A coupling model of water breakthrough time for a multilateral horizontal well in a bottom water-drive reservoir

Accepted Manuscript A coupling model of water breakthrough time for a multilateral horizontal well in a bottom water-drive reservoir Ping Yue, Bingyi Jia, James sheng, Tao Lei, Chao Tang PII:

S0920-4105(19)30170-6

DOI:

https://doi.org/10.1016/j.petrol.2019.02.033

Reference:

PETROL 5790

To appear in:

Journal of Petroleum Science and Engineering

Received Date: 8 October 2018 Revised Date:

31 December 2018

Accepted Date: 11 February 2019

Please cite this article as: Yue, P., Jia, B., sheng, J., Lei, T., Tang, C., A coupling model of water breakthrough time for a multilateral horizontal well in a bottom water-drive reservoir, Journal of Petroleum Science and Engineering (2019), doi: https://doi.org/10.1016/j.petrol.2019.02.033. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

A coupling model of water breakthrough time for a multilateral horizontal well in a bottom water-drive reservoir

Abstract: A series of complex production wells have been applied in challenging

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reservoirs to maximize oil recovery. Multilateral horizontal well has received plentiful attentions in recent years. The multilateral horizontal well technology is especially

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beneficial in expanding formation drainage area, reducing water cresting and coning, increasing productivity, and postponing water breakthrough time in bottom water

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reservoir. Since Muskat and Wyckoff introduced the concept of water coning into petroleum engineering field, different models have been proposed to forecast the critical rate and water breakthrough time for vertical and horizontal wells. However, the studies of water breakthrough time for complex structured multilateral horizontal

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wells have rarely been reported. This paper proposes a new coupling model for estimating the water breakthrough time of complex structured multilateral horizontal

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wells. The proposed model took multiple factors into account, including the wellbore 3D structure, wellbore vertical location, and reservoir and fluids properties. In

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addition, a field case study was conducted by utilizing the methods that were proposed in this paper. The practical production data was compared with the results obtained from the analytical model, and this coupling model shows an excellent agreement with the results from the actual scenarios. The proposed methods in this paper can be utilized to forecast the water breakthrough time and breakthrough locations of multilateral wells in bottom water reservoirs. Keywords: Multilateral well; Coupling model; Bottom water reservoir; Water

ACCEPTED MANUSCRIPT breakthrough time; Water coning 1. Introduction Nowadays, a series of complex production wells have been applied in challenging reservoirs to maximize oil recovery. As one of the popular methods,

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multilateral horizontal well has received plentiful attentions in recent years. [1-4]. In addition to the advantage of expanding drainage area [5-8], the multilateral horizontal well technology can avoid the disadvantages by the transportation of large volumes of

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fluid along a single horizontal borehole, which will result in the pressure loss in the wellbore, well productivity decrement [9-11] and potential early water breakthrough

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[12,13]. Therefore, complex structured branch wells usually have higher production rate with smaller production pressure difference that will delay the bottom water breakthrough time. The water breakthrough time correlates to the water coning or cresting, which is a rate-sensitive phenomenon that generally associates with high

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production rate and pressure difference. It develops once the pressure difference overcomes the natural buoyancy forces that segregate water from oil [12-15]. As a result, the production pressure difference of multilateral horizontal wells can be set to

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a lower value than that of a vertical or single horizontal well. Since Muskat and

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Wyckoff [16] introduced the concept of water coning into the petroleum engineering field, many researchers have been investigated extensively on analytical solutions and empirical correlations to forecast the critical rate [17-26] and water breakthrough for vertical [27-30] or horizontal wells [31-38]. Sobocinski and Cornelius [28] presented a correlation to predict the behavior of a water cone as it built from the static water-oil contact (WOC) to breakthrough conditions. The correlation is partly empirical and involves dimensionless groups of reservoirs, fluid properties, and well characteristics. Bournazel and Jeanson [29] proposed a new relationship for breakthrough time in

ACCEPTED MANUSCRIPT horizontal wells by applying the same dimensionless groups of Sobocinski and Cornelius. Ozkan and Raghavan [32-33] modeled the breakthrough time in horizontal wells from dimensionless groups theoretically. A semi-analytical solution was proposed by Papatzacos et al. [34] to estimate the water breakthrough time in an

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anisotropic, infinite reservoir with a horizontal well. Ahmadi et al. [38] proposed a method to predict the breakthrough time of water coning in the fractured reservoirs by implementing low parameter support vector machine. However, the study of water

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coning and cresting dynamics for complex structured multilateral horizontal wells has rarely been reported.

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An accurate prediction of coning and cresting dynamics consists of the estimation of breakthrough time and location. The accuracy of the prediction is critical for the treatment and prevention processes. The traditional treatment and prevention methods are listed as following, adjusting the perforation distance from the

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original water-oil contact (WOC) [17-23, 39], controlling liquid production below the critical rate [14, 40-43], producing oil and water separately with Downhole Water Sink (DWS) or Downhole Water Loop (DWL) technologies [44-48], utilizing Inflow

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52-55].

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Control Devices (ICD) [49-51], and injecting some types of fluid to form a barrier [15,

In a bottom water reservoir, the critical rate and critical pressure difference is the

fundament of investigating the dynamics of water coning and cresting. Based on previous works on critical parameters of horizontal wells [14,15,53,54], the dynamics of water coning and cresting of multilateral horizontal wells are investigated in this study. This paper proposed a theoretical coupling model to calculate the water breakthrough time and location by depicting the bottom water rising and breaking dynamic processes. The branched horizontal wells distribute and extend in

ACCEPTED MANUSCRIPT three-dimensions and the critical rate and pressure drop that affect breakthrough time around the wellbore are considered as 3D problems. Multiple effects could influence the critical rate, such as the mirror reflection on the formation boundaries, the mutual interference between the segments of wellbores, and the effect of pressure drop in the

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complex structured wellbore. Therefore, our new model took many factors into account, including the well structure, the length of main and branch wellbores, the space of main to branch wellbores, wellbore radius and preformation parameters. A

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field case study was conducted by the method proposed in this paper. The practical production data was compared with the results from the proposed analytical model,

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and the results show great agreement to the actual scenarios. The method can forecast the water breakthrough time and locations of multilateral wells in bottom water reservoirs. 2.Mathematical Model

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2.1 Model of Water Breakthrough Time for a Single Horizontal Well To compare the water breakthrough time of a single horizontal well to that of a multilateral well, the basic model of water breakthrough time for a single horizontal

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well is firstly introduced.

