Copyright 2007, Society of Petroleum EngineersThis paper was prepared for presentation at the European Formation Damage Conference held in Scheveningen, The Netherlands, 30 May–1 June 2007. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, Texas 75083-3836 U.S.A., fax 01-972-952-9435. Abstract In several places around the world, notably the North Sea and the Middle East, carbonate reservoirs are being accessed with very long horizontal wells (2000 to 20,000 feet of reservoir section.) These wells are often acid stimulated to remove drill- ing fluid filter cakes and to overcome formation damage effects, or to create acid fractures or deep matrix stimulation to enhance productivity. Good acid coverage with a relatively small acid volume is required to economically obtain the desired broad reservoir access. We have developed a model to predict the placement of injected acid in a long horizontal well, and to predict the subsequent effect of the acid in creating wormholes, overcoming damage effects, and stimulating productivity. The model tracks the interface between the acid and the completion fluid in the wellbore, models transient flow in the reservoir during acid injection, considers frictional effects in the tubulars, and predicts the depth of penetration of acid as a function of the acid volume and injection rate at all locations along the completion. We have used this model to simulate treatments that are typical of those performed in the North Sea and in the Middle East. We present a hypothetical example of acid placement in a long horizontal section and an example of using the model to history match actual treatment data from a North Sea chalk well. Introduction Horizontal wells are drilled to achieve improved reservoir coverage, high production rates, and to overcome water coning problems. Acid stimulation is a cost effective method to enhance the productivity of horizontal wells in carbonate reservoirs. Acid can be injected using many acid placement methods including bullheading down the production tubing, injection from coiled tubing, injection with or following a diverting material, injection into intervals isolated by packers, and injection from acid jetting tools. Effective stimulation requires that a sufficient acid volume be placed in all desired zones. The model presented here is aimed at predicting the acid distribution and subsequent stimulation for a variety of placement methods used in long horizontal wells. Eckerfield et al. 1 concluded in their work that movement of interfaces formed between acid and completion fluid is significantly affected by uneven reservoir flow distribution, which ultimately leads to nonuniform volume of acid injected into the formation. Wellbore hydraulics were found to have much less impact because of the small wellbore volume relative to the volume of acid injected. Gdanski 2 described recent advances in carbonate stimulation stating that zonal coverage of long carbonate sections remains a challenge and most of the acidizing treatments are designed on the basis of of rules devised on the basis of past experience. Davies and Jones 3 presented an acid placement model for horizontal wells. The model was for barefoot completions in sandstone formations and the simulator used a pseudo-steady state reservoir model. They concluded that variations in reservoir properties along the treatment interval significantly impacted the acid placement. The need to include wellbore phenomena was also emphasized in their work. A new model is presented in this paper to study the acid distribution and evolution of skin during acidizing treatments in horizontal wells in carbonate reservoirs. The acid placement model couples models of wellbore flow, including interface tracking, a wormhole model to predict the effect of the acid injection on local injectivity, a skin evolution model that combines the stimulation effect of the acid with other skin effects, and a transient reservoir inflow model. The model predicts the bottomhole pressure response during an acid treatment, the distribution of acid along the treated section, and the resulting distribution of stimulation. Model description In a typical matrix acidizing process, the acid is being injected into the wellbore through production tubing, coiled tubing, or drill pipe. The acid emanating from the tubing (whichever type), or from ports in the tubing, displaces the resident wellbore fluid, creating one or two interfaces between these fluids. The acid behind the front flows into the formation and creates wormholes in the reservoir rock, increasing the injectivity of the contacted portions of the formation. The effect of the acid on the formation injectivity at any location along the well is accounted for with a local skin