Prediction of water channeling along cement-aquifuge interface in CBM well: Model development and experimental verification

Accepted Manuscript Prediction of water channeling along cement-aquifuge interface in CBM well: Model development and experimental verification Jun Gu, Pin Gan, Lifei Dong, Shuai Wang, Naiqian Tang PII:

S0920-4105(18)30908-2

DOI:

10.1016/j.petrol.2018.10.040

Reference:

PETROL 5403

To appear in:

Journal of Petroleum Science and Engineering

Received Date: 2 November 2017 Revised Date:

31 August 2018

Accepted Date: 14 October 2018

Please cite this article as: Gu, J., Gan, P., Dong, L., Wang, S., Tang, N., Prediction of water channeling along cement-aquifuge interface in CBM well: Model development and experimental verification, Journal of Petroleum Science and Engineering (2018), doi: https://doi.org/10.1016/j.petrol.2018.10.040. 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

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Jun Gu∗, Pin Gan, Lifei Dong, Shuai Wang, Naiqian Tang

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Department of Petroleum Engineering, Faculty of Earth Resources, China University of Geosciences, Wuhan 430074, China

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ABSTRACT

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Controlling water channeling (e.g. water inrush) is vital for coalbed methane (CBM) development.

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In this work, the prediction models for bottom water channeling along cement-aquifuge interface

8

(CAI) in CBM well were deduced to describe the time variation of water channeling at different

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stages during production. The prediction is focused on the evolution process characteristics of the

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bottom water channeling along CAI. The models indicate that the water channeling time is related

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to the differential pressure of both ends of channeling pathway, and the tortuosity of the

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channeling pathway, the height of bottom water channeling, fracture size of CAI, bottom water

13

density, bottom water viscosity, bottom water surface tension, contact angle, bottom water

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dissolution time, permeability of mud cake before and after bottom water dissolution, and hole

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diameter. In order to verify the models, the simulation device was established independently. The

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experimental devices and methods are constructed to verify the prediction models. The results

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illustrate that the predicted results of the models are in good agreement with the experimental data.

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This study will provide a new insight into the solution of water channeling problem, which is

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common in the exploitation of CBM reservoirs.

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Keywords:

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Coalbed methane; Water channeling; Cement-aquifuge interface; Prediction model; Verification.

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1. Introduction

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Prediction of water channeling along cement-aquifuge interface in CBM well: Model development and experimental verification

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Coalbed methane (CBM) is usually considered as an unconventional natural gas

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(Moore, 2012; Reem, 2012; Kang et al., 2016; Peng et al., 2017). In recent years, the

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commercial development of CBM in high-rank coal reservoirs of the Qinshui basin in

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China has achieved significant success (Cai et al., 2011; Xu et al., 2012). However,

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the breakthroughs in the CBM industry have been accompanied by a number of

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difficulties (Hu et al., 2014). Controlling water channeling (e.g. water inrush) is one

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of the major concerns for the efficient exploitation of CBM (Li et al., 2011). Field * Corresponding author. Tel.: +86 27 6784 8569; fax: +86 27 6788 3051. E-mail address: [email protected] (J. Gu). 1

ACCEPTED MANUSCRIPT production practice has proved that excessive water influx in CBM wells has become

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a major restriction for gas production enhancement in CBM wells. For example, for

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477 wells in the Puchi block in China, 12% of wells produce only water without gas

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production, 59% of wells produce gas with water, and only 29% of the vertical wells

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produce gas without water. Moreover, five wells in the Puchi block have basically no

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gas output (Zhao et al., 2011). Among 399 wells in China's Zhengzhuang and

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Fanzhuang and other blocks, 105 wells produce less than 500 m3 gas per day,

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accounting for 26% of the total number of wells. And high water production wells

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have low gas production while low water production wells have high gas production

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(Hao and Duan, 2012). More than 20 vertical wells in Jiaozuo of China, after repeated

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fracturing, single well cumulative production period varies from 15 to 30 months and

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gas production of all wells are poor. Six wells were investigated but five of them

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produced water with no coal seam gas production (Yin and Jiao, 2012). In China's

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Jincheng area, daily water production for single well is tens of cubic meters, even to

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more than one hundred cubic meters, which is driven from aquifer below the aquifuge

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under pressure (Wang et al., 2012). The study also shows that if the cumulative

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displacement increases and the coal reservoir pressure is reduced in the process of

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CBM wells drainage decompression, it will be difficult to achieve the critical

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desorption pressure and results in no CBM output in the long term (Qin et al., 2012).

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The lower part of the coalbed gas reservoirs are mostly aquifer which can be called

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the bottom water. The coal-bearing strata are deposited layer by layer, so it has

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layered structure characteristics (Qian et al., 2003; Sun et al., 2005; Li et al., 2006; Yu

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et al., 2011; Zhang et al., 2013). Bottom water channeling is one of the biggest

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problems for high efficiency exploitation of CBM reservoirs (Yuan et al., 2007; Meng

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et al., 2011; Meng et al., 2012; Fan, 2012; Agarwal et al., 2013). Once the bottom

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water channels into CBM wells, a significant reduction in CBM extraction efficiency

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would be encountered (Meng et al., 2013; Wang et al., 2013), even no gas can be

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produced at all (Zhao et al., 2012). In general, average thickness of aquifuge is 20-30

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m (Qian et al., 2012). Research shows that the bottom water mainly flows up along

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the cracks which connect aquifer and aquifuge (Wang et al., 2012). However, in

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addition to channeling through the fracturing cracks (Wang et al., 2012), natural

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structural cracks and mining cracks (Miao et al., 2012), whether the bottom water can

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ACCEPTED MANUSCRIPT flow up along the cement-aquifuge interface (CAI, e.g. cementation plane between

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cement slurry and aquifuge) remains unknown. Many researches and studies have

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been carried out to solve the problem of water channeling into productive formation

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of mines, and more than ten theories and prediction methods for water channeling in

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mines have been developed (e.g. Li and Wu, 2008; Wang et al., 2008; Shi, 2009; Yin,

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2009; Li et al., 2009; Li and Ye, 2011; Fan, 2012; Han et al. 2012; Meng et al. 2012;

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Li et al., 2013; Zhang and Hu; 2013; Zhang et al., 2013; Jia Jet al., 2014; Liu et.al.,

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2014; Li et al., 2015; Sun et al., 2016; Chuai and Teng, 2017; Gui and Lin, 2017; Guo

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et al., 2017; Wu et al., 2017), such as the theory of floor relative to aquitard, the

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theory of water inrush coefficient of floor, the theory of hydraulic stress of rock mass,

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the theory of strong channel seepage, the theory of lower zone, theory of in-situ crack

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and zero damage, theory of critical layer of watertight floor, theory of progressive lift,

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theory of water inrush from closed borehole, nonlinear dynamic method, numerical

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simulation method and so on. These theories and methods are undoubtedly very

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useful to solve the water inrush problem and obtain satisfactory application effect.

