Preface

When the water flooding characteristic curve was first put forward, a statistical relationship appeared for its production data and had no theoretical significance. In 1959, the field data were first used to obtain the water flooding characteristic curve's statistical law, from which the water flooding characteristic curve can be used as an intuitive and straightforward statistical expression of dynamic reservoir law. At present, the water flooding characteristic curve method has been widely valued and applied (Muskat et al. 1935, Abass et al. 1988, Guo et al. 1992, Yuanqian 1994). The water flooding characteristic curve method can predict the water flooding oilfield's related development indexes and effectively predict and judge the water flooding oilfield's movable reserves and original geological reserves.

Because different types of oilfields have their characteristics, the realization of production performance of different types of oilfields would be inconsistent. Therefore, different types of oil fields would have the corresponding water flooding characteristic curve. Different reservoir types, reservoir physical properties, and crude oil properties show different dynamic characteristics. Based on statistical research, some scholars put forward various expressions of the water flooding characteristic curve. It reflects the particularity of oilfield production performance (Cunyou et al. 2011, Liu 2020, Recham 2000). Because of this particularity, some different expressions may be proposed in future. As long as several oilfield examples prove, it is the enrichment and enrichment of the research content. As a hot topic, many scholars are actively engaged in this research and exploration and have made many achievements, which is scientific progress.

Many scholars in the former Soviet Union have proposed various water flooding characteristic curves (Olabode et al. 2021; Hanqiao et al. 2006; Qitai 1998). At present, the widely used water flooding characteristic curves are Сазонов Б.Ф., Сипачев Н.В., and Назаров С. Н.. In 1978, Tong Xianzhang studied the water flooding characteristic curve expressed by the relationship between the logarithm of cumulative water production and cumulative oil production and established a complete application system. He began applying and researching the water flooding characteristic curve (Jackson et al. 2005; Wenjun et al. 2000; Qitai 1999). Chen Yuanqian combined with the Nonpiston theory of water flooding and made theoretical verification on Maкcимoв M. И. water flooding characteristic curve, Caзoнoв Б.Ф. water flooding characteristic curve, and Cипaчeв H.B. water flooding characteristic curve. The new water flooding characteristic curves proposed in the following research literature have been theoretically deduced or verified. With the deepening of application and research, it is found that different water flooding characteristic curves reflect different oil–water two-phase seepage characteristics.

The derivation of the water flooding characteristic curve is based on the semi-logarithmic linear relationship between the ratio of oil to water relative permeability and water saturation (Wenjun and Jun 2007; Yuanqian 1993; Jian et al. 2013; Chuanliang 2005). The relationship between oil and water relative permeability and water saturation only conforms to the linear relationship in the middle part but deviates from the linear relationship in the early development and ultra-high water cut stages. Therefore, the water flooding characteristic curve is generally applicable to the two parameters (water/oil ratio, cumulative oil production, oil recovery) in the rectangular coordinate system after the water cut reaches 40% cumulative water production. When there is a linear relationship or semi-logarithmic linear relationship, in the reservoir with strong bottom water and low amplitude, the water cut rises very fast, there is no water-free oil recovery period, and it would soon enter the high water cut period. When the oilfield enters the ultra-high water cut stage, the relationship between the ratio of oil and water relative permeability and water saturation deviated from the straight line, and the conventional water flooding characteristic curve would also deviate from the straight line. Simultaneously, applying the conventional water flooding characteristic curve to predict the development index would produce a significant error (Qi-tai 1999; Shaoxian.2012; Hua et al. 2013; Changfu et al. 2011). In this study, a new expression of the relationship between oil–water relative permeability and water saturation is proposed, and a water flooding characteristic curve suitable for low-amplitude reservoirs with strong bottom water is established. On this basis, a new type of water flooding characteristic curve is derived. Through the verification analysis, the new water flooding characteristic curve can reasonably predict the development index of the reservoir with strong bottom water and low amplitude, which is more in line with the actual reservoir development.

Deflection of conventional water flooding characteristic curve method in high water cut stage

In this study, the conventional water flooding characteristic curve (Table 1) predicts the oil recovery in the reservoir's medium and high water cut period with strong bottom water and low amplitude. It is found that there is a big gap with the actual situation, so it is not applicable.

