A numerical model for fractured horizontal well and production characteristics: Comprehensive consideration of the fracturing fluid injection and flowback

Journal Pre-proof A numerical model for fractured horizontal well and production characteristics: Comprehensive consideration of the fracturing fluid injection and flowback Zhongwei Wu, Li Dong, Chuanzhi Cui, Xiangzhi Cheng, Zhen Wang PII:

S0920-4105(19)31184-2

DOI:

https://doi.org/10.1016/j.petrol.2019.106765

Reference:

PETROL 106765

To appear in:

Journal of Petroleum Science and Engineering

Received Date: 8 July 2019 Revised Date:

29 November 2019

Accepted Date: 29 November 2019

Please cite this article as: Wu, Z., Dong, L., Cui, C., Cheng, X., Wang, Z., A numerical model for fractured horizontal well and production characteristics: Comprehensive consideration of the fracturing fluid injection and flowback, Journal of Petroleum Science and Engineering (2020), doi: https:// doi.org/10.1016/j.petrol.2019.106765. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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. © 2019 Published by Elsevier B.V.

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A numerical model for fractured horizontal well and production characteristics:

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comprehensive consideration of the fracturing fluid injection and flowback

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Zhongwei Wua,b , Li Dongc, Chuanzhi Cuia,b,∗, Xiangzhi Chengd, Zhen Wanga,b

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a. Key Laboratory of Unconventional Oil & Gas Development (China University of Petroleum

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(East China)), Ministry of Education, Qingdao 266580, P. R. China

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b. School of Petroleum Engineering, China University of Petroleum (East China), Qingdao

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266580, P. R. China

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c. Exploration & Production Research Institute SINOPEC, Beijing 100083, P.R. China;

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d. Petrochina Research Institute of Petroleum Exploration & Development, Department of

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Logging & Remote Sensing Technology, Beijing 100083, P. R. China

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Abstract: Currently, many investigations on fractured well productivity are

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conducted. However, all this productivity studies are conducted based on the initial

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pressure and fluid distribution. In this work, taking the reservoir damage caused by

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fracturing fluid and the effect of fracturing fluid injection on pressure and saturation

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into consideration, a flow model of multistage fractured horizontal well is built, which

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is composed of fracturing fluid injection model and fracturing fluid flowback and

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production model. Numerical solving method is utilized to solve the model and model

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verification are presented through comparing with the test results of 12 fractured well.

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The effects of fracture fluid injection rate and fracturing fluid viscosity on pressure

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and saturation at the end of fracturing fluid injection are analyzed. And the effects of

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the reservoir oil viscosity, porosity, permeability, heterogeneous and reservoir damage

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on flowback and production characteristic are analyzed. The results show: (1) The

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distances of saturation and pressure variation area from the fracture are respectively

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10m and 20m after the fracturing fluid injection is finished. When the fracturing fluid

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viscosity increases from 10 mPa·s to 30 mPa·s, the distance of pressure variation area

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from the fracture decrease. But the pressure and saturation near the hydraulic fracture

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increases. When the injection rate increases from 0.28m3/min to 28m3/min, the

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fracture length increases and the distances of pressure and saturation variation area ∗

Corresponding author: Chuanzhi Cui ([email protected]) 1

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from the fracture increase. The pressure near the fracture with a high injection rate is

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larger than that with a small injection rate. (2) With the development of reservoir, the

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water flow rate and water cut decreases, and the oil flow rate increases firstly and then

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decreases. When the reservoir oil viscosity increases from 1.5mPa·s to 2.5mPa·s,

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water cut increases. The maximum oil flow rate decreases from 16.4 to 6.2 m3/d. With

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the increase of reservoir permeability, the water cut decrease. When the reservoir

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porosity increases, the oil flow rate, water flow rate and cumulative fluid volume

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increase. While, the water cut decreases with the increase of reservoir porosity. When

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the residual resistance factor (RRF) increases from 1 to 1.08, the oil flow rate of when

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the RRF equals 1 is slight larger than that of when the RRF equals 1.08. The reservoir

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damage has slight effect on the water flow rate and water cut. (3) The development of

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heterogeneous reservoir is poorer than that of homogeneous reservoir when the

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development conditions are the same. This work provides a more accurate method to

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investigate the flowback and production characteristics of fractured well.

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Keywords: Fracturing horizontal well; production characteristic; flowback; fracturing

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fluid injection; reservoir damage caused by fracturing fluid

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1 Introduction

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Due to the decrease of conventional resources, many scholar’s attentions have

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been attached to the development of tight resources (Guo et al., 2018; Ren et al., 2019;

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Xu et al., 2019; Wang et al., 2014; Wu, et al., 2019a). Tight formation has a poor

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reservoir properties, which results in a poor ability of fluid flowing in the tight

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formation. So the hydraulic fracture technology, which can create one or more

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highway for fluid flowing in the tight formation, is widely used to high-effective

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develop tight resources (Guo et al., 2018, Sheng, et al., 2019), especially for

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multistage fractured horizontal well technology (Ren et al., 2019; Xu et al., 2019;

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Wang et al., 2014; Wu, et al., 2019a).

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The studies of fracture fluid flowback are great important to the hydraulic

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fracture optimization. Majid Ali Abbasi, et al. (2012; 2014) constructed basic

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diagnostic plots by using two-phase flowback data of three multi-fractured horizontal 2

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wells to understand the physics of flowback. Xu, et al. (2015) introduced a method to

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estimate effective fracture volume by assuming a simplistic two phase tank model for

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the fracture system. In this work, effective fracture volume is calculated using a

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modified material balance approach for the two-phase system. The material balance

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approach enables the estimation of effective fracture volume regardless of the fracture

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geometry. Based on a simplistic two phase tank model (Xu, et al. (2015)), Xu, et al.

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(2017) developed a closed-tank material balance model to estimate effective fracture

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volume using rates and pressure data measured during early-time water flowback. Fu,

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et al. (2017) recognized that flowback data from seven multi-fractured horizontal tight

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oil wells in Anadarko Basin show two separate regions during the single-phase water

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production. Region 1 shows a dropping casing pressure, and Region 2 shows a

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flattening casing pressure. Fu, et al. (2019) estimated the initial effective fracture pore

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volume and fracture volume loss for 21 wells completed in the Montney and Eagle

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Ford formations. The relationship between fracture volume loss and choke size is also

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evaluated. All those investigations are about the application of the flowback data, and

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few works related to the prediction of fracture fluid flowback are reported.

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The pressure and production characteristics analysis of fractured horizontal well

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has a great significance on the development and optimization of tight reservoirs.

