Introduction

Water flooding is an efficient method to maintain reservoir pressure, and it has been widely used to enhance oil recovery. Currently, most water-flooding reservoirs have been in a high water cut period in China. Due to the reservoir heterogeneity, different locations of the reservoir have different water flooding spread area, which results in that the distribution of remaining oil at high water cut stages are not the same.

At present, several major oil companies in China face with the problem of low permeability reservoir development. However, because of low porosity and low permeability, low permeability reservoirs have some differences with middle and high permeability reservoirs. A large number of studies have shown that (Deng and Liu 2001; Zhu et al. 2010; Yang et al. 2011; Sun 2010): the reservoir fluid in low permeability reservoirs does not follow Darcy’s law any more, and it is shown as a low-speed non Darcy percolation (Li et al. 2008; Zhang and Wang 2011; Fu and Ge 2002; Zhu 2007; Yang et al. 2007). Previous classical reservoir engineering methods have some limitations in the application and guidance of low permeability reservoir development. It is mainly showed in the following aspects:

  1. 1.

    The seepage model is too idealized, and only single-phase flow is considered, and the distribution of remaining oil in the reservoir and the effect of water flooding are not fully considered;

  2. 2.

    The calculation unit is often concentrated in one injection and production unit, and there is still a certain gap with the actual low permeability reservoir development well network;

  3. 3.

    The derivation process is complicated and inconvenient to calculate.

Therefore, it is necessary to establish an engineering calculation model for low permeability reservoir based on non-Darcy seepage.

At present, low permeability reservoirs are mainly exploited by area well pattern. In view of the productivity calculation of area well pattern, many researches have been done by predecessors. Using the principle of pressure superposition and the mirror inversion method, Luo et al. (2010) studied the pseudo-steady flow productivity and injection production pressure difference model was built for low permeability reservoirs with closed boundary. By combining the seepage characteristics of low permeability reservoirs and using the equivalent percolation resistance method, Du Dianfa et al. (2012) gives the expression of the area well network productivity. Based on the productivity formula of the area well network in the ultra-low permeability reservoir, Xu et al. (2014) used the superposition principle and the equivalent well diameter model to build the productivity expression for the joint area of horizontal well and straight well. He et al. (2009) adopt the flow pipe method to establish the production model of the low permeability rectangular reservoir fracturing straight well. Using the similarity principle of hydropower Zhao et al. (2008) derived the productivity model of different types of well pattern.

Previous studies are summarized in three aspects: starting pressure gradient model, productivity splitting model and quasi starting pressure gradient model. Based on the previous research results, this paper uses the streamline integration method (Munseok 2000; Higgins and Leighton 1961; Ji et al. 2008; He et al. 2009a, b), based on the non-Darcy flow and the two-parameter model, an area well network productivity model considering the starting pressure gradient is built, which makes the production calculation and the well network optimization closer to the actual production. By applying element analysis, an injection-producing unit of a fractured five-spot pattern is divided into four sub-units, each of which is divided into three calculation units in accordance with streamline distribution characteristics. On this basis, the injection production pressure difference and the well spacing are analyzed. The start-up coefficient is introduced to evaluate the utilization degree of low permeability reservoir.

Physical model for fractured five-spot patterns at high water cut stages

The schematic illustration in Fig. 1a shows an injection-producing unit of a fractured five-spot pattern. The fracture half-length of each injection well is the same which equals to \({L_{{\text{fw}}}}\). The fracture half-length of production well equals to \({L_{{\text{fo}}}}\). Well spacing and spacing of well array equal to \({L_1}\) and \({L_2}\), respectively. By applying element analysis, an injection-producing unit of a fractured five-spot pattern is divided into four SUs, each of which is divided into three CUs in accordance with streamline distribution characteristics. Therefore, an injection-producing unit of a fractured five-spot pattern is divided into 12 CUs totally, as shown in Fig. 1b.

Fig. 1
figure 1

The illustration of an injection-producing unit for a fractured five-spot pattern and its CUs

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The assumptions are as follows:

  • The saturation distributed in the injection-producing unit is not uniform at the high water cut stage. Various SUs have different saturation. However, Saturation for a SU is the same.

