A new water flooding characteristic curve at ultra-high water cut stage

A large number of field practices show that the water flooding characteristic curve will appear up-warping phenomenon in ultra-high water cut stage, and the conventional water flooding characteristic curve is difficult to accurately characterize. In this work, a new expression of oil–water relative permeability ratio (Kro/Krw) and water saturation (Sw) is proposed based on the statistical analysis of experimental data of oil–water relative permeability at high displacement multiples. The new expression has a simpler form and fewer unknown parameters. The results show that the expression can accurately fit the later section of the conventional relative permeability ratio curve, and the correlation coefficient is above 0.996. On this basis, a new type of water flooding characteristic curve suitable for the whole process of water flooding reservoir development is established by combining the reservoir engineering method. Numerical simulation and field application show that the new curve has higher accuracy and wider applicability than conventional curve. The prediction error of recoverable reserves calculated by the new curve is only 0.22%, and the error of geological reserves is less than 5%. According to the comparison between the actual data and the predicted data, the actual cumulative oil production is 2.579 × 106 t, the predicted by the new curve is 2.569 × 106 t, the actual ultimate oil recovery is 50.235%, and the predicted is 50.04%. The predicted value is consistent with the actual one. It provides a more reliable method to accurately predict reservoir development indexes and guides the oilfield’s subsequent decision-making.


Introduction
The water flooding characteristic curve has been widely used in water flooding oilfields, which can be used to predict the geological reserves, recoverable reserves and oil recovery of the oilfield, as well as future production performance of the oilfield (Zhifang et al. 1990;Chen 2002;Ke and Jianwen 2013;Yuanqian and Weirui 2014;. The theoretical basis of traditional water flooding characteristic curve derivation is the semi-log linear relationship between oil-water relative permeability ratio (K ro /K rw ) and water saturation (S w ) (Craft and Hawkins 1959;Wenjun and Zhengke 2000;Goda et al. 2007; Roghanian and Reza 2012;Can and Kabir 2014;Cuo and Chengfang 2017;Li et al. 2021). However, with the continuous development of water flooding, when the reservoir enters the ultra-high water cut stage, the semi-log relationship between oil-water relative permeability ratio and water saturation is no longer completely linear, and the curve will bend downward (Bondar and Blasingame 2002;Jian 2013;Zhaojie and Fengpeng 2013;Chunlei 2014;Cao et al. 2021). Besides, a large number of oilfield practices have shown that the water flooding characteristic curve will also occur upward warping (Zhaojie and Fengpeng 2013;Baohong 2015;Chuanzhi and Duanping 2015;Cuo 2017;Hongen and Sibo 2019;Qi et al. 2022), which results in the application of conventional water flooding characteristic curve to predict the development indexes will have a non-negligible error.
Based on the phenomenon, some experts derived the water flooding characteristic curve suitable for the ultra-high water cut stage by the method of reverse or fractal theory (Guan and Wen-rui 2019; Andersen et al. 2020;Liu et al. 2021). And most scholars proposed some special mathematical expression forms based on the conventional water flooding characteristic curve expression, such as polynomial function term (Zhaojie and Fengpeng 2013;Feng et al. 2014;Xiaolin and Zhiping 2015;Zhinbin 2015) or exponential function term (Dekang 2017;Jianwei and Yigen 2020). The water flooding characteristic curve generated in these ways will cause the left and right ends of the expression to be unequal when the water saturation approaches the extreme value. In addition, some experts have appended some special mathematical functions to the B-type water flooding characteristic curve, such as logarithmic function term (Chuanzhi and Duanping 2015;Zhinbin 2015;Zhibin 2016;Jiqiang and Shuhong 2017), trigonometric function form (Wang et al. 2013;Ke and Jinqing 2019) and infinite term rational polynomial form (Haohan 2019). Although the problem that the left and right ends of the expression are not equal has been solved, the "upward warping" phenomenon of the water flooding characteristic curve in ultra-high water cut stage cannot be accurately characterized. Meanwhile, there are also problems of complex relationship representation, low fitting accuracy and large prediction error.
In this study, through statistical analysis of the relative permeability experimental data of high displacement multiples, a new characterization formula describing the relationship curve of oil-water relative permeability ratio is proposed on the basis of the semi-log linear relationship between K ro /K rw and S w . On this basis, a new water flooding characteristic curve is also established combined with the reservoir engineering method, which has the advantages of simple expression form and high fitting accuracy. Through the verification analysis and field application, the new curve can be used to accurately predict the development index of the ultra-high water cut reservoir and is suitable for the whole water flooding development process.
Besides, this work is organized as follows. In Sect. 2.1, a new model to characterize the relationship between K ro /K rw and S w is proposed based on statistical experimental results, and then, according to the actual oilfield data, the new model is verified and compared with the traditional models (Sect. 2.2). In addition, based on reservoir engineering methods, a new water flooding characteristic curve is proposed (Sect. 3.1), and the new curve is justified by the numerical simulation result (Sect. 3.2). Field application of the type curve is presented in Sect. 4. In Sect. 5, we provide some concluding remarks.

