A New Method for Measuring the Effective Length of Acid-Fracturing Fractures

Abstract

Acid fracturing as an important stimulation technique, provides strong technical support for the exploration breakthrough and efficient development of carbonate oil and gas reservoirs. Accurately predicting the effective length of acid-fracturing fractures is of great significance for guiding the acid-fracturing design and improving the stimulation effect of acid fracturing. This article fully considers the essential requirement that the long-term conductivity of acid-fracturing fractures is not zero within the effective length segment. Based on the principle of the same acid concentration and acid dissolution amount, the long-term conductivity testing experiment of acid-fracturing fractures under different residual acid concentrations was designed and carried out with the consideration of the common ion effect. The critical acid concentration with long-term conductivity of 0 was obtained. This method overcomes the shortcomings of the existing methods that result in the overestimation of the effective length of acid-fracturing fractures due to inaccurate values of residual acid concentration or short-term conductivity as the determining criterion. The experimental results show that the higher the acid concentration, the deeper the acid etching groove, and the higher the initial conductivity of acid-fracturing fractures. The long-term conductivity decline rate of different acid concentrations is above 80%, which means that using short-term conductivity as an evaluation indicator alone will overestimate the effective length of acid-fracturing fracture and the yield-increasing effect of acid-fracturing treatment. In the case presented in this paper, the critical acid concentration for acid-fracturing fracture with long-term conductivity of 0 is 4%, and the effective length of acid-fracturing fractures is 120 m.

1. Introduction

China is rich in carbonate oil and gas resources. In recent years, large–medium-sized marine carbonate oil and gas fields have been discovered in Sichuan Basin, Ordos Basin, and Tarim Basin, which show great exploration and development prospects [1,2]. Acid fracturing, as a major production enhancement measure, provides strong technical support for the exploration breakthrough and efficient development of carbonate reservoirs. The accurate prediction of the effective length and conductivity of acid-fracturing fracture is of great significance in guiding the design of acid fracturing and improving the production enhancement effect [3,4].

The conductivity of the acid-fracturing fracture is the product of the fracture width and permeability, which can be measured by laboratory experiments. The effective length of the acid-fracturing fracture refers to the length of the fracture segment that still has conductivity after the acid-fracturing treatment is completed and the fracture is closed. Due to the high-temperature and high-pressure environment of the reservoir, the conductivity of acid-fracturing fracture will decrease rapidly in the initial stage of fracture closure [5]. Therefore, the above conductivity should be the long-term conductivity. During field construction, micro seismic monitoring data are usually used for rough estimation, but the micro seismic monitoring can only obtain the dynamic fracture length during the acid fracturing process, resulting in an overestimation of the effective length of acid-fracturing fracture [6,7,8,9,10]. Based on the hydraulic fracture propagation model, some scholars have introduced the acid-rock reaction model to establish several numerical models for predicting the effective length of acid-fracture fractures [6,11,12]. However, the accuracy of the numerical model predictions depends on the acid-rock reaction parameters measured by laboratory experiments. Therefore, some scholars directly adopt experimental methods to test the effective length of acid-fracturing fracture [13,14,15]. Liu Fei et al. [13] measured the effective consumption time of acid from fresh acid to residual acid by using the rock slab flow experiment. The effective length of acid-fracturing fracture was calculated by the product of the effective consumption time and the acid flow rate. The effective consumption time refers to the time experienced from the fresh acid concentration to the residual acid concentration. The residual acid concentration was taken as 10% of the fresh acid concentration according to the engineering experience, and the value of residual acid concentration of the same acid is bound to be different in different reservoirs, which indicates that the traditional calculation method of the effective length of acid-fracturing fracture is not accurate enough. Gu et al. [14] measured the conductivity of the gelled acid fracture by using the acid fracture conductivity device according to API industry standards and calculated the dissolution rate to deduce the acid rock reaction rate and residual acid concentration, which improved the accuracy of residual acid concentration value. However, this method believes that if the acid has a corrosion effect on the fracture wall, the acid-fracturing fracture will certainly have conductivity after the acid-fracturing treatment is completed and the fracture is closed. In fact, at the end of acid consumption, the ability to etch rocks is very limited. Under the continuous action of high closing stress, these fracture walls with insufficient etch degree will be flattened and lose conductivity. This method will also overestimate the effective length of acid-fracturing fractures.

