Steam Cavity Expansion Model for Steam Flooding in Deep Heavy Oil Reservoirs

Abstract

Steam flooding is crucial for the development of heavy oil reservoirs, and the development of the steam cavity significantly determines the efficiency of steam flooding. Previous studies have elucidated the concept of steam overburden and pseudomobility ratio; however, the thermal energy loss in deep heavy oil reservoirs during steam injection needs further investigation. Therefore, in this study, the vapour–liquid interface theory and mathematical integration were used to establish a steam cavity expansion model. The wellbore heat loss rate coefficient, steam overlay, and pseudomobility ratio were used to accurately describe the development of the steam cavity in deep heavy oil reservoirs. The proposed model was experimentally validated, and it was observed that the model could accurately reflect the actual mine conditions. In addition, the pressure gradient distribution of the steam belt and the heat dissipation areas of the top and bottom layers of the steam cavity were evaluated. The results showed that the influence of the wellbore heat loss rate coefficient on the pressure gradient of the oil layer was primarily in the range of 5–20 m away from the steam injection well. Furthermore, it was observed that the pseudomobility ratio is inversely proportional to the development of the steam cavity. As the wellbore heat loss rate coefficient increased, the wellbore heat loss increased. The larger the area ratio, the more pronounced the steam overlay phenomenon, and the large area ratio does not meet the development requirements of the steam chamber. The research closely combines theory with production, and the results of this study can help actual mines by providing theoretical support for the development of deep heavy oil reservoirs.

1. Introduction

Owing to the depletion of conventional oil reservoirs, heavy oil reservoirs have gradually become crucial for oil field development. Steam flooding, which is a significant contributor to the development of heavy oil reservoirs, has the advantages of high recovery, quick effect, and low pollution. During steam flooding, when the steam is injected into the reservoir along the wellbore, heat is released. Consequently, the heat reduces the interfacial tension of the crude oil, and the fluidity of the heated heavy oil increases, making it easier to displace and develop. However, the development and expansion of the steam cavity is a significant factor affecting the efficiency of steam flooding; therefore, the expansion law of the steam cavity should be investigated.

In the 1950s, Marx and Langenheim [1] established a steam-flooding, front-propulsion model based on the relationship between the growth of the steam zone and the dissipation rate of the top and bottom caprocks. Their research formed the basis for one of the fundamental theories of steam flooding and elucidated the relationship between measurable parameters and crude oil volume production. Furthermore, it provided a theoretical basis for the subsequent steam flooding and steam chamber expansion models. However, they assumed that the thickness of the vapour zone remained constant, which is not an accurate representation of the actual experimental conditions. Therefore, to mitigate the shortcomings of the Marx–Langenheim model, Neuman [2] proposed a steam flooding theoretical model based on the temperature distribution of the heat transfer in an infinite reservoir. The theoretical model was established using the total enthalpy balance equation, condensate water, and crude oil mass flow equation. However, it was assumed that the main expansion direction of the steam zone was vertically downward, and gravity was the dominant factor affecting the steam overlay. Myhill and Stegemeier [3] proposed a new steam cavity, leading-edge propulsion model. In their study, the steam cavity leading edge was not considered to be vertical, the displacement effect of condensed water on crude oil before the condensation front was ignored, and a description of the thermal efficiency of the steam zone was proposed. The proposed thermal efficiency function can be used to determine the volume of the steam zone; however, their model assumes that the oil displacement is equal to the oil production, and the research was based on the premise of a high injection rate.

Van-Lookeren [4] analysed the shape of the vapour–liquid interface and proposed that a maximum shape factor value can ensure optimal steam injection parameters. The shape factor value is a dimensionless parameter, which is used to characterize the shape of the steam cavity with the radial growth of steam flooding. However, in actual reservoirs, owing to the high viscosity of crude oil, the pseudomobility ratio is often not zero and sometimes even approaches one. Consequently, the pseudomobility ratio affects the shape of the front edge of the steam cavity; therefore, for a nonzero pseudomobility ratio, the maximum shape factor cannot optimise the steam injection parameters [5,6,7,8]. Vogel [9] identified the slow descending behaviour of the steam zone as the primary thermal oil recovery mechanism and established a steam flooding model. The model could accurately represent the various energy relationships of the reservoir during steam flooding; however, it did not consider the upper oil layer, which resulted in an overestimation of the rock heat loss at the bottom of the reservoir.

