Introduction

Buckley–Leverett theory and the Welge equation, as fundamental equations in percolation mechanics, establish the theoretical foundation of reservoir engineering (Ahmed 2018; Buckley & Leverett 1942; Ershaghi & Omorigie 1978; Jordan 1958; Terry et al. 2015; Welge 1952). Based on these theories, numerous scholars have developed practical reservoir engineering methods (Craig 1971; Dake 2001; Yang 2009, 2017). Currently, clean energy research is a global priority. However, fossil fuels continue to serve as the world’s primary energy source for the foreseeable future. Oil and gas resources, especially oil and its derivatives, will remain essential. Therefore, research in oil and gas field development remains crucial. In the majority of cases, waterflooding is the primary development method employed in over 80% of oilfields worldwide (Adegbite & Al-Shalabi 2020; Muther et al. 2022). Through well pattern infilling, injection–production optimization, and other techniques, more than 80% of recoverable reserves can be extracted (Li et al. 2023, 2022; Snosy et al. 2020). Dynamic performance prediction serves the vital purpose of forecasting production profiles for waterflooding development, determining well counts, timing and extent of dynamic adjustments, assessing the need for implementing adjustment wells, and establishing the launch timeline for adjustment development plans. Hence, it is imperative to enhance the prediction accuracy of waterflooding performance prediction models.

After years of development in the petroleum industry, various enhanced oil recovery (EOR) approaches have emerged, including low salinity water drive (Kumar Saw & Mandal 2023; Saw et al. 2023), thermal recovery (Liu et al. 2023; Sugama & Pyatina 2022), chemical flooding (Dean et al. 2022), carbon dioxide flooding (Chen et al. 2022; Yao et al. 2023), and microbial-enhanced oil recovery (MEOR) (Du et al. 2022; Pavan et al. 2022). These methods have been developed based on the foundation of waterflooding. It is crucial to compare the enhanced oil recovery results from these methods with those generated by traditional waterflooding development. The accuracy of performance prediction in waterflooding development directly impacts the prediction accuracy of subsequent enhanced oil recovery techniques. This, in turn, determines the feasibility and success of implementing these advanced recovery methods. Notably, enhanced oil recovery technologies typically require significant investments. Overestimating the potential results of enhanced oil recovery can lead to challenges in recovering the initial investment. Therefore, there is a pressing need to enhance the prediction accuracy of waterflooding. Doing so will establish a solid theoretical foundation for decision-making in oilfield development and its precise management.

To predict the waterflooding effect of oilfields, scholars have developed the basic theory of filtration mechanics, drawing upon the Buckley–Leverett theory (Spanos et al. 1986) and the Welge equation (Welge 1952), which serve as the theoretical foundation for reservoir engineering. Convenient reservoir engineering methods have been proposed to predict waterflooding performance (Bai et al. 2020). These methods are often employed for dynamic performance forecasting in annual and long-term planning (Huang et al. 2022; Mogollon et al. 2022; Salehi et al. 2019). In recent years, researchers have delved into displacement theories, analyzed viscous fingering during waterflooding, explored capillarity effects in formations with varying permeabilities, investigated fines migration, studied changes in reservoir properties over extended waterflooding periods, examined relative permeability characteristics, and analyzed emulsion flow through porous media (Al-Yaari et al. 2023; De Marchi et al. 2020; Ismail et al. 2022; Liu et al. 2022; Shahsavari et al. 2021). Their focus lies in understanding the fine displacement mechanism of water drive, representing displacement features, and comparing their effects with standard waterflooding development. However, there has been limited recent research on water/oil displacement theory and performance prediction models. Researchers focus on using seepage experiments or simulation methods to reveal the oil–water seepage mechanism (Yang et al. 2021, 2019). Although their results can explain the flow of oil–water fluids in the pore space of rocks by characterizing the distribution of remaining oil, their study scale is far too small, usually in the micron to millimeter scale (Blunt et al. 2013). Obviously, such a small scale cannot lead to a practical knowledge of seepage mechanism, which can be used for oilfield development. Commonly used models require further refinement due to certain limitations and issues, as outlined below: 1) The coefficients utilized in most models lack clear physical interpretations and theoretical foundations (Song et al. 2013; Wang et al. 2013). This deficiency may limit their applicability and adoption. 2) Many methods are only suitable for specific oil reservoirs and certain developmental stages (Liu et al. 2021; Lyu et al. 2019), making them unsuitable for high water cut periods, as noted by several scholars. Consequently, these methods exhibit poor universality.

This paper systematically analyzed the adaptability of the oil–water relative permeability characterization model. We introduced a new theoretical water drive curve by combining the analytical model of relative permeability with the improved Welge equation. Additionally, we explored the relationship between the water drive curve and production decline curve, while delving into the theoretical foundations of the commonly used water drive curve in the industry. Furthermore, we provided clarity on the physical interpretation of the water drive curve and Tong’s chart coefficient. We also elucidated the inherent connection between the relative permeability model and the water drive state prediction model. These findings hold significant theoretical value and practical importance.