In summary, the existing methods for predicting the effective length of acid-fracturing fractures in carbonate reservoirs do not accurately consider the essential requirement that the long-term conductivity within the effective length of acid-fracturing fracture is greater than zero. For this reason, firstly the calculation model of acid concentration distribution in acid-fracturing fracture is established. Secondly, based on the principle of equal acid concentration and acid dissolution amount, the common-ion effect was designed and carried out to test the long-term conductivity of acid-fracturing fractures under different residual acid concentrations, and the critical acid concentration with a long-term conductivity of 0 was obtained. Finally, the effective length of the acid-fracturing fracture is determined by combining the prediction results of the acid concentration distribution in fracture and the critical acid concentration value.

2. Calculation Model of Acid Concentration Distribution in Acid-Fracturing Fracture

As shown in Figure 1, the acid-fracturing fracture is assumed to be an ideal vertical fracture of equal width and height. Under the conditions of constant temperature and pressure, the acid flows stably between two rock plates, and the acid is incompressible and of uniform density. Neglecting the leak-off of acid in the vertical wall direction, the velocity component of acid in the vertical wall direction is 0, and the flow rate in the parallel fracture wall direction is u₀, and the flow rate of the whole section of the fracture is constant. The initial acid concentration at the entrance of the fracture is C₀, and the acid concentration decreases gradually as the acid flows in the fracture.

According to the law of mass conservation, we can establish a convection–diffusion partial differential equation to describe the flow and reaction of the acid solution along the fracture, as follows [6,16,17]:

$$u_0\frac{\mathit{\partial }C}{\mathit{\partial }x}=D_{H^{+}}\frac{{\mathit{\partial }}^{2}C}{\mathit{\partial }y}$$

(1)

The boundary conditions for the reaction of the acid with the chert are:

$$\left\{\begin{array}{l}C(x,y)\left|{}_{x=0}=C_0\right.\\ C(x,y)\left|{}_{y\pm \frac{\overline{w}}{2}}=0\right.\\ \frac{\mathit{\partial }C}{\mathit{\partial }y}\left|{}_{y=0}=0\right.\end{array}\right.$$

(2)

The acid convection–diffusion equation was solved by the separated variable method and Fourier series to obtain the formula for acid concentration at any x point in the direction of the fracture length of the acid fracture [6,16,17]:

$$C_x=\frac{8C_0}{{\pi }^2}{\displaystyle \sum _{n=0}^{\infty }\frac{1}{(2n+1{)}^2}·{\mathrm{e}}^{-(2n+1{)}^2\cdot S}}$$

(3)

Among these:

$$S=\frac{2{\pi }^2{L}_{x}h}{Q_f\overline{w}}{D}_{H^{+}}$$

(4)

In the equation, C_x is the concentration of acid at point x in the direction of the fracture length, %; C₀ is the concentration of fresh acid, %; S is a dimensionless frequency group; L_x is the distance flowed by acid from the mouth of the fracture to point x, m; h is the height of the dynamic fracture, m; Q_f is the discharge volume of acid injection in the field construction, m³/min; w is the average fracture width of the dynamic fracture, m; $D_H{}_{{}^{+}}$ is the mass-transfer coefficient of H⁺, m²/min. $D_H{}_{{}^{+}}$ is selected according to the real reservoir temperature, which can be measured by the acid rock reaction kinetics experiment. Generally, the higher the temperature, the greater the value of $D_H{}_{{}^{+}}$.

