New SAGD model for oil production using a concave parabola steam chamber geometry

Accepted Manuscript New SAGD model for oil production using a concave parabola steam chamber geometry L.C. Igbokwe, C.M. Obumse, M. Enamul Hossain PII:

S0920-4105(18)31156-2

DOI:

https://doi.org/10.1016/j.petrol.2018.12.052

Reference:

PETROL 5612

To appear in:

Journal of Petroleum Science and Engineering

Received Date: 6 February 2018 Revised Date:

17 December 2018

Accepted Date: 20 December 2018

Please cite this article as: Igbokwe, L.C., Obumse, C.M., Hossain, M.E., New SAGD model for oil production using a concave parabola steam chamber geometry, Journal of Petroleum Science and Engineering (2019), doi: https://doi.org/10.1016/j.petrol.2018.12.052. 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 New SAGD Model for Oil Production Using a Concave Parabola Steam Chamber Geometry L.C. Igbokwe1, C.M. Obumse2 and M. Enamul Hossain3* 1

Memorial University of Newfoundland, John’s Canada

2

Institute of Petroleum Studies, Nigeria, 3American

University in Cairo, Cairo, Egypt

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*Email: [email protected]

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Abstract

In this paper, a new steam chamber growth model using a concave parabola interface is presented. Steam

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chamber shape is crucial in describing the expansion of injected steam and the propagation pattern of the temperature of the steam front towards the producing well. Steam chamber evolution is particularly important for predicting oil production during steam assisted gravity drainage (SAGD) processes. The new model is derived based on the orientation of the temperature isoline obtained for the underground test facility (UTF) project, the results of several experiments, and the numerical simulation of SAGD

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processes for major field projects. The analytical description of this steam chamber growth pattern will assist in evaluating and developing fields using SAGD scheme. The model employs the concept of heat conduction involving a moving solid to estimate the steam chamber interface temperature profile and, eventually, the oil

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production rate and associated steam injection rate required based on an energy balance approach. This study builds upon the assumptions of Butler and Reis’ linear interface by introducing a parabolic steam interface. It

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is demonstrated that the new model predicts oil production and steam injection rates with higher accuracy than existing models. It also showed the strong the variation of oil rate with time unlike the constant

rate assumption. The results of the model are compared with the experimental data from Cheng and Butler, simulation with a CMG STARS simulator and the results of the UTF phase B project, which show better agreement and robustness. Keywords: Thermal Recovery SAGD, EOR, Steam Chamber Growth, Parabola, Steam Interface

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ACCEPTED MANUSCRIPT Nomenclature

Steam width factor, 1/

/ /

Specific heat of formation,

℃

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Acceleration due to gravity,

Height of reservoir,

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Permeability,

Length of the wellbore, /

Latent heat of steam,

Dimensionless viscosity coefficient

Oil drainage rate,

/℃

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Formation heat capacity

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Interface length,

/ ∙

Rate of Latent heat injection,

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inj

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Effective permeability of oil,

/ ∙

Rate of energy loss through overburden,

/

∙

Enthalpy required to heat oil ahead of the steam interface, Steam injection rate, !

∆#

3/

Enthalpy rate needed for expansion of steam chamber,

Initial oil - residual oil saturation of system (# % − # ' )

/ ∙

/ ∙

2

ACCEPTED MANUSCRIPT #)*

Steam–Oil ratio

+∗

Dimensionless temperature profile

-.

Steam-oil temperature difference (+ − + ), ℃

Maximum horizontal velocity,

/

-/

Velocity perpendicular to the steam chamber edge,

1

Half-width of steam chamber,

3

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Steam quality Thermal diffusivity of reservoir,

η

Coordinate parallel to interface

9

Porosity

=

?

?

? @

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<

/

Instantaneous angle of interface with horizontal, (°) /

∙

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:

Flow potential gradient, 78/

Dynamic oil viscosity,

Kinematic oil viscosity at steam temperature,

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5∇

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2

Reservoir thickness, m

/

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∆+

Density of formation, Density of oil,

Density of water,

/

/

/

/

Coordinate perpendicular to interface 3

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A

Time,

1.0

Introduction

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The formation of a steam chamber as steam is injected through the perforations of a completed well has long been established by a series of experiments conducted in laboratory, pilot tests and industrial project for SAGD processes. Steam chamber shape is used to describe the expansion of injected hot steam and the

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propagation pattern of the interface temperature of the steam front towards the producing well. Steam chamber shape evolution is particularly important in evaluating SAGD processes. The basic theories and

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fundamental concepts of SAGD development and analysis have been formulated by Butler (R. M. Butler & Stephens, 1981). Despite many investigations of the pattern and the nature of the steam chamber growth, there remains a lack of adequate prediction of the geometry of the steam chamber progression towards the production well. Consequently, underestimation or overestimation of oil production rates are common issues in many existing models (N. Edmunds & D. Gittins, 1993).

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In a typical configuration of the SAGD process (Figure 1), a pair of horizontally drilled wells are placed parallel to each other within the reservoir thickness. The upper injector well is located about 5 meters above the producing well that is placed slightly above the bottom of the reservoir thickness (usually 2-3

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meters from the bottom of the reservoir) (R. M. Butler & Stephens, 1981). The operational mechanism of SAGD is initiated by injecting hot streams of steam into the injector well. During SAGD, the injected high

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temperature steam contacts the highly viscous oil and cold bitumen, causing heat exchange to occur between the heavy oil and steam resulting in several magnitudes of viscosity reduction. The lightened oil and condensed steam then flows more readily into the lower horizontal well (producer) supported by the gravitational effect while the steam vapor continues to move upward to fill the void spaces (R. M. Butler, & Stephens D. J., 1980; R. M. Butler, McNab, & Lo, 1981; R. M. Butler & Stephens, 1981; RM, 1997; RM., 1997). Enhanced oil recovery by the thermal method has benefited from the invention of SAGD by Roger Butler and his team at Canada’s Imperial Oil. Since then, SAGD has been successfully used to recover heavy 4

ACCEPTED MANUSCRIPT and extra heavy and bituminous oil deposits, including those of the Athabasca oil sands (N. R. Edmunds, Haston J. A., & Best D. A., 1989; N. R. e. a. Edmunds, February 1991. ; Goobie, Aprii 1993; Good, Scott, & Luhning, 1994; O’Rourke, 1991), Peace River (Falk, Nzekwu, Karpuk, & Pelensky, 1996; Gates & Chakrabarty, 2008; Good et al., 1994; Dharmeshkumar R. Gotawala & Gates, 2011; Shell, 2013.; J. Sheng,

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2013,; K. Sheng, Okuno, & Wang, 2017), Liaohe, China (Peng, Yuan, Jiang, & Zhang, 2010; Wang, Han, & Zou, 2017), Cold Lake and Lloydminster(Roger M. Butler, 2001; Vásquez, 1999). The recovery of over 50% of the initial oil in-place (OOIP) using the SAGD technique are linked to enhanced sweep efficiency that is

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stabilized by gravity forces (Alvarez & Han, 2013; Roger M. Butler, 2001).