Further study on the oil/water relative permeability ratio model

The commonly used oil/water relative permeability ratio models were summarized in this paper firstly, and then, the advantages and disadvantages of the published models were discussed, and the universality of analytical relative permeability model and the internal relationship with the models proposed by other scholars were further established, thus laying a foundation for the proposal and analysis of subsequent performance prediction models.

Limitations of common oil/water permeability model

The most commonly used model is proposed by Craft et al. (Terry et al. 2015) who observed the linear relationship of relative permeability ratio Kro/Krw versus water saturation Sw on semi-log coordinate in the intermediate water saturation stage, seeing Eq. (1)

$$\ln \frac{{K_{{{\text{ro}}}} }}{{K_{{{\text{rw}}}} }} = \ln a - bS_{{\text{w}}}$$
(1)

Many scholars have derived the corresponding waterflooding prediction models through mathematical derivation based on the linear relationship mentioned above (Can & Kabir 2014; de França Corrêa, 2007; Dou et al. 2019; Feng et al. 2017; Liu et al. 2011; Song et al. 2013; Wang et al. 2013; Xu et al. 2014; Yortsos et al. 1999).However, during low and high water cut stage, especially extra-high water cut stage, the relationship deviates from linearity (seeing Fig. 1). As a result, the deduced waterflooding prediction model accordingly showed deviation during the high water cut stage (seeing Figs. 1 and 2), which led to larger error in dynamic development index prediction, and the recoverable reserves are often overestimated. It can be seen from the relationship between ln WOR and R commonly used by scholars, this model is proper only during the middle-high water cut stage; ln(WOR) does not maintain a linear relationship with R (or cumulative oil production) during extra-high water cut stage. Using the linear relationship equation to predict dynamic performance results in the recovery factor overestimation (seeing Fig. 2). Therefore, it is necessary to further study the oil/water relative permeability ratio model suitable for high water cut stage.

Fig. 1
figure 1

The relationship between ln(Kro/Krw) and Swe

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Fig. 2
figure 2

The relationship between ln(WOR) and R

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Comparative analysis of improved oil/water permeability ratio model

Actually, the core flooding experiments demonstrated that the traditional linear model of ln(Kro/Krw) versus Sw fails to characterize the nonlinear seepage behaviors during high water cut stage, as mentioned by many scholars (Can & Kabir 2014; Feng et al. 2017; Liu et al. 2011; Song et al. 2013; Wang et al. 2013; Warner 2015; Xu et al. 2014). Recently, many scholars tried to improve the nonlinear correlation of ln(Kro/Krw) versus Sw, and several representative models are proposed as follows (Table 1).

Table 1 Representative relative permeability ratio models
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Some scholars are also exploring some approximate functions, which have little difference from the models above, or are just the deformed or simplified forms. In order to analyze the advantages and disadvantages of the models in Table 2, the relative permeability data in published papers (see Fig. 3, three recently published literature) are used for comparative analysis, with the calculation results of different models shown in Table 2.

Table 2 Fitting results with different relative permeability ratio models
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Fig. 3
figure 3

Relative permeability curves from three published papers

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In conclusion, the representative relative permeability ratio models can enhance the fitting accuracy, especially during the high water cut stage. Nonetheless, it is important to note that the coefficients in these models are fitting parameters lacking clear physical interpretations. This can result in a significant proliferation of solutions when the coefficients take excessively large or small values.

Theoretical analysis of relative permeability analytical model

In order to solve the above mentioned problems, this research was conducted based on the most widely used analytical model by recent:

$$K_{{{\text{rw}}}} = K_{{{\text{rw}}}} (S_{{{\text{or}}}} )S_{{{\text{wd}}}}^{{n_{{\text{w}}} }}$$
(8)
$$K_{{{\text{ro}}}} = K_{{{\text{ro}}}} (S_{{{\text{wi}}}} )(1 - S_{{{\text{wd}}}} )^{{n_{{\text{o}}} }}$$
(9)

with

$$S_{{{\text{wd}}}} = \frac{{S_{{{\text{we}}}} - S_{{{\text{wi}}}} }}{{1 - S_{{{\text{wi}}}} - S_{{{\text{or}}}} }}$$
(10)

Based on Eq. (8)–Eq. (10)

$$\ln \frac{{K_{{{\text{ro}}}} }}{{K_{{{\text{rw}}}} }} = \ln \frac{{K_{{{\text{ro}}}} (S_{{{\text{wi}}}} )}}{{K_{{{\text{rw}}}} (S_{{{\text{or}}}} )}} + n_{{\text{o}}} \ln (1 - S_{{{\text{wd}}}} ) - n_{{\text{w}}} \ln S_{{{\text{wd}}}}$$
(11)

In order to verify the accuracy of the model, the comparative analysis based on the same data was performed, as shown in Fig. 4 (Table 3).

