Round 1

Reviewer 1 Report

In this manuscript, the permeability of a porous media is calculated using mathematical modelling of the non-Darcy flow which is solved using commercial software. The authors found a small difference from the results using Darcy seepage. The results and method are interesting and should be useful to the community involved in oil extraction. But the description of the method and results should be improved. I list below a few suggestions and questions to be addressed in the manuscript.

 

When the pressure is increased in your model, does it consider viscous fingering instabilities? How is it included in a macroscopic model like yours?

 

These curves for relative permeabilities were obtained before using other methods. How do your results compare to those in refs Phys. Fluids 34, 023102 (2022) and Physics of Fluids 34, 103303 (2022) for instance? In particular, why you do not have jumps in figure 2? The authors should compare their results with those in the literature.

 

In all those colour maps (figs 6, 7, 8, etc), what does the y-axis mean? The authors write the meaning of the x-axis, but not that of the y-axis.

 

What is the information provided by figures 4 and 5? They are equal, with uniform colour and the colour bar is useless.

 

In equations from 4 to 8, I suggest changing gradP by the actual gradient symbol (nabla).

 

Since the gradient of P is a vector and the argument of the exponential functions in equations 4 to 8 must be a scalar, the parameter b must be a vector. If this is the case, please use a vector symbol in b. Or maybe you want to replace gradP by a pressure drop.

 

The authors should provide a list with all the values of parameters used in their simulations.

Author Response

Main comments

REVIEWER #1:

In this manuscript, the permeability of a porous media is calculated using mathematical modelling of the non-Darcy flow which is solved using commercial software. The authors found a small difference from the results using Darcy seepage. The results and method are interesting and should be useful to the community involved in oil extraction. But the description of the method and results should be improved. I list below a few suggestions and questions to be addressed in the manuscript.

Response: Thank you very much for your valuable comments. Authors discussed the comments of reviewer specifically and revised them very carefully. Detailed suggestions for revision are in the manuscript.

 

1. When the pressure is increased in your model, does it consider viscous fingering instabilities? How is it included in a macroscopic model like yours? 

Response: Thank you for your valuable comments. Authors have proved the viscous fingering instabilities being ignored as the pressure increasing. Viscous fingering has been included in the macro model, but it has not been thoroughly considered in this paper, which may be mainly carried out in the following research work..

 

2. These curves for relative permeabilities were obtained before using other methods. How do your results compare to those in refs Phys. Fluids 34, 023102 (2022) and Physics of Fluids 34, 103303 (2022) for instance? In particular, why you do not have jumps in figure 2? The authors should compare their results with those in the literature.

Response: Thank you very much for finding this flaw in our manuscript. In the introduction, we have introduced the key points of this paper in detail, and the permeability variation based on the thickness of different layers is a major feature of this paper. In this paper, the discussion part as shown in section 4 is added to explain the characteristics of the model based on the actual case of oil field.

 

4 Discussions

Taking a certain reservoir as an example, the reservoir has a burial depth of 3523m, average permeability of 16.7mD, average porosity of 14.3%, and original formation pressure of 36.4MPa. By comparing the calculation results of Darcy flow and non-Darcy flow in Figure 17, the difference in formation pressure between oil wells and water wells is small (<3MPa) in Darcy flow, while the difference in formation pressure between oil wells and water wells is large in non-Darcy flow, resulting in the pressure around water Wells is serious. The large pressure drop around oil wells indicates that the pressure transfer of water injection is slow. Compared with the actual monitoring results, the calculation results of the non-Darcy seepage model are more accurate as shown in Table 1.

  

(a) Darcy flow               (b) non-Darcy flow

Fig. 17 Pressure profile in April 2020

Table 1 Calculation results of Darcy flow and non-Darcy flow

Time	Well	Formation Pressure MPa	Darcy Calculation MPa	Error %	Non-Darcy Calculation MPa	Error %
2020.4	Y120	25.6	39.1	52.7	27.2	6.2
2020.4	Y4-2-14	30.5	40.2	31.8	32.3	5.9
2020.4	Y4-6-15	53.4	42.2	-21.0	51.3	-3.9
2020.4	Y4-4-17	50.7	41.1	-18.9	49.9	-1.6

The test results of Table 2 and Table 3 show that the relative water absorption calculated by non-Darcy in Well Y4-6-15 in July 2009 and Well Y4-4-17 in May 2020 has a good agreement with the actual testing results.

