spe-63183-ms.pdf

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AbstractWhen a naturally fractured carbonate formation is treated withacid at pressures below the fracturing pressure, the acidizingprocess will likely be different from either a matrix treatmentor an acid fracturing treatment. Our previous experimentalstudy shows three kinds of acid etching patterns after acidizingnaturally-fractured carbonates and also illustrates theirrelationships with the fracture properties and acidizingconditions. Based on the experimental observations, amathematical model of acidizing naturally-fracturedcarbonates has been developed. The model includes bulksolution transport, acid transport and reaction, and the changeof fracture width by acid dissolution. A new approach wasused to treat leakoff acid and acid-rock reactions at thefracture walls. The new acidizing model for numericallygenerated rough-surfaced fractures predicts the same kinds ofacid etching patterns and the same relationships between acidetching patterns and the fracture width, surface roughness andleakoff rate as observed in experiments.

IntroductionMatrix acidizing is a stimulation method to improve the wellproductivity by pumping acid at a pressure lower thanformation fracture pressure, and usually creates wormholes inan un-fractured carbonate. Acid fracturing is another methodto improve the well productivity in carbonate reservoirs, inwhich acid flows through the relatively wide hydraulicfracture and etches the fracture walls. When a naturallyfractured carbonate formation is treated with acid at pressuresbelow the formation fracture pressure, the acidizing process isneither like matrix acidizing nor like acid fracturing. Ourprevious experimental studies1 show that in acidizingnaturally-fractured carbonates with a fracture width on the

order of 10-3 cm, most acid flows through and reacts with rockinside the fracture, not the matrix. When the fracture width issmaller than 2×10-3 cm, wormholes are created along thefracture surface from the inlet to the outlet, which is likematrix acidizing. When the fracture width is between 3×10-3

cm and 8×10-3 cm, a channel which is broad near the inlet andnarrower farther away from the inlet is created along thefracture surface. When the fracture width is larger than 1×10-2

cm, acid etches most of the fracture surface like in acidfracturing. When the fracture surface becomes rougher,wormholes are more easily created. Leakoff has little effect onthe etching pattern along the fracture surface but createswormholes perpendicular to the fracture surfaces.

We developed a mathematical model to describe theacidizing process in naturally-fractured carbonates. The modelis based on mass conservation for the acid solution, acidtransport and the change of fracture width by acid dissolution.Pressure, fracture width and acid concentration as functions ofposition and time can be predicted by the model. Thesimulation results compared well with the experimentalobservations.

Model DevelopmentWe initially formulated the model for the laboratory scale inorder to make direct comparisons with our experimentalresults. The model domain is a block of carbonate core samplewith a length, l, a thickness, 2wm and a height, h, as shown inFig. 1. A single fracture is placed in the middle and throughthe entire core sample. Acid is introduced from one side, flowsalong the fracture, and flows out of the fracture from theopposite side. The leakoff acid penetrates the matrixperpendicular to the fracture. The coordinate system is definedsuch that the x direction is the effluent acid flow direction,which is aligned with the length of the fracture, the z directionis aligned with the height of the fracture and the y direction isperpendicular to the fracture surface. The fracture plane is thex-z plane and the leakoff is in the y direction. The controlvolume is a parallelepiped with dimensions of ∆x, ∆z and b,where b is the fracture width at the point (x, z) (Fig. 2).

Mass Conservation. A mass balance for acid solution insidethe fracture is

SPE 63183

Modeling of the Acidizing Process in Naturally-Fractured CarbonatesC. Dong, D. Zhu and A. D. Hill, University of Texas at Austin

2 C. Dong, D. Zhu, A. D. Hill SPE 63183

( ) ( )t

bv

z

bv

x

bvl

zx

∂∂=−

∂∂−

∂∂

− 2 (1)

where vx and vz are the average velocities in the fracture atthe point (x, z) in the x and z directions respectively, vl is theleakoff velocity and b is the fracture width at the point (x, z).

If laminar flow is assumed, then the permeability of the

fracture at a certain position is 122b , and the average

velocities in the fracture width vx and vz can be calculated by

x

pbvx ∂

∂−=µ12

2

(2)

z

pbvz ∂

∂−=µ12

2

(3)

where µ is acid viscosity and p is the pressure at the point(x, z).

