^{1}

^{2}

^{3}

^{4}

^{1}

^{1}

^{1}

^{2}

^{3}

^{4}

Stimulated reservoir volume (SRV) which is generated by horizontal drilling with multistage hydraulic fracturing governs the production in the shale gas reservoirs. Although microseismic data has been used to estimate the SRV, it is high-priced and sometimes overestimated. Additionally, the effect of stress sensitivity on SRV is not considered in abnormal overpressure areas. Thus, the objective of this work is to characterize subsurface fracture networks with stress sensitivity of permeability through the shale gas well production data of the early flowback stage. The flowback regions are first identified with the flowback data of two shale gas wells in South China. Then, we measured the permeability stress sensitivity of the core after fracturing, coupled to the dynamic relative permeability (DRP) calculation to obtain an accurate and simple DRP curve. After that, a comprehensive model is built considering dynamic two-phase relative permeability function and stress sensitivity. Finally, we compared the calculated results with the microseismic data. The results show that the proposed model could reasonably predict the SRV using the flowback data after fracturing. Additionally, compared with the microseismic data, the stress sensitivity should be included, especially in the abnormal overpressure block. It is believed that this mathematical model is accurate and useful. The work provides an efficient approach to estimate stimulated reservoir volume in the shale gas reservoirs.

The development of shale gas has gained increasing attention with the decline of the production from conventional reservoirs [

While microseismic data has been extensively used to validate the stimulated reservoir volume, the SRV obtained directly from the microseismic data is generally overestimated. For example, SRV would include large unstimulated void regions by the convexity assumption [

Two-phase flow appears to be a classic and representative flow behavior during the flowback process after hydraulic fracturing [

To take advantage of the two-phase flow characteristics during the early gas production (EGP) stage, research studies have endeavored to estimate the SRV by coupling rock physics with various types of reservoir models. For example, Ezulike et al. [

While the aforementioned models provide insights in calculating SRV through multiflow data quantitatively, challenges of applying these models to the field remain because the Langmuir volume needs to be increased in the calculation process, which will lead to some extent uncertainty and inaccuracy. To overcome the complexity of applying these models in fields, Clarkson et al. [

However, due to some defects in the hypothesis of DRP function, a nonlinear phenomenon occurs when the model is applied to some field cases. And the phenomenon of the nonzero intercept leads researchers to question the physical properties of these problems. It may be due to the inaccuracies caused by stress sensitivity in some areas. In particular, there is abnormal overpressure in shale gas reservoirs in southern China [

We thus aimed to develop a stress-dependent two-phase relative permeability, which can be used to extend the existing single-phase model. Also, we proposed a new multiphase model to estimate SRV and fracture characteristics. Moreover, to verify our model, we compared the results with the microseismic data.

In this work, we first processed the production data of shale gas wells and obtain the gas-water ratio characteristics. Then, we calculated the simple and practical two-phase dynamic relative permeability (DRP) based on the production data collected in the field belonging to shale gas wells in southern Sichuan. Subsequently, the stress-strain curves of permeability are obtained by using the laboratory experiments. Finally, the material balance equation and diffusion equation are used to process the production data to derive the model, thus calculating fracture parameters. We compared the calculation results with the microseismic data to verify the rationality and accuracy of the calculation results.

In this work, we selected the flowback data from two production wells located in southern China: the shale gas formations of the Lower Silurianare, a typical marine shale gas reservoir. A large number of micron-nanoscale pores are developed in the mineral grains and organic matter [

The east-west anticline structure belt with few faults is arranged in the left echelon row as the main geological structure of this area [

Well pad schematic. Seven MFHWs drilled in the southern Sichuan Basin.

In this work, we used data from Well 3 and Well 4 to test our model for the following two reasons. Firstly, the two wells were drilled through the shale gas reservoir in different directions and are located in two different formations. Secondly, microseismic monitoring is conducted in the two wells together with a comprehensive drilling, completion data, and well test data. Figure

Diagnostic plots for two wells in shale gas formations belonging to southern Sichuan. (a) Production rate plot for Well 3. (b) Production rate plot for Well 4.

