10 1.0 90 10 4542 12 1.0 90 10 45

49

Fig. 1 Schematics of the physical system solved

PARAMETERS: P . & m. 1, n .1

1.0

0.5

00 0.1 0,2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

DIMENSIONLESS DISTANCE, L~

——.

.—. -

..

Fig. 2 Comparison of the 2-D model with Roberts and Guinfor dimensionless parameter P=2 and reaction order m=l

50

1.E+OO

1.E-01

1.E-02

#~“ 1.E-03

3z 1.E-04

21.E435

1.E-08

1.E4171.E-09 1.E-08 1.E-07 1.E-08 1.E-05 1.E-04

Diffusion coefficient De

1.E-01

1.E-02

Z 1.E-04

ii

&.E4)8

1.E-07

1.E-08

1.E-Q5 1.E-04 1.E-03 1.E-02

Leak-off velocity VI (ft/min)

Fig. 3 CASE 1 lab data - comparison of 1-D Fig. 4 CASE 1 lab data - comparison of 1-Dand 2-D models - sensitivity to De and 2-D models - sensitivity to leak-off

1.E+OO

.E-01

.E-02

i?~a 1.E44

‘%E4)8 +//d

1.E4)7

1.E-05 1.E-C)4 1.E-03 1.E432 1.E-01

Leak-off velocity VI (ft/min)

‘ml.EOf3

1.E4)7 ~

1.E45 I,E4?4 1.E-03 1E-(I2 1.E-01

Leak-off veiocity VI (ft/min)

‘ig. 5 CASE 2 SXE data - comparison of 1-D Fig. 6 CASE 2 gelled acid - comparison of 1-Dand 2-D models - sensitivity to leak-off and 2-D models - sensitivity to leak-off

51

,-- -o--

//

1.E4)5 1.E-04 1.E4J3 1.E~2 1.E4)1

Leak-off velocity VI (ft/min)

11- treatment BHP] I -+6-- 2-D model match I

6ocnj I “ - * -- -l-D model match I

o 103 203 300 400

time (reins)

Fig. 7 CASE 2 plain acid - comparison of 1-D Fig. 8 Field example - match of the BHP forand 2-D ‘modeis - sensitivity to leak-off

0.18

g 0.14

“= 0.12%!$ 0.1

“ij 0.08

$ 0.06

g 0.04

0.02

o 200 400 600 800

distance from weil (ft)

—

the re-frac treatment—=—.- .-

6000

#ooo

2

s4000>,-g3000

f

1 [ +2-D modei

-1 — 1-D modei

M0 200 400 600 800

distance from weii (ft)

Fig. 9 Comparison of 1-D and 2-D etched Fig. 10 1-D and 2-D frac conductiviteswidths for Cottonwood Creek well for Cottonwood Creek well

52

la)

n

# ‘0

10 500 1Om Im 20CKI 2503 3000 3500 4000 4502

Time Since Completion, days

Fig. 11- Cottonwood Creek well: Production history match

120 ...........................m-.T..............c%..cc<..~.......%

100

g 80u)a

~ 60

g

k 40

20

0

3830 3835 3840 3845 3850 3855 3860 3865 3870

Time Since Completion, days

Fig. 12 Cottonwood Creek well: buildup match

53