Table 1 Ordinary water flooding characteristic curve
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The conventional water flooding characteristic curve method considers that the relative permeability ratio of oil–water two-phase is expressed as \(\frac{{k_{ro} }}{{k_{rw} }} = ce^{{ - d \cdot S_{w} }}\). Through actual data discovery, the relationship between \({\text{ln}}\left( {\frac{{k_{ro} }}{{k_{rw} }}} \right){\text{ and }}S_{w}\) is only linear for some time (Shuhong et al. 2011; Qiaoyun et al. 2011; Yuanqian and Dang 2004; Welge 1952; Goda et al. 2007; Wang et al. 2013). There is a downward tilt in the later period (see Fig. 1). During the derivation of the conventional water flooding characteristic curve method \(\overline{{S_{w} }} \approx S_{we} + \frac{1}{{mf_{w} }}\). It is assumed that after the oil field enters the high water cut stage (\(f_{w} > 80\%\)), \(\overline{{S_{w} }} \approx S_{we}\). In other words, the water saturation at the high water cut the average saturation of formation can replace stage outlet. But in the actual reservoir, \(\overline{{S_{w} }} \ne S_{we}\). The formula \(\frac{{k_{ro} }}{{k_{rw} }} = ce^{{ - d \cdot S_{w} }}\) needs to be modified according to the actual oilfield.

Fig. 1
figure 1

Traditional water flooding characteristic curve method would deflect in the high water cut stage

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Establishment of water flooding characteristic curve for low-amplitude reservoir with strong bottom water

Derivation of water flooding characteristic curve for low-amplitude reservoir with strong bottom water

According to Buckley–Leverett, water flooding front advancing equation, the functional relationship of \(\overline{{S_{w} }}\) and \(S_{we}\) is as follows (Yuanqian and Cunyou 2014; Can and Kabir 2014):

$$S_{we} = \left( {\frac{1}{1 + k}\overline{{S_{w} }} + \frac{k}{1 + k}S_{wi} } \right)\left[ {1 + \left( {\frac{{\overline{{S_{w} }} - S_{wi} }}{m}} \right)^{ - 1/k} } \right]^{ - 1}$$
(1)

By deriving the two ends of formula (1):

$$\frac{{dS_{we} }}{{d\overline{{S_{w} }} }} = - \frac{1}{mk}\left[ {\frac{{\frac{1}{1 + k}\overline{{S_{w} }} + \frac{k}{1 + k}S_{wi} }}{{1 + \left( {\frac{{\overline{{S_{w} }} - S_{wi} }}{m}} \right)^{ - 1/k} }} - \overline{{S_{w} }} } \right]\frac{{\left( {\frac{{\overline{{S_{w} }} - S_{wi} }}{m}} \right)^{1/k - 1} }}{{1 + \left( {\frac{{\overline{{S_{w} }} - S_{wi} }}{m}} \right)^{ - 1/k} }}$$
(2)

Replace (1) with (2):

$$\frac{{dS_{we} }}{{d\overline{{S_{w} }} }} = - \frac{1}{mk}\left( {S_{we} - \overline{{S_{w} }} } \right)\frac{{\left( {\frac{{\overline{{S_{w} }} - S_{wi} }}{m}} \right)^{1/k - 1} }}{{1 + \left( {\frac{{\overline{{S_{w} }} - S_{wi} }}{m}} \right)^{1/k} }}$$
(3)

From the Welge equation, it can be concluded that:

$$\frac{{df_{oe} }}{{dS_{we} }} = \frac{{df_{oe} }}{{d\overline{{S_{w} }} }}\frac{{d\overline{{S_{w} }} }}{{dS_{we} }} = \frac{{f_{oe} }}{{S_{we} - \overline{{S_{w} }} }}$$
(4)

By substituting (3) into (4):

$$\frac{{df_{oe} }}{{d\overline{{S_{w} }} }} = - \frac{1}{mk}\frac{{\left( {\frac{{\overline{{S_{w} }} - S_{wi} }}{m}} \right)^{1/k - 1} }}{{1 + \left( {\frac{{\overline{{S_{w} }} - S_{wi} }}{m}} \right)^{1/k} }}f_{oe}$$
(5)

Solve the above differential equation:

$$f_{oe} = C_{1} /\left[ {1 + \left( {\frac{{\overline{{S_{w} }} - S_{wi} }}{m}} \right)^{1/k} } \right]$$
(6)

Then (6) the initial conditions are satisfied \(\overline{{S_{w} }} = S_{wi}\), \(f_{oe} = 1\) (Replace \(\overline{{S_{w} }} = S_{wi}\) into (1). The initial conditions can be determined \(\overline{{S_{w} }} = S_{wi}\), then \(f_{oe} = 1\)), \(C_{1} = 1\), substitution it into (6):

$$f_{oe} = 1/\left[ {1 + \left( {\frac{{\overline{{S_{w} }} - S_{wi} }}{m}} \right)^{1/k} } \right]$$
(7)

then

$$\left( {1 - f_{oe} } \right)/f_{oe} = \left( {\frac{{\overline{{S_{w} }} - S_{wi} }}{m}} \right)^{1/k}$$
(8)