Fig. 4
figure 4

Fitting results comparison with analytical relative permeability model

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Table 3 Fitting results with analytical relative permeability model
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The correlation of analytical relative permeability model with the commonly used and improved oil/water permeability ratio characterization models is analyzed below.

By taking the derivative of Eq. (11), it could be obtained

$$\frac{{{\text{d}}\ln \frac{{K_{{{\text{ro}}}} }}{{K_{{{\text{rw}}}} }}}}{{{\text{d}}S_{{{\text{wd}}}} }} = - \left( {n_{{\text{o}}} \frac{1}{{1 - S_{{{\text{wd}}}} }} + n_{{\text{w}}} \frac{1}{{S_{{{\text{wd}}}} }}} \right)$$
(12)

By taking the second derivative of Eq. (12), it could be obtained

$$\frac{{{\text{d}}^{2} \ln \frac{{K_{{{\text{ro}}}} }}{{K_{{{\text{rw}}}} }}}}{{{\text{d}}S_{{_{{{\text{wd}}}} }}^{2} }} = - \left[ {\frac{{n_{{\text{o}}} }}{{(1 - S_{{{\text{wd}}}} )^{2} }} - \frac{{n_{{\text{w}}} }}{{S_{{{\text{wd}}}}^{2} }}} \right]$$
(13)

It could be obtained by setting \(\frac{{{\text{d}}^{2} \ln \frac{{K_{{{\text{ro}}}} }}{{K_{{{\text{rw}}}} }}}}{{{\text{d}}S_{{_{{{\text{wd}}}} }}^{2} }}{ = }0\)

$$S_{{{\text{wd}}}} { = }\frac{1}{{1 + \sqrt {n_{{\text{o}}} /n_{{\text{w}}} } }}$$
(14)

Because of no and nw are usually between 1 and 5, so \(\frac{1}{{1 + \sqrt {n_{{\text{o}}} /n_{{\text{w}}} } }}\) is usually between 0.31 and 0.69. While \(0.31 \le S_{{{\text{wd}}}} \le 0.69\), we could obtain that \(\left[ {\frac{{n_{{\text{o}}} }}{{(1 - S_{{{\text{wd}}}} )^{2} }} - \frac{{n_{{\text{w}}} }}{{S_{{{\text{wd}}}}^{2} }}} \right] \to 0\), so \(\frac{{{\text{d}}^{2} \ln \frac{{K_{{{\text{ro}}}} }}{{K_{{{\text{rw}}}} }}}}{{{\text{d}}S_{{_{{{\text{wd}}}} }}^{2} }} \to 0\). In this case, \(\frac{{{\text{d}}\ln \frac{{K_{{{\text{ro}}}} }}{{K_{{{\text{rw}}}} }}}}{{{\text{d}}S_{{{\text{wd}}}} }}\) is close to a constant (seeing Fig. 5).

Fig. 5
figure 5

Oil/water permeability ratio curve and its first derivative, second derivative curves

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Equation (11) could be simplified to a linear equation by using Taylor expansion, setting the second derivative to be 0, and ignoring the third power and higher order terms.

$$\ln \frac{{K_{{{\text{ro}}}} }}{{K_{{{\text{rw}}}} }} = \ln a - bS_{{{\text{wd}}}}$$
(15)

with

$$a = \frac{{K_{{{\text{ro}}}} (S_{{{\text{wi}}}} )}}{{K_{{{\text{rw}}}} (S_{{{\text{or}}}} )}}\left( {\frac{{n_{{\text{o}}} }}{{n_{{\text{w}}} }}} \right)^{{\frac{{n_{{\text{o}}} }}{2}}} \left( {1 + \sqrt {\frac{{n_{{\text{o}}} }}{{n_{{\text{w}}} }}} } \right)^{{n_{{\text{w}}} - n_{{\text{o}}} }} \cdot e^{{n_{{\text{w}}} \left( {1 + \sqrt {\frac{{n_{{\text{o}}} }}{{n_{{\text{w}}} }}} } \right)}}$$
(16)
$$b = \left( {\sqrt {n_{{\text{o}}} } + \sqrt {n_{{\text{w}}} } } \right)^{2}$$
(17)

In essence, through appropriate simplification of the analytical relative permeability model within a specific range, we can derive the linear relationship between the relative permeability ratio and water saturation on a semi-logarithmic coordinate. This linear relationship serves as a foundational assumption in many reservoir engineering methods (Wang et al. 2013).

When the water saturation is sufficiently high, it becomes evident that as water saturation continues to increase, the second derivative of the oil/water relative permeability ratio no longer approaches zero. Consequently, the first derivative ceases to resemble a constant, and the deviation grows increasingly pronounced. This observation effectively explains the upward trend of primary-type water drive curves during the high water cut stage (Fig. 6).