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Based on the fluid and formation properties, and development measures of tight

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reservoirs, the researchers established different models of fractured horizontal well

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through the methods of reservoir engineering and numerical simulation (Wang et al.,

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2015; Su et al., 2016; Zhao, 2012). Fan et al. (2015) built a composite model of

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fracturing horizontal well, which is solved by the finite element method and verified

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by the classic analytical solution in dual porosity reservoir. Chen et al. (2019) and

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Zhang et al. (2015) presented a semi-analytical model of fractured horizontal well

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with considering crushed region, effected region and unstimulated region. Due to a

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poor reservoir properties, the flowing in the tight reservoir obeys the low-velocity

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non-Darcy flow, instead of Darcy flow (Liu et al., 2019). Considering low-velocity

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non-Darcy flow and stress sensitivity of tight reservoir medium, Wu et al. (2019b) 3

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established a multi-linear transient pressure model of multistage fracturing horizontal

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well. Furthermore, with considering the effect of stress-sensitivity of natural fractures

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and variable conductivity of artificial fractures, a well test model of fractured

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horizontal well in tight gas reservoirs is proposed (Wu et al., 2018). All those

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investigations stated above ignore the effect of the injection process of fracture fluid

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on reservoir pressure and saturation, and reservoir properties.

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During the fracturing fluid injection, in addition to flowing toward the fracture

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toe, the fracturing fluid displaced by fracturing fluid injection pressure and capillary

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pressure is leaked to the reservoir. That results in the change of reservoir pressure and

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saturation. When the flowback of fracturing fluid is finished, the reservoir pressure

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and saturation are further changed. So, the process of hydraulic fracture injection

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makes reservoir pressure and saturation a change and a different from initial pressure

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and saturation. In addition, the fracturing fluid contains surfactant and polymer. The

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reservoir properties, such as reservoir permeability and relative permeability, become

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poor due to the effect of surfactant and polymer after the fracturing fluid is leaked in

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the reservoir (Al-Ameri et al., 2018).

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In this paper, a multilinear flow model for fractured horizontal well with

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considering the effect of the injection process of fracture fluid on reservoir pressure

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and saturation, and reservoir damage caused by fracture fluid is proposed in section 2.

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The multilinear flow model is composed of the fracturing fluid injection model and

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flowback/production model. In section 3, a numerical method is utilized to solve the

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multilinear flow model. In section 4, results and discussions are conducted. Finally,

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the conclusions are drawn in section 5. This work combines the processes of

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fracturing fluid injection, fracturing fluid flowback and production, and have a

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significance on the development of tight reservoir by multistage fractured horizontal

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well.

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2 A flow model for fracturing horizontal well

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2.1 Flow physical model for fracturing horizontal well

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In this work, the horizontal well is interrupted by nf hydraulic fractures. The 4

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fracture is perpendicular to the horizontal well. All hydraulic fractures have the same

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length and conductivity, and are spaced uniformly along the horizontal well.

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According to the work of E. Stalgorova et al. (2012), the reservoir can be divided into

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nf flow units, and a quarter of each flow unit can be divided into three linear flow

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region (seen in Fig.1), which are respectively fracture region, inner region and outer

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region. The inner region is stimulated reservoir volume.

122 123

Fig.1 Schematic of a quarter of flow unit in the work of E. Stalgorova, et al (2012)

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In our work, the model is composed of fracturing fluid injection model, and

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flowback and production model. Inspired by the work of E. Stalgorova et al. (2012),

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when built the fracturing fluid injection model, a quarter of flow unit is composed of

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fracture and reservoir region (seen in Fig.2a). The fluid linearly flows to the fracture,

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and then linearly flows to the reservoir. When built the flowback and production

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model, a quarter of flow unit is composed of the fracture, fracturing fluid invasion

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region and reservoir region (seen in Fig.2b). The fluid linearly flows to the fracturing

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fluid invasion region and then linearly flows to the fracture, finally linearly flows to

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the wellbore.

5

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Fig.2 Schematic of a quarter of the flow unit ((a) fracturing fluid injection model; (b) flowback

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and production model)

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During the fracturing fluid injection, the reservoir properties equal to the initial

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value of which. The length of hydraulic fracture is related to the injection volume of

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fracturing fluid and can be approximately calculated by the mass conservation of

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injection fracture fluid. The reservoir properties of the fracturing fluid invasion region

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become poor when simulates the process of fracturing fluid flowback and production.

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This is due to the damage caused by fracturing fluid. The connection condition

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between fracturing fluid injection model and flowback and production model is that

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the reservoir pressure and saturation at the end of fracturing fluid injection equal to

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that at the initial moment of fracturing fluid flowback and production.

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2.2 Flow mathematical model for fractured horizontal well

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2.2.1 Fracturing fluid injection

147 148

The mass conservation equation in the hydraulic fracture is

−

∂ ∂ ( ρ f v ff ) + ρ f q fM + ρ f Qf = ( ρ f φ f ) ∂y ∂t

(1)

149

where, ρ f is the density of fracturing fluid, g/cm3; v ff denotes the velocity of

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fracturing fluid, m/d; q ffM is the fracturing fluid flow rate per unit volume flowing

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from fracture to reservoir, 1/d; Q f denotes the fracturing fluid injection rate per unit

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volume, 1/d; φ f is the porosity of hydraulic fracture, y is the distance along with 6

153 154 155 156

the hydraulic fracture, m; t denotes the time, d. The mass conservation equations for describing fluid flow in the reservoir are ∂ ∂ ( ρ f v fM ) + ρ f q ffM = ( s fM ρ f φ M ) ∂x ∂t ∂ ∂ − ( ρ o voM ) + ρ o qofM = ( soM ρ oφ M ) ∂x ∂t

−

(2a) (2b)

157

where, v fM and voM respectively denote the velocity of fracturing fluid and oil

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in the fracture, m/d; s fM and soM respectively denote the saturation of fracturing fluid

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and oil in the reservoir; qofM is the oil flow rate per unit volume flowing from

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fracture to reservoir, the value of qofM is zero during the fracturing fluid injection,

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1/d; φM is the reservoir porosity; ρ o is the oil density, g/cm3.

