In conventional natural fractured reservoirs, ω and λ could be detected from pressure buildup testing; however, it is not feasible in tight reservoirs, because it needs too much time to accomplish the pressure buildup test. Thus, we estimate the natural fractures property based on hydraulic fracturing data from microseismic interpretation. The average of original length, width and height of hydraulic fractures are defined as L_{FO}, W_{FO} and H_{FO} separately. The results of microseismic interpretation would provide what we need (L_{FO}, W_{FO} and H_{FO}). The original length, width and height (L_{FO}, W_{FO} and H_{FO}) of hydraulic fractures are detected from microseismic interpretation. The affected length of microseismic events is usually regarded as the original length of hydraulic fractures to calculate ω and λ (Lian et al. 2011; Wang 2015).

Due to the fact that the peak of hydraulic fractures may not be effectively supported by proppants, the original fracture length might be much longer than actual supported fracture length (X_{f} derived from square root of time analysis). Likewise, the detected height of microseismic events (H_{FO}) might be larger than the thickness of pay zone (h). The original width of hydraulic fractures (W_{FO}) may be little greater than the supported fracture width according to the size of proppant and the injected amount of sand. Thus, during the calculation of effective stimulated volume for an individual MFHW (effective SRV), we took the calculated effective fracture length of SRV zone (from Eq. 6) as L_{FO} which is based on the actual production performance analysis from MPA.

The detected height of microseismic events could be used to determine H_{FO}. If the height is larger than the thickness of pay zone, the thickness of pay zone (h) will be taken as the effective height of SRV zone (H_{FO}). Otherwise, the detected height is going to be regarded as the height of hydraulic fractures. The width of hydraulic fractures was replaced by the supported fracture width which could be estimated according to the size of proppant and the injected amount of sand.

It is assumed that the stimulated reservoir volume (SRV) of an MFHW is a cuboid and the effective well length (l_{e}), original length (L_{FO}) and height (H_{FO}) of fracturing stage detected from microseismic interpretation are regarded as the length, width and height for this stimulated cuboid, and then, the total volume of SRV (V_{SRV}) and hydraulic fractures (V_{F}) are estimated as follow:

Volume factor of natural fracture:

In fact, when ω and λ are used to characterize dual-porosity reservoir, secondary fracture network and the original natural fractures are regarded as a unified system (natural fracture system). Network parameters (ω and λ) are determined by the properties of natural fractures (k_{f,in} and R_{f}) in nature. Combining with volume factor of natural fracture, the permeability of natural fractures (k_{f,in}) can be used to characterize the permeability of natural fracture system which is the same as the average permeability of SRV(k_{SRV}). On the contrary, if k_{SRV} has been obtained, the inherent permeability of natural fractures (k_{f,in}) can be estimated. The average permeability of SRV can be figured out from Eq. (5), and then, the inherent permeability of natural fractures (k_{f,in}) can be derived from Eq. (26).

It is worth to mention that Eqs. (27) and (28) are used to estimate ω and λ for a MFHW as a whole. L_{FO}, W_{FO}, H_{FO} and η are the average values of all fracturing stages for a MFHW. V is the total volume of injected fracturing fluid for the whole well. When calculating ω and λ for an individual fracturing stage (Eqs. 29 and 30), those parameters should be replaced by actual data from microseismic interpretation for a particular fracturing stage (L_{FOi}, W_{FOi}, H_{FOi} and η_{i}). What’s more, not the effective length of horizontal well section (l e) and total volume of injected fracturing fluid for the whole well (V), but fracturing spacing (d= l_{e}/n_{f}) and injected fracturing fluid for a particular fracturing stage (V_{i}) should be used here. Then, ω and λ for each individual fracturing stage is given by:

### Example application

In order to demonstrate the application and feasibility of developed methods in detail, 5 field cases from an actual tight oil reservoir have been chosen to carry out the production performance analysis and OOIP_{SRV} evaluation. As a validation, 2 simulated cases have been analyzed as well. In simulated cases, the synthetic data are provided by a semi-analytical productivity prediction model for multistage fractured horizontal wells in naturally fractured reservoirs (Wang et al. 2015; Wang 2015). This productivity model, which can be used to predict the productivity of MFHWs in tight oil reservoirs, is based on the volumetric source methods.