Relative permeability characterization
At present, the expression commonly used to describe the quantitative relationship between oil-water relative permeability ratio and water saturation is (Craft and Hawkins 1959;Chen 2002) as follow: where K ro and K rw are the oil phase relative permeability and the water phase relative permeability, respectively; S w is the water saturation; a and b are the fitting parameters.
Equation (1) shows that oil-water relative permeability ratio (K ro /K rw ) and water saturation (S w ) have a linear relationship under the semi-log coordinate. Yu et al. (Chunlei 2014;Baohong 2015) carried out relative permeability experiments with high displacement multiple. The relative permeability experiment is improved, and 124 sets of core tests with injection volume up to 1000 PV are carried out.
The ln(K ro /K rw ) ~ S w and dln(K ro /K rw )/dS w ~ S w variation of one core is presented in Fig. 1. The experiment showed that the worse the flow capacity of crude oil, the smaller the ratio of oil-water relative permeability, and the change rate is getting faster and faster at the ultra-high water cut stage. When exceeding the changing trend described by the logarithmic function, the curve of ln(K ro /K rw ) ~ S w will bend downward (see Fig. 1).
(1) ln (K ro ∕K rw ) = a + bS w Therefore, Eq. (1) can only describe the straight section at the middle stage of water flooding development, but cannot describe the downward curve section at the ultra-high water cut stage.
Through analyzing the oil-water relative permeability experiment data under high displacement multiples (Chunlei.Yu 2014), a new characterization relation between oil-water relative permeability ratio and water saturation is proposed based on Eq. (1): where c is the fitting parameter; a, b, and c can be estimated from a nonlinear least-square fit to the measured relative permeability data.
According to relative permeability experiments (Chunlei 2014), 11 groups of oil-water relative permeability ratio curves are chosen to be fitted by Eq. (2), and the fitting relationship between K ro /K rw and S w is shown in Table 1.
Meanwhile, residual oil saturation values of these rock samples measured experimentally are also given in Table 1.
As can be seen from Table 1, the new characterization (Eq. (2)) is used to fit the oil-water relative permeability ratio and water saturation. The fitting accuracy is extremely high, and the average correlation coefficient R 2 is more than 0.996.
Furthermore, the residual oil saturation S or measured in 124 experiments and the parameter c obtained by fitting the relative permeability curve are summarized, and the relationship between c and 1-S or is shown in Fig. 2.
It can be seen from Fig. 2 that each scatter point is distributed near the 45 line, which indicates that the difference between c and 1-S or is very small. Therefore, when the residual oil saturation S or is known, the fitting parameter c can be replaced by 1-S or . Therefore, Eq. (2) can be further expressed as: where S or is the residual oil saturation.
Compared with Eq.
(1), a logarithmic function term of 1-S or -S w is added to Eq. (3), which can increase the fitting adaptability of the curve segment. The law reflected by Eq. (3) has obvious physical meaning in water flooding development. That is, when S w infinitely approaches 1-S or , K ro will also approach 0, and then, ln(K ro /K rw ) will be close to − ∞, which is similar to the water flooding law consistency. Thus, Eq. (3) describes the relationship between K ro /K rw and S w can be applied to the whole process of water flooding development.

Model verification
In order to further verify the general applicability and accuracy of Eq. (3), the typical ln(K ro /K rw ) ~ S w curves of four actual blocks (Jiqiang and Shuhong 2017)  (  (3) has the highest fitting accuracy, followed by Eq. (4) and Eq. (1). Among them, Eq. (1) is a linear equation, which has the worst adaptability to the curved line segments at the ultra-high water cut stage. In addition, although Eq. (4) can describe the downward curve to a certain extent, the fitting accuracy is significantly lower than that of Eq. (3). Besides, Eq. (4) is difficult to obtain the type-B characteristic curve through derivation. However, Eq. (3) is a simpler form and fewer unknown parameters, which can well characterize the relationship between K ro /K rw and S w in the whole process of water flooding development.

