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# Open Physics

### formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

Managing Editor: Lesna-Szreter, Paulina

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Volume 15, Issue 1

# The method of the spatial locating of macroscopic throats based-on the inversion of dynamic interwell connectivity

Aimin Lv
• School of Petroleum Engineering, China University of Petroleum (East China), Qingdao, Shandong, China
• Other articles by this author:
/ Xuyan Li
• Corresponding author
• School of Petroleum Engineering, China University of Petroleum (East China), Qingdao, Shandong, China
• Email
• Other articles by this author:
/ Miao Yu
• School of Petroleum Engineering, China University of Petroleum (East China), Qingdao, Shandong, China
• Other articles by this author:
/ Gangzhu Li
• School of Petroleum Engineering, China University of Petroleum (East China), Qingdao, Shandong, China
• Other articles by this author:
/ Shoulong Wang
• School of Petroleum Engineering, China University of Petroleum (East China), Qingdao, Shandong, China
• Other articles by this author:
/ Ruigang Peng
• School of Petroleum Engineering, China University of Petroleum (East China), Qingdao, Shandong, China
• Other articles by this author:
/ Yawen Zheng
• School of Petroleum Engineering, China University of Petroleum (East China), Qingdao, Shandong, China
• Other articles by this author:
Published Online: 2017-05-20 | DOI: https://doi.org/10.1515/phys-2017-0035

## Abstract

This paper presents a practical technique to quantitatively locate macroscopic throats between injector/producer pairs in a reservoir, considering the problems of extensively developed macroscopic throats and the low sweep efficiency of waterflooding on high water cut stage. The method combines dynamic and static data, based on the results of geological research and the inversion of dynamic interwell connectivity. This technique has implemented the spatial locating of macroscopic throats, using the data of injection/production profiles and tracer test over the years, considering the sedimentary facies of each small layer and the permeability of each sand body. The results of this work show that this method is more convenient and less expensive than previous ones. It is able to locate macroscopic throats in a reservoir accurately and quantitatively. Multiple materials ensure the accuracy of results, and this method is convenient to be applied in the oilfield.

PACS: 83.50.Ha

## 1 Introduction

After a long waterflooding in a reservoir, permeability and pore throat radius obviously increase in waterflooding area, and macroscopic throats widely spread due to geological factors and dynamic development [1]. In this case, macroscopic throats will result in poor or even noneffective circulation of waterflooding, dramatically reducing its work efficiency and sweep efficiency [2]. Therefore, the method of the spatial locating of macroscopic throats is of great importance.

Several methods have been formed by now to identify macroscopic throats. One uses the dynamic monitor of tracer’s data to identify the formation parameters [3, 4]. This method can qualitatively judge the existence of high permeability zone in the formation. But it is of high cost and cannot monitor the reservoir at any time. The second approach builds the fuzzy-recognition model of macroscopic throats [5]. This approach figures out the values and their percentages of dynamic/static parameter which can reflect the formation of the macroscopic throats. Values obtained from the oilfield will be used to get a composite indicator value. According to that value, we can fuzzily estimate the existence of the macroscopic throats. However, this method needs abundant datum, so it is hard to ensure the integrity of each parameter. Also, the weighted value of each parameter and the division of the scope of composite indicator value need rich experience. These lead to the lack of accuracy of its result. Another approach identifies the macroscopic throats based on grey relational analysis theory [6, 7]. This method defines the apparent water injectivity index of injectors as the sequence, and defines the apparent fluid productivity index of producers as the subsequence. The existence of the macroscopic throats is judged by the degree of association between factors and sub-factors. The closer the degree of association is to 1, the greater the possibility of the existence of macroscopic throats will be.

The degree of association between factors and sub-factors is given by: $ri,0=1n∑t=0nLt(i,0)$(1)

And the correlation coefficient between the sequence and the subsequence is given by: $Lt(i,0)=(Δmin−ρΔmax)/[Δt(i,0)+ρΔmax]$(2) where the resolution ratio ρ is used to weaken the influence of distortion coming from large absolute deviation. And ρ can make the difference between the correlation coefficient more obvious. However, ρ cannot avoid manmade subjectivity. It is hard to ensure that the correlation coefficient does reflect the real connectivity between injectors/producers.