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2.1.1 Case 1: Production rate is greater than critical rate When the production rate is set higher than the critical rate, it is obvious that the

bottom water will rise and generate water crest, as shown in Figure 1 [12]. In the YZ sections of horizontal well in bottom water reservoir, the top boundary is sealed, and the bottom boundary is the initial WOC. The bottom boundary pressure is constant at pe, MPa. The thickness of the pay zone is h, m. The distance of the horizontal well from the WOC is zw, m. The horizontal well is L, m in length and rw, m in radius. Assumptions for the model are as follows: 1) oil and water layers are distributed

ACCEPTED MANUSCRIPT horizontally; 2) the properties of fluids and reservoir rocks are homogeneous and isotropic; 3) fluids flow steadily with constant density and viscosity. According to the Mirror Image Reflection law, the well in finite zone of the YZ section may be reflected as a set of vertical wells that lines in infinite space with

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two-sources and two-sinks in an interlaced arrangement. As shown in Figure 2, the well pattern and well location of well lines in infinite space can be generally divided into the following four types:

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Two types of injection wells: （0, 2h+4nh+zw ）and（0, 4nh-zw ）

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Two types of production wells: （0, 2h+4nh-zw ） and（0, 4nh+zw ） where n=0, ±1, ±2, ±3, ……

Assume the potential function Φ = p + ρ o gz , with the initial WOC as the reference plane, the potential distribution of pay zone generated by the

Eq. 1.

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superimposition potential of the aforementioned four types wells can be obtained by

πy

+ cos(

π ( z − zw )

)] • [ch

πy

− cos(

π ( z + zw )

)] µo BoQ 2h 2h 2h 2h ln 4π KL [ch π y + cos( π ( z + zw ) )] • [ch π y − cos( π ( z − zw ) )]

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Φ ( y, z ) = Φ e −

[ch

2h

2h

2h

(1)

2h

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where Φ( y, z ) is the potential of point (y, z), MPa; K is the formation permeability, mD; ρ o is the Oil density, kg/m3; µo is the Oil viscosity, mPa·s; Bo is the crude oil volume factor, dimensionless; Q is the production rate of well, m3/ks; ch is the hyperbolic cosine function; cos is the cosine function. Set y=0, the potential function below the wellbore axis can be obtained by Eq. 2. (1 + cos(

π ( z − zw )

)) • (1 − cos(

π ( z + zw )

)) µ o Bo Q 2 h 2 h Φ = Φe − ln π ( z + zw ) π ( z − zw ) 4π KL (1 + cos( )) • (1 − cos( )) 2h

2h

(2)

ACCEPTED MANUSCRIPT By taking the partial derivative of potential distribution function Eq. 2, the potential gradient function can be obtained, then the seepage velocity of the oil phase can be written as Eq. 3.

π( z + zw ) π( z − zw ) ] − sin[ ] K ∂Φ BoQ h h 2 2 Vo = − = µo ∂z 4hL sin[ π( z + zw ) ] ⋅ sin[ π( z − zw ) ] 2h 2h sin[

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(3)

According to Eq. 2, the potential distribution in YZ section can be drawn as

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Figure 3. In Figure 3, the minimum and maximum value of potential gradient is at point M and N, respectively. According to Eq. 3, the relation between the potential

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gradient and z on the line M-N can be drawn as shown in Figure 4.

Due to the sweep efficiency and oil displacement efficiency, the oil seepage flow velocity Vo and its true seepage velocity υ o in porous medium satisfy the following relationship as shown in Eq. 4.

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Vo = φ S oi ERυo =φ S oi ER

dz dt

(4)

where ER is the oil recovery by waterflooding, ER=EVED; Ev is the volume sweep efficiency, EV=EAEZ; ED is the displacement efficiency by water flooding,

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ED =(Soi − Sor ) / Soi ;

EA

is

the plane

sweep efficiency;

Ez

is

the vertical

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sweep efficiency; Soi is the initial oil saturation; Sor is the residual oil saturation. With the separation variables, the Eq. 4 can be written as Eq. 5 d t=

φ S oi ER Vo

dz

(5)

Combine Eq. 3 and Eq. 5, then integrate Eq. 5. The formula for water breakthrough time, t, which the water cresting to wellbore can be obtained by Eq. 6. t

t = ∫ dt = ∫ 0

zw − rw

0

φ Soi ER Vo

4φ Soi ER Lh2 dz = [1 − ctg θ cos θ ln(sec θ + tan θ )] BoQ

(6)

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π zw 2h

2.1.2 Case 2: Production rate is not higher than critical rate. When the well is produced lower than the critical rate, the oil in the

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homogeneous formation is displaced by water flooding evenly. With the time of oil production, WOC rises uniformly. When the WOC rose to the depth of wellbore, the well will finally face water breakthrough problem. The production of this scenario can

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be calculated based on the material balance in the following steps.

In the pay zone of a bottom water reservoir, the drainage area of a single

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horizontal well in the XY section is A = 2Lre , and the thickness is zw , so the drainage volume controlled by the single horizontal well is Eq. 7.

V = 2Lre zw

(7)

where L is the horizonal wellbore length, m; re is the well drainage radius, m; zw is the

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distance from wellbore to WOC, m.

When the WOC rises to the wellbore and the oil recovery by water flooding is ER, according to material balance, the Eq. 8 can be used to determine the cumulative oil

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recovery, Qp, at water breakthrough.

QP =Vφ Soi ER / Bo

(8)

So, the water breakthrough time can be obtained by Eq. 9. t = QP /Q =

2 Lre zwφ Soi ER QBo

(9)

2.2 Coupling Model of Water Breakthrough Time for a Multilateral horizontal well 2.2.1 Physical Model and Assumptions

Figure 5 shows a schematic diagram of a complex structured multilateral horizontal

ACCEPTED MANUSCRIPT well in a bottom water reservoir. In this model, the branches of this multilateral horizontal well are randomly distributed in a three-dimensional space. The basic assumptions of the physical model are: 1) the pay zone thickness is h; 2) the top and bottom boundaries are parallel and the top boundary is a closed boundary and the

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bottom boundary is a constant pressure boundary; 3) slightly compressible single-phase oil with constant compressibility and viscosity; 4) the formation properties are independent from pressure.

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Based on the results of water breakthrough time from the single horizontal well mentioned above, the multilateral horizontal wells were processed into several micro

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segments, and each segment is treated as a short single horizontal well. Then, the coupling model of water breakthrough time in multilateral horizontal wells is studied by the method of Mirror Images Law and the Potential (gradient) Superimposition theory. The coupling model takes many factors into account, such as the

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corresponding reservoir physical properties (reservoir permeability, porosity, the oil recovery by waterflooding, the initial and residual oil saturation etc.), the mutual interference of each segment by the wellbore parameters (borehole diameter, skin,

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location, etc.) and the effect of wellbore pressure drop (friction pressure drop,

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acceleration pressure drop, mixed pressure drop, confluent pressure drop, etc.).