1.1

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Figure 1 Schematic configuration of SAGD Process

Steam Chamber Growth

Butler was the first to identify the formation of steam chamber when steam is injected in a completed well in a series of experiments he conducted on SAGD. He observed the formation of a steam saturated zone or steam depletion zone, referred to as a steam chamber, around the injector well when steam is injected as illustrated in Figure 2 5

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Figure 2: Shape of Steam Chamber Growth, UTF test, (Ito & Suzuki, 1996)

In theoretical studies on SAGD in heavy oil reservoirs, Butler postulated that the steam chamber is formed above the producer to an infinite horizontal length at constant injection pressure. In addition, the temperature

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at the horizontal interface is maintained at a constant value (+ = + CDE. ) as it propagates downward. Beyond

the interface, Butler observed steady state heat expansion by downward conduction. He expressed the oil

= F where

GHIJ∆ K L ./M

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production rate by the steam chamber expansion rate as a function of reservoir characteristics as follows:

(1)

is the oil drainage rate per unit length of well per unit time for each chamber side in

the difference between the initial and the residual oil saturation, =

is the oil dynamic viscosity in

is the coefficient of viscosity, ℎ is the reservoir thickness in meters,

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steam temperature,

, 3 is the thermal diffusivity of the reservoir in

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permeability to flow in

N , ∆# is

/ at

is the effective

/ , and P is the porosity. He

obtained a closed form solution of the model. Due to the density differential of steam and condense steam-oil emulsion, the residual oil saturation in the overburden is further decreased as the steam fills the drained upper pore space while the water saturation increases downwards. This mechanism further explains the steam chamber shape (tapering downwards) since the amount of heat loss relates to the diffusivity of the interacting medium.

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ACCEPTED MANUSCRIPT 1.2

Evolution of Geometrical Study of Steam Chamber Growth

Following Butler’s theory on steam chamber development (R. M. Butler et al., 1981), the experimental investigations of Butler and Stephen, (R. M. Butler & Stephens, 1981), Chung and Butler (Al-Bahlani & Babadagli, 2009; Chung & Butler, 1988) and the Underground Test Facility (UTF) field test by the Alberta

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Oil Sands Technology Research Authority (AOSTRA) (Apos et al., 1997, 1999; Apos, Rourke, Chambers, Suggelt, & Good, 1994; Boyle, Gittins, & Chakrabarty, 2003) the shape of the steam chamber development at different stages of the SAGD process has been modelled analytically by different researchers (Akin, 2005;

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Alali, Pishvaie, & Jabbari, 2009; Ali Azad & Chalaturnyk, 2012; Chung & Butler, 1988; Dianatnasab, Nikookar, Hosseini, & Sabeti, 2016; Edmunds N., 2007.; Gates & Chakrabarty, 2008; Dharmesh R.

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Gotawala & Gates, 2008; Guo, Zan, Ma, & Shi, 2013; Heidari, Pooladi-Darvish, Azaiez, & Maini, 2009; Irani & Ghannadi, 2013; Liang, Goodman, Tummala-Narra, & Weintraub, 2005; Ma et al., 2013; Nukhaev, Pimenov, Shandrygin, & Tertychnyi, 2006; Rabiei Faradonbeh, Harding, & Abedi, 2016; Reis, 1992, 1993; Sharma & Gates, 2011; Tian, Mu, Wu, & Xu, 2015; Lijuan Zhu, Zeng, & Huang, 2015). Virtually all studies agree that the steam chamber growth diminishes downward towards the producer and the fundamental

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concept of these studies follows Butler’s assumption that the steam chamber create fingers and touches the cap rock (i.e., reservoir top) and expands laterally sideways (R. M. Butler et al., 1981). However, this assumption does not capture the initial chamber growth stages (Ali, 1997; Dianatnasab et al., 2016; Shaolei,

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Linsong, Wenjun, Shijun, & Shuai, 2014; Shijun, Hao, Shaolei, Chenghui, & Yang, 2016; Yuan, Liu, Zhang, Jiao, & Li, 2017; Lijuan Zhu et al., 2015; L. Zhu, Zeng, Zhao, & Duong, 2012). Many experimental and

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numerical studies suggest a concentric eclipse or parabola tapering downward to the producer (Joshi, 1985; Shako & Rudenko, 2007; Shaolei et al., 2014). However, there are very few studies in the literature that describe this process analytically or semi-analytically for preliminary evaluations. From Butler’s model, it is assumed that the moving linear interface is maintained at constant temperature until the interface breaks through at the producer. Since the oil recovery is estimated based on the chamber expansion, the oil recovery is overestimated (R. M. Butler, 1985; R. M. Butler & Stephens, 1981) further improved other aspects of the initial theories in separate studies including the TANDRAIN for locating the 7

ACCEPTED MANUSCRIPT interface of the steam chamber half to the adjacent well and rising steam chamber (Chung & Butler, 1988). They validated these formulations with experimental data; however, the fundamental concepts and assumptions were not changed. Butler (Roger M. Butler, 2001)also showed the heat distribution for a typical SAGD project (Athabasca oil sands) as shown in Table 2. Surrounding heat chamber and overburden

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accounts for 19.2% and 17.5%, respectively, of the total heat distributed in reservoir formation. One of the most accepted models for predicting oil rates based on steam zone expansion in the SAGD process was developed by Reis (Reis, 1992). He formulated a model for predicting steam chamber interface

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using an inverted triangle to describe the steam chamber shape with the vertex pointed to the producer. Although the theoretical methodology of Reis’ model is similar to Butler’s, Reis’ model demonstrated the chamber diminished towards the producer by applying linear geometry (Reis, 1992) and that heat transport

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width diminished radially (Reis, 1993). Reis also used a multiplier ( 8 = 0.4 ) obtained experimentally to

simplify the analytical integration over the reservoir thickness. However, Reis did not account for heat losses on the curved sides of the inverted triangle, as presented in the UTF test and 3D simulation studies. Hence, any expanded heat over the curvatures were neglected. Reis’ model is too optimistic, similar to Butler’s

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model (A. Azad & Chalaturnyk, 2010; Reis, 1993), which is shown in the present study. Reis estimated the areal coverage of the total chamber based on the UTF isoline plots (O’Rourke, 1991). Azad and Chalaturnyk (Ali Azad & Chalaturnyk, 2012) offered a model called circular steam chamber geometry which regards the

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steam chamber as a constantly expanding circle. This new insight into the chamber's geometry as well as Butler's and Reis' models, allowed them to define the flow potential function (∆Φ) such that the increase in