Table 2 Comparison of the relative water absorption in Y4-6-15 Well (2009.7)

Layer	Monitoring results	Darcy Calculation	Relative Error	Non-Darcy Calculation	Relative Error
61	26.3	29.3	11.4	24.8	-5.7
62	14.5	21.4	47.7	15.1	4.4
71	5.6	11.5	105.4	5.3	-6.1
72	53.6	37.8	-29.5	54.8	2.2

Table 3 Comparison of the relative water absorption in Y4-4-17 Well (2020.5)

Layer	Monitoring results	Darcy Calculation	Relative Error	Non-Darcy Calculation	Relative Error
61	23.6	27.3	15.8	22.4	-5.0
62	11.7	18.7	59.7	12.1	3.4
71	22.4	19.4	-13.2	20.9	-6.5
72	42.4	34.6	-18.3	44.6	5.3

As can be seen from Figure 18, the large hierarchical difference of the remaining oil saturation in Well Y120 and the high remaining oil saturation of 7¹ and 7²² layers showed the weak submerged layers and the consistency with the test results of liquid production profile. The results of relative liquid production and water cut stratification show that the non-Darcy calculation method is more consistent with the actual monitoring results, as shown in Table 4.

 

Fig.18 The well profile of remaining oil saturation in Y120 Well and Y4-7-1 Well

Table 4 The testing results of liquid production in Y4-5-14 Well

Layer	Thickness/m	Testing Results		Darcy Calculation				Non-Darcy Calculation
		Liquid Production	Water Content	Liquid Production	Errors	Water Content	Errors	Liquid Production	Errors	Water Content	Errors
61	15.5	35.4	87.6	31.1	-12.1	85.3	-2.6	34.4	-2.8	88.7	1.3
62	8.8	38.1	92.4	17.6	-53.8	91.7	-0.8	38.6	1.3	91.7	-0.8
71	7.7	6.7	10.4	12.5	86.7	54.7	426.0	5.9	-11.9	11.5	10.6
721	13.8	14.5	72.1	26.2	80.6	84.4	17.1	15.5	6.9	71.1	-1.4
722	6.9	5.3	9.6	12.6	137.6	60.4	529.2	5.6	5.7	11.1	15.6

 

 

 

3. In all those colour maps (figs 6, 7, 8, etc), what does the y-axis mean? The authors write the meaning of the x-axis, but not that of the y-axis. 

Response: Thank you very much for finding this flaw in our manuscript. Figures 6-8 are only the comparison results of the oil saturation of the two models at different times, especially the representation of the displacement front interface。

 

4. What is the information provided by figures 4 and 5? They are equal, with uniform colour and the colour bar is useless.

Response: Thank you very much for finding this flaw in our manuscript. Figure 4 and Figure 5 show the porosity and permeability models, respectively. Authors have modified the contents as shown in the line 8 of page 7.

 

5. In equations from 4 to 8, I suggest changing gradP by the actual gradient symbol (nabla). Since the gradient of P is a vector and the argument of the exponential functions in equations 4 to 8 must be a scalar, the parameter b must be a vector. If this is the case, please use a vector symbol in b. Or maybe you want to replace gradP by a pressure drop.

Response: Thank you for your valuable comments. Authors have modified the symbol as ΔP of equations from 4 to 8.

 

7. The authors should provide a list with all the values of parameters used in their simulations.

Response: Thank you very much for finding this flaw in our manuscript. We have marked the meaning of all the symbols below the formula.

 

 

We sincerely thank you for your comments on this research, which has greatly improved the quality of our research. Thank you.

 

 

Attachment File

Author Response File: Author Response.docx

Reviewer 2 Report

A novel equivalent numerical simulation method for non-Darcy seepage flow in low permeability reservoirs Authors Hui Xu , Nannan Liu * , Yan Chen , Yapeng Tian , Wanjun Jiang , Yanfeng He

 

Derived results are correct and interesting. However, in order to publish in a reputed journal, authors need to incorporate the following suggestions to improve the quality of the manuscript.

1-Authors should improve the English as well as the presentation of this paper. There are many typos in the paper and the punctuations after each equation must be revised.

2- What is the physical meaning of Eqs. (3) and (4), why the author investigated these equations.

3) The authors are requested to add more details regarding their original contributions in this manuscript.
4) The article needs to be checked in grammar.
5) Authors need to improve the manuscript's novelty, and I suggest they present the merits of the considered method over techniques available in the literature.