Stegemeier, Farouq Ali, and Peake et al. [10,11,12] have also made significant contributions to the literature. In recent years, Cheng [13] established a steam chamber expansion model considering the pseudomobility ratio and shape factor and reported that the shape of the front edge of the steam chamber becomes ideal as the pseudomobility ratio increases. In addition, the study demonstrated that a higher degree of influence has a more significant effect on the sweep efficiency of steam flooding. Furthermore, the study improved the method used in previous studies to optimise steam injection parameters by making the pseudomobility ratio a non-negligible parameter [14,15,16]. By characterising the upper boundary of the steam front, Ding [17] established a mathematical model of the steam front that considered the start-up pressure gradient and the change in the upper boundary of the steam front; the results showed that the upper boundary of the steam front decreases with an increase in the starting pressure gradient.

Huang [18] established a prediction model of the steam cavity leading edge by studying the characteristics of the steam cavity leading edge submerged by horizontal steam; the model was used to analyse the position of the steam cavity leading edge under different production conditions and the shape of the steam cavity leading edge. The analysis results demonstrated that the linear steam cavity front is the ideal shape. In contrast, the convex shape was observed to be the worst shape because it aggravates the steam overburden phenomenon and affects the efficiency of steam flooding. However, the research was primarily conducted on thin heavy oil reservoirs and did not explain the characteristics of the vapour cavity front of deep heavy oil reservoirs.

Considering that the increase in asphaltene deposition will block the throat and inhibit the formation of the vapor cavity, Tian [19] proposed a steam cavity expansion model for thin heavy oil reservoirs. The IMPES (Implicit Pressure Explicit Saturation) method was used to establish the model. Previous studies have shown that the development and expansion of the steam chamber are proportional to the concentration of butane; however, if the concentration of injected butane is too high, it will lead to a steam overburden during the early stage in the steam chamber [20,21,22], and a front breakthrough of the steam chamber will occur between the injection and production wells. Farzain [23] and others used numerical simulation software to analyse the influence of steam quality and steam injection temperature on the expansion of the steam cavity. The results showed that a high steam injection temperature and medium-quality steam are the optimum configurations. Under this combination, the development of the steam chamber is optimal and cost effective.

Pang [24] quantitatively analysed the expansion of the steam cavity and water intrusion at the bottom of the steam cavity during the steam flooding process. The analysis results were used to establish a novel method to study the expansion law of the steam cavity by utilising the energy conservation rate. It was demonstrated that the steam mainly migrates to the top of the oil layer. During steam flooding, the content of petroleum components gradually changes, owing to distillation. The light hydrocarbon content is higher at the front of the steam cavity, heavy components mainly occupy the sweeping area flooded by the steam, and the front of the steam cavity reaches half the distance from the production well at the end of the steam flooding.

Owing to the large burial depth and high viscosity of deep heavy oil reservoirs, wellbore heat loss will occur during steam injection. As well depth increases, heat dissipation in the wellbore and the negative effects of wellbore friction will reduce the amount of heat injected into the reservoir by reducing steam dryness. The existing steam cavity expansion models are not applicable to the development of medium-deep heavy oil reservoirs or deep heavy oil reservoirs steam flooding.

Therefore, in this study, the wellbore heat loss rate coefficient was used to describe the heat loss of steam along the wellbore during the development of deep heavy oil reservoirs. A steam cavity expansion model was developed based on the wellbore heat loss rate coefficient of deep heavy oil reservoirs. After verifying the model, the pressure distribution at the front of the steam belt and the relationship between the steam-dissipated heat area were calculated. Furthermore, the influence of the wellbore heat loss rate on the expansion of the steam cavity was analysed. The steam cavity expansion model was used to analyse the sensitivity of parameters such as the shape factor and pseudomobility ratio and describe the effects of the steam cavity on the development of the deep heavy oil reservoirs. The model developed in this study can quickly and accurately predict the position of the front edge of the steam cavity and calculate the volume of the steam cavity without the assistance of traditional numerical software models. Thus, the novel model is fast, cost effective, and provides theoretical support for the development of deep heavy oil reservoirs.