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the chamber's diameter in each time step influences the results of Darcy's equation and the production rate. In addition, they considered the oil reservoir as a series of circular slices and thus, could replace the constant relative permeability by varying relative permeability in Darcy's equation. Despite a huge success in production prediction, this circular model fails to present a real curved chamber shape as it naturally occurs to be conical. Recently, Shaolei et al (Shaolei et al., 2014) derived the oil production rate from the volume of steam chamber expansion per unit times a function of reservoir and injection parameters. And the analysis agrees with numerical simulations. Although Shaolei’s models improved the optimistic evaluation of Butler’s theory significantly, a chunk of the chamber curved area is neglected by the inverted triangle 8

ACCEPTED MANUSCRIPT assumption. Instead of an inverted triangle, Sabeti et al. (Sabeti, Rahimbakhsh, & Mohammadi, 2016) represented the location of the oil–steam interface using an exponential function to eliminate the linear assumption made in Reis' model. Although the model closely matched the experimental results, the exponential curve extends beyond the steam zone as shown by their plots. This is concerning, as is the trial

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and error approach required for selecting a suitable scale factor, especially at early times. More recently, Dianatnasab et al. (Dianatnasab et al., 2016) varied key reservoir properties to investigate the steam chamber development process numerically. They compared the model accuracy with Chung and Butler’s experiment and reported parabolic plots from their numerical studies. The models reviewed above are summarized in the

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Table 1. It is shown that initial studies on steam chamber growth assumed linear chamber geometry until recently when field results and larger scale pilot tests were carried out. The analytical modeling technique is

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a choice method for describing the steam chamber growth characteristics, as it provides a sufficient preliminary evaluation guide and it is this, that is utilized in this study

Table 1: Summary of Major Steam Chamber Development Models Reporting Model type

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Author

Steam Chamber Shape

Chamber

Year

1981

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Butler and Stephen

1981

Experimental

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Butler et al

Analytical+

Heat Transfer Ahead interface

to Reservoir Straight Line

Steady state

Tangential Curve

Unsteady state

Experimental

Chung and Butler

1988

Experimental

Isosceles triangle

Steady State

Reis

1992

Analytical

Inverted Triangle

Steady State

Reis

1993

Analytical

Inverted Cone

Steady State

Azad and Chalaturnyk

2001

Semi Analytical

Linear

slices

of Steady State

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ACCEPTED MANUSCRIPT trapezoids 2007

Analytical

Inverted Triangle

Steady State

Alali et al

2009

Analytical

Curve

Transient

Azad and Chalaturnyk

2012

Semi Analytical

Circular Slices

Steady State

Miura and Wang

2012

Analytical

Inverted triangle

Steady state

Irani and Ghannadi

2013

Analytical

Barbell shape

SteadyState

Wi et al

2014

Analytical

Inverted triangle

Steady State

Ji et al

2015

Analytical

Straight line

SteadyState

Dianatnasab et al

2016

Semi Analytical

Triangular

Sabeti et al

2016

Analytical

Exponential function of Steady State

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Edmunds and Peterson

SteadyState

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Reis Geometry

Table 2: Heat Loss Distribution for SAGD in Athabasca, (Roger M. Butler, 2001) GJ/m3 of produced oil

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Produced Fluids

0.43

6%

2.33

32.5%

Steam Chamber

1.77

24.7%

Chamber surrounding

1.38

19.2%

Overburden

1.26

17.5%

Total

7.5

100

Water

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Oil

%

10

ACCEPTED MANUSCRIPT 1.3. Objective of the study Due to the importance of steam chamber growth prediction in estimating oil production in SAGD, an improved steam chamber growth model would mimic field performance of SAGD processes more accurately. Following the results of the UTF and several SAGD experiments and simulations discussed, a

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parabolic steam chamber interface has been suggested which differs from a rigid steam chamber interface as postulated by Reis. Proper modeling of the interface is important to capture the steam growth zone which would influence the result of the oil production rate model. In this study, Reis’ SAGD model is modified to

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include curvatures created by steam expansion over the Reis’s triangular steam chamber. A concave parabola steam interface function is generated, which is used to obtain the oil production rate and the equivalent steam

2.0

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injection rate model. Development of Model

A new model of the shape formed by a growing steam chamber is presented. When streams of hot steam are injected in the upper horizontal well, an envelope (otherwise called steam chamber) is observed at the

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perforations of the injection well. The chamber is then propagated into the formation by contacting the highly viscous formation. The formation is heated up by the progressing steam chamber driving the heated oil towards the producing well.

Underlying Model Assumptions i.

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2.1

We assume that communication has been established between the injector and the producer for

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SAGD start-up (Shijun et al., 2016; L. Zhu et al., 2012) during which the steam chamber has reached the producer. However, the steam width (or axial thickness) is nearly insignificant. ii. Oil saturation in the entire reservoir is at initial condition. However, zones invaded by injected steam (called steam chamber) are gradually changed from the initial oil saturation (# % ) to residual oil saturation (# ' ) as the steam chamber front advances

iii. An average saturation is assumed in the steam chamber to obtain a desired relative permeability needed to fit the analytical model before prediction 11

ACCEPTED MANUSCRIPT iv. A parabola, tapering down to the producer is assumed with width, 1 defined from the center of the injector. The assumption is made based on the temperature isoline plots (Figure 2) from the UTF phase A report (N. Edmunds & D. Gittins, 1993; Ito & Suzuki, 1996). v. The parabola is assumed concave due to the preferential direction of formation deformation in

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establishing communication between the two parallel horizontal wells. Ito and Suzuki (Ito & Suzuki, 1996) demonstrated the growth of the steam chamber by plotting the temperature isolines observed at the UTF (Figure 2). The figure shows a series of concentric parabolic

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shapes in contrast to an advancing horizontal straight line in most previous studies. It can be identified that concave parabola tapering towards the lower horizontal well can better describe the steam chamber

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development process in the start-up before the chamber is fully developed compared to a linear shape of the chamber interface. A linear shaped interface also does not capture the diminishing stages of the SAGD process as the inverted triangle apparently neglects curved sections of the chamber shape (Akin,

2.2

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2005; Reis, 1992, 1993; Sabeti, Ehsani, Nikookar, & Mohammadi, 2015; Sabeti et al., 2016).

Derivation of the Steam Chamber Interface Length

Figure 3 shows a 2-D parabolic function, T(U) which expresses the growth of the steam interface from the

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injection well towards the production well, where its derivative is a linear function of the local steam chamber width. Figure 4 shows the steam chamber in 3-D, which is bound above by the caprock and below

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by the under-burden below the lower of the well horizontal pair which is the oil producer, with the steam injector well completed above.

