Tight sandstone reservoirs play an important role in the oil industry. The permeability of tight sandstone reservoir generally has stronger stress sensitivity than that of conventional reservoir because of the latter’s poor physical properties. However, the production analysis of tight sandstone reservoir did not fully considered the stress-sensitive permeability yet.
This paper proposed a production analysis method considering the stress- sensitive permeability.
This paper firtstly investigated the stress sensitivity characteristics and the effect of stress-sensitive permeability on a tight reservoir. Decline-type curves that consider stress-sensitive permeability are then established, and a systematic analysis method was built for the production analysis to obtain the single-well controlled dynamic reserves and reservoir physical properties.
A field analysis was performed in combination with Block Yuan-284 of Changqing Oilfield. Results show that with the reduction of reservoir pressure, stress sensitivity leads to the decline in reservoir permeability and the increase in seepage resistance, thus reducing the actual single-well controlled reserve and radius.
By utilizing the analysis method based on the decline curves, we can effectively predict the single-well controlled dynamic reserves of such reservoirs and evaluate the characteristic parameters of reservoirs.
Unconventional oil and gas resources have become an important aspect of current oil and gas production [1R.W. Bentley, "Global oil & gas depletion: an overview", Energy Policy, vol. 30, no. 3, pp. 189-205, 2002.
[http://dx.doi.org/10.1016/S0301-4215(01)00144-6] , 2S. Sorrell, J. Speirs, J.R. Bentley, A. Brandt, and R. Miller, "Global oil depletion: A review of the evidence", Energy Policy, vol. 38, no. 9, pp. 5290-5295, 2010.
[http://dx.doi.org/10.1016/j.enpol.2010.04.046] ], and stress sensitivity is an important feature of unconventional oil and gas reservoirs, such as tight gas reservoir, tight oil reservoir, and shale gas [3S.A. Holditch, "Tight gas sands", SPE Journal of Petroleum Technology, vol. 58, no. 6, pp. 86-93, 2006.
[http://dx.doi.org/10.2118/103356-JPT] -8L. Zhu, Y. Ji, T. Yang, and X. Li, "Analysis on production of coal bed methane considering the change in permeability of coal rock", Open Petroleum Engineering Journal, vol. 9, pp. 289-298, 2016.
[http://dx.doi.org/10.2174/1874834101609010289] ]. These reservoirs have poor physical properties, and the effect of stress sensitivity is also relatively more significant. Production analysis using decline-type curves is efficient in obtaining single-well controlled reserve and reservoir physical properties. This method is more advanced than the volumetric method because of the strong heterogeneity, very complex pore structure, and certain particularity in the seepage mechanism of unconventional reservoir [9S. Wang, Research on Predication Method for Dynamic Reserves of Low-permeability Gas Reservoir., China University of Petroleum (Beijing): Beijing, 2009.-13C.L. Cipolla, E. Lolon, J. Erdle, and V. Tathed, "Modeling well performance in Shale-Gas Reservoirs", Proceeding of SPE/EAGE Reservoir Characterization and Simulation Conference, Society of Petroleum, Engineers, Abu Dhabi, UAE, 2009.
[http://dx.doi.org/10.2118/125532-MS] ]. The accurate evaluation and prediction of dynamic reserves is the basis of making a reasonable development program, and it plays a vital role in the reasonable and efficient exploitation of unconventional oil and gas resources.
The current production analysis methods do not consider the effect of stress sensitivity, and thus their calculation results produce a large deviation. This article brings stress-sensitive permeability into decline-type curves to obtain a more accurate production analysis, including single-well controlled dynamic reserves, to make the calculation result correspond with the actual situation of reservoir, and to effectively guide the development of stress-sensitive oil–gas reservoirs.
Pore pressure reduces constantly in the production of oil and gas, and thus effective stress increases accordingly. The corresponding relation is expressed in the following equation [14V.M. Dobynin, "Effect of overburden pressure on some properties of sandstones", SPE Journal, vol. 2, no. 4, pp. 360-366, 1962.