2. Expansion of Steam Flooding Chamber

2.1. Assumptions

Various factors influence each other during steam flooding production, and the heat transfer and seepage mechanisms are intricate. Therefore, in this study, to simplify the production process, the following assumptions were made regarding the characteristics of deep heavy oil reservoirs:

(1)

Homogeneous reservoirs and fractures were assumed to be absent.

(2)

The thermal conductivity values of the top and bottom rocks were the same, and the heat conduction in the horizontal direction was zero in top and bottom rocks.

(3)

The injected steam parameters were constant, and the steam did not undergo a phase change.

(4)

The wellbore friction loss was ignored.

(5)

The physical properties and fluid saturation of the oil layer were not affected by temperature.

(6)

The streamline is in the horizontal direction, and the flow only occurs in all planes perpendicular to the horizontal injection and production well; the cross-flow between vertical planes does not occur.

From the above assumptions, the three-dimensional problem can be converted into a two-dimensional problem, and the movement of the vapour–liquid interface can be studied. In addition, a steam cavity expansion model of a deep heavy oil reservoir can be established according to the actual situation. A schematic of the movement of the vapour–liquid interface is shown in Figure 1.

2.2. Vapor-Liquid Interface Shape

When the bottom end of the steam interface reaches the bottom end of the steam injection well, the pressure at each point on the steam–liquid interface at radial distances r₁ and r₂ from the steam injection well can be converted to the plane of Y = 0. The pressure and flow potential at the steam–liquid interface are shown in Figure 2.

$$\{\begin{array}{c}{\varphi }_{s1}=P_{s1}-{\rho }_{s}g{h}_{s1}\\ {\varphi }_{s2}=P_{s2}-{\rho }_{s}g{h}_{s2}\\ {\varphi }_{o1}=P_{o1}-{\rho }_{o}g{h}_{s1}\\ {\varphi }_{o2}=P_{o2}-{\rho }_{o}g{h}_{s2}\end{array}$$

(1)

where ${\varphi }_{s1}$ and ${\varphi }_{s2}$ are the potentials of the steam at points 1 and 2, respectively, at the vapour–liquid interface converted to the Y = 0 surface; ${\varphi }_{o1}$ and ${\varphi }_{o2}$ are the potentials of the heavy oil at points 1 and 2, respectively, at the vapour–liquid interface converted to the $Y=0$ surface; $P_{s1}$ and $P_{s2}$ are the pressures of the steam at points 1 and 2, respectively, at the vapour–liquid interface; $P_{o1}$ and $P_{o2}$ are the pressures of the heavy oil at points 1 and 2, respectively, at the vapour–liquid interface; $h_{s1}$ and $h_{s2}$ are the thicknesses of the steam belt at points 1 and 2, respectively; ${\rho }_s$ is the steam density; ${\rho }_o$ is the oil density; and $g$ is the gravitational acceleration.

Thus,

$$\{\begin{array}{c}△{\varphi }_s={\varphi }_{s1}-{\varphi }_{s2}=(P_{s1}-P_{s2})-{\rho }_{s}g(h_{s1}-h_{s2})\\ △{\varphi }_o={\varphi }_{o1}-{\varphi }_{o2}=(P_{o1}-P_{o2})-{\rho }_{o}g(h_{s1}-h_{s2})\end{array}$$

(2)

Because the pressures on both sides of the vapour–liquid interface are equal:

$$△{\varphi }_s-△{\varphi }_o=({\rho }_o-{\rho }_s)g(h_{s1}-h_{s2})=({\rho }_o-{\rho }_s)g△h_s$$

(3)