6. Results and discussion must be improved more.

1.     International Journal of Modern Physics B,Vol. 33, No. 24 (2019) 1950283 (24 pages)

7. The following latest studies are very relevant to present article. The authors must read and provide complete information on this topic through including these studies:

1.     Advances in Difference Equations (2018) 2018:66, 1-15.

1.     Mathematical Methods in the Applied Sciences, 44, Issue7 (2021) Pages 5265-5279.  

8. Some graphs for the some utilized examples should be given

 

 

 

 

Author Response

Main comments

REVIEWER #1:

A novel equivalent numerical simulation method for non-Darcy seepage flow in low permeability reservoirs Authors Hui Xu , Nannan Liu * , Yan Chen , Yapeng Tian , Zhenghuai Guo, Wanjun Jiang , Yanfeng He

 Derived results are correct and interesting. However, in order to publish in a reputed journal, authors need to incorporate the following suggestions to improve the quality of the manuscript.

Response: Thank you very much for your valuable comments. Authors discussed the comments of reviewer specifically and revised them very carefully. Detailed suggestions for revision are in the manuscript.

 

1. Authors should improve the English as well as the presentation of this paper. There are many typos in the paper and the punctuations after each equation must be revised. 

Response: Thank you for your valuable comments. Authors have modified the grammar mistakes and typos carefully. The detailed modifications with red color can be clearly seen in resubmitted manuscript.

 

2. What is the physical meaning of Eqs. (3) and (4), why the author investigated these equations.

Response: Thank you very much for finding this flaw in our manuscript. We had supplemented the meaning of parameters in the formula, as shown on the lines 7 and 12 of page 4 in the section 2.1. K represents the absolute permeability; R represents the radius of the throat; r represents radius of the bulk fluid; n represents the number of capillaries per unit area, as shown in lines 1-2 of page 4. The pressure gradient gradP and the thickness of the boundary layer δ were established.

According to previous studies, the nonlinear seepage in low permeability reservoirs is mainly related to the existence of boundary layer in low porosity media. The boundary layer does not flow, so the actual effective seepage radius r in the pores is smaller than the pore diameter R. Therefore, according to equation (3), the correction relationship between the apparent permeability of oil-water seepage and the absolute permeability measured by air can be determined. However, the thickness of boundary layer decreases with the increase of pressure gradient. When the boundary layer is reduced to a certain extent, the influence on seepage can be ignored, and the seepage conforms to the linear characteristics. Therefore, the relationship between apparent permeability and boundary layer thickness can be established through equation (3) and equation (4).

 

3. The authors are requested to add more details regarding their original contributions in this manuscript. 

Response: Thank you very much for finding this flaw in our manuscript. We have added the credit author statement into the page 13, as shown in lines 12-19.

“As the corresponding author，Nannan Liu has conceived the conception/design of the work and supervised the whole experiments. As the first author, Hui Xu wrote the main manuscript text and has approved the final version to be published; AND Xu agree to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated. Following the design, Liu and Jiang guided the writing of the article. Zhenghuai Guo and Yanfeng He provided all of experimental materials and sites in the laboratory. Yan Chen provided assistance with numerical simulation. Yapeng Tian gave the supports of language revision in my manuscript text. All authors discussed the results and critically reviewed the manuscript.”

 

4. The article needs to be checked in grammar.

Response: Thank you for your valuable comments. Authors have modified the grammar mistakes carefully. The detailed modifications with red color can be clearly seen in resubmitted manuscript.

 

5. Authors need to improve the manuscript's novelty, and I suggest they present the merits of the considered method over techniques available in the literature.

Response: Thank you for your valuable comments. Authors have added the section 4 about discussion of the model application from line 7 of page 12 to line 8 of page 14.

4 Discussions

Taking a certain reservoir as an example, the reservoir has a burial depth of 3523m, average permeability of 16.7mD, average porosity of 14.3%, and original formation pressure of 36.4MPa. By comparing the calculation results of Darcy flow and non-Darcy flow in Figure 17, the difference in formation pressure between oil wells and water wells is small (<3MPa) in Darcy flow, while the difference in formation pressure between oil wells and water wells is large in non-Darcy flow, resulting in the pressure around water Wells is serious. The large pressure drop around oil wells indicates that the pressure transfer of water injection is slow. Compared with the actual monitoring results, the calculation results of the non-Darcy seepage model are more accurate as shown in Table 1.