3. Experimental Design

3.1. Experimental Setup

In this experiment, the HXDL-2C long-term conductivity evaluation device (Figure 2) was used, with a maximum closure pressure of 130 MPa, a maximum experimental temperature of 150 °C, and a maximum injection flow rate of 500 mL/min.

3.2. Experimental Materials

The materials used in this experiment include an acid solution and a rock slab. The acid formula and concentration used in acid fracturing construction in oilfield was selected as an experimental acid solution.

Experimental rock slab size is selected based on the oil and gas industry standard “Test Method for Fracturing Proppant Flow Conductivity SY/T6302-2019”. The dimension of the experimental slab is 38 mm × 178 mm × 20 mm (Figure 3), which can be cut from full-diameter downhole cores of the target well. Two rock slabs are required for each set of experiments.

3.3. Test Methods

Acid corrosion experiments are carried out first, and the rock plate is put into the infusion chamber before the test, and several rubber particles (chloro-fluoro rubber material, high temperature and acid resistance, made of 2 mm diameter rubber ring cut) of 2 mm diameter are uniformly placed between the upper and lower rock plates, and a set fracture width is reserved, and the rock plate is not subjected to the closure pressure. Due to the small number of rubber particles, the contact area with the rock plate is small, and the effect on the acid-rock reaction face volume ratio is negligible. After the rock slab and the acid are heated to a specified temperature, the acid flows through the rock slab gap for a certain time at a set flow rate. After acid etching, the gel particles are removed, and the long-term inflow experiment is carried out under the real formation temperature and closure pressure conditions, and the test method is referred to the oil and gas industry standard “Test Method for Fracturing Proppant Flow Conductivity SY/T6302-2019”. The difference with the above industry standard is that there is no need to add proppant when the rock slab is put into the inflow chamber, and the conductivity is tested directly after the upper and lower rock slabs are affixed to the acid-etched fracture surface to record the conductivity every 1 h.

3.4. Experimental Parameter Design

In the process of acid fracturing, due to the non-homogeneity of the mineral distribution in the reservoir and the change of acid concentration in the fracture, which resulted in the formation of uneven grooves by the non-uniform etching of the fracture wall by the acid, the acid fracture still has a high flow-conducting capacity after the end of the construction. Therefore, the flow-conducting capacity of the acid fracture depends on the amount of acid dissolution in the formation of rock minerals and the degree of non-uniform etching. To ensure that the measured conductivity of acid-fracturing fracture is consistent with that of the formation, the parameter design of acid fracture experiments is carried out based on the principle that the acid concentration is the same as the amount of dissolution and corrosion. The amount of acid erosion on rock is equal to the product of acid rock reaction rate and acid erosion time, and the expression of acid rock reaction rate is [6]:

$$-\frac{\mathit{\partial }C}{\mathit{\partial }t}=D_{H^{+}}·\frac{S}{V}·\frac{C}{\delta }$$

(5)

In the equation, C is the instantaneous reaction acid concentration, mol/L; $\frac{\mathit{\partial }C}{\mathit{\partial }t}$ is the acid-rock reaction rate, mol/(L·s); is the reaction surface-to-volume ratio, cm⁻¹; $\delta$ is the reaction boundary layer thickness, cm.

According to Equation (5), the reaction rate of acid rock mainly depends on H⁺ mass transfer coefficient, reaction facies ratio and reaction boundary layer thickness. The boundary layer thickness is related to the acid viscosity and flow rate, the acid etching experiment simulates the real formation temperature and pressure conditions and keeps the experimental acid flow rate the same as the on-site construction flow rate, which can ensure that the acid viscosity and boundary layer thickness of the experiment and the site are equal. Based on the principle of the same flow rate, the formula for calculating the acid injection flow rate of the acid etching experiment is derived as follows:

$$Q=5\times {10}^5\frac{Q_f{w}_e{h}_e}{\overline{w}h}$$

(6)

In the equation, Q is the experimental acid injection flow rate, mL/min; w_e is the experimental slit width, m; h_e is the experimental slit height, m.

