Theoretical and Experimental Study on Cementing Displacement Interface for Highly Deviated Wells

Abstract

An effective drilling fluid removal is necessary to achieve an efficient cementing in oil and gas industry, i.e., it is ideal that all the drilling fluid is displaced by the cement slurry. The displacement efficiency is closely related to the stability and development of the displacing interface between the cementing slurry and drilling fluid. Thus, an effective cementing requires a validated theoretical model to describe the displacing interface to guide cementing applications, especially for highly deviated wells. The current studies suffer from a lack of experimental validation for proposed models. In this study, a theoretical model of cementing interfacial displacement in eccentric annulus is established. An experimental study is conducted to examine effects of well inclination, eccentricity and fluid properties on the stability of displacement interface to verify the theoretical model. The model is found to well describe the interface in the eccentric annulus, and it is applicable to the wellbore annulus with different inclination angles. The results show that: the displacement interface gradually extends (i.e., length is increased) with the increase of well inclination; the cement displacement effect became worse with deviation angle under the same injection and replacement conditions. Increasing the apparent viscosity of cement slurry is beneficial to improve the stability of displacement interface. In highly deviated wells, a certain casing eccentricity can inhibit the penetration of cement slurry in the wide gap of the low side of the annulus, which is conducive to maintaining the stability of the displacement interface.

1. Introduction

Successful cementing requires a sufficient displacement of drilling fluid by cement slurry in oil and gas industry. Otherwise, the retained drilling fluid (i.e., drilling fluid not completely displaced by cement slurry) will adversely affect the quality of cement stone. In this case, the service life of the oil well can be shortened, and the oil, gas, and water layers cannot be effectively blocked, resulting in interlayer sealing failure and cementing failure [1,2,3]. A typical design for cementing application consists of the following aspects. First, there is a preference of turbulent regimes for cementing displacement. If full turbulence condition cannot be achieved, then a series of conditions should be satisfied: (i) each displacing fluid should be heavier and more viscous than the fluid it displaces; (ii) the pressure gradient should be high enough to displace fluid on the narrow side of the annulus. In addition to these conditions, the displacement efficiency is closely related to the stability and development of the displacing interface between the cementing slurry and drilling fluid [4]. A high displacement efficiency is especially difficult for highly deviated wells due to the instability of the displacement interface strengthened by the asymmetric geometry and eccentricity of wells. Thus, an effective cementing requires a validated theoretical model to describe the displacing interface to guide cementing applications, especially for highly deviated wells.

The development of the displacement interface in the annulus has several different forms as shown in Figure 1 [3]. Among them, the “steady-state” displacement interface can effectively reduce the pollution of drilling fluid to cement slurry, prevent drilling fluid from channeling, and improve the displacement efficiency. Under actual conditions with certain well deviation angle and casing eccentricity, optimizing the performance parameters (e.g., rheological properties and density of cement slurry and drilling fluid, injection displacement) of cement slurry and drilling fluid to obtain a “steady-state” displacement interface is necessary for ensuring cementing quality.

Highly deviated wells and horizontal wells yield important applications in the exploration and development of oil and gas resources, especially for offshore oil and gas [5]. However, the prominent cement displacement problems such as serious casing eccentricity and displacement interface instability restricts their application [1,2]. At present, the design for cement injection and adjustment of fluid property mainly refer to the well construction experience from vertical wells or the qualitative results of indoor experiments [6,7]. However, there are some differences between cement displacement of highly deviated wells (or horizontal wells) and that of vertical wells [8,9]. Researchers have made some efforts to develop theoretical models to describe the displacing interface, most of which are conducted for vertical wells. A minority of authors made some efforts into the theoretical models for deviated wells. For example, Frigaard et al. [4] developed a new model for displacement of cementing fluids along eccentric annuli; they stated that, for certain combinations of the physical properties, it is possible to yield a steady state displacement front. In addition, it is possible to obtain an analytical expression for the shape of the displacement front. Pelipenko et al. [10] tackled the question of whether or not stable displacements can be realized for a given set of cementing parameters by developing a new theoretical model for inclined wells. Correspondingly, they have obtained explicit analytic expressions for steady-state solutions. Moyers-Gonzalez et al. [11] gave explanations for instability analysis of the two-layer eccentric annular Hele–Shaw flows by adopting a transient version of the Hele–Shaw approach, where fluid acceleration terms are retained. They demonstrated that negative azimuthal buoyancy gradients are typically stabilized in deviated wells, but buoyancy may exert a destabilizing effect through axial buoyancy forces which affects the base-flow interfacial velocity. In addition, Moyers-Gonzalez et al. [12] used a similar method to quantify the occurrence of static mud channels (it is a main reason for cementing failure) in primary cementing. They provided a simple semi-analytic expression for the maximal volume of residual fluid left behind in the annulus. Although there have been several studies for theoretical modelling of the displacing interface in inclined wells, rare experiments have been carried out to verify the theoretical models, which largely weakens the persuasiveness of the theoretical models. Thus, experiments need to be performed to provide efficient guidance for cementing in inclined wells [13].

