Understanding the temperature – resistance performance of a borate cross-linked hydroxypropyl guar gum fracturing ﬂ uid based on a facile evaluation method †

The evaluation of the temperature – resistance performance of fracturing ﬂ uid is essential for choosing suitable fracturing ﬂ uids during fracturing treatment. In this work, related parameters for characterizing the temperature – resistance performance of fracturing ﬂ uids have been analysed systematically. The maximum temperature, T max ( h 0 , t 0 ), which satis ﬁ es both the minimum viscosity requirement ( h $ h 0 ) of fracturing ﬂ uid and the time requirement of fracturing treatment ( t $ t 0 ), was used to characterize the temperature – resistance performance of fracturing ﬂ uids. A facile procedure for evaluating T max ( h 0 , t 0 ) has been proposed based on a step-by-step numerical-search method. The search for T max ( h 0 , t 0 ) starts from the upper limit temperature T max , which is the maximum temperature where the apparent viscosity meets the minimum viscosity requirement, i.e. , h $ h 0 . Using a borate cross-linked hydroxypropyl guar gum fracturing ﬂ uid as an example, the e ﬀ ects of pH, and the concentrations of thickening agent (hydroxypropyl guar, HPG) and crosslinking agent (Na 2 B 4 O 7 ) on the crosslink and temperature – resistance properties are also investigated. It has been found that the crosslinking time t c decreases with the increase of HPG concentration or Na 2 B 4 O 7 concentration. However, t c was found to increase when pH increases. The variation tendencies of T max ( h 0 , t 0 ) are di ﬀ erent from that of T max , viz. , T max ( h 0 , t 0 ) gradually increases with Na 2 B 4 O 7 concentration, whereas T max increases signi ﬁ cantly at low Na 2 B 4 O 7 concentration and remains almost unchanged at high Na 2 B 4 O 7 concentration. The possible mechanisms are proposed for interpreting the above phenomena according to related crosslinking-reaction kinetics and thermodynamics. Our work demonstrates a facile method for evaluating the temperature – resistance performance of the fracturing ﬂ uid and provides useful insights into the understanding of the temperature tolerance performance of the borate cross-linked hydroxypropyl guar gum fracturing ﬂ uid.


Introduction
Hydraulic fracturing is a commonly used technique to stimulate hydrocarbon production by creating a network of highly conductive fractures in the area surrounding a wellbore. 1 The network of fractures can not only improve the hydraulic conductivity of the reservoir rock, but also increase the surface area contributing to enhanced hydrocarbon production. 1,2 Thus, hydraulic fracturing is used in conventional hydrocarbon reservoirs to increase permeability in damaged formations or in formations that exhibit signicantly lower production owing to reservoir depletion. It is also used in unconventional reservoirs where the intrinsic permeability is too low to yield economical production. [3][4][5] Fracturing uid is of great signicance for hydraulic fracturing treatment. The main functions of fracturing uid include opening the fracture, and suspending and transporting proppants. 1 So, a fracturing uid is supposed to provide sufficient viscosity to ensure proper proppant transport and the even distribution of proppants along the fractures. Over the past few decades, linear or cross-linked polymer solutions, 6-9 energized uids, [10][11][12] and viscoelastic surfactants (VES) [13][14][15][16][17][18] have been developed as water-based fracturing uids. Biopolymers such as guar gum and cellulose derivatives, synthetic polymers such as polyacrylamide, are introduced in the water-based fracturing uids for improving the rheological performance. 2 Crosslinking improves the rheological properties of the polymers for fracturing purposes, e.g. borate, Ti(IV), Zr(IV), and Al(III) ions are oen used to crosslink water soluble polymers. [19][20][21] Guar gum and its derivatives, such as hydroxypropyl guar (HPG), carboxymethyl guar (CMG) and carboxymethyl hydroxypropyl guar (CMHPG), are the most-common polymers used in fracturing uid, accounting for up to 90% of all gelled fracturing uids. 1,22 Meanwhile, to satisfy the criterion to fracture deeper and hotter wells, synthetic terpolymer of 2acrylamido-2-methylpropanesulfonic acid (AMPS), acrylamide and acrylate cross-linked by Zr has been developed and used as a fracturing uid. 23,24 The evaluation on the temperature-resistance performance of fracturing uid is essential for choosing the suitable fracturing uid with excellent performance. Generally, two main methods were used to evaluate the temperature-resistance performance of fracturing uid. [25][26][27][28][29][30][31][32][33][34][35] One is to monitor the changes of the apparent viscosity at a constant shear rate with increasing temperature by continuously heating the