Fracturing fluid diversion plays a crucial role in maximizing the well productivity of multistage fractured wells. Proper sizing and material design of diverting agents are key elements to effectively bridge and plug perforations and fractures during treatment. In this study, we present an improved design for a water-soluble diverting agent that can cover a range of fracture widths. In addition, a wellbore flow model that predicts the swelling and dissolution behaviors of diverting agents flowing from the surface to the fractures was developed for field applications.

Butenediol vinyl alcohol copolymer (BVOH), which has elastic and sticking properties in water, is used as a diverting agent. Cylindrical pellets and smaller sized powder, made from the polymer, were mixed to bridge and plug hydraulic fractures. BVOH diverting agents were evaluated for slit widths of 1–4 mm with different pellet geometries using a high-pressure and high-temperature filtration apparatus. The swelling and dissolution rates depend on many parameters such as the temperature, dissolution time, crystallinity degree, and geometry. In this study, empirical correlations that predict the swelling and dissolution rates of BVOH polymers for various formulations were developed and implemented in a wellbore flow model that simulates fluid flow and heat transfer during pumping operations. A theoretical case study of a multistage hydraulic fracturing treatment was also presented to demonstrate the applicability and effectiveness of the treatment.

The filtration test results with various diverting agent designs indicate that the length and diameter of the pellets affect the performance and effectiveness of the bridging and plugging fracture-like slits. Moreover, an optimum pellet size exists for different slit sizes. With modified pellet size and diameter ratios, wider slits of 3–4 mm can be effectively plugged by diverting agents with reduced leakoff volumes. The swelling and dissolution models correlated well with the experimental data, considering the temperature, dissolution time, and crystallinity degree. The case studies presented in this study illustrate that the models can predict the time required for the diverting agents to dissolve under field conditions and determine whether the diverting agents pumped into the well provide sufficient conditions for diversion. Furthermore, the study results indicate that the pump rate and injection conditions before pumping the diverting agent are key controlling factors in determining the dynamic downhole temperatures and thus affect the time required for the degradation of the diverting agent. In addition, a field trial result is presented to demonstrate the effectiveness of the swellable, BVOH diverting agent at low downhole temperatures, and hydraulic fracturing treatment in the Permian Basin.

Swelling diverting agents exhibit a more elastic behavior than existing particulate diverting agents. Swelling polymers are less abrasive and thus reduce the risk of equipment damage during preparation and pumping. The wellbore flow simulator developed in this study helps stimulation engineers optimize material types, particle-size distribution, and concentration of diverting agents for various field applications.

Fluid diversion technology has recently advanced significantly, utilizing degradable particulate materials to improve stimulation efficiency (Centurion et al. 2014; Lolon et al. 2016; Rassenfoss 2017; Weddle et al. 2017; Zhu and Furui 2018). Thus, productivity in various stimulation operations, including newly fractured wells and refracturing of existing wells (Potapenko et al. 2009; Huang et al. 2018) as well as high-rate matrix acidizing treatments, has enhanced.

Barraza et al. (2017) presented a case study of Wolfcamp horizontal shale wells stimulated by degradable diverting agents. Their analysis results indicated that the wells stimulated with diverting agents provided an average production increase of 10% compared to that of offset wells stimulated without diverting agents. Zhang et al. (2020) analyzed the treatment and production data for 72 refractured wells in a North American shale play. They concluded that the choice of the diverting agent can significantly influence treatment results. Moreover, wells using a suitable diverting agent experienced a 40% greater average incremental estimated ultimate well recovery ratio than that of comparable wells refractured with no diverting agents. Spurr et al. (2016), Williams et al. (2016), and Vidma et al. (2018) also discussed the successful application of particulate diverting agents for far-field diversion. Diversion technology can be used to reduce the completion costs in horizontal wells; however, improperly placed or prematurely pumped diverting material may negatively affect overall well productivity, posing significant challenges in the optimization of diverting agent designs (Senters et al. 2017, 2018).

Particulate diverting agents have been successfully implemented in high-rate matrix acidizing treatments to improve the effectiveness of acid diversion in various carbonate reservoirs (Golenkin et al. 2017; Shirley and Hill 2019; Medina et al. 2021). Moid et al. (2020) presented a combination of a sequenced degradable diverter and fiber-laden viscoelastic self-diverting acid systems providing post-stimulation skin factors of −3.7 to −3.9 in a high-pressure and high-temperature gas reservoir. Caillon et al. 2021 also reported that the use of particulate diverting agents along with ball sealers provided better acid coverage, with poststimulation skins of −2.5 to −4.2.

Polylactic acid (PLA) is widely used as a soluble diverting agent in fracturing and acidizing treatments. PLA can be degraded by hydrolysis, and the rate of the hydrolysis reaction is significantly affected by temperature. Van Domelen (2017) examined the degradation data of PLA, and the results indicated that it took 10 days to achieve 74% degradation at 176°F, whereas it took only 10 hours for 100% degradation at 245°F. These results suggest that alternative diverting materials are required for multistage hydraulic fracturing treatments in low-temperature applications.