Therefore, the author takes the laminar flow displacement for cementing in the eccentric annulus of highly deviated wells as the research object, and the theoretical model for the displacement interface is established. A novel experiment is designed to verify the proposed theoretical model. This paper provides a certain reference and a theoretical as well as an experimental basis for the design of displacement parameters and of fluid properties, which has important scientific significance and application value for highly deviated wells.

2. Theoretical Model

2.1. Theoretical Basis

The flow characteristics of non Newtonian fluid in wellbore annulus are the theoretical basis for studying the displacement of two-phase fluids. The plate flow model is often used to analyze the flow of non-Newtonian fluid in the wellbore annulus, and relevant studies for the displacement of drilling fluid by cement slurry have been published [14,15,16,17,18]. The plate flow model assumes the flow in the annulus as a one-dimensional flow along the axial direction of the wellbore. In the annulus of a vertical well, the gravity of the fluid is consistent with the direction of fluid flow or cement slurry displacement, and the plate flow model has good applicability. However, in the annulus of highly deviated wells or horizontal wells, the component of fluid gravity along the axial direction of the wellbore is very small, and the component in other directions of the annulus cannot be ignored [19,20].

The Hele–Shaw model studies the two-dimensional flow characteristics of fluid between two infinite plates with a small spacing. Bittleston, Pelipenko and Frigaard [19,20] studied the fluid flow in the wellbore annulus, and they selected the characteristic quantity and defined the dimensionless variables. Thus, they obtained the modified Hele–Shaw model for describing fluid velocity and flow function fraction in the annulus [4,20]. Based on the research of Bittleston et al. [19,20], the author analyzed the characteristics of fluid velocity profile between two plates at a certain circumferential angle of the annulus, compared the velocity difference at the wide and narrow gap of the eccentric annulus, calculated the distribution of the flow function along the entire eccentric annulus, and discussed the effects of annulus eccentricity, fluid properties, and other factors on the flow characteristics in fluid annulus [9]. The basic model used is written as follows:

$$\{\begin{array}{l}\nabla \cdot \overrightarrow{S}=-\overrightarrow{f}\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \sigma >0\\ |\nabla \psi |=0\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \sigma \le 0\end{array}$$

(1)

where:

$$\overrightarrow{S}=\frac{\sigma (\left|\nabla \psi \right|)+{\tau }_Y/H}{\left|\nabla \psi \right|}\nabla \psi$$

(2)

$$\overrightarrow{f}=\nabla \cdot (\frac{\rho \cos \beta }{S{t}^{\ast }},\frac{\rho \sin \beta \sin \pi \varphi }{S{t}^{\ast }})$$

(3)

$$\left|\nabla \psi \right|=\{\begin{array}{l}0\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \sigma \le 0\\ \frac{H^{m+2}}{{\kappa }^m(m+2)}(\sigma +\frac{m+2}{m+1}\frac{{\tau }_Y}{H})\frac{{\sigma }^{m+1}}{{(\sigma +{\tau }_Y/H)}^2}\cdot \cdot \cdot \cdot \sigma >0\end{array}$$

(4)

$$\frac{\partial \psi }{\partial \varphi }=H\overline{w},\frac{\partial \psi }{\partial \xi }=-H\overline{v}$$

(5)

The details regarding the basic model can also be found in Pelipenko et al. [10]. In the equations mentioned above, $\nabla$ is the Hamiltonian operator, $(\varphi ,\xi )$ is the dimensionless coordinate system established for the axial and circumferential directions of the annulus; $\psi$ is stream function; $(\overline{v},\overline{w})$ is the average velocity of the fluid in the direction of $(\varphi ,\xi )$; and $H$ represents the dimensionless half width of the annulus. In the direction of $\xi$, there is $H=1+e\cdot \cos \pi \varphi$, where $e$ is eccentricity, while $\varphi =0$ and $\varphi =1$ represent wide clearance and narrow clearance in eccentric annulus, respectively.; $\overrightarrow{S}$ represents the flow characteristics of a certain Hershel–Bulkley fluid along a rectangular groove under a certain pressure gradient. Formula (1) in the model denotes the Poiseuille flow in a rectangular groove with a spacing of $2H$. The fluid does not flow when the pressure gradient is less than the yield value of the fluid, where $\left|\nabla \psi \right|=0$; $\sigma$ represents the force used to drive the fluid after overcoming the fluid yield stress, i.e.,