When r₁ and r₂ are infinitely close, the following can be concluded:

$$\frac{\partial {\varphi }_s}{\partial r}-\frac{\partial {\varphi }_o}{\partial r}=({\rho }_o-{\rho }_s)g\frac{\partial h_s}{\partial r}$$

(4)

Using Darcy’s formula for the two phases, we obtain the following:

$$\frac{\partial h_s}{\partial r}=\frac{1}{\left({\rho }_o-{\rho }_s\right)g}\left(\frac{{\mu }_s{w}_s}{2\pi r{h}_s{k}_s{\rho }_s}-\frac{{\mu }_o{w}_o}{2\pi r{h}_o{k}_o{\rho }_o}\right)$$

(5)

where ${\mu }_s$ is the viscosity of steam, ${\mu }_o$ is the viscosity of heavy oil, $w_s$ is the radial velocity of steam in the steam belt, $w_o$ is the radial velocity of heavy oil in the steam belt, $k_s$ is the steam permeability, $k_o$ is the oil permeability, $h_s$ is the thickness of the steam belt, and $h$ is the thickness of the oil layer.

The large burial depth and high crude oil viscosity observed in deep heavy oil reservoirs can be attributed to the nonzero pseudomobility ratio. The pseudomobility $M^{\prime }$ is obtained as follows:

$$M^{\prime }=\frac{{\mu }_o^{\prime }{k}_s}{{\mu }_s{k}_o}\times \frac{{\rho }_s{w}_o\left(r_e\right)}{{\rho }_o{w}_s\left(r_b\right)}$$

(6)

where ${\mu }_o^{\prime }$ is the crude oil viscosity during formation when heated by steam; $w_s(r_b)$ is the steam velocity at $r_b$ in the steam belt; and $w_o(r_e)$ is the oil velocity at $r_e$ in the steam belt.

From Van-Lookeren’s research, the dimensionless shape factor is obtained as follows [4]:

$$A_{rD}^2=\frac{{\mu }_s{i}_s}{\pi \left({\rho }_o-{\rho }_s\right){gh}^2{k}_s{\rho }_s}$$

(7)

where, $i_s$ is the steam injection rate.

We can obtain

$$\frac{\partial h_s}{\partial r}=-A_{rD}{}^2\frac{h^2{w}_s}{2r{h}_s{i}_s}[1-M^{\prime }\frac{h_s}{h-h_s}\frac{w_o{w}_s(r_b)}{w_s{w}_o(r_e)}]$$

(8)

2.3. Wellbore Heat Loss Rate

Owing to the large burial depth of deep heavy oil reservoirs, heat loss occurs during the injection of steam from the surface into the reservoir [25,26,27].

A microelement structure in the wellbore was selected, as shown in Figure 3.

$$dq=\pi dK[T-(t_0+mx)]dx=-GCdT$$

(9)

where $dq$ is the heat dissipation loss from the wellbore; $K$ is the total heat transfer coefficient from the outside of the tubing to the formation; $d$ is the inner diameter of the oil pipe or insulated pipe; $T_{}$ is the temperature of the fluid; t₀ is the surface temperature; $m$ is the geothermal gradient; $G$ is the injected fluid flow; and $C$ is the specific heat capacity of the injected fluid.

When $W=GC$, $K_l=\pi dK$, the following is obtained:

$$\frac{dT}{dx}=\frac{K_l}{W}[(t_0+mx)-T]$$

(10)

Thus,

$$\frac{dT}{dx}+\frac{K_l}{W}T=\frac{K_l}{W}(t_0+mx)$$

(11)

$$T=C{e}^{-\frac{K_l}{W}x}+(t_0+mx)-\frac{Wm}{K_l}$$

(12)

When $x=0$, $T=T_0$, the following is obtained:

$$C=T_0-t_0+\frac{Wm}{K_l}$$

(13)

Thus,

$$T=\frac{Wm}{K_l}(e^{-\frac{K_{l}x}{W}}-1)+(T_0-t_0)e^{-\frac{K_{l}x}{W}}+(t_0+mx)$$

(14)

The heat dissipation loss when there is no phase change during the injection of hot fluid along the wellbore is calculated as follows:

$$q=GC(T_0-T)$$

(15)

The wellbore heat loss rate coefficient is obtained:

$$n=\frac{q}{q_0}=\frac{GC(T_0-T)}{GC{T}_0}$$

(16)

where $q_0$ is the heat at wellhead injection.