factor that is changing in response to the acid injected at that point. Local injectivity is simultaneously affected by the transient nature of the process – injection of any fluid will cause a pressure build SPE 107780 An Acid-Placement Model for Long Horizontal Wells in Carbonate Reservoirs Varun Mishra, SPE, D. Zhu, SPE, and A.D. Hill, SPE, Texas A&M U., and K. Furui, SPE, ConocoPhillips SPE 107780 2 _______________________________________________________________________________________________________________________________________ up in a porous medium. The transient pressure build up due to injection and the acid stimulation that is increasing injectivity are competing effects that must both be considered to properly predict acid placement. This acidizing model for a long horizontal well integrates several sub models which are coupled. These include a wellbore model which handles the pressue drop and material balance in the wellbore; an interface tracking model to predict the movement of interfaces between different fluids in the wellbore; a transient reservoir flow model; a skin factor model accounting for partial penetration and well completion effects; and, an acid stimulation model that predicts wormhole growth and the effects these have on local injectivity. Each model is discussed in this paper separately. Wellbore flow model This model incorporates the wellbore material balance and wellbore pressure drop. Fig. 1 Schematic of a wellbore during an acidizing process Figure 1 shows a part of the wellbore during an acid injection process. For flow of an incompressible fluid in a horizontal wellbore, we have 5 2 )] , ( [ 2 ) , ( d t x q f x t x p w f w ρ − = ∂ ∂ (1) ) , ( ) , ( t x q x t x q R w − = ∂ ∂ (2) Equation 1 describes the frictional pressure drop in the wellbore. The material balance, Eq. 2 shows that the change in the wellbore flow rate is equal to the flow rate into the formation at that point in the well. Model for tracking fluid interfaces A model to track the interfaces created between various injected fluids was presented by Eckerfield et al. 1 Our acid placement model uses a discretized solution approach which is integrated with the reservoir flow, wormhole, and skin models. Figure 2 depicts a part of the wellbore where the interface created between injected acid and wellbore fluid is traveling to the right. The velocity of an interface located at x int is simply, int int x x w A q dt dx = = (3) Injected acid X i nt | t=t X int | t= t+ ∆t A q w ∆ t Fig. 2 Interface movement inside the wellbore We solve this equation by discretizing the wellbore into small segments and assuming constant q w over each segment. Reservoir flow model During the acidizing process, the wellbore rate and the reservoir inflow at any location are changing with time so transient effects are occurring in the reservoir. A transient inflow equation with variable injection rate is 4 n n j D n n j j R n w s q t t p q p i p kl D D + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − ∑∆ = − − − = ) ( ) ( 2 1 1 µ π (4) where 1 − − = ∆ j R j R j R q q q (5) ) 80907 . 0 (ln 2 1 + ≈ D D t p (6) 2 6 10 395 . 4 w t D r c kt t φµ − × = (7) After dividing through by l, the length of a reservoir segment, and rearranging, q R n , the transient injection rate per unit length of wellbore at time t n can be written as Jx n w R Jx n R b p p a q − − − = ) ( (8) where ] ) ( [ 10 91816 . 4 1 6 n n D n D D Jx s t t p k a + − × = − − µ (9) ] ) ( [ ) ( ) ( 1 1 1 1 1 1 n n D n D D n D n D D n R n j j D n D D j R Jx s t t p t t p q t t p q b + − − − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − ∆ = − − − − = − ∑ (10) The constant in Eq. 9 is for oilfield units of bpm/ft for injection rate, md for permeability, and cp for viscosity. Wormhole model We have implemented two empirical models of the wormholing process that occurs in carbonate acidizing. The first of these is the volumetric model 5, 6 , which is based on the assumption that a constant fraction of the rock volume is dissolved in the region penetrated by wormholes. For radial flow, the volumetric model is SPE 107780 3 _________________________________________________________________________________________________________________________ bt w wh hPV V r r πφ + = 2 (11) The key parameter in this model is PV bt , the number of pore volumes of acid needed to propagate a wormhole through a core sample. The PV bt can vary from as low as one, or even slightly lower, when acid is injected at near the optimal rate in limestone, to as high as 50 when the wormholing process is not efficient. We also note that the wormholing model presented by Gdanski 2, 7, 8 , which is presented by Glasbergen et al. 8 as h V r wh φ 25 . 0 35 . 