A new approach for post-fracturing evaluation and productivity prediction based on a reservoir fracability index model in shale gas reservoirs

Multistage fracturing technology is the primary means of reservoir stimulation in shale gas wells. However, the productivity contribution of each stage varies greatly. It is essential to evaluate the fracturing effect in order to make an optimized treatment design. In this study, we adopted an integrated workflow to assess the main control factors of geological and engineering parameters and a novel approach was proposed for post-fracturing evaluation. For this purpose, the H block in Zhaotong shale gas demonstration zone in Sichuan, China, has been taken as an object of study. The production predicting model was built based on the reservoir fracability index (RFI) which took both fluid type and proppant size differences into consideration. The results demonstrated that (1) if the reservoir quality index (RQI) in the target zone is greater than 5.0, then the area has good reservoir quality and development potential. (2) The RFI of H Block is generally at 4.0–6.0, it can be used as the key parameter to screen out the sweet spot. This method not only serves as a set of practical fracturing evaluation methods but also as a set of productivity prediction and fracturing optimization methods, which can provide strong support for the development of shale gas reservoirs.


Introduction
With the revolution of shale oil and gas technology in the USA, shale gas has become an important resource to replace conventional oil and gas sources (Zou et al. 2016a, b;Bellani et al. 2021). As a key technology of shale gas development, the multistage fracturing of horizontal wells has been widely recognized and applied (Jiang and Wang 2021;Zhu et al. 2021). However, this technology faces a common problem: the fracturing effect of each well varies greatly. Even in the same well, the productivity contribution of each stage can be very different. Nearly a quarter of the perforation clusters have no productivity contribution (Miller et al. 2011;Yu et al. 2015;Nikolskiy and Lecampion 2020). It is because that we are not able to get enough information from the reservoir due to the high cost of shale gas development, many auxiliary measures (microseismic monitoring, production logging, etc.) are rarely adopted (Guimaraes et al. 2020;Shu et al. 2022). Therefore, obtaining a better understanding of a shale formation based on limited data has always been a challenging problem for most scientists and engineers. Postfracturing evaluation as one of the most efficient method is of great significance for fracturing design optimization and productivity prediction.

State of the art
Post-fracturing evaluation is a complex process that includes the dynamic monitoring of fracturing operations, the geometrical parameters of fractures, the coincidence rate between fracturing design and operations, and longterm productivity analysis. These methods can be roughly divided into two categories: direct methods and indirect methods. Direct methods include the far-field method and near-wellbore method. Microseismic monitoring technology, a far-field method, collects microseismic events caused by fracturing on the ground, analyzes and determines fracture parameter information, describes the fracture extension process and evaluates the fracturing effect, which can provide the true geometry and direction of the fractures underground (Maulianda et al. 2019;. Production logging technology, a near-wellbore method, measures the fluid flow profile of production wells and discerns the properties and flow rates of produced or inhaled fluids in each perforation section to determine the fracture height and gas production profile of each stage in the horizontal well (Meng et al. 2019). More accurate data can be obtained by the methods mentioned above, but the technical cost is high, so few wells can adopt these direct methods. At present, indirect methods include fracturing model fitting technology, which is mainly based on a three-dimensional fracturing model. Through simulation and fitting, parameters such as fracture geometry and flow conductivity can be obtained (Zou et al. 2016a, b;Zhou et al. 2020). In addition, treatment curve analysis (shut-in decline curve analysis, net pressure match) is a popular method that is widely used for post-fracturing evaluation. This method is based on a G-function analysis of the pump shut-in pressure decline curve to obtain accurate reservoir and fracture parameter information. However, if the test time is short, the results cannot be explained. In addition, the result of the evaluation is directly related to the experience of the engineer, which introduces greater uncertainty (Bian et al. 2016;Liu and Economides 2019). Reservoir numerical simulation (history match) is another method that many scholars have studied; building a seepage mathematical model between the reservoir and fractures, the production performance data under different operation parameters can be obtained, and then the optimized fracturing design can be screened out. This method has been considered comprehensively; although the optimization process is complex, the result is relatively accurate, but it requires many parameters, which limits its field application (Soliman et al. 2010;Kim et al. 2019).
To solve these problems, this paper proposes a new method for post-fracturing evaluation based on a dynamic reservoir fracability index (RFI) model with the support of geologyengineering integration. The concept combines modern mathematical analysis methods with reservoir engineering. The main controlling factors are selected based on a large amount of data, and a mathematical model is established to evaluate the fracturing effect. First, based on geophysical, geological and logging data, a fine geological model is established through the integrated software Petrel. Second, the influence of various geological factors on production is assessed, and the reservoir quality index (RQI) is proposed to characterize geological "sweet spots." Then, according to the treatment data of the fractured well, the impacts of the operational parameters affecting production are analyzed, and the main control factors affecting the post-fracturing productivity are determined. Finally, the RFI evaluation model considering both geological and engineering impacts is established by multivariate analysis. This method is useful not only for fracturing evaluation but also for productivity prediction and fracturing design optimization.
The remainder of this paper is organized as follows. In the second part, the static and dynamic factors affecting postfracturing production are studied, and a comprehensive model of the RFI is established. The third part and the fourth part examine the H Block of the Sichuan Basin as a research case to analyze and discuss the application of the model. Conclusions are summarized in Sect. "Conclusions".