  • Before seeing water in the oil well, the permeability is constant;

  • In an injection unit, the starting pressure gradient is constant;

  • The two-phase unstable percolation occurs in the fluid flow in the stratum, and the fluid viscosity is constant.

  • The effects of gravity and capillary forces are ignored;

  • In the injection unit, no gas is dissolved or escaped.

  • Fractures of injection well and production well have infinite conductivity

Productivity of an injection–production unit for a fracture five-spot pattern equals to the sum of productivities for all CUs. Method of productivity for every CU is introduced in the next part.

There are many methods for describing mathematical models of nonlinear seepage at present (Ruina and Xiaodong 2011; Baoquan et al. 2011; Xu et al. 2018), as shown in Table 1. The seepage model in Table 1 is mainly divided into the proposed pressure gradient model, segmented model and continuous model. At present, the nonlinear seepage is mainly based on phenomenological method. The reason is that the classical Darcy’s law cannot describe the percolation characteristics of the low-permeability reservoirs. The low-permeability reservoirs have the starting pressure, boundary layer and micro-scale flow effects, and the starting pressure is not a constant which increases the difficulty and accuracy of describing the equation of motion of the true seepage law of low-permeability reservoir fluids.

Table 1 Nonlinear mathematical models
Full size table

Where \(v\) is the seepage velocity for a flow tube, sm/day; \(\nabla p\)is the displacement pressure between the injection well and the production well, MPa; \(k\) is the effective permeability for fluid flow, 10− 3 µm2; \({a_1},{a_2},a,b\) are the model parameters and their values are related to the actual reservoir; \(\mu\) is the fluid viscosity, mPa s; \(G\) is the starting pressure gradient, MPa/m; \(L\) is the distance from injection well along the centerline for a flow tube, m.

Productivity for every CU

The well pattern productivity equals to the sum of the production for all flow tubes which are distributed in the CU. Therefore, it is necessary for the production well to calculate the production of a flow tube.

From Table 1, the two-parameter continuous model reflect the seepage flow of underground fluid well, and it is also very easy to get the two-parameter in this model, so this paper chooses the two-parameter continuous model as the basic seepage model to build the production model for the flow tube.

Production for a flow tube

The injection unit interface diagram is shown in Fig. 2. By rewriting the two-parameter continuous model and taking the oil–water two-phase flow into consideration, the production of a cross-sectional of a flow tube is given by the following equation,

Fig. 2
figure 2

Single flow tube sketch diagram

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$${q_{\text{o}}}=c\frac{{{k_{\text{o}}}({{\overline {s} }_{\text{w}}})}}{{{\mu _{\text{o}}}}}A(\xi )\left( {1 - \frac{1}{{a+b\frac{{{\text{d}}{{\text{p}}_\xi }}}{{{\text{d}}\xi }}}}} \right),$$
(1)

where \({q_{\text{o}}}\) is the production of a single flow tube, sm3/day; \(c\) is unit conversion factors; \({k_{\text{o}}}({\overline {s} _{\text{w}}})\) is the effective permeability of oil, 10− 3 µm2; \({\overline {s} _{\text{w}}}\) is the average water saturation; \({\mu _{\text{o}}}\) is the oil viscosity, mPa s; \(\xi\) is the distance between the injection well and the cross section, m; \(A(\xi )\) is the cross-sectional area, m2; \({\text{d}}{{\text{p}}_\xi }\) is the displacement pressure of the cross section, MPa; \(\lambda\) is the starting pressure gradient, MPa/m; a, b are the parameters of the model, the values are related to the actual reservoir.

Equation (1) can be rewritten as:

$${\text{d}}{{\text{p}}_\xi }=\left( {\frac{{1 - a}}{{2b}}+\frac{M}{2}+\frac{{\sqrt {{{(a+1+Mb)}^2} - 4a} }}{{2b}}} \right){\text{d}}\xi ,$$
(2)

where \(M=\frac{{\Delta q}}{{\sum\nolimits_{{i=1}}^{2} {A({\xi _i})} }}\frac{\mu }{k}\).