Sato et al. (2020) presented a novel degradable diverting agent based on BVOH, which has controllable water solubility and is suitable for low-temperature reservoir applications. Their results showed that degradable BVOH materials can be used as effective plugging agents for fracture-like narrow slits. In this study, the aspect ratio of the pellets was modified to improve the sealing capability of the diverting agent.

Polyvinyl alcohol is a crystalline polymer whose degree of crystallinity can be controlled by the degree of saponification. BVOH, which is a type of modified polyvinyl alcohol, can increase hydrophilicity and reduce crystallinity, giving it has many unique properties such as water solubility at low temperatures and high salinity, and film forming properties, swelling property, and extrudability. The swellability of BVOH leads to a more efficient plugging property with less loading volume and fewer flowback issues compared to conventional solid particle diverting agents made of PLA or polyglycolic acid (PGA). Its film-forming property can provide a low-permeability temporary plug even when it is contaminated with proppants, sand, and so on. Furthermore, its water solubility can reduce the time required for degradation, and it can be dissolved at saline and acidic conditions (Sato et al. 2020). In addition, BVOH is biodegradable, compostable, and is less harmful to the environment.

The swelling and dissolution mechanisms of the BVOH particles are illustrated in Fig. 1 . The BVOH particle starts to swell from the particle surface by water absorption and the size of the core in the particle decreases gradually. BVOH molecules are peeled off from the particle surface and dissolved in the solution when the swelling reaches equilibrium. This reduces the amount of BVOH available for plugging. Therefore, it is important to understand the relationship between the swelling and dissolution of BVOH to clarify how the diverting agent works during stimulation.

Fig. 1—

Swelling and dissolution mechanisms of BVOH.

Fig. 1—

Swelling and dissolution mechanisms of BVOH.

Close modal

This study aims to provide detailed swelling and dissolution behaviors of BVOH diverting agents and develop a wellbore flow model that predicts the swelling and dissolution behaviors of diverting agents that seal a wide range of fracture widths considering different material types, particle-size distributions, and concentrations for various field applications.

Definitions and Measurements of BVOH Swelling and Residual Rates

Polyvinyl alcohol polymers have been used in many household, medical, personal care, and industrial products, including food packaging, paper products, and water treatment chemicals. Naitou et al. (1957) investigated the dissolution and swelling behavior of BVOH films. In this section, the experimental results are presented to establish a specific relationship between the dissolution and swelling of BVOH pellets and powder.

The key factors that control the dissolution and swelling behaviors of the BVOH polymer are temperature, dissolution time, and degree of crystallinity. Static swelling tests were conducted to measure the residual and swelling rates of the pellets and powders with different degrees of crystallinity. The residual and swelling rates were defined as follows.

The residual rate is the ratio of the number of polymer particles that have not been dissolved to the number of initial polymer particles in the solution (expressed in % by weight), which can be computed by

(1)

where A is the weight of a cup (g), B is the weight of a cup and a mesh bag (g), C is the weight of the initial polymer sample (g), D is the weight of a cup and sample passed through mesh bag (g), E is the weight of dried mesh bag and cup (g), Vmesh is the ratio of nonvolatile components of the mesh bag, and Vs is the ratio of nonvolatile components in the sample.

The swelling rate is the ratio of the absorption and swelling of the polymer in the solution and is expressed as a weight percent:

(2)

where F is the weight of the dried cup and mesh bag with the sample (g), and Wwater is the weight of water captured in the mesh bag (g).

Test Results

Table 1 summarizes the residual and swelling rate measurements for various BVOH powder and pellet samples at ambient temperature (23°C). In this study, three different powder samples with different degrees of crystallinity were prepared by saponification of a polyvinyl ester-based resin. The degrees of saponification were set at 99.3, 98.9, and 98.5 mol% for Powders A, B, and C, respectively, with an average polymerization degree of 450 for all the samples (Mandai et al. 2021).

Table 1—

Swelling and residual rate measurements for BVOH pellet and powder samples.

Sample TypeTime(minutes)Residual Rate(%)Swelling Rate(%)
Powder Powder A(crystallinity: high) 81 328 
15 75 418 
30 76 485 
60 69 522 
90 66 492 
120 64 539 
Powder B(crystallinity: medium) 73 351 
15 68 491 
30 63 525 
60 54 618 
90 47 576 
120 50 597 
Powder C(crystallinity: low) 70 332 
15 54 501 
30 53 570 
60 44 734 
90 42 736 
120 38 700 
Pellet Pellet G 96 101 
15 92 110 
30 89 145 
60 90 140 
90 88 191 
120 87 200 
Sample TypeTime(minutes)Residual Rate(%)Swelling Rate(%)
Powder Powder A(crystallinity: high) 81 328 
15 75 418 
30 76 485 
60 69 522 
90 66 492 
120 64 539 
Powder B(crystallinity: medium) 73 351 
15 68 491 
30 63 525 
60 54 618 
90 47 576 
120 50 597 
Powder C(crystallinity: low) 70 332 
15 54 501 
30 53 570 
60 44 734 
90 42 736 
120 38 700 
Pellet Pellet G 96 101 
15 92 110 
30 89 145 
60 90 140 
90 88 191 
120 87 200 

The experimental results are shown in Fig. 2 . Powder samples with low residual rates (i.e., high swelling rates) can dissolve faster than those with a high residual rate (i.e., low swelling rate) during the fracturing process. Therefore, the polymer type (i.e., degree of crystallinity) should be optimized for specific reservoir conditions and operational requirements.