fracturing uid. [25][26][27] The apparent viscosity decreases with increasing temperature and when it drops to a certain value, the indicated temperature is used as the thermal stability temperature of fracturing uid. For example, Baruah et al. found that the VES fracturing uid developed from Tween 80/NaOA/2-ethyl hexanol/clove oil/water system presented an apparent viscosity at shear rate 100 s À1 greater than 90 mPa s when the temperature is below 116.3 C. The insertion of 500 ppm ZnO nanoparticle further improved the thermal stability temperature to 119.5 C. 25 The other method is to measure the changes of the apparent viscosity with time under constant temperature. Generally, the temperature-resistance performance of the fracturing uid is obtained by either analysing the apparent viscosity aer heating or measuring the stability time when the apparent viscosity of the uid is above criterion value at different temperatures. [28][29][30][31][32][33][34][35] For example, Holtsclaw et al. measured the apparent viscosity of zirconium cross-linked terpolymer gel as a function of time at 177, 191, 204 and 218 C. It is found that the apparent viscosity at shear rate 40 s À1 of the gel was greater than 2000 mPa s aer shearing for 4 hours at 177 C, almost 500 mPa s aer shearing for 4 hours at 204 C, and approximately 300 mPa s aer shearing for 2 hours at 218 C. 29 So they concluded the synthetic zirconium crosslinked terpolymer gel with good stability at temperature larger than 177 C. Wang et al. found that the apparent viscosity of organic zirconium cross-linked terpolymer gel can reach 130 mPa s aer 90 min of shearing at shear rate 170 s À1 at 220 C, and but it is concluded that the fracturing uid resists at high temperature of 220 C. 35 The duration time when the viscosity of fracturing uid meets the required apparent viscosity for fracturing at the formation temperature should be larger than the time of fracturing treatment. 1 Otherwise, it will affect the efficiency of fracturing uid during fracturing treatment. Consequently, when evaluating the temperature-resistance performance of fracturing uid, not only the required apparent viscosity, but also the duration time should be considered. For this reason, the second evaluation method is used more oen. However, most of previous tests are not conducted under uniform standard conditions, e.g. the measurement of apparent viscosity is not at a same shear rate, so the apparent viscosities have to be measured and compared again when developed new thickening agent, crosslinking agent and other chemical additive agents. Furthermore, so far as we know, there is no evaluation method for accurately determining the maximum tolerance temperature of the fracturing uid, which leads to the fact that the effectiveness of temperature stabilizer and chelating agent can only be indirectly reected by measuring the apparent viscosity changes of the fracturing uid at a certain temperature. 36,37 Thus, it is still necessary to explore a standard method on evaluating the temperature-resistance performance of fracturing uid, which will remarkably benet researchers and users for the screening and developing of fracturing uids.
The key point of the evaluation on the temperature-resistance performance of fracturing uid is to establish a method for measuring the maximum tolerance temperature which meets the criteria for the apparent viscosity and the time requirement of the fracturing treatment. Our previous work demonstrated a way of evaluating the effectiveness of temperature stabilizer by measuring the difference of the maximum tolerance temperature in satisfying the criteria of fracturing treatment before and aer adding the temperature stabilizer. 38 However, no systematic study on the inuence of main components in fracturing uid on the temperature resistance performance has been published yet. Herein, in this work, related parameters for characterizing the temperature-resistance performance of fracturing uid have been analysed. And the maximum tolerance temperature is determined by a step-bystep numerical search method starting from the upper limit temperature. By using this method, inuence factors such as pH, the concentration of thickening agent (e.g., HPG) and crosslinking agent (e.g., Na 2 B 4 O 7 ) on the crosslink and temperature-resistance properties of borate cross-linked hydroxypropyl guar gum fracturing uid are investigated and the inuence mechanisms are also discussed.

Materials
Hydroxypropyl guar gum with an average molecular weight of 3.2 Â 10 6 Da is commercially available and supplied by Shandong Dongying Xinde Chemical Co., Ltd. Borax (Na 2 B 4 O 7 -$10H 2 O, 99%) and sodium hydroxide (NaOH, 99%) are all A.R. grade and from Beijing Chemical Company. The water was triply distilled by a quartz water purication system. All reagents are used without further treatment.