2.4. Steam Cavity Expansion Model for Steam Flooding

Previous studies have reported that after the steam rises to the top of the oil layer, it maintains a steady balance in the radial and longitudinal directions of the steam belt. Subsequently, the steam expands outwards in both radial and longitudinal directions simultaneously. In the longitudinal direction of the steam zone, the steam rate decreases uniformly, and the steam rate ratio is defined as the ratio of the thickness of the steam zone to the thickness of the formation.

$$\frac{w_s}{w_s(r_b)}=\frac{h_s}{h}$$

(17)

$$\frac{w_o}{w_o(r_e)}=1-\frac{h_s}{h}$$

(18)

Van-Lookeren [4] and Cheng [13] have shown that the ratio of the bottom hole steam mass flow rate to the steam radial velocity on the steam belt is equal to the ratio of the front radius of the steam cavity, whereas the bottom hole mass flow rate is equal to the ratio of the steam cavity leading-edge radius.

However, in deep heavy oil reservoirs, due to the increase of the well depth, the steam will lose heat energy in the wellbore, and the wellbore heat loss rate cannot be ignored. Therefore, the concept of wellbore heat loss rate was proposed based on the characteristics of deep heavy oil reservoirs. The wellbore heat loss rate coefficient is calculated by considering the kinetic energy loss of steam in the wellbore and brought into the relationship between the steam zone rate and the bottom hole steam rate. The coefficient values range from 0 to 0.5.

$$\frac{w_s}{i_s}=(1-n)(\frac{r_e{}^2-r^2}{r_e{}^2-r_b{}^2})=(1-n)(1-\frac{r^2}{r_e{}^2})$$

(19)

where $r$ is the radial distance from the steam injection point, and $i_c$ is the bottom hole steam mass rate.

Substituting Equations (17)–(19) into Equation (8), we obtain the following:

$$\frac{\partial h_s}{\partial r}=A_{rD}{}^2\frac{h^2(1-n)}{2r{h}_s}(1-M^{\prime })(1-\frac{r^2}{r_e{}^2})$$

(20)

Integrating Equation (20), we obtain the steam chamber leading-edge expansion model as follows:

$$h_s=A_{rD}h\sqrt{(\ln \frac{r_e}{r}-\frac{1}{2}+\frac{1}{2}\frac{r^2}{r_e{}^2})\times (1-M^{\prime })\times (1-n)}$$

(21)

From the steam flooding front shape equation, it can be inferred that the shape of the steam flooding cavity front in deep heavy oil reservoirs is mainly determined by the shape factor, wellbore heat loss rate coefficient, and pseudomobility ratio.

3. Model Validation

To verify the accuracy of the proposed model, the steam flooding test area of a deep heavy oil reservoir in an oil field in northeastern China was selected as the research object. The basic parameters of the reservoir are presented in

Table 1. Basic data table of steam flooding test area.

Item	Value	Item	Value
Reservoir density/kg·m−3	980	Oil layer thickness/m	15
Steam density/kg·m−3	20	Well spacing/m	70
Steam viscosity/mPa·s	1.44 × 10−2	Steam injection rate/kg·s−1	1.5
Oil viscosity/mPa·s	7.5 × 105	Wellbore heat loss rate coefficient/f	0.15

. The parameters were used in the Van-Lookeren steam chamber expansion model, Cheng steam chamber expansion model, modified steam chamber model, and compared with the steam chamber in the actual field. The actual data is measured by the observation well temperature method. The temperature distribution of the oil drainage zone can be used to infer the position of the front edge of the steam chamber, and then the shape of the front edge of the steam chamber can be obtained by fitting the curve [28].