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However, even though CBM wells experience the same strata and share the

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similar strata structure in the vertical direction with coal mines, most of CBM wells

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must be isolated by running casing, and the annular space between the casing and the

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formation must be sealed off by cement. So in fact, the structure of CBM well

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resembles more to the wellbore structure of conventional oil and gas wells. Therefore,

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using theory and prediction models for water channeling in coal mines to solve the

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similar case in CBM well may cause major deviation, let alone to enhance

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exploitation efficiency of CBM. The reasons for water channeling in CBM wells is

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that the aquifuge and cement paste is difficult to form an interface without any

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clearance (Liu et al., 2008; Jing et al., 2010; Gu et al., 2011), since the mud cake

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embedded in the gap between the aquifuge and cement cannot be removed completely,

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which is necessary for drilling safety and reservoir protection. No matter how thin the

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cake is, it will result in a great decrease of interfacial layer adhesive strength (Zhao et

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al., 1996; Gu et al., 2008; Liu et al., 2008; Gu and Chen, 2010; Lyons, 2011; Gu et al.,

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2012). And once the production pressure is too high or control method is improper

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(Liu et al., 2008), the bottom water will break through CAI, causing high water

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content of CBM wells. Moreover, as shown in Fig.1, after immersion, erosion,

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washing and other effects of bottom water, the channeling pathway will be enlarged

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(Liu et al., 2008; Guo et al., 2009).

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No water channeling

Mud cake

Mud cake

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Aquifuge

Aquifuge

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Water flooding

Casing

CBM production

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104 105

Reservoir Cement sheath

No dissolution

Dissolution

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Perforation

Perforation

Reservoir

Aquifuge

Aquifuge

Aquifer

Aquifer

Bottom water channeling

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Water channeling

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Fig. 1. Schematic diagram of physical model of water channeling at CAI in CBM well

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In this paper, a prediction models for water channeling in CBM wells is deduced

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and established. Taking the samples from CBM well T5-293 of Qinshui basin in

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China as an example, the prediction models are verified.

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2. The models

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According to the principle of fluid mechanics, all the particles in laminar flow

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movement stream flows along a certain order rather than being mixed together.

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Typically, the tube inner diameter less than a millimeter is called a capillary.

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According to the practice of the oil industry, the thickness of mud cake is less than 0.5

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mm. And it is usually no more than a millimeter in the oil field application. So the

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flow of the bottom water along CAI can be regarded as laminar flow in a capillary. On

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this basis, the process of bottom water channeling along CAI can be divided into three

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stages:

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● Capillary seepage stage.

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● Corrosion and dissolution stage.

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● Crack pipe flow stage. 4

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2.1. The stage of capillary seepage The main pathway for bottom water channeling is the tiny cracks at CAI and the

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pores in mud cake matrix. Both of them can be deemed as the capillary tubes duo to

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their scales. Therefore, the water migration through these cracks and pores can be

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treated as capillary flow. In this stage, the bottom water channeling is mainly

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determined by the following four forces, i.e., capillary force (Fcap), viscosity

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resistance of capillary wall (Fvis), hydrodynamic force of bottom water (Fw) and

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gravity of bottom water (G). However, these forces make different contributions in

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migration development. In the initial stage of bottom water channeling, the seepage

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flows upward very slowly, thereby the effect of viscous resistance and gravity, which

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depend on the amount of liquid, is limited and can be negligible. Most of the pressure

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is supported by the solid particles and the rest of the pressure put on the fluid column

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is very little and can be negligible.

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Capillary force is a major power for bottom water channeling at this stage, that is

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to say, the water channeling is mainly affected by the surface tension (Zhang and

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Wang, 1989):

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Fcap = f ⋅ 2πR cos θ

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(1)

Where f is the surface tension of the bottom water, θ is contact angle and R is the radius of micro-cracks or pores, the maximum of which is the thickness of mud cake. When a steady flow along CAI or in the mud cake matrix is established, the

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amount of liquid keeps increasing in micro-cracks or pores, so the viscosity resistance

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of capillary wall and bottom water's gravity is becoming more and more prominent.

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These forces controls bottom water upward seepage against capillary force. As the

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bottom water flowing up, the viscosity resistance and its own gravity can balance the

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capillary force, and the bottom water cannot go upward any more as a result. At this

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time, the bottom water reaches the maximum height.

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Fcap = Fvis + G

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(2)

According to the law of Newton fluid internal friction calculation (Yang, 1989), the total viscosity resistance in capillary side wall can be calculated as:

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ACCEPTED MANUSCRIPT Fvis = 2πRh ⋅ τ = 2πRhη

rate of bottom water, r is the coordinates of the diameter direction. The gravity of the bottom water can be expressed as:

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(3)

Where τ is internal friction shear stress, η is bottom water viscosity, v is the flow

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dv dr r = R

G = mg = πR2hρw g

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(4)

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Where h is the height of the bottom water channeling (maximum value is the

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thickness of aquifuge), ρw is the density of bottom water, g is gravitation acceleration.

f ⋅ 2πR cosθ = 2πRhη

dv + πR 2hρw g dr r = R

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Eqs. (1), (4) and (3) to (2), the following equation can be obtained:

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Based on Hagen-Poiseuille equation (Zhao, 2011), the velocity gradient of bottom

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water seepage in micro-cracks or pore at CAI can be expressed as:

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1 d dv 1 ∂p (r )= r dr dr η ∂z

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micro-cracks coordinates, positive direction is upwards.

dv = 0 ；r=R，ν=0 dr

According to the boundary conditions, r=0，

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Assume that ∂p is independent of r, for Eq. (6) integral: ∂z

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dv 2 r ∂p = ⋅ dr 4η ∂ z

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v=

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1 ∂p (r 2 − R 2 ) 4η ∂z

(7) (8)