[http://dx.doi.org/10.2118/461-PA] ]:
(1) |
where S is the total stress, σ is the effective stress (matrix stress and pressure between particles), a is the stress sensitivity correction coefficient, and is the fluid pressure or pore pressure.
Formula 1 shows that under the condition of unchanged total stress, the change in fluid pressure leads to the change in effective stress, which will correspondingly result in the change in rock physical property.
A permeability stress sensitivity experiment was conducted for the ultra-low permeability sandstone from Block Yuan-284 of Changqing Oilfield. The core belongs to the Triassic Chang-6 Reservoir Group of Ordos Basin. The main fragment component of the sandstone is feldspar lithic sandstone, which is fine-medium grain and very fine-fine grain. The stress sensitivity characteristics of sandstone were analyzed in the research area. The confining pressure was maintained, and the displacement pressure was changed to simulate the process in which the formation rock was compressed during exploitation. The stress sensitivity experiment was conducted using eight cores. Fig. (1) presents the curves of k/k_{i} changing with pore pressure, where k/k_{i} is the ratio of permeability and initial permeability under different pore pressure states. Fig. (1) also illustrates the experiment results. We can fit the results with the exponential law curves in which the fitting compliance degrees are all above 0.9.
Fig. (1) Permeability stress-sensitive experiment results for chang-6 tight sandstones. |
Blasingame’s decline-type curve [15T.A. Blasingame, T.L. McCray, and W.J. Lee, "Decline curve analysis for variable pressure drop/variable flowrate systems", In: Proceeding of SPE Gas Technology Symposium, Society of Petroleum Engineers: Houston, Texas, 1991.
[http://dx.doi.org/10.2118/21513-MS] , 16J. Palacio, and T. Blasingame, "Decline curve analysis using type curves: analysis of gas well production data", In: Proceeding of SPE Joint Rocky Mountain Regional and Low Permeability Reservoirs Symposium, Society of Petroleum Engineers: Denver, USA, 1993.
[http://dx.doi.org/10.2118/25909-MS] ] is an important method developed on the basis of Fetkovich’s decline-type curve [17M.J. Fetkovich, "Decline curve analysis using type curves", SPE Journal of Petroleum Technology, vol. 32, no. 06, pp. 1065-1077, 1980.
[http://dx.doi.org/10.2118/4629-PA] -19D. Yin, D. Wang, C. Zhang, and Y. Duan, "Shale gas productivity predicting model and analysis of influence factor", Open Petroleum Engineering Journal, vol. 8, pp. 203-207, 2015.
[http://dx.doi.org/10.2174/1874834101508010203] ] for production data analysis to evaluate single-well controlled dynamic reserves and reservoir parameters. However, to the best of our knowledge, this method currently does not consider the effect of stress sensitivity. In this study, we established a physical and mathematical model that considers stress-sensitive permeability to obtain a new series of decline-type curves that allow fr the stress-sensitivity of permeability.
Assuming the presence of a vertical well with a circular reservoir (Fig. 2), the formation and the fluid satisfy the following conditions:
Fig. (2) Schematic Diagram of a Vertical Well in a Circular Closed Formation. |
The basic flow equation of a homogeneous reservoir is
(2) |
When the permeability’s change in pressure follows an exponential law, the permeability modulus is effective in characterizing the permeability variation law. The relation of permeability change with effective stress can be expressed as follows [20O.A. Pedrosa, "Pressure transient response in stress-sensitive formations", Proceeding of SPE California Regional Meeting, , 1986
[http://dx.doi.org/10.2118/15115-MS] ]:
(3) |
where γ is the permeability modulus of 1 MPa.
The integral for Equation (3) is
(4) |
where k_{i} is the reservoir permeability under the original formation pressure, mD.
Equation (4) is the expression of the exponential change law of permeability with pressure.
By bringing (4) into (2), we obtain the following eq:
(5) |
The above formula is the basic percolation differential equation of a tight sandstone reservoir that considers the stress-sensitive effect of permeability. The following dimensionless variables are introduced:
(6) |
(7) |
(8) |
The following dimensionless permeability modulus is defined as:
(9) |
According to flow mechanics theory, the mathematical model of a homogeneous closed reservoir that considers stress sensitivity can be established.