From the definition,

$$\left( {1 - f_{oe} } \right)/f_{oe} = Q_{w} /Q_{o}$$
(9)
$$Q_{w} = dW_{p} /dt$$
(10)
$$Q_{o} = dN_{p} /dt$$
(11)

Under the condition that the formation pressure is maintained by water injection and the reservoir is relatively homogeneous, it can be seen from the material balance equation that:

$$\overline{{S_{w} }} = \frac{{N_{p} \left( {1 - S_{wi} } \right)}}{N} + S_{wi}$$
(12)

Combining (9) (10) (11) (12) with (8), we can get:

$$dW_{p} /dN_{p} = \left[ {\left( {1 - S_{wi} } \right)\frac{{N_{p} }}{mN}} \right]^{1/k}$$
(13)

(13) When the initial conditions are satisfied \(W_{p} = 0\), \(N_{p} = N_{p0}\). By solving the above differential equations, we can obtain:

$$lnN_{p} = A + Bln\left( {W_{p} + C} \right)$$
(14)

where

$${\text{A}} = \frac{1}{1 + k}ln\left[ {\frac{1}{1 + k}\left( {\frac{mN}{{1 - S_{wi} }}} \right)^{1/k} } \right]$$
$${\text{B}} = \frac{1}{1 + k}$$
$${\text{C}} = \frac{1}{1 + k}\left( {\frac{{1 - S_{wi} }}{mN}} \right)^{1/k} N_{p0}^{1 + 1/k}$$

There is no water-free oil recovery period in low-amplitude structural reservoir with strong bottom water, so \(N_{p0}\) = 0, \({\text{C}}\) = 0, from (14)

$$ln\frac{{W_{p} }}{{N_{p} }} = a + blnN_{p}$$
(15)

where \(b = k\); \(a = \ln \left[ {\frac{1}{1 + k}\left( {\frac{mN}{{1 - S_{wi} }}} \right)^{1/k} } \right]\)

According to the partial flow formula:

$$f_{w} = 1 - f_{oe} = 1/\left[ {1 + K_{ro} /\left( {\mu_{r} K_{rw} } \right)} \right]$$
(16)

The oil–water relative permeability ratio formula is:

$$K_{ro} /K_{rw} = \mu_{r} /\left[ {\left( {\overline{{S_{w} }} - S_{wi} } \right)/m} \right]^{1/k}$$
(17)

where

\(f_{oe} ,f_{w}\)—They are oil content and water content at the outlet, \(f\);

\(k\)—Constant;

\(K_{ro} ,\;K_{rw}\)—They are relative permeability of oil and water, dimensionless;

\(L_{p}\)—Cumulative liquid production, \(10^{4} m^{3}\);

\(m\)—Constant coefficient;

\(N,N_{p}\)—They are geological reserves and cumulative oil production, \(10^{4} m^{3}\);

\(Q_{L} ,Q_{O} ,Q_{W}\)—They are liquid production, oil production, and water production, \(m^{3} /d\);

\(\overline{{S_{w} }} ,S_{wi}\)—They are average water saturation and irreducible water saturation, \(f\);

\(S_{we}\)—Water saturation at outlet, \(f\);

\(W_{p}\)—Cumulative water production, \(10^{4} m^{3}\);

\(\mu_{r}\)—Oil–water viscosity ratio, dimensionless.

Substitute the actual relative permeability curve (see Table 2 and Fig. 2) into Formula 17. The relationship between \(\frac{{k_{ro} }}{{k_{rw} }}\) and \(S_{w}\) is obtained. The correlation coefficient of \(\frac{{k_{ro} }}{{k_{rw} }} and S_{w}\) has reached 95.92% (Fig. 3). The higher the water content, the better the fitting degree. It is suitable for low-amplitude reservoirs with strong bottom water.

Table 2 Relative permeability under different water saturation
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Fig. 2
figure 2

Oil–water relative permeability of oilfield

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Fig. 3
figure 3

Relation curve of \(K_{ro} /K_{rw}\) and \(1/\left( {\overline{{S_{w} }} - S_{wi} } \right)\)

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Through the formula derivation in this section, the relationship between \(\frac{{k_{ro} }}{{k_{rw} }}\) and \(S_{w}\) is obtained. The correlation coefficient of \(\frac{{k_{ro} }}{{k_{rw} }}{\text{ and }}S_{w}\) has reached 95.92%. The water flooding characteristic curve suitable for a low-amplitude reservoir with strong bottom water is obtained.