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(6)

Where P is fluid pressure, Z is the CAI (or porosity) extension direction of

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(5)

From Eq. (8), the flow velocity (v) may change if any of the variables R, r or ǝp/ǝz

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varies. But at the stage of capillary seepage, the velocity of seepage along CAI in

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CBM well is very slow. Thus, this equation could be solved under the assumption of

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steady state (Kong, 2010). The velocity that liquid flows along the radius direction is

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changing in the micro-cracks or pores at CAI, so the average flow velocity is used to

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represent the stages of the flow rate of bottom water:

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Q 1 1 ν = 2 = 2 ∫ 2πrvdr = 2 πR πR 0 πR

R

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1 ∂p 2 R 2 ∂p 2 π ( R − r ) 2 rdr = ∫0 4η ∂z 8η ∂z R

So: 6

(9)

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Eqs. (10) to (7), the following equation can be obtained:

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dv 2 r ∂p 2 r 8η ν = ⋅ = ⋅ dr 4η ∂ z 4η R 2

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dv dr

= r=R

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At r = R at the wall of the micro-cracks on CAI, Eq. (11) can be expressed as:

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(11)

4ν R

(12)

Eqs. (12) to (5), in this stage, the flow rate of bottom water can be calculated:

ν=

2 fR cosθ − hρw gR2 8ηh

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(10)

(13)

And the model of bottom water channeling time at this stage can be expressed as: tA =

h

ν

=

8ηh 2 2 fR cosθ − hρ w gR 2

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∂p 8ην = 2 ∂z R

(14)

For the derivation of equations, the channel pathway along CAI is considered as a

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cylindrical tube. In fact, the channel pathway is complex in shape, which has certain

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effect on the bottom water channeling time. So, the introduction of a tortuosity (ε) as a

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correction factor (the ratio of actual distance that fluid flow and the apparent distance).

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Typically, the tortuosity increases as the thickness of the barrier layer increases.

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Therefore, the amount related to the distance of the bottom water channeling at this

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stage should be multiplied by this factor.

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So, the revised model of bottom water channeling time in this stage is as follows: 8ηε 2 h 2 2 fR cosθ − εhρ w gR 2

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tA =

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2.2. The stage of corrosion and dissolution

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The mud cake is the deposition of solid particles in the drilling fluid. Crosslinking

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material is wrapped around the cake particles thus occupies parts of the pore volume

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of mud cake and forms a mesh structure, which connects the particles in the cake. In

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the previous stage, the bottom water has channeled to a certain height, and then it will

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maintain a certain balance. 7

ACCEPTED MANUSCRIPT Firstly, some of the bound water inside the mud cake will dissolve some

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crosslinking material by a long period of water invasion. Dissolution-precipitation

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remains a relative balance, but with bottom water channeling under capillary force,

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the concentration of the crosslinking substances in the liquid phase was diluted.

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Consequently, the equilibrium is broken and dissolution process is promoted.

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Moreover, the bottom water contains more mineral chemicals thereby corresponding

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corrosion reactions can take place during long-term contact, further communicating

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the network inside the mud cake matrix, expanding fracture scale and peeling particle.

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Eventually, a continuous channel form along CAI under the action of the pressure,

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resulting in intensified water channeling or even explosive flooding.

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Second, as the contact time increases, the bottom water continues to infiltrate the

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crosslinking material around the cake particles, resulting in the loss of local capillary

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force. According to the study of sand grains (Yang et al., 2008), sometimes an upright

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pit can be dug in wet sand, the short-term stability of which derives from capillary

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force caused by water between the contact surfaces of particles. The capillary force

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inside the pore increases resistance among the sand grains thus prevents relative

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movement. Once completely submerged by water, the capillary force will be equal to

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zero, and this pseudo-cohesive (Fig. 2) to stabilize wet pit disappears, leading to

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dislocation of particles. By the same token, with the unceasing invasion and corrosion

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of bottom water, the particles in the mud cake will be completely immersed, making

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the inter particle capillary force disappear. Original water inside the mud cake

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shortens this process. When subjected to differential pressure or bottom water

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hydrodynamic force, the particles will be dislocated and the channel will be enlarged.

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Thus, the bottom water can go up again.

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Capillary force

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Particle

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Fig. 2. Schematic diagram of capillary force (pseudo-cohesive). 8

ACCEPTED MANUSCRIPT 235

However, the damage caused by the infiltration is relatively weak, and the

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communication of pores or micro-fractures in the mud cake is poor. Therefore, the

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time required for the bottom water to completely immerse the cake particles is very

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long. Therefore, compared to dissolution, the infiltration just play a guiding role. In

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other words, to establish the mathematical model of this stage, the corrosion reactions

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must be studied. The dissolution mechanism includes chemical dissolution and

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physical erosion. Before the channeling pathway is connected, the bottom water is

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relatively stable so physical erosion is relatively mild. Therefore, the dissolution

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process mainly depends on chemical corrosion, manifesting as the increase of pore

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size and growth of micro-cracks.

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From the scanning electron microscope image of the field drilling mud cake (Fig.

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3), it can be seen that the space between cake particles (mainly clay particles) is filled

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with the crosslinking substances. The drilling fluid for the preparation of the mud

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cake is from CBM well T5-293 in Qinshui basin in China, which is composed mainly

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of 0.5 wt.% encapsulating agent (Na-HPAN), 0.7 wt.% fluid loss agent (SDS), 1.5

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wt.% anti-sloughing agent (SDFT-2) and 1.0 wt.% anti-sloughing agent (SDFT-1).