$$\sigma =G-{\tau }_Y/H$$

(6)

$$G={[{(\partial p/\partial \varphi -\rho /S{t}^{\ast }\sin \beta \sin \pi \varphi )}^2+{(\partial p/\partial \xi +\rho /S{t}^{\ast }\cos \beta )}^2]}^{1/2}$$

(7)

where ${\tau }_Y$ denotes the dimensionless fluid yield stress; $m$ is the reciprocal of fluid fluidity index; $\kappa$ is the dimensionless consistency coefficient; $\beta$ is the inclination angle of the well; $\rho$ is the dimensionless density of fluid. For the dimensionless factor $S{t}^{\ast }$, which is composed of characteristic quantities obtained in the dimensionless treatment of fluid motion equation, there is $S{t}^{\ast }={\tau }^{\ast }/({\rho }^{\ast }g{d}^{\ast })$; select a large fluid density as the reference density ${\rho }^{\ast }$ for cementing displacement; $g$ is the acceleration of gravity. Under the average shear rate, the drilling fluid and cement slurry are subject to shear stress, and the larger one of the two shear stress is selected as ${\tau }^{\ast }$; $d^{\ast }$ is half of the average annulus clearance of the whole well section.

2.2. Model Development for Displacement Interface

Targeted at cementing displacement in eccentric annulus of highly deviated wells, the authors investigate the continuity conditions of two-phase fluid at the displacement interface and establish a mathematical model for cementing a displacement interface. The contact and replacement process between two different non Newtonian fluids is very complex. In addition to the non Newtonian properties and thixotropy of cement slurry and drilling fluid, it is more important that there are extremely complex physical, chemical and mechanical effects on the displacement interface. For the convenience of theoretical research, the following analysis and assumptions are made:

(1)

The displacement interface completely separates the drilling fluid and cement slurry, and there is no obvious mixing and diffusion between them.

(2)

The shape of the displacement interface gradually changes from unstable to stable ones after the cement slurry is pumped into the annulus. The length of annulus is much larger than the extension length of displacement interface, so the object of study is determined as the stable displacement interface after long enough displacement time.

(3)

Take a wellbore annulus section with a length of $L$. In the annulus area of this section, the borehole is regular; the well deviation angle and casing eccentricity are constant (i.e., $\beta$ and $e$ are constant), and the flow rate of cement slurry in annulus is constant.

(4)

In the practice of on-site cementing, especially in highly deviated wells and horizontal wells, it is difficult for the flow state of cement slurry to achieve turbulent flow. In addition to the pressure bearing capacity of on-site equipment, it is easy for turbulent displacement to cause formation fracture. Thus, on-site cementing displacement is still dominated by laminar flow. Thus, the displacement interface model studied by the author is aimed at the situation of laminar flow.

Take the entire annulus zone $\mathrm{\Pi }=(\xi ,\varphi )=(0,1)\times (0,L)$, as shown in Figure 2. The cementation displacement interface divides the area $\mathrm{\Pi }$ into two parts, i.e., ${\mathrm{\Pi }}_1$ and ${\mathrm{\Pi }}_2$, representing the annulus area occupied by cement slurry and drilling fluid, respectively. $(\xi ,\varphi )$ denotes the fixed coordinate system, and the coordinate origin is established at the bottom of the well; $(z,\varphi )$ is the reference coordinate system and advances with the displacement interface. Assuming that the flow rate of cement slurry in the annulus is constant, the relative velocity of $(\xi ,\varphi )$ and $(z,\varphi )$ is 1 after parameter non-dimensionalization, as shown in Figure 2. Based on the basic assumption (1) and $\sigma >0$, the stream function distribution is expressed as follows for zone ${\mathrm{\Pi }}_1$ and zone ${\mathrm{\Pi }}_2$, respectively [21]:

$$\nabla \cdot {\overrightarrow{S}}_1=0,(\varphi ,\xi )\in {\mathrm{\Pi }}_1,\nabla \cdot {\overrightarrow{S}}_2=0,(\varphi ,\xi )\in {\mathrm{\Pi }}_2$$

(8)