 

(a) Darcy flow               (b) non-Darcy flow

Fig. 17 Pressure profile in April 2020

Table 1 Calculation results of Darcy flow and non-Darcy flow

Time	Well	Formation Pressure MPa	Darcy Calculation MPa	Error %	Non-Darcy Calculation MPa	Error %
2020.4	Y120	25.6	39.1	52.7	27.2	6.2
2020.4	Y4-2-14	30.5	40.2	31.8	32.3	5.9
2020.4	Y4-6-15	53.4	42.2	-21.0	51.3	-3.9
2020.4	Y4-4-17	50.7	41.1	-18.9	49.9	-1.6

The test results of Table 2 and Table 3 show that the relative water absorption calculated by non-Darcy in Well Y4-6-15 in July 2009 and Well Y4-4-17 in May 2020 has a good agreement with the actual testing results. As can be seen from Figure 1, the large hierarchical difference of the remaining oil saturation in Well Y120 and the high remaining oil saturation of 7¹ and 7²² layers showed the weak submerged layers and the consistency with the test results of liquid production profile. The results of relative liquid production and water cut stratification show that the non-Darcy calculation method is more consistent with the actual monitoring results, as shown in Table 4.

Table 2 Comparison of the relative water absorption in Y4-6-15 Well (2009.7)

Layer	Monitoring results	Darcy Calculation	Relative Error	Non-Darcy Calculation	Relative Error
61	26.3	29.3	11.4	24.8	-5.7
62	14.5	21.4	47.7	15.1	4.4
71	5.6	11.5	105.4	5.3	-6.1
72	53.6	37.8	-29.5	54.8	2.2

Table 3 Comparison of the relative water absorption in Y4-4-17 Well (2020.5)

Layer	Monitoring results	Darcy Calculation	Relative Error	Non-Darcy Calculation	Relative Error
61	23.6	27.3	15.8	22.4	-5.0
62	11.7	18.7	59.7	12.1	3.4
71	22.4	19.4	-13.2	20.9	-6.5
72	42.4	34.6	-18.3	44.6	5.3

 

Fig.18 The well profile of remaining oil saturation in Y120 Well and Y4-7-1 Well

Table 4 The testing results of liquid production in Y4-5-14 Well

Layer	Sand Thickness /m	Testing Results		Darcy Calculation				Non-Darcy Calculation
		Liquid Production	Water Content	Liquid Production	Errors	Water Content	Errors	Liquid Production	Errors	Water Content	Errors
61	15.5	35.4	87.6	31.1	-12.1	85.3	-2.6	34.4	-2.8	88.7	1.3
62	8.8	38.1	92.4	17.6	-53.8	91.7	-0.8	38.6	1.3	91.7	-0.8
71	7.7	6.7	10.4	12.5	86.7	54.7	426.0	5.9	-11.9	11.5	10.6
721	13.8	14.5	72.1	26.2	80.6	84.4	17.1	15.5	6.9	71.1	-1.4
722	6.9	5.3	9.6	12.6	137.6	60.4	529.2	5.6	5.7	11.1	15.6

 

6. Results and discussion must be improved more.

Response: Thank you for your valuable comments. Authors have modified the results and conclusion, and added the discussions. The modified contents is shown.

4 Discussions

Taking a certain reservoir as an example, the reservoir has a burial depth of 3523m, average permeability of 16.7mD, average porosity of 14.3%, and original formation pressure of 36.4MPa. By comparing the calculation results of Darcy flow and non-Darcy flow in Figure 17, the difference in formation pressure between oil wells and water wells is small (<3MPa) in Darcy flow, while the difference in formation pressure between oil wells and water wells is large in non-Darcy flow, resulting in the pressure around water Wells is serious. The large pressure drop around oil wells indicates that the pressure transfer of water injection is slow. Compared with the actual monitoring results, the calculation results of the non-Darcy seepage model are more accurate as shown in Table 1.

 

(a) Darcy flow               (b) non-Darcy flow

Fig. 17 Pressure profile in April 2020

Table 1 Calculation results of Darcy flow and non-Darcy flow

Time	Well	Formation Pressure MPa	Darcy Calculation MPa	Error %	Non-Darcy Calculation MPa	Error %
2020.4	Y120	25.6	39.1	52.7	27.2	6.2
2020.4	Y4-2-14	30.5	40.2	31.8	32.3	5.9
2020.4	Y4-6-15	53.4	42.2	-21.0	51.3	-3.9
2020.4	Y4-4-17	50.7	41.1	-18.9	49.9	-1.6

The test results of Table 2 and Table 3 show that the relative water absorption calculated by non-Darcy in Well Y4-6-15 in July 2009 and Well Y4-4-17 in May 2020 has a good agreement with the actual testing results. As can be seen from Figure 1, the large hierarchical difference of the remaining oil saturation in Well Y120 and the high remaining oil saturation of 7¹ and 7²² layers showed the weak submerged layers and the consistency with the test results of liquid production profile. The results of relative liquid production and water cut stratification show that the non-Darcy calculation method is more consistent with the actual monitoring results, as shown in Table 4.