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ACCEPTED MANUSCRIPT 1

−U

+U

a

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U

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−a

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Figure 3: Illustration of Progressing Concave Parabolic Steam Chamber Interface

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Figure 4: Downward Tapering Concave Steam Chamber Expansion The length of one wing of the steam chamber interface derived from the parabolic function can be written as: (1 ) = 1 F1 + X [ + M sinh`V X [ Z ]Y Z V

Y M

Z\

Y M

(2)

The detailed derivation of the steam chamber interface length is provided in the Appendix A. Equation (2) differs from Reis’ triangular model which uses its hypotenuse as the steam chamber interface expressed as: 13

ACCEPTED MANUSCRIPT (1 )

2.3

D% =

F1 +

(3)

Flow of Heated Oil

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As the steam chamber front continues to advance under sustained injection, oil drains with the condensed water into the producing well. Within the steam zone oil saturation is reduced to the residual oil saturation (R. M. Butler, & Stephens D. J., 1980; R. M. Butler et al., 1981; Heidari et al., 2009; Reis, 1992; Shaolei et al., 2014). The drainage rate per unit length on the interface is obtained using Darcy’s equation: =

cdK c (? eK

− ? )∇5 @

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b

(4)

and

'

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where ∇5, is the gradient of potential in the direction of flow, < is the oil viscosity in the direction of flow is the relative permeability of oil in the flow direction.

The density difference can be approximated to be the density of only the oil due to the small magnitude of the steam density. i.e. ? ≫ ? and (? − ? ) ≈ ? , where ? is the oil density and ? is the density of the ∇5 = .

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injectant. The flow potential can be expressed as: Y h

is the height of the bed from the producing well and

where

(5) is the length of the steam interface. The

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kinematic viscosity at a distance ξ from the interface is obtained using the temperature model developed in

follows:

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equation 6, which was proposed by Butler (R. M. Butler, 1985; Reis, 1992) and adjusted by Reis (1992) as

+ ∗ = jUk X /KM

/(n)

`Elm n [ I

= (+ ∗ ).

(6)

(7)

14

ACCEPTED MANUSCRIPT is the dimensionless coefficient of viscosity. -/ is the velocity perpendicular to the motion of the

Where

interface with medium of thermal diffusivity, 3. The kinematic viscosity is related to the viscosity by the expression: e(n) oK

and =

=

eKM oK

(8)

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=(@) =

Substitution of equation (8) into equation (4) yields the rate of oil production b

=

cdK c eKM

p?

qj h

Y

rstum v w

(9a)

b

=

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heated oil drainage by the progressing steam chamber interface:

Xh[

IcdK coK H Y EteKM lm

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Substituting into the Darcy’s oil drainage equations and integrating from @ = 0 A @ = ∞ gives the rate of

(9b)

2.4 Instantaneous Velocity of the Steam Front

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The maximum instantaneous velocity of the steam zone interface (i.e., normal to the interface) is defined at

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the top, and it is related to the local interface velocity (Figure 5).

-.

(90 − :)

:

Figure 5: Steam Chamber Interface

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ACCEPTED MANUSCRIPT In Figure 5, the instantaneous angle of the steam interface is related to the maximum velocity and the velocity normal to the interface by equation (10) -/ = -. cos(90 − :) = -. sin :

(10)

derivative (tangent) of the steam chamber function. (tan : = T ~ (U)) It can be shown that sin : = ±F

V V€•‚ƒ\ „

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To obtain sin :, where : is the instantaneous angle the interface makes with the horizontal, we build on the

(11)

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By substituting the resulting angle the interface makes with the horizontal into equation (9) and evaluating at U = 1 gives: =

IcdK coK HY F1 + EteKM l…s† h

X YM [ Z

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b

(12)

The oil drainage rate along one face of the steam (invaded) zone (per unit length along the horizontal well) is equal to the time derivative of the cross-sectional area of the steam chamber. Thus, =

‡

‡C

ˆP∆# (A) ∙ ‰Š

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b (A)

(13)

The cross-sectional area, A is obtained from the parabolic steam chamber function along one face of the

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steam chamber, while the maximum velocity is obtained as the derivative of the steam width as shown in equations (14a) and (14b) respectively: ‰ = ‹Œ • T(U) U = ‹b Œ

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Ž

ZM

‡ZM ‡C

= -.EŒ

X1 − Z \ [ U = Œ\ M

1

(14a)

(14b)

The change in oil saturation is evaluated, we obtain as follows (see Appendix C): ∆# (A) = # % − #

'

(15)

Substituting equation (14a) into equation (13) we obtain:

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ACCEPTED MANUSCRIPT b

=

P∆# ∙

‡ZM ‡C

(16)

Combining equations (12) and (14b) and substituting equation (16) yields the analytical solution for the rate of oil production from the well: • \

∙ •1 + X [ ‘ Y ZM

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JcK ∆ K ∙HI Et/KM h

(17)

To adjust the model to a Reis-type equation as follows:

D% – — ‡D˜

•

•] Y •1 + XZM [ ‘\ h Y ’““ “““”“““ ““•

— ‡%™%šEC% › C D% – — ‡D˜

The adjustment factor to the Reis’ model œ

• • \ \

(18)

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=

J∆ K ∙cK HYI ‘∙ •F Et/ KM ’“““”“““•

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=

b

∙•

]Y h

•1 + X YM [ ‘ • accounts for the departure of the interface Z

from a rigid diagonal to a curve and demonstrates the time dependency of the model.

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Practically, the oil production estimated by this approach will vary with time. Reis (Reis, 1992) pointed that the underlying assumption of the Butler-like models (e.g., Reis, this Study, etc) is that there is total removal of mass at the interface. During gravity drainage, material is not removed. It

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simply flows down the interface where it inhibits the contact of the steam with cold bitumen at the lower level, thus violating the assumptions used to derive the temperature profile, upon which the present model

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was developed. Hence it leads to slight over-estimation of the actual oil production. To compensate for this, the derivative of the production rate with respect to steam width is equated to zero at the corresponding interface length, because the growth of the steam zone is mainly accounted for by the steam width growth. ‡ž

‡ZM

=2∙

`

∙ •1 + X M [ ‘ ]Y \ ] Y ZM

V

Z

¡

.F

JcK ∆ K ∙HI Et/KM h

=0

(19)

17

ACCEPTED MANUSCRIPT Because, all the parameters in equation 19 are non-negative, the only condition for setting 1 as zero. To achieve this, the interface length

‡ž ‡ZM

to be zero, is by

is first determined based on equation (2), then a

zero 1 is entered in equation 18. This adjustment is in agreement with experimental results demonstrated in the model validation. Because the length of the steam chamber interface

continues to increase with time or

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increasing rate, the adjustment factor (hence the proposed model) varies inversely with time. It is expected that the rate of production will be on the decline after communication is established between the injector and

Evaluation of the Interface Temperature

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2.5

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the producer if all relevant parameters and/or production constraints should remain unaltered.