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These agents are consist of high molecular polymers, which have turned into the

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crosslinking substances filling between clay particles. The adsorption groups (–CN

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and –CONH2) and hydration group (–COONa) in the molecular chain of

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Na-hydrolyzed polyacrylonitrile (Na-HPAN) can form hydrogen bond with O2 on the

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surface of clay particles, and coordination bonds with Al3+ on the edge of broken bond

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of clay particles. Therefore, the Na-HPAN can be absorbed on the surface of clay

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particles (Ma, 2011). The –SO4Na in the molecular chain (C12H25SO4Na) in sodium

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dodecyl sulfate (SDS) can form coordination bond with Al3+ on the edge of broken

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bond of clay particles. Meanwhile, the –SO4- in C12H25SO4Na can combine with Al3+

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at the edge of clay particles. So, the SDS can also be absorbed on the surface of clay

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particles (Fan et al., 2013). The SDFT-2 and SDFT-1 are both the modified

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polyalcohol anti-sloughing agent. The modified polyols can be absorbed on the

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surface of clay particles by the hydrogen bonding, and also would enhance the

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adsorption ability by the electrostatic force (Qiu et al., 2006; Muherei et al., 2009). It

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is observed that the crosslinking material is wrapped around the cake particles. It

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occupied parts of the pore volume of mud cake, and formed a netlike structure which

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connects the particles in the cake. Therefore, the crosslinking material among the mud

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cake particles will always coat the mud cake particles, as shown in Fig. 4. In order to establish a mathematical model of bottom water channeling, an

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abstract model reflecting the inner structure of mud cake should be established to

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simplify the analysis. Fig. 3 shows the structure of mud cake at CAI can be described

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as dense packing structure (Fig. 4). Thus in fact, the dissolution of mud cake can be

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simplified as dissolution of crosslinking substances. However, the crosslinking

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material composition is complicated. Assume A is crosslinking material, xi represents

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the amount of dissolved crosslinking material by bottom water after a certain period

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time (ti), and then dissolution of crosslinking material can be expressed as:

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Time

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t=0

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t = t1

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t = tB（channeling）

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n0

0

n-x1

x1

n-xB

x

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Crosslinking material

Fig. 3. ESEM pattern of mud cake

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Small unit

Mud cake particle

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a

Mud cake particle

Fig. 4. Microstructure structural model of mud cake

Taking the small unit in Fig. 4 as a research object, assuming that the length of the

283

unit cell is a, the aquifuge thickness is H, the mud cake thickness is d, the molar mass

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of crosslinking material is M, and the diameter is D, the cake volume can be

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calculated as follows:

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VMud

2  D2  D    = πH − − d  = πHd ( D − d )  4  2   

10

(16)

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The volume of the tiny unit at CAI is: VUuit = a 3

(17)

Therefore, the number of the tiny units in the mud cake can be expressed:

N=

VMud πHd = 3 （(D − d ) VUnit a

(18)

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When t = 0, the bottom water has not begun to dissolve crosslinking material, and the initial amount of crosslink material can be expressed as:

n0 = N

(19)

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m ρV πHρVFilld (D − d ) = N Fill = M M a3M

Where m is the mass of the crosslinking material in the tiny unit, ρ is the density

295

of the crosslinking material, and VFill is the volume of the crosslinking material in the

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tiny unit.

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As the bottom water dissolves crosslinking material, the porosity of mud cake

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increases. When t = t1, the amount of dissolved crosslinking substances can be

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represented by the porosity of mud cake:

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x1 = N

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(20)

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m ρV πHρVFill d = N Fill = ( D − d（ ) (φ1 − φ0 ) M M a3M

Where Ø1 is the porosity of the mud cake after t1, and Ø0 is the initial porosity of the mud cake.

Bottom water dissolution is essentially a chemical reaction. Combined with the

304

chemical reaction rate equation (Jin, 2014), relationship between the dissolution rate

305

and the time can be expressed as:

306

vr =

308

=

1 dn A γ AV dt

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ξ

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V

(21)

Where vr is the reaction rate, ζ is the conversion rate, γA is the stoichiometric

number, V is the reaction volume, and nA is the amount of the reactant substance.

309

Assuming that the temperature of the aquifuge section is constant, the

310

stoichiometric number ζ is 1 and the dissolved quantity of crosslinking substance is x,

311

according to Eq. (21), dissolution rate can be calculated as:

312

vA =

1 dx V dt

(22)

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ACCEPTED MANUSCRIPT 313

Where vA is dissolution rate.

314

The dissolution rate equation is integrated from the initial time t0 to the time t1: x1

315

t

1 1 ∫ V dx = t∫ vA dt x0 0

(23)

If the temperature is constant, the dissolution rate is only determined by

317

concentration. Since the crosslinking substance diffuse rapidly in the bottom water

318

and its concentration in the bottom water is almost constant, i.e., the dissolution rate

319

does not change with time. Therefore, the vA within the integral is independent of t.

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1 ( x1 − x0 ) = vA (t1 − t0 ) V And then the dissolution rate is:

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vA =

1 ( x1 − x0 ) V (t1 − t0 )

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The integration is performed on Eq. (23), and it can be obtained:

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(24)

(25)

Assuming that the channeling pathway is formed when t = tB and the channeling

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pathway is tubular pipe with a radius of R and a tortuous degree of ε, the number of

326

micro-unit which has been dissolved at tB can be expressed as:

327

NB =

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of channeling pathway and R is the radius of channeling pathway.

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Therefore, the amount of dissolved crosslinking substance can be expressed as:

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m ρVFill επHR2 ρVFill xB = N B = NB = M M a3 M

xB

t

333

B 1 ∫ V dx = t∫ vA dt x0 0

334

vA =

336

(27)

Similarly, the dissolution rate equation is integrated from the initial time t0 to tB:

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335

(26)

Where NB is the number of micro-unit which has been dissolved, VR is the volume

328 329

επHR2 VR = VUnit a3

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(28)

1 ( xB − x0 ) V (t B − t 0 )

(29)

Where xB is the amount of dissolved crosslinking substance in the time tB, and x0 is the amount of dissolved crosslinking substance at the initial time t0. 12

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338

339 340

From Eqs. (25) and (29), the following equation can be obtained:

1 ( x1 − x0 ) 1 ( xB − x0 ) = V (t1 − t0 ) V (t B − t0 )

(30)

Take the initial moment: t0 = 0, x0 = 0. It is possible to obtain the time required for the bottom water to form a channeling channel:

επHR2 ρVFill

341

tB =

RI PT

337

( )t1 ( xB − x0 ) xt εt1R2 a3 M (t1 − t0 ) + t0 = B 1 = = πHρVFilld ( x1 − x0 ) x1 ( D − d )(φ1 − φ0 ) d ( D − d )(φ1 − φ0 ) a3 M

(31)

In Eq. (31), the measurement of porosity Ø0 and Ø1 are relatively difficult, but

343

they can be converted to the permeability according to the relationship between

344

porosity and permeability in the ideal soil model.

