A New Correlation of Acid-Fracture Conductivity Subject to Closure Stress

Transcription

1 A New Correlation of Acid-Fracture Conductivity Subject to Closure Stress J. eng SPE Texas A&M University; Jianye Mou SPE China University of Petroleum (Beijing); and A.. Hill SPE and. Zhu SPE Texas A&M University Summary The conductivity of an acid-etched fracture depends strongly on void spaces and channels along the fracture resulting from uneven acid etching of the fracture walls. In this study we modeled deformation of the rough fracture surfaces acidized in heterogeneous formations based on synthetic permeability distributions and developed a new correlation to calculate the acid-etched fracture conductivity. In our previous work we modeled the dissolution of the fracture surfaces in formations having small-scale heterogeneities in permeability. The characterization of the correlated permeability fields of rock includes the average permeability normalized correlation lengths in both horizontal and vertical directions and normalized standard deviation. These statistical parameters have a significant influence on the fracture-etching profiles obtained from the model. Beginning with this fracture-width distribution we have modeled the deformation of the fracture surfaces as closure stress is applied to the fracture. The elastic properties of the rock such as Young s modulus and Poisson s ratio have effects on the size of the spaces remaining open after fracture closure. After the model yields the width profile under closure stress the overall conductivity of the fracture is then obtained by numerically modeling the flow through this heterogeneous system. In this paper we introduce our models and investigate the effects of permeability and mineralogy distributions and rock elastic properties on the overall conductivity of an acid-etched fracture. A new acid-fracture conductivity correlation is developed on the basis of many numerical experiments. Introduction In an acid-fracturing treatment plain acid gelled acid foamed acid or an acid-containing emulsion is injected into a hydraulically created fracture. The acid flows along the fracture leaks off into the formation and is transported to the fracture walls by diffusion and convection to react nonuniformly with the rock. As hydraulic pressure dissipates after acidizing the fracture props itself open with the relatively undissolved regions acting as pillars that leave more-dissolved regions as open channels. The uneven etching along the fracture walls yields the lasting conductivity after closure (Ruffet et al. 1997). Several correlations have been developed to predict the conductivity of rough-walled fractures. Some empirical correlations were based on experiments (Nierode and Kruk 1973; Nasr-El-in et al. 28; Pournik 28) and some correlations were developed theoretically (Gangi 1978; Walsh 1981; Swan 1983; Gong et al. 1999). The theoretical models are not straightforward to apply in practice because the fracture information after acidizing in these models is difficult to obtain. The empirical correlations based on experiments such as Nierode and Kruk s correlation are widely used in industry because they are in simple form and the parameters are attainable. However the core plugs in Nierode and Kruk s experiments were 1 in. in diameter and 2 3 in. in length. On this scale only random roughness can be observed on the fracture Copyright 212 Society of Petroleum Engineers This paper (SPE 1442) was accepted for presentation at the SPE Hydraulic Fracturing Technology Conference and Exhibition The Woodlands Texas USA January 211 and revised for publication. Original manuscript received for review 18 November 21. Revised paper received for review 26 August 211. Paper peer approved 31 August 211. surfaces after acidizing. Strongly correlated roughness yields channels that are not detected because they are on the same scale as the core plugs. Hakami and Larsson (1996) pointed out that all fractures exhibited channeling to some extent. But the current correlations do not account for the contribution from channels because of their small scale. Additionally they do not explicitly incorporate the effects of heterogeneities on fracture conductivity. Even though Nierode and Kruk did acid-fracture conductivity tests on many kinds of rock with different heterogeneities the least-squares fit of data only predicts the median of all kinds of surface-etching profiles. Therefore the empirical correlations provide a conservative method to predict fracture conductivity (Williams et al. 1979). In addition the grid dimension in an acid-fracture simulator is several feet to tens of feet. A scalar gap exists between the grid size in simulation and the size of core plugs in experiments. In this study we developed new acid-fracture conductivity correlations for an intermediate scale. The calculation domain is 1 1 ft which is the same order of magnitude as one simulator gridblock. The numerical simulation also incorporates the small-scale features observed in experiments for each gridblock with gridblock size being similar to that of a core plug. The new correlations act as a bridge between the acid-fracture simulator and the small-scale experiments. On the intermediate scale the numerical simulations consider the effect of the spatial distribution of formation properties and heterogeneities that determine the roughness distribution and the etching pattern after acidizing. Therefore the new acid-fracture correlations capture not only the small-scale features such as roughness but also the