In order to explore the trend of GWR of production data, this section simply processes and analyzes the production data to observe whether there is an immediate gas breakthrough after the production of shale gas wells [

This is largely because the effective fracture network system is saturated with both the gas and water phases after two shut-ins. The gas source here is assumed to be from three aspects: (1) the originally existing initial gas in the active natural fracture, (2) the gas displaced by fracturing fluid under the influence of the strong countercurrent water imbibitions into the shale matrix due to the huge pressure difference during the first shut-in period, and (3) the gas accumulation that resulted from spontaneous imbibitions of fracturing fluid during the second shut-in period [

Similarly, before establishing the model, we also need to use the gas-water yield ratio to gain the V-shaped trend. Thus, we processed the initial two-phase production data of Well 3 and Well 4 and finally obtained the trend of the gas-water ratio (GWR). Figure

Diagnostic plots for two wells in shale gas formations belonging to southern Sichuan. (a) GWR plot for Well 3. (b) GWR plot for Well 4.

The gas-water ratio decreases and then increases, and we can study it from the most basic theory. Assume that the gas-water two-phase flow satisfies Darcy’s law:
^{3}/s; ^{2}; ^{2};

After ignoring the capillary force in the fracture network system, the gas-water ratio becomes
^{3}/d; ^{2};

After the well opening for production, the gas viscosity decreases with the decrease of pressure, while the water viscosity remains relatively unchanged. According to the change of the gas-water ratio over time, the ratio of gas-water relative permeability decreases. Therefore, on the basis of the relationship between permeability and saturation, it can be inferred that the saturation ratio has a corresponding variation trend, meaning that the initial gas saturation (

In general, the negative slope on the GWR diagnostic plot is called early gas production (EGP) and the rise of the GWR diagnostic plot is called late gas production (LGP). It is generally believed that the EGP region is the stage of wellbore storage effects and both the gas and water productions come from the effective fracture connected with the horizontal well. Also, the increase of water relative permeability is greater than the reduction of gas viscosity. The LGP phase is the result of the matrix gas transfer to the fracture network after wellbore effects becoming negligible [

In order to facilitate the establishment of the mathematical model for the EGP phase, the fracture network around the shale gas fractured well is simplified into the SRV region composed of the matrix system and fracture system. As shown in Figure

Schematic diagram of the horizontal well with multistage hydraulic fracturing for the development of material balance equation. Dashed arrows show fluid flow direction, which is sequentially from the matrix to fractures and fractures to the wellbore.

This model simplifies complex, active natural and secondary fractures, as well as artificial hydraulic fractures, into a simple fracture system. The length of the artificial hydraulic fracture is used as the width of the entire stimulated reservoir volume, and the length of the horizontal wellbore is used as the length. In the whole stimulated reservoir area, the height of the major fracture, including the matrix part connected to the fracture system, participates in the flow. In the equivalent fracture system, the fracture is saturated with fracturing fluid (water phase) and natural gas (gas phase). It is assumed that no matrix gas is involved in the flow during the early gas production (EGP).

It is assumed that the fracture system can be approximated as a homogeneous/closed/tank system. And the fluid flow from the fracture to the horizontal well is assumed to be linear. The mechanism driving the gas-water flow includes two aspects: (1) fracture closure and (2) expansion of the fluid (gas-water phase). Kuchuk et al. [

In general, for the purpose of facilitating the establishment and solution of the model, we made the following assumptions: (1) capillary pressure in fracture systems is ignored; (2) the gas from the matrix is negligible in the EGP stage; (3) the fracture system is approximated as a homogeneous/closed system; (4) Darcy’s law applies to fluid flow; and (5) the effective fracture system is saturated with fracturing fluid (water phase) and natural gas (gas phase) initially.

Effective compressibility terms are defined using simplified gas material balance equations (MBEs) [^{3}; ^{3}/m^{3}; ^{-1}; ^{3}; ^{3}; and

Ignoring the gas flow rate from the matrix system to the fracture, the gas phase material balance equation is
^{3}; and ^{3}.

According to the relationship between gas volume ^{3}.

By substituting the gas production ^{3}/m^{3}.

In the early gas production (EGP), it is assumed that ^{3}/d.