M AN U

SC

342

According to the research on the microstructure of mud cake at CAI, combined

346

with the microstructure model of mud cake (Fig. 4) and the ideal soil model (Fig. 5),

347

the unit of mud cake can be regarded as a cubic which is composed of same size

348

particles, and the center of each particle is located in the cubic lattice vertex. Each

349

particle is in contact with other six particles to form a cube arrangement. Different

350

particle arrangement patterns and tightness will form different shapes of pores, and its

351

pore distribution and size are also very different.

TE D

345

As mentioned previously, the micro-cracks or pores in mud cake at CAI can be

353

equivalent to capillary (Fig. 6). According to Poisenille’s law, the flow rate of the

354

fluid in a single capillary is:

355

π r1 4 ∆ p Qs = 8η l

357 358

(32)

AC C

356

EP

352

Where r1 is the radius of a single capillary and l is the length of the same single

capillary.

13 2r l

ACCEPTED MANUSCRIPT

Fig. 5. Schematic diagram of ideal soil model

Fig. 6. Schematic diagram of capillary model

359

The micro-cracks and pores can be viewed as n capillaries with the same diameter.

361

By the capillary seepage law (Zhang and Wang, 1989), the flow of bottom water

362

through CAI can be expressed as:

363

nπr ∆p Q= 1 8ηl

RI PT

360

4

SC

The porosity can be defined as:

VØ nπr1 l = VMud AH 2

φ=

M AN U

364 365

(33)

(34)

366

Where VØ is the volume of pore, A is the sectional area of mud cake, n is the

367

number of capillary tube, ∆Ρ is the differential pressure of the both ends of channeling

368

pathway.

According to the definition of tortuosity which associated with complexity of pore

TE D

369 370

structure of mud cake at CAI, the following equation can be obtained:

371

l = εH

374 375 376 377 378 379

φ =

nπr1 ⋅ εH nπr1 ε = AH A 2

So:

EP

373

Therefore: 2

AC C

372

(35)

(36)

n π e 2ε = φ A

(37)

Eqs. (37) to (33), the following equation can be obtained:

2 2 nπr1 ⋅ r1 ∆p φ A ⋅ r1 ∆p Q= = 8ηεH 8ηε 2 H 2

(38)

The Darcy's law (Zhang and Wang, 1989): Q=

KA ∆p ηH

(39)

14

ACCEPTED MANUSCRIPT 380

Where K is the permeability of mud cake.

381

From Eqs. (38) and (39), the following equation can be obtained:

KA∆p φ A ⋅ r1 ∆p = ηH 8ηε 2 H 2

384

simplifying Eq. (40):

385

8Kε 2 φ= 2 r1

RI PT

The relationship equation of porosity and permeability can be obtained by

383

(41)

tB =

t1R 2 K K 8εd ( D − d )( 21 − 20 ) r1 r0

SC

Eqs. (41) to (31), the function of corrosion time at this stage can be obtained:

386 387

(40)

(42)

M AN U

382

388

Where K0 is the initial permeability of mud cake, K1 is the permeability of mud

389

cake after corroded for time t1, r0 is the radius of the pore of mud cake or micro-crack

390

at the initial time t0, and r1 is the radius of the pore of mud cake or micro-crack at time

391

t1 .

Channeling pathway at CAI has not run through after t1. Although corrosion has

393

widened pores in the mud cake, the permeability of mud cake is dominated by the

394

radius of the smallest pores or micro-cracks at this moment. Therefore, r1 is

395

equivalent to r0:

396

r1 = r0

EP

(43)

Eqs. (43) to (42), the time required to form a channeling pathway at CAI can be

AC C

397

TE D

392

398

obtained:

399

t1r0 R 2 tB = 8εd ( D − d )( K1 − K 0 )

400

2.3. The stage of crack pipe flow

2

(44)

401

The particles lose protection and restriction when bottom water dissolves

402

crosslinking material away. Consequently, the particles are surrounded by water

403

molecules (Fig. 7) and isolated from the other adjacent particles. With migration of

404

bottom water, the particles are gradually separated from the mud cake and flow away 15

ACCEPTED MANUSCRIPT 405

with bottom water (Fig. 8).

Water molecule

RI PT

Water molecule

Mud cake particle

Mud cake particle

Fig. 8. Schematic diagram of mud cake peeled

corrosion at CAI

and corroded by the bottom water

SC

Fig. 7. Schematic diagram of mud cake

M AN U

406

After the stage of corrosion and dissolution, the crosslinking substance is

408

separated from mud cake particles and the channeling pathway forms. Once this

409

happens, the mud cake particles are continuously washed away under the scour of

410

bottom water, leading to quick expansion of channeling pathway and intensified

411

bottom water channeling. Often, oil wells will appear high water content or even

412

explosive flooding as a consequence.

TE D

407

With the dislocation of the mud cake skeleton and the destruction of structural

414

strength, the pressure that solid particles withstand decreases while the pressure that

415

pores water hold increases, and the hydrodynamic force of bottom water become the

416

main power of bottom water channeling. Thus, the original balance is disturbed and

417

the bottom water continues to channel. At this time, the capillary fore is trying to

418

restore to the original balance. So the capillary force becomes resistance for water

419

migration. It is assumed that the bottom water flows up evenly along CAI and the

420

flow regime of bottom water is laminar flow. The force analysis of bottom water in

421

channeling pathway is shown in Fig. 9.

AC C

EP

413

422 Fw

423 424

f

f

425

θ

426 16

G Fvis

ACCEPTED MANUSCRIPT 427 428 429 430 431

Fig. 9. Force analysis diagram of bottom water in channeling pathway of bottom water

433 434

RI PT

432

According to the Fig. 9, the balance can be re-established if:

Fcap + Fw = Fvis + G

(45)

The radius of capillary tube is the radius of channeling pathway (R) at this

436

moment because the channeling pathway of bottom water has formed, so the

437

hydrodynamic force of bottom water can be calculated as:

438

Fw = πR 2 ∆p

M AN U

SC

435

(46)

The expression equation of the capillary force (Fcap), the viscosity resistance of

440

capillary sidewall (Fvis), and the liquid’s gravity (G) is basically the same as the

441

equations deduced in capillary seepage stage. Substituting these equations into Eq.