Using new water flooding characteristic curve to predict cumulative oil production \(N_{P}\)

From Eq. (15), we can get the following results:

$$\ln W_{p} = a + \left( {b + 1} \right)\ln N_{p}$$
(18)

The derivative of Eq. (18) to \(N_{p}\) is obtained:

$$\frac{1}{{W_{p} }}\frac{{dW_{p} }}{{dN_{p} }} = \left( {b + 1} \right)\frac{1}{{N_{p} }}$$
(19)

By \({\text{WOR}} = \frac{{dW_{p} }}{{dN_{p} }}\):

$$W_{p} = \frac{{N_{p} WOR}}{b + 1}$$
(20)

Substituting (20) into (18) yields:

$$N_{p} = e^{{\frac{{ln\frac{WOR}{{b + 1}} - a}}{b}}}$$
(21)

Through (21), recoverable reserves under a given water–oil ratio (which can be converted into water content) can be obtained.

This section uses the water flooding characteristic curve in Sect. 3.1 to derive the formula for calculating cumulative oil production \(N_{p}\).

Prediction of future water cut based on relative permeability

From the yield equation (Zhaojie et al. 2013):

$$Q_{o} = \frac{{0.543KK_{ro} h\left( {P_{e} - P_{wf} } \right)}}{{\mu_{o} B_{o} \left( {ln\frac{{r_{e} }}{{r_{w} }} + S} \right)}}$$
(22)
$$Q_{w} = \frac{{0.543KK_{rw} h\left( {P_{e} - P_{wf} } \right)}}{{\mu_{w} B_{w} \left( {ln\frac{{r_{e} }}{{r_{w} }} + S} \right)}}$$
(23)

(22) and (23), the:

$$\frac{{Q_{w} }}{{Q_{o} }} = \frac{{\mu_{o} B_{o} K_{rw} }}{{\mu_{w} B_{w} K_{ro} }}$$
(24)

(17) By substituting (24), the:

$$\frac{{Q_{w} }}{{Q_{o} }} = \frac{{B_{o} }}{{B_{w} }}\left[ {\left( {\overline{{S_{w} }} - S_{wi} } \right)/m} \right]^{1/k}$$
(25)
$$N_{o} = 100Ah\emptyset \left( {1 - S_{wi} } \right)/B_{o}$$
(26)
$$N_{or} = 100Ah\emptyset \left( {1 - \overline{{S_{w} }} } \right)/B_{o}$$
(27)

From (26) and (27), the:

$$\overline{{S_{w} }} = \left( {1 - S_{wi} } \right)R_{o} + S_{wi}$$
(28)

By substituting (28) into (25), the:

$$\frac{{Q_{w} }}{{Q_{o} }} = \frac{{B_{o} }}{{B_{w} }}\left[ {\left( {1 - S_{wi} } \right)R_{o} /m} \right]^{1/k}$$
(29)
$$f_{w} = \frac{{Q_{w} }}{{Q_{o} + Q_{w} }} = \frac{1}{{\frac{{Q_{o} }}{{Q_{w} }} + 1}} = \frac{1}{{\frac{1}{{\frac{{Q_{w} }}{{Q_{o} }}}} + 1}}$$
(30)

By substituting (29) into (30), the:

$$f_{w} = \frac{1}{{\frac{1}{{\frac{{B_{o} }}{{B_{w} }}\left[ {\left( {1 - S_{wi} } \right)R_{o} /m} \right]^{1/k} }} + 1}}$$
(31)

According to formula (31), the future water cut under constant liquid volume can be obtained.

where

\(K\)—Effective permeability of reservoir, mD.

\(\mu_{o} ,\mu_{W}\)—They are the viscosity of oil and the viscosity of water, mPa·s.

\(a,b\)—Coefficient,\({ }f\).

\(B_{o} ,B_{W}\)—They are the volume coefficient of oil and the volume coefficient of water, \(f\)

\(R_{o}\)—Oil recovery, \(f\)

\(A\)—Reservoir cross section, \(m^{2}\).

\(h\)—Valid thickness, \(m\)

\(\emptyset\)—Porosity, \(f\)

According to the relative permeability productivity formula, this section deduces the formula for calculating water cut when the cumulative oil production \(N_{p}\) is known (Sect. 3.2).