Given that single-phase, steady-state flow can be described using the continuity equation and Darcy’s law, the single-phase gas diffusion equation in the fracture system is given by Zhang and Winter [

Define the pseudopressure and pseudotime functions [^{2}; and ^{-1}.

Then, the governing equation of single-phase gas flow in the fracture system is

In Equation (

The method in this paper is similar to that in Ezulike and Dehghanpour’s study [

Procedure to estimate the dynamic relative permeability (DRP) curve considering stress sensitivity.

In Figure

Gas-water relative permeability curve of the cores from Well 3 and Well 4.

Casing pressure and effective stress changed with time for two wells in shale gas formations belonging to southern Sichuan. (a) Well 3 and (b) Well 4.

Take the stress sensitivity experiments of cores belonging to the stimulated area of Well 3 as an example. Both the matrix and the fractures are most likely contained in these cores. We had required as many experiments as possible to obtain the stress-sensitive test data in the stimulated region after fracturing as accurately as possible. However, subject to the insufficient samples or the different distances of samples from the horizontal well, these data maybe cannot represent the whole area in fact but still have considerable reference and research value.

The specific experimental procedures are as follows: (1) the initial confining pressure was set as the original formation pressure which is 38 MPa, and the internal pressure was 23 MPa; (2) the confining pressure was increased to 58 MPa slowly which is the formation pressure before well opening for flowback, and the internal pressure was increased to 43 MPa at the same time to keep the effective stress constant; and (3) the internal pressure was reduced to different pressure points to increase the effective stress, and the gas permeability of the sample was measured after each pressure point was stabilized.

Dimensionless permeability is defined as
^{2}; ^{2}; and

The experimental results are reported in Figure

Stress-strain experiment results of Well 3 and Well 4.

Dynamic relative permeability function over time considering stress sensitivity for two wells in shale gas formations belonging to southern Sichuan. (a) DRP fitting curve for Well 3. (b) DRP fitting curve for Well 4.

Since the above equation is obtained based on flowback data and core experiment relative permeability,

By substituting Equation (

In order to establish the relationship between pseudopressure function and pseudotime function, we make the following transformation:

And

Define the equivalent gas rate as [

Substituting

By substituting

Then, the following inner and outer boundary conditions can be solved:

Define the fracture storage coefficient as

Then, the following relationship is gained:

Substitute Equation (

By combining Equation (

Theoretically, a plot of the rate-normalized pseudopressure (RNP) vs. the pseudotime should yield a straight line relationship. With the pseudotime function as an independent variable and the left side of the equation as a dependent variable, the slope and intercept can be obtained according to the fitting curve, and the relationship between the equivalent fracture porosity/half-length of the effective fracture system and effective fracture system permeability can be described as follows:

We propose the following analysis procedure:

Obtain and process water and gas flowback data to explore a V-shaped gas-water ratio trend (see Figures

The early flowback period (EGP) is distinguished from the late flowback period (LGP) according to the V-shaped trend of the GWR curve (see Figure

Conduct a simpler fracture network system model for the EGP (see Figure

Calculate effective compressibility by Equation (

Define the pseudopressure and pseudotime functions (Equations (

Plot the gas-water relative permeability curve (see Figure

Calculate gas DRP coupling stress sensitivity (see Figure

Modify the final two-phase flow mode (Equation (

Plot rate-normalized pressure change with pseudotime (see Figure

Calculate SRV by Equation (

Verify the analytical model against microseismic data (see Figure

Analysis of EGP data of two wells belonging to southern Sichuan: rate-normalized pressure change with pseudotime. (a) Well 3 and (b) Well 4.

(a) Renderings of microseismic detection technology for Well 3 and Well 4. (b) Comparison between the analytical model and the microseismic data.

The flowback data we need to obtain include production rates and pressure and cumulative production data profiles. Then, we got a V-shaped trend in the gas-water ratio curve by processing the flowback data. Thus, an analytical model was established for the EGP stage. Finally, the calculated results were compared with the microseismic data to validate the mathematical results.

