**Table 9** OOIP_{SRV} evaluation results of APDA

With the assumption that the stimulated reservoir volume (SRV) for an MFHW is a cuboid, SRV of field examples and simulated wells have been estimated (Table 10). Taking the effective well length (l_{e}), original length (L_{FO}) and height (H_{FO}) of fracturing stage detected from microseismic interpretation as the length, width and height for this stimulated cuboid, detected SRV from microseismic (DSRV) has been figured. From APDA, SRV for an MFHW can be calculated as well (Table 2).

It is certain that this estimated volume can be treated as the effective SRV (ESRV) for an individual MFHW, because oil resource contained in ESRV (dynamic reserve) can potentially be extracted from underground. As shown in Table 10, DSRV and ESRV for 5 field wells have been calculated. Comparing the DSRV from microseismic with ESRV from APDA (Table 10), it has been found that the DSRV is obviously larger than estimated ESRV. It just confirms that the effective SRV is far less than the volume controlled by microseismic events (Wang et al. 2015). It has been found that the effective SRV is about 40.74% of DSRV in average for this tight oil reservoir.

**Table 10** Estimation of stimulated reservoir volume for field examples and simulated wells

### Relationship between fracturing network parameters and productivity

Calculation of ω and λ has been accomplished for 5 field wells completed in tight oil reservoir as mentioned above. After counting the SRV, ω and λ of all the fracture stages for 5 field wells (including Table 4 and more calculated results for other 4 cases), it has been found that there exists exponential relation between fracturing network parameters (ω and λ) and SRV. As shown in Figs. 13 and Fig. 14, there exists apparent exponent relation (Eq. 31) between fracturing network parameters (ω, λ) and SRV, and the relationship parameters are presented in Table 11.

**Fig. 13** Relationship between ω and SRV for an individual fracture stage

**Fig. 14** Relationship between λ and SRV for an individual fracture stage

**Table 11** Relationship parameters of ω and λ

As a statistical result, Eq. (31) reflects the relationship between fracturing network parameters and detected SRV of each individual fracturing stage in this particular tight oil field. Based on Eq. (31), it is applicable to estimate ω and λ for more new wells even if some reservoir parameters are not known because of that SRV can be estimated from microseismic interpretation directly. Certainly, those new wells should belong to the same oilfield block with 5 field wells. It makes sure the relationship between fracturing network parameters and detected SRV complies with this exponent relation. Although Eq. (31) is available for this particular tight oil field, the methods to obtain this equation are applicable for other reservoirs and worth to be promoted. As for a new tight oil reservoir, a new equation (similar to Eq. 31) can be figured out with help of the developed methods, which are used to calculate ω and λ (Eqs. 27–30). Then, ω and λ for more wells in this new tight oil field block can be easily estimated as well.

**Fig. 15** Relationship between ω and λ

It is easy to find obvious power relation between ω and λ (Fig. 15). This power function relationship just corroborated that more underground oil resource connected by hydraulic or secondary fractures (higher ω) can be much easily extracted (higher λ) from tight rock whose permeability is ultra-low.

From Fig. 16, on the basis of production performance analysis results, it has been found that there exists dramatic negative linear relationship between the ratio of ω to λ (ω/λ) and the slope of characteristic straight line on square root time plot (m). It is a fact that a smaller m indicates higher productivity for an individual MFHW (From Figs. 4, 9). The average daily production (Q_{ave}), which was collected from metering station for filed wells, and calculated ω/λ are presented in Table 12.

**Fig. 16** Relationship between ω/λ and m

As shown in Fig. 17, apparent positive linear relationship between ω/λ and Q_{ave} has been observed. It happens to validate the negative correlation of m and productivity (smaller m implies higher productivity). In other words, for the purpose of improving productivity of a horizontal well in tight oil reservoir, network fracturing techniques should be taken to enhance the storativity firstly. It also means that the key point of fracturing stimulation in tight reservoir is not to establish hydraulic fractures with larger scale but to connect more natural or secondary fractures with hydraulic fractures. This derived linear relationship is not only applicable for 5 field examples but also applicable for 2 simulated wells in this tight oil reservoir (Fig. 17).