Integrating of Eq. (2) over \(\xi\), the production for a flow tube can be obtained, as the following:

$${q_{\text{o}}}=c\frac{{\frac{{{k_{\text{o}}}({{\overline {s} }_{\text{w}}})}}{{{\mu _{\text{o}}}}}({P_{{\text{in}}}} - {P_{{\text{pro}}}} - \lambda l)}}{{\int_{l} {\frac{{d\xi }}{{A(\xi )}}} }},$$
(3)

where \({P_{{\text{in}}}}\) is the injection well pressure, MPa; \({P_{{\text{pro}}}}\) is the production well pressure, MPa; \(l\) is the length of a single flow tube, m.

Production for a triangular CU

When the two phases of oil and water reach pseudo-steady state flow, the production of each triangular CU is the same. To simplify the model, this paper just takes the CU 1 as an example to calculate the production of a triangular CU, which is shown in Fig. 3.

Fig. 3
figure 3

The illustration of CU 1

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By rewriting the Eq. (3), the production of a single flow tube of CU 1 must meet:

$${q_{{\text{o}}1}}=c\frac{{\frac{{{k_{\text{o}}}({{\overline {s} }_{\text{w}}})}}{{{\mu _{\text{o}}}}}({P_{{\text{in}}}} - {P_{{\text{pro}}}} - \lambda {l_1})}}{{\int_{{{l_1}}} {\frac{{d\xi }}{{A(\xi )}}} }},$$
(4)

where \({l_1}\) is the length of a single flow tube of CU 1, m.; \({q_{{\text{o}}1}}\) the production of a flow tube of CU 1, sm3/day.

In Fig. 3, the following relationship can be easily obtained:

$$\frac{\alpha }{\beta }=\frac{{\Delta \alpha }}{{\Delta \beta }}=\frac{{{\alpha _m}}}{{{\beta _m}}}=n,$$
(5)
$${x_i}=\frac{{\omega d}}{{\sin (\alpha +\beta )}}\sin \left[ {\frac{{\alpha +\beta }}{2} - {{( - 1)}^i}\frac{{\alpha - \beta }}{2}} \right],\quad {\kern 1pt} i=1,2,$$
(6)
$${A_i}(\xi )=2h{\xi _i}tg\left[ {\frac{{\Delta \alpha +\Delta \beta }}{4} - {{( - 1)}^i}\frac{{\Delta \alpha - \Delta \beta }}{4}} \right],\,{r_w} \leqslant {\xi _i} \leqslant {x_i},\quad i=1,2,$$
(7)

where \({r_w}\) is the well radius, m; \(\alpha\), \(\beta\) is the angle of the CU1 between the selected point of the main streamline from the production well and the injection well, respectively, °; \({\alpha _m}\), \({\beta _m}\) is the angle of flow tube between the selected point of the main streamline from the production well and the injection well, respectively, °; \(\Delta \alpha\), \(\Delta \beta\) is the angle increment of flow tube between the selected point of the main streamline from the production well and the injection well, respectively, °; \(n\) is the shape factor of CU 2;

Substitute Eqs. (5)–(7) into Eq. (8), it can be obtained that:

$${l_1}=\sqrt {L_{1}^{2}+{{\left( {\frac{{{L_2}}}{2} - {x_1}} \right)}^2}} \frac{{(\sin {\alpha _m}+\sin {\beta _m})}}{{\sin ({\alpha _m}+{\beta _m})}} - {r_{\text{w}}}.$$
(8)

By the way, \(A(\xi )\) is also can be obtained:

Substitute Eqs. (7)–(8) into Eq. (4), it can be obtained that:

$${q_{{\text{o}}1}}=c\frac{{\frac{{{k_{\text{o}}}({{\overline {s} }_{\text{w}}})}}{{{\mu _{\text{o}}}}}\left( {{P_{{\text{in}}}} - {P_{{\text{pro}}}} - \lambda \left( {\sqrt {L_{1}^{2}+{{\left( {\frac{{{L_2}}}{2} - {x_i}} \right)}^2}} \frac{{(\sin {\alpha _m}+\sin {\beta _m})}}{{\sin ({\alpha _m}+{\beta _m})}} - {r_{\text{w}}}} \right)} \right)}}{{\frac{1}{{2h\tan \left( {\frac{{\Delta \alpha }}{2}} \right)}}\ln \left( {\frac{{\sqrt {L_{1}^{2}+{{\left( {\frac{{{L_2}}}{2} - {L_{{\text{fw}}}}} \right)}^2}} \sin {\beta _m}}}{{\sin ({\alpha _m}+{\beta _m})M}}} \right)+\frac{1}{{2h\tan \left( {\frac{{\Delta \beta }}{2}} \right)}}\ln \left( {\frac{{\sqrt {L_{1}^{2}+{{\left( {\frac{{{L_2}}}{2} - {L_{{\text{fw}}}}} \right)}^2}} \sin {\alpha _m}}}{{\sin ({\alpha _m}+{\beta _m})M}}} \right)}}.$$
(9)

The production of CU 1 can be obtained:

$$\begin{gathered} {Q_{{\text{o}}1}}=\int\limits_{0}^{\alpha } {{q_{{\text{o}}1}}} {\text{d}}\alpha \hfill \\ \quad =c\int\limits_{0}^{\alpha } {\frac{{\frac{{{k_{\text{o}}}({{\overline {s} }_{\text{w}}})}}{{{\mu _{\text{o}}}}}\left( {{P_{{\text{in}}}} - {P_{{\text{pro}}}} - \lambda \left( {\sqrt {L_{1}^{2}+{{\left( {\frac{{{L_2}}}{2} - {x_i}} \right)}^2}} \frac{{(sin{\alpha _m}+sin{\beta _m})}}{{\sin ({\alpha _m}+{\beta _m})}} - {r_{\text{w}}}} \right)} \right)}}{{\frac{1}{{2h\tan \left( {\frac{{\Delta \alpha }}{2}} \right)}}\ln \left( {\frac{{\sqrt {L_{1}^{2}+{{\left( {\frac{{{L_2}}}{2} - {L_{{\text{fw}}}}} \right)}^2}} sin{\beta _m}}}{{\sin ({\alpha _m}+{\beta _m})M}}} \right)+\frac{1}{{2h\tan \left( {\frac{{\Delta \beta }}{2}} \right)}}\ln \left( {\frac{{\sqrt {L_{1}^{2}+{{\left( {\frac{{{L_2}}}{2} - {L_{{\text{fw}}}}} \right)}^2}} sin{\alpha _m}}}{{\sin ({\alpha _m}+{\beta _m})M}}} \right)}}} {\text{d}}\alpha , \hfill \\ \end{gathered}$$
(10)

where \({Q_{{\text{o}}1}}\) is the production of CU 1, sm3/day.

Production for a quadrilateral CU

When the two phases of oil and water reach pseudo-steady state flow, the production method for each quadrilateral CU is the same. To simplify the model, this paper just takes the CU 1 as an example to calculate the production of a triangular CU, which is shown in Fig. 4.

Fig. 4
figure 4

The illustration of CU 2 and its flow tube in the situation where \({L_{{\text{wf}}}} \ne {L_{{\text{wo}}}}\)

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By rewriting the Eq. (3), the production of a single flow tube of CU 2 must meet:

$${q_{{\text{o}}2}}=\frac{{\frac{{{k_{\text{o}}}({{\overline {s} }_{\text{w}}})}}{{{\mu _{\text{o}}}}}({P_{{\text{in}}}} - {P_{{\text{pro}}}} - \lambda {l_2})}}{{\int\limits_{{{l_2}}} {\frac{{d\xi }}{{A(\xi )}}} }},$$
(11)

where \({l_2}\) is the length of a single flow tube of CU 1, m; \({q_{{\text{o}}2}}\) the production of a flow tube of CU 1, sm3/day.