We apply the analytical model presented above to analyze the flowback data of Well 3 and Well 4. However, there are several issues that need to be addressed and discussed:

It is difficult to gain an appropriate initial gas saturation of the fracture system from actual field data. Unlike conventional numerical simulation, this parameter is unknown in actual field data. After the fracturing operation is completed, the effective fracture system connecting the wellbore is filled with fracturing fluid approximately. Thus, after the well is opened for flowback operation and before production, the volume of the recovered fracturing fluid under the ground is filled with gas renewedly. Therefore, in this paper, we take a reasonable value which is the recovery percentage of the total injection amount of fracturing fluid as the initial fracture system gas saturation in calculation. According to Xu’s paper [

Due to that, fracturing fluid may leak off into the existing inactive natural fractures and into the matrix during injection; the fluid flowback percentage decreases, but the initial gas saturation in the fracture system increases. Hence, when a significant portion of the injected fluid volume does not contribute to create fracture volume, this value of the initial gas saturation for calculation is actually low. One direction of future work is to consider using the results to iteratively optimize the initial gas saturation value.

The value of the fracture closure term in the total compressibility cannot be accurately expressed. Fortunately, it is found that this value has no obvious influence on the results in the calculation process. Since this value refers to the inverse of fracture stiffness when dealing with fractures, it is considered to give a reasonable value in the subsequent work from the perspective of rock fracture mechanics

The final model calculation results are shown in Figure

As shown in Table

Different results of the analytical model and microseismic data.

Parameter name | Well 3 | Well 4 |
---|---|---|

SRV calculated by the analytical model | ||

SRV estimated by the microseismic data | ||

The absolute deviation | ||

The relative deviation | 3.2% | 6.8% |

Effective fracture system permeability | 0.301 mD | 0.244 mD |

Average half-length of hydraulic fractures | 320 m | 285 m |

Effective fracture system porosity | 15.5% | 13.8% |

If the SRV estimated by microseismic data is taken as a reference for comparison, the specific calculation process is as follows:
^{3}; and ^{3}.

If the stress sensitivity effect is not considered in the process of the DRP calculation, the deviation of the DRP curve will occur, as shown in Figure

Dynamic relative permeability function over time without stress sensitivity for two wells belonging to southern Sichuan. (a) DRP for Well 3. (b) DRP for Well 4.

Analysis without considering stress sensitivity of EGP data of two wells in shale gas belonging to southern Sichuan: rate-normalized pressure change with pseudotime. (a) Well 3 and (b) Well 4.

Comparison between the analytical model with or without considering stress sensitivity and the microseismic data.

It can be seen that the relative deviation calculated without considering the stress sensitivity effect is greater than that calculated with considering the stress sensitivity effect. Moreover, because the real SRV has been overestimated by the microseismic data, the calculation results of the model without considering the stress sensitivity effect are higher than those of the microseismic data, which further indicates that the calculation results without considering the stress sensitivity effect in the high-pressure area will have a large error, which cannot be ignored. We also calculated the permeability of the effective fracture system, and the effective permeability of Well 3 and Well 4 was 0.432 mD and 0.283 mD, respectively. It can be seen that the calculation results are 43.50% and 15.75% more than those considering the stress sensitivity effect.

In this paper, a comprehensive model is developed to predict SRV in the shale gas reservoirs using the flowback data after fracturing. The stress sensitivity is included as well. The results of SRV are compared with the microseismic data. The following conclusions can be drawn:

The flowback data show that the gas-water ratio is V-shaped, i.e., the early descending stage and the late ascending stage in this field, which can be used to estimate the volume of the effective fracture system

The stress sensitivity is a key factor affecting the permeability of the effective fracture system as well as the SRV in the shale gas reservoirs. Once it is neglected, the estimation will be overestimated

Stress sensitivity is taken into account to forecast SRV in this typical block. Results show that the relative deviation of stimulated reservoir volume calculated by this proposed model and the microseismic data is less than 10%, indicating that this method could provide reasonable prediction

The test data used to support the findings of this study are included within the article. Readers can obtain data supporting the research results from the test data table in the paper.

The authors declare that there is no conflict of interest regarding the publication of this paper.

This research was funded by the National Science and Technology Major Project of China (2017ZX05009-005), Fundamental Research Funds for the Central Universities (2652018209), and National Natural Science Foundation of China (51804282).