Case analysis

The geological reserve of a strong bottom water sandstone reservoir is 2000 × 104m3; the adequate thickness is 6 m, the viscosity of crude oil is 50 mPa·s, the viscosity of water is 0.51 mPa·s, the density of oil is 0.923 g/cm3, and the volume coefficient of oil is 1.13. Take A2H well in this oilfield as an example (see Table 3). According to Eq. (15), there is an excellent linear relationship between well \(ln\frac{{W_{p} }}{{N_{p} }}{\text{ and }}lnN_{p}\) (Fig. 4). The corresponding a and b values can be obtained, and the water content calculated by the formula (31) is compared with the actual water content (Fig. 5). The calculated cumulative oil production is in good agreement with the actual cumulative oil production (Fig. 6). In this way, the recovery degree of a single well under different water cuts can be obtained.

Table 3 Actual production data and forecast data of A2H
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Fig. 4
figure 4

Relation curve of \(ln\frac{{W_{p} }}{{N_{p} }}{\text{ and }}lnN_{p}\)(the data of middle and high water cut stage were selected for linear regression)

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Fig. 5
figure 5

Comparison between calculated water content and actual water content

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Fig. 6
figure 6

Comparison between calculated cumulative oil production and actual cumulative oil production

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The recoverable reserves under the given water–oil ratio (which can be converted into water content) are obtained.

Figure 4 shows \(a = - 21.175,b = 1.8649\). It can be concluded from Eq. 15:

$$ln\frac{{W_{p} }}{{N_{p} }} = - 21.175 + 1.8649lnN_{p}$$
(32)

By \(W_{p} + N_{p} = L_{P}\), Eq. 32 shows that:

$$ln\frac{{L_{P} - N_{p} }}{{N_{p} }} = - 21.175 + 1.8649lnN_{p}$$
(33)

Through Eq. 33, under the condition of constant liquid production (see Table 3. The actual production data from 2021/3, 48405t), in future, the accumulated liquid production \(L_{p}\) at any time in future can be obtained; thus, the accumulated oil production \(N_{p}\) at any time in future can be obtained. Through the accumulated oil production \(N_{p}\), the monthly oil production can be obtained, and the corresponding water content can be obtained. That is to say, through Eq. 33, knowing \(L_{P}\) and finding \(N_{p}\), we can get the output of different water content at any time in future under constant liquid quantity.

According to the comparison between the actual data and the calculated data, in 2021/3, the actual water cut is 97.29%, the water cut predicted by the formula is 97.27%, the actual cumulative oil production is 31.19 × 104t, and the predicted cumulative oil production is 31.31 × 104t. The predicted value is consistent with the actual value.

Conclusions

In this study, the derivation process of the conventional water flooding characteristic curve is studied. At the same time, according to the Buckley–Leverett water flooding front advancing equation, combined with the characteristics of strong bottom water and low-amplitude structural reservoir without a water-free oil recovery period, the water flooding characteristic curve is modified:

(1)The conventional water flooding characteristic curve method considers that the relative permeability ratio of oil–water two-phase is expressed as \(\frac{{k_{ro} }}{{k_{rw} }} = ce^{{ - d \cdot S_{wo} }}\). Through the actual data, it is found that the relationship between \(\ln \left( {\frac{{k_{ro} }}{{k_{rw} }}} \right)\) and \(S_{w}\) is only a straight line in a period, and there is a downward tilt in the later period.

(2) During the derivation of the conventional water flooding characteristic curve method, \(\overline{{S_{w} }} \approx S_{we} + \frac{1}{{mf_{w} }}\). It is assumed that after the oil field enters the high water cut stage (\(f_{w} > 80\%\)), \(\overline{{S_{w} }} \approx S_{we}\). In other words, the water saturation at the high water cut the average saturation of formation can replace stage outlet. But in the actual reservoir, \(\overline{{S_{w} }} \ne S_{we}\).

(3)After correction, the correlation coefficient between \(\frac{{k_{ro} }}{{k_{rw} }}\) and \(S_{w}\) is 95.92%.

(4)According to the comparison between the actual data and the calculated data, in 2021/3, the actual water cut is 97.29%, the water cut predicted by the formula is 97.27%, the actual cumulative oil production is 31.19 × 104t, and the predicted cumulative oil production is 31.31 × 104t. The predicted value is consistent with the actual value.

(5)The existing water flooding characteristic curve is corrected, and the improved water flooding characteristic curve in the high water cut period of low-amplitude structural reservoir with strong bottom water is obtained. The practical application is in line with the oilfield's actual situation, which provides a more reliable method for predicting the recoverability of low-amplitude structural reservoirs with strong bottom water and provides guidance for the subsequent decision-making of the oilfield.