After the cementing displacement interface stabilizes, and under the fixed coordinate system, the displacement interface $h$ is only a function of $\varphi$ and $t$. Then, the equation of motion for the displacement interface is obtained as follows:

$$\frac{\partial h}{\partial t}+\overline{v}\frac{\partial h}{\partial \varphi }=\overline{w}$$

(9)

At a certain time $t_0$ after the displacement interface is stable, the displacement interface is only a function of the circumferential angle $\varphi$. At the displacement interface, the fluid pressure in the annulus is continuous [10,21]. Therefore, the directional derivative of pressure $p$ along the tangent direction of the displacement interface satisfies:

$${[\frac{\partial p}{\partial \varphi }+\frac{\partial h}{\partial \varphi }\frac{\partial p}{\partial \xi }]}_1^2=0$$

(10)

Derived from literature [4,20], it can be obtained that:

$$\frac{\partial p}{\partial \varphi }=\frac{\sigma (\left|\nabla \psi \right|)+{\tau }_Y/H}{\left|\nabla \psi \right|}\frac{\partial \psi }{\partial \xi }+\frac{\rho \sin \beta \sin \pi \varphi }{S{t}^{\ast }} \frac{\partial p}{\partial \xi }=\frac{\sigma (\left|\nabla \psi \right|)+{\tau }_Y/H}{\left|\nabla \psi \right|}\frac{\partial \psi }{\partial \varphi }-\frac{\rho \cos \beta }{S{t}^{\ast }}$$

(11)

Substitute Equation (11) into Equation (10) to obtain:

$${[\frac{{\sigma }_k+{\tau }_{k,Y}/H}{\left|\nabla \psi \right|}(\frac{\partial \psi }{\partial \xi }-\frac{\partial h}{\partial \varphi }\frac{\partial \psi }{\partial \varphi })+\frac{{\rho }_k\sin \beta \sin \pi \varphi }{S{t}^{\ast }}-\frac{{\rho }_k\cos \beta }{S{t}^{\ast }}\frac{\partial h}{\partial \varphi }]}_1^2=0$$

(12)

where $k=1$ refers to cement slurry; $k=2$ represents drilling fluid. Formula (10) includes cementation displacement interface $h(\varphi ,t)$ and ${\psi }_k$, which can be obtained from Formula (6), and the solution conditions required [10,21]) are:

$$\psi (0,\xi ,t)=0 \psi (1,\xi ,t)=1$$

(13)

$$\frac{\partial \psi }{\partial \xi }(\varphi ,L,t)=0 \frac{\partial \psi }{\partial \xi }(\varphi ,0,t)=0$$

(14)

Equation (13) is obtained based on the definition Equation (5) of the stream function and the symmetry of the distribution of the stream function in the annulus. Assuming that at the inlet 0 and outlet $L$ of the annulus, cement displacement only occurs in the axial direction, Formula (12) is obtained.

In order to solve the cement displacement interface model, a reference coordinate system is established. The mutual transformation relationship of relevant variables between the two coordinate systems $(\xi ,\varphi )$ and $(z,\varphi )$ are written as follows:

(1)

The displacement interface expression for $(z,\varphi )$: $z=g(\varphi ,t)=h(\varphi ,t)-t$;

(2)

The stream function $\mathrm{\Phi }$ for $(z,\varphi )$ and the stream function $\psi$ for $(\xi ,\varphi )$ satisfy the following relations: $\psi =\mathrm{\Phi }+\varphi +e/\pi \cdot \sin \pi \varphi$. This is due to the following relations: $\partial \mathrm{\Phi }/\partial \varphi =0$, $\partial \psi /\partial \varphi =H=1+e\cos \pi \varphi$;

(3)

The motion equation for $(z,\varphi )$ is: $(1+e\cos \pi \varphi )\frac{\partial g}{\partial t}-\frac{\partial \mathrm{\Phi }}{\partial z}\frac{\partial g}{\partial \varphi }=\frac{\partial \mathrm{\Phi }}{\partial \varphi }$;

(4)

The continuity equation for $(z,\varphi )$ is:

$${\left\{\frac{{\sigma }_k+{\tau }_{k,Y}/H}{\left|\nabla (\mathrm{\Phi }+\varphi +\frac{e}{\pi }\sin \pi \varphi )\right|}\left[\frac{\partial \mathrm{\Phi }}{\partial z}-\frac{\partial g}{\partial \varphi }(\frac{\partial \mathrm{\Phi }}{\partial \varphi }+1+e\cos \pi \varphi )\right]+\frac{{\rho }_k\sin \beta \sin \pi \varphi }{S{t}^{\ast }}-\frac{{\rho }_k\cos \beta }{S{t}^{\ast }}\frac{\partial g}{\partial \varphi }\right\}}_1^2=0$$

(15)

(5)

The boundary condition for $(z,\varphi )$:$\mathrm{\Phi }(0,z)=\mathrm{\Phi }(1,z)=0$; $\frac{\partial \mathrm{\Phi }}{\partial z}(\varphi ,0)=\frac{\partial \mathrm{\Phi }}{\partial z}(\varphi ,Z)=0$.