Establishment of the three-dimensional geological model
A three-dimensional structural model mainly characterizes the distribution pattern of an underground formation by grid technology (Chen et al. 2020). To build the structural model, a fault model and a formational structural plane model are typically established along with a structural grid model. The structural model in this paper adopts the overall structural framework method. The use of this method can ensure the regularity of the grid shape and reasonably characterize the formation structure. This fine geological model takes the H Block of the Sichuan Basin as the research object, as shown in Fig. 1. Vertically, the block includes the Longmaxi and Wufeng Formations as well as the formation 50 m below the Wufeng Formation. The plane accuracy of the modeling is 40 × 40 m, the average vertical accuracy is 0.5 m ~ 122 grids, and the total grid number is 455 × 285 × 1202 = 15,820,350. The YS block is located in the southern margin of the southern Sichuan low-steep fold zone of Taiao, Sichuan, adjacent to the northern Yunnan-Guizhou depression in the south. The sedimentary cover is approximately 6000 m to 7000 m thick from the Sinian to the Jurassic. The main sedimentary assemblages consist of marine facies and continental facies. Marine black shale is the main gas source rock in the study area. There is a thick layer from the Longmaxi Formation of the Lower Silurian to the Wufeng Formation of the Upper Ordovician, which has high abundance of organic carbon and good preservation and is the key target strata of this block. Figure 1 shows the structural map of the bottom boundary of the Silurian Longmaxi formation in the YS block.

Total organic carbon (TOC)
Shale gas production is determined by the abundance in source rocks and reservoir space, which has a close relationship with TOC. First of all, shale gas is mainly composed of the free gas and adsorbed gas. The free gas is stored in the pores or fractures while the adsorbed gas is within the organic-rich formation. The adsorption state of natural gas accounts for more than 20% to 85% of the total occurrence of shale gas, thus TOC determines the shale gas potential production (Liang et al. 2017;Zhang et al. 2020). As shown in Fig. 2, the TOC model had been built for further analysis.

Brittleness
Brittleness index (BI) is one of the most important factors in fracability evaluation (Guo et al. 2021). It is a key parameter for selecting the perforation zone and making frac proposal before fracturing. The higher the BI is, the higher the complexity degree of fractures formed by fracturing and the larger stimulated reservoir volume (SRV) (Sun et al. 2022). The BI is calculated according to the brittle mineral content and mechanical properties. Rickman et al. (2008) proposed the BI as a function of the Young's modulus and Poisson's ratio for shale as: where E BRIT is the normalized Young's modulus and υ BRIT is the normalized Poisson's ratio, both of which are dimensionless. See in Fig. 3, the BI model was built based on Eq. 1 in geological model.

Pore pressure
Abnormal pressure usually occurs in shale gas reservoirs. Thermogenic shale gas reservoirs are generally dominated by high pressure, while biogenic shale gas reservoirs are mainly characterized by low pressure (Guo et al. 2015). Thermogenic gas is generated under the combined action of temperature and pressure. This kind of gas reservoir usually has experienced sufficient burial, compaction, overburden pressure and fluid thermal pressurization. In the process of conversion from organic matter to hydrocarbon, the volume expansion leads to high abnormal pressure (Wang et al. 2013). Block H belongs to this type of reservoir, and its pressure coefficient (C p ) varies from 1.6 to 2.0 (Nie et al. 2015). According to the field data, a pore pressure model was set up, as shown in Fig. 4.
Geological parameters are mostly thought as reservoir intrinsic properties which reflect the quality of reservoirs, and we can use linear relationship to describe the correlation between those parameters and production. The RQI, which integrate those three parameters mentioned above, is introduced. It can be expressed as:  where ξ(x i ) is the gray relational coefficient, Δ(min) is the second-order minimum difference, Δ(max) is the secondorder maximum difference, Δ(x i ) is the absolute difference between the standard data array (production data) and the correlation data array (geological parameters), γ i is the correlation coefficient, and N is the number of samples. Based on the gray relational analysis, the weighting factors in this case are 1, 1 and 0.5.