larger-scale features such as channels. The new correlations contain two parts: fracture conductivity at zero closure stress and the conductivity change with closure stress. Fracture surface-etching profiles control the conductivity at zero closure stress while the conductivity change with closure stress is also a function of rock elastic properties. On the basis of the spatial distributions of formation properties Mou et al. (21b 211) have developed new correlations of acid-fracture conductivity at zero closure stress. In turn on the basis of that development this paper studies the conductivity subject to closure stress and develops new correlations for the overall acid-fracture conductivity. Methodology The calculation domain which represents part of a fracture is 1 ft high and 1 ft long and gridded to approximately.5 in. by 2 in. for each gridblock. Its total dimension is comparable to the gridblock size in an acid-fracture simulator while its grid size is comparable to core samples in experiments. Many parameters such as acid-injection condition acid-diffusion coefficient acid concentration and temperature have an impact on the treatment process. But this study focuses on the effects of formation-property distributions that determine etching patterns. Therefore we set other parameters to typical field values and generate spatial distributions of permeability and mineralogy geostatistically as the inputs to the simulation. Two statistical parameters characterize the degree of correlation for either permeability or mineralogy distribution: correlation lengths in fracture-length and fracture-height directions x and z respectively. Another statistical parameter the standard deviation of the natural logarithm of permeability represents the degree of permeability heterogeneity. In this study a formation only consists of calcite and dolomite so the standard 158 May 212 SPE Production & Operations

2 Permeability distribution Mineralogy distribution Acid-flow/reaction simulation Etched fracture surface profiles Conductivity at zero closure stress Fracture-closure simulation Fracture width distribution Overall fracture conductivity Fig. 1 Computational procedure of models. deviation does not mean much for the mineralogy distribution. For the convenience of comparison and analysis the dimensionless parameters are defined as x = (1) x L z = (2) z H k = (ln ) (3) ln k where L is the domain length H is the domain height and k is average permeability. When the average permeability is 1 md k = (ln ) ln (4) Beatty (21) investigated the acquisition of these geostatistical parameters and presented some case studies. As hydraulic pressure dissipates after acidizing the contacting fracture surfaces deform. Even under high closure stress some apertures remain open to create conductivity for fluid flow. For the intermediate scale eng et al. (211) modeled the deformation of fracture surfaces as closure stress is applied to the fracture. At each cross section along the fracture the fracture shape is approximated as being a series of elliptical openings. Under the assumption of elastic behavior of the rock elliptical gaps remaining open and their sizes are a function of the applied stress. By modeling numerous cross sections along the fracture we observed that many channels were open when permeability and mineralogy were strongly correlated horizontally. The overall conductivity of the fracture results from both channels and roughness and it can be obtained by numerically modeling the flow through this heterogeneous system. Our previous papers (Mou et al. 21a b; eng et al. 211) have presented the models in detail. Fig. 1 shows the computational procedure of the simulation. In this paper we performed many numerical experiments and studied the effects of the input parameters on overall fracture conductivity subject to closure stress. A new set of correlations have been developed to predict acidfracture conductivity for the intermediate scale. Effects on Conductivity In order to develop the correlation of acid-fracture conductivity we need to study the effects of the key parameters on the overall conductivity. uring acid injection permeability and mineralogy distributions determine the etching profile. After acidizing rock properties and closure stress have significant impacts on the fracture-width distribution that decides fracture conductivity. The following discussion focuses on these aspects. Effect of Permeability istribution. ifferent correlation lengths of permeability distributions result in different etching patterns. Fig. 2 shows two examples of etching profiles for intermediatescale fractures after acidizing. The average permeability for each case is.1 md. The only difference in these two simulations is the normalized correlation length in the horizontal direction. The first synthetic fracture shown in Fig. 2a has a low normalized correlation length x =.156 in the horizontal direction. The openings in the fracture created are isolated as a result of the weakly correlated permeability distribution. In the second synthetic example shown in Fig. 2b many long straight channels go through the whole fracture because the normalized correlation length in the horizontal direction is x =.5 which is higher than the first case. The apertures are connected laterally to form long channels. Under the same closure stresses (3 psi) the closure models predict the width distributions for both cases shown in Fig. 3. Fig. 3a shows the width profile for the low x. More asperities touch and deform as the closure stress applies. The openings become