Carslaw and Jaegar (Carslaw H S, 1959) provided a solution for heat conduction in a moving solid where the

region @ > 0 moves with velocity (-) with initial temperature profile (+ ∗ + 8@). The surface region @ > 0 is maintained at temperature profile (+V∗ + £A) for A > 0, and heat is generated in the solid at the rate ‰ per

unit time per unit volume.

¤\ ¥ ∗ ¤n \

−

l ¤¥ ∗ I ¤n

−

V ¤¥ ∗

I ¤C

=

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The differential equation for the above situation is shown in equation (20) as (Carslaw H S, 1959) `¦K c

(20)

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They obtained the solution of the temperature distribution as equation (21) by applying the following boundary conditions (BC) in equation (20) for the conduction of heat through the moving solid.

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+ ∗ = + ∗ + 8@, @ > 0, A = 0 § ¨ + ∗ = +V∗ + £A, @ = 0, A = 0 + ∗ = + ∗ + 8@ + X3 +

V X£ l

¦K C [ c

+ 8- −

(21)

− 8-A + (+V∗ − + ∗ ) ©jªT«

I¦K [ ®(@ c

V

+ -A)j \ jªT« uv

n€lC √IC

n`lC √IC

+ (-A − @)jªT«

n`lC ¯ √IC

(22)

18

ACCEPTED MANUSCRIPT During steam injection, steam zone will expand laterally as well as drain along the steam/oil interface to the producing well. The drainage by gravity influence can be modelled as a solid moving along the lateral axis (R. M. Butler & Stephens, 1981; Reis, 1992), @ at a fixed velocity - and having a field temperature at its surface, thus heating up the surrounding formation.

¤\ ¥ ∗ ¤n \

−-

¤¥ ∗ ¤n

−

V ¤¥ ∗

I ¤C

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The resulting differential equation yields =0

(23)

Equation (23) is obtained from equation (20) by setting, ‰ = 0 i.e neglecting heat generation. Equation (23)

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is similar to equation (20) by simply setting: + ∗ = 0 , = 0 , +V∗ = 1 and £ = 0. The temperature at the +∗ =

¥(°) `¥± ¥M `¥±

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interface +(C) can therefore be estimated from the temperature profile as shown in equation (24).

(24)

where +% is the initial temperature of the steam chamber and + is the temperature of injected steam. The BC for the process described is given as:

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+ ∗ = 0, at A = 0, ² > 0 + = +% § ¨ + ∗ = 1, at A = ∞, ² ≥ 0 + = +

(25)

+ ∗ = ®jªT«

n`´C √IC

+ j w ∙ jªT« µv

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V

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Substituting the value of the constants into equation (24) yields the temperature profile model as follows: n€´C ¯ √IC

(26)

According to Reis (Reis, 1992), matter is either lost or added at the surface to maintain the interface at a fixed location (@ = 0). For the case of gravity drainage, matter is lost as it drains away from the surface. In

the upper section of the steam zone, the thickness over which the temperature varies, @ is small compared to

the approximate steam zone width, -A. After noting that U is negative for the removal of mass and changing its sign so that the absolute value of - can be used, equation (26) can be written for long times by applying the mathematical relationships jªT(∞) = 1 and jªT«(∞) = 0, as:

19

ACCEPTED MANUSCRIPT

+∗ = j

ruv w

(27)

To correct for the discrepancies of equation (27), a velocity adjustment factor 8 is introduced as suggested by Reis (Reis, 1992) to obtain equation (6). Estimation of the Steam Chamber Width ¶·

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2.6

During SAGD, the injected steam rises to the cap rock or overburden and then moves laterally. The steam

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width defines the maximum lateral excursion of the injected steam in the steam zone, which is obtained by integrating the steam expansion rate of the parabola cross-section with respect to time.

= ‹b

ZM

P ∆# .

‡ZM ‡C

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The cumulative oil production per unit length is written as: ∙ A = P ∆# . 1

(28)

Equating (26) and time integral of (28), the equation is simplified to obtain the expression below: V Y¸ √

¹1 + √1 + 64

Where JEt»KM h∆ K cK HIC \

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= ∙

]

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1 =

(30)

From equations (29) and (30), we notice the implicit behavior of the terms; ,

and 1 . As such, 1 ,

varies inversely with time and the production rate

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requires an iterative solution till convergence. Because

(29)

varies partly with , the model will predict a declining oil rate over time. The solution approach for the production rate is given in the flowchart in Figure 6.

20

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2.7

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Figure 6: Flow diagram for Model Simulation

Energy Balance

The models presented for the oil production rate, the interface temperature profile ahead of the steam front and the instantaneous velocity were derived independent of heat balance. The energy balance concept is used to define heat loss requirement for oil production at the rate predicted by the analytical model. An appropriate steam injection rate is needed to compensate the heat losses via: a) conduction to the overburden formation, b) heat loss to the oil zone as the steam interface progresses towards the production well, and c) 21

ACCEPTED MANUSCRIPT the heat required to maintain the growing steam chamber temperature and raise the temperature of the produced oil from the reservoir temperature to the steam temperature. The energy balance can be written in terms of the three energy consumption components (Reis, 1992): =

~

!

+

~

+

~ h

(31)

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%›¼

Because the shape of the steam chamber growth interface tapers towards the producing well (which behaves like a sink), the oil-steam interface is assumed to remain constant at the bottom and near the producing well. Therefore, the steam chamber does not develop below this boundary, resulting in negligible energy loss to

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the under-burden. However, for heterogeneous dip systems like fractured reservoirs, the under-burden heat loss must be incorporated (Kamari, Hemmati-Sarapardeh, Mohammadi, Kiasari, & Mohagheghian, 2015; %›¼ ,

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Kiasari, Hemmati-Sarapardeh, Mighani, Mohammadi, & Sola, 2014).

!,

and

h

indicate the

latent heat injection per unit length along the horizontal well; the enthalpy associated with the spreading steam zone; the enthalpy required to preheat the formation ahead of the steam zone; and the enthalpy loss to the overburden formation respectively. The required steam chamber enthalpy is obtained from the derivation of the enthalpy of the bulk (Reis, 1992). The bulk enthalpy is expressed as: ½ ∆+

=

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!

where ½ is the bulk per unit length,

(32)

is the formation heat capacity (

=?

). For the parabolic steam

=

‡¾¿À = ‡C

«∆+

‡»Á ‡C

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!

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chamber model, it can be stated that: =

∆+

‡»Á ‡C

=

∆+ -.