**Fig. 17** Relationship between ω/λ and average production rate

The consistency of derived relationship (between ω/λ and Q_{ave}) from the analysis field cases and simulated case has just verified the applicability of developed model to calculate ω and λ for MFHW in tight oil reservoirs. The significance of the derived linear relationship between ω/λ and Q_{ave} is that the productivity for more new wells completed in the same oil filed can be estimated approximately according to microseismic interpretation, because ω and λ can be easily estimated (using Eq. 31) on the basis of detected SRV. Indeed, this linear relationship between ω/λ and Q_{ave} (Fig. 17) may be not applicable for other tight oil reservoirs. However, the method developed to calculate network parameters (ω and λ) is worth to be applied to more tight oil reservoir. A new relationship between ω/λ and Q ave could be established for other reservoirs, and then, it is going to be convenient to predict productivity for more MFHWs completed in other tight oil reservoirs.

**Table 12** Parameters of production analysis

## Conclusions

We have developed a model to estimate OOIPSRV and a solution to calculate fracturing network parameters (ω and λ); all the developed models were run for single-phase case. The results of analysis and calculation for 7 cases (5 field cases and 2 simulated cases) have validated the developed models and solutions.

Modern production analysis method (MPA) and advanced production decline analysis (APDA) approach are available for production performance analysis of MFHW completed in tight oil reservoirs. Log–log normalized rate vs time plot and square root time plot are convenient methods to perform flow regime identification.

It proves to be applicable and accurate when the developed model is used to perform OOIP

_{SRV}estimation of individual MFHW completed in tight oil reservoirs due to the fact that the calculated results of OOIP_{SRV}from MPA are closed to the outcomes of APDA.It has been found that there exists exponential relationship between fracturing network parameters (ω and λ) and SRV. Fracturing network parameters (ω and λ) decrease with increase of SRV. Furthermore, the derived exponential relation between fracturing network parameters (ω and λ) and SRV makes it feasible to obtain network parameters for new wells on the basis of SRV estimation.

In order to improve the productivity of horizontal tight oil wells, enhancement of ω should be given first priority. Thus, the key point of fracturing stimulation in tight reservoir is not to establish longer hydraulic fractures, but to connect more natural or secondary fractures with hydraulic fractures.

For target tight oil reservoir, it has been found that there exists linear relationship between ω/λ and Q

_{ave}. On the basis of derived linear relationship between ω/λ and Q_{ave}, the productivity for more individual MFHW can be directly estimated according to microseismic interpretation.

Notes

#### Acknowledgements

The authors wish to thank State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Chengdu for the technical support of this study.

### Appendix 1: Equations of MPA plot

Dimensionless rate and dimensionless are defined as follow:

### Appendix 2: Equations of square root time analysis method

As shown in Fig. 18, there are two sides for an individual hydraulic fracture. The total matrix surface area contacted by a hydraulic fracture (A c) is defined as (Clarkson and Beierle 2010; Clarkson 2013):

There are two sides of an individual fracture. With the assumption that hydraulic fractures of a MFHW are identical and the number of hydraulic fractures equals the number of stage, the contacted matrix surface area for a single side of fracture is (A_{c1}):

### Appendix 3: Equations of APDA analysis method

Material balance time, dimensionless material balance time, dimensionless material balance pseudo-time, dimensionless borehole radius and dimensionless pseudo-rate are defined as follow (Palacio and Blasingame 1993; Sun 2013):

The dimensionless borehole radius (r_{eD}) is determined from the fit of production data on standard chart (Fetkovich 1980; Palacio and Blasingame 1993). In standard chart, there are a series of standard curves and each standard curve corresponds to a r_{eD}. On the basis of best fit of production data, the particular standard cure can be detected for a single well, and then, r_{eD} is figured out. It is assumed that all the hydraulic fractures contribute to flow and the individual fractured horizontal well is located in a circular reservoir zone; with the best fit of production data and the knowledge of r_{eD}, reservoir parameters and reserve can be estimated (Liu et al. 2010; Sun 2013).

Selecting an actual data point (t_{ca}, q/△p) arbitrarily from actual production data points, the selected actual production data point (t_{ca}, q/△p) corresponds to a theoretical fitting point on standard chart (t_{c}D_{a}, q_{Dd}). With the combination of detected r_{eD}, the effective permeability (k_{SRV}) and effective borehole radius (r_{wa}) can be calculated:

From previous studies (Liu et al. 2010; Sun 2013), the total pore volume can also be directly estimated from the fitting of production data using arbitrary matched data points:

the controlled radius for a horizontal

### Appendix 4: Equations of fracturing network parameters

With the combination of Eqs. (20) to (26), the storativity (ω) can be calculated:

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Hongwen Luo: [email protected]

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