Sample preparation
The desired amount of HPG is added to the water in a 500 ml beaker and stirred at 6000 rpm for 5 min. Then 5 wt% NaOH is added dropwise to adjust the pH to desired value. These HPG solutions are continuously stirred for 5 min and sealed in a 30 C thermostatic bath for above 4 hours, which is used as the base uid for the preparation of fracturing uid. The base liquid is added and stirred at 300 rpm in a 250 ml beaker. Aer the addition of the desired amount of cross-linked agent Na 2 B 4 O 7 , the mixture is continuously stirred to form a uniform fracturing uid system. The fracturing uid is then sealed in a 30 C thermostatic bath for more than 6 hours before rheology measurements.

Crosslinking time measurements
The crosslinking time of the borate cross-linked hydroxypropyl guar gum fracturing uid is measured by macroscopic appearance observation at 30 C. Typically, 100 ml base liquid is added and stirred at 300 rpm in a 250 ml beaker, and the desired amount of borate is added to the base liquid. Continue stirring until the vortex disappears and the liquid surface has just risen, and the time is dened as the crosslinking time.

Rheology measurements
The rheological properties of samples are measured with a high temperature and high pressure rheometer (HAAKE MARS-III, Thermo Fisher Scientic). For measuring the viscositytemperature curve of the fracturing uid, the sample was continuously sheared at the shear rate of 170 s À1 and heated at the rate of 3.0 AE 0.2 C min À1 , and then the apparent viscosity was measured at the corresponding temperature. For measuring the viscosity-time curve of the fracturing uid, the sample was continuously sheared at the shear rate of 170 s À1 and heated to the setting temperature. Then the apparent viscosity was measured as a function of time.
3. Evaluation method for the temperature-resistance performance 3.1. Temperature-resistance performance of fracturing uid Table 1 shows the parameters for characterizing the temperature-resistance performance of fracturing uid according to the technical requirement of fracturing treatment. h 0 , t 0 and T 0 are the minimum viscosity requirement of fracturing uid, time requirement and formation temperature for fracturing treatment, respectively. And h is the apparent viscosity of fracturing uid at a certain shear rate. The two temperatures, viz., T max and T max (h 0 , t 0 ) and the duration time t(T) are the most important parameters to characterize the thermal stability of the fracturing uid. Firstly, T max is the maximum temperature where the apparent viscosity meets the minimum viscosity requirement of fracturing uid, and T max is also the highest temperature to satisfy the condition of h[T] $ h 0 . A convenient way to determine T max is through measuring the apparent viscosity as a function of temperature. Normally, the apparent viscosity decreases with increasing temperature and when it drops to h 0 , the corresponding temperature is dened as T max . Secondly, t(T) is the duration time where the apparent viscosity meets the minimum viscosity requirement of fracturing uid at temperature T, and t(T) is also the longest time to satisfy the condition of h[T, t] $ h 0 . The way to determine t(T) is by measuring the apparent viscosity as a function of time at temperature T. And the difference between the time when h drops to h 0 and the time when the temperature rises to the setting temperature is dened as t(T). Thirdly, T max (h 0 , t 0 ) is the maximum temperature where the apparent viscosity meets both the minimum viscosity requirement of fracturing uid and the duration time requirement of fracturing treatment. In other words, T max (h 0 , t 0 ) is also the highest temperature to satisfy the condition h[T, t 0 ] $ h 0 . It can be experimentally determined by measuring the t(T) of fracturing uid at several temperatures, and T max (h 0 , t 0 ) is the highest temperature to satisfy the condition t(T) $ t 0 . Furthermore, it is worth noting that T max is always larger than T max (h 0 , t 0 ) because of the degradation of fracturing uid accompanied by a decrease in apparent viscosity with the increase of heating time.