Derivation of new type curve
The conventional B-type water flooding characteristic curve has universal applicability in water flooding development oilfields, and its expression is: where WOR is water-oil ratio; N p is the cumulative oil production, 10 4 t.
Equation (5) shows a linear relationship between water-oil ratio (WOR) and cumulative oil production (N p ) under the semilog coordinate. The method of linear segment extrapolation can be (4) ln(K ro ∕K rw ) = a + bS w + ce S w (5) lnWOR = a + bN p used to determine the ultimate recovery and recoverable reserves (Shuhua 2001). However, during actual oilfield production and laboratory tests, it was found that the B-type water flooding characteristic curve is no longer applicable at the ultra-high water-cut stage, and the curve of this stage will be up-warping.
In order to accurately describe the up-warping phenomenon, a new water flooding characteristic curve is established on the above expression Eq. (2) or Eq. (3) and combined with the reservoir engineering method.
In the stable seepage condition of water flooding, regardless of the gravity and capillary force influence, the relationship between the oil-water relative permeability ratio and oil-water production is as follows (Hanqiao et al. 2006): where Q o and Q w are the output of surface crude oil and surface water, respectively, t·d −1 ; µ o and µ w are the viscosity of formation crude oil and formation water, respectively, mPa·s; B o and B w are, respectively, the volume coefficients of formation crude oil and formation water, without dimensionality; γ o and γ w are the relative density of surface degassed crude oil and surface water, without dimensionality.
Equation (2) can be transformed into the following form: Combining Eqs. (6) and (7), we can get: Besides, water saturation can be expressed as (Buckley and Leverett 1942): where S wc is irreducible water saturation; N p is the cumulative oil production, 10 4 t; N is geological reserves, 10 4 t, R is oil recovery: Substitute Eq. (9) into Eq. (8) and take the logarithm of both sides to obtain that: Then, substituting Eq. (10) into Eq. (11) and sorting lnWOR out, it can be obtained:  where A, B, C are the fitting parameters, respectively. The expression of the new water flooding characteristic curve can be obtained as: we can get that: where f w is water cut.
Substituting Eq. (17) into Eqs. (16), (16) can also be expressed as: Equation (16) is the new type of water flooding characteristic curve equation, which reflects the relationship between water-oil ratio (WOR) and cumulative oil production (N p ) at the later stage of water flooding development, so it is suitable for the ultra-high water-cut stage.
As the oil recovery (R) is low, the cumulative oil production (N p ) will also be very low, that is, ln(C-N P ) tends to 0. In this case, Eq. (16) can be written as lnWOR = A-BN p , which is the familiar Eq. (6) of B-type water flooding characteristic curve. Therefore, the B-type water flooding characteristic curve is a special case of the new type water flooding characteristic curve when the oil recovery is low. Besides, when the oilfield development reaches the economic limit water-oil ratio (In general, water-oil ratio is 49 or water cut is 98%) under the abandonment condition, the recoverable reserves and oil recovery can be predicted by Eq. (16) or Eq. (18). Furthermore, Eq. (16) can also be used to predict the future development of the oilfield and calculate the geological reserves of the oilfield.

Model verification
In order to verify the accuracy and applicability of the new water flooding characteristic curve, a five-spot well pattern model is established on the actual reservoir geological characteristics. The traditional type curve (Eq. (5)) and the new water flooding characteristic curve (Eq. 16) are used to fit the production performance, respectively.
The grid number of the model is 49 × 49 × 4, the grid length is 15 m, the vertical grid length is 8 m, and the balance between injection and production. The basic parameters are presented in Table 2, and the relative permeability of oil and water is shown in Fig. 5.
Basis on numerical simulation results, the new water flooding characteristic curve is applied and compared with the conventional B-type water flooding characteristic curve. In the case of known residual oil saturation, data in Fig. 5 are fitted based on the new oil-water relative permeability ratio characterization Eq. (3), and fitting parameters in Eq. (3) are obtained as follows: a = 6.1802, b = -5.2134.
The calculation results of N p and lnWOR are fitted by Eqs. (5) and (16), and the fitting results are shown in Fig. 6 and Table 3.
It can be seen from Fig. 6 and Table 3 that the conventional B-type water flooding characteristic curve can only fit the straight line segment at the early stage, and the upwarping curve segment in the later period could not be fitted effectively. However, the application of the new curve can realize the high precision fitting of the whole water flooding development process, and the fitting correlation coefficient is more than 0.9993.
By extrapolating to the limit water-oil ratio (WOR = 49) by the B-type water flooding characteristic curve, the calculated recoverable reserves of the reservoir are 33.9521 × 10 4 t, which has an error of more than 20% compared with the actual recoverable reserves of 27.0513 × 10 4 t (see Figs. 6, 7). However, the calculated recoverable reserves by the new curve are 27.1101 × 10 4 t, and the prediction error is only 0.22%.  Fig. 4 The correlation coefficients of the four oilfields, respectively, by using Eqs.
(1), (3), and (4) Besides, it can be seen from Table 3 that the fitting parameters of the new water flooding characteristic curve are: A = 7.6860, B = -0.0177, C = 35.5337. Therefore, according to Eq. (14), the deduction can be obtained: And then, the geological reserves of the reservoir can be obtained by substituting the above parameters into Eq. (19): N = 201.4670 × 10 4 t. Furthermore, compared with the actual geological reserves of 210.6459 × 10 4 t, the relative error between them is less than 5%. This shows the new water flooding characteristic curve has higher accuracy and wider applicability, and it is suitable for the whole process of water flooding development.