2.3. Solution of Model

After the cement displacement interface is stable, it meets $\partial g/\partial t=0$. In addition, the fact that the displacement interface between cement slurry and drilling fluid coincides with a flow function isoline can be obtained based on $(1+e\cos \pi \varphi )\frac{\partial g}{\partial t}-\frac{\partial \mathrm{\Phi }}{\partial z}\frac{\partial g}{\partial \varphi }=\frac{\partial \mathrm{\Phi }}{\partial \varphi }$; from the boundary conditions under $(z,\varphi )$, the contour of this stream function is $\mathrm{\Phi }=0$.

For cemented displacement in concentric annulus, substitute $\mathrm{\Phi }=0$ and $e=0$ into model (6), and the stream function distribution equations for zone ${\mathrm{\Pi }}_1$ and ${\mathrm{\Pi }}_2$ can be satisfied automatically [10,21]. Therefore, the calculation of cementation displacement interface in concentric annulus only needs to meet Equation (15):

$$[{\sigma }_1(1,1)+{\tau }_{1,Y}-{\sigma }_2(1,1)-{\tau }_{2,Y}]\frac{\partial g_0}{\partial \varphi }-\frac{({\rho }_2-{\rho }_1)\cos \beta }{S{t}^{\ast }}\frac{\partial g_0}{\partial \varphi }=\frac{({\rho }_1-{\rho }_2)\sin \beta \sin \pi \varphi }{S{t}^{\ast }}$$

(16)

where $g_0$ denotes displacement interface for concentric annulus cementing for $(z,\varphi )$; when $\sigma >0$, and $\sigma =\sigma (\left|\nabla \psi \right|,H)$ can be obtained based on Equation (4); then, ${\chi }_k(1,1)$ can be solved inversely by Equation (4). Equation (16) can easily calculate the cement displacement interface in the concentric annulus at different well inclination angles, and is the basis for solving the displacement interface in this study.

A perturbation method is used to calculate the cement displacement interface in the eccentric annulus. To put it simply, the solution of the differential equation is expanded into a power series and represented by the first few terms of the asymptotic expansion. According to the established interface model for cement slurry displacement, the selected small parameter is eccentricity $e$, the expanded parameters are stream function $\mathrm{\Phi }$, and eccentric annulus displacement interface $g$ for $(z,\mathrm{\varphi })$. The Expansion form for $\mathrm{\Phi }$ and $g$ are:

$$\mathrm{\Phi }(z,\varphi )={\mathrm{\Phi }}_0(z,\varphi )+e{\mathrm{\Phi }}_1(z,\varphi )+e^2{\mathrm{\Phi }}_2(z,\varphi )+\dots \dots g(\varphi )=g_0(\varphi )+e{g}_1(\varphi )+e^2{g}_2(\varphi )+\dots \dots \dots$$

(17)

where $\mathrm{\Phi }(z,\varphi )$ and $g(\varphi )$ are the expressions of flow function and displacement interface in eccentric annulus under reference coordinates, respectively; ${\mathrm{\Phi }}_0(z,\varphi )$ and $g_0(\varphi )$ are the expressions of flow function and displacement interface when casing is centered under reference coordinates, respectively.

According to the corresponding relationship of related variables between $(\xi ,\varphi )$ and $(z,\varphi )$, and substitute expanded form of $\mathrm{\Phi }$ and $g$ into Equations (6) and (15), and with the help of Taylor formula, the results are obtained as follows:

(1)

Stream function expressions for zone ${\mathrm{\Pi }}_1$ and ${\mathrm{\Pi }}_2$ are converted into:

$$\frac{{\partial }^2{\mathrm{\Phi }}_1}{\partial {\varphi }^2}{\sigma }_1’(1,1)+[{\sigma }_1(1,1)+{\tau }_{1,Y}]\frac{{\partial }^2{\mathrm{\Phi }}_1}{\partial z^2}=[{\sigma }_1’(1,1)+{\sigma }_{1,H}(1,1)-{\tau }_{1,y}]\cdot \pi \sin (\pi \varphi )\hspace{1em}(\varphi ,\xi )\in {\mathrm{\Omega }}_1$$