Fig. 2—

Relationship between the residual and swelling rates.

Fig. 2—

Relationship between the residual and swelling rates.

Close modal

The results indicate that the relationship between the residual and swelling rates is different for the pellet and powder samples; however, the powder samples with varying degrees of crystallinity may be approximated by a single straight line.

Based on these observations, the following empirical equations for the powder and pellet were obtained, assuming a linear relationship between the residual rate and swelling rate.

(3)
(4)

Because of the abundance of experimental data and the fact that the theory of polymer dissolution is too complex to formulate, approximate equations were developed rather than theoretical equations. An empirical equation was created for BVOH dissolution based on the observations that residual amounts of BVOH materials exist at particular temperatures, the dissolution of BVOH progresses with time, and the rate of dissolution increases with temperature.

(5)

where Cr is the residual rate of BVOH (wt%), C0 is the initial residual rate of BVOH (wt%), C is the final residual rate of BVOH (wt%), a is the degradability of BVOH (1/ hourb/°C), b is a model parameter, t is the dissolution time (hours), and T is the temperature (°C).

The final residual rate of BVOH, C is the residual rate measured for a sufficiently long time (t), which depends on the temperature and degree of crystallinity, as summarized in Table 2. The model parameters a and b for the pellets and powder are listed in Table 3.

Table 2—

Empirical correlations of C for pellet and powder samples.

Sample TypeFinal Residual Rate (%)
Powder A (crystallinity: high) C=-1.673T+94.67 
Powder B (crystallinity: medium) C=-1.211T+65.28 
Powder C (crystallinity: low) C=-0.750T+38.90 
Pellet G C=-2.364T+121.05 
Sample TypeFinal Residual Rate (%)
Powder A (crystallinity: high) C=-1.673T+94.67 
Powder B (crystallinity: medium) C=-1.211T+65.28 
Powder C (crystallinity: low) C=-0.750T+38.90 
Pellet G C=-2.364T+121.05 
Table 3—

Empirical coefficients a and b for the powder and pellet.

PowderPellet
a −0.055 −0.018 
b 0.356 0.327 
PowderPellet
a −0.055 −0.018 
b 0.356 0.327 

The empirical parameters (such as the degradability coefficient, a and the model parameter, b) are determined by minimizing the error, R computed from the mean squared errors for the powder and pellet, respectively, as follows:

(6)

where n is the number of samples.

Based on the above consideration, the residual rate prediction model for powder and pellets is formulated as follows:

(7)
(8)

Fig. 3 shows a comparison of the experimental data and the residual rates predicted by Eqs. 7 and 8. As shown in the figure, the predicted values match the experimental data very well.

Fig. 3—

Comparison between the experimental results and predicted values.

Fig. 3—

Comparison between the experimental results and predicted values.

Close modal

In this study, a high-pressure and high-temperature filter press was used to evaluate the fluid loss properties of the BVOH diverting agents. The base fluid was prepared by mixing 400 mL of water with 1.92 g of a guar gum. A diverting agent, consisting of pellets and powder, was added to the base fluid and stirred at ambient temperature. For the filtration test, steel plates with slit widths of 1–4 mm, which simulate a fracture opening, were used in this study. The steel plate used in the filtration test was stiffer than the rock. However, unlike far-field diverting agents that block fluid flow inside a fracture, the diversion mechanism for near-wellbore diverting agents is quite simple. Therefore, the use of a steel plate is a reasonable analog for field conditions in this particular case. The filtration pressure was set at 145 psi, and the fluid loss from the outlet of the test apparatus was measured for 30 minutes. Backpressure was not applied. The detailed test procedure can be found in Sato et al. (2020).

The pellet samples used in this study are shown in Fig. 4a,. The powder consisted of BVOH particles of various sizes with an average diameter of 1 mm (Fig. 4b). The pellet samples had cylindrical shapes with nine different length-to-diameter ratios labeled A–I.

Fig. 4—

(a) BVOH pellets with various aspect ratios and (b) BVOH powder with an average particle size of 1 mm.

Fig. 4—

(a) BVOH pellets with various aspect ratios and (b) BVOH powder with an average particle size of 1 mm.

Close modal

The detailed dimensions (length and diameter) of the different pellet types are summarized in Table 4.

Table 4—

BVOH pellets with various aspect ratios (lengths and diameters).