The condition for fracturing uids to meet the requirements for fracturing treatment is h[T 0 , t $ t 0 ] $ h 0 , which means that at the formation temperature T 0 , the duration time t(T 0 ) shall be larger than the time requirement for fracturing treatment t 0 . Thus, once t 0 , h 0 and T 0 are determined in the designed fracturing project, there are three possible situations according to the relationship between T max , T max (h 0 , t 0 ) and T 0 . When T 0 # T max (h 0 , t 0 ), the temperature-resistance performance of Duration time when the apparent viscosity meets the minimum viscosity requirement of Maximum temperature where the apparent viscosity meets the minimum viscosity requirement of fracturing uid and the duration time meets the time requirement of fracturing treatment fracturing uid can satisfy the requirements of fracturing treatment. When T max (h 0 , t 0 ) < T 0 # T max , the fracturing uid cannot meet the treatment requirements, but the initial apparent viscosity is larger than h 0 . Therefore, it is possible to consider delaying the reduction of viscosity by adding additives such as temperature stabilizer to meet the treatment requirements. When T 0 > T max , the fracturing uid has a low initial apparent viscosity thus it cannot meet the treatment requirements. So it is necessary to change the main-component concentration or the type of fracturing uid to increase the thickening ability of fracturing uid.

Evaluation procedure
Based on the above analysis, it can be seen that T max (h 0 , t 0 ) is the parameter that directly reects the temperature-resistance performance of the fracturing uid in principle. Therefore, the key point for the evaluation method is to skilfully design experiments to determine T max (h 0 , t 0 ) and avoid the boundless search. Owing to T max > T max (h 0 , t 0 ), the T max (h 0 , t 0 ) can be determined by a step-by-step numerical search method starting from the upper limit temperature T max , where the T max is easily determined by measuring the viscosity-temperature curve. Inspired by this idea, we designed a facile method on evaluating the temperature-resistance performance of the fracturing uid and the specic experimental procedures are brief illustrated and shown in Fig. 1, and this method mainly includes the following steps.
(1) Setup the minimum viscosity requirement of fracturing uid h 0 and the time requirement t 0 based on the requirements for fracturing treatment.
(2) Determination of T max by measuring the viscositytemperature curve of fracturing uid.
(3) Determination of T max (h 0 , t 0 ) by numerical search method (Fig. 1a): (i) Setup the initial search length S 0 and the precision k of the numerical search; (ii) Measure the t(T) of the fracturing uid at (T max À S 0 Â n) C, where n is a natural number. The test was stopped when t(T) $ t 0 , and the current temperature was recorded as T(t); (iii) If t(T) ¼ t 0 , stop search and execute step (vi); if t(T) > t 0 , order T(t) ¼ T m , where m is the number of times for execution steps (ii) and (iii), execute step (iv) until the step (v) is satised and then execute step (vi); (iv) Shorten the search length to S m and repeat steps (ii) and (iii) at (T m + S mÀ1 À S m Â n) C in the temperature range of T m $ T m + S mÀ1 ; (v) Order a # k, the measurement has been completed of t(T m ) and t(T m + 2a), and meanwhile, the condition of t(T m ) > t 0 and t(T m + 2a) < t 0 is also satised; (vi) If t(T) ¼ t 0 , the current T(t) is T max (h 0 , t 0 ) that conforms to the precision of numerical search; if t(T m ) > t 0 and t(T m + 2a) < t 0 , the (T m + a) is T max (h 0 , t 0 ) that conforms to the precision of numerical search.
It is noted that the rigorous mathematical formulation is given in the above evaluation method. However, there are still many experimental steps that can be carried out exibly. Firstly, the upper limit temperature T max,1 can be chose near T max , satisfying the condition of t(T max,1 ) < t 0 . This will facilitate the subsequent temperature setting in the test, because T max may be a fraction depending on the preset heating rate in the viscosity-temperature curves measurement. Secondly, the binary search method is recommended as the numerical search method. Although the golden section method, Fibonacci method and other onedimensional numerical search method are better strategy compared to the binary search method, but the programs are relatively complex and difficult to master. 39,40 In the binary search method, the step length of numerical search is reduced by half during each step, which is easy to master. Fig. 1b shows the scheme of binary search method where T max,1 is the upper limit temperature and 24 C is the initial search length.