442

(45), the following equation can be obtained:

TE D

439

f ⋅ 2πR cos θ + πR 2 ∆p = 2πRhη

444

dv 4v = dr r =R R

445

dv + πR 2 hρ w g dr r = R

(48)

Eqs. (48) to (47), the rate of bottom water channeling at this stage can be obtained:

447

R 2 ∆p + 2 Rf cos θ − R 2 hρ w g v= 8hη

AC C

446

448

(47)

EP

443

(49)

Considering the tortuosity, the model of bottom water channeling time at this

449

stage can be obtained:

450

tC =

εh v

=

8ηε 2h2 R 2∆p + 2Rf cosθ − R2εhρ w g

(50)

451

Bottom water migration along CAI is dominated by corrosion and dissolution

452

process. Chemical dissolution is the main reason for the channeling pathway 17

ACCEPTED MANUSCRIPT development and physics destruction is important factor for channeling pathway

454

expansion. The three stages are interconnected and have no fixed boundaries. Early,

455

the capillary seepage provides conditions and lays a fundamental for the stage of

456

corrosion and dissolution, and the stage of crack pipe flow is the evolution of the

457

stage of corrosion and dissolution.

458

3. The verification

459

3.1. Experimental

460

3.1.1. Development of experimental simulation system

SC

RI PT

453

In order to verify the accuracy of mathematical models, a simulated experimental

462

device is built independently (Fig. 10). The pressure range of hydraulic pump is from

463

1 MPa to 25 MPa. The accuracy of pump is 0.05 MPa. It can increase and maintain a

464

constant pressure without surging. The highest temperature of heating device can

465

reach 260 °C. The temperature can be controlled automatically and displayed on the

466

screen. The flowmeter is used for measuring the amount of fluid channeling along

467

CAI.

TE D

M AN U

461

468 469 470 Outlet pipe

473

477

Heating and holding device

AC C

476

Sample chamber

475

Mud cake （CAI）

Cement paste

Pressure gage

472

474

Fluid flowout

Flowmeter

EP

471

Valve

Sealing glue

Cylinder block

Valve

SWB Hydraulic pump

478 Inlet pipe 479

Flowmeter

Pressure gage Hydraulic pressure

(a) Schematic diagram of experimental device

(b) Schematic diagram of sample

Fig. 10. Simulating apparatus for evaluating bottom water channeling at CAI

18

ACCEPTED MANUSCRIPT 480

3.1.2. Preparation of Samples The preparation of samples is divided into six steps:

482

1. According to the physical properties of aquifuge and the compaction laws, the

483

materials composition for simulated wellbore (SWB) and the pressure value of

484

hydraulic pump can be determined after many simulation experiments.

485

Permeability and porosity of SWB are close to those of aquifuge. The

486

preparation procedures of SWB was described in the literature (Gu et al., 2012).

487

2. Seal the bottom of SWB with grease in case of the leakage of drilling fluid from

490 491 492

SC

489

gap between glass sheet and SWB.

3. Inject drilling fluid into SWB and hold for a while, then the mud cake will be naturally formed due to the filtration of drilling fluid.

4. Take off the glass sheet and scrape the false mud cake with glass rod to obtain

M AN U

488

RI PT

481

mud cake with certain thickness.

5. Seal the bottom of SWB with grease and glass sheet again, then inject prepad

494

fluid into SWB and immerse for 2-3 minutes. Afterwards, pour out prepad fluid

495

and inject cement slurry into SWB. Finally, put SWB into the constant

496

temperature water curing box.

TE D

493

6. Seal SWB with the sealing glue so that liquid can only flow through CAI. Then

498

put SWB into cylinder block and ensure lateral seal. Eventually SWB for

499

bottom water channeling experiment will get ready after glue consolidation.

501 502 503 504

3.1.3. Experimental Schedule

The experimental schedule is divided into three steps:

AC C

500

EP

497

1. Air emission: put the cylinder block and SWB into water horizontally so that the left space of cylinder block is filled with water. Then cover the lids at both ends of cylinder block to ensure air in the gaps of cylinder block let out.

505

Afterwards add water into water inlet and water outlet with hydraulic pump in

506

order to discharge air in pipeline.

507

2. Record time when the first water drop penetrated along CAI: close the water

508

inlet valve, exert a certain pressure to cylinder block with hydraulic pump and

509

take record of the pressure value. Open the water inlet valve and start timing. 19

ACCEPTED MANUSCRIPT 510

Record the changes of pressure gage and flowmeter and measure the time when

511

the first water drop penetrated out of water outlet pipe. 3. Measure changes laws of the bottom water migration along CAI versus time:

513

connect water outlet pipe and the device beside it. Record the variation of

514

flowmeter readings with time, and record the time when crosslinking materials

515

are totally corroded by bottom water, that is, when the bottom water channeling

516

pathway completely forms at CAI.

517

3.1.4. Experimental materials and conditions

RI PT

512

The drilling fluid is from CBM well T5-293 in Qinshui basin in China. The well

519

depth is 782 m. The cement slurry is composed of API class G oil well cement, 100%

520

microsphere, 0.4% dispersant, 3% shrinkage agent, 3% early strength agent and 91%

521

tap water. The permeability and porosity of SWB are 1.5×10-3 µm2 and 8%,

522

respectively. The SWBs are cured in water. The experimental temperature is 35 °C

523

and the curing time is 2 days. The thickness of mud cake is 0.5 mm.

524

3.2. Results and analysis

525

3.2.1. Verification of Eq. (15)

526

3.2.1.1. Determination of contact angle (θ). The mud cake is the deposition of solid

527

phase particles in drilling fluid. At present, the water-based drilling fluid is widely

528

used in oilfields. Besides, the mud cake contacts with bottom water in a long period,

529

so mud cake particles have good hydrophilicity, shown as Fig. 11. Put the drilling

530

fluid from Qinshui Basin into water loss instrument. Keep it under 0.75 MPa. Two

531

hours later, take out the filter paper and remove the false mud cake. Drip a drop of

532

water on the surface of mud cake, then water drop spreads on the surface rapidly,

533

which proves that the mud cake has good hydrophilicity. The extended water drop is a

534

spherical segment, and the computational formula of its volume can be calculated by

535

Eq. (51):

536

Vs =

537

AC C

EP

TE D

M AN U

SC

518

πhs 2 3

(3Rs − hs )

(51)

Where Vs is the volume of spherical segment, hs is the height of spherical segment, 20

ACCEPTED MANUSCRIPT 538

and Rs is the radius of spherical segment.