2.2.2 Case 1： ：Production rate is greater than critical rate When the well is produced above the critical rate, it will cause water coning and

cresting. For a single horizontal well, when other factors are not considered, the potential gradient function below the wellbore can be directly solved by the analytical Eq. 6. Then, the seepage velocity of the oil phase can be obtained. For multilateral horizontal wells, when wellbore pressure drops, wellbore parameters, and influence of micro segments mutual interference are considered, the potential gradient distribution

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function

∂Φ ij

below each micro segment of wellbore cannot be estimated directly

∂z

by the analytical method. However, it can be obtained by the method of Mirror Images for the mixed boundary and the theory of Potential Gradient Superimposition

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which is shown in the Eq. A14 in Appendix A. The oil phase seepage flow velocity can be solved by Eq. 10 when the potential gradient value at any position below the micro segment of the wellbore are given,

Kij ∂Φij

µo ∂z

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Vo,ij = −

(10)

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where i=0 represents the main wellbore, i>0 represents the ith branch wellbore, j represents the jth micro segment of ith branch; ∂Φij ( xij , yij , zij ) / ∂z is the potential gradient caused by the jth micro segment at ith branch.

Based on ER and the oil displacement efficiency, the relation between the oil

expressed by Eq. 11.

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seepage flow velocity Vo,ij and its true seepage velocity υ o,ij in porous media can be

d tij =

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Thus,

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Vo,ij = φij S oi ERυo,ij =φij Soi ER

φij Soi ER Vo,ij

dz d tij

dz

(11)

(12)

With the integration for both sides of Eq. 12, the Eq. 13 can be obtained to

calculate the water breakthrough time for each micro segment of multilateral horizontal well. tij

tij = ∫ d tij = φ So ER ∫ 0

zw, ij − rw, ij

0

1 dz Vo, ij

(13)

The Trapezoidal or Simpson numerical integration can be used to get the value of

ACCEPTED MANUSCRIPT Eq. 13.

2.2.3 Case 2: Production rate is not higher than critical rate Similar to the single wellbore horizontal well, the crude oil in a homogeneous formation can be evenly displaced by water flooding when the well is produced at the

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rate lower than the critical rate. When the WOC rises to the lowest point at the wellbore, the no water production period can be calculated based on the material balance as shown in Eq. 14.

2 Lre zw,ijφ Soi ER QBo

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t = QP /Q =

(14)

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where zw,ij is the distance from jth micro segment on ith branch to WOC, m. In Eq.14, the ij micro segment is the one which has minimum distance to WOC.

3. Coupled Model Solution and Comparative Analysis

In order to do the comparative analysis, a single horizontal well is used as a

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contrast sample. Therefore, four evenly distributed branches were added to a single horizontal well to construct a complex structured multilateral horizontal well. The reservoir parameters and fluid properties parameters are listed in Table 1.

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3.1 Well production rate is not higher than critical rate Based on the Appendix A, the calculated critical rate of the multilateral well is

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116 m3/d. If the well produces at a rate less than the critical rate, the time for water breakthrough can be calculated by Eq. 14 as shown in Figure 6. Figure 6 illustrates that there will be a long period of no water production while

the well production rate is lower than the critical rate. Since the well is producing crude oil below the critical rate, and the WOC rises uniformly. The water breakthrough dynamics shows that the wellbore flooding is originated from the lowest position of micro segment of wellbore which has minimum distance to WOC.

ACCEPTED MANUSCRIPT 3.2 Well production rate is greater than critical rate Take the multilateral horizontal well production rate of 116m3/d which is just greater than the critical rate as an example, the potential gradients and the reciprocals

calculated by Eq. A13 as shown in Figure 7 and Figure 8.

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of velocity at different heights (distance from WOC) below each segment were

The physical meaning of Eq.13 is that the water breakthrough time for each segment equals to that of the area enclosed by each velocity reciprocal curve and the

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coordinate axis in Figure 8 multiplied by the constant coefficient φi Soi ER . The water breakthrough time for each micro segment can be obtained by integrating the

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reciprocal velocity by Simpson numerical integration to and then multiply by the constant coefficient φi Soi ER , as shown in Figure 9.

In Figure 9, the earliest water breakthrough time is 170.27 days at the location of 47th segment, 2rd segment of the 2rd branch of the multilateral horizontal well. After

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that, the water breakthrough followed by the 16th micro segment on the main wellbore is 170.66 days. Sequentially from low to high, the water breakthrough time shows the

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water coning dynamics of wellbore. Using the method above, when the production rate is greater than the critical rate, the calculated water breakthrough time is shown in

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Figure 10.

The results of Figure 6, Figure 10 and the single horizontal well were put in one

figure as Figure 11 for comparison and analysis. The critical rate has a strong impact on the water breakthrough time. It can be noticed that, when the production rate is greater than the critical rate, the water breakthrough time is significantly decreased for both single and multilateral horizontal wells. Figure 11 also illustrated that the critical rate has an important effect on the oil recovery before breakthrough. The production rate shown as the blue line filled area in Figure 11

ACCEPTED MANUSCRIPT shows a greater critical rate and late water breakthrough time for multilateral horizontal well, which has better effect in preventing water coning and water cresting than that of the single horizontal wells. So, the application of complex structure branch wells in bottom water reservoir has a bright future to reduce the suffering of

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water cresting or coning. Also, it can be expected to delay the water breakthrough time while with a higher production rate than that of single horizontal wells.

Water breakthrough time in a multilateral horizontal well closely correlates to the

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critical rates. At the point that is close to the critical rate, if the actual production rate changes from below the critical rate to higher than the critical rate, there will be a

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jump mutation at the time of water breakthrough. When the production rate is greater than the critical rate, the water breakthrough in oil wells is significantly ahead of time compared to production rate below the critical rate. The critical production is affected by the reservoir properties (reservoir permeability, porosity, oil saturation), wellbore

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parameters (well diameter, skin, location), wellbore pressure drop (friction, acceleration, mixing, combined pressure drop), development dynamic parameters (flow pressure, oil well distribution, water drive efficiency) and other factors.

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above.

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Therefore, the water breakthrough time in the oil well is also related to the factors

4. Case verification and discussion 4.1 Case verification of multilateral wells 4.1.1 Basic parameters For the field case analysis of multilateral wells, the well FZ1 in an offshore oil field located in the Liaodong Bay area of the Bohai Sea is selected. The average porosity of this reservoir is 0.31. The average permeability of the formation is 1711mD. The initial pressure of the formation is 10.65 MPa. The initial average oil

ACCEPTED MANUSCRIPT saturation is 0.75. The recovery from water flooding is about 40%. The pay zone thickness of the formation is 18.2 m. The distance between the wellbore and WOC is 12m. The formation crude oil viscosity is 401mPa·s. The dissolved gas-oil ratio for this oil is 15 m3/m3. The formation water and oil densities are 1100kg/m3 and

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900kg/m3, respectively. The main wellbore of the FZ1 well is 396.8m and it is completed with a high-quality sieve tube. There are 4 branches and the lengths are 153, 100, 142, and 100m, respectively. All the branches are completed with open-hole

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completion method. The radius of wellbores is 0.10m. The well completion data and

4.1.2 Productivity analysis

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wellbore structure are shown in Table 2 and Figure 12.

Figure 13 shows the daily production and differential pressure data of the FZ1 well. It shows that the earliest water breakthrough time is 270 days. The relationship between the production rate and the production pressure difference is obtained by

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selecting the data before water cut exceeds 10% as shown in Figure 13. By performing regression analysis on the production rate and pressure difference, the obtained liquid (mainly is oil) production index was 84.51m3/(MPa·d) shown in Figure 14.