Fig. 6
figure 6

Prediction results comparison between improved model and common Tong’s model

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Other scholars (Can & Kabir 2014; Feng et al. 2017; Liu et al. 2011; Song et al. 2013; Wang et al. 2013; Xu et al. 2014) have attempted to correct deviation values through mathematical approximations, but the fundamental essence of their approaches remains consistent. However, the introduced coefficients in their methods lack clear physical interpretations, introducing uncertainty into the fitting process and comparative analysis. Consequently, the relative permeability analytical model offers distinct advantages. It not only maintains fitting accuracy but also provides clear physical meanings for the parameters, minimizing the presence of multiple solutions. This clarity is beneficial for elucidating the physical significance of coefficients in subsequent dynamic prediction models and enhancing the accuracy of water drive state prediction models.

New theoretical waterflooding performance prediction model

To enhance prediction accuracy during the high water cut stage, it is imperative to minimize reliance on the oil/water permeability ratio function and explore the possibility of developing a waterflooding performance prediction model through theoretical derivation.

Theoretical derivation

Without considering the influence of gravity and capillary pressure, the fractional flow equation could be expressed as follows:

$$f_{{\text{w}}} = \frac{1}{{1 + \frac{{\mu_{{\text{w}}} B_{{\text{w}}} }}{{\mu_{{\text{o}}} B_{{\text{o}}} }}\frac{{K_{{{\text{ro}}}} }}{{K_{{{\text{rw}}}} }}}}$$
(18)

Welge derived the average water saturation function related to outlet water saturation, water cut, and the derivative \(f^{\prime}_{{{\text{w}}{\kern 1pt} }}\).

$$\overline{S}_{{\text{w}}} = S_{{{\text{w}}{\kern 1pt} {\text{e}}}} + \frac{{1 - f_{{{\text{w}}{\kern 1pt} }} }}{{f^{\prime}_{{{\text{w}}{\kern 1pt} }} }}$$
(19)

Welge equation is in differential form, and the differential form of Welge equation is transformed into linear form through theoretical derivation and practical data verification, shown as follows (Zhang & Yang 2018).

$$\overline{{S_{{\text{w}}} }} = \omega S_{{{\text{we}}}} + (1 - \omega )(1 - S_{{{\text{or}}}} )$$
(20)

\(\omega\) is Welge coefficient, related with the relative permeability analytical model parameters and oil/water viscosity ratio.

Based on Eqs. (19) and (20), the correlationship between the cumulative oil production and the cumulative liquid production could be obtained as follows.

$$N_{{\text{p}}} = N_{{\text{R}}} - \frac{{N_{{\text{R}}} - N_{{{\text{pbt}}}} }}{{\left[ {1 + \frac{1}{{N_{{{\text{pbt}}}} }}(L_{{\text{p}}} - L_{{{\text{pbt}}}} )} \right]^{{\frac{1 - \omega }{\omega }}} }}$$
(21)

where

$$N_{{{\text{pbt}}}} = N_{{\text{R}}} - \omega N\frac{{(1 - S_{{{\text{or}}}} - S_{{{\text{wf}}}} )}}{{(1 - S_{{{\text{wi}}}} )}}$$
(22)
$$\omega = 1 - \frac{{1 - f_{{{\text{wf}}}} }}{{f_{{{\text{wf}}}} }}\frac{{N_{{{\text{pbt}}}} - N_{{\text{R}}} + \omega N_{{\text{R}}} }}{{N_{{\text{R}}} - N_{{{\text{pbt}}}} }}$$
(23)
$$N_{{\text{R}}} = \frac{N}{{(1 - S_{{{\text{wi}}}} )}}(1 - S_{{{\text{or}}}} - S_{{{\text{wi}}}} )$$
(24)

Assuming that

$$A = N_{{{\text{pbt}}}}^{{^{{\frac{1}{\omega } - 1}} }} (N_{{\text{R}}} - N_{{{\text{pbt}}}} )$$
(25)
$$C = N_{pbt} - L_{pbt}$$
(26)
$$m = \frac{1 - \omega }{\omega }$$
(27)

So, a certain relationship could be obtained

$$N_{{\text{p}}} = N_{{\text{R}}} - \frac{A}{{(L_{{\text{p}}} + C)^{m} }}$$
(28)

In the previous derivation, Eq. (28) presents a theoretical water drive curve with highly specific coefficient implications. This equation elucidates the functional interplay among various parameters, including cumulative production, recoverable reserve, relative permeability, and the coefficients within Welge’s equation. Notably, the parameter ω is intricately connected to irreducible water saturation, residual oil saturation, the oil phase index, the water phase index, and the oil–water viscosity ratio. Consequently, the newly introduced model is inherently linked to these aforementioned parameters.

Furthermore, the derivation process of the new model did not incorporate the oil/water permeability ratio characterization model, nor did it rely on the approximate Ebbinghaus formula. As a result, the theoretical foundation of this model is more robust, justifying its classification as a theoretical waterflooding prediction model or theoretical waterflooding curve.

Extended models based on the new proposed model

Based on the theoretical waterflooding prediction model derived from 3.1, the following series of waterflooding prediction models are established through mathematical derivation.