(18)

$$\frac{{\partial }^2{\mathrm{\Phi }}_1}{\partial {\varphi }^2}{\sigma }_2’(1,1)+[{\sigma }_2(1,1)+{\tau }_{2,Y}]\frac{{\partial }^2{\mathrm{\Phi }}_1}{\partial z^2}=[{\sigma }_2’(1,1)+{\sigma }_{2,H}(1,1)-{\tau }_{2,y}]\cdot \pi \sin (\pi \varphi )\hspace{1em}(\varphi ,\xi )\in {\mathrm{\Omega }}_2$$

(19)

(2)

The continuity equation at the displacement interface is transformed into:

$$\begin{array}{l}{\left[\frac{{\rho }_k\cos \beta }{S{t}^{\ast }}\frac{\partial (g_0+e\cdot g_1)}{\partial \varphi }-\frac{{\rho }_k\sin \beta \sin \pi \varphi }{S{t}^{\ast }}\right]}_1^2=\{\frac{[{\sigma }_k(1,1)+{\tau }_{k,Y}]\cdot e\frac{\partial {\mathrm{\Phi }}_1}{\partial z}}{1+e\cos \pi \varphi +e\frac{\partial {\mathrm{\Phi }}_1}{\partial \varphi }}-\\ \left\{{\sigma }_k(1,1)+{\tau }_{k,Y}+e[(\cos \pi \varphi +\frac{\partial {\mathrm{\Phi }}_1}{\partial \varphi }){\sigma }_k’(1,1)+\cos \pi \varphi \cdot {\sigma }_{H,k}(1,1)-{\tau }_{k,Y}\cos \pi \varphi ]\right\}\frac{\partial (g_0+e\cdot g_1)}{\partial \varphi }{\}}_1^2\end{array}$$

(20)

(3)

Transformation form of definite solution condition is: ${\mathrm{\Phi }}_1(0,z)={\mathrm{\Phi }}_1(1,z)=0$; $\frac{\partial {\mathrm{\Phi }}_1}{\partial z}(\varphi ,0)=\frac{\partial {\mathrm{\Phi }}_1}{\partial z}(\varphi ,z)=0$.

Where $\sigma ’$ and ${\sigma }_H$ denote partial derivative of $\sigma$ to $|\nabla \psi |$ and $H$, respectively.

To obtain the cementing displacement interface in eccentric annulus, $g_0$ and $g_1$ need to be calculated. $g_0$ can be determined as Equation (16). Solve Equations (18) and (19), and then use Equation (20) to calculate the displacement interface in the coordinate system. Substituting $g_0$ and $g_1$ into Equation (17), the expression of cemented displacement interface in eccentric annulus in coordinate system $(z,\varphi )$ can be obtained. Then, convert the replacement interface of the coordinate system $(z,\varphi )$ to a fixed coordinate system $(\xi ,\varphi )$.

3. Experiment Verification of Model

Since the 1980s, experimental research on cement displacement has been carried out; In 1991, Jakosen conducted an experimental study on cementing displacement in the wellbore annulus with a well deviation angle of 60° and a casing eccentricity of 0.55, and analyzed the influence of fluid viscosity, displacement flow pattern, and fluid density difference on displacement efficiency [22]; Southwest Petroleum University of China (1993) and the University of British Columbia (2009) have successively established a set of cementing displacement experimental devices for vertical wells, directional wells, and horizontal wells, and preliminarily conducted research on the influence of different parameters on cementing displacement efficiency [23,24]. The experimental method to study the displacement pattern of two-phase non-Newtonian fluid plays an important role in investigating the displacement mechanism and guiding the production practice. With the help of an experimental device for cement displacement simulation, experimental research is performed to verify the theoretical model of cement displacement interface in this study.

3.1. Experimental Research Device and Method

Set up an experimental simulation device for cement displacement for highly deviated wells, as shown in Figure 3. In the lying transparent PVC pipe, there is a hollow steel pipe which can rotate. It can simulate the influence of casing rotation on the cement displacement interface. Drilling fluid and cement slurry are pumped into the steel pipe and then into the annulus through the liquid pump successively.