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The flow equation in the hydraulic fracture is

v ff = −

163 164 165 166

0.0864k f dp f

µf

dy

(3)

where, k f is fracture permeability, mD; p f is the pressure in the fracture, MPa;

µ f is the fracture viscosity, mPa·s. The flow equations in the reservoir are

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v fM = −

0.0864k M krf , M ( s fM ) dpMf ( − G) µf dx

(4a)

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voM = −

0.0864 k M k ro , M ( s fM ) dpMo ( − G) µo dx

(4b)

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where, k M is reservoir permeability, mD; pMf is the water phase (fracturing fluid)

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pressure in the reservoir, MPa; pMo is the oil phase pressure in the reservoir, MPa;

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krf , M and kro , M are the relative permeability; µo is the oil viscosity, mPa·s; G is the

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threshold pressure gradient, MPa/m;

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State equations and normalization conditions are as follows,

ρ f = ρ fi (1+ c f (p - pi )) 7

(5)

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ρo = ρoi (1+co (p - pi ))

(6)

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φM = φMi (1+cM (p - pi ))

(7)

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φ f = φ fi (1+ c ff (p - pi ))

(8)

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s fM + soM = 1

(9)

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where, ρ fi and ρ oi are respectively fracturing fluid density and oil density at

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initial pressure, g/cm3; pi is the initial pressure, MPa; p is the pressure, MPa; φ fi

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and φMi are respectively the porosity of fracture medium and reservoir;

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c o are respectively the compressibility of fracturing fluid and oil, MPa-1; cM and

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c ff are respectively the compressibility of reservoir and hydraulic fracture, MPa-1;

c f and

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The connection condition between fracture and reservoir is that the reduction of

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fluid in the hydraulic fracture equals to the increment of fluid in the reservoir. The

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initial conditions of the flow model are that the pressure and saturation in the reservoir

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respectively equal to the initial pressure and saturation. The initial pressure and

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saturation of fracture equal to reservoir initial pressure and 1, respectively.

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Eqs. (1) to (9) form the fracturing injection flow model.

2.2.2 Fracturing fluid flowback and production

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During the fracturing fluid flowback and production, the fluid flows to fracture

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firstly, and then flows to the wellbore along with the fracture. The mathematical

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model of fluid flow in the reservoir of fracturing fluid flowback and production is the

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same to that of fracturing fluid injection. The difference between them is the initial

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condition, which is presented in the next. So, the next content mainly presents the

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flow equations for describing fracturing fluid flowback and production in the

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hydraulic fracture.

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Before the fracturing fluid flowback, the fracturing fluid has been degraded and

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become to water. The mass conservation equations in the hydraulic fracture are as

200

follows, 8

−

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∂ ∂ ( ρwvwf ) + ρwqwfM = (swf ρwφ f ) ∂y ∂t

(10a)

∂ ∂ ( ρo vof ) + ρo qofM = (sof ρoφ f ) ∂y ∂t

(10b)

−

202 203

where, ρ w denotes the density of water, g/cm3; qwfM is water flow rate per unit

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volume flowing from reservoir to fracture, 1/d; vwf and vof are respectively water

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flow rate and oil flow rate at hydraulic fracture, m/d; swf and sof are respectively

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water saturation and oil saturation at hydraulic fracture.

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The flow equation for fluid flowing in the hydraulic fracture is

208

vwf = −

0.0864k f krw, f (swf ) dp fw

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vof = −

0.0864k f kro, f (swf ) dp fo

210

where, krw, f and kro, f

µw µo

dy

dy

(11a)

(11b)

are respectively the water and oil phase relative

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permeability of fracture, mD; swf is the water saturation in the fracture; µw is the

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water phase viscosity, mPa·s; p fw and p fo are respectively water phase pressure

213

and oil phase pressure, MPa.

214

The connection condition between fracture and reservoir is that the increment of

215

fluid in the hydraulic fracture equals to the reduction of fluid in the reservoir. The

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initial conditions of fracturing fluid flowback and production are that the pressure and

217

saturation in the fracture and reservoir equal to that of when the fracturing fluid

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injection is finished.

219

3 Model solution method

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In order to solve the flow model, we firstly mesh the fracture and reservoir into

221

N×(NF+1) grids and then number every discrete grid (Seen in Fig.3). N is the discrete

222

grid number along the reservoir. NF is the discrete grid number along the hydraulic

223

fracture. 9

224 225 226 227 228

Fig.3 Schematic of the discrete grid and its number in a quarter of a flow unit

3.1 Fracturing fluid injection From Eqs. (1) (3) (5) and (8), we can obtain the flow controlling equation of the fracture as following,

0.0864k f dp f ∂p ∂ (ρ f ) − ρ f q ffM + ρ f Qf = ρ fiφMi (c f + c ff ) f µf ∂y dy ∂t

229 230 231 232

According to the discrete grid in the hydraulic fracture (Fig.3), the equations of describing fluid flow in each discrete grid are as follows, The (N×NF+1) grid: − ρ f φ f c ft ∆t

233

+ 234

The

4 ∆x × w f × h

=

( ∆y ) − ρ f φ f c ft ∆t

2

p nf +,(1N × NF + 2) − ρ f q ffM ,( N × NF +1)

p

n f ,( N × NF +1)

(13)

vv grid (the value of vv is from (N×NF+2) to (N×NF+ NF-1)): (∆y )

2

p nf +,vv1 −1 − (

− ρ w q ffM ,vv 236

λ1

p nf +,(1N × NF +1) +

Qf

λ f ,vv −1 235

(12)

λ f ,vv −1 + λ f ,vv

(∆y ) − ρ f φ f c ft n = p f ,vv ∆t 2

The (N×NF+ NF) grid:

10

+

ρ f φ f c ft ∆t

) p nf +,vv1 +

λ f ,vv (∆y )

2

p nf +,vv1 +1 (14)

λ f , N × NF + NF −1 (∆y )

237

2

p nf +, N1 × NF + NF −1 − (

− ρ f q ffM , N × NF + NF =

where, λ f =

238

− ρ f φ f c ft ∆t

0.0864ρ f k f

µf

λ f , N × NF + NF −1 + λ f , N × NF + NF (∆y ) p

2

+

ρ f φ f c ft ∆t

) p nf +, N1 × NF + NF

(15)

n f , N × NF + NF

; c ft = c f + c ff ; w f denotes the fracture width, m; h

239

denotes the reservoir height, m. Subscript n and n+1 denote the time step. Superscript

240

N×NF+1, N×NF+ NF and

241 242

Eqs. (13) to (15) are the discrete form of flow controlling equation of hydraulic fracture.

243 244

vv denote the grid number.

From Eqs. (2) (4) (5) (6) and (7), the flow controlling equations in the reservoir are

245

0.0864kM krw,M (swM ) dpM ∂s ∂ ∂p (ρw ( − G)) + ρwqwfM = swM ρwiφMi (cM + cw ) M + ρwφM wM (16) ∂x dx ∂t ∂t µw

246

0.0864kM kro,M (swM ) dpM ∂s ∂ ∂p ( ρo ( − G)) + ρoqofM = soM ρoiφMi (cM + co ) M + ρoφM oM (17) ∂x dx ∂t ∂t µw

247

Combining Eqs. (9) (16) and (17), we can eliminate saturation and obtain an

248

equation of reservoir pressure as following, AA

249

0.0864kM kro, M (swM ) dpM 0.0864kM krw,M (swM ) dpM ∂ ∂ ( ρo ( − G)) + ( ρw ( − G)) ∂x dx ∂x dx µw µw

+ AAρo qofM + ρw qwfM

∂p ∂p = AAsoM ρoiφMi (cM + co ) M + swM ρwiφMi (cM + cw ) M ∂t ∂t

(18)

250

where, AA = ρw / ρo .