The leakoff velocity vl is an important parameter in Eq. 1. Inour experiments, the leakoff acid penetrates the rock matrix inthe y direction, and depends on the pressure drop appliedacross the core. Thus, in the laboratory scale, the leakoffvelocity is proportional to the pressure difference at the point(x, z) in the y direction, which is calculated by Darcy’s law as

m

el w

ppkv

−=µ

(4)

where k is the permeability of the core matrix, pe is the backpressure and wm is the half thickness of the core.

According to Eq. 4, the leakoff velocity changes withposition and time. Since the pressure near the inlet is higherthan farther away from the inlet, the leakoff rate near the inletshould be higher than other areas. Thus, wormholes in theleakoff direction are most probably created near the inlet,which is consistent with the experimental observations.

Combining Eqs. 1 - 4 yields

t

b

w

ppk

z

pb

yx

pb

x

m

e

∂∂=

−

−

∂∂

∂∂+

∂∂

∂∂

µ

µµ

2

12

1

12

1 33

(5)

Equation 5 represents transport of the acid solution.

Acid transport. Acid transport in the fracture is given by

( ) ( ) ( )t

bCkCvC

z

bvC

x

bvCgl

zx

∂∂=−−

∂∂−

∂∂

− 22 (6)

where C is the average acid concentration across thefracture width at the point (x, y) and kg is the apparent mass-transfer coefficient, defined by Roberts and Guin2 as

( )wgb

y

CCky

CD −=

∂∂

−=

2

(7)

where C is the acid concentration, D is acid diffusioncoefficient and Cw is the acid concentration at the fracturewalls. kg has the same units, m/s, as velocity, and accounts forthe rate of acid flux to the fracture walls by diffusion. Whenhydrochloric acid is used to acidize carbonates as in ourexperiments, the reaction rate is so high that Cw can be treatedas zero.

The first two terms of Eq. 6 are the acid transport caused bythe acid flow in the x and z direction. The last two terms arethe acid transport out of the fracture in the y direction byleakoff and diffusion respectively.

Substituting Eqs. 2 - 4 into Eq. 6 yields

( )t

bCkC

w

ppkC

z

pbC

zx

pbC

x

gm

e

∂∂=−−

−

∂∂

∂∂+

∂∂

∂∂

22

12

1

12

1 33

µ

µµ(8)

Equation 8 represents acid transport during the acidizingprocess.

Change of the fracture width. The conventional way to treatthe acid transport in the y direction is shown in Fig. 3. Acid istransported to the fracture walls by leakoff and diffusion, andbefore leaving the fracture, all leakoff acid and diffusion acidcontacts and reacts with the fracture walls. No wormhole iscreated perpendicular to the fracture. If this is true, the etchingof the fracture wall increases as leakoff rate increases.However the experimental results showed that there was nosignificant difference in acid etching between with andwithout leakoff. Furthermore, almost all leakoff acid flowsthrough wormholes not through the fracture walls uniformly,which implies that most leakoff acid flows out of the fracturewithout reacting with the fracture walls. Thus, a new approachwas used in the model as shown in Fig. 4.

Leakoff acid is highly unevenly distributed. Most leakoffacid flows through wormholes perpendicular to the fracture,and does not react with the fracture walls, as shown in Fig. 4.Only a small part of the leakoff acid reacts with the fracturewalls before leaving the fracture. Since diffusion acid does notcreate wormholes, all acid transported by diffusion reacts withthe formation at the fracture walls. With these assumptions,the change of the fracture width is represented by

( )β

φρη −∂∂=+ 1

22t

bvCCk lg (9)

SPE 63183 MODELING OF THE ACIDIZING PROCESS IN NATURALLY-FRACTURED CARBONATES 3

where ρ is the rock density, φ is the rock matrix porosity, βis the dissolving power of acid and η is the fraction of leakoffacid that reacts with rock at the fracture walls. The two termson the left hand side of Eq. 9 represent the rate of acidtransported to the fracture walls and reacted with rock at thefracture walls by diffusion and by leakoff. All the acidtransported by diffusion reacts with rock on the fracturesurface, while only a small part of the leakoff acid reacts at thefracture surface. For our experimental conditions, η << 1.

Substituting Eq. 4 into Eq. 9 and rearranging yields

( ) t

bCkC

w

ppkg

m

e

∂∂=

+−

−22

1 µη

φρβ

(10)

Equation 10 represents the change of the fracture width byacid dissolution.