Based on Eq. (21), we can get

$${\text{WOR}} = \frac{{f_{{\text{w}}} }}{{1 - f_{{\text{w}}} }} = \frac{1}{{(1 - f_{{{\text{wf}}}} )\left[ {\frac{{1 - S_{{{\text{wi}}}} }}{{\omega (1 - S_{{{\text{wf}}}} - S_{{{\text{or}}}} )}}} \right]^{{\frac{1}{1 - \omega }}} }}\left( {\frac{{1 - S_{{{\text{wi}}}} - S_{{{\text{or}}}} }}{{1 - S_{{{\text{wi}}}} }} - R} \right)^{{ - \frac{1}{1 - \omega }}} - 1$$
(29)

Equation (29) indicates that \({\text{WOR}}\) is not a simple linear relationship.

Based on Eq. (28)

$$\ln (W_{{\text{p}}} { + }C) = \ln \left[ {\left( {\frac{A}{{N_{{\text{R}}} - N_{{\text{p}}} }}} \right)^{\frac{1}{m}} - N_{{\text{p}}} } \right]$$
(30)

The relationship between water cut and cumulative oil production:

$$f_{{\text{w}}} = 1 - \frac{m}{{A^{\frac{1}{B}} }}(N_{{\text{R}}} - N_{{\text{p}}} )^{{\frac{m + 1}{m}}}$$
(31)

The relationship between water cut increasing rate and cumulative oil production:

$$f^{\prime}_{{\text{w}}} = \frac{{(m + 1)N_{{\text{R}}} }}{{A^{\frac{1}{m}} }}(N_{{\text{R}}} - N_{{\text{p}}} )^{\frac{1}{m}}$$
(32)

The relationship between water cut increasing rate and water cut:

$$f^{\prime}_{w} = \frac{{(m + 1)N_{R} }}{{(Am)^{{\frac{1}{m + 1}}} }}(1 - f_{w} )^{{\frac{1}{m + 1}}}$$
(33)

Universality analysis for commonly used models waterflooding models

Compared with conventional waterflooding prediction models, the new theoretical prediction model has fewer assumptions and stronger theoretical universality. The universality of the new model is theoretically confirmed below.

While \(0.31 \le S_{{{\text{wd}}}} \le 0.69\), we could obtain that \(\left[ {\frac{{n_{{\text{o}}} }}{{(1 - S_{{{\text{wd}}}} )^{2} }} - \frac{{n_{{\text{w}}} }}{{S_{{{\text{wd}}}}^{2} }}} \right] \to 0\), so \(\frac{{{\text{d}}^{2} \ln \frac{{K_{{{\text{ro}}}} }}{{K_{{{\text{rw}}}} }}}}{{{\text{d}}S_{{_{{{\text{wd}}}} }}^{2} }} \to 0\).

Based on Eq. (15)

$$\frac{{{\text{d}}W_{{\text{p}}} }}{{{\text{d}}N_{{\text{p}}} }} = \frac{{f_{{\text{w}}} }}{{1 - f_{{\text{w}}} }} = \frac{{\mu_{{\text{o}}} B_{{\text{o}}} }}{{\mu_{{\text{w}}} B_{{\text{w}}} }} \cdot \frac{{K_{{{\text{rw}}}} }}{{K_{{{\text{ro}}}} }} = \frac{{\mu_{{\text{o}}} B_{{\text{o}}} }}{{\mu_{{\text{w}}} B_{{\text{w}}} \cdot a}}e^{{bS_{{{\text{wd}}}} }}$$
(34)

Based on the material balance equation,

$$S_{{{\text{wd}}}} { = }\frac{{1 - S_{{{\text{wi}}}} }}{{\omega \left( {1 - S_{{{\text{wi}}}} - S_{{{\text{or}}}} } \right)}}\frac{{N_{{\text{p}}} }}{N} - \frac{1 - \omega }{\omega }$$
(35)

It could be obtained

$$W_{{\text{p}}} + \frac{{\mu_{{\text{o}}} B_{{\text{o}}} }}{{\mu_{{\text{w}}} B_{{\text{w}}} \cdot a}}e^{{ - b\frac{1 - \omega }{\omega }}} \frac{{\omega N_{{\text{R}}} }}{b}e^{{\frac{b}{{\omega N_{{\text{R}}} }}N_{{{\text{pf}}}} }} - W_{{{\text{pf}}}} = \frac{{\mu_{{\text{o}}} B_{{\text{o}}} }}{{\mu_{{\text{w}}} B_{{\text{w}}} \cdot a}}e^{{ - b\frac{1 - \omega }{\omega }}} \frac{{\omega N_{{\text{R}}} }}{b}e^{{\frac{b}{{\omega N_{{\text{R}}} }}N_{{\text{p}}} }}$$
(36)