In order to further explain the determination of the specic parameters in the designed evaluation method, the fracturing uid prepared by 0.3 wt% HPG/0.8 wt% Na 2 B 4 O 7 cross-linked at pH 9 is chosen to investigate the temperature resistance performance. Since the prepared fracturing uid is a water-based gelling fracturing uid, the minimum viscosity requirement of fracturing uid h 0 and the time requirement t 0 are set as 50 mPa s and 120 min, respectively, and the apparent viscosity is measured at shear rate 170 s À1 according to the general technical specications of fracturing uid in China. 41 Fig. 2 shows the viscosity-temperature curve of the fracturing uid. It can be seen that the T max of the fracturing uid is 104.9 C. And then the upper limit temperature T max,1 , the initial search length S 0 and the precision k for the binary search method are set as 105 C, 24 C and 0.5 C, respectively. When the temperature is reduced by 24 C from T max,1 to 81 C, the temperature reaches the set point aer 12 min heating, and the apparent viscosity is greater than 50 mPa s within the overall test time of 150 min (Fig. 3a). It is indicated that the t(T ¼ 81 C) of the fracturing uid is larger than 138 min, which is higher than 120 min. Thus, the T max (h 0 , t 0 ) of the fracturing uid is between 81 C and 105 C. Fig. 3b shows the viscosity-time curve at 93 C, and the t(T ¼ 93 C) is 31 min by calculating the difference between the time when h drops to h 0 and the time when the temperature is raised to the setting temperature, which further limits the search range to 81-93 C. Similarly, the viscosity-time curves of the fracturing uid at 87 C, 90 C, 88 C, 89 C are also measured in turn and were shown in Fig. 3c-f. It is observed that the t(T ¼ 88 C) and t(T ¼ 89 C) satised the condition of t(T ¼ 88 C) > 120 min and t(T ¼ 89 C) < 120 min and the T max (h 0 , t 0 ) is 88.5 C which conforms to the preset precision of this numerical search method.

Inuencing factors
The polymer concentration is a key factor affecting the structure and properties of cross-linked gel fracturing uid system. Fig. 4 shows the effect of HPG concentration W HPG on the crosslinking time t c , T max and T max (h 0 , t 0 ). It can be seen from Fig. 4a that the crosslinking time t c is 480 s when the HPG concentration is 0.2 wt%. The crosslinking time t c will decrease with the increasing of HPG concentration and t c reaches 25 s when the HPG concentration is 0.8 wt%. Considering that the friction is too large at the initial stage of fracturing if the crosslinking is too fast, the HPG concentration isn't further increased in the cross-linked gel fracturing uid. The experiment results on the evaluation of the temperature-resistance performance of the fracturing uid at different HPG concentrations are shown in Fig. S1-S7 and Tables S1-S7 at the ESI. † And the effect of HPG concentration on T max and T max (h 0 , t 0 ) are summarized in Fig. 4b. It can be seen that the T max and T max (h 0 , t 0 ) of the fracturing uid all increase with the HPG concentration. The T max and T max (h 0 , t 0 ) are 51.1 C and 39 C when the HPG concentration is 0.2 wt%. When the concentration of HPG increased to 0.45 wt%, the T max and T max (h 0 , t 0 ) are 150.9 C and 99.5 C, respectively, which increase by 99.8 C and 60.5 C as compared to those at 0.2 wt% HPG. Further increasing the concentration of HPG to 0.8 wt% will result in the increase of T max for 16.4 C to 167.3 C, whereas the T max (h 0 , t 0 ) has gone up 34 C to 133.5 C compared to those at 0.45 wt% HPG. Another inuencing factor on the crosslink and temperature-resistance properties of the fracturing uid is the concentration of crosslinking agent. Fig. 5 shows the plots of the crosslinking time t c , T max and T max (h 0 , t 0 ) as a function of Na 2 B 4 O 7 concentration, W C , respectively. It can be seen from Fig. 5a that the crosslinking time t c decreases with the increase of Na 2 B 4 O 7 concentration. And the crosslinking time t c is 720 s and 30 s for 0.2 wt% Na 2 B 4 O 7 and 2.0 wt% Na 2 B 4 O 7 , respectively. The experiment results on the evaluation of the temperature-resistance performance of the fracturing uid at different Na 2 B 4 O 7 concentrations are shown in Fig. S8-S18 and Tables S8-S18 at the ESI. † And the effect of Na 2 B 4 O 7 concentration on T max and T max (h 0 , t 0 ) are summarized in Fig. 5b. It can be seen that both T max and T max (h 0 , t 0 ) of the fracturing uid increase Fig. 4 (a) The crosslinking time t c , (b) T max and T max (h 0 , t 0 ) as a function of HPG concentration (W HPG ) of the fracturing fluid. The fracturing fluid is prepared by HPG/0.8 wt% Na 2 B 4 O 7 cross-linked at pH ¼ 9. with Na 2 B 4 O 7 concentration, but the variation tendencies are different. Within the concentration range of 0.2-0.8 wt%, T max increases by 56.8 C, whereas in the concentration range from 0.8 wt% to 2.0 wt%, T max only increases by 4.6 C. In comparison, T max (h 0 , t 0 ) gradually increases with the increase of Na 2 B 4 O 7 concentration, e.g. T max (h 0 , t 0 ) increases by 26 C when the Na 2 B 4 O 7 concentration changed from 0.2 wt% to 0.8 wt%, meanwhile, the T max (h 0 , t 0 ) increases by 22 C when the Na 2 B 4 O 7 concentration increases from 0.8 wt% to 2.0 wt%.