even smaller and fewer openings are left. The width profile for high x is shown in Fig. 3b. The channels become shorter and narrower. Compared with the short isolated openings in Fig. 3a these channels contribute more to the fracture conductivity because the fluid flow in the channels is easier than that between two rough surfaces. From Fig. 3 the widths at some grids are zero and the apertures are closed completely in the numerical simulations. In this case the isolated channels do not result in any conductivity. However that is not the case in reality. The intermediate simulator misses some information of roughness at a small scale during the closure process. Every gridblock in our intermediate-scale model has dimension of inches which is at the same order as in experiments. But the roughness observed in the laboratory also contributes to the fracture conductivity. Therefore we assign a base conductivity in those closed grids accounting for small-scale roughness features when calculating the conductivity for intermediate-scale fractures. We use the Nierode and Kruk (1973) correlation to determine the base conductivity. Fig. 4 shows the conductivity curves with respect to closure stresses in semilog plots for these two cases. The only difference between these two cases is the horizontal correlation length. The base conductivities by Nierode and Kruk s correlation are close to each other. However when the horizontal correlation length for permeability field is high the long channels result in higher May 212 SPE Production & Operations 159

3 Fig. 2 Width profiles after acidizing with different horizontal correlation lengths. Low horizontal correlation length; high horizontal correlation length. Fig. 3 Width profiles under closure stress of 3 psi with different horizontal correlation lengths. Low horizontal correlation length; high horizontal correlation length. 1 1 Low λ_x Low λ_x Base High λ_x High λ_x Base Fig. 4 Conductivity comparison with different horizontal correlation lengths. 16 May 212 SPE Production & Operations

4 Fig. 5 Width profiles after acidizing with different vertical correlation lengths. Low vertical correlation length; high vertical correlation length. conductivities. Considering the logarithm scale we observed that the channel flow dominates in the fracture. In contrast the low horizontal correlation length gives fewer channels and the fracture conductivity is close to the base conductivity. The vertical correlation length is another key parameter that influences the etching pattern and the closure behavior. The following two examples (Figs. 5 and 6) show the difference between two cases with low and high vertical correlation lengths. The average permeability for each case is.1 md and the horizontal correlation length is.25. The first case (Fig. 5a) has a low vertical correlation length z =.4. The channels etched by acid are very narrow because of the weakly correlated permeability in the vertical direction. The narrow channels are difficult to close. Therefore many channels are still open under the closure stress of 3 psi as shown in Fig. 6a. In contrast the second case has a permeability distribution with a high vertical correlation length z =.25. Even though the horizontal correlation length for both cases is good the acid-etched fracture (Fig. 5b) has more unconnected void space than channels. Only a few openings are observed under the same closure stress of 3 psi as shown in Fig. 6b. In this case the roughness is the main contributor to the conductivity while the isolated large void space plays an insignificant role. The conductivity comparison of these two synthetic cases is shown in Fig. 7. Especially under low closure stress the conductivity of the low-vertical-correlation length case decreases slower than that of the high-vertical-correlation length case because the narrow channels in the first case remain open. The large isolated openings in the second case crush even under low closure stress. The fracture conductivity of the high- z case is even lower than the base conductivity of the low- z case. Also it is very close to the base conductivity of the high- z case because the deformation behavior in this case with few channels is similar to the deformation behavior of a fracture with small-scale roughness only. Normalized standard deviation is another one of the statistical parameters that characterize the permeability distribution. It represents the permeability variation from the average permeability. The following two examples (Figs. 8 and 9) illustrate the effect of normalized standard deviation on fracture-width profiles after closure. Except for the standard deviation they have the same average permeability of.1 md vertical correlation length of.16 and horizontal correlation length of.5. The case shown in Fig. 8a has a low standard deviation =.1. Fig. 9a presents the width distribution after closure under closure stress of 3 psi. Compared with the case with high standard deviation ( =.5) shown in Fig. 8b the greener profile of the first case demonstrates that the width does not vary much from the average width of approximately.1 in. The high standard deviation yields the bluer width profile for the second case because of greater width variation. As a Fig. 6 Width profiles under closure stress of 3 psi with different vertical correlation lengths. Low vertical correlation length; high vertical correlation length. May 212 SPE Production & Operations 161