27 = (12) for units of cm for r wh and m 3 /m for V/h, can be approximated as a special case of the volumetric model with PV bt set to 1.1. This can be derived by equating the right hand sides of Eqs. 11 and 12 and neglecting the r w 2 term in Eq. 11. An improved empirical model of the wormholing process is that presented by Buijse and Glasbergen 9 . In this model, the wormhole propagation rate varies with the acid flux in a manner based on the commonly observed “optimal flux” behavior. The user of this model supplies the optimal acid flux and the optimal PV bt based on laboratory tests. We have implemented both the volumetric and the Buijse models of wormhole propagation in our acid placement simulator. If the PV bt input to the volumetric model is close to the value determined by the Buijse model at the acid flux occurring in the simulated acid treatment, the results from these two models are similar. Skin factor and well completion model The changing injectivity during acid injection is accounted for with a local skin factor, s(x), which includes the effects of the completion, possible formation damage, and the stimulation effect of the acid. In addition, injectivity of individual zones along a long horizontal well are affected by a partial penetration effect which can be treated as a skin effect. This partial penetration effect is described separately in the next section. The effects of the completion, formation damage, and stimulation are all coupled and depend on the completion type. For a cased, perforated completion, we used the perforation skin factor model of Furui et al. 10, 11 For this type of model, we assume that wormholes propagating from the tips of perforations can be considered as extensions of the effective lengths of the perforations. For an openhole or slotted/perforated liner completion, we assume radial flow of the acid through a possibly damaged zone that extends to a radial distance, r d . For this case, if the pressure drop in the region penetrated by wormholes is small, the evolving skin factor is For r wh <r d : ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = w d wh d d r x r x r x r x k k x s ) ( ln ) ( ) ( ln ) ( ) ( (13) And for r wh >r d : ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = w wh r x r x s ) ( ln ) ( (14) The radius of the region penetrated by wormholes, r wh , is obtained from the wormhole model. Partial penetration skin model Acid injection in long horizontal wells is often into relatively short, isolated sections of the well. Because the section treated is connected to the entire reservoir, the injectivity is higher than it would be if the reservoir ended at the end of the completion interval. A partial penetration skin factor, which will be negative, can be used to account for this effect. This partial penetration effect is important when injecting into relatively small intervals of horizontal wells and is not widely recognized, so a brief review is in order. Fig. 3 A partially completed vertical well The effect on productivity of completing a vertical well in only a portion of the reservoir has been described numerous times, beginning with Muskat 12 . For a well completed along a thickness, h w , in a reservoir of thickness h (Fig. 3), and in the absence of any other skin effects, the steady-state productivity index is ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = c w e s r r B kh J ln 2 . 141 µ (15) where s c is the partial completion (also called partial penetration) skin factor. When h w is less than h, s c is positive, accounting for the lessened productivity of the partially completed well. Models to calculate s c have been presented in many studies, including those of Cinco-Ley et al. 13 , Odeh 14 , and Papatzacos 15 . The productivity index could also be written using the completed thickness in the inflow equation: ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = ' ln 2 . 141 c w e w s r r B kh J µ (16) h Z w h w h w = Completion thickness Z w = Elevation r w SPE 107780 4 _______________________________________________________________________________________________________________________________________ If h w is less than h, s c ’ must necessarily be negative to give the same productivity index as Eq. 15. When h w is relatively small compared with h, these partial completion effects are large. For example, when hw/h is 0.25, s c is 8.8 using the Papatzacos model when the completion is centered in an isotropic reservoir. If ln(re/r w ) is 8, a typical value, the corresponding sc’ is -3.8. Thus, when calculating productivity or injectivity based on the completion zone thickness, the well appears to be stimulated because the reservoir is thicker than the completed interval. The corresponding situation for acid injection into a short interval of a horizontal well is shown in Fig. 4. Because we are assuming radial flow from the completed interval in our reservoir flow model, there will be a large partial penetration effect which we can account for with a negative skin factor. Fig. 4 Horizontal well partially open to the reservoir Fig. 5 Ellipsoidal flow geometry We have developed a simple model to calculate this type of skin factor as follows. Consider a horizontal well partially open to the reservoir as in Fig. 4. Ellipsoidal flow exists due to the partial opening of the wellbore in the reservoir as in Fig. 5. The ellipsoidal inflow equation is ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + = ∆ 1 1 ln ) 2 ( 2 . 