Table 2 Comparison of the relative water absorption in Y4-6-15 Well (2009.7)

Layer	Monitoring results	Darcy Calculation	Relative Error	Non-Darcy Calculation	Relative Error
61	26.3	29.3	11.4	24.8	-5.7
62	14.5	21.4	47.7	15.1	4.4
71	5.6	11.5	105.4	5.3	-6.1
72	53.6	37.8	-29.5	54.8	2.2

Table 3 Comparison of the relative water absorption in Y4-4-17 Well (2020.5)

Layer	Monitoring results	Darcy Calculation	Relative Error	Non-Darcy Calculation	Relative Error
61	23.6	27.3	15.8	22.4	-5.0
62	11.7	18.7	59.7	12.1	3.4
71	22.4	19.4	-13.2	20.9	-6.5
72	42.4	34.6	-18.3	44.6	5.3

 

Fig.18 The well profile of remaining oil saturation in Y120 Well and Y4-7-1 Well

Table 4 The testing results of liquid production in Y4-5-14 Well

Layer	Sand Thickness /m	Testing Results		Darcy Calculation				Non-Darcy Calculation
		Liquid Production	Water Content	Liquid Production	Errors	Water Content	Errors	Liquid Production	Errors	Water Content	Errors
61	15.5	35.4	87.6	31.1	-12.1	85.3	-2.6	34.4	-2.8	88.7	1.3
62	8.8	38.1	92.4	17.6	-53.8	91.7	-0.8	38.6	1.3	91.7	-0.8
71	7.7	6.7	10.4	12.5	86.7	54.7	426.0	5.9	-11.9	11.5	10.6
721	13.8	14.5	72.1	26.2	80.6	84.4	17.1	15.5	6.9	71.1	-1.4
722	6.9	5.3	9.6	12.6	137.6	60.4	529.2	5.6	5.7	11.1	15.6

 

5. Conclusions

The apparent permeability and relative permeability are employed to describe the non-Darcy flow in low permeability reservoirs. The simulation results showed the difference between Darcy flow and non-Darcy flow.

(1) The boundary layer is the key parameter for the nonlinear characteristics of oil-water two-phase seepage in low permeability reservoirs. The smaller pressure gradient and the larger boundary layer thickness will result in the smaller effective seepage radius and the smaller apparent permeability. Therefore, the equation between apparent permeability and pressure gradient constructed in this paper can effectively characterize the nonlinear seepage characteristics of low permeability reservoirs.

(2) Based on the relationship between apparent permeability and pressure gradient, the fractional flow equation is also modified in this paper. The calculation results show that the water cut calculated by the new fractional flow equation is higher than that calculated by Darcy seepage, which is consistent with the actual development of low permeability reservoirs.

(3) The numerical simulation of low permeability reservoir is carried out, which proves that the calculated results are in good agreement with the test results. The key parameters of equation (4) need to be obtained through laboratory experiments.

 

7. The following latest studies are very relevant to present article. The authors must read and provide complete information on this topic through including these studies:

①International Journal of Modern Physics B,Vol. 33, No. 24 (2019) 1950283 (24 pages)

②Advances in Difference Equations (2018) 2018:66, 1-15.

③Mathematical Methods in the Applied Sciences, 44, Issue7 (2021) Pages 5265-5279.

Response: Thank you very much for finding this flaw in our manuscript. We have read these articles carefully, which are indeed related to the theoretical model of our study. In the article, we clarify and draw on the content of these studies, and cite these references.

Kamel Mohamed and Aly Seadawy. Finite volume scheme for numerical simulation of the sediment transport model. International Journal of Modern Physics B, 2019, 33(24):1950283.

Kashif Ali Khan, Aly R. Seadawy & Adil Jhangeer. Numerical appraisal under the influence of the time dependent Maxwell fluid flow over a stretching sheet. Mathematical Methods in the Applied Sciences, 2021, 44(7):5265-5279.

Lu, D., Seadawy, A.R. & Khater, M.M.A. Structure of solitary wave solutions of the nonlinear complex fractional generalized Zakharov dynamical system. Adv. Differ. Equ. 2018, 266:1-18. https://doi.org/10.1186/s13662-018-1734-4

 

8. Some graphs for the some utilized examples should be given

Response: Thank you for your valuable comments. Authors have added the section 4 about discussion of the model application from line 7 of page 12 to line 8 of page 14.