5 1 1 Low λ_z Low λ_z Base High λ_z High λ_z Base Fig. 7 Conductivity comparison with different vertical correlation lengths. Fig. 8 Width profiles after acidizing with different standard deviations. Low standard deviation; high standard deviation. Fig. 9 Width profiles under closure stress of 3 psi with different standard deviation. Low standard deviation; high standard deviation. 162 May 212 SPE Production & Operations

6 Low Standard eviation Low Standard eviation Base High Standard eviation High Standard eviation Base Fig. 1 Conductivity comparison with different standard deviation. result the channel depth for the second case is greater. So under the same closure stress the channels in the first fracture are easier to close while more openings are left for the fluid flow in the second fracture (Fig. 9b). Higher overall conductivity can be expected for the second case with the high standard deviation. The conductivity with respect to the closure stress for the preceding cases is also plotted in Fig. 1. The conductivity of the low-standard-deviation case declines dramatically under low closure stress and trends very closely to the base conductivity. The relatively even fracture surfaces become flat quickly. In contrast the high-standard-deviation case maintains the fracture conductivity quite well with many channels remaining open. In addition the two standard deviations lead to different fracture widths that yield different base conductivities. From the comparison for the three statistical parameters characterizing the permeability distribution the normalized horizontal correlation length x and the normalized vertical correlation length z have opposite feedbacks on the fracture-width profile after closure. Low vertical correlation length and high horizontal correlation length favor the formation of long and narrow channels that lead to high fracture conductivity. The roughness highly depends on the standard deviation of the permeability field. High standard deviation gives greatly uneven surfaces that yield high conductivity. Effect of Mineralogy istribution. Mineralogy distribution is another principal input in the model. Acid fracturing is a stimulation method implemented in carbonate reservoirs. The carbonate rock consists mainly of calcite and dolomite. So in this study we consider just calcite and dolomite and neglect other kinds of mineralogy. Mineralogy determines the acid/rock reaction rate. Calcite reacts with acid faster than dolomite so rock is dissolved more for calcite than dolomite. The variable reaction rates yield rough fracture surfaces. According to Blatt et al. (198) most carbonate sediments show laminations. This indicates that the mineralogy distribution has a high correlation length in the bedding direction. The mineralogy will be changed abruptly without transition. Therefore we use a high horizontal correlation length to generate mineralogy distributions. As a result channels form easily when mineralogy distribution dominates the surface-etching patterns. Mou et al. (21b) suggested that the vertical correlation length of mineralogy distribution does not affect conductivity at zero closure stress. Instead the percentage of calcite and dolomite affects the etching patterns because of the variation of the reaction rates under a typical temperature 21 F which is used in the simulations. A higher percentage of calcite leads to a larger average fracture width for a certain acid-contact time. In this paper we followed the same judgment and investigated the effect of calcite percentage on fracture overall conductivity. Two examples shown next have calcite percentages of 2 and 4% respectively. Other than calcite the rest of the rock is composed of dolomite. Fig. 11 shows the width profiles after acid treatments while Fig. 12 is the width profiles after the fractures are closed under a closure stress of 3 psi. Fig. 11 Width profiles after acidizing with different calcite percentages. Low calcite percentage; high calcite percentage. May 212 SPE Production & Operations 163

7 Fig. 12 Width profiles under closure stress of 3 psi with different calcite percentages. Low calcite percentage; high calcite percentage. In Fig. 13 conductivity for both cases is presented with respect to closure stress. As in the previous estimation higher calcite percentage yields higher fracture conductivity at the same closure stress. Observe that both conductivity curves are higher than the base conductivity curves. This indicates that the channels are the main contributors to the conductivity for both cases. The amount of channels highly depends on the mineralogy percentage. The variation of reaction rates initiates the formation of channels. The intense etching happens where calcite is present because it reacts faster than dolomite. As the calcite percentage increases more channels are created. Consequently more channels are open for fluid flow after deformation caused by the closure stress. Therefore higher conductivity can occur for the case with higher calcite percentage. However it can be imagined that the conductivity may reach a peak and decrease after the calcite percentage is greater than 5%. The etching pattern will be more even when the fracture surface consists of calcite only (1%). In this case no channels will appear and contribute to the overall conductivity unless they are caused by permeability heterogeneities. This study investigated the cases with calcite percentages of less than 5%. Effect of Rock Properties. With the assumption of elasticity two parameters decide the rock properties: Poisson s ratio and Young s modulus. In this section we studied the effects of these two parameters respectively on overall