In Fig. 4, the following relationship can be easily obtained:

$${l_2}=\sqrt {0.5({L_2} - {L_{{\text{fw}}}} - {L_{{\text{fo}}}})} ,$$
(12)
$$\frac{{{L_{{\text{fo}}}}}}{{{L_{{\text{fw}}}}}}=\frac{{\Delta {L_{{\text{fo}}}}}}{{\Delta {L_{{\text{fw}}}}}}=m,$$
(13)

where \(A^{\prime}F^{\prime}\) equals to \(\Delta {L_{{\text{fo}}}}\), m; \(C^{\prime}E^{\prime}\) equals to \(\Delta {L_{{\text{fw}}}}\), m; \(m\) is the shape factor of CU 2.

Substitute Eqs. (15)–(17) into Eq. (14), it can be obtained that:

$${q_{{\text{o}}2}}=c\frac{{{k_{\text{o}}}({{\overline {s} }_{\text{w}}})}}{{{\mu _{\text{o}}}}}\left[ {\frac{{\Delta q}}{{\Delta a}}\frac{1}{{2mh}}\ln \frac{{{x_1}}}{{{r_{\text{w}}}}}+\frac{{\Delta q}}{{\Delta a}}\frac{1}{{2mh}}} \right].$$
(14)

The production of CU 2 can be obtained:

$$\begin{gathered} {Q_{{\text{o}}2}}=\int\limits_{0}^{\alpha } {{q_{{\text{o}}2}}} {\text{d}}\alpha \hfill \\ \quad =c\frac{{{k_{\text{o}}}({{\overline {s} }_{\text{w}}})}}{{{\mu _{\text{o}}}}}\left[ {\int\limits_{{{r_{\text{w}}}}}^{{{x_1}}} {\frac{{\sqrt {{{\left( {a+1+\frac{{\Delta q}}{{\Delta a}}\frac{{\mu b}}{{kh{\xi _1}}}} \right)}^2} - 4a} }}{{2b}}} {\text{d}}{\xi _1}+\int\limits_{{{r_{\text{w}}}}}^{{{x_2}}} {\frac{{\sqrt {{{\left( {a+1+\frac{{\Delta q}}{{\Delta a}}\frac{{\mu b}}{{nkh{\xi _2}}}} \right)}^2} - 4a} }}{{2b}}} {\text{d}}{\xi _2}} \right], \hfill \\ \end{gathered}$$
(15)

where \({Q_{{\text{o}}2}}\) is the production of CU 2, sm3/day.

Above these two situation, the production for a five-spot pattern can be obtained:

$${Q_{\text{o}}}=\sum\limits_{{i=1}}^{{12}} {{Q_{{\text{o}}i}}} .$$
(16)

Transient productivity method for fractured five-spot patterns

According to the material conservation theory, it can be obtained that:

$$\overline {s} _{{{\text{w}}j}}^{{n+1}}=\overline {s} _{{{\text{w}}j}}^{n}+\frac{{{Q_{{\text{os}}j}}\Delta t{B_{\text{o}}}}}{{{V_{\phi j}}}},$$
(17)

where \(j\) is the number of SU; \({\overline {s} _{_{{{\text{w}}j}}^{n}}}\) and \({\overline {s} _{_{{{\text{w}}j}}^{{n+1}}}}\) are the average water saturation of SU \(j\) at the time \({t_n}\) and \({t_{n+1}}\), respectively. \({V_{\phi j}}\) is the pore volume of SU \(j\). \({Q_{{\text{os}}j}}\) is the production of SU \(j\) at the time \({t_n}\), \({\text{s}}{{\text{m}}^{\text{3}}}{\text{/day}}\), \(\Delta t={t_{n+1}} - {t_n}\).