As shown in Figure 4, the shape of the leading edge of the steam cavity in the Van-Lookeren model is more ideal than the shape of the leading edge of the steam cavity of the latter two, and the phenomenon of steam overlap is not as obvious as the protruding phenomenon of the leading edge of the steam cavity in the Cheng model. Compared with the Van-Lookeren model and Cheng model, the front edge of the steam cavity of the modified model is further back, the longitudinal distance between the tops of the steam belts is larger, the steam overlap phenomenon is more obvious, and it is also closer to the actual site steam cavity.

Because the Van-Lookeren model and Cheng model are based on steam flooding development of thin heavy oil reservoirs, the thermal energy loss caused by wellbore heat loss is ignored, which seriously affects the development and expansion of steam chambers. Therefore, on the basis of Cheng’s model, a wellbore heat loss rate coefficient was introduced to describe the phenomenon of heat loss due to the increase of steam along the well depth. Compared to the Cheng model, the accuracy of the calculation results was improved by approximately 10%. The position of the leading edge of the steam drive cavity obtained in this study was more backward, and the steam overburden phenomenon was more notable.

4. Model Application

4.1. Gradient Analysis of Steam Belt Pressure

The differential formula for radial flow in the Darcy plane is:

$$Q=\frac{AK}{\mu }\frac{dp}{dr}$$

(22)

where $Q$ is well flow rate, $A$ is seepage area of plane radial flow, and $K$ is the permeability.

Due to the different properties of the fluid near the injection well and the production well, the pressure gradients of the production well and the injection well are calculated separately:

$${\eta }_1=\frac{Q_1{\mu }_1}{2\pi K{h}_s}\times \frac{1}{r}$$

(23)

$${\eta }_2=\frac{Q_2{\mu }_2}{2\pi K{h}_s}\times \frac{1}{(L-r)}$$

(24)

where $Q_1$ is the output of the production wells, $Q_2$ is the steam injection volume, ${\mu }_1$ is the fluid viscosity near production wells, ${\mu }_2$ is the fluid viscosity near injection wells, and $L$ is the spacing between the injection and production wells.

According to the superposition principle of the potential, the pressure gradient of the steam belt on the main flow line of the injection-production well can be obtained:

$$\frac{dp}{dr}=\frac{Q_1{\mu }_1}{2\pi K{h}_s}\times \frac{1}{r}+\frac{Q_2{\mu }_2}{2\pi K{h}_s}\times \frac{1}{(L-r)}$$

(25)

Table 2. Reservoir parameters and injection parameters.

Item	Value
Production well output/m3·d−1	150
Steam injection volume/m3·d−1	200
Fluid viscosity near production wells/mPa·s	500
Fluid viscosity near injection wells/mPa·s	10
Permeability/10−3μm2	1500

shows the reservoir and injection parameters for the test area of the medium-deep heavy oil reservoir. The data were used as input for Equation (25), and the relationship between the oil layer pressure and steam zone distance in the deep heavy oil reservoir was obtained.

From Figure 5, it can be inferred that at the same position, as the wellbore heat loss rate coefficient increased, the pressure gradient increased. The steam zone was close to the steam injection well, and the wellbore heat loss rate coefficient had a significant influence on the reservoir pressure gradient. As the steam zone moved farther away from the steam injection well, the effect of the wellbore heat loss rate coefficient on the reservoir pressure gradient became less marked.

After steam injection, heat loss occurs in the wellbore, and a large amount of heat energy is consumed. When the pressure gradient is large, the steam belt pressure difference per unit distance changes significantly. It can be inferred that the steam belt pressure drops significantly, and the development of the steam cavity is not ideal.

According to the prediction, it is found that in the reservoir in this study, the optimal heating radius of steam flooding is about 35 m. Similarly, when the radius of the steam belt increased again, the pressure gradient also increased. Therefore, it can be concluded that the optimal heating radius of the steam zone in deep heavy oil reservoirs is approximately 35 m. When the steam zone radius exceeds this range, severe steam overburden is observed, the oil layer pressure drops rapidly, and the efficacy of steam flooding decreases.

