Introduction

Unconventional resources becomes more important in present petroleum industry (Tang and Wu 2018; Xiong et al. 2018, 2017b, 2019). The heat and mass transfer process that happens in an engineering case can be a study topic for engineers (Sheikholeslami 2018; Fengrui et al. 2019a, 2020a, b; Xiong et al. 2017a; Huang 2016, 2017; Feng et al. 2018; Fengrui et al. 2019c, d, 2020c, d; Sun et al. 2017a, b, c, d; Xindi and Baojun 2017; Yu et al. 2019; Sun et al. 2019, Fengrui et al. 2019b). The mass transfer can be found in nearly every aspect in the petroleum industry (Meng et al. 2018; Fengrui et al. 2019e, f, g; Sun et al. 2017e, f, g, h; Jichao et al. 2018), while the heat transfer always happens during the thermal recovery process of heavy oil (Zhanghua et al. 2015; Huang et al. 2018a, b, c; Sun et al. 2018a, b, c; Mohammad et al. 2018; Ezeuko and Gates 2018). In this paper, the thermal recovery process of cyclic superheated steam stimulation using a horizontal well is studied.

When it comes to the enhanced oil recovery, horizontal well has been proved effective in increasing contact area (Yi et al. 2018; Yang et al. 2018; Sun et al. 2018d, e, f). On the other hand, newly developed thermal fluid, such as multi-component thermal fluid and superheated steam, have been proved successful in increasing oil recovery efficiency (Sun et al. 2018g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v; Fengrui et al. 2018). However, it comes with the difficulty that the steam channeling may become more serious when there is a long horizontal wellbore or the horizontal heterogeneity is serious.

Given the fact the heterogeneity always happens along the horizontal well, which may have an influence on the performance of the production well. As a result, we conducted a numerical study to investigate the effect of horizontal heterogeneity on productivity of cyclic superheated steam stimulation horizontal wells using a numerical model.

Effect of different combination of permeability in the horizontal direction is studied. First, a numerical model is established using the commercial numerical simulation software. Then, different combinations of permeability in the horizontal direction are input into the model. The effect of different combinations of permeability in the horizontal direction on the productivity of the horizontal well is studied.

Numerical model

Corner grid is used to build the structure of the model. The reservoir size is 450 × 200 × 20 m. The grid size is 10 × 10 × 2 m. The length of the horizontal well is 250 m. The location of the wellbore in the reservoir is shown in Fig. 1 below.

Fig. 1
figure 1

The I-J and I-K profiles of the numerical model

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Some basic parameters used for calculation are shown in Table 1 below. Note that the permeability in the horizontal direction is 1000 mD.

Table 1 Basic parameters of the numerical model
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Based on the above model, the permeability along the horizontal direction is modified to study the effect of horizontal heterogeneity on the performance of the production well. The permeability distribution pattern is shown in Fig. 2 below.

Fig. 2
figure 2

The permeability distribution pattern along the horizontal wellbore

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Totally nine permeability distribution patterns are studied in this paper. The detailed permeability distribution patterns are shown in Fig. 3 below. The permeability level has been classified as high, medium and low. The high level is equal to 1700 mD. The medium level is equal to 1000 mD. The low level is equal to 300 mD.

Fig. 3
figure 3

The detailed permeability distribution pattern along the horizontal wellbore

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Results and discussion

Based on the above model, these non-permeability distribution patterns are input into the model. The calculated results are shown below.

Figure 4 shows the effect of permeability distribution pattern in the horizontal direction on the recovery degree. It is observed that the curves of different permeability distribution patterns are close to each other. The effect of permeability distribution pattern in the horizontal direction on the recovery degree can be neglected. This may be caused by the infinity of fluid fluidity in the wellbores.

Fig. 4
figure 4

The effect of permeability distribution pattern on the recovery degree

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Figure 5 shows the effect of permeability distribution pattern in the horizontal direction on the cyclic oil production. It is observed that the cyclic oil production is close to each other under all cycles. The effect of permeability distribution pattern in the horizontal direction on the cyclic oil production can be neglected.

Fig. 5
figure 5

The effect of permeability distribution pattern on the cyclic oil production

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Figure 6 shows the temperature fields under various permeability distribution patterns. It is observed that the temperature is the highest in the high permeability section, while the temperature is the lowest in the low permeability section. However, given the fact that the wellbore conductivity is assumed to be infinity, the effect of permeability distribution pattern on the performance of the production well can be neglected.

Fig. 6
figure 6

The temperature fields under various permeability distribution patterns

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Conclusion

The horizontal heterogeneity is always the issue that bothers the practical engineers. The existence of horizontal heterogeneity leads to the low recovery efficiency and serious steam fingering phenomenon. In this paper, a numerical model is built to study the effect of horizontal heterogeneity on the productivity of production well during the cyclic superheated steam stimulation process. Some meaningful conclusions are listed below:

  1. a.

    The effect of permeability distribution pattern in the horizontal direction on the recovery degree can be neglected. This may be caused by the infinity of fluid fluidity in the wellbores.

  2. b.

    The effect of permeability distribution pattern in the horizontal direction on the cyclic oil production can be neglected.

  3. c.

    The temperature is the highest in the high permeability section, while the temperature is the lowest in the low permeability section. However, given the fact that the wellbore conductivity is assumed to be infinity, the effect of permeability distribution pattern on the performance of the production well can be neglected.