Equations 5, 8 and 10 form the theoretical basis of the

model. Three unknowns p, b and C change with both positionand time, and can be calculated by solving the system ofequations.

Initial and boundary conditions. In our laboratoryexperiments, before acidizing, inside the fracture the pressuresare the back pressure and acid concentrations are zero. Whenacidizing begins, acid is introduced into the fracture with aconstant injection rate from the inlet side at x = 0. There is noflow at the top and bottom sides where z = 0 and z = h. At theoutlet side where x = l, the pressure is the back pressure. Theacid concentration at the inlet side is Ci. Eqs. 11 - 18 representthe initial and boundary conditions for our laboratoryexperiments.

0 ,, 0),( =∀= tzxzxp (11)

0 ,, 0),( =∀= tzxzxC (12)

0at ),(),( == tzxfzxb (13)

0 12 0

0

3

>=∂∂

=∫ tqdz

x

pb

x

h

µ(14)

0 ,z ),( >∀= tpzlp e (15)

0 , 00

>∀=∂∂

=tx

z

p

z

(16)

0 , 0 >∀=∂∂

=tx

z

p

hz

(17)

0 ,z ),0( >∀= tCizC (18)

where q is the injection rate.

Generation of Rough-Surfaced FracturesTo solve the system of equations, we numerically generate therough-surfaced fracture with unevenly distributed fracturewidths as the one of the initial conditions. First, rough fracturesurfaces are generated by a method called the random mid-

point displacement method whose algorithm is based on thefractional Brownian motions proposed by Saupe3. Twoparameters, fractal dimension and initial standard deviation ofthe rough surface are needed to generate a rough surface. Eachrough surface is divided into rectangular elemental areas.Within each elemental area, the elevation over the surfacebase is the same, and the surface roughness is caused bydifferent elevations of different elemental areas. Figs. 5-7show the numerically generated rough surfaces with the samestandard deviation of 0.01 cm but different fractal dimensionsranging from 2.0 to 2.5.

Two surfaces are combined together to form a rough-surfaced fracture with a certain hydraulic width. In most cases,the hydraulic width is not equal to the real average fracturewidth. Brown4 analyzed the fluid flow in rough-surfacedfractures and found that the relationship between the hydraulicwidth and the real average fracture width can be expressed by

20.1

3

6.01

1−

+

=

ap

avav

h

bb

b

σ

(19)

where bh is the hydraulic width, bav is the real averagefracture width and σap is the standard deviation of the fracturewidth.

We use the method provided by Amadei and Illangasekare5

to generate rough-surfaced fractures. Two rough surfaces withthe same fractal dimensions and initial standard deviations butdifferent random surfaces are placed parallel to each other andthe distance between the two surfaces is set to the hydraulicwidth. The fracture width (void space) can be calculated fromthe difference in topographic elevation between the twosurfaces. At contact points, any overlap of the surface isremoved and the local width is set to the value correspondingto the permeability of the rock matrix without fractures. Thereal average fracture width bav and the standard deviation ofthe fracture width are calculated. By checking the relationbetween bav and bh with Eq. 19 and adjusting the distancebetween the two surfaces, the initial fracture width distributionwith a certain surface roughness and a hydraulic width isobtained. This fracture width distribution is input into thesimulation as one of the initial conditions.

SimulationResults of Eqs. 5, 8, and 10 were solved numerically tosimulate the acidizing process in naturally-fracturedcarbonates.

In the simulations, the hydraulic widths were in the range of1.7 × 10-3 cm to 1.2 × 10-2 cm, which were the same as in theexperiments. We measured the fracture surface roughnessfrom the core samples and used these values in thesimulations. In both the experiments and the simulations, theinitial leakoff rate was controlled by the rock matrixpermeability and the back pressure. All other parameters such


as the rock properties and the acidizing conditions were thesame as used in the experiments. The input parameters used inthe simulations are listed in Table 1.

For each simulation, the fracture was numerically generatedwith a different set of random numbers. The dimension of thefracture was 6.5×6.5 cm, and the grid blocks were 65×65. Thefracture width distribution before and after acidizing wereplotted. Other data like pressure, flow rate and acidconcentration were also recorded, but here only the fracturewidth data are presented.