4.2. Calculate the Heat Dissipation Area

The steam flooding front equation can be obtained from Equation (21):

$$h_s=A_{rD}h\sqrt{(\ln \frac{r_e}{r}-\frac{1}{2}+\frac{1}{2}\frac{r^2}{r_e{}^2})\times (1-M^{\prime })\times (1-n)}$$

When the steam reaches the bottom of the steam injection well, $r=r_b$, $h=h_s$, and the following is obtained:

$${(\frac{1}{A_{rD}})}^2=(\ln y-\frac{1}{2}+\frac{1}{2}{y}^2)\times (1-M^{\prime })\times (1-n)$$

(26)

where $y$ is the ratio of the heat dissipation areas of the top and bottom layers.

Thus, the following is obtained:

$$y=f(\frac{1}{A_{rD}{}^2(1-n)(1-M^{\prime })})$$

(27)

The relationship between the heat dissipation area between the top and bottom cover layers is as follows:

$$A_1(t)=\pi r_e{}^2=y^2\pi r_b{}^2=y^2{A}_2(t)$$

(28)

where $A_1(t)$ is the heat dissipation area for the top layer, and $A_2(t)$ is the heat dissipation area for the bottom layer.

The shape factor is also called the steam overlay coefficient, which has a significant influence on the development of steam flooding and steam chambers. The relationship between the heat dissipation area of the top and bottom layers of steam flooding and the shape factor is shown in Figure 6. When the shape factor was 0–1.2, the ratio of the heat dissipation area of the top layer to that of the bottom layer changed significantly. Thus, it can be concluded that the heat dissipation area of the top layer was much larger than that of the bottom layer. In addition, severe steam overburden was observed. Furthermore, the steam inrush of the top layer was severe, which is not conducive to the efficacy of steam flooding; when the shape factor was greater than 1.2, the ratio of the heat dissipation area of the top layer to that of the bottom layer was small. The two ends of the steam chamber were at an equally advanced position towards the production well, and the oil recovery factor increased. Under the same shape factor, the wellbore heat loss rate coefficient increased, and the ratio of the area of the top layer to that of the bottom of the steam zone increased. Note that the front edge of the steam cavity protruded significantly, and the steam overburden had an adverse effect on the development of steam flooding [29].

4.3. Sensitivity Analysis of Steam Chamber Expansion

4.3.1. Influence Analysis of Shape Factor

Figure 7 shows that the shape factor increased, the angle between the steam cavity and the horizontal direction tended to be 90°, the shape of the front edge of the steam cavity became steeper, the longitudinal thickness of the steam belt increased, and the steam overlay phenomenon became less marked. In deep heavy oil reservoirs, the lower part of the steam cavity could reach the bottom of the well only for a shape factor of approximately 1.2, which formed a steam zone over the entire oil layer. Only when the front edge of the steam cavity reaches the bottom of the oil layer can the steam belt of the whole oil layer be formed. Thus, the range of steam flooding will increase, and the reservoir recovery will be improved.

4.3.2. Influence Analysis of Pseudomobility Ratio

Figure 8 shows the shape of the steam flooding front under different pseudomobility ratios. From the figure, it can be inferred that the pseudomobility ratio had an adverse effect on the shape of the steam front. With increasing pseudomobility ratio, the steam front became severely inclined, and the longitudinal thickness of the steam belt became smaller. In actual reservoirs, the pseudomobility ratio cannot be ignored, due to reservoir heterogeneity and excessive crude oil viscosity. Therefore, when $M^{\prime }\ge 0.8$, the steam overburden phenomenon intensifies, the difference between the radius of the upper part of the steam chamber and the radius of the lower part of the steam chamber increases. The aggravation of the steam overlay hinders steam flooding, resulting in reduced oil recovery.