The main features of the experimental device for cement displacement interface are:

(1)

The inner diameter of PVC pipe is 70 mm, the outer diameter of steel pipe is 30 mm, the annular space gap is 20 mm, and the total length is 4.2 m. The PVC pipe simulates wellbore, and steel pipe simulates casing;

(2)

Through the lifting system on the support, the deviation angle of the well can be adjusted between 80–90°;

(3)

The maximum displacement of the liquid pump is 10 L/min. By reasonably adjusting the rheological parameters of the fluid, the flow state of the fluid in the annulus can be obtained as laminar flow;

(4)

The eccentricity of the steel pipe can be changed by adjusting the flange combination on the plexiglass pipe.

The experimental procedure adopted is written as follows: prepare polyacrylamide aqueous solution with certain rheological properties to replace drilling fluid and cement slurry, and carry out a cement injection experiment. Dye the simulated cement slurry to effectively distinguish the position and shape of the two-phase interface. Use inorganic salts to regulate the density of simulated solution. The Hershel-Bulkley rheological model is used to describe the rheological properties of simulated fluids of cement slurry and drilling fluid. During the experiment, after the transparent drilling fluid is pumped into the annulus, the red cement slurry fluid enters the annulus. With the help of a high-definition video recorder, the development process and morphological characteristics of the two-phase displacement interface in the annulus are recorded.

3.2. Results and Discussion

The experimental data can be found in

Table 1. The basic parameters of the cementing displacement experiment.

Borehole Parameters		Length of Videoed Well Section/m	Well Deviation Angle/°		Eccentricity
		0.84	83°/80°/85°		0/0.4
Displacement parameters		Displacement capacity/L/min
		10
Drilling fluid parameters		Drilling fluid density/kg/m3	Liquidity index	Consistency coefficient/Pa·sn	Yield stress/Pa
		1030	0.8562	0.0250	0.1180
Cement slurry parameters	Formulation number	Cement slurry density/kg/m3	Liquidity index	Consistency coefficient/Pa·sn	Yield stress/Pa
	#1	1150	0.9563	0.0459	1
	#2	1150	0.8969	0.0541	0.7132
	#3	1150	0.9038	0.1073	1.4277

and Figure 4. The experimental data in Table 1 are dimensionlessly treated and substituted into the cement displacement interface model to solve the equation. Figure 5 shows the theoretical results of the influence of well inclination on the displacement interface under experimental conditions. As can be seen, the cement displacement direction is from left to right, and each curve represents the development of the displacement interface in the annulus with time. In addition, the displacement interface morphology, which is affected by casing eccentricity and cement slurry formula, is also theoretically analyzed.

Both the experimental and theoretical results of cement displacement in highly deviated wells show that the cement displacement interface gradually extends with the increase of well deviation angle.

3.3. Model Validation

Combining the experimental results with the theoretical results, the effects of well deviation angle, casing eccentricity and cement slurry formulations on the stability of the displacement interface are consistent: i.e., the extension length of cement displacement interface in the annulus increases with the increase of well deviation angle. Increasing the apparent viscosity of cement slurry is beneficial to improve the stability of the displacement interface. In highly deviated wells, casing eccentricity to some extent inhibits the breakthrough of the displacement interface at the narrow gap on the low side of the annulus, and reduces the extension length of the displacement interface [10]. In order to further verify the theoretical model of cement displacement interface, the interface length obtained from the cement displacement experiment is compared quantitatively with the result of the theoretical model. The specific method is as follows:

During the cement displacement experiment in highly deviated wells, the camera photographed the wellbore annulus with a length of 0.81 m, and the corresponding dimensionless annulus axial length is 7.4. Measure the length of the displacement interface in the experimental annulus, and calculate the ratio of the displacement interface to the length of the wellbore annulus of 0.81 m. At the same time, the results of the theoretical model for cement displacement interface are also converted, and the ratio of dimensionless displacement interface length to dimensionless annulus axial length is 7.4. The comparison is shown in

Table 2. Certification of theoretical model for cementing displacement interface.

Experimental Conditions (Model Solving Conditions)	Experimental Result (%)	Theoretical Results (%)	Relative Error (%)
Well deviation angle = 85°, Casing centered, Cement slurry: Formula 1	42	40.88	2.667
Well deviation angle = 83°, Casing centered, Cement slurry: Formula 1	37.7	34.94	7.409
Well deviation angle = 80°, Casing centered, Cement slurry: Formula 1	31.3	28.6	8.626
Well deviation angle = 83°, Casing eccentricity = 0.4, Cement slurry: Formula 1	28.3	29.61	4.622
Well deviation angle = 83°, Casing centered, Cement slurry: Formula 2	45	39.86	11.422
Well deviation angle = 83°, Casing centered, Cement slurry: Formula 3	31.9	25.12	20.254

and Figure 6.