Pellet TypeLength (mm)Diameter (mm)
A
B
C
D
E
F
G
H
1.5
1.5
1.5
1.5
3.0
3.0
3.0
3.0
4.0 
1.6
2.5
3.4
4.3
1.6
2.5
3.4
4.3
4.0 
Pellet TypeLength (mm)Diameter (mm)
A
B
C
D
E
F
G
H
1.5
1.5
1.5
1.5
3.0
3.0
3.0
3.0
4.0 
1.6
2.5
3.4
4.3
1.6
2.5
3.4
4.3
4.0 

The cumulative leakoff volumes measured from the filtration test were plotted against the square root of time. The filtration coefficient and spurt loss values were estimated from this plot, as illustrated in Fig. 5 . Consistent filtration coefficients were obtained by drawing an approximate straight line for the early test data. Notably, in some tests, the filtration coefficients (i.e., slope of the leakoff curve) changed significantly during the course of the test period. The filtration coefficient is a parameter that evaluates the plugging of the filter cake formed by a diverting agent. A low filtration coefficient indicated the formation of a low-permeability filter cake. In contrast, spurt loss represents the leakoff volume of the fluid from the start of the test until the pellets are deposited on the lower part of the apparatus to form a filter cake. For field applications, significantly larger volumes of fracturing fluids and diverting agents are available. Therefore, the filtration coefficient is considered more important than the spurt loss when assessing the quality and effectiveness of the diverting agent.

Fig. 5—

Filtration coefficient and spurt loss calculated from the filtration test for diverting agents.

Fig. 5—

Filtration coefficient and spurt loss calculated from the filtration test for diverting agents.

Close modal

First, filtration tests were performed for diverting agents with pellet types A–H using a 1-mm wide slit. Powder A (high degree of crystallinity) was used in all the tests. The filtration coefficients measured in the tests are shown in Fig. 6 . Filtration tests were conducted for each diverting agent with mixing times of 30 and 60 minutes. The test with 60-minute mixing time was performed twice to confirm the repeatability of the results. For the tests with the 1-mm wide slit, the slit width was smaller than the length and diameter of the tested pellets. Under these conditions, all pellet samples showed relatively low filtration coefficients, indicating good plugging effects for diversion. The filtration coefficient tended to decrease at diameters and lengths close to the slit size (Pellet A). This suggests that the pellets can effectively bridge at the slit opening, resulting in a superior plugging effect. Variations were observed in the filtration coefficients in some tests when the initial and subsequent tests were compared at a mixing time of 60 minutes. This suggests that the filtration performance of the diverting agent must be understood statistically.

Fig. 6—

Filtration coefficients measured for different pellet sizes (1-mm slit width).

Fig. 6—

Filtration coefficients measured for different pellet sizes (1-mm slit width).

Close modal

As discussed in the previous section, the swelling and residual rates of the BVOH materials change over time. In the tests with a mixing time of 30 minutes, both the pellets and powder swelled less than those in the 60-minute tests. The effect of mixing time was significant, especially for the powder, as shown in Fig. 2. Overall, it appears that somewhat better plugging effects are observed with a shorter mixing time. However, the effects of mixing time should be further investigated in future studies.

Fig. 7 summarizes the filtration tests performed for diverting agents with pellet types A–H using a 2-mm wide slit. As shown in the figure, Pellets A and E could not plug the 2-mm wide slit. Notably, in these tests, several pellets were recovered from the effluents (i.e., the pellets that passed through the slit opening). These pellets had diameters less than the slit opening. These results suggest that the pellet diameter should be designed to be larger than the target fracture width. In addition, Pellets B, C, and F exhibited relatively good plugging effects. As discussed in the test with a 1-mm wide slit, when the pellet diameter approached the target slit width, the filtration coefficients tended to decrease. In addition, Pellets D, G, and H showed relatively larger filtration coefficients than that of Pellets C and F. This indicates that, when the pellet diameters become too large, the filtration coefficients increase owing to the larger pore size of the filter cake formed by the mixture of the pellets and powder.

Fig. 7—

Filtration coefficients measured for different pellet sizes (2-mm slit width).

Fig. 7—

Filtration coefficients measured for different pellet sizes (2-mm slit width).

Close modal

For the tests with a 3-mm wide slit, Pellets C, D, G, H, and I, whose diameters were larger than 3 mm, were used (Fig. 8) . Pellets C and D were unable to plug the slit because of complete dehydration events at mixing time of 60 minutes. These pellets had a length of 1.5 mm, which was smaller than the slit width. The results suggest that the pellet length should be designed to be longer than the target fracture opening for field applications. Pellets G, H, and I, which have both length and diameter greater than the slit width, show excellent plugging effects, as shown in the figure.

Fig. 8—

Filtration coefficients measured for different pellet sizes (3-mm slit width).

Fig. 8—

Filtration coefficients measured for different pellet sizes (3-mm slit width).

Close modal

For the tests with a 4-mm wide slit, Pellets G, H, and I were tested (Fig. 9) . Pellets D and H were unable to plug the wide slit at a mixing time of 60 minutes, whereas Pellet H successfully plugged the slit at a mixing time of 30 minutes. In contrast, Pellet I (at 4 mm in length) showed superior plugging results for the 4-mm slit test. These observations are consistent with those reported in 2- and 3-mm slit width tests.

Fig. 9—

Filtration coefficients measured for different pellet sizes (4-mm slit width).

Fig. 9—

Filtration coefficients measured for different pellet sizes (4-mm slit width).