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According to the principle of micro segments division, the main wellbore can be

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divided into 40 micro segments, and the branch 1 to 4 can be divided into 15, 10, 14 and 10 micro segments, where each micro segment length is 10m. The liquid production index calculated by the theoretical coupling model is 92.18m3/(MPa·d) as shown in Figure 15. The error between the calculation result (92.18) and the dynamic production data result (84.51) is less than 8%. The results calculated in this paper is a single-phase production index, which is slightly different from the liquid production index by the statistical production data. There are two main reasons for the error. First, it is assumed that the production dynamic data with water cut lower than 10% is

ACCEPTED MANUSCRIPT considered as single oil phase flow. In fact, the liquid flow in this situation is two phases flow, which will increase the flow seepage resistance, and will cause the actual liquid production index become slightly lower than the one that was calculated by the theoretical coupling model. On the other hand, the result of theoretical coupling

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model is on this premise that all segments of the branched well are involved in the production. However, in actual situation, there may be an unconsolidated sandstone reservoir, which will result in partial segment of the open hole completion branch

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wellbore collapse. So, the actual production length of the wellbore is shorter than the theoretical coupling model, and the actual liquid production index is slightly lower

4.1.3 Critical parameter analysis

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than that of the theoretical coupling model.

Using the critical parameter calculation workflow proposed in Appendix A, the highest potential gradient at WOC below each micro segment can be estimated as

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0.002MPa/m by Eq. A18. Through the trial calculation, while the production rate is 42.8 m3/d, the corresponding potential difference is 0.44MPa, the calculated potential gradient value for each point on the WOC below the micro segment is below the

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0.002MPa/m (Red line) as shown in Figure 16.

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4.1.4 Analysis of water breakthrough time From the daily production data, the no water production period of the

multilateral well is about 270 days. The average liquid (oil) volume at this period was 138.8 m3/d, which exceeds the critical rate of 42.8 m3/d. So, Eq. 13 can be utilized to calculate the water breakthrough time for each segment. The forecasting result of the earliest water breakthrough time is 261 days and shows an excellent agreement with the actual water breakthrough time, 270 days, as shown in Figure 17. The water breakthrough time of the top 10 micro segments is shown in Table 3. According to the

ACCEPTED MANUSCRIPT relationship between the micro segment numbers and locations, the dynamic of water breakthrough can be predicted (Figure 18). The water dynamics in this multilateral horizontal well shows that the main branch is the first to see the water near the adjacent position of the multilateral horizontal well. Then the water breakthrough

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develops into the middle section of the main wellbore, and the heel position of each branch wellbore. The water breakthrough at the heel and toe of main wellbore and the toe of the branch is relatively late. The results show the location and time for the

dynamics.

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4.2 Case verification of single horizontal wells

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water cresting in the wellbore, which can a reasonably explain the production

After setting the length of branch wells to zero, the multilateral horizontal well model was degraded into single horizontal well model, where the method proposed in

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this paper can also be applied. We added additional two examples of horizontal wells in Tahe filed, Xinjiang province of China, to validate our model. The Triassic reservoir in area 9 of Tahe oil field is a typical bottom-water reservoir with sufficient

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bottom water energy. The initial average oil saturation is 0.68. The formation crude oil

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viscosity is 2.8 mPa·s. The oil volume factor is 1.34. The formation water and oil densities are 1141kg/m3 and 757kg/m3 respectively. The trajectory of TK906H and TK909H are shown in Fig.19 and Fig.20 respectively. The daily production rate and water cut are shown in Fig.21 and Fig.22. The other parameters of single-well model of TK906H and Tk909H are shown in Table 4. The average daily production of TK906H well before water breakthrough is 79.41 m3/d, which is less than the calculated critical rate of 105.9 m3/d. Therefore, the

ACCEPTED MANUSCRIPT water breakthrough time calculated by Eq. 14 is 573 days, which is close to the actual water breakthrough time of 446 days. In addition, the production rate reached 170 m3/d after 55 days of production, and experienced 28 days where the actual

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production exceeded the critical rate. Therefore, the actual water breakthrough time was slightly earlier than the water breakthrough time estimated based on the average production rate. The average daily production of TK909H before water breakthrough

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is 80.1 m3/d which is greater than the critical rate of 45.6 m3/d. Therefore, the water

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breakthrough time, 55 days, estimated by Eq. 13 is consistent with the actual water breakthrough time of 55 days. The water breakthrough time of the two single horizontal wells mentioned above can also be reasonably predicted by the proposed method in this paper. Theoretically, this method also can be utilized in a situation that

branch distribution.

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the multilateral horizontal well consists of any number of branches and any structural

4.3 Basic guidance to use the new method

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A basic guidance of three steps to use our tool mentioned in this paper is shown

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in the following. First, the critical rate shall be calculated based on the reservoir properties, the fluid properties and the wellbore information. Second, the actual production data of oil wells needs to be processed to determine the relationship between the actual average production rate and the critical production rata before water breakthrough. Third, if the actual production rate of a well is greater than the critical rate, we can use the method of Mirror and Potential Gradient Superimposition, which is shown in the Eq. A14 in Appendix A, to estimate the potential gradient

ACCEPTED MANUSCRIPT distribution function,

ij

∂Φ ij ∂z

, for each wellbore micro segment. If the actual

production rate of well is greater than critical rate, we can apply the Trapezoidal or Simpson numerical integration based on Eq. 13 to estimate the water breakthrough

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time for each micro segment. Otherwise, we can use Eq. 14 to estimate the water breakthrough time while the actual production rate of a well is less than the critical

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rate.

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5. Conclusions

(1) Based on the Mirror Images method and the Potential (gradient) Superimposition theory, a coupling model is proposed to forecast the time and location of water breakthrough of a multilateral horizontal well in a bottom water reservoir, which combines the water cresting startup condition, the wellbore segment

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mutual interference and pressure drop factors into account. The solution is obtained by a MATLAB program.

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(2) The new model takes many factors into account, including the wellbore 3D structure, wellbore vertical location, mutual interference of each micro segment,

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reservoir and fluids properties which have a significant impact on the water cresting or coning dynamic. For a given bottom water reservoir, the application of complex structure branch wells has a bright future to reduce the water cresting or coning to the wellbore. It can delay the water breakthrough time with a higher production rate than that of single horizontal wells. (3) The comparison analysis shows that the critical rate has an important effect on the oil recovery before water breakthrough. There will be a sudden change in the water breakthrough time at the critical rate point. When the production rate is greater

ACCEPTED MANUSCRIPT than the critical rate, the water breakthrough time is significantly decreased for both single and multilateral horizontal wells. The reason is that if the production rate is less than the critical rate, the WOC rises uniformly, otherwise, the water coning will cause early water breakthrough.