141 ξ ξ µ e e a k q p (17) where ) ( sinh 1 D r − = ξ (18) a r r D / = (19) The radial flow equation based on a completed interval of length 2a is ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ∆ c w s r r a k q p ' ln ) 2 ( 2 . 141 µ (20) Equating the pressure drops given by Eqs. 17 and 20 gives the horizontal well partial penetration skin factor as ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + + − + + + = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − + = ) 1 1 ( ) 1 1 ( 2 ln ) 1 ( ) 1 ( 2 ln ' 2 2 D D D D w w c r r h r r r e h e r s ξ ξ (21) Solution approach The models for wellbore flow, partial penetration and completion skin factor, front tracking, reservoir inflow, wormhole growth, and skin evolution were incorporated into a numerical simulator. The solution method for these coupled models is described in the Appendix. Results We illustrate the predictions of the horizontal well acid placement model presented here with two contrasting examples. In the first example, acid is injected at a relatively low rate into a long section of a horizontal well. This is the situation where wellbore flow conditions are most likely to be significant. The second example, the simulation of an actual North Sea acid treatment, is a case of high rate injection into a very short interval. Example 1 – Small volume injection into a long interval. In this case, we investigate the effects of acid volume and acid injection rate on the placement of injected acid and the resulting distribution of acid along the well. The conditions for this case are presented in Table 1. The volumetric model of wormhole growth was used in this example. Table 1 Data for Case 1 Well length 1000 ft Number of grid blocks 50 Grid block length 20 ft Completion Open hole Damage radius 0.5 ft Permeability 2 md Index of anisotropy 1 Permeability impairment ratio 0.5 Reservoir rock Limestone Acid 15 % Hcl Reservoir pressure 3200 Psi Wormhole model Volumetric Pore volume for breakthrough (PV bt ) 2 Injection rate 2 bpm Duration of pumping 100 Min Assuming that the acid is being injected from a tubing tail located at one end of the completed interval, the progression of acid placement with time is shown in Fig. 6. By the end of 200 barrels of acid injection at 100 minutes of pumping time, acid has not yet reached the far end of the completed interval. SPE 107780 5 _________________________________________________________________________________________________________________________ For better acid coverage with this small volume treatment (the total volume pumped in 100 minutes is only 8.4 gal/ft), some method of diversion is required. 0.00 0.05 0.10 0.15 0.20 0.25 0 200 400 600 800 1000 Position along well (ft) A c i d v o l u m e ( b b l / f t ) 10 mi n 4 0 mi n 8 0 mi n 10 0 mi n Fig. 6 Acid coverage over the entire length of wellbore 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 0 200 400 600 800 1000 Position along well (ft) W o r m h o l e l e n g t h ( i n ) 10 min 40 min 80 min 100 min Fig. 7 Wormhole length distributions at different times 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0 200 400 600 800 1000 Position along well (ft) A c i d v o l u m e ( b b l / f t ) 500 bbls acid (21 gal/ft) 200 bbls acid (8.4 gal/ft) Fig. 8 Acid placement profiles for 200 and 500 bbls of acid The distribution of wormhole lengths along the wellbore created by this acid injection is shown in Fig. 7. By 100 minutes of acid injection, wormholes had extended 6 inches into the formation at the heel of the completed interval. Injection of larger volumes of acid improves the coverage of acid in this long interval. With 500 bbl of acid injected, the far end of the completed interval has received a significant amount of acid injection, with good acid coverage along most of the interval (Fig. 8). For a well with only minor damage, as was assumed for this case, although the acid is increasing the local injectivity, and thus retarding the progress of the acid down the wellbore, the injectivity is changing slowly, and thus does not have a strong effect on the acid placement. Another illustration of this is obtained by changing the efficiency of the acid treatment by changing the PV bt parameter used in the volumetric model. Figure 9 compares the acid placement for cases ranging from PVbt of 0.5 (very rapidly propagating wormholes) to inert fluid (no wormholes, hence no change in injectivity during injection). The acid coverage changes a little depending on how efficiently the acid is increasing injectivity, but it is not a large effect. 