*4.2. Synthetic Case*

Based on the simulator Eclipse, the numerical model of a MFHW in the unconventional gas reservoir is established. The numbers of grids are X × Y × Z = 80 × 260 × 3 and the grid steps are X × Y × Z = 5 m × 5 m × 5 m, respectively. A dual-porosity media model is used to describe the SRV (DUALPORO keyword), and a single-porosity medium model is used to describe the USRV. Hydraulic fractures are described by LGR keyword as shown in Figure 7. If only the gas viscous flow with slippage effect is considered, then the FMFM can be simplified as a semianalytic model for a MFHW with the homogeneous SRV in the tight gas reservoir. We set the same modeling parameters of FMFM with those of numerical model and then discuss the applicability of FMFM with ES-MDA.

**Figure 7.** Schematic diagram of a MFHW numerical model (*yl* = *ye*).

Firstly, according to the dimensionless parameters listed in Table A1, the production rate of the numerical model can be non-dimensionalized as *qD* = *pqf* /<sup>2</sup>π*kwhci*ϕ − ϕ*w f*.

Then, the *Nd*—dimensional vector of observed data can be chosen as:

$$d\_{\rm obs} = \left[ d\_s^T, \mathbf{q}\_o^T \right]\_{N\_d \times 1'}^T \tag{23}$$

where *ds* donates the *Nd*-dimensional vector of static parameters of the reservoir and MFHW; **q***o* donates the dimensionless production rate of numerical model.

The *Nd* × *Nd* covariance matrix of observed data measurement errors is given by

$$\mathbf{C}\_{D} = \begin{bmatrix} \sigma\_{d\_{1}}^{2} & & & \\ & \ddots & & \\ & & \sigma\_{d\_{N\_{d}}}^{2} \end{bmatrix}\_{N\_{d} \times N\_{d}} \prime \tag{24}$$

where σ*d* is equal to 0.3 for real production rate.

The matching results of dimensionless production rate by ES-MDA after 4 iterations and error analysis are shown in Figure 8. The values of some variables are listed as *Np* = 23, *Nd* = 115, *Ne* = 100, and √<sup>α</sup>*i* = 2. Meanwhile, the root-mean-square error and the average objective function in Figure 7 can be expressed as, respectively,

$$RMSE = \sqrt{\frac{\sum\_{j=1}^{N\varepsilon} \left(d\_{\text{sobs}} - d\_{sj}\right)^2}{N\_p}},\tag{25}$$

$$O(m) = \frac{1}{N\_d} [\mathbf{g}(m) - d\_{\rm obs}]^T \mathbb{C}\_D^{-1} [\mathbf{g}(m) - d\_{\rm obs}].\tag{26}$$

**Figure 8.** The matching results of dimensionless production rate by ensemble smoother with multiple data assimilation (ES-MDA) after 4 iterations and error analysis.

Figure 8 shows that the error between the ensemble mean model and that of true model is large with one iteration, and the distribution range of ensemble members is large. All models of the ensemble (light blue curves) represent the results calculated by FMFM, the ensemble mean (blue curve) represents the calculated results matching with unperturbed observation data (average value of ensemble members), the true value (red curve) is the true dimensionless production rate obtained by FMFM, and the numerical results (red point) is the observed data in Figure 8. We can see that with two iterations, the distribution range of ensemble members becomes small, and ensemble mean model basically coincides with the true model. With three and four iteration, the distribution range

of ensemble members is further reduced around the true model, and the ensemble mean model and the observed data are well-matching, which is almost the same with the true model. In addition, the RMSE and objective function values of the prior model are large, indicating that the model errors and data errors are large. After one iteration, the error decreases. The RMSE and objective function values becomes small and basically the same after three and four iterations, which indicates that the matching results are good as shown in Table 2. Therefore, the discussion mentioned above suggests that parameters of FMFM can be obtained by automatically matching production data of the numerical model by ES-MDA method. The technique can also obtain reservoir physical properties and fracturing parameters by matching the actual production data of the oilfield.


**Table 2.** Comparison of matching results of FMFM and numerical model parameters by ES-MDA.

## *4.3. Field Case*

The presented FMFM model with ES-MDA history matching method was applied based the shale gas production data of a MFHW in western China [36]. Compared with the presented models based on different simulating conditions, the matching results are shown in Figure 9. Noting that the actual shale gas production rate of a MFHW in western China is represented by red points; when *dfs* = *dfa* = 2, θ = 0, *yfD* = *yeD*, the FMFM can be assumed as a typical trilinear model with homogenous SRV (black line); the results of Sheng's FMPM model [20] with fractal SRV are drawn by the light green line, and they calculated the *dfs*= 1.90 by the box-counting method and the fractal random-fracture-network algorithm and θ = −0.05 by the random walk method; based on the presented FMFM with ES-MDA, the fractal dimensions, tortuosity index, and SRV size can be obtained as shown by the blue line. Figure 9 shows that the fractal dimension and tortuosity index of induced-fracture system matched by FMFM model based on ES-MDA approximate the results of Sheng's FMPM model. The unmatched early-time data was caused by the early-time flow back process. In addition, the results dimensionless production rate calculated by FMFM were smaller but matched better with actual data than Sheng's FMPM model when the SRV size was taken into account. This section mainly provides an application case of our presented approach.

**Figure 9.** The matching results of actual shale gas production rate and different models.