 

We sincerely thank you for your comments on this research, which has greatly improved the quality of our research. Thank you.

 

Author Response File: Author Response.docx

Reviewer 3 Report

Title: A novel equivalent numerical simulation method for non-Darcy seepage flow in low permeability reservoirs

1. It is important to justify equation 4 with references

2. In equation 4 all variables must be described

3. It is important to justify the inclusion of equation 9 with references.

4. All variables should be described immediately after their first occurrence.

5. It would be appropriate to give a better physical explanation before giving the equation 12

6. In the paragraph before equation 14, it is important to include references that justify the incorporation of equation 14.

7. In the results section, it is necessary to include other works or references that justify and strengthen the results obtained by the authors.

8. Regarding the conceptual model of line 7 on page 7, its description should be improved.

Author Response

Main comments

REVIEWER #1:

In this manuscript, the permeability of a porous media is calculated using mathematical modelling of the non-Darcy flow which is solved using commercial software. The authors found a small difference from the results using Darcy seepage. The results and method are interesting and should be useful to the community involved in oil extraction. But the description of the method and results should be improved. I list below a few suggestions and questions to be addressed in the manuscript.

Response: Thank you very much for your valuable comments. Authors discussed the comments of reviewer specifically and revised them very carefully. Detailed suggestions for revision are in the manuscript.

 

1. It is important to justify equation 4 with references 

Response: Thank you for your valuable comments. The pressure gradient and the thickness of the boundary layer δ are measured by using the equal diameter of micro-tubes at laboratory conditions, and the exponential relationship between the thickness of the boundary layer and the pressure gradient are concluded as follows (Liu et al., 2015; Li et al., 2018).

                               （4）

Liu, H., Wu, S. The Numerical Simulation for Multi-Stage Fractured Horizontal Well in Low permeability reservoirs Based on Modified Darcy's Equation. Paper presented at the SPE/IATMI Asia Pacific Oil & Gas Conference and Exhibition, Nusa Dua, Bali, Indonesia, October 2015. doi: https://doi.org/10.2118/176269-MS.

Li, H., Guo, H., Yang, Z., et al. Boundary retention layer influence on permeability of tight reservoir. Journal of Petroleum Science and Engineering, 2018, 168:562-568.

 

2. In equation 4 all variables must be described

Response: Thank you very much for finding this flaw in our manuscript. We had supplemented the meaning of parameters in the formula, as shown on the lines 7 and 12 of page 4 in the section 2.1. K represents the absolute permeability; R represents the radius of the throat; r represents radius of the bulk fluid; n represents the number of capillaries per unit area, as shown in lines 1-2 of page 4. The pressure gradient gradP and the thickness of the boundary layer δ were established.

According to previous studies, the nonlinear seepage in low permeability reservoirs is mainly related to the existence of boundary layer in low porosity media. The boundary layer does not flow, so the actual effective seepage radius r in the pores is smaller than the pore diameter R. Therefore, according to equation (3), the correction relationship between the apparent permeability of oil-water seepage and the absolute permeability measured by air can be determined. However, the thickness of boundary layer decreases with the increase of pressure gradient. When the boundary layer is reduced to a certain extent, the influence on seepage can be ignored, and the seepage conforms to the linear characteristics. Therefore, the relationship between apparent permeability and boundary layer thickness can be established through equation (3) and equation (4).

 

3. It is important to justify the inclusion of equation 9 with references 

Response: Thank you very much for finding this flaw in our manuscript. In order to facilitate the calculation, this paper adopts the low-velocity non-Darcy flow formula established:

                                                             （9）

Among them， , b, b₁ are constant (Lei et al., 2019; Wang et al., 2020.

Lei, H., He, L., Li, R., et al. Effects of boundary layer and stress sensitivity on the performance of low-velocity and one-phase flow in a shale oil reservoir: Experimental and numerical modeling approaches. Journal of Petroleum Science and Engineering, 2019, 180:186-196.

Wang, H., Tian, L., Gu, D., et al. Method for Calculating Non-Darcy Flow Permeability in Tight Oil Reservoir. Transport in Porous Media, 2020, 133: 357-372.

 

4. All variables should be described immediately after their first occurrence.

Response: Thank you for your valuable comments. We have marked the meaning of all the symbols below the formula.