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0 200 400 600 800 1000 Position along well (ft) A c i d v o l u m e ( b b l / f t ) PVbt=0.5 PVbt=2 PVbt=10 Inert fluid Fig. 9 Acid placement profiles for different values of PV bt One of the interesting predictions of this model is the downhole pressure response during acid injection. Bottomhole pressure measurements are becoming more and more common during acid injection and can provide very useful diagnostic information about the treatment. The predicted pressure responses for a wide range of PV bt are shown in Fig. 10. When an inert fluid is injected, the pressure builds up because of the transient nature of the reservoir flow. With acid injection, the simultaneous stimulation is tending to decrease the injection pressure. Thus, depending on how efficiently the acid is increasing the near-well permeability, the injection pressure may rise or fall, as shown in Fig. 10. Comparison of actual treatment response with predictions like these provide a means of diagnosing the effectiveness of acid stimulation, and if done in real time can be used to optimize a treatment on the fly. The final aspect of this hypothetical case that we studied was the effect of the wormhole model on the predicted acid placement. Figure 11 shows the wormhole length distribution from the volumetric model with PV bt set to 2.5 compared with the predicted placement using the Buijse model with the PV bt- opt equal to 1.5. With the Buijse model, the wormhole propagation is varying with acid flux, with the maximum wormhole propagation being at the optimal injection condition. In this particular case, the acid fluxes are near the SPE 107780 6 _______________________________________________________________________________________________________________________________________ optimum, but somewhat higher. For the range of acid fluxes occurring in this treatment, the PV bt from the Buijse model varies from about 2 to about 2.5. The volumetric model, which assumes a constant PV bt independent of acid flux, gives a similar prediction of acid placement, and hence, wormhole distribution, when a value of 2.5 was used for PV bt . 3100 3200 3300 3400 3500 3600 3700 3800 0 20 40 60 80 100 120 Time (min) P r e s s u r e ( p s i ) PVbt=0.5 PVbt=2 PVbt=10 Inert fluid Fig. 10 Pressure response during acid injection 0.00 1.00 2.00 3.00 4.00 5.00 6.00 0 200 400 600 800 1000 Position along well (ft) W o r m h o l e l e n g t h ( i n ) Buijse's Model for PV bt-opt = 1.5 Volumetric model for PV bt = 2.5 Fig. 11 Comparison of wormhole distributions from the volumetric and Buijse’s models Example 2 – North Sea short interval, high volume acid treatment. In this case, we present predictions for an actual North Sea horizontal well completed in a chalk formation. The 6000 ft-long horizontal well was completed with sixteen individual 10 foot-long perforated intervals spaced along the well. Each interval is perforated with one shot per foot with the perforations oriented downward. In this stimulation treatment, each zone was isolated with packers and individually treated with 15% HCl. The treating string was equipped with pressure gauges between the packers and on either side of the packers enabling the operator to monitor the downhole treating pressure and to determine if the packers were set and not leaking. We used our acid placement model to history match the treating pressure response for one of the zones treated. The pressure records from the three downhole gauges are shown in Fig. 12. There is a clear indication of the packers being set. The pressure gauge on the heel side of the first packer shows no pressure response to injection, indicating that it is set. Then at about 22 minutes, the second packer is set, as indicated by the rapid pressure falloff recorded by the gauge beyond the second packer. We began simulation of the treatment at the 22 minute time, when both packers were set and acid injection into the isolated interval began. 