The flow control equation is:
(10) |
The initial conditions are expressed as:
(11) |
The internal boundary are expressed as:
(12) |
(13) |
The external boundary conditions are expressed as:
(14) |
To calculate the above model, we used the perturbation technique for calculation and introduced the transformation relation as follows:
(15) |
The following perturbation techniques are applied as follows:
(16) |
(17) |
(18) |
(19) |
Given the small dimensionless permeability modulus, the zero-order perturbation solution is used. Thus, Equations (10–14) are expressed as follows:
(20) |
η_{OD}(r_{D},0) = 0 | (21) |
(22) |
(23) |
(24) |
By conducting the Laplace transform for Equation (20), its Laplace spatial solution can be obtainedas:
(25) |
(26) |
(27) |
In the above functions, K_{O} and K_{1} are a zero-order and a first-order modified Bessel function, respectively, g is the Laplace Variable, L^{-1} is the Laplace inverse transform, and O(γ_{D}), η_{wD} is the allowance above zero-order solution.
By transforming the above analytical solution into a real spatial solution through the Stehfest inversion algorithm, we obtained P_{wD}~t_{D}. A typical decline-type curve mainly includes the decline-type curve dimensionless production proposed by Fetkovich, q_{Dd}, the dimensionless material balance time t_{Dd} proposed by Blasingame, and the dimensionless production integral and its derivative proposed by McCray, q_{Ddi} and q_{Did}.
The decline-type curve dimensionless production and time are defined as follows:
(28) |
(29) |
Here, r_{we}=r_{w}·e^{-s}.
The auxiliary drawing function generally includes the dimensionless production integral and its derivative proposed by McCray. It is introduced for practical application and is conducive to improving the efficiency and accuracy of field engineers to explain and analyze production data. In actual use, the method of fitting three groups of data simultaneously is generally adopted to obtain the best fitting effect.
The following is the dimensionless production integral:
(30) |
The following is the dimensionless production integral derivative:
(31) |
By drawing q_{Dd} t_{Dd}, q_{Ddi} t_{Dd} and q_{Ddid} t_{Dd} on the log-log plot, decline-type curves for vertical wells in homogeneous reservoirs of stress-sensitive reservoir are formed.
Fig. (3) presents the decline-type curve under different dimensionless permeability moduli γ_{D} when the dimensionless reservoir radius r_{eD} is 5000, and the values of γ_{D} are 0, 0.0015, 0.003, and 0.0045 respectively. The difference between curves in the unsteady stage in the figure is small, and it is mainly in the pseudo steady stage. The curves are shown when γ_{D} is 0, 0.0015, 0.003, and 0.0045 from right to left.
Fig. (3) Decline-type Curves for Vertical Wells in Stress-sensitive Reservoirs (Different Permeability Moduli). |
Fig. (4) illustrates the decline-type curves under different dimensionless reservoir radii r_{eD} when the dimensionless permeability modulus γ_{D} is 0.003 and the values of r_{eD} are 200, 500, 2000, and 5000, respectively. The difference exists both in the unsteady stage and in the quasi-steady stage. The curves are shown when r_{eD} is 200, 500, 2000, and 5000 from top to bottom.
Figs. (3) and (4) summarize the following basic laws of the decline-type curves for vertical wells in stress-sensitive reservoirs:
Fig. (4) Decline-type Curves for Vertical Wells in Stress-sensitive Reservoirs (Different Reservoir Radii). |
An instance analysis was conducted by selecting a well in Block Yuan 284 that belongs to Ordos Basin. Block Yuan 284’s purpose of horizon is the Chang-6 formation. The effective thickness of this well was 10.94 m, with porosity being 0.13, irreducible water saturation being 43%, and well radius being 0.1 m. Its production history is illustrated in Fig. (5). The history of this well was studied first. No other operation measures wereadopted after the fracturing production of this well. Its production and pressure data were monitored daily that are representative of relatively high accuracy and good correlation (Fig. 5). The water injection enhancement productivity effect was poor, pressure and production declined with time, and the oil well remained at the depletion development state.