There have been many other approaches that identify the macroscopic throats. However, they have some disadvantages such as high cost, limited accuracy and lacking of ability for real-time monitoring. Also, these techniques can only qualitatively or semi-quantitatively identify the macroscopic throats, and fuzzily estimate the existence. Nowadays, there has been no method that can systematically identify and accurately find out the spatial location of macroscopic throats.

Recently, the method of inversion of dynamic connectivity between injector/producer pairs based on production and injection rates in the oilfield has gained wide attention [812]. In order to implement the identification and the location of macroscopic throats in terms of time and space, this paper presents an innovative technique to locate macroscopic throats between each injector-producer pair in a reservoir. The method combines qualitative and quantitative research making use of static description and dynamic testing analysis together with dynamic inversion. Ultimately, the final objective is accuracy identification and location of macroscopic throats at any time.

This paper uses the inversion model of dynamic connectivity between wells, which considers diffusivity filters, production response and stress constraints. And the model inverses the connectivity based on production/injection rates and pressure data, using the adaptive genetic algorithm. Through the analysis of quantitative dynamic connectivity between injector/producer pairs, the preferential seepage channels of injected water can be positioned in plane [13]. And then with the combination of dynamic monitoring data and static geological data obtained from the oilfield, we can identify the spatial location of macroscopic throats between injector/producer pairs.

## 2 The inversion of dynamic interwell connectivity

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]: $αm=Δnτexp(m−n)/τ$(3) where n = the time of sampling, Δn = 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]: $iijc(t)=∑n−0mαij(n)ii(t−n)$(4) where $\begin{array}{}{\alpha }_{ij}^{\left(n\right)}\end{array}$ is the n-th filter coefficient between injector i affecting producer j, ii (tn) is the observed injection rate at (tn)-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]: $Δp=C1×Ei(−dr2t)t≤1C1×Ei(−dr2t)−Ei(−dr2(t−1))t>1$(5) where C1 = a constant, Ei = 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]: $CtVpdp¯dt+qj(t)=∑i=1i=Iλijii(t)$(6) where Ct = the total compressibility, Vp = the drainage pore volume, $\begin{array}{}\overline{p}\end{array}$ = the average pressure in Vp, ii(t) = the injection rate of injector i, qj(t)= the production rate at producer j and λij = 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]: $qj(t)=qj(t0)exp−(t−t0)τj+∑i=1i=Iλijexp⁡(−t/τj)τj∫ξ=t0ξ=texp⁡(ξ/τj)iij(ξ)dξ+Vjpwf(t0)exp[−(t−t0)τj]−pwf(t0)+exp⁡(−t/τj)τj∫ξ=t0ξ=texp⁡(ξ/τj)pwf(ξ)dξ$(7) where τj = the time constant, where τj = CtVp/J, J = productivity index, qj(t0) = the initial production rate, Vj = the coefficient of bottom-hole pressure term and pwf = 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 λij, τj and Vj 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 (λij), 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 (τj) 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 (λij) and the coefficient of bottom-hole pressure term (Vj) 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 (R2) is introduced here to analyse the quality of models. R2 is determined by [11]: $R2=1−∑m=0M(qj(m)−q^j(m))2∑m=0M(qj(m)−q¯j)2$(8) where $\begin{array}{}{q}_{j}^{\left(m\right)}\end{array}$ = the real production rate, $\begin{array}{}{\overline{q}}_{j}\end{array}$ = the average real rate, and $\begin{array}{}{\stackrel{^}{q}}_{j}^{\left(m\right)}\end{array}$ = the modeled production rate.

The closer R2 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. R2 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.

## 3 The spatial locating of macroscopic throats

The existing macroscopic throats will worsen the interlayer and in-layer contradiction [12]. Taking this into account, spatial locating of macroscopic throats can be decomposed into two aspects, in plane and in vertical. First, this approach uses the dynamic inter-well connectivity model based on diffusivity filters, the response of production and the constraint of pressure. With the analysis of the interwell connectivity, we can determine the preferential channels of injected water in plane. Then, it identifies the layer of macroscopic throats based on dynamic monitoring data and static geological data. The data include injection/production profiles, tracer test data, sedimentary facies of each small layer and permeability of each sand body. Figure 3 shows the procedure of spatial locating of macroscopic throats.