The pH when the crosslinking reaction occurs also affects the properties of fracturing uid. Fig. 6 shows the effect of pH on the crosslinking time t c , T max and T max (h 0 , t 0 ). It can be seen from Fig. 6a that the crosslinking time increases with the increase of pH. The crosslinking time t c is 40 s at pH 7 and 220 s at pH 13. The experiment results on the evaluation of the temperature resistance performance of the fracturing uid at different pH are shown in Fig. S19-S24 and Tables S19-S24 at the ESI. † And the effect of pH on T max and T max (h 0 , t 0 ) are summarized in Fig. 6b. It can be seen that the T max increases from 98.7 C to 159.3 C when the pH increases from 7 to 11. And then the T max gradually decreases aer reaching the maximum. The T max reduced to 150.8 C when the pH goes up from 11 to 13. The T max (h 0 , t 0 ) increases from 77.5 C to 99.5 C with the pH rising from 7 to 9. However, the T max (h 0 , t 0 ) almost unchanged in the pH range from 9 to 11. When the pH increases to 13, the T max (h 0 , t 0 ) continually increases to 116.5 C and increases by 17 C compared with that at pH 11.

Inuencing mechanisms
Through the designed evaluation method, the crosslink and temperature-resistance properties of Na 2 B 4 O 7 cross-linked HPG fracturing uid are studied in detail. It is interesting to analysis and discuss the dependence of t c , T max and T max (h 0 , t 0 ) on the HPG concentration, the Na 2 B 4 O 7 concentration and the pH. It is helpful to further clarify the relationship between the measured parameters and gel properties. Firstly, the crosslinking time t c reects the rate at which the crosslinking of fracturing uid reaches the gel state. This denition is proposed from the viewpoint of fracturing treatment. If the gel state is achieved, the friction of fracturing uid in the initial injection process will be remarkably increased. 42 It is worth noting that the gel forms in a short time when HPG mixed with Na 2 B 4 O 7 as shown in Fig. 4a, 5a and 6a. However, the crosslinking reaction is still in progress with time and the gel properties such as strength are also changing. The time when the gel reaches thermodynamic equilibrium (e.g., strength does not change) is called gelation time. Experimental results indicated by Rietjens et al. show the gelation time could go up to 750-22 000 s when the pH changes from 10.6 to 12.85 for the borate cross-linked HPG fracturing uid. 43 That is why the prepared fracturing uids are sealed for above 6 hours in this study to obtain the relatively stable gel for evaluating the temperature-resistance performance. Secondly, it can be seen from Table 1, the measured T max is the maximum temperature where the apparent viscosity meets the minimum viscosity requirement of fracturing uid. Thus, the T max reects the viscosifying ability, which is closely related to the molecular weight and distribution of the crosslinking system of the fracturing uid. 44 As the temperature increases, the molecular thermal motion increases in the cross-linked gel system, so the intermolecular distance in the cross-linked gel network increases, which induces the decreases of ow resistance as well as viscosity. Thirdly, the high temperature treatment of the cross-linked gel will lead to the gel degradation owing to the cleavage of crosslinking and chemical bond induced by thermal hydrolysis, the thermal oxidation and other decomposition pathways. [45][46][47] Because the gel degradation is a gradual process, the reduction of viscosity is highly related to the heating time, in other words, the longer the action time, the higher the degree of degradation and the more serious the viscosity decreases. Thus the measured T max (h 0 , t 0 ) comprehensively reects both the viscosifying ability and the degradation tolerance ability, which is not only closely related to the molecular weight and distribution of the crosslinking system, but also with the degradation process.