The results of the theoretical model for cement displacement interface are quantitatively compared with the experimental results, and the average error of the theoretical model is 8.85%. Thus, the established theoretical model for cement displacement interface is verified [4].

4. Application of Theoretical Model

Casing eccentricity is a significant factor affecting cement displacement. In a vertical well, whether the casing is centered is an important factor for cementing displacement. However, preliminary research shows that, in highly deviated wells or horizontal wells, the influence of casing eccentricity on the stability of displacement interface is different from that in vertical wells. The influence of casing eccentricity on the stability of cemented displacement interface in highly deviated wells is calculated by using the theoretical model developed in this study [4].

Basic calculation parameters are listed as follows: borehole size is 215.9 mm; casing size is 177.8 mm; well deviation angle is 80°; displacement capacity is 1 m³/min; drilling fluid density is 1200 kg/m³; cement slurry density is 1600 kg/m³; the plastic viscosity of drilling fluid is 22.5 mPa·s, and the yield stress is 9.5 Pa; The fluidity index of cement slurry is 0.712, and the consistency coefficient is 0.36 Pa·sⁿ. The calculated $S_t{}^{\ast }=0.086$, and the casing eccentricity is taken as 0–0.5, respectively. Figure 6 shows the displacement interface morphology when the casing eccentricity is 0, 0.3, 0.4, and 0.5, respectively.

In Figure 7, the annulus narrow gap is at the dimensionless circumferential angle of 1, while the annulus wide gap is at the dimensionless circumferential angle of 0. In highly deviated wells, it is very common for the casing to lean towards the well wall due to the gravity of the casing. Analyze the variation of displacement interface with casing eccentricity in Figure 7 to obtain that: when the casing is in centered, the cement displacement interface advances faster along the lower side of the annulus. With the increase of casing eccentricity, the displacement interface gradually advances evenly in the whole annulus, such as for the case of eccentricity equal to 0.3. However, when the eccentricity of the casing is further increased, the displacement interface begins to lunge along the wide gap on the upper side of the annulus, such as when the eccentricity is equal to 0.5. The theoretical model shows that, in highly deviated wells, a certain casing eccentricity can inhibit the uneven penetration of cement slurry in the annulus, which is conducive to maintaining the stability of the displacement interface.

In the eccentric annulus, the flow resistance of the fluid at each gap of the annulus is different. Relatively, the flow resistance at the wide gap is small, while the flow resistance at the narrow gap is large. In addition, the density of cement slurry is generally higher than that of drilling fluid during cementing displacement, resulting in the trend of stratified flow of two-phase fluid in the annulus of highly deviated wells. The combined effects of flow resistance and stratified flow lead to the above variation pattern of cement displacement interface. When the casing is in the middle or the eccentricity is small, the flow resistance between the gaps in the annulus does not show much difference, and the stratified flow of two-phase fluid plays a major role. Therefore, the cement displacement interface lunges along the lower side of the annulus; When the casing eccentricity is large, the fluid flow resistance at the wide and narrow gap of the annulus show a big difference, which becomes the main influencing factor. In this case, the displacement of drilling fluid at the narrow gap at the lower side of the annulus is restrained, and the displacement interface begins to lunge along the wide gap at the upper side of the annulus. When these two effects are balanced, a stable displacement interface can be obtained [11].

For a given set of well parameters, such as borehole size, casing size, and well deviation angle, and casing eccentricity, the proposed theoretical model can provide a guidance for the selection of a series of cementing parameters listed below: displacement capacity, drilling fluid density, cement slurry density, plastic viscosity, the yield stress, the fluidity index, and the consistency coefficient of cement slurry to achieve a stable displacing interface for a high displacement efficiency.

5. Conclusions

(1)

A theoretical model for cement displacement interface is established, and the theoretical model is verified by the experiments which investigate influence of well deviation angle, casing eccentricity, and cement slurry formulations on the displacement interface. The new model is not limited to the well deviation angle, and can simulate the cement displacement interface morphology in concentric annulus and eccentric annulus.

(2)

The finite difference method can effectively solve the theoretical model. The analysis shows that: with the increase of well deviation angle, the cement displacement interface length gradually extends, and under the same injection conditions, the well with a larger well deviation angle has a poor cement displacement effect. Increasing the apparent viscosity of cement slurry is beneficial to improve the stability of displacement interface. In highly deviated wells, a certain casing eccentricity can inhibit the uneven penetration of cement slurry in the annulus, which is conducive to maintaining the stability of the displacement interface.