Figure 3

The procedure of spatial locating of macroscopic throats

The layers of macroscopic throats have obvious characteristics of static geology and dynamic development [1417].

## 3.1 Injection/production profiles

In the formation where macroscopic throats exist, the difference degree of injection/production profiles is greater. Layers with macroscopic throats have greater injection/production ability. Therefore, if the injection/production ability sharply increases and the value of the capability becomes larger, the possibility of existing macroscopic throats will be greater in the layer.

## 3.2 Sedimentary facies

Macroscopic throats mostly occur in distributary channels or river mouth bars where there is a long-time scouring by water flow. Besides, macroscopic throats are formed more easily following the sedimentary direction of watercourse. Therefore, if physical quality of sedimentary facies between injector/producer pairs is good and the orientation between wells follows the sedimentary orientation, the possibility of existing macroscopic throats will be greater.

## 3.3 Conditions of macroscopic throats forming

The thickness of formation has the influence on macroscopic throats forming. If the thickness is small and the water absorption is poor the amount of absorbed water is not enough, macroscopic throats will hardly form in the formation. Macroscopic throats are mainly distributed in the layer whose thickness is larger than 3 m and the thickness of water-absorbing section is over 2 m. Porosity of macroscopic throats should be bigger than 0.2 and permeability should be greater than 100 mD [18]. According to these characteristics, quantitatively comparing the physical properties of each layer, we can identify layers where macroscopic throats may exist.

## 4 Application

The method of spatial locating of macroscopic throats was applied to JZ oilfield in China. We analyzed the well group E5 of the field and chose the time period from 2005 to 2012. After long-time waterflooding development, there evidently exist macroscopic throats in this block. Overall, the dynamic injection/production rate observed in the well group E5 has a certain fluctuation and maintains continuity. The well group E5 meets the condition of using the inversion of dynamic interwell connectivity. Due to the changes of operations in E5, the 7 years need to be divided into 3 periods to analyze the dynamic connectivity. The basic data of the well group E5 is provided at the end of the article.

Figure 4 shows a map of the well group E5 in JZ oilfield and the representation of dynamic interwell connectivity in each period. In Figure 4, the connectivity coefficients (λij) are represented by inverted arrows that start from the injector i and point to the producer j. The larger the arrow, the larger the value of the connectivity coefficient between the two wells. Table 1 shows the values of the connectivity coefficients λij in each period.

Figure 4

Representation of the connectivity coefficients λij. (a) 2005.2-2007.1; (b) 2007.2-2008.11; (c) 2011.5-2012.8

Table 1

The values of the connectivity coeflcients λij in Figure 4. (a) 2005.2-2007.1; (b) 2007.2-2008.11; (c) 2011.5-2012.8

From Figure 4, we know there is constantly-changing interwell connectivity during the long-time waterflooding development. With the analysis of the interwell connectivity, we can identify preferential channels of injected water, and then locate the macroscopic throats in plane.

Figure 4 shows the connectivity between injector E2-2 and adjacent producers (producer E2-3, and producer W3-2). Since the values of connectivity coefficients between injector E2-2 and producer E2-3 are bigger, macroscopic throats quite possibly form between these wells.

This paper identified the spatial location of macroscopic throats between injector E2-2 and producer E2-3 as an example. Figure 5 shows the values of injection capability of injector E2-2 in each sand group over years. From Figure 5, we know that the injection capability in sand group I sharply increased from April, 2012 to January, 2013. Therefore, there were obviously formed macroscopic throats in this period. Similarly, the macroscopic throats formed in sand group II from September, 2011 to April, 2012. And then, considering more datum such as sedimentary facies and permeability, we achieved more accurate spatial location of macroscopic throats. The objective is to find the sand body where the preferential seepage channels of injected water are most easily formed.