251

According to the grid in the reservoir (Fig.3), we can obtain the discrete Eq. (18)

252 253

as following, The vv1 grid (the value of vv1 is 1, N + 1 , 2 N + 1 , 3 N + 1 ,····, ( NF − 1) × N + 1 ): −ρwφM cMt n+1 T −ρ φ c pM ,vv1 + vv1 2 pMn+,1vv1+1 + ρwqwfM ,vv1 = w M Mt pMn ,vv1 ∆t (∆x) ∆t

254 255 256

(19)

The vv2 grid (the value of vv2 is from ii × N + 1 to ii × N + N − 1 . ii is from 0 to NF-1): 11

Tvv 2−1 n+1 T +T ρφ c T −ρ φ c pM ,vv 2−1 − ( vv 2−1 2 vv 2 + w M Mt ) pMn+,1vv 2 + vv 2 2 pMn+,1vv 2+1 = w M Mt pMn ,vv 2 (20) 2 (∆x) (∆x) ∆t (∆x) ∆t

257 258

The vv3 grid (the value of vv1 is N , 2N , 3N ,····, NF × N ): Tvv3−1 n+1 T +T ρφ c −ρ φ c pM ,vv3−1 − ( vv3−1 2 vv3 + w M Mt ) pMn+,1vv3 = w M Mt pMn ,vv3 2 (∆x) (∆x) ∆t ∆t

259

0.0864kM kro, M ( swM )

260

where, λoM = ρo

261

cMt = swM cw + soM co + cM .

µo

, λwM = ρ w

kM krw, M ( swM )

µw

(21)

, T = AA × λoM + λwM ,

262

Eqs. (15) to (21) form a closed matrix on pressure. Solving the matrix by the

263

Gaussian elimination method, we can get the pressure at a time step. Then we need to

264

get the saturation at this time step. The saturation equations can be get by discretizing

265

Eq. (16) as following,

266 267

The

vv1 grid (the value of

S

=S

n wM ,vv1

268

λwM ,vv1+1 × ( pMn+,1vv1+1 − pMn+,1vv1) 2ρwkM ,vv1k f ,(vv1−1)/ N + + × ( φM ρw µw∆x(∆x + wF ) ∆x2 ∆t

( pnf +,(1vv1−1)/2 − pMn+,1vv1 ) −

270

∆t

=S

n wM ,vv 2

+

∆t

( pMn+,1vv1 − pMn ,vv1 ) + ρwqwfM )

φM ρw

(

λwM ,vv 2+1 × ( pMn+,1vv 2+1 − pMn+,1vv 2 ) λwM ,vv 2+1 × ( pMn+,1vv 2 − pMn+,1vv 2−1 ) −

∆x2

∆x2

S n ρ φ (c + c ) − wM w M Mt w ( pMn+,1vv 2 − pMn ,vv 2 )) ∆t

(23)

The vv3 grid (the value of vv1 is N , 2N , 3N ,····, NF × N ): n+1 n SwM ,vv 3 = SwM ,vv 3 +

273

− 274

(22)

0 to NF-1):

S

272

n SwM ,vv1ρwφM (cMt + cw )

The vv2 grid (the value of vv2 is from ii × N + 1 to ii × N + N − 1 . ii is from

n+1 wM ,vv 2

271

1 , N + 1 , 2 N + 1 , 3 N + 1 ,····,

( NF − 1) × N + 1 ): n+1 wM ,vv1

269

vv1 is

∆t

φM ρw

(−

λwM ,vv3 × ( pMn+,1vv3 − pMn+,1vv3−1 )

n SwM ,vv3 ρwφM (cMt + cw )

∆t

∆x2

(24)

( pMn+,1vv3 − pMn ,vv3 ))

Eqs. (19) to (24) are the discrete flow model of reservoir at the stage of fracture 12

275

injection.

276

3.2 Fracturing fluid flowback and production

277

The solving method of reservoir flow model at the stage of fracturing injection is

278

the same to that at the stage of fracturing fluid flowback and production. So, we just

279

present the solving method of fracture flow model at the stage of fracturing fluid

280

flowback and production below.

281 282

Combining Eqs. (5) (6) (8) (10) and (11), we can obtain the flow controlling equation of the hydraulic fracture as following

283

0.0864k f krw, f (swf ) dp f ∂p f ∂swf ∂ ( ρw ) + ρwqwfM = swf ρwiφ fi (c ft + cw ) + ρwφ f ∂y dy ∂t ∂t µw

(25)

284

0.0864k f kro, f (swf ) dp f ∂p f ∂sof ∂ ( ρo ) + ρo qofM = sof ρoiφ fi (c ft + co ) + ρoφ f ∂y dy ∂t ∂t µw

(26)

285 286 287 288

where, krw, f , kro, f are respectively the relative permeability of hydraulic fracture. By normalization condition ( sof + swf = 1 ), we can eliminate saturation and obtain an equation of fracture pressure as following, AA

289

0.0864k f kro, f (swf ) dp f 0.0864k f krw, f (swf ) dp f ∂ ∂ ( ρo ) + ( ρw )+ µw µw ∂y dy ∂y dy

AAρo qofM + ρwqwfM = AAsof ρoiφ fi (c ft + co ) 290 291 292

∂t

+ swf ρwiφ fi (c ft + cw )

∂p f

(27)

∂t

According to the fracture discrete grid, we discretize Eq. (34) and obtain a matrix. The detailing of matrix is as follows, The (N×NF+1) grid: − ρ wφ f c ft ∆t

293

294

∂p f

p nf +, N1 × NF +1 +

T f , N × NF +1 ( ∆y ) 2

pMn +,1N × NF + 2 + ρ w ( qwfM , N × NF +1 + qofM , N × NF +1 )

(28)

− ρ wφ f c ft n QI w + = p f , N × NF +1 w f h∆x ∆t

The

vv grid (the value of vv is from (N×NF+2) to (N×NF+ NF-1)):

13

T f ,vv −1 ( ∆y )

295

296

p nf +,vv1 −1 − (

T f , vv −1 + T f ,vv ( ∆y )

2

( ∆y )

2

p nf +, N1 × NF + NF −1 − (

where, λ fo = ρo

0.0864k f kro,f (swf )

µo

c ft = swf cw + sof co + c f ;

300

flow rate of production well, m3/d.