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(4) This method can also be utilized in a situation that the multilateral horizontal well consists of any number of branches and any structural branch distribution, such as: single horizontal well, two, three and n branch horizontal wells in an irregular

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distribution with different completion. To develop tractable analytical solutions, some assumptions may limit the practical applicability, but certain analytical solutions of

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this paper can be helpful to serve as a preliminary guide and provide a valuable tool to forecast the water coning dynamic for the complex structure wells in bottom water reservoirs.

Acknowledgements

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This work was supported by the National Major Projects of China (No.2016ZX05048-002), Fund of Oil & Gas Reservoir Geology and Exploitation Engineering of Southwest Petroleum University (No. PLN20190X), center for

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References

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post-doctoral studies of China University of Petroleum (Beijing).

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for horizontal wells. SPE Reservoir Engineering, 4(04), 409-416. [32] Ozkan E, Raghavan R. 1989. "Performance of horizontal wells subject to bottom water drive." SPE Reserv Eng 5 (1989): 375-383.

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horizontal wells. In European Petroleum Conference. Society of Petroleum Engineers. https://doi.org/10.2118/20964-MS [34] Papatzacos, P., Herring, T. R., Martinsen, R., & Skjaeveland, S. M. 1991. Cone breakthrough

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time for horizontal wells. SPE reservoir engineering, 6(03), 311-318. [35] De Souza, A. L. S., Arbabi, S., & Aziz, K. 1998. Practical procedure to predict cresting behavior of horizontal wells. SPE journal, 3(04), 382-392.

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Insight in Coning Control. Oil & Gas Science and Technology 45(1):51-62.

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Oil Recovery in Reservoirs with Water Coning. Carbon Management Technology Conference. doi:10.7122/502487-MS

[46] Qin, W., Luo, P., Meng, J., Li, H., Mao, T., Wojtanowicz, A. K., & Feng, M. 2017. Successful Field Trials of Water Control in High Water Cut Wells Using an Improved Downhole Water Sink/Drainage System. Society of Petroleum Engineers. doi:10.2118/188958-MS [47] Radwan, M. F. (2017, March 6). Feasibility Evaluation of Using Downhole Gas-Water Separation Technology in Gas Reservoirs with Bottom Water. Society of Petroleum Engineers. doi:10.2118/183739-MS

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[48] Riazi, M., Banaki, R., Malayeri, M. R., & Salmanpour, S. (2017, January 1). Simulation of Gel Injection, Oil Barrier and DWS Methods in Water Coning Prevention in a Conventional and Tight Reservoir. World Petroleum Congress. [49] Aljubran, M. J., & Al-Yami, A. S. (2017, March 14). Drilling the World’s Longest 6-1/8 in.

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Offshore Sidetrack Completed with ICD Screen Partially Cemented System. Society of Petroleum Engineers. doi:10.2118/184595-MS [50] Carpenter, C. (2016a, February 1). Optimization of Upper Burgan Reservoir Multilateral Well

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with Inflow-Control Device. Society of Petroleum Engineers. doi:10.2118/0216-0068-JPT. [51] Pakdeesirote, A., Ackagosol, S., Lewis, K., Kitvarayut, N., Pritchett, J., & Goh, K. F. G. (2017, October 17). Pilot In-Situ Inflow Control Device ICD Waterflood Injector Transformation- A Case

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doi:10.2118/186201-MS.

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ACCEPTED MANUSCRIPT [55] Siddiqi, S.S. and Wojtanowicz, A.K. 2002. A Study of Water Coning Control in Oil Wells by Injected or Natural Flow Barriers Using Scaled Physical Model and Numerical Simulator. Paper SPE 77415 presented at the SPE Annual Technical Conf. and Exhibition, San Antonio, Texas, 29 September-2 October. http://dx.doi.org/10.2118/77415-MS. [56] Yue, P., Xie, Z., Huang, S., Liu, H., Liang S., and Chen, X. 2018. The application of N2 huff and puff for IOR in fracture-vuggy carbonate reservoir, FUEL, 234 (2018) 1507-1517.

A1. Physical Model and Assumptions

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Appendix A: The critical rate coupling model of multilateral horizontal well

The Physical Model and Assumptions are same to the Figure 5 in this paper.

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A2. Reservoir Seepage Model

Assuming that the main wellbore and branches are all involved in production, for

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the potential distribution of horizontal wells in an infinite formation, the method of Mirror Images and the theory of Potential Superimposition can be utilized to get the potential distribution. Subdividing the ith branch wellbore into Ni segments (Figure A1), superposing the potentials generated by each micro segment of wellbore, the

as shown in Eq. A1. N

Ni

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total potential generated by the whole branch well at point M (x, y, z) can be obtained

Φ ( M ) = ∑ ∑ Φ ij ( M ) i=0 j =0

(A1)

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 Φ ij [( xij , yij , 4 nh + z ij ), ( xi ( j +1) , yi ( j +1) , 4 nh + z i ( j +1) ), α ij , M ]    N N i +∞  +Φ ij [( xij , yij , 4 nh − z ij ), ( xi ( j +1) , yi ( j +1) , 4 nh − z i ( j +1) ), α ij , M ]  = ∑∑ ∑   , M ] +Φ [( x , y , 4 nh + 2 h − z ), ( x , y , 4 nh + 2 h − z ), α i = 0 j = 0 n = −∞ ij ij ij ij i ( j +1) i ( j +1) i ( j +1) ij    +Φ ij [( xij , yij , 4 nh + 2 h + z ij ), ( xi ( j +1) , yi ( j +1) , 4 nh + 2 h + z i ( j +1) ), α ij , M ]

where i=0 represents the main wellbore; i>0 represents the ith branch wellbore.

Mij(xij, yij, zij) represents the material point (particle) coordinate of jth segment on

ith branch wellbore as shown in Figure A1. The Eq. A2, which is the weighted average method, is used to get the coordinate of material point Mij(xij, yij, zij). aij represent the angle between the X-axis and the direction of the segment of jth segment on ith branch.

ACCEPTED MANUSCRIPT xij

∫ =

yE

f ( x , y ) dy

yS

y E − yS

,y

ij

∫ =

xE

xS

f ( x, y) dx xE − xS

, zij

∫ =

yE yS

f ( z, y)dy

2 yE − yS

∫ +

xE

xS

f ( z, x )dx

2 xE − xS

(A2)

The relationship between each micro segment (jth wellbore segment on ith branch) induced potential at the M point and the inflow rate of the micro segment can be

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written in Eq. A3.

Φij ( xij , yij , zij ,αij , M )= qijξij (M )

(A3)

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where ξij ( M )=Φij ( xij , yij , zij , αij , M )/qij is the induced potential at the M point while the

jth wellbore segment on ith branch with unit inflow rate. When the point M (x, y, z) is

WOC can be obtained as Eq. A4.

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located at the bottom boundary of the bottom water reservoir, the potential on the

Φ ij e ( xij , yij , zij , α ij , M e )=qijξ ij e

(A4)

The result of superposition of the potential on the bottom boundary, WOC,

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generated by each micro segment production can be obtained by4. N

Ni

Φ e = Φ ( M e )= ∑ ∑ qijξ ij e

(A5)

i = 0 j =1

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where Φe is the potential of the bottom constant pressure boundary. ξije is the value of

ξij (Me) when M is at the bottom constant pressure boundary.