Figure 5

The values of injection ability of injector E2-2 in each sand group

Figure 6 shows the permeability distribution of injector E2-2 and producer E2-3. In Figure 6, the values of the permeability and the thickness of injector E2-2 and producer E2-3 are all greater in the sand body I31 and the sand body II21. There possibly exist a high-permeability zone or macroscopic throats.

Figure 6

The permeability distribution of injector E2-2 and producer E2-3

Figure 7 shows the permeability distribution in the sand body I31 and the sand body II21. The purple red area has the highest permeability which is over 5000 md .The permeability of other area is lower. Comparing the sand body I31 with II21, we know that the permeability of formation between injector E2-2 and producer E2-3 is higher in the sand body I31 and its value is more than 5000 md.

Figure 7

The permeability distribution; (a) I31; (b) II21

Figure 8 shows the distribution of sedimentary facies in the small layer I3 and the small layer II2. The green area represents distributary channel, and the orange area represents river mouth bar.

Figure 8

The sedimentary facies distribution; (a) I3; (b) II2

From Figure 8, we can find that there is distributary channel between these two wells in the small layer I3, where the physical property is good. Also, the direction between wells follows the sedimentary direction. All of these are benefit for the forming of high permeability zone or macroscopic throats. However, a river mouth bar between injector E2-2 and producer E2-3 in the small layer II2 exists, together with the change of sedimentary facies and physical quality. Taken together, it can be concluded that macroscopic throats form in the sand body I31 between injector E2-2 and producer E2-3.

Based on the method above, we have achieved spatial location of macroscopic throats between other injector/producer pairs and obtained the spatial position distribution of macroscopic throats in the well group E5 in Figure 9.

Figure 9

The spatial position distribution of macroscopic throats in the well group E5.

In Figure 9 macroscopic throats have been developed in the most formations between injector/producer pairs in the well group E5, and they are all formed in the sand body I31. It is due to the high permeability and good physical quality in the sand body I31.

## 5 Conclusions

By combination of dynamic and static data, this paper can accurately identify and spatially locate the macroscopic throats. The method is based on the results of geological research, the inversion of dynamic interwell connectivity and the dynamic monitoring data. This technology is considerably simple, efficient, low-cost and adaptable. Combined with inversion of dynamic connectivity, this method can achieve the identification of macroscopic throats at any time and it can be widely used in oilfield development.

This paper adds the constraint of pressure to the model of dynamic interwell connectivity. The modeled production rate is closer to the real production rate after considering the influence of BHP. The accuracy of inversion of dynamic connectivity has been improved.

The technique was applied to the well group E5 in JZ oilfield, and our results agree with the present known oilfield features. Using this method to achieve the spatial location of macroscopic throats has the guiding significance in oilfield and provides beneficial basis for the maximization of oil recovery of existing waterfloods.

Nomenclature

ri,0 = degree of association

Lt = correlation coefficient

ρ = resolution ratio

αm = diffusivity filter coefficient

Δn = selected discretization interval

$\begin{array}{}{\alpha }_{ij}^{\left(n\right)}\end{array}$ = diffusivity filter coefficient

ii = observed injection rate (m3/d)

$\begin{array}{}{i}_{ij}^{c}\end{array}$ = convoluted injection rate (m3/d)

C1 = proportionality constant

Δp = pressure change (MPa)

Ei = exponential function

r = distance from the point to the well (m)

d = dissipation constant

Ct = total compressibility (MPa−1)

Vp = drainage pore volume (m3)

$\begin{array}{}\overline{p}\end{array}$ = average pressure (MPa)

qj = production rate (m3/d)

λij = weighting factors

τj = time constant

J = productivity index (m3/(d⋅m))

Vj = coefficient of bottom-hole pressure term

pwf = bottom-hole flowing pressure (MPa)

Subscripts and Superscripts

i = injector index

j = producer index

m = observed data point

t = time

## A Appendix-Basic data

Table 2

The basic data of the well group E5; (a) 2005.2-2007.1; (b) 2007.2-2008.11; (c) 2011.5-2012.8

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Accepted: 2016-11-28

Published Online: 2017-05-20

Citation Information: Open Physics, Volume 15, Issue 1, Pages 313–322, ISSN (Online) 2391-5471,

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