The productivity calculating method of a quadrilateral CU is the same. Therefore, taking CU 2 (Figure 4) as an example to present the method for calculating productivity of the quadrilateral CU.

Figure 4 Illustration of CU 2 and its flow tube in the situation where*L*_{wf} ≠ *L*_{wo}

From Eq. (3), the production of one flow tube in CU 2 can be obtained by

$$\begin{array}{}{\displaystyle \mathit{\Delta}{q}_{o2}=\gamma \frac{\frac{{k}_{o}({s}_{w})}{{\mu}_{o}}({P}_{in}-{P}_{pro}-G{l}_{2})}{\underset{{l}_{2}}{\int}\frac{d\xi}{A(\xi )}}\frac{1}{{B}_{o}}}\end{array}$$(16)

where *Δq*_{o2} is the production of one flow tube in CU 2, sm^{3}/d; andd *l*_{2} is the centerline length of one flow tube, m.

Here are two situations in which to calculate *A*(*ξ*). One situation is that the fracture length of the injection well does not equal the fracture length of the production well (seen in Figure 4). From the geometric relationship in Figure 4, the following expressions are obtained:

$$\begin{array}{}{\displaystyle {l}_{2}=\sqrt{0.5({L}_{2}-{L}_{fw}-{L}_{fo})}}\end{array}$$(17)

$$\begin{array}{}{\displaystyle A(\xi )=2h\left({A}^{\prime}{F}^{\prime}-\frac{({A}^{\prime}{F}^{\prime}-{\mathrm{C}}^{\prime}{E}^{\prime})}{{l}_{2}}\xi \right)}\end{array}$$(18)

$$\begin{array}{}{\displaystyle \frac{{L}_{fo}}{{L}_{fw}}=\frac{\mathit{\Delta}{L}_{fo}}{\mathit{\Delta}{L}_{fw}}=ratio2}\end{array}$$(19)

Here *A*′*F*′ is the distance between *A*′ and *F*′, and is equal to *ΔL*_{fo}, m; C′*E*′ is the distance between *C*′ and *E*′, and is equal to *ΔL*_{fw}, m; and *ratio*2 is a constant which depends on the shape of CU 2.

Substituting Eqs. (17) and (18) into Eq. (16), then simplifying, the production calculating expression of one flow tube is as follows:

$$\begin{array}{}{\displaystyle \mathit{\Delta}{q}_{o2}=\gamma}\\ {\displaystyle \frac{\frac{{k}_{o}({s}_{w})}{{\mu}_{o}}\left({P}_{in}-{P}_{pro}-G\left(\sqrt{0.5({L}_{2}-{L}_{fw}-{L}_{fo})}\right)\right)}{\frac{\sqrt{0.5({L}_{2}-{L}_{fw}-{L}_{fo})}}{2h(\mathit{\Delta}{L}_{fo}-\mathit{\Delta}{L}_{fw})}\mathrm{ln}(\frac{\mathit{\Delta}{L}_{fo}}{\mathit{\Delta}{L}_{fw}})}\frac{1}{{B}_{o}}}\end{array}$$(20)

The productivity of CU 2 is equal to the sum of the production of all flow tubes in CU 2, so the productivity of CU 2 can be obtained by

$$\begin{array}{}{\displaystyle {Q}_{o2}=\underset{0}{\overset{{}_{{L}_{fw}}}{\int}}\left(\underset{\mathit{\Delta}{L}_{fw}\to 0}{lim}\frac{\mathit{\Delta}{q}_{o1}}{\mathit{\Delta}{L}_{fw}}\right)dL}\end{array}$$(21)

where *Q*_{o2} is the productivity for CU 2 in the situation where *L*_{wf} ≠ *L*_{wo}, sm^{3}/d. *L* is the distance from injection well along the fracture of injection well, m.

Substituting Eq. (20) into (21), and then simplifying by Eq. (19), the expression of *Q*_{o2} can be rewritten as follows:

$$\begin{array}{}{\displaystyle {Q}_{o2}=}\\ {\displaystyle \gamma \frac{\frac{{k}_{o}({s}_{w})}{{\mu}_{o}}\left({P}_{in}-{P}_{pro}-G\left(\sqrt{0.5({L}_{2}-{L}_{fw}-{L}_{fo})}\right)\right)}{\frac{\sqrt{0.5({L}_{2}-{L}_{fw}-{L}_{fo})}}{2h(ratio2-1)}\mathrm{ln}(ratio2)}\frac{1}{{B}_{o}}{L}_{fw}}\end{array}$$(22)

Another situation is that the fracture length of the injection well is equal to the fracture length of the production well. Thus, the value of *ratio*2 is as follows:

$$\begin{array}{}{\displaystyle ratio2=1}\end{array}$$(23)

Substituting Eq. (23) into Eq. (22), we can find that the denominator (*ratio*2 – 1) is zero, so the expression is not a legal equation. Therefore, under the situation where the fracture length of the injection well is equal to that of the production well (Figure 5), Eq. (22) can’t be used and the whole of CU 2 can be considered as a flow tube.

Figure 5 Illustration of CU 2 in the situation where *L*_{wf} = *L*_{wo}

The following expressions for describing the geometric relationship in Figure 5 are as follows:

$$\begin{array}{}{\displaystyle A(\xi )=h{L}_{fo}}\end{array}$$(24)

$$\begin{array}{}{\displaystyle {l}_{2}=\sqrt{{L}_{1}^{2}+{\left(\frac{{L}_{2}}{2}-{L}_{fo}\right)}^{2}}}\end{array}$$(25)

Substituting Eqs. (24) and (25) into Eq. (16), the productivity of CU 2 can be calculated by

$$\begin{array}{}{\displaystyle Q{}_{o2}^{\prime}=\mathit{\Delta}{q}_{o2}}\\ {\displaystyle \phantom{\rule{2em}{0ex}}=\gamma \frac{\frac{{k}_{o}({s}_{w})}{{\mu}_{o}}\left({P}_{in}-{P}_{pro}-G\left(\sqrt{{L}_{1}^{2}+{(\frac{{L}_{2}}{2}-{L}_{fo})}^{2}}\right)\right)}{h{L}_{fo}\sqrt{{L}_{1}^{2}+{\left(\frac{{L}_{2}}{2}-{L}_{fo}\right)}^{2}}}\frac{1}{{B}_{o}}}\end{array}$$(26)

where
$\begin{array}{}{\displaystyle Q{}_{o2}^{\prime}}\end{array}$ is the productivity of CU 2 in the situation where *L*_{wf} = *L*_{wo}, sm^{3}/d.

The productivity for a fractured five-spot pattern is equal to the sum of the productivity of all CUs, so the productivity for a fractured five-spot pattern can be obtained by

$$\begin{array}{}{\displaystyle {Q}_{o}=\sum _{i=1}^{12}{Q}_{oi}}\end{array}$$(27)

where *Q*_{o} is the productivity for a fractured five-spot pattern, sm^{3}/d; *i* is the number of CU; and *Q*_{oi} is the productivity of CU *i*, sm^{3}/d.

Equation (27) is used to calculate the steady state productivity. How to obtain the transient productivity by using the steady state productivity mentioned above will be introduced below.

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