Close modal

In summary, based on the filtration tests presented above, the length and diameter of the pellets used for diverting agents should be designed to be larger than the width of the target fracture. However, the use of large pellets may result in higher filtration coefficients. Therefore, the optimum size and aspect ratios of the pellets should be selected based on the expected fracture widths and the minimum filtration coefficients required for fluid diversion.

Model Descriptions

In this study, a wellbore flow model was developed to predict the dissolution and swelling characteristics of diverting agents flowing in a wellbore for field applications. The model is based on GEOTEMP2, a thermal simulator developed by Mitchell (1982) that was originally designed to allow the thermal simulation of the complex drilling and completion processes of typical oil, gas, and geothermal wells. Here, the model was used to calculate wellbore temperature profiles that changed during the stimulation treatment. Because a water-based fracturing fluid is considered in this study, single-phase liquid flow is assumed. In addition, diverting agents that consist of pellets and powder are assumed to flow together with the injection fluid (i.e., no slip velocity is considered between the injection fluid and diverting agents). The interface tracking method is based on the algorithm implemented in GEOTEMP2, which enables the tracking of the diverting agents pumped from the surface and flowing to the fracture depths. The BVOH diverting agents continue to dissolve and swell according to the downhole temperature, which varies during the pumping operations. The model developed in this study can be used to assess whether the diverting agents pumped into the well prevail in sealing the perforation and fracture inlets and dissolving without interfering with production.

The wellbore temperature was calculated using the first law of thermodynamics or energy conservation. The temperature of the flowing fluid in the wellbore was determined from the energy balance equation, as follows:

(9)

where Ec accounts for the energy exchange in the radial direction caused by the temperature difference between the wellbore fluid and the surrounding formation, ΔEf accounts for the energy exchange in the flow direction (i.e., convective heat flow), Q̇ is the heat generated at the depth of interest, and ΔEa is the energy accumulation.  Appendix A provides a detailed description of each term.

The solution of Eq. 9 gives the temperature of the wellbore fluid as a function of the depth and time. The incremental form of Eq. 5 or Eqs. 7 and 8 is shown to be,

(10)

The flow of calculations used in the wellbore flow model is shown in Fig. 10 .

Fig. 10—

Flow chart of the wellbore flow model for predicting diverter performance.

Fig. 10—

Flow chart of the wellbore flow model for predicting diverter performance.

Close modal

This section presents a case study to demonstrate the applicability of the wellbore flow model developed in this study. The reservoir properties, well geometry, and pump data used in the analysis are summarized in Table 5.

Table 5—

Simulation conditions.

Model ParametersCase 1Case 2Case 3Case 4Case 5
Well type Horizontal well 
Pump rate (bbl/min) 50 
Tubing size (in.) 4.5 (ID = 3.958) 2 7/8 (ID = 2.441) 
Measured depth (ft) 13,000 13,000 16,500 21,500 16,500 
True vertical depth (ft) 10,000 10,000 11,500 11,500 11,500 
Horizontal length (ft) 3,000 3,000 5,000 10,000 5,000 
Surface temperature (°C) 25 25 25 25 30 
Reservoir temperature (°C) 125 150 150 150 150 
Model ParametersCase 1Case 2Case 3Case 4Case 5
Well type Horizontal well 
Pump rate (bbl/min) 50 
Tubing size (in.) 4.5 (ID = 3.958) 2 7/8 (ID = 2.441) 
Measured depth (ft) 13,000 13,000 16,500 21,500 16,500 
True vertical depth (ft) 10,000 10,000 11,500 11,500 11,500 
Horizontal length (ft) 3,000 3,000 5,000 10,000 5,000 
Surface temperature (°C) 25 25 25 25 30 
Reservoir temperature (°C) 125 150 150 150 150 

In Cases 1 and 2, a large size tubing of 4.5-in. OD was assumed, whereas a smaller size tubing of 2 7/8-in. OD was used in the other cases. The well length, surface temperature, and reservoir temperature were varied, while the pump rate was set at 50 bbl/min in all cases. In all cases, the fracturing fluids were pumped for 1 hour before injecting the diverting agents. Then, the diverting agents were mixed with the fracturing fluids and pumped from the surface for 1 hour. Once the diverting agents arrived at the perforation depth, the well was shut in for 2 weeks. Notably, all the calculations presented in this section are based on the diverting agent with Powder B and Pellet G.

Wellbore Temperature Changes during and after Fracture Treatment

The wellbore temperature changes during and after fracture treatment are presented in Fig. 11 . Case 1 represents a reservoir temperature of 125°C (or 0.01°C/ft), whereas a slightly higher geothermal gradient of 0.0125°C/ft was assumed in Case 2. As shown in the figure, the wellbore temperature dropped shortly after pumping began. At a pumping time of 1 hour, the bottomhole temperature was less than 40°C. The pumping operation was stopped at 2 hours and the wellbore temperatures slowly returned to the initial formation temperatures in both cases.

Fig. 11—

Wellbore temperatures for Cases 1–5.

Fig. 11—

Wellbore temperatures for Cases 1–5.