### Field cases

As for field cases, 5 MFHWs from an actual tight oil reservoir were chosen to carry out the production performance analysis and OOIP_{SRV} evaluation. The average thickness of reservoir pay zone is about 34.8 m. The porosity is 10% in average, and permeability is about 0.012 mD. The content of clay in the reservoir rock with indistinct sensitivity is low (1.96%). The density of crude oil is 0.89 g/cm^{3}, and the viscosity is 13 mPa s. The pressure system appears to be normal, and pressure coefficient of target oil layers is 1–1.2. Those basic reservoir features and reservoir physical parameters are from reservoir test or well sampling test. In this tight oil reservoir, multistage fracturing stimulation has been performed for every horizontal well to achieve economic development.

### Flow regime identification

The characteristics of linear flow after fracturing are the basis of MPA approach. The first step of production performance analysis is to figure out two key parameters (T_{elf} and m) which are essential to the calculation of reservoir parameters and OOIP_{SRV}. Taking Well 1 as an example, we have demonstrated how to identify the flow regimes for a fractured horizontal well completed in tight oil reservoir.

**Fig. 3** Log q/(P_{i} − P_{wf}) versus log time plot of Well 1

As shown in Fig. 3, the 1/2 slope straight line (blue line) represents linear transient flow regime and the unit slope line (yellow line) characterizes the boundary dominated flow. The production history was divided into two sections by the green vertical line. The flow regime just changed at this intersection point (T_{elf}). We have found that all of the 5 actual wells have reached the boundary dominated flow, and the results of diagnosis are presented in Table 1.

**Table 1** Results of square root time plot analysis for 5 field cases

On the square root time plot (Fig. 4), a satisfactory fitting straight line indicates the linear flow regime. The data points begin to deviate from this characteristic straight line at T_{elf} (143^{th} day) that is coincident with Fig. 3. The most significant meaning of this chart is to determine the end time of linear flow (T_{elf}) and the slope of characteristic straight line (m). With knowledge of T elf and m, not only the OOIP_{SRV} can be easily figured out according to Eq. (12), but also the effective permeability of SRV (k_{SRV}) and effective half fracture length (X_{f}) can be estimate from Eqs. (5) and (6), respectively. From Fig. 4, the derived end time of linear flow (T_{elf}) is 143 days and the slope of the square root time plot (m) is 0.17. The calculated OOIP_{SRV} for Well 1 is 102.63 × 104 m^{3} (Table 1).

**Fig. 4** Square root time plot of Well 1

Following the analysis steps mentioned above, 5 field examples have been analyzed and the comprehensive calculated results of are shown in Table 1.

### Dynamic reserve evaluation

Considering the above flow regimes diagnosis results (as for all of the 5 actual field examples, boundary dominated flow regime have been observed), Blasingame, Agarwal–Gardner (A–G) and normalized pressure integration (NPI) TCM techniques of APDA were utilized to analyze the cases studied in this work. Instead of using a single TCM technique, a collective of TCM approaches were adopted to corroborate the analysis outcomes.

**Fig. 5** Blasingame-type curve matching

From Figs. 5, 6 and 7, we can see that good fittings of actual production data and standard chart for all 3 kinds of APDA methods (Blasingame, A–G and NPI TCM techniques) have been achieved. On the basis of good fitting of rate data, accurate reservoir parameters and OOIP_{SRV} have been figured out for Well 1 (Table 2). This OOIP_{SRV} determined by the daily oil production and pressure performance only is regarded as the dynamic reserve controlled by Well 1. The main purpose of this section is to find out the OOIP_{SRV} for field wells; not all of the calculation results of reservoir parameters have been presented in this paper.

**Fig. 6** A–G-type curve matching

**Fig. 7** NPI-type curve matching

**Table 2** Advanced production decline analysis results of Well 1

The final calculation results of OOIP_{SRV} for 5 field examples are shown in Table 3. Comparing Table 3 with Table 1, it is easy to find that the results of OOIP_{SRV} estimation using MPA method are close to the results from APDA.

