This paper views each injector/producer pair together with the interwell formation as an integrated system. In this system, injection rate is considered as stimulus signal while production rate as response signal. The changes of injection rate at injectors result in the fluctuation of liquid production rate. This fluctuation reflects the characteristics of connectivity between injector/producer pairs. Particularly, the amplitude of fluctuation of production rate is related to the connectivity degree.

There will be the attenuation of stimulus (injection)signal in the reservoir, when stimulus (injection) signal is transmitted in the formation between wells. Therefore, we need to modify injection rate. Taking this into account, we use diffusivity filter to account for the time lag and attenuation that occurs between stimulus and response. Sampling the data of injection rate, we define the filter coefficient of the discrete filter function as [10]:
$$\begin{array}{}{\alpha}_{m}={\displaystyle \frac{\mathit{\Delta}\mathit{n}}{\tau}\mathrm{exp}\left[(m-n)/\tau \right]}\end{array}$$(3)
where *n* = the time of sampling, *Δ**n* = the selected discretization interval. Usually, the filters are discretized by sweep the effects of the most recent 12 months of injection. The convoluted injection rate of injector *i* affecting producer *j* at time *t* is given by [8]:
$$\begin{array}{}{i}_{ij}^{c}(t)={\displaystyle \sum _{n-0}^{m}{\alpha}_{ij}^{(n)}{i}_{i}(t-n)}\end{array}$$(4)
where
$\begin{array}{}{\alpha}_{ij}^{(n)}\end{array}$
is the *n*-th filter coefficient between injector *i* affecting producer *j*, *i*_{i} (*t* − *n*) is the observed injection rate at (*t* − *n*)-th time point.

Meanwhile, the pulse fluctuation of injection rate at injectors results in the pressure change. When pressure drop spreads to the location of producer, the production rate will change. Using the superposition principle, the pressure change can be expressed as [8]:
$$\begin{array}{}\mathit{\Delta}p=\left\{\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\begin{array}{ll}{C}_{1}\times {E}_{i}(-d\frac{{r}^{2}}{t})& \phantom{\rule{2em}{0ex}}t\le 1\\ {C}_{1}\times \left\{{E}_{i}(-d\frac{{r}^{2}}{t})-{E}_{i}(-d\frac{{r}^{2}}{(t-1)})\right\}\phantom{\rule{1em}{0ex}}& t>1\end{array}\right.\end{array}$$(5)
where *C*_{1} = a constant, *E*_{i} = the exponential function, *r* = the distance from the point to the well, *t* = time, and *d* = the dissipation constant of the medium where *d* = 1/*η*.

In this paper, the capacitance model is built based on a total mass balance with compressibility. Then, by making use of superposition in space, the governed material balance equation for producer *j* and *I* injectors is [10]:
$$\begin{array}{}{C}_{t}{V}_{p}\frac{d\overline{p}}{dt}+{q}_{j}(t)=\sum _{i=1}^{i=I}{\lambda}_{ij}{i}_{i}(t)\end{array}$$(6)
where *C*_{t} = the total compressibility, *V*_{p} = the drainage pore volume,
$\begin{array}{}\overline{p}\end{array}$
= the average pressure in *V*_{p}, *i*_{i}(*t*) = the injection rate of injector *i*, *q*_{j}(*t*)= the production rate at producer *j* and *λ*_{ij} = the weighting factors (the connectivity coefficient).