302 303 304 305 306 307

) p nf +,vv1 +

∆t − ρ wφ f c ft ∆t

T f , N × NF + NF −1 + T f , N × NF + NF

299

301

ρ wφ f c ft

p

T f , vv ( ∆y )

2

p nf +,vv1 +1

(29)

n f , vv

The (N×NF+ NF) grid:

( ∆y ) − ρ wφ f c ft 2

+ ρ w (qwfM , N × NF + NF + qofM , N × NF + NF ) = 298

+

QI o + ρ w ( qwfM ,vv + qofM ,vv ) + = w f h ∆x

T f , N × NF + NF −1 297

2

∆t

p

λ fw = ρw

;

+

ρ wφ f c ft ∆t

) p nf +, N1 × NF + NF (30)

n f , N × NF + NF

k f krw,f (swf )

µw

; T f = AA × λ fo + λ fw ;

QIw and QIo are respectively the water flow rate and oil

The expression of QIw is as follows,

QI w =

k f , N ×NF +1krw, f ( swf , N × NF +1 ) p f , N ×NF +1 − pwf

µw

ln(re / rw )

(31)

Similarly, the expression of oil flow rate is as follows,

QI o =

k f , N × NF +1kro, f (swf , N ×NF +1 ) p f , N × NF +1 − pwf

µw

ln(re / rw )

(32)

where, re is the equivalent out boundary of the N × NF + 1 grid and its expression is as follows, 1

1 1  2  k 2  k y  2  re = 0.28   ∆ x 2 +  x  w f 2  k  kx  y     

1

1

 ky 4  k 4 /  + x     kx   k y 

(33)

308

where, kx = ky = k f ,N×NF+1 , mD.

309

Eqs. (28) to (33) are the fracture pressure discrete matrix of flow mathematical

310

model at the stage of fracturing fluid flowback and production.

311

In next part, we introduce a method of calculating water saturation at the stage of

312

fracturing fluid flowback and production. The reservoir saturation calculation method

313

in the fracturing fluid flowback and production period is the same to that in fracturing 14

314

fluid injection period. The fracture saturation calculation method are as follows.

315

Discretizing Eq. (25) as following,

316

The (N×NF+1) grid:

Swfn+1,N ×NF +1 = Swfn , N ×NF +1 + 317

318

λwf , N ×NF +2 × ( p nf +, N1 ×NF + 2 − p nf +, N1 ×NF +1 ) ∆y 2

n

The

(34)

vv grid (the value of vv is from (N×NF+2) to (N×NF+ NF-1)):

319

+ ρ w qwf ,vv −

∆t

φ f ρw

(

λwf ,vv +1 × ( p nf +,vv1 +1 − p nf +,vv1 ) λwf ,vv +1 × ( p nf +,vv1 − p nf +,vv1 −1 ) −

∆y 2

S wfn ρ wφ f (c ft + cw ) ∆t

∆y 2

(35)

( p nf +,vv1 2 − p nf ,vv ))

The (N×NF+ NF) grid:

S

n +1 wf , N × NF + NF

=S

n wf , N × NF + NF

321

+ ρ w qwf , N × NF + NF − 322

φ f ρw

(

ρ φ (c + c ) S QI w + ρwqwf , N ×NF +1 + ρw − wf , N ×NF +1 w f ft w ( p nf +, N1 × NF +1 − pnf , N ×NF +1 )) w f h∆x ∆t

S wfn +1,vv = S wfn ,vv +

320

∆t

+

∆t

φ f ρw

(−

λwf , N × NF + NF × ( p nf +, N1 × NF + NF − p nf +, N1 × NF + NF −1 )

n S wM , N × NF + NF ρ wφ f (c ft + cw )

∆t

∆x 2

(36)

( p nf +, N1 × NF + NF − p nf , N × NF + NF ))

Eqs. (34) to (36) are the solving method of water saturation in the fracture at the

323

stage of fracturing fluid flowback and production.

324

3.3 Parameter preparation

325

During fracturing fluid injection, the hydraulic fracture is infinite conductivity,

326

namely the dimensionless conductivity of hydraulic fracture is larger than 300. And

327

the reservoir permeability, pressure and saturation are initial value. However, due to

328

the fracturing fluid invasion into the reservoir, the fluid flow ability become poor. The

329

size of invasion region, the permeability and relative permeability are characterized at

330

this section.

331

3.3.1 The size of invasion region

332

In our work, we recognize the invasion process as a piston flooding process. Due

333

to the infinite conductivity of hydraulic fracture at the stage of fracturing fluid

334

injection, we introduce a parameter named invasion depth to describe the size of 15

335

invasion region (Fig.4). According to the mass conservation principle, we can obtain

336

the expression of invasion depth ( llk ) as following,

llk =

337

V f ,injection − V f φ 2φ hL f

(37)

338 339

Fig.4 Fracturing fluid invasion depth

340

3.3.2 Permeability and relative permeability with considering the fracturing fluid

341

damage

342

The residual resistance factor (RRF) has been widely utilized to describe the

343

permeability harm and is the ratio of permeability before and after reservoir damage

344

(Al-ameri et al. (2018)). Based on the definition of RRF, we can get

kd =

345 346 347 348 349

k RRF

(38)

According to the work of Al-ameri et al. (2018), the value of RRF is range from 1.02 to 1.08. The next words introduce the method for calculating relative permeability. Burdine (2012) equations are as follows sw

k rw = sw2

350

∫ 0 1

dsw pc2

ds ∫0 pc2w

(39a)

1

dsw pc2 sw

∫

kro = (1 − sw ) 2

351

352

where, sw =

sw − swc . 1 − swc 16

1

ds ∫0 pc2w

(39b)

353

Corey (1954) capillary force formula is as follows

1 = csw pc2

354 355 356

(40)

Combining Eqs. (39) and (40), we can obtain the relative permeability expression as following,

krw =

357

(1 − swc ) sw4 + 2swc sw3 swc + 1

krnw = (1 − sw )2 (1 −

358

krw ) sw2

(41a)

(41b)

359

According to Eqs. (41), when the irreducible water saturation is known, we can

360

obtain the relative permeability. Fig.5 is the relative permeability under different

361

irreducible water saturation. After hydraulic fracture, a part of the reservoir

362

irreducible water becomes movable. The irreducible water saturation decreases.

363 364

Fig.5 Relative permeability curve

365

4 Results and discussions

366

4.1 The effect of fracturing fluid injection on pressure and saturation

367

The parameters utilized to investigate the effect of fracturing fluid injection on

368

pressure and saturation are shown in table 1. We use those parameters and fracturing

369

fluid injection model to investigate pressure and saturation distribution at end of

370

fracturing fluid injection. The relative permeability used is the red curve in Fig.5. And

371

the dimensionless fracture conductive is 300, which means that the fracture is infinite

372

conductivity. 17

373

Table 1 The properties of reservoir, fracture and fluid No.