Case 1. Base case. The hydraulic fracture width beforeacidizing was 6.4 × 10-3 cm. The standard deviation and thefractal dimension of the fracture surface were 1.0 × 10-2 cmand 2.0 respectively. The initial leakoff rate was 0%. Fig. 8shows the initial fracture width distribution before acidizing,illustrating the random distribution of the fracture width. InFig. 8 and all the following pictures, the bottom side is theacid inlet side and the topside is the outlet side.

The simulated width distribution after acidizing is shown inFig. 9. A channel was created, which was broad and deep nearthe inlet and narrow and shallow near the outlet. The depth ofthe channel is around 0.3 cm and the surface of the channelchanged because of the acid etching. Other fracture areasbeside the channel had little change.

This simulation result is very close to the experimentalresult with the same acidizing conditions, as shown Fig. 10. Inboth the simulation and the experiments, we observed thesame kind of etching patterns with the same characteristics.

Case 2. Increased fracture width. In case 2, the fracturewidth before acidizing was increased to 1.2 × 10-2 cm, and allother parameters were similar to case 1. The fracture beforeacidizing contained more large fracture widths, which weredistributed unevenly as shown in Fig. 11.

After acidizing, a very broad channel was predicted, and theentire inlet side was etched by acid as shown in Fig. 12. Asacid flowed away from the inlet, the channel converged andbecame narrower at the outlet, but the average channel breadthwas larger than in case 1. Other areas beside the channel hadlittle change. In case 2, as the fracture width increased, thechannel became broader, and the initial fracture widthdistribution had less effect on the final fracture widthdistribution. Compared with the experimental result with thesame conditions shown in Fig. 13, the simulator predicted thesame kind of etching pattern as the fracture width increased.

Case 3. Decreased fracture width. In case 3, the fracturewidth was decreased to 1.7 × 10-3 cm with all other parametersthe same as case 1. The fracture before acidizing containedfew large fracture widths, and the contact area became muchlarger than before as shown in Fig. 14. Large fracture widthswere concentrated in the top-left and the lower-center parts ofthe fracture.

Figure 15 shows the simulation result of case 3. A narrowchannel was created, which became gradually narrower farther

away from the inlet, and finally split to a channel and a narrowwormhole. The average channel breadth was smaller than incase 1, and the channel was along the initial large fracturewidths before acidizing. In this case, when the fracture widthdecreased, the channel became narrower and the initialfracture width distribution before acidizing had more effect onthe final fracture width distribution, very similar to theexperimental results shown in Fig. 16.

The simulation results of cases 1, 2 and 3 predict the samerelationship between the acid etching pattern and the initialfracture width as formed in the experiments.

Case 4. Increased leakoff. In case 4, the leakoff rate wasincreased to 19.2%, and the other parameters were the same ascase 1. The initial fracture width distribution is shown in Fig.17, which is like case 1.

After acidizing, a channel was created as shown in Fig. 18.The channel depth was about 0.3 cm. The depth and theaverage channel breadth were close to both the experimentalresult shown in Fig. 19 and case 1. Thus in both simulationand experiments we observed that increasing the leakoff ratehas little effect on acid etching patterns.

Case 5. Increased standard deviation of fracture surface.In case 5, we increased the fracture surface roughness byincreasing the standard deviation of the fracture surfaces to 3× 10-3 cm and keeping the other parameters the same as in case1. The fracture contained larger contact area and the fracturewidth distribution became more non-uniform as shown in Fig.20.

After acidizing, a “y” shaped wormhole was created asshown in Fig. 21. The average wormhole breadth was smallerthan in case 1. Other areas besides the wormhole had littlechange. In this case, as the standard deviation of the fracturesurfaces increased, the channel became narrower andwormholes were easily created. This was very close to theexperimental result shown in Fig. 22.

Case 6. Increased fractal dimension. In case 6, the fractaldimension of fracture surfaces was increased to 2.5, and otherparameters were unchanged. The initial fracture widthdistribution is shown in Fig. 23. The contact area becamemuch larger, and the fracture width distribution became morenon-uniform.

After acidizing, a narrow channel was created as shown inFig. 24. The channel followed the center part of the fracturewhich contained the largest fracture width before acidizing.The breadth of the channel did not change much and wassmaller than in case 1. Other areas besides the channel hadlittle change. In this simulation, as the fractal dimensionincreased, the channel became narrower.

Combining the simulation results of cases 5 and 6, when thefracture surface roughness increased, the channel becamenarrower and a wormhole was more easily created. Also theinitial fracture width distribution before acidizing had moreeffect on the final fracture width distribution. Thus in both


simulations and experiments, we observed the samerelationship between the channel breadth and the fracturesurface roughness.