Under conditions of identical temperature, pressure, and acid concentration, the mass transfer coefficient of H⁺ is significantly affected by the common ion effect. Therefore, when conducting acid corrosion experiments, an appropriate amount of calcium chloride or an equivalent substance should be added to the acid solution to simulate the ion pairing effect. The concentration of acid flowing in the fracture gradually decreases, and the mass of calcium chloride produced by the reaction with the rock is half of the mass of hydrochloric acid consumed. So, the calculation formula of calcium chloride mass addition is as follows.

$$m=\frac{111\left(C_0-C_{dc}\right)}{73}\rho V$$

(7)

In the equation, m is the mass of calcium chloride to be added, g; C_dc is the concentration of the acid to be tested, %; ρ is the density of the acid, g/cm³.

When acid pressure construction occurs on site, the amount of acid is basically more than one hundred cubic meters. In the experimental process, by reducing the width of the fracture and increasing the face-to-face ratio, the acid rock reaction rate is enhanced, the experimental over-acid time is reduced, and then the acid dosage is reduced to reduce the cost of the experiment. Based on the principle of the same amount of acid erosion, the formula for calculating the amount of acid required for the experiment is deduced as:

$$V=Q\left(t-t_x\right)\frac{w_e}{\overline{w}}$$

(8)

Among these:

$$t_x={\left(\frac{2\pi h{L}_{cx}{C}_L}{Q_f}\right)}^2$$

(9)

where V is the amount of experimental acid, mL; t is the total construction time, min; t_x is the time required for the acid to reach point x in the fracture, min; L_cx is the fracture distance flowed by the acid from the fresh acid concentration down to the concentration of the acid to be tested, which can be obtained from Equations (3) and (4), m; and C_L is the acid’s filtration loss coefficient in the formation, m/min^0.5.

Acid concentration to be tested based on engineering experience from the fresh acid concentration of 50% to start testing, if the long-term conductivity is not zero, reduce the acid concentration to continue testing, until a certain concentration of acid corrosion long-term conductivity of 0 when the test is stopped. (There is a lower limit for the flow capacity tested by the experimental equipment, and if the flow capacity is lower than 0.01 D·cm, it defaults to 0. At the same time, according to engineering experience, if the flow capacity is lower than this value, it also loses production significance.)

4. Case Studies

Carbonate rock oil well THX in Tahe Oilfield was selected to carry out an example study. The closure pressure of the reservoir section of this well is 50 MPa, the temperature is 120 °C, and 20% concentration of ground crosslinking acid is used for acid pressure reforming, and the specific construction parameters are shown in

Table 1. The THX well acid-fracturing parameters.

Parameter Class	Value
Fresh acid concentration	20%
Hydrogen ion mass transfer coefficient	6.532 × 10−9 m2/min
Experimental fracture width	0.002 m
Experimental fracture height	0.0365 m
Dynamic fracture height	50 m
Average fracture width of dynamic fracture	0.008 m
Construction capacity	5 m3/min
Construction time	60 min
Acid density	1.136 g/cm3
Combined filtration coefficient	1.3 × 10−3 m/min1/2

. The formula for ground cross-linking acid is as follows: hydrochloric acid (according to the experimental design to determine the mass fraction of acid) + 0.8% ground gelling agent DMJ-A + 2.0% high-temperature acidification corrosion inhibitor + 1.0% iron ion stabilizer + 1.0% auxiliary discharging agent + 1.0% emulsion breaker + 0.3% ground crosslinking conditioner DMJ-B2.