While

$$C = \frac{M}{{\left( {\frac{{n_{{\text{o}}} }}{{n_{{\text{w}}} }}} \right)^{{\frac{{n_{{\text{o}}} }}{2}}} \left( {1 + \sqrt {\frac{{n_{{\text{o}}} }}{{n_{{\text{w}}} }}} } \right)^{{n_{{\text{w}}} - n_{{\text{o}}} }} \cdot e^{{n_{{\text{w}}} \left( {1 + \sqrt {\frac{{n_{{\text{o}}} }}{{n_{{\text{w}}} }}} } \right)}} }}e^{{ - \left( {\sqrt {n_{{\text{o}}} } + \sqrt {n_{{\text{w}}} } } \right)^{2} \frac{1 - \omega }{\omega }}} \frac{{\omega N_{{\text{R}}} }}{{\left( {\sqrt {n_{{\text{o}}} } + \sqrt {n_{{\text{w}}} } } \right)^{2} }}e^{{\frac{{\left( {\sqrt {n_{{\text{o}}} } + \sqrt {n_{{\text{w}}} } } \right)^{2} }}{{\omega N_{{\text{R}}} }}N_{{{\text{pf}}}} }} - W_{{{\text{pf}}}}$$
$$B_{{1}} = \frac{{\left( {\sqrt {n_{{\text{o}}} } + \sqrt {n_{{\text{w}}} } } \right)^{2} }}{{\omega N_{{\text{R}}} }}$$
$$A_{{1}} = \ln \frac{M}{{\left( {\frac{{n_{{\text{o}}} }}{{n_{{\text{w}}} }}} \right)^{{\frac{{n_{{\text{o}}} }}{2}}} \left( {1 + \sqrt {\frac{{n_{{\text{o}}} }}{{n_{{\text{w}}} }}} } \right)^{{n_{{\text{w}}} - n_{{\text{o}}} }} \cdot e^{{n_{{\text{w}}} \left( {1 + \sqrt {\frac{{n_{{\text{o}}} }}{{n_{{\text{w}}} }}} } \right)}} }}\frac{{\omega N_{{\text{R}}} }}{{\left( {\sqrt {n_{{\text{o}}} } + \sqrt {n_{{\text{w}}} } } \right)^{2} }} - \left( {\sqrt {n_{{\text{o}}} } + \sqrt {n_{{\text{w}}} } } \right)^{2} \frac{1 - \omega }{\omega }$$

After taking the logarithm of Eq. (36), it could be obtained

$$\ln (W_{{\text{p}}} + C^{\prime}) = A^{\prime} + B^{\prime}N_{{\text{p}}}$$
(37)

This is the commonly used Type-A waterflooding curve model.

Based on Eq. (37)

$$\frac{{{\text{d}}W_{{\text{p}}} }}{{{\text{d}}N_{{\text{p}}} }} = \frac{{f_{{\text{w}}} }}{{1 - f_{{\text{w}}} }} = B^{\prime } e^{{A^{\prime } + B^{\prime } N_{{\text{p}}} }}$$
(38)

After taking the logarithm of Eq. (38) in both sides, it could be obtained

$$\ln {\text{WOR}} = \ln B^{\prime} + A^{\prime} + B^{\prime}N_{{\text{p}}}$$
(39)

So

$$\ln {\text{WOR}} = \ln \frac{M}{{\left( {\frac{{n_{{\text{o}}} }}{{n_{{\text{w}}} }}} \right)^{{\frac{{n_{{\text{o}}} }}{2}}} \left( {1 + \sqrt {\frac{{n_{{\text{o}}} }}{{n_{{\text{w}}} }}} } \right)^{{n_{{\text{w}}} - n_{{\text{o}}} }} \cdot e^{{n_{{\text{w}}} \left( {1 + \sqrt {\frac{{n_{{\text{o}}} }}{{n_{{\text{w}}} }}} } \right)}} }} - \left( {\sqrt {n_{{\text{o}}} } + \sqrt {n_{{\text{w}}} } } \right)^{2} \frac{1 - \omega }{\omega } + \frac{{\left( {\sqrt {n_{{\text{o}}} } + \sqrt {n_{{\text{w}}} } } \right)^{2} }}{{\omega N_{{\text{R}}} }}N_{{\text{p}}}$$
(40)

Because of \(R = \frac{{N_{{\text{p}}} }}{N}\)

$$\ln {\text{WOR}} = \ln B^{\prime} + A^{\prime} + B^{\prime}NR$$
(41)

So

$$\ln {\text{WOR}} = \ln \frac{M}{{\left( {\frac{{n_{{\text{o}}} }}{{n_{{\text{w}}} }}} \right)^{{\frac{{n_{{\text{o}}} }}{2}}} \left( {1 + \sqrt {\frac{{n_{{\text{o}}} }}{{n_{{\text{w}}} }}} } \right)^{{n_{{\text{w}}} - n_{{\text{o}}} }} \cdot e^{{n_{{\text{w}}} \left( {1 + \sqrt {\frac{{n_{{\text{o}}} }}{{n_{{\text{w}}} }}} } \right)}} }} - \left( {\sqrt {n_{{\text{o}}} } + \sqrt {n_{{\text{w}}} } } \right)^{2} \frac{1 - \omega }{\omega } + \frac{{\left( {\sqrt {n_{{\text{o}}} } + \sqrt {n_{{\text{w}}} } } \right)^{2} }}{{\omega E_{{\text{d}}} }}R$$
(42)