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From Eq. A2 to Eq. A5, when the well produces in fixed yield, the potential drop

at any point M (x, y, z) can be obtained by Eq. A6. N

Ni

Φe −Φ(M )=-qij ∑∑ξij (M ) − ξij (Me )

(A6)

i =0 j =1

Setting the M point to the material point Mij of each micro segment of the wellbore, using above Eq. A6 for each micro segment, the reservoir seepage flow equations which takes the micro segments interference and wellbore pressure drop into account can be obtained as Eq. A7.

ACCEPTED MANUSCRIPT ξ01 (0,1)-ξ01e (0,1) ξ02 (0,1)-ξ02e (0,1) L ξij (0,1)-ξije (0,1) L ξmnm (0,1)-ξmnm e (0,1)  q01   pe − p01   z01 − rw01        z − r  ξ ξ ξ ξ ξ ξ ξ ξ (0,2 ) (0,2) (0,2 ) (0,2) L (0,2 ) (0,2) L (0,2 ) (0,2) q p − p 01e 01 01e ij ije mnm mnm e  01   02   e 02   02 w02  M  M  M  M  K     = − ρo g     ξ01 (i,j)-ξ01e (i,j) ξ01(i,j)-ξ01e (i,j) L ξij (i,j)-ξije (i,j) L ξmnm (i,j)-ξmnm e (i,j)  qij  µo Bo  pe − pij   zij − rwij      M  M M  M        q  pe − pmn   zmn − rwmn  ξ01 (m,n)-ξ01e (m,n) ξ01 (m,n)-ξ01e (m,n)L ξij (m,n)-ξije (m,n)L ξmnm (m,n)-ξmnm e (m,n)  mn 

(A7)

the production in unit inflow of jth segment on ith branch.

A3. Wellbore Pressure Drop Model

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where ξij(m, n) represents that the potential at nth segment on mth branch generated by

A8.

8ρ f t,ij

π d 2

5 ij

Qij2 ∆Lij

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∆pij =

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The pressure drop of each perforation segment can be calculated as shown in Eq.

(A8)

where ft,ij is the appearance frictional coefficient of jth segment on ith branch. The total pressure drop of micro segments of main wellbore which connected branch can be written as Eq. A9.

8 ρ f t,ij

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π 2 d ij5

Qij2 ∆Lij + ∆pcon,ij

(A9)

where the convergence pressure drop can be calculated by Eq. A10.

ρ

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∆pcon,0i =

A

(qw,0i +1vw,0i +1 − qw,0i vw,0i − qw, j 0vw, j 0 cos ai )

(A10)

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Due to the cumulate flow rate of each micro segment section at different

positions of the wellbore is the sum of the inflow rates of all the upstream micro segments, the total flow rate of the multilateral wells can be written as Eq. A11. N

Ni

Q = Q01 = ∑ ∑ qij

(A11)

i = 0 j =1

where qw,ij = qij /∆Lij, is the wall inflow rate of jth segment on ith branch with unit length, m3/(s·m). Qij is the cumulate flow rate of wellbore section at the middle point of jth segment on ith branch, m3/s. When subscript i=0, represents the main wellbore. Q

ACCEPTED MANUSCRIPT is the total flow rate, yield, of the branched well, m3.

A4. Coupling Model and Solution According above wellbore pressure drop model, each micro segment pressure drop vector is Eq. A12. (A12)

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T

∆p = ∆p01, ∆p02 ,L…, ∆p11,L… , ∆pij ,L… , ∆pmn 

Based on the principle of continuous pressure, the pressure of each micro

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segment of the borehole can be written in Eq. A13.

P =  p01 , p02 ,L… , p11 ,L… , pij ,L… , pmn 

T

(A13)

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Put the pressure Eq. A13 into the seepage flow Eq. A7 to obtain coupled nonlinear equations. The unknowns are the flow rate and the pressure of each micro segment, and the Gaussian iteration can be used to solving the problem by computer programming. The flow vector Q and the pressure vector P of the micro segment are

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solved to obtain the inflow and the corresponding wellbore pressure of the micro segment of the horizontal well. By the partial derivative of potential distribution function Eq. A1, the potential gradient for each segment vertical potential gradient at

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any location M (x, y, z) in the reservoir can be obtained as is shown in Eq. A14.

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∂Φ(S, E,α, n, h, M ) +∞ N ∂Φi (Sin , Ein ,αi , M ) = ∑∑ ∂z ∂z n=−∞ i=0 ∂Φij [(xij , yij ,4nh + zij ),(xi( j +1) , yi( j +1) ,4nh + zi( j+1) ),αij , M ]/ ∂z    +∞ N Ni +∂Φ [( x , y ,4nh − z ),( x ij ij ij ij i ( j +1) , yi ( j +1) ,4nh − zi ( j +1) ),αij , M ]/ ∂z   = ∑∑∑  n=−∞ i=0 j =1 +∂Φij [( xij , yij ,4nh + 2h − zij ),( xi ( j +1) , yi ( j +1) ,4nh + 2h − zi ( j +1) ),αij , M ]/ ∂z   +∂Φij [(xij , yij ,4nh + 2h + zij ),(xi( j+1) , yi( j+1) ,4nh + 2h + zi( j+1) ),αij , M ]/ ∂z

(A14)

where ∂Φ ij ( xij , yij , zij ) / ∂z is the potential gradient caused by the jth micro segment at ith branch. Based on the single well’s potential distribution formula, the potential of any location M (x, y, z) caused by the jth micro segment at ith branch can be obtained by Eq. A15.

ACCEPTED MANUSCRIPT Φij ( xij , yij , zij ) =

µo qij 4πKLij

ln

( x − xij 2 )2 + ( y − yij 2 )2 + ( z − zij 2 )2 + ( x − xij 2 ) ( x − xij1 )2 + ( y − yij1 )2 + ( z − zij1 )2 + ( x − xij1 )

+C

(A15)

The above formula can be transformed into Eq. A16.

µ o qij 4 πKLij

ln

rij1 + rij 2 + Lij rij1 + rij 2 − Lij

+C

(A16)

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Φ ij = −

where rij = ( x − xijk )2 + ( y − yijk ) 2 + ( z − zijk )2 ，k=1, 2; Lij is the micro segment length,

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m. (xij1, yij1, zij1), (xij2, yij2, zij2) is the heel and toe point coordinate of micro segment respectively.

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Computing the partial derivative to z from the above Eq. A16, the potential gradient caused by the jth micro segment at ith branch can be got as is shown in Eq. A17. ∂Φ ij ( x, y , z ) ∂z

=

µo qij 4 πKLij

f ( xij1 , yij1 , zij1 , xij 2 , y ij 2 , zij 2 , x, y , z )

(A17)

The value of Eq. A17 can be solved by MATLAB programing. Sum the Eq. A17

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of each segment can get the Eq. A14, which can be used to evaluate the potential gradient distribution at any position M (x,y,z) in the reservoir. When the yield of a complex structure well is given such that the potential gradient at the WOC below all

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micro segments does not exceed the startup gradient, another word, all the points’

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potential gradient function satisfies the Eq. A18, then this yield is the critical rate of the branch well and the corresponding pressure difference is the critical difference of the well.