4.3.3. Influence Analysis of Wellbore Heat Loss Rate Coefficient

Figure 9 shows that the smaller the wellbore heat loss rate is, the smaller the angle between the steam zone and the plane is, and the front edge of the steam cavity is linear. It can be inferred that the coefficient of the wellbore heat loss rate was inversely proportional to the shape of the steam cavity front. In actual fields, because of the geological characteristics of deep heavy oil reservoirs, after steam is injected along the wellbore, energy loss occurs with increasing well depth; this results in changes in steam dryness and steam mass flow rate. The increase of the wellbore heat loss rate will lead to the reduction of the heat energy carried by the steam injected into the reservoir, and the steam mass flow rate at the steam zone will decrease accordingly. Thus, the effective sweeping range of steam flooding and the oil displacement efficiency are reduced significantly; it is not conducive to the development of steam flooding.

5. Discussion

Heavy oil is widely distributed and rich in resources in the world, and countries regard heavy oil as an important strategic energy reserve. Compared with other recovery methods, steam flooding can realize the complete process of steam cavity formation, expansion, and breakthrough, which is in line with the development law of heavy oil thermal recovery. However, the success of steam flooding depends on the development and expansion of steam chambers, so it is very important to accurately grasp the development and expansion laws of steam chambers.

In the current research, the research on the expansion of steam cavities in shallow and thin heavy oil reservoirs is relatively complete, and many influencing factors are considered, such as steam overlay, pseudomobility ratio and other parameters. However, the existing steam cavity models cannot be perfectly applied to the development of steam flooding in deep heavy oil reservoirs. Due to the characteristics of large burial depth and high viscosity in deep heavy oil reservoirs, after steam is injected along the wellbore, thermal energy loss will occur with the increase of well depth, resulting in the reduction of the heat carried by the injected steam into the reservoir.

Therefore, in this study, the vapour–liquid interface theory and mathematical integration were used to establish a steam cavity expansion model. The wellbore heat loss rate coefficient, steam overlay, and pseudomobility ratio were used to accurately describe the development of the steam cavity in deep heavy oil reservoirs. The proposed model was experimentally validated, and it was observed that the model could accurately reflect the actual mine conditions.

This research considers the thermal energy loss phenomenon of steam in the injection process and expounds the macroscopic expansion law of steam displacement chamber in deep heavy oil reservoirs. Compared with the traditional numerical model, this research can quickly and accurately predict the position of the front edge of the steam cavity and calculate the volume of the steam cavity, thereby saving time and costs, and improving the economic benefit of steam flooding development.

6. Conclusions

In this study, a steam cavity expansion model for deep heavy oil vertical well steam flooding was established considering the wellbore heat loss rate coefficient, steam overlay, and pseudomobility ratio. The proposed model was analysed and validated. The following conclusions were made:

(1)

During the steam flooding development of deep heavy oil reservoirs, after the steam is injected from the wellhead, energy loss occurs with increasing wellbore depth. The wellbore heat loss rate coefficient can accurately reflect the heat energy loss phenomenon during the steam injection process and provide a theoretical basis for the development of steam flooding in deep heavy oil reservoirs.

(2)

The influence of the wellbore heat loss rate coefficient on the pressure gradient of the oil layer was primarily observed at a distance of 5–20 m from the steam injection well. At a greater distance, the influence of the wellbore heat loss rate becomes negligible, and deep heavy oil is obtained. Therefore, the optimum heating radius for reservoir steam flooding is approximately 35 m. When the radius of the steam zone exceeds the optimum range, the phenomenon of steam overburden becomes significant.

(3)

As the wellbore heat loss rate coefficient increased, the heat carried by the steam injected into the reservoir increased. The reduction of the heat dissipation area ratio of the top and bottom layers of the steam chamber means that the steam overlay effect is not obvious; the steam zone can maintain a vertical shape and advance toward the production well.

(4)

From the sensitivity analysis of the steam cavity expansion model, it was concluded that with increasing shape factor, the shape of the steam cavity is better developed. The pseudomobility ratio is inversely proportional to the shape of the steam cavity. In addition, with increasing well depth, the loss of steam heat energy becomes more severe. Thus, the greater the coefficient of heat loss rate of the wellbore, the less developed the steam chamber is, resulting in lower ultimate recovery.