(33)

Considering Reis analysis, the total enthalpy of the preheated formation ahead of the steam chamber can be determined from the energy of the oil zone: (+ − +% ) @ Â

=

(34)

By combining the temperature distribution model and, integrating from 0 to 1 , it can be shown that: = El Á ÃÄÅ „ Â — ∆¥I …

(35)

22

ACCEPTED MANUSCRIPT The enthalpy required to preheat the tar sand ahead of the steam zone is given as: ~

=

‡¾Á ‡C

= El Á ÃÄÅ „ ‡C — ∆¥I

‡Â

…

Á = E ÃÄÅ

~ h

The heat loss rate to the overburden, = −Æ ‹b

ZM ‡¥(C`Ç) ‡È

. 1

(36)

is given by Reis as:

(37)

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~ h

— ∆¥I „

Assuming that a stationary boundary exists between the steam zone and the overburden in the direction É, the

+ ∗ = erfc (U) = •

‘

È

¹I(C`Ç)

‡ ˆerf (U)Š ‡Œ

= −

√Í

j `Π= \

‡¥ ∗ ‡Œ

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The derivative of the complimentary error function is given as:

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resultant temperature profile can be expressed in terms of the complimentary error function as:

(38)

(39)

Because, É is usually small, it can be shown that equation (37) approximates to: ∆+-. F

≅ 2

(C`Ç)I Í

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~ h

(40)

The overall heat equation is given as: ∆+. . -. +

—Á ∆¥I E ÃÄÅ „

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=

+2

∆+-. F

(C`Ç)I Í

(41)

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~ %›¼ (A)

2.8 Steam Oil Ratio (SOR)

The SOR is an important parameter for assessing the recovery efficiency of any steam project, defined as the amount of steam (in terms of cold water equivalent CWE) required to produce one barrel of oil. The total steam injection rate (CWE) is obtained, by converting the latent heat injection rate to the total steam injection rate and dividing the result by the oil production rate. This is expressed as:

23

ACCEPTED MANUSCRIPT ~

(A) =

– ¾±ÏÐ (C)

oÑ hM Ò



(42)

At any given time, t, during the steam injection, the SOR can be determined as:

3.0

¾M– (C) žK (C)

(43)

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#)*(A) =

Results and Discussion

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The models developed in this study for the oil production rate and equivalent steam injection rate were validated with an experimental data, numerical simulation, and field results. The project data used for the

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validation have proven to be reliable for SAGD evaluation. This study has been compared with the results of data obtained from the Underground Test Facility (UTF) phase B (Apos et al., 1997, 1999; Apos et al., 1994; O’Rourke, 1991), Chung and Butler experiment (Chung & Butler, 1988) and numerical simulation with

3.1.

Case 1

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CMG STARS simulator (Mozaffari et al., 2013; Sabeti et al., 2016).

Model Comparison with Chung and Butler Experiments

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The model is tested against the results of the oil production rate from the Chung and Butler experiment.

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Table 3 shows the physical properties for the experiment that are used in this study. Table 3: Experimental data obtained from Chung and Butler Experiment (Chung & Butler, 1988., Reis 1992; 1993) Properties

Value

Absolute permeability

2390 D (2360µm2)

Oil density

0.98gm/cc

24

Residual oil saturation

5%

Thermal diffusivity

0.0507m2/d

Porosity

39%

Model height

0.21m

Model thickness

0.03m

Oil viscosity at steam temperature

9m2/d

Constant “a”

0.4

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100%

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Initial oil saturation

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ACCEPTED MANUSCRIPT

3.6

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EP

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Constant “m”

Figure 7: Comparison of Model Results with Chung and Butler Experiments

25

ACCEPTED MANUSCRIPT The result from this case study is calculated based on the approach used by Reis. The measured rate was considerably lower than that predicted by the models of Reis and this Study when permeability was not adjusted, as both models predicted oil production rate of ~200gm/hr. Reis (Reis 1992) explained that the lower measured oil production rate was because oil and water along the steam zone interface were not

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completely segregated. This means that there was multiphase flow of both oil and water through intertwining pore channels along the interface. The intertwining of oil and water channels through the interface was also reflected in Butler and Chung experiment (Reis, 1992). Chung and Butler (1988) reported that the fluid produced consisted of free water and a water-in-oil emulsion. Because of this, an appropriate factor for

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relative permeability to oil was used instead of the absolute permeability (Reis, 1992, 1993). This model calculated the oil rate to capture the intertwining pore channels (Reis, 1992) by using the effective oil

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permeability rather than the absolute permeability to obtain a good match. Relative to Reis model that used relative permeability factor of 0.48, the proper match for this Study was achieved when a 0.4 compensation factor for relative permeability was used, which is same as what was assumed by Butler and Chung experiment (Butler et al, 1981., Reis, 1992). Figure 7 shows a declining production rate over time, which is

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in agreement with Butler’s data. However, Reis’ model predicted a constant rate at all times. It is evident that the model is a close match to the experimental results obtained by Butler et al. (R. M. Butler et al., 1981). In the theory of SAGD, the steam chamber usually grows elliptically until the cap rock is reached beyond

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which the growth pattern would change to a parabolic form. This could explain the incremental recovery realized in the new model compared to Reis estimations. In the SAGD process, the wellbore heat efficiency

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and formation heat capacity significantly affects the rate of heat loss in the early times (Gu et al., 2015; Hasan & Kabir, 2012; Ji, Zhong, Dong, & Chen, 2015). The curvature of the steam chamber interface is thus not accounted for by Reis’ inverted triangular shaped steam model which would lead to under-estimation at early time, and over-estimation at mid to late time, mainly due to the fact that it assumes a constant incremental steam chamber width (and production) up until infinite time. Reis (Reis, 1992)(1992) reported that for the experiment conducted by Chung and Butler (Chung & Butler, 1988), the steam zone reached the sidewalls after about one hour of steam injection. Therefore, the

26

ACCEPTED MANUSCRIPT comparison between the present model (and Reis model) and the experimental data is not valid after that time. The accuracy of the present model is compared with the Reis’ model for data-points predicted within the one-hour period of injection based on Butler’s experiment, using the statistical percentage absolute error, and

lower than the outcome by Reis model which gave up to 13% error. Table 4: Error Analysis of the Present Model and Reis’ model

0.19

139.90

0.33

144.26

0.50

140.83

0.68

136.29

0.84

140.64

Reis Model

This Study

Reis Model

This Study

125.65

144.52

10%

3%

125.65

143.57

13%

0%

125.65

142.13

11%

1%

125.65

140.26

8%

3%

125.65

138.53

11%

2%

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3.2 Case 2

Butler data

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Time (Hours)

Absolute Error (%)

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Rate (gm/hr)

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it is shown in Table 4. The maximum percentage error reported for the present model is 3% which is far

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Comparison of Oil Production with the CMG (STARS)

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CMG (STARS), a recognized commercial reservoir simulation software for advanced reservoir studies is used to test the model developed in the study using the data (Table 5) obtained from the numerical studies on an Iranian field (Mozaffari et al., 2013). Ninety days was used to preheat the space between well pairs for wells to be connected, thereafter the chamber expanded, and the well was ready for SAGD process. The assumption of most SAGD analytical or semi-analytical models’ development (including the presented model) is that communication is already established between the steam injector and the oil producer, therefore, production and injection performance before 90 days (although predicted) will not be considered while evaluating the performance of the developed models. The plot in Figure 8 reveals that the decline in oil 27

ACCEPTED MANUSCRIPT rate was captured by the developed model compared to Reis’ (constant oil rate). The model also shows better agreement with the numerical simulator results. Table 5: Properties of the Iranian Reservoir: Iranian oil reservoir (Mozaffari et al., 2013; Sabeti et

Value

Absolute permeability

√0.85 × 1 µm2

Oil density

0.98gm/cc

Thermal diffusivity

5.86×10-7 m2/s

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Properties

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al., 2016).