539 Mud cake

540

Water drop

541 542

RI PT

543 544 545 546

SC

547

Fig. 11. Hydrophilic experiment of mud cake at CAI

548

Measure the diameter ds of water drop on the mud cake by vernier caliper. Then

550

the following formulas can be obtained according to the geometric relation of

551

segment:

552

ds = Rs sin θ 2

553

hs = Rs (1 − cosθ )

TE D

554

M AN U

549

(52) (53)

From Eqs. (51), (52) and (53), the following formulas can be obtained according

555

to the transformation of triangle function:

556

tan 3

θ

θ

557

6V =0 (54) 2 2 π ( d )3 2 Generally speaking, the volume of one water drop is approximately 0.05 ml, while

558

the average diameter of the extended water drop on the mud cake is about 16.4 mm.

559

Substituting these data into Eq. (54), the half of contact angle can be calculated as 6.6

560

degree.

561

3.2.1.2. Determination of pore radius (R). Take a certain amount of on-site drilling

562

fluid and dry it. Then smash and get the solid powder. Afterwards，analyze the particle

563

size of solid powder by JL-1155 type laser particle size analyzer. The result is as

564

follows: D10=3.930 µm, D50=9.621 µm, D90=21.112 µm. In fact, the width of micro

565

fracture is under 100 µm in geoscience. Based on ideal soil model (Fig. 12) and

EP

−

AC C

+ 3 tan

21

ACCEPTED MANUSCRIPT 566

regularities of random probability distribution, the pore radius can be calculated:

R = [( 2 − 1) D90 + 100] / 2 = 54.37 µm.

567 568

D90

569

RI PT

R

570 571 572

Fig. 12. Ideal soil model

3.2.1.3. Determination of other parameters. It is assumed that the experimental

574

temperature is 20 degree, and therefore the surface tension of water f is equal to

575

72.75×10-3 N/m, the density of bottom water ρw is 1070 kg/m3, the viscosity of bottom

576

water η is 1.005 mPa·s, g is 9.81 m/s2, and the tortuosity ε is between 2.05 and 2.55

577

(Boudreau and Meysman, 2006). The tortuosity ε is set equal to 2.3.

578

3.2.1.4. Results and Analysis. According to the characteristics of capillary pore, the

579

fluid is driven by a certain differential pressure rather than the gravity. Therefore, the

580

gravity term in Eq. (15) can be ignored for simplified calculation. Regard the time of

581

first water drop penetrated along CAI as the bottom water channeling time in capillary

582

laminar flow stage. The verification result is listed in Table 1.

TE D

M AN U

SC

573

583 584 Table 1

EP

585 586

Verification results of Eq. (15).

1 2 3 4

Inner diameter of

Height of Simulated

Experimental

Computation

Relative

SWB (mm)

Aquifuge h (mm)

Time (s)

Time tA (s)

Error (%)

21

54

19.17

18.56

3.18

21

36

7.92

8.25

4.17

33

46

14.47

13.47

6.91

33

28

4.27

4.99

16.86

AC C

No.

587 588

Table 1 shows that:

589

1. The channeling time of bottom water increases with the thickness of

590

cement-aquifuge while decreases with hole size. Besides, former’s experimental

591

error is smaller than the latter, which accords with the actual situation of field 22

ACCEPTED MANUSCRIPT site.

592 593

2. The prediction error of Eq. (15) is less than 17%, which indicates that the theoretical model of this stage is reasonable and practical.

594

3.2.2. Verification of Eq. (44)

596

3.2.2.1. Determination of permeability of mud cake (Ki) at CAI. It is assumed that the

597

inner structures of both cement-aquifuge and mud cake are homogeneous but different

598

from each other. So CAI can be seen as composed by two parallel layers with

599

different transverse permeability, shown as Fig. 13.

600 601 Q1

602

Aquifuge

Mud

Ki

Ks

604

L1

605 606

611 612

613 614

TE D

equation can be obtained:

ηLQ KA

=

ηL1Q1 Ks A

+

ηL2Q2 Ki A

EP

610

L2

According to Darcy's Law, Q = KA∆P /(ηL) and ∆p=∆p1=∆p2，the following

(55)

And Q=Q1=Q2, L=L1=L2: L L1 L2 = + K Ks Ki

(56)

AC C

609

cake

Fig. 13. Schematic diagram of stratified sample

607 608

Q2

M AN U

603

∆p2

SC

∆p1

RI PT

595

So:

Ki =

L2

(57)

L L ( − 1) K Ks

615

Where ∆p1 and ∆p2 are differential pressure of both ends of cement-aquifuge and

616

mud cake, respectively, Ks, Ki, L2 and L1 are permeability and length of

617

cement-aquifuge and mud cake, respectively, K and L are permeability and length of

618

combined sample, respectively, Q, Q1 and Q2 are the flow through sample, inlet flow 23

ACCEPTED MANUSCRIPT 619

and outlet flow, respectively, and A is the area of cross section of sample.

620

Considering that the value of K can be measured directly, the value of Ks can be

621

measured before injecting drilling fluid. Therefore, the permeability of mud cake can

622

be computed according to Eq. (57). The results of repeated experiments are listed in

623

Table 2.

Table 2 Obtained experimental results on permeability parameter. Before No.

-3

2

Ks (10 µm )

Corrosion K

K (10-3µm2)

(10-3µm2)

Corrosion

After corrosion

Time t1 (h)

K (10-3µm2) 90.80

21.34

231.53

60.76

288.04

84.07

1272.12

45.47

10.33

1

2

1368.21

91.29

21.59

3

3

1478.13

93.42

22.87

5

627

M AN U

1

SC

625 626

RI PT

624

K (10-3µm2)

3.2.2.2. Determination of flow pore throat radius (r1) in mud cake. The mud cake

629

particles will be surrounded by water molecules and isolated from other particles after

630

the crosslinking material is corroded by bottom water. Thus, due to the function of

631

bottom water, the mud cake particles are peeled off from main body and dissolved

632

into bottom water, as shown in Fig. 14.

634 635

640

AC C

636

Water molecule

EP

633

TE D

628

641

The process of water molecules surrounding mud cake particles is spontaneous and

642

transitory, so the mud cake internal flow pore throat radius will increase after the mud

643

cake particles is corroded by bottom water. The test result of the above mentioned

644

drilling fluid from field site shows that, D90 is 21.112 µm. Thus, D90 is viewed as the

645

mud cake internal flow pore throat radius, namely r0=r1=21.112 µm.