Parameter names

Values

Units

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Reservoir thickness Distance between two adjacent fracture Fracture width Reservoir initial pressure Fracturing fluid compressibility Oil compressibility Reservoir compressibility Fracture compressibility Reservoir fracturing fluid viscosity Reservoir oil viscosity Reservoir porosity Fracture porosity Reservoir permeability Fracturing fluid injection rate Fracturing fluid density Oil density Fracturing injection time

5 400 0.002 17 0.00051 0.005 0.00042 0.00042 10 1.8 15.5 21.7 5 2.8 1 0.833 0.0926

m m m MPa MPa-1 MPa-1 MPa-1 MPa-1 mPa·s mPa·s % % mD m3/min g/cm3 g/cm3 d

374

The pressure and saturation are in Fig.6. Fig. (6a) is the schematic of a flow unit

375

of fractured horizontal well (The midpoint between two adjacent fracture is the closed

376

boundary. The region enclosed by two adjacent closed boundary is a flow unit). We

377

take a quarter of a flow unit namely red box area as an example. The calculated

378

saturation and pressure at the end of fracturing fluid injection are respectively Fig.(6b)

379

and (6c). From Fig.(6b) and (6c), we can get that the injection of fracturing fluid

380

causes the variation of reservoir pressure and saturation. When the fracturing fluid

381

injection is finished, the distances of saturation and pressure variation area from the

382

fracture are respectively 10m and 20m.

383 18

384

(a)

385 386

(b)

(c)

387

Fig.6 The pressure and saturation distribution at the end of fracturing fluid injection: (a) the flow

388

unit of a fracture; (b) The water saturation; (c) The pressure

389

In the next part, we investigate the effect of fracture fluid viscosity, injection rate

390

on the pressure and saturation at the end of fracturing fluid injection. The effects of

391

fracture fluid viscosity on the pressure and saturation are seen in Fig.7.

392

When the fracturing fluid viscosity increases from 10 mPa·s to 30 mPa·s, the

393

distance of pressure variation area from the fracture decrease. But the pressure and

394

saturation near the hydraulic fracture increases. This is due to an increase in the

395

fracturing fluid viscosity corresponding to an increase in the flow resistance. The

396

increase of the flow resistance results in the fracturing fluid retention in the reservoir

397

near the fracture.

398 399

(a-1)

(a-2)

19

400 401

(b-1)

(b-2)

402 403

(c-1)

(c-2)

404

Fig.7 The pressure and saturation at the end of fracturing fluid injection: (a-1) and (a-2) are

405

respectively the saturation and pressure when the viscosity is 10mPa·s; (b-1) and (b-2) are

406

respectively the saturation and pressure when the viscosity is 20mPa·s; (c-1) and (c-2) are

407

respectively the saturation and pressure when the viscosity is 30mPa·s;

408

The effects of fracturing fluid injection rate on the pressure and saturation are

409

seen in Fig.8. When the injection rate increases, the fracture length increases and the

410

distances of pressure and saturation variation area from the fracture increase. The

411

pressure near the fracture with a high injection rate is larger than that with a small

412

injection rate. The reason for this scenario is that an increase in fracturing fluid

413

injection rate corresponds to the increase of cumulative injection volume when the

414

time of fracturing fluid injection is a constant. That results in the increase of the

415

distances of pressure and saturation variation area from the fracture increase. While,

416

the pressure and saturation near the fracture increase. 20

417 418

(a-1)

(a-2)

419 420

(b-1)

(b-2)

421 422

(c-1)

(c-2)

423

Fig.8 The pressure and saturation at the end of fracturing fluid injection: (a-1) and (a-2) are

424

respectively the saturation and pressure when fracturing fluid injection rate is 0.28m3/min; (b-1)

425

and (b-2) are respectively the saturation and pressure when fracturing fluid injection rate is

426

2.8m3/min; (c-1) and (c-2) are respectively the saturation and pressure when fracturing fluid 21

427

injection rate is 28m3/min.

428

4.2 Fracturing fluid flowback and production characteristic

429

Based on the pressure and saturation at the end of fracturing fluid injection, we

430

investigate the flowback and production characteristic by the fracturing fluid

431

flowback and production model built above. RRF and the bottom hole flow pressure

432

are respectively 1.03 and 8.5 MPa. The relative permeability of invasion region is the

433

blue red curve in Fig.5. The other inputting parameters are in table 1. The effects of

434

the reservoir oil viscosity, reservoir permeability, reservoir porosity, and reservoir

435

damage on flowback and production characteristic of fractured well are presented

436

below.

437

Fig.9 shows the effect of the reservoir oil viscosity on production characteristics

438

of fractured horizontal well. In this study, the reservoir oil viscosity is from 1.5 to 2.5

439

mPa·s. Seen from Fig.9, with the development of reservoir, the water flow rate and

440

water cut decreases, and the oil flow rate increases firstly and then decreases. When

441

the reservoir oil viscosity increases from 1.5mPa·s to 2.5mPa·s, water cut increases.

442

However, the maximum oil flow rate decreases from 16.4 to 6.2 m3/d. There are a

443

cross between the curves of flow rate with time under different reservoir oil viscosity.

444

This is due to an increase of reservoir oil viscosity indicating the increase of flow

445

resistance, which results in the decrease of oil flow rate and water flow rate at the

446

initial stage of flowback and production. However, the reservoir resources is a

447

constant, more oil and water are produced at the initial stage, so the water flow rate

448

and oil flow rate of when the viscosity is 1.5 mPa·s is less than that of when the

449

viscosity is 2.5 mPa·s at later stage.

450 22

451

(a)

(b)

(c)

(d)

452 453 454

Fig.9 The effect of reservoir oil viscosity: (a) water flow rate; (b) oil flow rate; (c) water cut; (d)

455

cumulative fluid volume.

456

Fig.10 shows the effect of the reservoir permeability on production characteristics

457

of fractured horizontal well. In this study, the reservoir permeability is from 5 to 50

458

mD. Seen from Fig.10, with the increase of reservoir permeability, the water cut

459

decrease. While, there are a cross between the curves of flow rate with time under

460

different reservoir permeability. This is due to an increase of reservoir permeability

461

showing a decrease in flow resistance, which results in an increase in oil flow rate and

462

water flow rate at the initial stage. However, the reservoir resources is a constant,

463

more oil and water are produced at the initial stage, so the water flow rate and oil flow

464

rate of when the reservoir permeability is 50mD is less than that of when the reservoir

465

permeability is 5mD at later stage.

466 467

(a)

(b)

23

468 469

(c)

(d)

470

Fig.10 The effect of reservoir permeability: (a) water flow rate; (b) oil flow rate; (c) water cut; (d)

471

cumulative fluid volume.

472

Fig.11 shows the effect of the reservoir porosity on production characteristics of

473

fractured horizontal well. In this study, the reservoir porosity is from 15% to 18%.

474

Seen from Fig.11, when the reservoir porosity increases, the oil flow rate, water flow

475

rate and cumulative fluid volume increase. While, the water cut decreases with the

476

increase of reservoir porosity. This is due to an increase of reservoir porosity showing

477

the increase of reservoir resource, which indicates an increase of reservoir elastic

478

energy.