0.25

Porosity×saturation (φ×∆S) Height of bed

20m

Formation width

50m

43oC

Steam temperature at surface

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Reservoir temperature

227oC 6.55*10-6 m2/s

Oil density

961kg/m3

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Oil viscosity at steam temperature

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Coefficient of velocity “a”

0.4

Steam quality

0.7

Formation heat capacity

1955J/kgoC

Constant “m”

3.6

28

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Time (Days)

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ACCEPTED MANUSCRIPT

Figure 8: Comparison of Oil Production and Steam Injection with the CMG STARS Simulation

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From the production profile, the present model has better match with the linear and the radial models proposed by Reis (Reis, 1992, 1993). Although Reis’ radial model varies with time, it is negatively correlated with the simulator results, as it predicts an increasing oil rate instead of an overall declining oil

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rate. Similarly, Reis’ linear oil rate model does not show alteration of oil rates with time. In spite of this, the presented model over-predicted oil and steam rates at early times and at late time (beyond 1200 days) from

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the result of the numerical simulator. The discrepancy at early times, as mentioned earlier is because the model assumes that the injection has been strong enough to have established communication between the injector well and the producer well, which required about 90 days to be fully developed. The discrepancy at late time is suspected to result from the simplification of the heat transfer and energy balance equations in the model development compared to the numerical simulator with smaller grids. And the assumption of thin transition zone used in the model development. However, the match is within good accuracy with the lab, field and numerical data.

29

ACCEPTED MANUSCRIPT From the steam injection profile in Figure 8, the developed model had a very close match with the outcome of the numerical simulator, while Reis’ model was overly inconsistent. 3.3 Case 3

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Comparison of Oil Production with the UTF Phase B AOSTRA’s Underground Test Facility (UTF) is one of the most successful SAGD projects. The UTF is located 70km northwest of Fort McMurray in Alberta, Canada (apos et al., 1999; Li, Chalaturnyk, & Tan, 2006; O’Rourke, 1991) Its first SAGD was first done in 1998. Following the success reached when

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producing from pair of wells in phase A, phase B was initiated by drilling three pair of wells each of which having a length of 500m and located 70m away from each other in a depth of 22m in 1992. The SAGD

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process was started by continuous steam injection in 1993, lasting almost a year. During this period, the production rate of wells increased before reaching a steady decline in the daily production rate (Yee & Stroich, 2004) Following the reservoir parameters presented in the Table 6, the Reis’ model and plot was used to calibrate the data-set which shows a well length of about 630m, instead of 500m specified. The

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adjusted value was due to possible error originating from any of the petro-physical parameters. The plot

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generated is presented in figure 9:

30

ACCEPTED MANUSCRIPT

Table 6: Experimental Data Obtained from Chung and Butler Experiment (Chung & Butler, 1988;

Absolute permeability

7 µm2

Oil density

0.98gm/cc

Thermal diffusivity

6 * 10-7m2/s

Porosity-saturation product

0.21

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Value

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Properties

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Sabeti et al., 2016)

Height of bed

21m

Formation width

500m 7oC

Reservoir temperature

220oC

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Steam temperature at surface

7.95*10-6 m2/s

Oil density

882kg/m3

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Oil viscosity at steam temperature

0.4

Steam quality

0.95

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Coefficient of velocity “a”

Formation heat capacity

1955kJ/m3oC

Constant “m”

4

31

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Figure 9: Comparison of Oil Production Rate for this Study with UTF Results

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Figure 9 demonstrates that the model performed better than both the Reis’ linear and radial models. It shows a more accurate match to the experimental UTF pilot project and captures a declining oil rate with respect to time. The over-prediction of the model just after the one year that was used to establish communication

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between the producers and the injector shows that communication is yet to be fully established as targeted. In practice for most SAGD pre-heating stage, the two adjacent well drilled are preliminarily for

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interchangeable injection-production patter until communication is established between them, then one of the wells is converted to only production. Shijun (Shijun et al., 2016; You, Yoon, & Lee, 2012) experiment established the existence of an interlayer between the injector and producer, which inhibits the oil flow and hinders establishing the needed communication. They recommended that steam must be injected on both ends to overcome this. When the steam injection pressures and steam temperature are increased, steam or oil breakthrough at the producer may be shortened (Chan, Chen, & Dong, 2012; Mozaffari et al., 2013). The heat propagation and oil flow parameters may be obtained under unsteady state condition by determining the 32

ACCEPTED MANUSCRIPT time for the steam to breakthrough at the producer. Although this aspect is not covered in this study. It is therefore conclusive to state that the model is best suited for using beyond the critical time required to establish full communication between the wells. Figure 9 shows that the new model developed predicts oil production with improved accuracy. Although Reis’ radial model varies with time, it is negatively correlated

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to the experimental results. The linear model on the other hand, highly over-predicts oil production rate. The reason for the over-prediction can be explained by the chart in Figure A-1, which shows the progression of the steam chamber interface over time for the UTF Phase B project case. This reveals that Reis’ oil rate will ]

4.

Conclusion

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because of larger steam width reached by the Reis’ model.

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exceed this study’s model rate at an early time (when (A) > H ) and will continue same until infinite time

1. A new analytical steam chamber growth model using a concave parabola was presented for evaluating SAGD operations. 2.

The contribution of this study is a proposed improvement on the current analytical models used to

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predict the oil rate during SAGD operations. The improvements are made in a better representation of the steam chamber boundary shape and its evolution as a function of time. 3. The new steam chamber growth model developed from a diminishing concave parabola was based on the results of the Underground Test Facility, the laboratory experiment data and the numerical

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studies from the various SAGD operational fields. 4. The temperature of the steam interface profile during the steam injection is estimated using the concept of moving solid. Based on the expansion of the injected steam, the oil production rate is

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evaluated.

5. The new model accounts for the departure of the interface from a rigid linear interface to a curve and depicts time dependency, which are omitted in the previous models for SAGD evaluation 6. The results of the model gave high accuracy compared with experimental data from the Cheng and Butler’s study, a CMG STARS simulator and the results of the UTF phase B projects. 7. The developed model captures the overall declining oil rate corresponding to the results of these experiments as demonstrated unlike in previous models like the Reis’ linear model. 8. Even with the modified Reis’ (radial) model including rate variation with time, it was negatively correlated with the simulator results, as it predicts an increasing oil rate instead of an overall declining oil rate. 33

ACCEPTED MANUSCRIPT

5.