fracture conductivity. Poisson s ratio is the ratio of the contraction or transverse strain (perpendicular to the applied load) to the extension or axial strain (along the applied load). Gercek (27) investigated Poisson s-ratio values for rocks. A typical range of Poisson s ratio for dolomite is from.1 to.35 while the normal range for calcite is from.1 to.33 which is very close to dolomite. Therefore the Poisson s ratio for carbonates in this study is higher than.1 but lower than.4. In order to study the effect of Poisson s ratio we selected an example and kept all the parameters the same except the Poisson s ratio. The permeability distribution for this case has a high horizontal correlation length a low vertical correlation length and a high standard deviation. For a constant Young s modulus of 3 Mpsi (million psi) fracture overall conductivity is plotted with respect to closure stress for four values of Poisson s ratio (Fig. 14). Although the Poisson s ratio changes from.1 to.4 all conductivities are very close to each other. Slight difference can be found under high closure stress such as 6 psi in this case. From the analysis Poisson s ratio of carbonate does not affect fracture overall conductivity significantly. In this study we ignored the effect of Poisson s ratio and set a typical value.3 for it in the numerical experiments. Similar to rock embedment strength in experiments Young s modulus is a measure of the stiffness of rock in the quantitative analysis and calculation. The definition of Young s modulus is the ratio of the uniaxial stress over the uniaxial strain when the mate- Low f calcite Low f calcite Base High f calcite High f calcite Base Fig. 13 Conductivity comparison with different calcite percentages. 164 May 212 SPE Production & Operations

8 v=.1 v=.2 v=.3 v= Fig. 14 Effect of Poisson s ratio on fracture conductivity. rial is elastic. It can be determined from the slope of a stress/strain curve created during tensile tests conducted on a rock sample. Young s modulus plays an important role during the fractureclosure process. When the Young s modulus is high the rock is strong and difficult to deform under closure stress. Then many openings lead to high fracture conductivity. For demonstration one acid fracture described next deforms with different Young s moduli. It is the same example as shown in the Poisson s-ratio discussion but with the Young s modulus changing instead of the Poisson s ratio. In Fig. 15 the Young s modulus ranges from 2. to 4.5 Mpsi. The y-axis of conductivity is on a logarithmic scale. When the closure stress is low the difference does not look large and the results are of the same order of magnitude. As the closure stress increases the curves diverge and the variation becomes large. Especially for the soft rock with the Young s modulus of 2 Mpsi the conductivity drops much faster than for the other cases. To further clarify the effect of Young s modulus Fig. 16 plots the relationship between the conductivity and the Young s modulus for the same closure stress of 7 psi. The convex curve implies the significant effect of Young s modulus. In particular a high Young s modulus which means strong rock gives more openings for fluid flow and yields high fracture conductivity. Effect of Closure Stress. Closure stress is one of the most important parameters in this study. The relationship between conductivity and closure stress decides the form of the correlations. We calculated the conductivity for the case shown in Fig. 2b and plotted with the base conductivity in the same figure (Fig. 17) as an example. The empirical correlation by Nierode and Kruk (1973) was used to calculate the base conductivity. Rock-embedment strength represents the rock stiffness in Nierode and Kruk s correlation while Young s modulus is used in the numerical simulations. To compare these correlations on the same basis we used a linear relationship between Young s modulus and rock-embedment strength that was derived from a limited set of laboratory data for Texas cream chalk Indiana limestone and San Andres dolomite. More details of this procedure are given by eng (21). In the semilog plot the conductivity curve is essentially a straight line especially when closure stress is higher than 1 psi. This indicates that the relationship between conductivity and closure stress can be represented as an exponential function. As discussed previously the base conductivity accounts for the contribution of roughness. The difference between the fracture conductivity and base conductivity is caused by the fluid flow in the channels after closure (Fig. 3b). For a case such as this one in which distinct channels are formed the contribution to conductivity from channels is more significant than that from roughness. Correlations and iscussions We performed a large number of numerical experiments using the models introduced previously. Then the best least-squares fit for the data led to the new set of acid-fracture-conductivity correlations. In this study the correlations of acid-fracture conductivity will use an exponential function as the essential model as Young s Modulus 2. Mpsi 2.5 Mpsi 3. Mpsi 3.5 Mpsi 4. Mpsi 4.5 Mpsi Fig. 15 Fracture conductivity with different Young s moduli. May 212 SPE Production & Operations 165