Analysis of main control factors-operation parameters
Conventional fracturing theory believes that the larger the frac scale, the longer the fracture length. For volume fracturing in shale gas reservoir, frac scale has a good relationship with fracture geometry, the more fluids and proppants pumped in the well, the more complex fracture geometry it is (Wang 2015;Wu et al. 2020). Thus, operation parameters, such as frac fluids, proppants, pump rate and net pressure, should be considered in studying the fracability issues.

Fracturing Fluids and Proppants
The viscosity of frac fluids has an important impact on the complexity of fracture propagation. High viscosity fluids can easily make fractures resist the effects of natural fractures and in-situ stress, and extend along the original direction. However, low viscosity fracturing fluids can make the hydraulic fractures turn to natural fractures. In shale gas fracturing, the composition of "slickwater + linear gel" is widely adopted, slickwater is used for the initiation and propagation of the fractures, while linear gel can help improve the sand concentration and fracture conductivity. Since fluids vary in their viscosity, the volume of linear gels should be converted to adapt the slickwater volume system ): where V f ' is the converted volume of linear gels, m 3 ; V lg is the volume of linear gels, m 3 ; (conc) lg is the average sand concentration in linear gel, kg/m 3 ; and (conc) sw is the average sand concentration in slickwater, kg/m 3 . V ftotal is the total volume of fracturing fluids, m 3 , V fsw is the actual volume of slickwater, m 3 . Similarly, the average viscosity μ' can be expressed as: Fluids with high viscosity can carry more concentrated proppants. Considering the influence of proppant size and ignoring its type impact, in the H Block, the compound mode of proppant size with 100mesh + 40/70mesh + 30/50mesh is often used. According to Eq. 7, the converted volume can be expressed as: where ν s ' is the converted volume of proppants, m 3 ; ν s100 is the volume of the 100mesh proppant, m 3 ; d 100 is the average diameter of the 100mesh proppant, mm; and d 40/70 is the average diameter of the 40/70mesh proppant, mm.

Pump rate
The pump rate has an important influence on the post-fracturing effect. On the one hand, if the rate is too high, natural fractures easily form horizontal fractures, which will narrow the fracture width and be sensitive to sand concentration and a high risk of sand screening out. Meanwhile, the treating pressure is high, and there is a risk of casing damage. On the other hand, if the rate is too low, the operation will take a long time. In addition, this will lead to insufficient crack complexity, and due to the low viscosity of slickwater, the low rate will lead to easier sand settlement in the sandcarrying fluids. For shale gas wells, the higher the pump rate is, the greater the net pressure is, and the net pressure in the fracture will be greater than the horizontal stress difference, which is beneficial for fracture diversion and the formation of a complex fracture network.

Net pressure
The higher the net pressure is, the easier it is for hydraulic fractures to deflect and distort from the expected propagation plane, and the more complex the fractures geometry is. For those naturally fractured reservoirs, if the net pressure is greater than the initiation pressure, it is conducive to form complex fracture network. The net pressure is supposed to be at least greater than the sum of the horizontal principal stress difference and the tensile strength. Therefore, the higher net pressure indicates the higher achieved production. On the field, the net pressure can be defined as the total pressure inside the fracture minus the closure stress:

Establishment of RFI model
The production is determined by both geological and engineering conditions. The better reservoir quality indicates a higher production. For those operating parameters, they are variable as the fracturing treatment continues. As mentioned above, SRV is one of the key factors that influence the shale gas production. And the fracturing scale, pump rate and net pressure are main control factors to determine the SRV. Thus, a new unified RFI model which take both geological and engineering conditions into consideration is proposed as follows: where C is the unit conversion coefficient; Q is the pumping rate; t is the fracturing time; conc is the average sand concentration; and L is the length of the horizontal well. a, b, c and d are the correlation.