Field application
The proposed type function has been employed in the northern part of west Gudao oilfield (GD oilfield), China.    The northern part of the west Gudao oilfield is located in the Neogene large draping belt in the east of Zhanhua sag, Jiyang depression, and it is a large drapery anticlinal integrated heavy oil reservoir with unconsolidated sandstone of Guantao Formation in tertiary. A set of thick oil reservoirs are developed in the upper Ng3 and lower Ng4 formations. The average thickness of sand bodies in Ng3 5 and Ng4 4 is 10-12 m, and the average effective thickness is 9 m. The target block has an oil-bearing area of 1.54 km 2 , and the average porosity and permeability are 0.341 and 3213 × 10 -3 μm 2 , respectively, belonging to a medium-high permeability reservoir. The original oil in place of the reservoir is 5.1342 × 10 6 t. After normalizing the oil-water relative permeability curve (see Fig. 8), the irreducible water saturation is 0.319 and the residual oil saturation is 0.242.
The actual development data of the block are shown in Fig. 9. The reservoir was transferred to water flooding development after 2002, from 2002 to 2006, the comprehensive water cut of the reservoir was lower than 90%, and it was in the stage of medium-high water cut. While in 2010, the comprehensive water cut rose to 95% and the reservoir entered the ultra-high water cut stage. At present, the water cut of the reservoir has reached 98.924%, and the oil recovery is 50.235%.
According to the actual reservoir production data, the production data before the water cut of 95.156% (WOR is 19.644) are fitted by using the new water flooding characteristic curve (Eq. (16)) and Eq. (20) (Jian 2013, Jiqiang. Wang 2017 derived from formula (4), respectively, while the subsequent production data are used as the validation data of the prediction segment.
As shown in Fig. 10, the new water flooding characteristic curve (Eq. (16)) is used to fit the previous data, and the results show that the fitting accuracy is high, and the correlation coefficient is above 0.999. Equation (20) also has a good fitting effect on early stage production data. Besides, the relationship of the new water flooding characteristic curve of GD oilfield obtained by fitting is as follows: In the prediction stage, the prediction results obtained by using Eq. (16) can coincide with the actual data. In addition, when the comprehensive water cut reaches 98.924% (the ultimate WOR is 92), the ultimate recoverable reserves predicted by Eq. (21) are 2.5691 × 10 6 t, and the corresponding ultimate oil recovery is 50.04%. The recoverable reserves and recovery calculated from the new model are very close to the actual values (the actual cumulative oil production is 2.579 × 10 6 t and the ultimate recovery is 50.235%), which (20) ln WOR = a + bN p + ce dN p Therefore, the new water flooding characteristic curve established in this paper can not only reflect the special oil-water seepage law in ultra-high water cut stage, but also has the advantages of simpler form and more accurate prediction results. It is suitable for the whole process of water flooding reservoir development and can provide a basis for oilfield development decision in high water cut stage. The actual development data of the target reservoir Conclusions 1) By statistical analysis of the experimental data of phase permeability with high displacement multiple, a new relationship between the ratio of oil-water relative permeability and water saturation is proposed. The expression is simple in form and has high fitting accuracy, which well solves the problem of characterizing the relationship between K ro /K rw and S w in the whole process of water flooding development. 2) Based on the improved model, a new type of water flooding characteristic curve is derived and established. Compared with the conventional water flooding char-acteristic curve, the new curve has higher accuracy and wider applicability, which can achieve a high-precision fitting of the whole process of water flooding development. And the fitting correlation coefficient is more than 0.9993. The prediction error of recoverable reserves calculated by the new curve is only 0.22%, and the error of geological reserves is less than 5%. Besides, the curve can accurately predict the recoverable reserves and oil recovery of ultra-high water cut reservoirs, which provides a basis for oilfield development decisions in high water cut period.