So the \({Q_{{\text{os}}j}}\) (j = 1, 2, 3 or 4) can be written as the following:

$${Q_{{\text{os}}1}}=\sum\limits_{{j=1}}^{3} {{Q_{{\text{o}}j}}} ,$$
(18)
$${Q_{{\text{os}}2}}=\sum\limits_{{j=4}}^{6} {{Q_{{\text{o}}j}}} ,$$
(19)
$${Q_{{\text{os}}3}}=\sum\limits_{{j=7}}^{9} {{Q_{{\text{o}}j}}} ,$$
(20)
$${Q_{{\text{os}}4}}=\sum\limits_{{j=10}}^{{12}} {{Q_{{\text{o}}j}}} .$$
(21)

By combining the Eqs. (18)–(21), and by following the operation of ‘six step method’, the transient productivity of fractured five-spot patterns can be obtained.

The method is present as the following (Yao et al. 2014; Yoa and Wang 2004):

  1. 1.

    ① The initial reservoir parameters and saturation for each SU are known;

  2. 2.

    ② Calculate the production of every CU at the beginning moment using Eqs. (10) and (15);

  3. 3.

    ③ Calculate the production of every SU at the beginning moment using Eqs. (18)–(21);

  4. 4.

    ④ Calculate the production for fractured five-spot patterns at the beginning moment using Eq. (16);

  5. 5.

    ⑤ Calculate the saturation for each SU at the next moment using Eq. (17);

  6. 6.

    ⑥ Update the saturation for each CU in the steps ①, Repeat steps ②–⑥, one can get the production for fractured five-spot patterns at the next moment; Finally, we get the transient productivity.

Reservoir basic parameters and productivity method validation

Reservoir basic parameters

The Lasa Xing oil field is located in the Changshu anticlinal structural belt in the Central Depression of the Songliao Basin, and it is also located in the most favorable area for oil production and oil storage in the Songliao basin. The sedimentary system is a large river-delta body, and the lithology is mainly fine sand and siltstone. The air permeability of the main oil layer is 0.5 µm− 2 or less, and the permeability of the thin sand layer is between 0.02 and 0.08 µm− 2, and the permeability of the off-sheet reservoir is between 0.001 and 0.05 µm− 2, which is an anticline structured reservoirs and also a low permeability reservoir. The Lasa Xing oil field is now in the high water cut stage. This paper takes the Lasa Xing oil field as an example to show the method of calculating productivity for five-spot well pattern.

The input value which are necessary for the model to calculating the productivity is shown in Table 2.

Table 2 The value of the input parameters
Full size table

The starting pressure gradient is given as the following:

$$\lambda =\varepsilon {\left( {\frac{k}{{\mu _{{\text{w}}}^{{{S_{{\text{w}}n}}}}\mu _{{\text{o}}}^{{1 - {S_{{\text{w}}n}}}}}}} \right)^{ - n}},$$
(22)

where \(\varepsilon\) and \(n\) are regression coefficients, and the value of \(\varepsilon\) and \(n\) are given as 1.2116 and 0.9428 in this paper. \({{\text{S}}_{{\text{w}}n}}\) is the normalized water saturation, which can be obtained by the following:

$${S_{{\text{w}}n}}=\frac{{{S_{\text{w}}} - {S_{{\text{wc}}}}}}{{1 - {S_{{\text{wc}}}} - {S_{{\text{or}}}}}},$$
(23)

where \({S_{{\text{wc}}}}\) is the irreducible water saturation and \({S_{{\text{or}}}}\) is the residual oil saturation of the reservoir.

The relative permeability of water phase can be obtained using the Parker–Lenhard model, that is:

$${k_{{\text{rw}}}}={(1 - {s_{\text{o}}})^{0.5}}{\{ 1 - {[1 - {(1 - {s_{\text{o}}})^{\frac{1}{m}}}]^m}\} ^2}.$$
(24)

The relative permeability of oil phase can be obtained by using the Brooks–Corey–Burdine model, that is

$${k_{{\text{ro}}}}={s_{\text{o}}}^{{0.5}}{[1 - {(1 - {s_{\text{o}}})^{\frac{1}{m}}}]^{2m}}.$$
(25)

\({k_{{\text{ro}}}}\), \({k_{{\text{rw}}}}\) is the relative permeability of the oil phase and the water phase, respectively; \({s_{\text{o}}}\) is the oil saturation; \(m\) is the fitting parameter.