4.1. Experimental Parameters

In the initial experiment, the acid solution concentration is set at 10%, which is half of the fresh acid concentration. If the long-term conductivity of the acid-fractured fissure is not zero at this concentration, the acid concentration is incrementally reduced in 2% intervals for further testing until the long-term conductivity of the acid-fractured fissure becomes zero. Based on the data in Table 1, parameters such as the required acid injection rate, the amount of acid solution used, and the quantity of calcium chloride added under different acid concentration conditions are calculated using Equations (6)–(9). The specific results are shown in

Table 2. Experimental parameters of acid etching fracture.

Acid concentration (%)	10	8	6	4	2
Acid injection flow (mL/min)	456	456	456	456	456
Acid dosage (mL)	6539	6158	5469	4104	211
Calcium chloride mass (g)	1130	1277	1323	1134	680

.

4.2. Experimental Results

As can be seen from Figure 4 and Figure 5, the higher the acid concentration, the deeper the acid etching groove, the higher the flow conductivity of the acid fracture at the beginning of the acid fracture, with the increase in the time of the closed stress effect, the flow conductivity decreases. After 0–8 h, the contact area between the fractures smaller bumps under the action of the closed stress is extremely unstable, deformation and fragmentation caused by the acid etching fracture grooves become shallower, some debris moves with the fluid to block the channel, and the flow capacity declines rapidly [18,19,20,21]; after 8–40 h, the weak contact between the fractures bumps Basic broken, high strength contact bumps in the high temperature and high-pressure conditions of creep, the average fracture width gradually reduced, and the conductivity slowly decreased [18]; after 40 h, the contact between the fractures tends to stabilize, the conductivity basically no longer change. Fit the data of different acid concentration conductivity changes over time, and the fitting formulas for 10%, 8%, 6%, and 4% concentration acid solutions are y = 10.744 x^−0.552, y = 9.741 x^−0.737, y = 7.9428 x^−1.02, and y = 6.419 x^−1.424, respectively. As time increases, the conductivity decreases rapidly in the initial stage, and then the decline rate gradually decreases, and almost no changes occur in the later stage [22]. Overall, 10%, 8%, 6%, and 4% concentrations of acid fracture conductivity resulted from the initial 7.3 D·cm, 5.2 D·cm, 3.8 D·cm, and 3.3 D·cm, and eventually decreased to 1.2 D·cm, 0.5 D·cm, 0.2 D·cm, and 0, with the decrease rate above 80%. This suggests that taking short-term conductivity as an evaluation index will overestimate the effect of acid pressure on increasing production.

The experimental results show that the critical acid concentration for the acid fracture long-term conductivity of 0 is between 4% and 6%. To further determine the critical acid concentration, the test results of long-term conductivity of 10%, 8%, and 6% acid concentration were used to fit the power function relationship equation between acid concentration and long-term conductivity of acid fracture using the least squares method. As shown in Figure 6, the long-term conductivity of acid-fracturing fracture corresponding to 4% acid concentration is about 0.02 D·cm, which is basically consistent with the experimental test results. In summary, the critical acid concentration for the long-term conductivity of acid-fracturing fracture in the target block to be reduced to 0 is about 4%.

According to the parameters in Table 1 and Equation (3), the values of acid concentration at each point in the fracture during the acid pressure construction were calculated (Figure 7). As shown in Figure 7, the acid flowed from the mouth of the fracture to the deeper part of the fracture, and the acid concentration gradually decreased, and under the influence of the common ion effect, the acid-rock reaction rate was reduced, and the rate of acid concentration decrease was also gradually slowed down. Taking the critical acid concentration of 4% as the vertical coordinate, a horizontal line is drawn in Figure 7, and the value of the fracture distance corresponding to the intersection of the horizontal line and the curve (60 m) is the effective half-length of the acid fracture of the present embodiment, so the effective length of the acid fracture of the present embodiment is 120 m.