9 Young s Modulus (Mpsi) Fig. 16 Effect of Young s modulus on fracture conductivity. ( ) = exp c (5) where is conductivity in md-ft. The parameter contains conductivity at zero closure stress ( ). The parameter incorporates Young s modulus and other influential factors. When developing the correlations of conductivity at zero closure stress Mou et al. (21b) classified fracture surface-etching patterns into three categories: permeability-distribution-dominant cases mineralogy-distribution-dominant cases and competing effect of permeability and mineralogy distributions. The correlations for overall fracture conductivity follow the same categorization and include the fracture closure behavior. For all of the simulations supporting the correlation development the temperature was assumed to be a moderate value of 21 F. Permeability-istribution-ominant Cases. When the leakoff coefficient is greater than approximately.4 ft/(min).5 the leakoff distribution dominates the etching patterns. The mineralogy distribution has a negligible effect for the cases with a high leakoff rate. The correlation of conductivity at zero closure stress for permeability-distribution-dominant cases (Mou et al. 21b) is 9 3 ( wk f ) = w 1+ ( a1erf ( a2( x a3)) a4erf ( a5( z a6)) ) ( e 1) a = 182. a = a = a = 131. a = 671. a = (6) where ( ) is conductivity at zero closure stress in md-ft and w is average fracture width in inches. The average width is the fracture width at zero closure stress. In practice ideal fracture width w i which is defined as dissolved rock volume divided by fracture surface area is easier to obtain than average fracture width. Mou et al. (211) provided the definitions and relationship between them. For the high leakoff coefficient [>.4 ft/(min).5 ] 83. w= 56. erf ( 8. ) w (7) i For the medium leakoff coefficient [~.1 ft/(min).5 ] with uniform mineralogy distributions 1 1 Base Conductivity Fracture Conductivity Fig. 17 Fracture conductivity with respect to closure stresses. 166 May 212 SPE Production & Operations

10 81. w= 2. erf ( 78. ) w (8) i Then the correlation for overall fracture conductivity is [ ] = exp c (9a) (( ) ) x z. = ( ). 22( ) (9b) = ln ( ) 6. 81ln( ) 1 4 E (9c) Ιn Eq. 9 c is closure stress in psi is normalized standard deviation and E is Young s modulus in Mpsi. The unit of Young s modulus is not compatible with the conductivity unit but the regression for these empirical correlations is based on it. In addition Young s modulus is required to be greater than 1 Mpsi. In general soft rock with Young s modulus less than 2 Mpsi is not a good candidate for acid fracturing. The restrictions on E also apply to the following correlations. Our results show that fracture conductivity decreases very rapidly with increasing closure stress for stresses below approximately 5 psi and then decreases exponentially with increasing closure stress above this value as described by Eq. 9a. The parameter in Eq. 9a is obtained by extrapolating the exponential relationship between conductivity and closure stress (the linear portion of the response on a semilog plot such as Fig. 17) to zero closure stress. The correlations presented are thus valid for closure stresses above approximately 5 psi. Often the vertical correlation length of permeability distribution is low because the sedimentary carbonates are laminated. When the dimensionless vertical correlation length is low enough (for example z <.2) Eq. 9 can be simplified as [ ] = exp c (1a) 12 x (1b) =. ( ) ( ) 1. = ln ( ) 7. 8ln( ) 1 4 E (1c) Mineralogy-istribution-ominant Cases. When the leakoff coefficient is less than approximately.4 ft/(min).5 mineralogy distribution dominates etching patterns and corresponding conductivity. In this scenario permeability is low and horizontal correlation length for the mineralogy distribution is high because of the laminated character of sedimentary carbonates. Calcite and dolomite are the only two types of minerals considered in this study. According to a discussion by Mou et al. (21b) the calcite percentage in the rock is the key parameter that affects average width and conductivity of an acid fracture. Thus the only parameter about mineralogy distribution included in the correlation is mineralogy percentage. The correlation of conductivity at zero closure stress for mineralogy-distribution-dominant cases is ( wk f ) = ( f ). f calcite calcite w i (11) where f calcite is the percentage of calcite (.1 f calcite.5). Instead of the average fracture width w the ideal width w i is used in this correlation. The correlation for overall fracture conductivity is [ ] = exp c (12a) = ( wk ) f ( f ).. calcite (12b) = 1. 2exp (. 952 f )+ 1. 5E (12c) calcite Competing Effects of Permeability and Mineralogy istributions. If the leakoff coefficient is medium [approximately.1 ft/(min).5 ] both permeability and mineralogy distributions have competing effects on etching patterns and corresponding conductivity. Thus the correlation of conductivity for this scenario is ( ) = + + ( 9 wk f a a a x a 1 2erf ( 3( 4)) a9 a5erf ( a6( z a7)) ) ( e 1) a8fcalcite a + 1 a = 2. a = 1. a = 5. a = 12. a = 6. a = a = 3. a = 1. a = 43. a = 14. a = w a11 i (13) This correlation also uses the ideal fracture width w i instead of the average fracture width w. Compared with the correlation for the permeability-dominant cases the format for the competing effects cases is similar. The correlation for overall fracture conductivity is [ ] = exp c (14a) ln ( )+ 15. = ( ) x z (14b) = ln ( E) ln( ) (14c) The correlation has a similar form to the one for permeabilitydistribution-dominant cases. For all the correlations presented the normalized horizontal correlation length is in the range of.156 x 1 the normalized vertical correlation length is in the range of.4 y.5 and the standard deviation is in the range of.1.9. The correlations were derived by first determining the