Close modal

Fig. 12 shows temperature warmbacks at the perforation depth of 13,000 ft for Cases 1 and 2. The wellbore temperature warms back faster in Case 2 because of the higher reservoir temperature. For the well geometry and operating conditions used in this case study, the downhole temperatures increased to approximately 44°C and 56°C for Cases 1 and 2, respectively.

Fig. 12—

Temperature changes simulated at the perforation depth of 13,000 ft-MD.

Fig. 12—

Temperature changes simulated at the perforation depth of 13,000 ft-MD.

Close modal

Effect of Temperature on Diverting Agents under Field Conditions

Fig. 13 shows residual rates of BVOH powder and pellets calculated for Cases 1 and 2. As shown in the figure, the residual rates of both powder and pellets in Case 1 are higher than that for Case 2 because of the lower reservoir temperature. The analysis results indicate that the residual rates are about 50 and 77% for the powder and pellets, respectively, which may be sufficient to plug the inlet of the fractures during stimulation.

Fig. 13—

Short-term residual rates estimated for Cases 1 and 2.

Fig. 13—

Short-term residual rates estimated for Cases 1 and 2.

Close modal

Fig. 14 illustrates the long-term residual rates of the BVOH powder and pellets calculated for Cases 1 and 2. As shown in the figure, the pellets and powder dissolved completely at 140 and 215 hours in Case 1. The residual rates of the pellets were higher than those of the powder during the early dissolution stage; however, the pellets dissolved more quickly because the dissolution rate of the pellets was higher than that of the powder. Because the reservoir temperature was higher in Case 2, the pellets and powder dissolved completely after 100 and 160 hours, respectively.

Fig. 14—

Long-term residual rates estimated for Cases 1 and 2.

Fig. 14—

Long-term residual rates estimated for Cases 1 and 2.

Close modal

Effect of Well Length

The horizontal well lengths were increased to 5,000 and 10,000 ft for Cases 3 and 4, respectively. Longer well lengths resulted in higher downhole temperatures. The residual rates of the pellets and powder were calculated to be 78 and 51%, respectively, in Case 3, whereas the residual rates were estimated to be 72 and 44%, respectively, in Case 4 (Fig. 15) . This example illustrates that water-soluble diverting agents may perform differently at different treatment stages (i.e., plugging depths).

Fig. 15—

Short-term residual rates estimated for Cases 3 and 4.

Fig. 15—

Short-term residual rates estimated for Cases 3 and 4.

Close modal

For the long-term dissolution behavior of the diverting agents, the powder and pellets dissolved in 4.5 and 3 days, respectively in Case 3. In Case 4, the powder and pellets dissolved in 4 and 2.5 days, respectively (Fig. 16) .

Fig. 16—

Long-term residual rates estimated for Cases 3 and 4.

Fig. 16—

Long-term residual rates estimated for Cases 3 and 4.

Close modal

Effect of Surface Temperature on Performance of Diverting Agents

The surface temperature in Case 5 was assumed to be 5°C higher than that in Case 3. Cases 3 and 5 are compared in Figs. 17 and 18 to demonstrate the effect of surface temperature. As shown in the figure, the residual rate of the powder is below 50% in Case 5, whereas the residual rate of the pellets is expected to be still higher than 70%.

Fig. 17—

Short-term residual rates estimated for Cases 3 and 5.

Fig. 17—

Short-term residual rates estimated for Cases 3 and 5.

Close modal
Fig. 18—

Long-term residual rates estimated for cases 3 and 5.

Fig. 18—

Long-term residual rates estimated for cases 3 and 5.

Close modal

For the long-term residual rates of the powder and pellets, the effect of the surface temperature was not as significant as that in the previous cases (Figs. 14 and 16).

In all cases, until after 1 hour, nearly half of the powder and more than 70% of pellets remained in the fracturing fluids. This means that sufficient amounts of the diverting agent were present in the solid state and were available for plugging a fracture during pumping operations. On the other hand, regarding its solubility, both the pellets and powder in all the cases presented in this study were completely dissolved within 9 days. Therefore, the risk of formation damage due to the use of particulate diverting agents is very low. Notably, the number of the pellets and powder available for fluid diversion and the rate of dissolution are affected by many parameters such as surface and reservoir temperature, pump rate, and well length, as presented in this study. For fit-for-purpose stimulation designs, it is feasible to select and use pellets and powders with different degrees of crystallinity to enhance the diversion performance.

A field trial result of a hydraulic fracturing treatment with the BVOH diverting agent developed in this study is presented in this section. The field is located in the Permian Basin, and the treated well is a horizontal well with a cased hole and perforated completion. The bottomhole temperature was estimated to be 54°C at the time of fracturing. The well depth was 9,900 ft-TVD, and the lateral length was approximately 13,000 ft, as summarized in Table 6. The multistage fracturing treatment was performed with 66 stages of 200-ft stage length each. Each stage had 40 perforations, with a perforation diameter of 0.42 in. The effectiveness of the swelling diverting agent was evaluated by treating a zone with a conventional nonswellable diverting agent. The next zone was then treated with a swelling diverting agent for comparative analysis. Powder A and Pellet G were used during treatment with the BVOH diverting agent.