5. Discussion

To verify the accuracy of the new method presented in this paper, it is compared with two existing methods established in the literature [13,14]. In the literature [13], the effective length of acid-fracturing fractures is determined by residual acid concentration which is equivalent to a critical acid concentration with a conductivity of 0. Based on the engineering experience, the residual acid concentration was usually taken as 10% of the fresh acid concentration. Namely, the residual acid concentration is 2% for the case shown in Section 4. It can be seen from Figure 7 that the effective length of the acid-fracturing fracture is about 180 m (90 m × 2). In the literature [14], the residual acid concentration was defined as the critical concentration at which acid loses its ability to dissolve rock. Obviously, the effective length of acid-fracturing fractures is greater than 120 m for the case shown in Section 4. Experimental results show that the ground cross-linking acid still has the ability to dissolve rocks when the concentration is reduced to 4%. In summary, the new method can more accurately determine the effective length of acid-fracturing fractures in carbonate reservoirs.

Combining the definition of the effective length of acid fracture and the essential requirement of non-zero long-term hydraulic conductivity within the effective length of acid fracture, this paper establishes a more accurate method for determining the effective length of acid fracture in carbonate reservoirs, which overcomes the shortcomings of the existing methods that the effective length of acid fracture is overestimated due to the inaccurate value of the residual acid concentration or short-term hydraulic conductivity as the criterion of determination. Furthermore, based on the fundamental principles that the long-term conductivity of acid-fractured fissures depends on the dissolution volume of the acid solution on the reservoir rock minerals and the degree of uneven etching, an experimental parameter design method considering the ion pairing effect was proposed. The experiments were conducted using target reservoir rock samples under simulated reservoir temperature and pressure conditions. In the experimental process, the reaction rate of acid rock is enhanced, and the experimental time of over-acid is reduced by decreasing the width of the slit and increasing the face-to-face ratio, which realizes the effect of reducing the dosage of acid and lowering the cost of the experiment without affecting the accuracy of the test results. Compared with existing methods for testing the effective consumption time and flow rate of the acid solution, as well as using a 10% fresh acid concentration as the residual acid concentration to calculate the effective action distance of acid-fracturing fractures, this method’s experimental testing is closer to the construction environment of the formation, and the measured critical acid concentration is more accurate. However, there are some shortcomings in this method. The main problem is that the model for calculating the distribution of acid concentration in the fracture assumes that the loss of acid filtration is zero, which makes the model only applicable to the denser carbonate reservoirs, but not for the fracture-hole carbonate reservoirs, which will be the focus of our future research work.

6. Conclusions

(1) A new method for measuring the effective length of acid-fracturing fracture in carbonate reservoirs was proposed by considering the essential requirement that the long-term conductivity within the effective length segment of acid-fracturing fracture is not zero. Based on the principle of the same acid concentration and acid dissolution amount, the long-term conductivity experiment of acid-fracturing fractures under different residual acid concentrations was designed and carried out with consideration of the common ion effect. The critical acid concentration with long-term conductivity of 0 was obtained. The effective length of the acid-fracturing fracture is determined by combining the prediction results of the acid concentration distribution in fracture and the critical acid concentration value. The new method overcomes the shortcomings of the existing methods that result in the overestimation of the effective length of acid-fracturing fractures due to inaccurate values of residual acid concentration or short-term conductivity as the determining criterion.

(2) The higher the acid concentration, the deeper the etching groove of the acid solution, and the higher the initial conductivity of the acid-fracturing fracture. As the time of closure stress increases, the conductivity of acid-fracturing fractures first rapidly decreases, then slowly decreases, and finally, stabilizes. The long-term conductivity decline rate of acid-fracturing fractures in the example well is above 80%, and using short-term conductivity as an evaluation indicator alone will overestimate the effectiveness of acid-fracturing stimulation.

(3) The critical acid concentration of the acid-fracturing fracture of the example well with a long-term conductivity of 0 is 4%, and the effective length of the acid-fracturing fracture is determined to be 120 m by combining the prediction results of the acid concentration distribution in fracture and the critical acid concentration value.