etching pattern created in hundreds of simulations of acid transport and dissolution through a portion of a fracture domain and then calculating the resulting fracture conductivity at zero closure stress. Next for each of these patterns the fracture conductivity under closure stress was modeled to predict the final fracture conductivity at an elevated closure stress. More details on how the empirical acid-fracture correlations presented here were derived are given by Mou et al. (211) eng et al. (211) Mou (29) and eng (21). Example Calculation In order to apply our new correlations we need six parameters as inputs. Three of them are the horizontal correlation length the vertical correlation length and the standard deviation for the permeability distribution. The only parameter for the mineralogy distribution is the calcite percentage. The Young s modulus represents the rock strength. The ideal fracture width can be obtained either from each gridblock of an acid-etched profile in an acid-fracture simulator or from other theoretical models. We next show a permeability-dominant example to clarify the calculation process. An ideal fracture width w i =.4 in. was selected from an acid-etched width profile which was produced by a acid-fracture simulator (Beatty et al. 211). We assumed normalized vertical correlation length z =.156 normalized standard deviation =.7 and Young s modulus E = 4 Mpsi. The calcite percentage is not necessary because the permeability distribution dominates. The comparison will show the conductivity predicted with a high normalized horizontal correlation length x =.7 and with a low value x =.156. First Eq. 7 gives the average fracture width for the high leakoff case: May 212 SPE Production & Operations 167

11 λ x High Low N_K Fig. 18 Example of fracture-conductivity calculation. 83. w=. 56erf (. 8 ) w =. 22 in (15) i For the high horizontal-correlation-length case Eq. 6 gives the conductivity under zero closure stress: ( wk f ) = w 9 3 ( ) 1+ a1erf ( a2( x a3)) a4erf ( a5( z a6)) ( e 1) = md-ft (16) Then the and in Eq. 9 can be obtained by (( ) ) = ( wk ) f. 22( x ) z = md-ft (17) = ln ( ) 7. 8ln( E) 1 = (18) Thus the relationship between the conductivity and closure stress is ( ) = exp. 639 c (19) For the low horizontal correlation length the is the same because it is not a function of x. But the ( ) and are different. Then the fracture conductivity after closure is ( wk f ) = w a1erf ( a2( x a3)) a4erf ( a5( z a6)) ( e 1) = md-ft (2) 4 ( ) (( ) ) = ( wk ) f. 22( x ) z = md-ft (21) ( ) = exp. 639 c (22) Using Eqs. 19 and 22 we computed the fracture conductivities for these two cases for closure stresses up to 6 psi (Fig. 18). At lower closure stresses the channels that are predicted to be created for the high-horizontal-correlation-length case create substantially higher conductivity than that for the low-horizontal-correlationlength case. Regardless of the effect from correlation length the conductivity estimated by the Nierode and Kruk correlation is lower than the other two but it is close to the low-horizontalcorrelation-length case that has fewer channels. Conclusions The intermediate-scale simulation of acid fracturing captures the heterogeneous features such as channels that the experiments fail to see. It can close the gap between the macroscale simulator and the microscale experiments. We have modeled the acid transportation and the deformation of the fracture surfaces under closure stress. By assigning base conductivity to closed regions the overall fracture conductivity is obtained as a result of numerically calculating the flow through the heterogeneous system. On the basis of the discussion of influential effects and extensive numerical experiments this study reveals the following important conclusions: 1. A new set of correlations in three categories is developed to predict the acid-fracture conductivity on the intermediate scale. 2. Low vertical correlation length and high horizontal correlation length of the permeability distribution lead to narrow and long channels that are difficult to be closed and are favorable to overall conductivity. High standard deviation results in uneven etchings that also contribute significantly to the fracture conductivity. 3. On the basis of simulations at a temperature of 21 F high calcite percentage leads to more channels that result in high fracture conductivity when the mineralogy distribution dominates the etching patterns. 4. Strong rock with high Young s modulus is able to resist the closure stress effectively and allow much space for fluid flow. In contrast another rock property Poisson s ratio has negligible effect on conductivity calculation. Nomenclature E = Young s modulus Mpsi f calcite = percentage of calcite H = calculation-domain height ft k = permeability md k = average permeability md L = calculation-domain length ft w = average fracture width in. w i = ideal fracture width in. 168 May 212 SPE Production & Operations