Productivity method validation

Step 1 Input parameters.

Through the fracture construction design report and fracture monitoring data, the fracture parameters of the injection–production well, such as the length of the fracture, are obtained. It is noted that the fracture half-length of the injection well is 125 m, and the fracture length of the production well is 114 m, which is not equal, so the Eq. (19) is used to obtain the production for a quadrilateral CU.

Step 2 Saturation distribution.

The target block is produced from October 10 in 2012 to February 11 in 2015. Now select the production time period: October 10 in 2012 to February 10 in 2015. The producing time is 40 months, and the time step will be set to 1 month. Calculated 40 times. Substituting the known parameters into Eq. (21), the saturation distribution of the previous time step and the next time step can be obtained, that is:

$$\left\{ \begin{gathered} \overline {s} _{{{\text{w}}1}}^{{n+1}}=\overline {s} _{{{\text{w}}1}}^{n}+3.25\quad {\kern 1pt} (\overline {s} _{{{\text{w}}1}}^{0}=0.62) \hfill \\ \overline {s} _{{{\text{w}}2}}^{{n+1}}=\overline {s} _{{{\text{w}}2}}^{n}+4.11\quad (\overline {s} _{{{\text{w}}2}}^{0}=0.61) \hfill \\ \overline {s} _{{{\text{w}}3}}^{{n+1}}=\overline {s} _{{{\text{w}}3}}^{n}+2.94\quad (\overline {s} _{{{\text{w}}3}}^{0}=0.57) \hfill \\ \overline {s} _{{{\text{w}}4}}^{{n+1}}=\overline {s} _{{{\text{w}}4}}^{n}+1.07\quad (\overline {s} _{{{\text{w}}4}}^{0}=0.58). \hfill \\ \end{gathered} \right.$$
(26)

Step 3 Starting pressure gradient.

The saturation obtained in step 2 is normalized and substituted into Eq. (22) to obtain the starting pressure gradient at different time steps, that is:

$$\left\{ \begin{gathered} {G_1}=1.2427 \times {\left( {\frac{{32.3}}{{{{0.51}^{{S_{{\text{w}}n}}}}{{6.4}^{1 - {S_{{\text{w}}n}}}}}}} \right)^{ - 0.9753}} \hfill \\ {G_2}=1.2427 \times {\left( {\frac{{30.4}}{{{{0.51}^{{S_{{\text{w}}n}}}}{{6.4}^{1 - {S_{{\text{w}}n}}}}}}} \right)^{ - 0.9753}} \hfill \\ {G_3}=1.2427 \times {\left( {\frac{{25.3}}{{{{0.51}^{{S_{{\text{w}}n}}}}{{6.4}^{1 - {S_{{\text{w}}n}}}}}}} \right)^{ - 0.9753}} \hfill \\ {G_4}=1.2427 \times {\left( {\frac{{29.1}}{{{{0.51}^{{S_{{\text{w}}n}}}}{{6.4}^{1 - {S_{{\text{w}}n}}}}}}} \right)^{ - 0.9753}}. \hfill \\ \end{gathered} \right.$$
(27)

The phase-infiltration curve is obtained using the initial oil saturation and the combined formulas (24) and (25).

The relative permeability curves of the reservoir are shown in Fig. 5.

Fig. 5
figure 5

Relative permeability curves

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Step 4 The transient productivity can be obtained, which is shown in Fig. 6.