The cis-OH pairs on the galactose side chains in HPG molecules can undergo 1-1 and 2-1 crosslinking with borate and the 2-1 crosslinking reaction can be divided into intramolecular and intermolecular crosslinking as illustrated in effective for the formation of network structure, which signicantly increases the molecular weight of the cross-linked gel system and the corresponding viscosity of the fracturing uid (Fig. 7). With the increase of HPG concentration, more crosslinking sites will appear and higher chance of crosslinking reaction will happen. At the same time, the decrease of intermolecular distance between HPG is more favourable for intermolecular 2-1 crosslinking reaction. Thus, the rate of gelation increases, which leads to the decrease of crosslinking time t c as observed in Fig. 4a. The degree of intermolecular 2-1 crosslinking increases, which makes the increase of the molecular weight of cross-linked gels, and thereby enhances the viscosity. Hence, the T max of the cross-linked gel increase with the increase of HPG concentration (see Fig. 4b). When the concentration of crosslinking agent Na 2 B 4 O 7 is xed, the crosslinking reaction will approach the upper limit of crosslinking degree accompanied by the increase of HPG concentration. 48,49 Although more crosslinking sites at high HPG concentration appears, the excess HPG molecules can only exist as non-cross-linked state. Therefore, in low concentration range (e.g., 0.2-0.45 wt%), the increase of T max value caused by the same amount of HPG is greater than that in the high concentration range (e.g., 0.45-0.8 wt%) as shown in Fig. 4b. When the cross-linked gel degrades at high temperature, the non-crosslinked HPG molecules can react with the crosslinking agent Na 2 B 4 O 7 again. As a result, T max changed slightly, but the T max (h 0 , t 0 ) continued to increase in the high HPG concentration range (see Fig. 4b). However, once the cross-linked system degrades at low HPG concentration, the molecular weight of cross-linked gel would decrease signicantly. Therefore, the enhancement of T max is larger than that of T max (h 0 , t 0 ) by adding the same amount of HPG in the low concentration range (e.g., 0.2-0.45 wt%).
Similarly, when the concentration of thickening agent HPG is xed, the rate of gelation increases and the crosslinking time t c decreases with the increase of the Na 2 B 4 O 7 concentration (see Fig. 5a). Meanwhile, the increase of Na 2 B 4 O 7 concentration also promotes the intermolecular 2-1 crosslinking reaction, which results in the increase of the molecular weight and the viscosifying ability of cross-linked gels, and then the increase of T max . However, the degree of crosslinking almost reaches maximum when the Na 2 B 4 O 7 concentration is above 0.8 wt%. So the molecular weight of cross-linked gel does not signicantly increase by continue adding Na 2 B 4 O 7 , where T max behaves as almost a constant (see Fig. 5b). Meanwhile, those excessive Na 2 B 4 O 7 can crosslink again with low molecular weight HPG molecules produced by thermal degradation, which could result in the delayed degradation. Therefore, T max remains unchanged but the T max (h 0 , t 0 ) continues to increase aer the Na 2 B 4 O 7 concentration was higher than 0.8 wt%. Whereas before reaching the maximum crosslinking degree, the change of T max would be greater than that of T max (h 0 , t 0 ) (Fig. 5b).
The crosslinking agent Na 2 B 4 O 7 will be hydrolyzed in aqueous solution to yield boric acid B(OH) 3 and borate anion B(OH) 4 À . The B(OH) 3 exists as a pH dependent equilibrium with the B(OH) 4 À such that higher pH drives the reaction towards the formation of the B(OH) 4 À :

B(OH) 3 + H 2 O ¼ B(OH) 4 À + H +
According to the experimental results of 11 B NMR spectra, both boric acid and borate ion have been implicated in the crosslinking step to form the 1-1 complex as shown in Fig. 8. 50 The 1-1 crosslinking reaction products will continue to undergo intramolecular or intermolecular 2-1 crosslinking reactions with HPG molecules. In the aspect of reaction kinetics, the reaction rate constant of a borate ion with cis-OH pairs is much smaller than that of boric acid. [51][52][53][54][55] Therefore, the crosslinking reaction slows down, and the crosslinking time increases with the increase of pH and the corresponding amount of borate ion (Fig. 6a). In the aspect of reaction thermodynamics, it has been Fig. 7 The molecular structure of HPG (a), the simplified molecular structure of HPG (b) and the crosslinking reactions between borax and HPG (c).