12 = fracture conductivity md-in. ( ) = fracture conductivity at zero closure stress md-in. = correlation length ft = normalized correlation length = Poisson s ratio = standard deviation c = closure stress psi = normalized standard deviation Acknowledgments The authors gratefully acknowledge the sponsors of the Acid Fracture Conductivity JIP and the Crisman Institute in the epartment of Petroleum Engineering at Texas A&M University. References Beatty C.V. 21. Characterization of Small Scale Heterogeneity for Prediction of Acid Fracture Performance. MS thesis Texas A&M University College Station Texas (August 21). Blatt H. Middleton G. and Murray R Origin of Sedimentary Rocks second edition. Englewood Cliffs New Jersey: Prentice Hall. eng J. 21. Mechanical Behavior of Small-Scale Channels in Acid- Etched Fractures. Ph dissertation Texas A&M University College Station Texas. eng J. Hill A.. and Zhu A Theoretical Study of Acid-Fracture Conductivity Under Closure Stress. SPE Prod & Oper 26 (1): SPE PA. Gangi A.F Variation of whole and fractured porous rock permeability with confining pressure. Int. J. Rock Mech. Min. Sci. & Geomech. Abstracts 15 (5): Gercek H. 27. Poisson s Ratio Values for Rocks. Int. J. Rock Mech. Min. Sci. 44 (1): Gong M. Lacote S. and Hill A New Model of Acid-Fracture Conductivity Based on eformation of Surface Asperities. SPE J. 4 (3): SPE-5717-PA. Hakami E. and Larsson E Aperture Measurements and Flow Experiments on a Single Natural Fracture. Int. J. Rock Mech. Min. Sci. 33 (4): Mou J. 29. Modeling acid transport and non-uniform etching in a stochastic domain in acid fracturing. Ph dissertation Texas A&M University College Station Texas (August 29). Mou J. Zhu. and Hill A.. 21a. Acid-Etched Channels in Heterogeneous Carbonates a Newly iscovered Mechanism for Creating Acid-Fracture Conductivity. SPE J. 15 (2): SPE PA. Mou J. Zhu. and Hill A.. 21b. A New Acid Fracture Conductivity Model Based on the Spatial istributions of Formation Properties. Paper SPE presented at the SPE International Symposium and Exhibition on Formation amage Control Lafayette Louisiana USA 1 12 February. Mou J. Zhu. and Hill A New Correlations of Acid-Fracture Conductivity at Low Closure Stress Based on the Spatial istributions of Formation Properties. SPE Prod & Oper 26 (2): SPE PA. Nasr-El-in H.A. Al-riweesh S.M. Metcalf A.S. and Chesson J.B. 28. Fracture Acidizing: What Role oes Formation Softening Play in Production Response? SPE Prod & Oper 23 (2): SPE PA. Nierode.E. and Kruk K.F An Evaluation of Acid Fluid Loss Additives Retarded Acids and Acidized Fracture Conductivity. Paper SPE 4549 presented at the Fall Meeting of the Society of Petroleum Engineers of AIME Las Vegas Nevada USA 3 September 3 October. Oeth C. Hill A.. Zhu. and Sullivan R Characterization of Small Scale Heterogeneity to Predict Acid Fracture Performance. Paper SPE presented at the SPE Hydraulic Fracturing Technology Conference The Woodlands Texas USA January. org/1.2118/14336-ms. Pournik M. 28. Laboratory-scale fracture conductivity created by acid etching. Ph dissertation Texas A&M University College Station Texas (ecember 28). Ruffet C.S. Féry J.J. and Onaisi A Acid-Fracturing Treatment: A Surface-Topography Analysis of Acid-Etched Fractures To etermine Residual Conductivity. SPE J. 3 (2): SPE PA. Swan G etermination of stiffness and other joint properties from roughness measurements. Rock Mech. Rock Eng. 16 (1): dx.doi.org/1.17/bf Walsh J.B Effect of pore pressure and confining pressure on fracture permeability. Int. J. Rock Mech. Min. Sci. & Geomech. Abstracts 18 (5): Williams B.B. Gidley J.L. and Schechter R.S Acidizing Fundamentals 55. New York: SPE/AIME. Jiayao eng is a senior engineer with NSI Technologies in Tulsa. His research interest is in well stimulation. He holds BS and MS degrees in mechanical engineering from Tsinghua University China and a Ph degree in petroleum engineering from Texas A&M University. Jianye Mou is an associate professor in the Petroleum Engineering epartment of the China University of Petroleum in Beijing. His main research areas include production engineering and well stimulation. He holds a BS degree from the aqing Petroleum Institute; an MS degree from China University of Petroleum Beijing; and a Ph degree from Texas A&M University all in petroleum engineering. A.. Hill is a professor and holder of the Robert L. Whiting Endowed Chair in Petroleum Engineering and an associate department head at Texas A&M University. He taught for 22 years at The University of Texas at Austin after spending 5 years in the industry. Hill is an expert in the areas of production engineering well completions well stimulation production logging and complex well performance (horizontal and multilateral wells) and has presented lectures and courses and consulted on these topics throughout the world. He holds a BS degree from Texas A&M University and MS and Ph degrees from The University of Texas at Austin all in chemical engineering. He is the author of the SPE monograph Production Logging: Theoretical and Interpretive Elements coauthor of the textbook Petroleum Production Systems coauthor of the SPE book Multilateral Wells and author of more than 13 technical papers and five patents. He has been an SPE istinguished Lecturer has served on numerous SPE committees and was founding chairman of the Austin SPE section. He was named a distinguished member of SPE in 1999 and received the SPE Production and Operations Award in 28. He currently serves on the SPE editorial review committee and is chairman for the Hydraulic Fracturing Technology Conference. ing Zhu is an associate professor in the epartment of Petroleum Engineering at Texas A&M University. Her research areas are production engineering well stimulation and complex well performance. Zhu is the author of more than 8 technical papers and a coauthor of the book Multilateral Wells. She holds MS and Ph degrees in petroleum engineering from The University of Texas at Austin. Her research areas are production engineering well stimulation and complex well-performance. She has been a committee member and chairwoman for many conferences and events with SPE. May 212 SPE Production & Operations 169