−

d Φij dz

z =0

= ( ρ w − ρo ) g

(A18)

Eq. A18 illustrates that coning is a rate-sensitive phenomenon generally associated with high producing rates or pressure difference. Strictly a near-wellbore phenomenon, it only develops once the pressure forces drawing fluids toward the wellbore overcome the natural buoyancy forces that segregate water from oil. The

ACCEPTED MANUSCRIPT workflow to calculate the critical parameters which solved by MATLAB programing

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is shown in Figure A2.

ACCEPTED MANUSCRIPT Table 1 Basic parameters of formation, fluid properties, and wellbore parameters Parameters

Value

Pay zone thickness: h, m

32

Formation isotropic permeability: K, mD

150

Oil viscosity: µo, mPa·s

15

3

850

Formation water density: kg/m3

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Oil density: kg/m

1050

Main wellbore length: L0, m

350

Each branch well length: Li, m

100

Distance between wellbore and bottom boundary: zw, m

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Wellbore radius: rw, m

24

Distance between each branch: di, m

0.10 70

Serial number 1

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Table 2 Well completion data for FZ1 MD of top MD of bottom boundary, m boundary, m 1901.0 2134.3

Wellbore length, m 233.3

Completion style High quality sieve tube

2137.5

2301.0

163.5

3

2001.0

2154.0

153.0

4

2051.0

2151.0

100.0

5

2096.0

2238.0

142.0

6

2148.0

2248.0

100.0

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2

Remarks Main wellbore Branch 1

Open hole

Branch 2 Branch 3 Branch 4

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Table 3 Micro segment inflow rate and the top 10 of the water breakthrough time Segment

Segment inflow rate

Water breakthrough time

Number

3

(m /d)

(d)

19

0.144

259.52

2

24

0.144

260.55

3

23

0.141

261.16

4

18

0.141

261.99

5

68

0.143

264.70

6

20

0.136

265.34

7

22

0.133

266.28

8

67

0.134

267.20

9

14

0.149

269.01

10

17

0.135

269.22

No.

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1

ACCEPTED MANUSCRIPT Table 4 Parameters of two horizontal wells and Prediction results re

h

L

zw

K

Q

(m)

(m)

(m)

(m)

(mD)

TK906H

100

18.5

255

13.9

139

79.4

TK909H

100

15.5

157

13.3

110

80.1

Qc

Actual Theoretical

Well Name （m3/d） （m3/d）

t’(d)

105.9

446

573

45.6

55

52

RI PT

t(d)

Qc is the critical rate of well; t is the actual water breakthrough time of well; t’ is the

AC C

EP

TE D

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SC

theoretical water breakthrough time based on the model of this paper.

ACCEPTED MANUSCRIPT Oil zone

rw

z h

N

y

zw M C

RI PT

Bottom water zone

C

Fig. 1 Horizontal well YZ section in bottom water reservoir …

h

SC

zw

M AN U

h injection wells （ 0, 4nh-zw）

zw

h

injection wells

（ 0, 2h+4nh+zw）

（0, 0） ）

production wells （ 0, 2h+4nh-zw）

production wells

zw

TE D

（ 0, 4nh+zw）

h

h

h

EP

…

AC C

Fig. 2 Mirror Image Reflection law of four types of production and injection wells

N

M

re

Bottom water zone

Fig. 3 Potential distribution on YZ section

ACCEPTED MANUSCRIPT

z (m)

zw

z

N

0

−

dΦ dz

z =0

(MPa/m) = w−

o

RI PT

M

TE D

M AN U

SC

Fig. 4 Potential gradient changes at different points on line MN

3000

2000

EP

Water breakthrough 见水时间(d)time (d)

Fig. 5 Complex structure multilateral horizontal well in a bottom water reservoir.

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1000

0 0

20

40

60 80 3 3 油井产量(m /d) Well production rate (m /d)

100

120

Fig. 6 Water breakthrough time when the production rate is lower than the critical rate

SEG 微元1 SEG 微元4 SEG 微元7 SEG 微元10 SEG 微元13 SEG 微元16 SEG 微元19 SEG 微元22 SEG 微元25 SEG 微元28 SEG 微元31 SEG 微元34 SEG 微元37 SEG 微元40 SEG 微元43 SEG 微元46 SEG 微元49 SEG 微元52 SEG 微元55 SEG 微元58 SEG 微元61 SEG 微元64 SEG 微元67 SEG 微元70 SEG 微元73

0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 10

20

30

40

SC

0

SEG 微元2 SEG 微元5 SEG 微元8 SEG 微元11 SEG 微元14 SEG 微元17 SEG 微元20 SEG 微元23 SEG 微元26 SEG 微元29 SEG 微元32 SEG 微元35 SEG 微元38 SEG 微元41 SEG 微元44 SEG 微元47 SEG 微元50 SEG 微元53 SEG 微元56 SEG 微元59 SEG 微元62 SEG 微元65 SEG 微元68 SEG 微元71 SEG 微元74

距油水界面距离z(m) Distance from wellbore to WOC (m)

SEG 微元3 SEG 微元6 SEG 微元9 SEG 微元12 SEG 微元15 SEG 微元18 SEG 微元21 SEG 微元24 SEG 微元27 SEG 微元30 SEG 微元33 SEG 微元36 SEG 微元39 SEG 微元42 SEG 微元45 SEG 微元48 SEG 微元51 SEG 微元54 SEG 微元57 SEG 微元60 SEG 微元63 SEG 微元66 SEG 微元39 SEG 微元72 SEG 微元75

RI PT

Potential gradient of segments 各微元下方势梯度 (MPa/m)(MPa/m)

ACCEPTED MANUSCRIPT

M AN U

Fig. 7 Potential gradients at different heights below each micro segment 1200

SEG 微元11 SEG 微元44 微元77 SEG SEG 10 微元10 微元70 …. 微元73 SEG 73

800

600

400

200

0

5

10

EP

0

TE D

Velocity速度倒数 reciprocal (m)

1000

15

SEG 微元22 SEG 微元55 微元88 SEG SEG 11 微元11 微元71 …. SEG 微元74 74

20

25

SEG 3 微元3 SEG 6 微元6 微元9 SEG 9 SEG 12 微元12 微元72 …. SEG 微元75 75

30

35

Distance距油水界面距离(m) from wellbore to WOC (m)

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Fig. 8 Reciprocals of velocity at different heights below each micro segment

Water各微元段见水时间(d) breakthrough time (d)

500

400

300

200

2

100

1

0 1

6

11 16 21 26 31 36 41 46 51 56 61 66 71 微元段编号 Micro segments

ACCEPTED MANUSCRIPT Fig. 9 Prediction of water breakthrough time for each micro segment when the