 

5. It would be appropriate to give a better physical explanation before giving the equation 12

Response: Thank you for your valuable comments. The author uses Figure 2 and the corresponding text to show the meaning of the equation

Through the above method, the relative permeability curve of the studied sample was corrected.

 

   Fig. 2 Comparison of oil-water relative permeability curves of the two models

The results of Fig.2 showed that the newly calculated oil-water relative permeability were slightly higher than the previous results calculated according to the Darcy flow model.

 

6. In the paragraph before equation 14, it is important to include references that justify the incorporation of equation 14.

Response: Thank you for your valuable comments. Authors have modified the contents and added some references.

 

7. In the results section, it is necessary to include other works or references that justify and strengthen the results obtained by the authors.

Response: Thank you very much for finding this flaw in our manuscript. We have read these articles carefully, which are indeed related to the theoretical model of our study. In the article, we clarify and draw on the content of these studies, and cite these references. In addition, the discussion part as shown in section 4 is added to explain the characteristics of the model based on the actual case of oil field.

Kamel Mohamed and Aly Seadawy. Finite volume scheme for numerical simulation of the sediment transport model. International Journal of Modern Physics B, 2019, 33(24):1950283.

Kashif Ali Khan, Aly R. Seadawy & Adil Jhangeer. Numerical appraisal under the influence of the time dependent Maxwell fluid flow over a stretching sheet. Mathematical Methods in the Applied Sciences, 2021, 44(7):5265-5279.

Lu, D., Seadawy, A.R. & Khater, M.M.A. Structure of solitary wave solutions of the nonlinear complex fractional generalized Zakharov dynamical system. Adv. Differ. Equ. 2018, 266:1-18. https://doi.org/10.1186/s13662-018-1734-4.

4 Discussions

Taking a certain reservoir as an example, the reservoir has a burial depth of 3523m, average permeability of 16.7mD, average porosity of 14.3%, and original formation pressure of 36.4MPa. By comparing the calculation results of Darcy flow and non-Darcy flow in Figure 17, the difference in formation pressure between oil wells and water wells is small (<3MPa) in Darcy flow, while the difference in formation pressure between oil wells and water wells is large in non-Darcy flow, resulting in the pressure around water Wells is serious. The large pressure drop around oil wells indicates that the pressure transfer of water injection is slow. Compared with the actual monitoring results, the calculation results of the non-Darcy seepage model are more accurate as shown in Table 1.

  

(a) Darcy flow               (b) non-Darcy flow

Fig. 17 Pressure profile in April 2020

Table 1 Calculation results of Darcy flow and non-Darcy flow

Time	Well	Formation Pressure MPa	Darcy Calculation MPa	Error %	Non-Darcy Calculation MPa	Error %
2020.4	Y120	25.6	39.1	52.7	27.2	6.2
2020.4	Y4-2-14	30.5	40.2	31.8	32.3	5.9
2020.4	Y4-6-15	53.4	42.2	-21.0	51.3	-3.9
2020.4	Y4-4-17	50.7	41.1	-18.9	49.9	-1.6

The test results of Table 2 and Table 3 show that the relative water absorption calculated by non-Darcy in Well Y4-6-15 in July 2009 and Well Y4-4-17 in May 2020 has a good agreement with the actual testing results.

Table 2 Comparison of the relative water absorption in Y4-6-15 Well (2009.7)

Layer	Monitoring results	Darcy Calculation	Relative Error	Non-Darcy Calculation	Relative Error
61	26.3	29.3	11.4	24.8	-5.7
62	14.5	21.4	47.7	15.1	4.4
71	5.6	11.5	105.4	5.3	-6.1
72	53.6	37.8	-29.5	54.8	2.2

Table 3 Comparison of the relative water absorption in Y4-4-17 Well (2020.5)

Layer	Monitoring results	Darcy Calculation	Relative Error	Non-Darcy Calculation	Relative Error
61	23.6	27.3	15.8	22.4	-5.0
62	11.7	18.7	59.7	12.1	3.4
71	22.4	19.4	-13.2	20.9	-6.5
72	42.4	34.6	-18.3	44.6	5.3

As can be seen from Figure 18, the large hierarchical difference of the remaining oil saturation in Well Y120 and the high remaining oil saturation of 7¹ and 7²² layers showed the weak submerged layers and the consistency with the test results of liquid production profile. The results of relative liquid production and water cut stratification show that the non-Darcy calculation method is more consistent with the actual monitoring results, as shown in Table 4.