**Table 3** OOIP_{SRV} evaluation results for 5 field examples using APDA

### Calculation of fracturing network parameters

Combining the construction data (injected liquid volume) and microseismic interpretation results (L_{FOi}, W_{FOi} and H_{FOi}), ω and λ of each individual fracturing stage for Well 1 have been calculated (Table 4). From Eqs. (29) and (30), ω and λ of the whole well have been figured out as well (Table 5). As seen from Tables 4 and Table 5, for Well 1, the average values of ω and λ for every individual fracturing stages (ω = 0.003793, λ = 0.001142) are close to the calculated ω and λ of the whole well (ω = 0.003791, λ = 0.001015).

**Table 4** ω and λ of each individual fracturing stage for Well 1

In fact, all 5 field examples comply with this fact. Thus, it is convenient and feasible to take the average ω and λ of every individual fracturing stage as the ω and λ for the whole horizontal well. Due to the fact that there are too many calculated data of individual stages for the 5 field wells, to list them all (detailed fracturing network parameters and calculation for Well 1 are shown in Table 4), the calculated ω and λ for all 5 field wells are presented in Figs. 13 and 14.

**Table 5** Calculation results of fracturing network parameters for 5 field cases

### Simulated cases

In this section, two multistage fractured horizontal well completed in this tight oil reservoir are simulated using a semi-analytical productivity prediction model for multistage fractured horizontal wells in naturally fractured reservoirs (Wang et al. 2015 and Wang 2015). Input reservoir parameters are the average values from this actual tight oil reservoir (Table 6). For these two simulated MFHWs, it is assumed that the number of hydraulic fractures equals the number of stages and all fracturing stages are identical.

**Table 6** Input parameters for simulated cases

The production prediction for 300 days has been carried out using this semi-analytical productivity prediction model. Then, the synthetic data are used to carry out production performance analysis and OOIP_{SRV} estimation as what was done for 5 field examples above.

## Flow regime identification

As shown in Fig. 8, the 1/2 slope straight line (blue line) represents linear transient flow regime and the unit slope line (yellow line) characterizes the boundary dominated flow. It indicates that the boundary dominated flow has been observed for this simulated case. From Figs. 8 and 9, the derived end time of linear flow (T_{elf}) is 37 days and the slope of the square root time plot (m) is 0.22. The calculated OOIP_{SRV} for simulated case 1 is 62.29 × 104m^{3}. With the combination of square root time analysis results, reservoir parameters and OOIP_{SRV} have been calculated for those 2 simulated cases (Table 7).

**Fig. 8** Log q/(P_{i}−P_{wf}) versus log time plot of simulated case 1

**Fig. 9** Square root time plot analysis of simulated case 1

**Table 7** Results of square root time plot analysis for 2 simulated cases

### Dynamic reserve evaluation

From Figs. 10, 11 and 12, good fittings of predicted production data with standard chart for all 3 kinds of APDA methods (Blasingame, A–G and NPI TCM techniques) have been achieved. On the basis of good fitting of rate data, reservoir parameters and OOIP_{SRV} have been figured out for 2 simulated cases (Tables 8, 9). Comparing Table 7 with Table 9, it has been found that the results of OOIP_{SRV} estimation using MPA method are close to the results from APDA.

**Fig. 10** Blasingame-type curve matching

**Fig. 11** A–G-type curve matching

**Fig. 12** NPI-type curve matching

**Table 8** Advanced production decline analysis results of simulated case 1

The effectiveness of network fracturing stimulation is positively reflected by ω and λ, and the scale of SRV is the decisive factor of productivity improvement in tight oil reservoirs. Comparing Table 1 with Table 3 (or comparing Table 7 with Table 9 for simulated cases), the OOIP_{SRV} evaluation results from MPA are closed to the outcomes of APDA. It proves to be applicable and accurate when the developed model is used to perform OOIP_{SRV} evaluation in tight oil reservoirs.