This well does not undergo unstable pressure testing after being put into production. Therefore, we mainly used its production data to judge the model as shown in Fig. (6).
The basic model characteristics are as follows:
Fig. (5) Production History of the Target Well. |
Fig. (6) Curves for the Normalized Productivity Index Function vs. Material Balance Time. |
To sum up, the basic model of this well is a vertical well within a stress-sensitive closed reservoir. This well can be the model of a vertical well within a stress-sensitive closed reservoir. Through the exponential curve fitting, we obtained its permeability modulus of 0.0097 (1/MPa). The dimensionless permeability modulus of 0.0031 was calculated by Equation (7), to select a reasonable decline chart series from it.
To calculate the material balance time, the material balance time of the oil well is defined as follows:
(32) |
The calculated normalized production, production integral, and its derivative are defined as follows:
Normalized production
(33) |
Production integral (defined as the average production of well at a certain time)
(34) |
Production integral derivative
(35) |
A log-log plot was drawn for the curve of three production functions versus material balance time, with its coordinate scale, the same as that of the typical decline chart, and a typical decline chart with γ_{D} equal to 0.003 was used. Through fitting, the production curve can be better fitted with a typical decline-type curve with a dimensionless reservoir radius r_{eD} equal to 200 (Fig. 7).
Fig. (7) Fitting Curve of the Typical Decline-type Chart. |
The following fitting parameters can be obtained through fitting:
r_{eD}=200
(t_{Dd})_{MP}=1.0, (t_{c})_{MP}=6000(h)
(q_{Dd})_{MP}=1.0, (q_{O}/ p)_{MP}=0.38(m_{3}/ (dMPa))
The typical curve analysis and calculation result are as follows:
Single-well controlled dynamic reserves
(36) |
Single-well controlled dynamic reserves expressed by quality
(37) |
Reservoir discharge area
(38) |
Equivalent control radius
(39) |
Original effective permeability
(40) |
Skin coefficient
s = -ln[(r_{e} /r_{eD})/ r_{w}] = -2.01. | (41) |
After the model and the parameters were determined, the production history was required to be fit to verify the correctness of the model and the parameters. The well is a vertical well model within a stress-sensitive closed reservoir, and the basic parameters for fitting have been determined through the chart. The history fitting curve is presented in Fig. (8).
Fig. (8) Production History Fitting Curve. |
Fig. (8) indicates that the fitting degree is high, and thus the adopted model and parameters have high credibility and can correctly reflect the actual status of the oil well.
The authors confirm that this article content has no conflict of interest.
This work is supported by the National Natural Science Foundation of China (Grant No. 51604287, 51490652).
[1] | R.W. Bentley, "Global oil & gas depletion: an overview", Energy Policy, vol. 30, no. 3, pp. 189-205, 2002. [http://dx.doi.org/10.1016/S0301-4215(01)00144-6] |
[2] | S. Sorrell, J. Speirs, J.R. Bentley, A. Brandt, and R. Miller, "Global oil depletion: A review of the evidence", Energy Policy, vol. 38, no. 9, pp. 5290-5295, 2010. [http://dx.doi.org/10.1016/j.enpol.2010.04.046] |
[3] | S.A. Holditch, "Tight gas sands", SPE Journal of Petroleum Technology, vol. 58, no. 6, pp. 86-93, 2006. [http://dx.doi.org/10.2118/103356-JPT] |
[4] | F.O. Jones, and W.W. Owens, "A laboratory study of low-permeability gas sands", SPE Journal of Petroleum Technology, vol. 32, no. 9, pp. 1631-1640, 1980. [http://dx.doi.org/10.2118/7551-PA] |