479 480

(a)

(b)

24

481 482

(c)

(d)

483

Fig.11 The effect of reservoir porosity: (a) water flow rate; (b) oil flow rate; (c) water cut; (d)

484

cumulative fluid volume.

485

Fig.12 shows the effect of the reservoir damage caused by fracturing fluid on

486

production characteristics of fractured horizontal well. In this study, the RRF is from

487

1 to 1.08 (According to the work of Al-ameri et al. (2018), the maximum of RRF is

488

1.08). Seen from Fig.11, when the RRF increases from 1 to 1.08, the oil flow rate of

489

when the RRF equals 1 is slight larger than that of when the RRF equals 1.08. The

490

reservoir damage has a slight effect on the water flow rate and water cut. The reason

491

for this scenario is that the reservoir damage caused by fracturing fluid is weak.

492 493

(a)

(b)

25

494 495

(c)

(d)

496

Fig.12 The effect of reservoir damage: (a) water flow rate; (b) oil flow rate; (c) water cut; (d)

497

cumulative fluid volume.

498

In this sub-section, we investigate the effect of reservoir heterogeneous on the

499

flowback and production characteristics. Firstly, we generate a random permeability

500

matrix using the Matlab software (Seen in Fig.12), the maximum permeability and

501

minimum permeability are respectively 10 and 1mD. The average of this random

502

permeability is 5.48mD. We respectively calculate the flowback and production

503

characteristics by the random permeability and average of the random permeability.

504

The results are seen in Fig.13. From Fig.13, the oil flow rate and cumulative fluid

505

volume of the heterogeneous reservoir are less than that of homogeneous reservoir.

506

While, the water cut is higher than that of the homogeneous reservoir. In a word, the

507

development of heterogeneous reservoir is poorer than that of homogeneous reservoir

508

when the development conditions are the same.

509 26

510

Fig.12 Random permeability in a quarter of a flow unit

511 512

(a)

(b)

(c)

(d)

513 514 515

Fig.12 The effect of reservoir heterogeneous: (a) water flow rate; (b) oil flow rate; (c) water cut; (d)

516

cumulative fluid volume.

517

4.3 Model verification

518

In order to verify the proposed model, we collect oil test results of 12 fractured

519

vertical wells in the field. The detailing data of those 12 fractured well is seen in table

520

2. The proposed model can be simplified to simulate the production process of

521

fractured vertical well when the number of hydraulic fracture is one.

522

By the data in table 2 and the simplified model, the oil flow rate, water flow rate

523

and water cut are calculated. The comparing results between our calculated results and

524

test results are seen in table 3. From the comparing results in table 3, we can get that

525

the relative error of oil flow rate, water flow rate are respectively less than 8.55% and

526

8.39%. The relative error of water cut between calculated results and oil test results is

527

smaller than 2.61%. So, the proposed model can be accurately simulate the process of 27

528

hydraulic fracture.

28

529

Table 2 The data used in the model comparison Reservoir

Fracturing fluid

Fracturing fluid

Fracture

Irreducible water

Permeability

Porosity,

Oil viscosity,

Reservoir

thickness,m

injection time, min

injection rate, m3/min

length,m

saturation,%

,mD

%

mPa·s

pressure, MPa

Long 45

7.80

213.40

1.76

241.15

45.00

9.26

16.04

14.30

34.88

Long 58

1.00

84.33

4.80

320.14

47.00

3.29

14.11

9.00

18.30

Long 59

4.20

91.30

2.98

323.80

48.00

2.61

12.77

8.90

17.70

Long 60

3.00

69.40

3.00

348.00

40.00

4.50

14.56

21.90

18.95

Long 65

5.00

69.40

2.80

455.00

43.00

2.10

13.30

9.50

16.73

Long 66

1.25

97.90

2.85

116.25

40.00

10.81

16.24

11.80

15.90

Long x46

8.80

61.00

3.79

131.19

43.57

6.70

14.94

10.40

12.00

Long 48

7.40

117.40

2.65

210.47

43.00

8.00

16.14

26.60

14.20

Ta 86

0.80

77.40

2.73

659.38

41.00

9.70

16.87

30.00

14.40

Ta x80

5.00

76.50

2.05

257.00

45.79

23.67

14.91

24.40

14.60

Ta 87

2.40

55.50

2.21

255.60

37.81

16.94

17.99

33.20

14.54

Ta X73

2.40

56.80

2.21

217.70

39.36

17.11

17.68

39.30

13.80

Well No.

530 531 29

532

Table 3 The comparing results between our calculated results and test results 3

3

Oil，flow，rate,m3/d， /d，

Water，flow，rate,m3/d， /d，

Water，cut %，

， Well No.