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 10 20 30 40 50 Time (min) P r e s s u r e ( p s i ) BHP-Zone BHP-Below BHP-Above Isolation of bottom zone achieved Isolation of top zone achieved Start of acid injection in the formation Fig. 12 Pressure response of downhole gauges in Case 2 To history match the pressure response during this treatment, we input the actual injection rate schedule recorded to our model – Fig. 13 shows how we approximated the changing rate schedule as a series of discrete rate changes. Additional data used in the model is given in Table 2. Table 2 Input data for Case 2 Casing ID 6.625 inches Coiled tubing OD 2.55 inches Pipe roughness 0.0001 Zone length 10 ft Reservoir pressure in zone 5350 psi Reservoir compressibility 5E-06 psi -1 Permeability 5 md Porosity 0.38 Initial formation damage none Perforation length 7 inches Perforation diameter 0.264 inches Perforation spacing 1 spf Perforation phasing 0 degree Perforation orientation 90 degree Acid type HCl Acid density 69.91 lbm/ft 3 Acid viscosity 1 cp Acid concentration 15% Wormhole model Volumetric Number of grid blocks 10 Reservoir thickness 200 ft SPE 107780 7 _________________________________________________________________________________________________________________________ 0 2 4 6 8 10 12 14 16 0 10 20 30 40 50 Time (min) I n j e c t i o n r a t e ( b p m ) Treatment rate Simulated rate zonal isolation achieved at 6 bpm Pzone=9050 Psi Acid Injection Started at 5 bpm Response of the whole well (packers are not set) Acid injection stopped Fig. 13 Rate schedule for Case 2 From the data given about the well, we calculated the initial skin factor as follows. For the given perforating conditions, we obtained a perforation skin factor of 4.6 using the Furui et al. 10,11 model. For this very short interval in a large reservoir, we calculated a partial penetration skin factor with Eq. 21 of - 5.5. Combining these, and assuming no formation damage was present initially, we use an initial total skin factor of -.9. We then adjusted the reservoir permeability and the PV bt in the volumetric model to obtain a match of the actual treating pressure (Fig. 14). This match ws obtained by setting the PV bt to 4.5, which means the acid is propagating wormholes relatively slowly into the matrix and that a large volume of rock is being dissolved in the treated region. With PV bt of 4.5, the wormhole front is moving 4.5 times slower than the injected fluid (spent acid) front. 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 20 40 60 80 Time (min) P r e s s u r e ( p s i ) Treatment pressure Simulated pressure Fig. 14 History match of treatment pressure For the high rate injection into such a short interval, acid placement is not an issue, as shown in Fig. 15. More importantly for this type treatment is what the model can tell us about the effects of this large volume acid treatment from the predicted depth of acid penetration into the formation. Notice that this interval has received 120 barrels of acid, about 500 gal/ft. 0 2 4 6 8 10 12 0 2 4 6 8 10 Position along well (ft) A c i d v o l u m e ( b b l / f t ) 120 bbls acid injected (504 gal/ft) Fig. 15 Acid placement for Case 2 From the history-matched pressure response using a PV bt of 4.5, we predict that a radial region of wormholes has propagated about 40 inches into the formation. The volumetric model presumes that the acid is dissolving a fixed fraction of rock, given by 6 bt Ac PV N = η (22) Where the Acid Capacity No., N Ac , is ( ) rock HCl Ac N ρ φ ρ φβ − = 1 15 (23) For this high porosity chalk formation, η is 0.22, meaning that in the regions where wormholes have formed, 22% of the rock has been removed. With the initial porosity in this chalk formation being 38%, after this amount of dissolution, the porosity would be 0.52. It is likely that this amount of dissolution would result in the collapse of some of the remaining rock in this region, leaving a large cavern. Based on the dissolving power of 15 % HCl reacting with calcite, 12 bbl of acid injection into a single perforation will dissolve 5.5 ft 3 of solid. Assuming that the dissolution region extends 40 inches from the wellbore, as predicted by the volumetric model with PV bt = 4.5 as used in this history match, the acid has likely dissolved a sufficient amount of rock out to at least this distance to make the remaining rock unstable. Conclusions We have developed an acid placement model for horizontal wells in carbonate reservoirs which combines a wellbore flow model, including interface tracking, a wormhole model to predict the effect of the acid injection on local injectivity, a skin evolution model that combines the stimulation effect of the acid with other skin effects, and a transient reservoir inflow model. With this model, we find that • Small volume treatments in long horizontal intervals result in non-uniform acid placement, but that the placement improves with increasing acid volume; SPE 107780 8 _______________________________________________________________________________________________________________________________________ • Partial penetration effects are important when injecting into relatively short intervals of long horizontal wells; • The parameters in a wormholing model can be adjusted to history match (or predict) the pressure response of an acid treatment in a horizontal well; • History matching of an acid treatment in a North Sea well completed in a chalk formation required a relatively high value of the pore volumes to breakthrough parameter, suggesting that the acid is propagating slowly into the rock, creating a cavity around the wellbore. Acknowledgements The authors thank the sponsors of the Middle East Carbonate Stimulation joint industry project at Texas A&M University for support of this work. Nomenclature a = half length of open interval, ft a jx = parameter in inflow equation, bbl/min-psi A = cross-sectional area of wellbore, ft 2 A i = coefficients in solution matrix b jx = parameter in inflow equation, bbl/min B = formation volume factor, dimensionless B i = coefficients in solution matrix c t = total compressibility, psi -1 C i = coefficients in solution matrix d = internal diameter of wellbore, ft f f = fanning friction factor, dimensionless h = reservoir thickness, ft h w = length of completed interval, ft J = productivity index, bbl/day/psi J s = specific productivity index at any point in wellbore, bbl/day/psi/ft k = permeability of reservoir rock, md k d = permeability of damaged region, md l = length of reservoir segment, ft L = length of wellbore, ft N Ac = acid capacity number, dimensionless p D = dimensionless pressure p i = initial reservoir pressure, psi p w = pressure at any point in the wellbore, psi PV bt = pore volume for break through, dimensionless q R = reservoir inflow rate per unit length of wellbore, bbl/min/ft q w = wellbore flow rate at any point, bbl/min r d = radius of damaged zone, ft r D = dimensionless radius r e = reservoir drainage radius, ft r w = wellbore radius, ft r wh = radius of wormhole region, inches s = skin factor, dimensionless s c = partial completion skin factor, dimensionless s c ’ = partial completion skin factor using h w for thickness t = time, minutes t D = dimensionless time V = volume, ft 3 x = position of any point along the wellbore length, ft x int = location of interface from the heel of the well, ft Z w = elevation of completed interval, ft β 15 = gravimetric dissolving power of 15% HCl, dimensionless ζ = pressure drop function, psi/ft/bbl/min η = wormholing efficiency, dimensionless µ = viscosity of fluid, cp ξ = ellipsoidal coordinate dimension ρ = density of fluid in wellbore, lb m /ft 3 ρ HCl = density of HCl, lb m /ft 3 ρ rock = density of rock, lb m /ft 3 φ = porosity of the reservoir rock, fraction ∆q R = change in rate, bbl/min ∆t = time step, minute References 1. 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Fig. A-1 Schematic of a segmented wellbore Appendix A: Solution approach ) , ( ) , ( t x q x t x q R w − = ∂ ∂ (A-1) Jx n w R Jx n R b p p a q − − − = ) ( (A-2) ) , ( ) ( )] , ( [ 2 ) , ( 5 2 t x q q d t x q f x t x p w w w f w ζ ρ − = − = ∂ ∂ (A-3) Eq. A-1 is the wellbore material balance. Eq. A-2 is achieved from the reservoir flow model and Eq. A-3 represents the pressure drop in the wellbore, where ζ i is a function of q w . ] ) ( [ , , , 2 / 1 , 2 / 1 , i Jx i w R i Jx i i w i w b p p a x q q + − ∆ = − − + (A-4) For i=1, 2, 3, 4, 5 2 / 1 , 1 , 1 , 2 ) ( + + + ∆ + ∆ − = − i w i i i i w i w q x x p p ζ (A-5) For i=1, 2, 3, 4 Fig. A-1 is a schematic of segmented wellbore. These equations are to be discretized in this domain and will be solved simultaneously. Eq. A-1 and Eq. A-2 are coupled and can be written in discretized form as Eq. A-4. Eq. A-5 is written as discretized form of pressure drop equation, Eq. A-3. Initial and boundary conditions can be applied on this domain. When the injection rate is specified at the heel, 9 equations can be written for 5 segments. This set of 9 equations reduces to a tri-diagonal matrix system as in Eq. A-6, where coefficients A i , B i , and C i are defined by Eqs. A-7, A-8 and A- 9 respectively. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ + + + + + + = ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ − − − − − − − − 5 5 4 4 3 3 2 2 1 1 5 , 2 / 9 , 4 , 2 / 7 , 3 , 2 / 5 , 2 , 2 / 3 , 1 , 5 4 4 3 3 2 2 1 1 0 0 0 0 1 0 0 0 0 0 0 0 1 ) ( 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 ) ( 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 ) ( 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 ) ( 1 0 0 0 0 0 0 0 1 B p A B p A B p A B p A Q B p A p q p q p q p q p A q C A q C A q C A q C A i i i i w i w w w w w w w w w (A-6) i Jx i i a x A , ∆ = (A-7) i Jx i i b x B , ∆ = (A-8) 2 / ) ( 1 i i i i x x C ζ ∆ + ∆ = + (A-9)