Results analysis and discussion
The shale gas in the Upper Ordovician Wufeng Formation-Lower Silurian Longmaxi Formation has been assessed. Six representative platforms including 19 wells in the H Block were chosen. Table 1 shows the geological and operating parameters of the 19 wells in the H Block. Based on Eq. 2 and Eq. 10, the RQI and RFI can be calculated. The RQI can reflect the quality of a reservoir. As Fig. 5 shows, the RQI had a positive correlation with shale gas production, which means that high gas production was based on a high RQI value. There are obvious differences in reservoir quality among each platform in the same block, and (9) the reservoir quality of the high-yield platform is generally better than that of the low-yield platform.
In addition, the RQI varies in different wells on the same platform. For example, for the H2 and H6 platforms, the corresponding RQI for each well on the platform does not change much, indicating that the formation conditions of the platform are relatively stable. Correspondingly, for the H1 platform and H3 platform, the RQI is highly variable in the same platform, which suggests that the reservoirs of the area are highly heterogeneous. According to the research results, the RQI of H Block is generally at 3.5 ~ 5.0. When the RQI exceeds 5.0, it means the area has high production potential and can be taken as a key area for later stimulation. If the RQI is lower than 3.5, the optimal development plan should be determined based on the technical conditions and economic benefits and risks. The RFI  5 The correlation between the RQI and shale gas production can be calculated by Eq. 10, the linear correlation between the RFI and production and the fitting equation can be written as: where P d is the daily production per stage, 10 4 m 3 /d/stage; k is the constant coefficient determined by the target zone, m is the total frac stages, and n is production without frac. As shown in Fig. 6, k was set as 0.09 in this case; and n was set as 0.04 in this case. Thus, Eq. 11 can be expressed as follows: According to Eq. 12, the RFI is closely related to reservoir conditions and engineering measures. In general, the (11) P d = m × (k × RFI + n) (12) P d = m × (0.09 × RFI + 0.04) reservoir conditions are relatively stable; once the target zone is selected, the corresponding geological parameters will be determined accordingly. Thus, to achieve higher production, these engineering measures need to be improved. Based on the established RFI evaluation model, the following three strategies are proposed: (1) Reducing the interval distance between each stage, which means increase the frac stages, the unit section length decreases. (2) If the conditions allow for increasing the scale of fracturing treatment, then increase of the amount of liquids and sands; the fracture length and transverse sweep width will increase, and the stimulated reservoir volume will also increase. (3) Increase the sand concentration and improve the conductivity near the wellbore by properly increasing the sand ratio in the sand slurry stage. Fig. 6 The correlation between the RFI and daily production per stage

Case study
For the above-mentioned achievements, the post-fracturing effect of the H8 platform in the same block is evaluated. LI-1-1 and LI1-3 Formation are the main target zone (See in Fig. 7). Based on the fine geological model established in the previous study, the geological properties of each well in the H8 platform are determined, as shown in Table 2. Among these wells, the H8-1 well was the first well to frac on the platform, and the operation parameters are shown in Table 3. According to Eq. 2, the RQI of H8-1 is 3.73. The daily production per stage is 0.354 × 10 4 m 3 /d and he RFI calculated by Eq. 12 is 3.37, and the predicted production after frac is 0.354 × 10 4 m 3 /d.
H8-2 and H8-3 were the two wells fractured after H8-1. The RQIs of these two wells are 3.80 and 3.82, which is similar to H8-1. Based on the operational experience of the H8-1 well, the fracturing schemes of the H8-2 well and H8-3 well were adjusted. The number of stages originally designed in wells H8-2 and H8-3 was 20, but now they were increased to 21 stages. In addition, the pump rate was appropriately increased in the fracturing design; compared with the H8-1 well, the rate of the H8-2 well and H8-3 well increases to 14.0 m 3 /min. Finally, within the pressure limit, the sand concentration is increased in the last linear gel stage, and the average sand ratio of the H8-1 well is 3.35%, while the average sand ratios of the H8-2 and H8-3 wells are 5.50%, as shown in Table 3.
The RFIs of the H8-2 and H8-3 wells were 4.59 and 4.56, respectively, and the predicted production per stage were 0.453 × 10 4 m 3 /d and 0.450 × 10 4 m 3 /d, respectively. The actual production of the two wells is 0.48 × 10 4 m 3 /d and 0.47 × 10 4 m 3 /d, respectively, with errors of 5.6% and 4.2%, which can meet the needs of engineering prediction (see Table 4).

Conclusions
Post-fracturing evaluation plays a vital role in the optimization of hydraulic fracturing in shale reservoirs. This paper develops a new approach for evaluating the fracturing effects in shale reservoirs. The RFI model comprehensively considers the influence of static geological factors and dynamic engineering factors on production, and the fracturing effect can be quantified by the model. The accuracy and feasibility of this approach are proven by a field case study. The RFI model provides a practical method to screen formations with great potential and optimize fracturing design. The following conclusions are obtained: (1) In H Block, the RQI varies from platform to platform, when the RQI is greater than 5.0, this area has high development value and potential.
(2) The RFI can be the key parameters to predict the production, and the results show that the effectiveness of fracturing treatment correlates strongly with the RFI. If the RFI is greater than 6.0, the area or platform is highly recommended to take the stimulation treatment. While if the RFI is lower than 4.0, it means this region might not be the sweet spot for hydraulic fracturing. (3) Based on the RFI model, improvements can be made after the frac treatment. The interval distance between each stage can be appropriately reduced, the pump rate and the sand concentration can be increased if conditions permit. The above measures are beneficial to improve the effect of reservoir stimulation.
However, one should be cautious when calculating the weight coefficients of the RQI through gray relational analysis. Sampling errors may occur since the weight coefficients vary according to the number and quality of the