Acknowledgement

The authors would like to thank the financial, academic and technical support of process engineering

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department, Memorial University of Newfoundland. The authors would also like to thank the school of graduate studies, Memorial University of Newfoundland, Research and Development Cooperation of

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Newfoundland and Labrador and Statoil Canada Limited for funding this research.

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Irani, M., & Ghannadi, S., 2013, Understanding the Heat-Transfer Mechanism in the Steam-Assisted

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Appendix A

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145. doi:10.1016/j.petrol.2012.11.010

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Figure A-1 Steam Chamber Interface Advancement with Time

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The chart shows the propagation of the steam chamber interface over time, based on the UTF case presented in the main body for this model and Reis model. A hasty look at the resulting oil rate model developed by this study as presented in Eqn. 20 would suggest a higher rate to be predicted than this model over Reis’, which is however the opposite of the outcome. The over-estimation of Reis model can be seen from the

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figure above, as it assumes that the interface is propagating linearly, as opposed to a parabolic interface advancement, shown by this model.

Appendix B: Derivation of the Steam Interface Length (Refer to Figure 3)

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The derivative of the interface function over the U -direction is given as:

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T ~ (U) = 8b U + 8V

(B1)

The resulting function is obtained by integration wrt U as: Œ

‹b

T ~ (U) = ‹b (8b + 8V ) U

T(U) =

Œ

EŽ Œ \

+ 8V U + 8

(B2)

The length, of the interface can be obtained as follows:

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ACCEPTED MANUSCRIPT = U F1 + X

‡™(Œ) [ ‡Œ

V/

(U) = ‹b p1 + ØT ~ (U)Ù q Œ



= ‹b ¹1 + (8b U + 8V ) ∙ U

U

Œ

Let 8b U + 8V = sinh Ú, at U = 0, Úb = sinh`V (8V )

⇒ U =

(B4) 1 cosh Ú Ú 8b

⇒ (Ú) = ‹Û X¹1 + sinh Ú[ ∙ cosh Ú Ú E Û

V

Ž

V

EŽ

Ú

p sinh 2Ú + Úq V

= Û

=

ÛŽ

V Û V ‹ (cosh 2Ú EŽ ÛŽ V

EŽ

ˆ(sinh Ú cosh Ú + Ú)ŠÛÛ Ž

Taking Limits V psinh Ú ¹(1 + EŽ

=

V

EŽ

sinh Ú) + Úq

Û

ÛŽ

psinh Ú (1 + sinh Ú)V/ − sinh Úb (1 + sinh Úb )\ + (Ú − Úb )q

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(Ú) =

+ 1) Ú

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=

V Û ‹ cosh EŽ ÛŽ

Ž

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=

(B3)

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= U + ˆ T(U)Š ⇒

•

(B5)

By replacing Ú and Úb with Ú = sinh`V (8b U + 8V ) and Úb = sinh`V(8V ) EŽ

p(8b U + 8V )¹1 + (8b U + 8V ) − 8V ¹1 + 8V + sinh`V (8b U + 8V ) − sinh`V (8V )q(B6)

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V

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(U) =

The function T(U) models the steam chamber interface for the SAGD process which is initiated once communication is established between the upper injection well and the lower horizontal producer. We assume that the injected steam terminates in the producer well along the vertical direction but continues to advance in the lateral direction as steam injection continues. From the figure 3 and following the nature of symmetrical parabola, the following boundary condition imposed

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, U = 0 â Ý T(U) = 0 , U = 1 Þ á Ý Ý ÜT(U) = 0 , U = −1 à ß Ý

T(U) =

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(B7)

Solving equation (B2) following the conditions in equation (B7), we obtain Y

ZM \

, 0 and −

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8b , 8V and 8 are obtained as

respectively and substituting into equ. (B2) gives:

T(U) = ãX [ − 1ä Œ

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(B8)

The length of the steam chamber interface, L, Equation (B6) can be re-written in terms of the constants

(U) =

ZM \ YŒ F1 + å ]Y ZM \

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8 , 8V , 8 as:

X Z \ [ + sinh`V X Z \ [æ YŒ

YŒ

M

(U) = = å (U) ∙ F1 + X V

YΠ[ ZM\

M

+

ZM \ YŒ sinh`V X \ [æ ZM ]Y

is evaluated at maximum horizontal reach i.e at U = 1

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The length of the steam chamber interface,

(1 ) = 1 F1 + X [ + M sinh`V X [ Z Z ]Y V

Y M

(B9)

Z\

Y M

(B10)

Appendix C: Derivation of the Approximate Saturation Change in the Steam Chamber

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Following the underlying assumptions of the developed model, the Steam chamber of areal extent ‰ ,

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comprises of a thin transition zone with areal extent ‰ ¥ , and average saturation #̅ ¥ , at the front of the steam chamber, with the remainder of the steam chamber at residual oil saturation # ' .

The change in saturation of the steam chamber is therefore approximated following the expression: ‰. ∆# (A) = ‰ ¥ ∙ ∆#

¥

+ (‰ − ‰ ¥ ) ∙ ∆#

(C1)

= ‰ ¥ ∙ (# % − #̅ ¥ ) + (‰ − ‰ ¥ ) ∙ (# % − # ' ) = ‰ ∙ (# % − # ' ) − ‰ ¥ ∙ (#̅

¥

− # ')

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ACCEPTED MANUSCRIPT ∆# (A) = (# % − # ' ) p1 −

Because ‰ ¥ ≪ ‰, and (#̅

¥

̅ ` Kd ) ¦è ( Kè q ¦ ( K± ` Kd )

(C2)

− # ' ) ≪ (# % − # ' ), the fractional component of the above expression can be

neglected. ∆# (A) ≅ (# % − # ' )

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(C3)

The resultant approximation is necessary because of the difficulty [if not the impossibility] of evaluating the exact representative transition zone oil saturation. However, this approximation will lead to an over-

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estimation of the oil recovery, especially in formations with varying flow zonation.

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a. Concave parabola was used to model Steam Chamber propagation based on the Underground Test Facility (UTF, AOSTRA) experiment and the numerical studies in various SAGD operational fields.

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b. The temperature of the steam interface profile during steam injection is estimated using the concept of moving solid, and based on the expansion of the injected steam.

c. The new model accounts for the departure of the interface from a linear interface to a curve and

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accounts for the time dependency oil production rate.

d. The results of the model are compared with experimental data of the Cheng and Butler (1988)

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experiments, a CMG STARS simulator results and the UTF phase B projects. e. Developed model results clearly captures the overall declining oil rate corresponding to the

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results of these experiments.