200 150 100 50 0 100

120

140

160

RI PT

见水时间(d)time (d) Water breakthrough

production rate is greater than the critical rate

180

3

SC

油井产量(m Well production rate/d)(m3/d)

200

M AN U

Fig. 10 Water breakthrough time when the production rate is greater than the critical

3000

Multilateral well 分支水平井 分支水平井

Critical rate 临界产量

2500

单支水平井 Single单支水平井 horizontal well

2000

y1=75600/x R2=1

1500 1000 500 0 0

临界产量 Critical rate

TE D

Water见水时间(d) breakthrough time (d)

rate

20

40

60

80

y2=22070x-1.0189 R2=0.9999

100 120 140 160 180 200 3

EP

油井产量(m /d)3/d) Well production rate (m

AC C

Fig. 11 Compare water breakthrough time of single to multilateral horizontal well

ACCEPTED MANUSCRIPT Fig. 12 Well trajectory of FZ1 Liquid 产液

产液

Oil 产油 产油

Water 产水 产水

压差 Pressure difference 压差 5

600 4 400

3 2

200

1

0 0

500

1000

1500 生产天数 Time (d)

压差(MPa) Production pressure difference (MPa)

6

RI PT

3 rate (m3 /d) production (m /d)

Oil / Water / Liquid 日均产液量\油量\水量

800

2000

2500

0 3000

SC

600 y = 84.51x

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Production rate 3(m 日产液量(m /d)3/d)

Fig. 13 Production curves of FZ1 well

2

400

R = 0.47

200 0 0

1

2

3

4

5

Production pressure difference (MPa) 压差(MPa)

TE D

1000

y = -92.167x + 981.7

800 600

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EP

Production rate3 /d) (m3/d) 日产液量(m

Fig.14 Regression analysis of production and pressure difference of FZ1 well

400

Micro segments

200 0

0

3

6 9 12 p (MPa) wf Flowing bottomhole pressure (MPa)

Fig. 15 IPR curve of single-phase flow in FZ1by coupling model

ACCEPTED MANUSCRIPT

0.7

0.005

3/d) 各微元流量(m Inflow rate (m/d)

0.6

0.004

3

0.5 0.4

0.003

0.3

0.002

0.2 0.001

RI PT

0.1 0

0.000

1

11

21

31

41

51

61

Potential gradient (MPa/m) 势梯度(MPa/m)

各微元正下方势梯度 Potential gradient

各微元流量 Inflow rate

71

81

微元段 Micro segments

SC

Fig. 16 Inflow rate of each micro segment and the potential gradient at the WOC

M AN U

2.0

3

各微元流量(m Inflow rate (m/d)3/d)

各微元见水时间 Breakthrough time

各微元流量 Inflow rate

2.5

1.5 1.0

0.0 1

11

21

31

600

400

200

TE D

0.5

800

见水时间(d) time (d) Water breakthrough

below each micro segment

41 51 微元段

61

71

81

Micro segments

55

EP

Fig. 17 Inflow rate of each micro segment and water breakthrough time

78 77 6 7 75

50

74 3 7

280

72

47 44

70 9 6

45

71

46 42

68 7 6 6 6

43 41

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79

53 51

275

52

270

48

265

49

260

54

Water 见水时间（天） Breakthrough Time (d)

88 89

65

64

63

87

62

86

61

85

60

84

59

83

58

82

57

81

80

56

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Fig. 18 Dynamic of water breakthrough when bottom water cresting

ACCEPTED MANUSCRIPT -4540

TK906H 油水界面

Vertical depth (m) 深度(m) 深度(m)

井轨迹

沙层顶面

-4560

Upper boundary -4580

WOC

Well trajectory

-4600

-4620 100

200

300

400

500

600

700

800

RI PT

0

m) 位移( Horizontal departure (m) 位移(m)

Fig. 19 Reservoir boundary and well trajectory of well TK906H 油水界面

沙层顶面

SC

TK909H

Upper boundary

-4570

-4590

WOC

Well trajectory

M AN U

深度(m) 深度(m) Vertical depth (m)

井轨迹 -4550

-4610 0

100

200

300

400

500

600

700

800

位移(m) Horizontal departure (m) 位移(m)

TK906H TK906H井

200

100 80 60 40

09-02-15 2040

08-10-15 1920

08-06-15 1800

08-02-15 1680

07-10-15 1560

07-06-15 1440

Time (d)

07-02-15 1320

20 06-10-15 1200

06-02-15 960

05-10-15 840

TE D 05-06-15 720

05-02-15 600

04-10-15 480

EP

240 04-02-15

0

03-10-15 120

50

04-06-15 360

100

06-06-15 1080

日产油量 Oil rate Water 含水率cut

150

含水率（%） Water cut (%)

TK906H

03-06-15 0

3 3 Oil 日产油量（m production rate /d）(m /d)

Fig. 20 Reservoir boundary and well trajectory of well TK909H

0

TK909H井 TK906H

日产油量 Oil rate Water cut 含水率

120

100

TK909H

80 60

80 40

Fig. 22 Daily production of well TK909H

09-01-11 1260

08-10-11 1170

08-07-11 1080

08-04-11 990

08-01-11 900

Time (d)

07-10-11 810

07-07-11 720

07-04-11 630

07-01-11 540

06-10-11 450

06-07-11 360

0

06-04-11 270

0

06-01-11 180

20 05-10-11 90

40

含水率（%） Water cut (%)

160

05-07-11 0

Oil production3 /d） rate (m3/d) 日产油量（m

AC C

Fig. 21 Daily production of well TK906H

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

Fig. A1 Micro segment in 3D space

Input parameters: reservoir, fluid properties, wellbore parameters, etc.

TE D

Calculate the potential gradient on the WOC under each segment while each segment on unit inflow Set an initial potential difference on heel of main wellbore According to the potential difference, call the solving process of branch-horizontal well coupling model

EP

Calculate the potential gradient on the WOC under each ∂Φ segment while each segment with real inflow： ij z =0 ∂z

AC C

Find the maximum of potential gradient,

∂Φ max ∂z

z =0

< ( ρo − ρ w ) g

Yes

Set a higher potential difference

∂Φ max ∂z

∂Φmax ∂z

z=0

z =0

, compare to the startup condition of cresting

= (ρo − ρw )g

Yes End the calculation, output each micro segment inflow qij , output total well cumulative production, ie, critical rate, Output the pressure of heel of main wellbore to get the critical pressure difference.

∂Φ max ∂z

z =0

> ( ρo − ρw ) g

Yes Set a smaller potential difference

Fig. A2 Workflow to calculate the critical parameters

ACCEPTED MANUSCRIPT Highlights: (1) Fills the gap to forecast water breakthrough time for a multilateral horizontal. (2) Takes many factors including wellbore 3D structure and fluids properties into model.

AC C

EP

TE D

M AN U

SC

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(3) Guides operator to choose a reasonable production rate to delay the water coning.