Taking Eq. (42) with the logarithm with 10 as the base, it could be obtained

$$\lg {\text{WOR}} = \lg \frac{M}{{\left( {\frac{{n_{{\text{o}}} }}{{n_{{\text{w}}} }}} \right)^{{\frac{{n_{{\text{o}}} }}{2}}} \left( {1 + \sqrt {\frac{{n_{{\text{o}}} }}{{n_{{\text{w}}} }}} } \right)^{{n_{{\text{w}}} - n_{{\text{o}}} }} \cdot e^{{n_{{\text{w}}} \left( {1 + \sqrt {\frac{{n_{{\text{o}}} }}{{n_{{\text{w}}} }}} } \right)}} }} - \left( {\sqrt {n_{{\text{o}}} } + \sqrt {n_{{\text{w}}} } } \right)^{2} \frac{1 - \omega }{{\omega \ln 10}} + \frac{{\left( {\sqrt {n_{{\text{o}}} } + \sqrt {n_{{\text{w}}} } } \right)^{2} }}{{\omega E_{{\text{d}}} \ln 10}}R$$
(43)

The cumulative oil production when the water content is 98% is taken as the recoverable reserves, and the corresponding recovery factor is ER, then

$$\lg 49 = \lg \frac{M}{{\left( {\frac{{n_{{\text{o}}} }}{{n_{{\text{w}}} }}} \right)^{{\frac{{n_{{\text{o}}} }}{2}}} \left( {1 + \sqrt {\frac{{n_{{\text{o}}} }}{{n_{{\text{w}}} }}} } \right)^{{n_{{\text{w}}} - n_{{\text{o}}} }} \cdot e^{{n_{{\text{w}}} \left( {1 + \sqrt {\frac{{n_{{\text{o}}} }}{{n_{{\text{w}}} }}} } \right)}} }} - \left( {\sqrt {n_{{\text{o}}} } + \sqrt {n_{{\text{w}}} } } \right)^{2} \frac{1 - \omega }{{\omega \ln 10}} + \frac{{\left( {\sqrt {n_{{\text{o}}} } + \sqrt {n_{{\text{w}}} } } \right)^{2} }}{{\omega E_{{\text{d}}} \ln 10}}E_{{\text{R}}}$$
(44)

Equation (43) subtracts Eq. (44), it could be obtained

$$\lg {\text{WOR}} = 1.69 + \frac{{\left( {\sqrt {n_{{\text{o}}} } + \sqrt {n_{{\text{w}}} } } \right)^{2} }}{{\omega E_{{\text{d}}} \ln 10}}(R - E_{{\text{R}}} )$$
(45)

The commonly used Tong’s curve model:

$$\lg {\text{WOR}} = 1.69 + 7.5(R - E_{{\text{R}}} )$$
(46)

Equation (46) is a prediction model integrated into the global Daks database, known as Tong’s curve model. The Daks database is utilized by the world’s largest multinational oil companies, significant national oil and gas firms, as well as government energy agencies. However, Tong’s curve model was originally developed based on statistics from high-permeability oilfields, with a fixed coefficient of 7.5. The physical significance of Tong’s curve model has been theoretically established, relying on factors such as oil–water index, Welge coefficient, and oil displacement efficiency. This theoretical foundation illuminates why low-permeability oilfields may possess coefficients greater than 7.5. This phenomenon can be attributed to two key factors: 1) low oil displacement efficiency in low-permeability oilfields. 2) Generally low crude oil viscosity and w value in low-permeability oilfields.

After comparing the actual data from Bohai oilfield, it becomes evident that the prediction error between the new model and Tong’s curve can be as high as 6%. Consequently, it is advisable to utilize coefficients calculated based on the specific conditions for practical predictions, rather than adhering to the fixed value of 7.5.

The proposed model can be simplified under specific conditions to derive the frequently used water drive curve equation. Given the commonly used oil/water permeability ratio, we will not reiterate the derivation of other models. Furthermore, utilizing the improved oil/water relative permeability ratio models, various waterflooding prediction models can be generated, as illustrated in Table 4.

Table 4 Correspondence between oil/water relative permeability ratio model and waterflooding performance prediction model
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The traditional waterflooding characteristic curves are fundamentally based on the linear relationship between the relative permeability ratio and water saturation on a semi-logarithmic coordinate axis. In contrast, the improved water drive curves for predicting ultra-high water cut periods can be derived by simplifying the analytical model of relative permeability or approximating its function. The model presented in this paper is theoretically more versatile, offering a concise and physically meaningful representation.