Table 6—

Well data and reservoir properties.

LocationPermian Basin, Texas
Bottomhole temperature at treatment 54°C 
Well depth 9,900 ft 
Lateral length 13,000 ft 
Number of stages 66 stages (200 ft/stage ea) 
Clusters per stage 10 
Perforations per cluster 40 
Perforation diameter 0.42 in. (10.7 mm) 
LocationPermian Basin, Texas
Bottomhole temperature at treatment 54°C 
Well depth 9,900 ft 
Lateral length 13,000 ft 
Number of stages 66 stages (200 ft/stage ea) 
Clusters per stage 10 
Perforations per cluster 40 
Perforation diameter 0.42 in. (10.7 mm) 

Using the wellbore flow simulator presented in the previous section, the residual rates for pellets and powder were estimated under the treatment conditions. The results indicated that the residual rates were estimated to be 67.7% and 84.8% at a pumping time of 1 hour for the powder and pellets, respectively. This suggests that sufficient amounts of diverting agents were available for stage treatment. Additionally, the pellets and powder were predicted to completely dissolve in 7.98 and 11.65 days, respectively. This also suggests that diverting agents do not cause production issues.

Fig. 19 shows the treatment data (surface pressure, pump rate, and proppant concentration) for the stimulation with a nonswellable diverting agent. The red, green, and yellow curves represent the surface pressure, pumping rate, and proppant concentration, respectively. Van Domelen (2017) suggested that the amount of PLA diverting agent required per stage should be 0.8–1 lbs/perf for earlier stages and may be increased to up to 5 lbs/perf for cemented completions. These guidelines correspond to the required amount of diverting agent (32–200 lbs) in this treatment. A relatively large amount (250 lbs) of a conventional diverting agent was pumped because a smaller dosage of the diverting agent did not work effectively in previous attempts. As shown in the figure, treatment with 250 lbs resulted in 1,038 psi diversion pressure.

Fig. 19—

Treatment data for the fracturing stage with the conventional, nonswellable diverting agent.

Fig. 19—

Treatment data for the fracturing stage with the conventional, nonswellable diverting agent.

Close modal

For comparison, the swellable diverting agent was pumped into the next treatment zone. Fig. 20 shows the treatment data for the stimulation with the BVOH diverting agent. The BVOH diverting agent (50 lbs) was pumped at 40 bbl/min after the first fracturing treatment. Subsequently, an increase of approximately 945 psi in the surface pressure (i.e., diversion pressure) was observed along with an incremental pressure gain during the second fracturing treatment. The diversion pressure exceeded the minimum pressure response of 500 psi as suggested by Van Domelen (2017). These pressure behaviors indicate that the BVOH swellable diverting agent successfully plugged fractures, providing an effective fluid diversion downhole. In addition, the operator reported that there was no flowback issue related to the BVOH diverting agent, which confirmed its degradability. These observations qualitatively matched well with the simulation results predicted by the wellbore flow simulator mentioned above.

Fig. 20—

Treatment data for the fracturing stage with the BVOH diverting agent.

Fig. 20—

Treatment data for the fracturing stage with the BVOH diverting agent.

Close modal

The field examples presented above show that the swellable, BVOH diverting agent requires one-fifth of the dosage volume measured at the diversion pressure equivalent to that of the nonswellable diverting agent, indicating that the BVOH swellable diverting agent can provide more effective and efficient diversion during hydraulic fracturing treatment.

In this study, experimental and modeling analyses of swelling and degradable diverting agents were performed. The following conclusions were drawn:

  • The effects of the aspect ratio of BVOH pellets were investigated. The filtration test results indicate that, by changing the length and diameters of the pellets, the diverting agents can effectively plug fracture-like slits with widths ranging from 1 to 4 mm. In addition, for optimum diverting agent design, the length and diameter of the pellets should be equal to or slightly larger than the target fracture widths.

  • The swelling and water solubility characteristics of BVOH powder and pellets are also presented in this paper. The experimental results show that the powder has a higher swelling tendency than the pellets, whereas the pellets have a higher dissolution rate. Empirical correlations that account for dissolution time, degree of crystallinity, and temperature were also developed to predict the swelling and residual rates of the BVOH diverting agents, which can be used for various applications.

  • A thermal flow model was developed to compute wellbore pressures and temperatures during stimulation and track the diverting agents flowing from the surface to the fracture depth. An empirical model that predicts the swelling and residual rates of the diverting agents is implemented in the thermal flow model so that the dissolution behavior of BVOH pellets and powder can be predicted for field applications.

  • The case study results indicate that the pump rate and injection conditions before pumping the diverting agent are key controlling factors in determining the downhole temperatures and thus affect the time required for the degradation of the diverting agent. In addition, the simulation results indicate that a sufficient amount of the pellets and powder may remain undissolved during the pumping periods while they degrade completely within 9 days, making the well ready for production.

  • The wellbore flow simulator developed in this study helps stimulation engineers optimize material types, particle-size distribution, and concentration of diverting agents for various field applications.

  • The field treatment results showed that the BVOH diverting agent provided effective diversion as indicated by the surface treatment pressure responses, reducing the required diverting agent volume during hydraulic fracturing treatment.