By solving this model which consists of one producer and multiple injectors, the generalized capacitance model for producer *j* in a discrete form is given by [10]:
$$\begin{array}{}{q}_{j}(t)={q}_{j}({t}_{0})\mathrm{exp}\left[{\displaystyle \frac{-(t-{t}_{0})}{{\tau}_{j}}}\right]\\ +{\displaystyle \sum _{i=1}^{i=I}{\lambda}_{ij}\frac{\mathrm{exp}(-t/{\tau}_{j})}{{\tau}_{j}}\underset{\xi ={t}_{0}}{\overset{\xi =t}{\int}}\mathrm{exp}(\xi /{\tau}_{j}){i}_{ij}(\xi )d\xi}\\ +{V}_{j}\left\{{p}_{wf}({t}_{0})\mathrm{exp}{\displaystyle [\frac{-(t-{t}_{0})}{{\tau}_{j}}]-{p}_{wf}({t}_{0})}\right.\\ \left.+{\displaystyle \frac{\mathrm{exp}(-t/{\tau}_{j})}{{\tau}_{j}}\underset{\xi ={t}_{0}}{\overset{\xi =t}{\int}}\mathrm{exp}(\xi /{\tau}_{j}){p}_{wf}(\xi )d\xi}\right\}\end{array}$$(7)
where *τ*_{j} = the time constant, where *τ*_{j} = *C*_{t}V_{p}/*J*, *J* = productivity index, *q*_{j}(*t*_{0}) = the initial production rate, *V*_{j} = the coefficient of bottom-hole pressure term and *p*_{wf} = the bottom-hole flowing pressure (BHP).

From the Equation 7, it can be concluded that the production signal consists of three components. The first component is the response of the initial production rate. The second component is the contribution from the injection signal, which is the most important component. The last component is the output signal caused by change in the bottom-hole flowing pressure (BHP) of the producer.

In the Equation 7, many variables need solving. For each producer, characteristic parameters of *λ*_{ij}, *τ*_{j} and *V*_{j} are unknown. This paper solves the model based on adaptive genetic algorithm, transforming the solution of the model to the process of continuous parameter optimization. The whole procedure of solution is shown in Figure 1. Ultimately, the final objective is to determine the coefficient of dynamic connectivity between injector/producer (*λ*_{ij}), and then it’s the base of the spatial locating of macroscopic throats.

Figure 1 The whole procedure of solution of the model

This paper compiles a relevant program based on the adaptive genetic algorithm, in order to make the inversion of dynamic connectivity more convenient. In this program, the modeled production rate can be exported. In the first step, the initial parameters that we set are the characteristic parameters of interwell formation, including the distance from the point to the well, the reservoir permeability, compressibility coefficient, etc. Then we calculate time constants (*τ*_{j}) using these parameters. The adaptive genetic algorithm only optimizes time constants. The injection data need pre-processing using Equation 3 and 4. The connectivity coefficient (*λ*_{ij}) and the coefficient of bottom-hole pressure term (*V*_{j}) are both solved by multiple linear regression model. Finally, we can get the satisfied connectivity coefficients.

Comparation between the modeled production rate and the observed prodution rate can intuitively reflect the fitting degree between them in the solution of the model. Typically, in this paper, the inversion of connectivity uses both the injection/production rates and the bottom-hole flowing pressure (BHP) of the producers to ensure the accuracy of the model.

This paper compared the accuracy of inversion of production rate before and after considering the bottom-hole flowing pressure (BHP), using the data obtained from the well group E5 in JZ oilfield. Figure 2 shows the inversion results of the total production rate in the well group E5 from February, 2005 to January, 2007. The red line indicates the modeled production rate, while the blue line indicates the real one.

Figure 2 Inversion results of the total production rate in the well group E5. (a) Not considering the BHP; (b) Considering the BHP

Correlation coefficient (*R*^{2}) is introduced here to analyse the quality of models. *R*^{2} is determined by [11]:
$$\begin{array}{}{R}^{2}=1-{\displaystyle \frac{\sum _{m=0}^{M}({q}_{j}^{(m)}-{\hat{q}}_{j}^{(m)}{)}^{2}}{\sum _{m=0}^{M}({q}_{j}^{(m)}-{\overline{q}}_{j}{)}^{2}}}\end{array}$$(8)
where
$\begin{array}{}{q}_{j}^{(m)}\end{array}$
= the real production rate,
$\begin{array}{}{\overline{q}}_{j}\end{array}$
= the average real rate, and
$\begin{array}{}{\hat{q}}_{j}^{(m)}\end{array}$
= the modeled production rate.

The closer *R*^{2} is to 1, the closer the modeled production rate is to the real production rate. That is, the model is more accurate and the connectivity of inversion is closer to the real case. *R*^{2} is 0.649 before considering the BHP, while after is 0.916. Figure 2 shows that the modeled production rate is closer to the real production rate after considering the BHP. Therefore, the connectivity of inversion is closer to the real case considering the BHP.

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