637 638 639

Mud cake particle

Fig. 14. Schematic diagram of mud cake particles surrounded and isolated by water molecules

24

ACCEPTED MANUSCRIPT 3.2.2.3. Determination of other parameters. The part of CAI was transfixed after

647

capillary seepage stage and corrosion and dissolution stage. Therefore, set R of this

648

stage as half of thickness of mud cake. The thickness of mud cake d of drilling fluid

649

from field site is usually 0.8 mm, that is, R=0.4 mm, D=33 mm.

650

3.2.2.4. Result and analysis. Substitute above parameters into Eq. (44) and the bottom

651

water corrosion channeling time of sample 1, 2 and 3 are 41.7 h, 35.2 h and 27.5 h,

652

respectively. Combining the result in Table 2, it can be concluded that the

653

permeability of mud cake at CAI significantly increases with corrosion time, which in

654

turn shortens bottom water corrosion channeling time greatly. This is the same

655

discovery as actual situation of field site.

656

3.2.3. Verification of Eq. (50)

M AN U

SC

RI PT

646

657

In order to preliminarily verify Eq. (50), set R of this stage as thickness of mud

658

cake, 0.8mm. In this study, variation of bottom water channeling time versus the

659

differential pressure is used to present the channeling law in this stage. The

660

comparison of experimental value and calculated value is shown in Fig. 15.

661

AC C

664 665 666 667 668 669 670 671 672 673 674

tc (s)

663

Calculated value Experimental value

EP

TE D

662

∆P (103 Pa)

Fig. 15. Comparison of experiment value and calculated value of Eq. (50)

675 676

As shown above, the following results can be obtained:

677

1. The bottom water channeling time decreases with differential pressure, namely

678

the larger bottom water energy, the quicker the corrosion channel forms at CAI,

679

followed by more serious water channeling. 25

ACCEPTED MANUSCRIPT 680

2. Eq. (50) is basically correct, while there is a certain error between calculated

681

value of it and experimental value. Further research should be done to improve

682

it. 3. The channeling pathway at CAI becomes larger and larger due to the gradual

684

corrosion of bottom water. The radius of channeling pathway increases from the

685

original size of mud cake particles to the thickness of mud cake. And then, it

686

results in the serious water channeling even explosive flooding.

689

The models have been developed and validated to predict water channeling along

SC

688

4. Conclusions

CAI in CBM wells. The summary of the work and findings are as follows:

M AN U

687

RI PT

683

690

1. The process of bottom water channeling along CAI can be divided into three

691

stages: capillary seepage stage, corrosion and dissolution stage, and crack pipe

692

flow stage. Early capillary seepage provides conditions and lays a fundamental

693

for the stage of corrosion and dissolution, and the stage of crack pipe flow is the

694

evolution of the stage of corrosion and dissolution.

2. The time of bottom water flows up along CAI is related to the differential

696

pressure of both ends of channeling pathway, and the tortuosity of the

697

channeling pathway, the height of bottom water channeling, fracture size of CAI,

698

bottom water density, bottom water viscosity, bottom water surface tension,

699

contact angle, bottom water dissolution time, permeability of mud cake before

700

and after bottom water dissolution, and hole diameter.

702 703 704

705

EP

3. The models theoretically explained characteristics of the evolution process of

AC C

701

TE D

695

confined water channeling upward along CAI.

4. The prediction results of the models are in good agreement with the experimental data.

Acknowledgments

706

This work was supported by the National Natural Science Foundation of China

707

(grant Nos. 41572142 and 51774258), the National Science and Technology Major

708

Project

of

China

(grant

No.

2017ZX05009003-003), 26

the

Fundamental

ACCEPTED MANUSCRIPT Research Funds for the Central Universities,

710

(Wuhan) (grant No. CUGQYZX1710) and the Project of Experimental Technology

711

Research from China University of Geosciences (Wuhan) . The authors would like to

712

thank Mr. Ju Huang and Ke Li for his assistance in editing this manuscript. The

713

authors also would like to thank the editor and reviewers for their assistance in

714

publishing this manuscript.

715

References

716

Agarwal, A., Mandal, A., Karmakar, B., Ojha, K., 2013. Modeling and performance

717

prediction for water production in CBM wells of an Eastern India coalfield. J.

718

Petrol. Sci. Eng. 103, 115-120.

720

University

of

Geosciences

SC

Boudreau, B.P., Meysman, F.J.R., 2006. Predicted tortuosity of muds. Geology 34 (8),

M AN U

719

China

RI PT

709

693-696.

721

Cai, Y.D., Liu, D.M., Yao, Y.B., Li, J.Q., Qiu, Y.K., 2011. Geological controls on

722

prediction of coalbed methane of No. 3 coal seam in Southern Qinshui Basin,

723

North China. Int. J. Coal Geol. 88,101-12.

Chuai, X.Y., Teng, J.W., 2017. Water inrush mechanism research of strong conducting

725

(water including) karstic collapse column. Chin. J. Geophys. Acta Geophys. Sin.

726

60, 430-440.

TE D

724

Fan, H.M., Zhang, Y.N., Zhang, J., Wang, D.Y., Gao, J.B., Kang, W.L., Meng, X.C.,

728

Zhao, J., Xu, H. 2013. Dynamic surface adsorption properties of sodium dodecyl

729

sulfate aqueous solution. Acta Phys. Chim. sin. 29 (2), 351-357.

EP

727

Fan, S.K., 2012. The Coal Floor Water Bursting Evaluating and Countermeasure of

731

South Coalfields in North China., Master's Thesis, China University of Mining &

732

AC C

730

Technolgy, Xuzhou, China.

733

Gu, J., Chen, X.F., 2010. Study of relationship between interlayer thickness and

734

shearing strength at cement-formation interface. J. China Univ. Min. Technol. 39

735

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Highlights: ● Evolution process of bottom water channeling along CAI is divided into three stages. ● Models for different stages are developed to predict water channeling along CAI. ●The models can explain the evolution characteristics of water channeling along CAI.

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● Simulation device is established to verify the models of water channeling along CAI.

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● Prediction results of the models are in good agreement with the experimental data.