 

Fig.18 The well profile of remaining oil saturation in Y120 Well and Y4-7-1 Well

Table 4 The testing results of liquid production in Y4-5-14 Well

Layer	Thickness/m	Testing Results		Darcy Calculation				Non-Darcy Calculation
		Liquid Production	Water Content	Liquid Production	Errors	Water Content	Errors	Liquid Production	Errors	Water Content	Errors
61	15.5	35.4	87.6	31.1	-12.1	85.3	-2.6	34.4	-2.8	88.7	1.3
62	8.8	38.1	92.4	17.6	-53.8	91.7	-0.8	38.6	1.3	91.7	-0.8
71	7.7	6.7	10.4	12.5	86.7	54.7	426.0	5.9	-11.9	11.5	10.6
721	13.8	14.5	72.1	26.2	80.6	84.4	17.1	15.5	6.9	71.1	-1.4
722	6.9	5.3	9.6	12.6	137.6	60.4	529.2	5.6	5.7	11.1	15.6

 

8. Regarding the conceptual model of line 7 on page 7, its description should be improved.

Response: Thank you very much for finding this flaw in our manuscript. Figure 4 and Figure 5 show the porosity and permeability models, respectively. Authors have modified the contents as shown in the line 8 of page 7.

 

We sincerely thank you for your comments on this research, which has greatly improved the quality of our research. Thank you.

 

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

The authors did some work to improve the manuscrispt, but not all points have been appropriately addressed.

 

In figs 4 and 5, I suggest removing the color bar since it gives no information about the (uniform) colors.

 

The authors write in the reply letter that the references Phys. Fluids 34, 023102 (2022) and Physics of Fluids 34, 103303 (2022) are discussed in the introduction, but I could not find them. Please include them in the list of references.

 

In point 5 of the previous letter, the authors replaced the symbol of gradient with the symbol \delta, which is appropriate. However, in the text, it is still written pressure gradient as in line 10 of page 4. It should be pressure difference.

Author Response

Minor comments

REVIEWER #1:

The authors did some work to improve the manuscript, but not all points have been appropriately addressed.

Response: Thank you very much for your valuable comments. Authors discussed the comments of reviewer specifically and revised them very carefully. Detailed suggestions for revision are in the manuscript.

 

1. In figs 4 and 5, I suggest removing the color bar since it gives no information about the (uniform) colors. 

Response: Thank you for your valuable comments. Authors have modified the information in Figures 4 and 5, and importantly modify the color as shown in the revised manuscript.

     

Fig.4 Porosity model                 Fig.5 Permeability model

A conceptual model of 20×20×5 was established, and the reverse nine-point well pattern was used for water injection development (Fig.4-5).

 

2. The authors write in the reply letter that the references Phys. Fluids 34, 023102 (2022) and Physics of Fluids 34, 103303 (2022) are discussed in the introduction, but I could not find them. Please include them in the list of references.

Response: Thank you very much for finding this flaw in our manuscript. In the introduction, we have modified the contents and cited the important references as shown in lines 28-32 of page 8.

At present, the models for characterizing nonlinear seepage flow mainly include: threshold pressure gradient model (Ding et al, 2014; Hao et al, 2008; Zeng et al, 2018; Sedahmed et al., 2022) and nonlinear models based on equal diameter capillary bundle model or fractal model (Xu et al, 2013). Seepage fluid is the fluid in seepage environment, including bulk and boundary fluid (Sedahmed et al., 2022).

 

3. In point 5 of the previous letter, the authors replaced the symbol of gradient with the symbol \delta, which is appropriate. However, in the text, it is still written pressure gradient as in line 10 of page 4. It should be pressure difference. 

Response: Thank you very much for finding this flaw in our manuscript. We have modified the contents from lines 10-14 of page 4 in the revised manuscript。

The pressure difference ΔP and the thickness of the boundary layer δ were established by using microtubes of equal diameter under laboratory conditions, and found that the thickness of the boundary layer and the pressure difference are exponential relationship.

                                                                                                     （4）

Then the radius of the bulk fluid r=R-δ, that is, the radius of the bulk fluid and the pressure difference also have an exponential relationship. Can be expressed as:

                                                                                           （5）

 

 

We sincerely thank you for your comments on this research, which has greatly improved the quality of our research. Thank you.

 

Author Response File: Author Response.docx

Reviewer 2 Report

Accepted in this form

Author Response

Thank you very much

Reviewer 3 Report

The authors have attended to my recommendations and improved the quality of the work.

Author Response

Thank you very much