Results and discussion

When the oilfield development went into extra-high water cut stage, the relationship deviates from linearity. To verify the proposed theoretical model (Eq. (28)), sensitivity cases were analyzed based on different oil/water viscosity ratios and different relative permeability parameters.

From Eq. (28), we obtained

$$\frac{{{\text{d}}N_{{\text{p}}} }}{{{\text{d}}L_{{\text{p}}} }} = \frac{Am}{{(L_{{\text{p}}} + C)^{m + 1} }}$$
(47)

From Eq. (47) obtained

$$\frac{{N_{{\text{R}}} - N_{{\text{p}}} }}{{1 - f_{{\text{w}}} }} = \frac{{L_{{\text{p}}} + C}}{m}$$
(48)

From core relative permeability experiments, Swi, Sor, nw, no, and Kro (Swi) could be obtained. The related curves between \(\frac{{N_{{\text{R}}} - N_{{\text{p}}} }}{{1 - f_{{\text{w}}} }}\) and \(L_{{\text{p}}}\) were then calculated as follows. Because the oil and water phase index was usually between 2 and 4, sensitivity cases were discussed (see Figs. 7, 8, and 9).

Fig. 7
figure 7

The linear relationship curve when water phase index nw = 2 and oil phase index no = 4

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Fig. 8
figure 8

The linear relationship curve when water phase index nw = 3 and oil phase index no = 3

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Fig. 9
figure 9

The linear relationship curve when water phase index nw = 4 and oil phase index no = 2

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From Figs. 7, 8, and 9, it could be obtained that \(\frac{{N_{{\text{R}}} - N_{{\text{p}}} }}{{1 - f_{{\text{w}}} }}\) and \(L_{{\text{p}}}\) present a good linear relationship with all correlation coefficients close to 1, while the water cut is higher than fwf. Therefore, the new model could solve the problem of the prediction error increasing in the high water cut stage.

To verify the prediction accuracy of the new proposed model, the actual published example was chosen. Different prediction models could be obtained through actual data fitting. To verify the forecasting accuracy, data points (0.50 ≤ Swe ≤ 0.60) were used for fitting, and the points while fw = 0.98 were used to analyze the prediction errors for different models (Table 5).

Table 5 Waterflooding recoverable reserves predicted by different waterflooding characteristic curves
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For computational purposes, it is advisable to employ a designated function fitting method to estimate the model’s parameters. The independent variables within the function can be transformed, through suitable deformation, into cumulative oil production for prediction. Additionally, recoverable reserves can be calculated when the water cut reaches 98%, once the relationship between water cut (or oil cut, WOR) and cumulative oil production has been derived. Models that cannot be directly computed may be solved numerically using iterative algorithms.

Comparing the predicted results with the actual published data, the prediction results based on different models were obtained. It could be concluded from Table 6 as follows:

  1. (1)

    The first seven commonly used waterflooding performance prediction models are based on the empirical formula derived from data statistics, or based on oil/water relative permeability ratio linear characterization model, and the errors are larger compared with the modified models.

  2. (2)

    The accuracy of predicting waterflooding performance using the modified model, which is based on the adjusted oil/water relative permeability ratio, has indeed improved to a certain extent. However, it is worth noting that the physical meaning of the model coefficients proposed by many scholars remains unclear. This lack of clarity makes it impossible to determine their reasonability. In some cases, these coefficients even reach unrealistic extremes, which contradicts the principles of physical significance. In contrast, the model presented in this paper has demonstrated nearly the highest prediction accuracy, with an error margin of just 0.3%. This significant improvement in accuracy underscores the model’s effectiveness.

Table 6 Comparison of assumptions of waterflooding performance prediction models
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The two commonly used assumptions by scholars were not incorporated into the more rigorous derivation process. This omission serves to establish a stronger theoretical foundation for scholars as they delve deeper into the study of waterflooding performance prediction, recoverable reserves calibration, and the associated methods for optimization and adjustment.

At present, many scholars believe that the average water saturation and outlet water saturation show a linear relationship with the slope of 2/3, mainly based on the experimental conclusions of Зфpoc. But Зфpoc explicitly pointed out that the linear relationship with a slope of 2/3 was only applicable to the oil/water viscosity ratio less than 10. Welge coefficient was introduced into the new Welge equation, revealing that the average water saturation and outlet water saturation still showed a linear relation, but the slope was affected by the oil/water viscosity ratio and relative permeability parameters instead of 2/3. Theoretically, the new equation is applicable to the oil reservoirs of any viscosity, thus extending the application scope of the new model.

Conclusions

Based on the oil–water relative permeability model, the material balance method and Welge equation are used to derive a new theoretical waterflooding model. Compared with the traditional waterflooding model, the new model proposed in this research has a wider application range and can be applied to oilfield at high water cut. At the same time, the new model clarifies the coefficient of waterflooding curve A and the physical meaning of parameter 7.5 in Tong’s chart method for the first time. The new model proposed in this research further enriches the connotation of waterflooding theory and has certain application significance.