Long 45

Results，from，oil，

Calculated，

Relative，

test， test，

results， results，

error， error，

52.47， 52.47，

49.10， 49.10，

6.43%， 6.43%，

，

， Results，from，oil，

Calculated，

Relative，

test， test，

results， results，

error， error，

\，

\，

\，

， Long 58

1.92， 1.92，

1.80， 1.80，

6.25%， 6.25%，

5.64， 5.64，

6.02， 6.02，

6.12， 6.12，

5.77， 5.77，

5.72%， 5.72%，

17.95， 17.95，

8.39%， 8.39%，

， Long 60

33.65， 33.65，

35.82， 35.82，

6.45%， 6.45%，

10.47， 10.47，

9.57， 9.57，

\，

\，

\，

\，

\，

， Long 66

0.21， 0.21，

0.22， 0.22，

10.29， 10.29，

7.22%， 7.22%，

， Long x46

24.37， 24.37，

22.87， 22.87，

6.16%， 6.16%，

49.86， 49.86，

46.80， 46.80，

5.64， 5.64，

6.14%， 6.14%，

5.27， 5.27，

6.56%， 6.56%，

42.14， 42.14，

39.12， 39.12，

\，

7.17%， 7.17%，

\，

\，

16.96， 16.96，

16.23， 16.23，

3.84， 3.84，

4.30%， 4.30%，

3.56， 3.56，

7.29%， 7.29%，

8.34， 8.34，

7.89， 7.89，

8.88， 8.88，

5.40%， 5.40%，

8.28， 8.28，

6.81%， 6.81%，

8.58， 8.58，

8.99， 8.99，

1.72， 1.72，

4.78%， 4.78%，

76.22%， 76.22%，

0.13%， 0.13%，

74.59%， 74.59%，

74.89%， 74.89%，

0.39%， 0.39%，

\，

\，

\

\，

\，

\

97.89%， 97.89%，

97.93%， 97.93%，

0.04%， 0.04%，

18.79%， 18.79%，

18.73%， 18.73%，

0.35%， 0.35%，

\，

\，

\

8.35%， 8.35%，

8.34%， 8.34%，

0.12%， 0.12%，

34.37%， 34.37%，

33.77%， 33.77%，

1.77%， 1.77%，

17.10%， 17.10%，

17.55%， 17.55%，

2.61%， 2.61%，

\，

\，

\

，， ，，

，， ，，

，， ，，

， 1.68， 1.68，

2.33%， 2.33%，

， Ta X73

76.12%， 76.12%，

， ， ，，

，

， Ta 87

\

，

， Ta x80

\，

，

， Ta 86

\，

，

， Ta 48

error， error，

， 9.60， 9.60，

5.16%， 5.16%，

results， results，

， \，

8.55%， 8.55%，

oil，test， oil，test，

，

， Long 65

Relative，

， 16.56， 16.56，

6.74%， 6.74%，

Calculated，

，

， Long 59

，

Results，from，

， \，

\，

，

\， ，

533

30

，， ，，

534

5 Conclusions

535

Taking the fracturing fluid damage and the effect of fracturing fluid injection on

536

pressure and saturation into consideration, a flow model of multistage fractured

537

horizontal well is built, which is composed of fracturing injection model and

538

fracturing fluid flowback and production model. Numerical solving method is utilized

539

to solve the model and model verification are presented through comparing with the

540

test results of 12 fractured well. The effects of fracture fluid injection rate and

541

fracturing fluid viscosity on pressure and saturation at the end of fracturing fluid

542

injection are analyzed. And the effects of the reservoir oil viscosity, porosity,

543

permeability, heterogeneous and reservoir damage caused by fracturing fluid on

544

flowback and production characteristics are analyzed. From this work, some

545

conclusions are drawn as following,

546

(1) The distances of saturation and pressure variation area from the fracture are

547

respectively 10m and 20m after the fracturing fluid injection is finished. When the

548

fracturing fluid viscosity increases from 10 mPa·s to 30 mPa·s, the distance of

549

pressure variation area from the fracture decrease. But the pressure and saturation

550

near the hydraulic fracture increases. When the injection rate increases from

551

0.28m3/min to 28m3/min, the fracture length increases and the distances of pressure

552

and saturation variation area from the fracture increase. The pressure near the fracture

553

with a high injection rate is larger than that with a small injection rate.

554

(2) With the development of reservoir, the water flow rate and water cut

555

decreases, and the oil flow rate increases firstly and then decreases. When the

556

reservoir oil viscosity increases from 1.5mPa·s to 2.5mPa·s, water cut increases.

557

However, the maximum oil flow rate decreases from 16.4 to 6.2 m3/d. With the

558

increase of reservoir permeability, the water cut decrease. When the reservoir porosity

559

increases, the oil flow rate, water flow rate and cumulative fluid volume increase.

560

When the RRF increases from 1 to 1.08, the oil flow rate of when the RRF equals 1 is

561

slight larger than that of when the RRF equals 1.08. The reservoir damage has slight

562

effect on the water flow rate and water cut. 31

563

(3) The oil flow rate and cumulative fluid volume of the heterogeneous reservoir

564

are less than that of homogeneous reservoir. While, the water cut is higher than that of

565

the homogeneous reservoir. The development of heterogeneous reservoir is poorer

566

than that of homogeneous reservoir when the development conditions are the same.

567

(4) The proposed model can be utilized to simulate the process of hydraulic

568

fracture with taking the effect of fracturing fluid injection on the reservoir properties,

569

pressure and saturation into consideration. However, there two limitations of the

570

proposed model, which are

571

fractures are neglected;

572

development process of fractured horizontal well;

573

behavior of fracturing wells is ignored.

574

Acknowledgments

the changes in geomechanics and the role of natural the model just can be utilized to simulate the depletion The effect of proppant on the

575

This work is supported by the Fundamental Research Funds for the Central

576

Universities through grant number 18CX06011A, National Natural Science

577

Foundation of China with No. 51974343, National Massive Oil & Gas Field and

578

Coal-bed

579

2016ZX05010-002-007, National Science and Technology Major Demonstration

580

Project ‘Tight Oil Development Demonstration Project of Bohai Bay Basin Jiyang

581

Depression’ through grant number 2016ZX05072006-004.

582

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583

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35

1 2

(c)

(d)

3

Fig.12 The effect of reservoir damage: (a) water flow rate; (b) oil flow rate; (c) water cut; (d)

4

cumulative fluid volume.

5

In this sub-section, we investigate the effect of reservoir heterogeneous on the

6

flowback and production characteristics. Firstly, we generate a random permeability

7

matrix using the Matlab software (Seen in Fig.13), the maximum permeability and

8

minimum permeability are respectively 10 and 1mD. The average of this random

9

permeability is 5.48mD. We respectively calculate the flowback and production

10

characteristics by the random permeability and average of the random permeability.

11

The results are seen in Fig.14. From Fig.14, the oil flow rate and cumulative fluid

12

volume of the heterogeneous reservoir are less than that of homogeneous reservoir.

13

While, the water cut is higher than that of the homogeneous reservoir. In a word, the

14

development of heterogeneous reservoir is poorer than that of homogeneous reservoir

15

when the development conditions are the same.

16 1

17

Fig.13 Random permeability in a quarter of a flow unit

18 19

(a)

(b)

(c)

(d)

20 21 22

Fig.14 The effect of reservoir heterogeneous: (a) water flow rate; (b) oil flow rate; (c) water cut; (d)

23

cumulative fluid volume.

24

4.3 Model verification

25

In order to verify the proposed model, we collect oil test results of 12 fractured

26

vertical wells in the field. The detailing data of those 12 fractured well is seen in table

27

2. The proposed model can be simplified to simulate the production process of

28

fractured vertical well when the number of hydraulic fracture is one.

29

By the data in table 2 and the simplified model, the oil flow rate, water flow rate

30

and water cut are calculated. The comparing results between our calculated results and

31

test results are seen in table 3. From the comparing results in table 3, we can get that

32

the relative error of oil flow rate, water flow rate are respectively less than 8.55% and

33

8.39%. The relative error of water cut between calculated results and oil test results is

34

smaller than 2.61%. So, the proposed model can be accurately simulate the process of 2

35

hydraulic fracture.

3

1. A multilinear flow model for fracturing horizontal well is proposed with considering the effect of fracturing fluid injection. 2. The permeability and relative permeability are modified with considering the fracturing fluid damage. 3. The effects of key parameters on the pressure and production characteristics are investigated.

Zhongwei Wu builds and solves the model. Most part of revised work are done by him Chuanzhi Cui provides the idea of this paper and the funding for this work. Authors Li Dong and Xiangzhi Cheng analyze the results and part of revised work are done by them Zhen Wang draws the figures in the manuscript.

The authors declare that they have no conflicts of interest.