Fig. 6
figure 6

Calculating results between the new method and the actual data

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Through the 12 subdivided units, the capacity calculation formulas of the triangular injection unit and the quadrilateral injection unit can be obtained, that is:

$$\left\{ \begin{gathered} {Q_{{\text{o}}1}}=\left. {\frac{{{k_{{\text{o}}1}}(s_{{{\text{w}}1}}^{0})(8.01 - 0.372{G_1})}}{{7.296}}} \right|_{0}^{{31.4}} \hfill \\ {Q_{{\text{o}}2}}=\left. {\frac{{{k_{{\text{o}}2}}(s_{{{\text{w}}2}}^{0})(8.01 - 0.372{G_2})}}{{7.296}}} \right|_{0}^{{27.3}} \hfill \\ {Q_{{\text{o}}3}}=\left. {\frac{{{k_{{\text{o}}3}}(s_{{{\text{w}}3}}^{0})(8.01 - 0.372{G_3})}}{{7.296}}} \right|_{0}^{{39.5}} \hfill \\ {Q_{{\text{o}}4}}=\left. {\frac{{{k_{{\text{o}}4}}(s_{{{\text{w}}4}}^{0})(8.01 - 0.372{G_4})}}{{7.296}}} \right|_{0}^{{29.7}}. \hfill \\ \end{gathered} \right.$$
(28)

The saturation obtained in the above step and the sub-unit average permeability inversely combined with the phase-infiltration curve are sequentially substituted into the corresponding sub-unit productivity model (28), and then using the iterative method, the transient productivity can be obtained, which is shown in Fig. 6.

Figure 6 shows the calculating results between the new method and the actual data. Figure 6 shows that the data of new method and the actual data have a quite good agreement, which means that the new method in this paper is reliable. Therefore, it is suitable to use this method to predict the transient productivity of five-spot patterns.

Figure 7 shows that it takes only 21.282 s to run the program of transient productivity prediction for a fractured five-spot pattern once, when the computer’s processor model is Intel(R) Core(TM) i7-6700 CPU @3.4 GHz.

Fig. 7
figure 7

Running time of productivity prediction program

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When \(\lambda\) approaches zero, it becomes Darcy’s seepage. The pressure difference between the production well and the injection well is 15 MPa, and the permeability is 0.033 mD, and the thickness of the oil layer is 5 m, and the viscosity is 5 mPa s, and the radius of wellbore is 0.1 m. Calculate the single well output of the deduced five-point well network (this paper solution) and compare it with the Muskat (1937) calculation solution (reference solution), and the calculation results comparing between this paper solution and the reference solution is shown in Table 3. Table 3 shows that the relative error is within 0.5%, which shows the method in this paper is feasible and accurate.

Table 3 calculation results between this paper solution and the reference solution
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Both the seepage theory and the reservoir engineering illustrate the correctness of the new method in this paper.

Sensitivity analysis

In this section, using the single factor variable principle, the impacts of some relevant parameters on the transient productivity of five-spot patterns are analyzed. The relevant parameters are shown in Table 2.

Figures 8 and 9 illustrate the impacts of fracture length on production and cumulative production for a fractured five-spot pattern. Seen from Figs. 8 and 9, the length of fracture has a significant effect on productivity or cumulative production for fracture five-spot patterns. The bigger the fracture length is, the more the production rate will be. From the analysis of flow regime, the flow regime in the area of CU 2, CU 4, CU 6 and CU 8 is linear flow. The flow regime in other CUs is radial flow. The bigger the fracture length, the bigger the area of CU 2, CU 4, CU 6 and CU 8 will be. Therefore, the flow resistance for a fractured five-spot pattern will be small and the productivity rate will be larger, when the fracture is longer.

Fig. 8
figure 8

Effects of fracture length of injection well on production

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Fig. 9
figure 9

Effects of fracture length of production well on production

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Figure 10 shows the impacts of pressure difference between injection well and production well on production. Seen from Fig. 10, the bigger the pressure difference is, the more the productivity rate will be.

Fig. 10
figure 10

Effects of pressure difference between injection well and production well on production

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Conclusions

Using the methods of elemental analysis and flow tube integration, a new productivity of predicting five-spot well pattern is built, and this new method takes the remaining oil heterogeneity into account. This new method can accurately predict the productivity of the five-point model of fractured well in high water cut.

The fracture length and pressure difference between the injection well and the production well have a significant effect on the productivity of the fractured five-point well pattern. The greater the length of the fracture is, the higher the productivity of the fractured five-point well pattern is. The greater the pressure difference is, the higher the productivity of the fractured five-point well pattern is.