     
  • a

    degradability of BVOH, 1/ hourb/°C

  •  
  • A

    weight of a cup, g

  •  
  • b

    model parameter

  •  
  • B

    weight of a cup and a mesh bag, g

  •  
  • cp

    molar heat capacity at a constant pressure, J/K

  •  
  • C

    weight of the initial polymer sample, g

  •  
  • Cr

    residual rate of BVOH, wt%

  •  
  • C0

    initial residual rate of BVOH, wt%

  •  
  • C

    final residual rate of BVOH, wt%

  •  
  • D

    weight of a cup and sample passed through mesh bag, g

  •  
  • E

    weight of dried mesh bag and cup, g

  •  
  • ff

    friction loss factor

  •  
  • F

    weight of dried cup and mesh bag with the sample, g

  •  
  • g

    gravitational acceleration, ft/sec2

  •  
  • gc

    dimension conversion factor of gravity, lbm∙ft/(lbf·sec2)

  •  
  • Q̇

    heat generated at the depth of interest, ft∙lbf

  •  
  • r

    distance from the center in the radial direction of the point corresponding to the temperature, ft

  •  
  • t

    dissolution time, hours

  •  
  • T

    temperature, °C

  •  
  • u

    fluid flow velocity, ft/sec

  •  
  • U

    thermal conductivity

  •  
  • Vmesh

    ratio of nonvolatile components of the mesh bag

  •  
  • Vs

    ratio of nonvolatile components of the sample

  •  
  • V̇

    volumetric flow rate, ft3/sec

  •  
  • Wwater

    weight of water captured in the mesh bag

  •  
  • ΔH

    sum of enthalpy changes, ft∙lbf

  •  
  • ΔP

    pressure drop in the wellbore, psi

  •  
  • ΔT

    temperature change, °C

  •  
  • Δz

    cell length in the vertical (flow) direction, ft∙lbf

  •  
  • ΔEa

    energy accumulation, ft∙lbf

  •  
  • ΔEc

    energy exchange in the radial direction, ft∙lbf

  •  
  • ΔEf

    energy exchange in the flow direction, ft∙lbf

  •  
  • ΔEKE

    kinetic energy change, ft∙lbf

  •  
  • ΔEPE

    potential energy change, ft∙lbf

  •  
  • ΔWs

    changes in work caused by external forces, ft∙lbf/sec

  •  
  • ρ

    fluid density, lb/ft3

The present study was performed as part of the activities of the Research Institute of Sustainable Future Society at the Waseda Research Institute for Science and Engineering at Waseda University. The authors would like to thank Mitsubishi Chemical Corp., Soarus L.L.C., and FTS International for permission to publish this paper.

Appendix A—Governing Equations and Assumptions Used in Wellbore Flow Simulator

In GEOTEMP2, pressure calculations are ignored for single-phase liquid flow as the fluid properties are calculated independently of the fluid momentum equations. In this study, the following fluid momentum equation is implemented to calculate the pressure of the injected fluid as a function of depth.

(A-1)

If the fluid is incompressible and a shaft work device is not present in the well, the pressure drop in the wellbore, ΔP, can be obtained by

(A-2)

where the three terms on the right-hand side represent the pressure drops caused by kinetic energy, potential energy, and frictional contributions, respectively.

The wellbore temperature was calculated using the first law of thermodynamics or energy conservation. The temperature of the flowing fluid in the wellbore was determined from the energy balance equation as follows:

(A-3)

where ΔEc accounts for the energy exchange in the radial direction caused by the temperature difference between the wellbore fluid and the surrounding formation, ΔEf accounts for the energy exchange in the flow direction (i.e., convective heat flow), Q̇ is the heat generated at the depth of interest, and ΔEa is the energy accumulation. The grid system used in the wellbore flow simulator is shown in Figs. A-1 and A-2.

Fig. A-1

Energy exchange in a fluid cell.

Fig. A-1

Energy exchange in a fluid cell.

Close modal
Fig. A-2

Grid configuration of the wellbore cells for thermal calculation.

Fig. A-2

Grid configuration of the wellbore cells for thermal calculation.

Close modal

For the ith cell shown in Fig. 11, the conductive energy transfer in the radial direction can be computed as follows:

(A-4)
(A-5)

where z is the cell length in the vertical (flow) direction, Tk,i is the temperature difference in the radial direction, and Uk,i is the thermal conductivity calculated as follows:

(A-6)

The energy exchange due to heat convection is expressed by the sum of the enthalpy change H, potential energy change EPE, and kinetic energy change EKE to give:

(A-7)

where each term can be obtained by:

(A-8)
(A-9)
(A-10)

Finally, the accumulation term is calculated by:

(A-11)

where V is the cell volume.

This paper (SPE 209784) was revised for publication from paper URTEC 208338, presented at the SPE/AAPG/SEG Asia Pacific Unconventional Resources Technology Conference, Virtual, 16–18 November 2021. Original manuscript received for review 15 November 2021. Revised manuscript received for review 18 January 2022. Paper peer approved 21 February 2022.

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