CONCLUSIONSA mathematical model has been developed to describe theacidizing process in naturally-fractured carbonates. The modelis base on mass conservation of the acid solution, acidtransport and the change of fracture width by acid dissolution.A new approach is used to treat leakoff acid and acid-rockreactions at the fracture walls. Simulations of acidizingnumerically-generated, random-rough-walled fractures predictchannels, which are broad and deep near the inlet and narrowand shallow farther away from the inlet. As the fracture widthdecreases, or as the surface roughness increases, the averagebreadth of the channels decreases and wormholes are moreeasily created. When the initial fracture width increases, thechannel becomes broader and less affected by the initialfracture width distribution. Leakoff rate has little effect onetching patterns. In both experiments and the simulations, thesame relationships between acid etching patterns and fracturewidth, surface roughness and leakoff rate are observed,indicating that the description of acid transport and reactionand the method of numerically generating rough-surfacedfractures are correct.

NOMENCLATUREb = fracture width (cm)

bav = average fracture width (cm)bh = hydraulic fracture width (cm)C = acid concentration (wt.%)

C = average acid concentration across the fracturewidth at the point (x, y) (wt.%)

Cw = acid concentration at the fracture walls (wt.%)D = diffusion coefficient (m2/s)h = fracture height (cm)k = rock matrix permeability (md)

kg = mass transport coefficient (m/s)L = fracture length (cm)p = pressure inside the fracture (psi)

pe = back pressure (psi)q = acid injection rate (ml/min)t = time (s)

vl = fluid leakoff velocity (cm/s)vx = average velocity in the x direction across the

fracture width (cm/s)vy = average velocity in the y direction across the

fracture width (cm/s)wm = half thickness of the core sample (cm)

β = acid solving power, (kg rock/kg pure acid)φ = rock matrix porosityµ = acid viscosity (cp)ρ = rock density (g/cm3)η = fraction of leakoff acid that reacts with the

rock at the fracture wallsσap = standard deviation of the fracture width (cm)

REFERENCE1. Dong, C., Zhu, D. and Hill, A. D.: “Acid Etching Pattern

in Naturally-Fractured Formations,” SPE 56531 presentedat the 1999 SPE Annual Technical Conference, Houston,Texas, 3–6 October 1999.

2. Roberts, L. D. and Guin, J. A.: “A New Method forPredicting Acid Penetration Distance”, SPEJ (Aug. 1975)277-285.

3. Peitgen H. O. and Saupe D.: The Science of FractalImages, Springer, Berlin 1988.

4. Brown S.: “Fluid Flow Through Rock Joints: the Effect ofSurface Roughness,” J. Geophys. Res. 92, 1337-1347(1987).

5. Amadei, B. and Illangasekare, T.: “A MathematicalModel for Flow and Solute Transport in Non-Homogeneous Rock Fractures,” Int. J. Rock Mech. Min.Sci. & Geomech., vol. 29, November 1994.


Table 1. Input parameters for simulations

SimulationHydraulic fracture

width before acidizing( × 10-3 cm )

Fractal dimensionof the fracture

surface

Initial standarddeviation of thefracture surface

( ×10-2 cm )

Initialleakoff rate

Case 1 6.4 2.0 1.0 0%

Case 2 12.0 2.0 1.0 0%

Case 3 1.7 2.0 1.0 0%

Case 4 6.7 2.0 1.0 20%

Case 5 6.0 2.5 3.0 0%

Case 6 6.9 2.0 1.0 0%

Fig. 1. Model domain: a carbonate core sample with asingle fracture

Fig. 2. Control volume in the fracture


Fig. 3. Conventional treatment of leakoff anddiffusion acid

Fig. 4. New approach to treat leakoff anddiffusion acid

Fig.5. Numerically generated roughsurface with fractal dimension 2.0

Fig. 6. Numerically generated roughsurface with fractal dimension 2.2

Fig. 7. Numerically generated roughsurface with fractal dimension 2.5


Fig. 8. Initial fracture width distribution beforeacidizing in case 1

Fig. 9 Final fracture width distribution afteracidizing in case 1

Fig. 10. Acid etching pattern fromexperiments



Fig. 12. Final fracture width distribution afteracidizing in case 2





Fig. 16. Acid etching patternfrom experiments

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