[5] | J. Vairogs, C.L. Hearn, D.W. Dareing, and V.W. Rhoades, "Effect of rock stress on gas production from low-permeability reservoirs", SPE Journal of Petroleum Technology, vol. 23, no. 9, pp. 1161-1167, 1971. [http://dx.doi.org/10.2118/3001-PA] |
[6] | L. Wang, H. Liu, S. Zhen, X. Liu, J. Li, and Y. Song, "Research on quantitative interpretation for stress sensitivity of low-permeability reservoir", Acta Petrolei Sinica, vol. 30, no. 01, pp. 96-99, 2009. |
[7] | C. Jiao, S. He, Q. Xie, D. Gu, H. Zhu, L. Sun, and H. Liu, "Experiment for stress sensitivity of ultra-low permeability sandstone reservoir", Acta Petrolei Sinica, vol. 32, no. 3, pp. 489-494, 2011. |
[8] | L. Zhu, Y. Ji, T. Yang, and X. Li, "Analysis on production of coal bed methane considering the change in permeability of coal rock", Open Petroleum Engineering Journal, vol. 9, pp. 289-298, 2016. [http://dx.doi.org/10.2174/1874834101609010289] |
[9] | S. Wang, Research on Predication Method for Dynamic Reserves of Low-permeability Gas Reservoir., China University of Petroleum (Beijing): Beijing, 2009. |
[10] | W. Wang, P. Shen, X. Ma, L. Fan, and J. Tang, "Determination of dynamic reserves for heterogeneous complicated low-permeability gas reservoir", Natural Gas Industry, vol. 24, no. 07, pp. 80-82, 2004. |
[11] | Y. Hao, and X. Xu, "Correct calculation of dynamic reserves for low-permeability gas reservoir", Acta Petrolei Sinica, vol. 29, no. 5, pp. 66-68, 2002. |
[12] | D. Ilk, J.A. Rushing, and T.A. Blasingame, "Integration of production analysis and rate-time analysis via parametric correlations -- theoretical considerations and practical applications", Proceeding of SPE Hydraulic Fracturing Technology Conference, , 2011 [http://dx.doi.org/10.2118/140556-MS] |
[13] | C.L. Cipolla, E. Lolon, J. Erdle, and V. Tathed, "Modeling well performance in Shale-Gas Reservoirs", Proceeding of SPE/EAGE Reservoir Characterization and Simulation Conference, Society of Petroleum, Engineers, Abu Dhabi, UAE, 2009. [http://dx.doi.org/10.2118/125532-MS] |
[14] | V.M. Dobynin, "Effect of overburden pressure on some properties of sandstones", SPE Journal, vol. 2, no. 4, pp. 360-366, 1962. [http://dx.doi.org/10.2118/461-PA] |
[15] | T.A. Blasingame, T.L. McCray, and W.J. Lee, "Decline curve analysis for variable pressure drop/variable flowrate systems", In: Proceeding of SPE Gas Technology Symposium, Society of Petroleum Engineers: Houston, Texas, 1991. [http://dx.doi.org/10.2118/21513-MS] |
[16] | J. Palacio, and T. Blasingame, "Decline curve analysis using type curves: analysis of gas well production data", In: Proceeding of SPE Joint Rocky Mountain Regional and Low Permeability Reservoirs Symposium, Society of Petroleum Engineers: Denver, USA, 1993. [http://dx.doi.org/10.2118/25909-MS] |
[17] | M.J. Fetkovich, "Decline curve analysis using type curves", SPE Journal of Petroleum Technology, vol. 32, no. 06, pp. 1065-1077, 1980. [http://dx.doi.org/10.2118/4629-PA] |
[18] | M. Fetkovich, M. Vienot, M. Bradley, and U.G. Kiesow, "Decline curve analysis using type curves: case histories", SPE Formation Evaluation, vol. 2, no. 4, pp. 637-656, 1987. [http://dx.doi.org/10.2118/13169-PA] |
[19] | D. Yin, D. Wang, C. Zhang, and Y. Duan, "Shale gas productivity predicting model and analysis of influence factor", Open Petroleum Engineering Journal, vol. 8, pp. 203-207, 2015. [http://dx.doi.org/10.2174/1874834101508010203] |
[20] | O.A. Pedrosa, "Pressure transient response in stress-sensitive formations", Proceeding of SPE California Regional Meeting, , 1986 [http://dx.doi.org/10.2118/15115-MS] |