Introduction

The contradiction between the high carbon emission and energy consumption of oil sand bitumen recovery and the high-quality development of oilfields is increasing. Steam-assisted gravity drainage (SAGD), as one of the important ways to develop oil sand reservoirs, consists of two main processes: steam circulation (or startup) stage and production stage (Huang et al. 2018). Steam circulation heats the medium between the pair horizontal wells by simultaneously circulating steam in a bi-horizontal well to reduce the viscosity of oil sand bitumen, which aims to quickly and uniformly establish inter-well medium mobility, and facilitating the expeditious attainment of the target peak oil production rateduring production stage (Duong et al. 2008). The steam injection into formation is an extremely complex process involving multiphase flow, phase shift heating, and microfine porous media interactions. Within this process, phenomena such as phase change, gravity separation (with steam moving upward and the liquid phase moving downward), and viscosity change are intertwined. In field, the non-homogeneity of the reservoir, injection pressure, and wellbore fluctuations can lead to uneven inter-well connectivity, and localized development of steam chambers. Although many scholars have now proposed many processes such as Steam And Gas Push (SAGP) (Bulter 1999), Expanding Solvent Steam Assisted Gravity Drainage (ES-SAGD) (Nasr 2003), Solvent Aided Process(SAP) (Gittins et al. 2005) and so on to reduce carbon emissions and enhance oil recovery of bitumen. However, for these SAGD-based development methods, the steam circulation stage is more or less the same, and the main difference lies in the components of injected fluids and so on. Given that the preheating effect of startup stage near the pair horizontal wells profoundly influences subsequent reservoir development and production, studying the distribution of key parameters along the wellbore and the preheating effect under different steam operational parameters is of great significance, in order to achieve connectivity between injection and production wells more rapidly and uniformly.

The operational parameters of SAGD steam circulation generally include injection pressure, steam dryness, pressure difference, steam circulation rate and reservoir properties, etc., and the specific parameters are generally formulated according to the actual conditions of the oilfield after analytical solution and numerical simulation. For the first method, the flow and heat transfer patterns of the fluid within the wellbore and formation are typically analyzed to establish a heat transfer model for the preheating phase of SAGD. For example, using discrete wellbore or coupling wellbore and reservoir modeling to analyze the effects of steam operational parameters on the circulating preheating effect(Vanegas et al. 2005; Yuan and McFarlane 2011). This type of analysis aims to investigate the variation and distribution of key parameters along the wellbore under different operational conditions. As for the second method, performing numerical simulations using CMG STARS for preheating can be considered as a preliminary procedure before production, which is no substantive difference between two procedures (Parmar et al. 2009; Yongbin et al. 2012; Anderson et al. 2012; Zhang et al. 2016). However, in the research on the operational parameters, the conventional strategy of SAGD steam circulation has the steam injection pressure of injection (I) well greater than that of production (P) well, while the current study of the operational parameter lacks the case where the steam injection pressure of P well is greater than that of I well.

The prediction of reservoir temperature during the startup stage is one of the most important issues for field engineers, because it can evaluate the efficiency of steam circulation and estimate the optimal opportunity of switching to SAGD production stage from startup stage. Nevertheless, due to the unique nature of the oil sand bitumen, making accurate predictions is challenging. Up to now, many researchers have studied the heat transfer problem in the whole process of SAGD. Currently, two main heat transfer mechanisms are considered in the study of heat transfer in the SAGD process (Butler 1991; Al-Bahlani and Babadagli, 2008, 2009), of which heat conduction is considered to be the only mechanism, while heat convection is usually negligible (Edmunds 1999; Duong 2008; Irani and Ghannadi 2013; Ghannadi et al 2016). Nonetheless, there exist some scholars who believe that heat convection induced by the mobile condensation flows has a considerable effect (Farouq-Ali 1997; Ito and Suzuki 1999; Edmunds 1999). At present, the research models for the prediction of temperature propagation in the reservoir near the pair horizontal wells while preheating are mainly divided into analytical models and numerical simulation, where the fundamentals of the latter are the same as those of the former. In general, the analytical models of temperature propagation in the formation around a pair horizontal wells upon preheating were studied, given that reservoir thermo-physical properties are constant and the boundary condition is either constant heat flux or constant temperature (Duong 2008; Moini and Edmunds 2013; Ghannadi et al. 2016; Duong, 2008). The published analytical models are only applicable to either the production sage when the steam chamber has already developed and the considerable heat convection induced by condensate flows occurs (Butler and Stephens 1981; Reis 1992; Azad and Chalaturnyk 2010), or only to the startup stage, where the heat conduction is only considered incorporating the factors like the startup strategies (Zhu et al. 2012; Moini and Edmunds 2013). Normally, the steam can hardly infiltrate into the reservoir and therefore no mobile condensate flow exists before switching to production under a adequate pressure difference between injector and producer. However, where the steam injection pressure of the P well is greater than that of the I well, the steam chamber near the P well may increase mobile condensation flows during development, thereby intensifying heat convection, with the potential to further shorten the time of steam circulation. And it is unfortunate that none of those models have mentioned heat convection effects occurring during startup stage.

Existing literature lacks quantitative approaches to describe the effects of injection pressure difference on steam circulation when the steam injection pressure of producer exceeds than that of injector. For this purpose, the effects of various injection pressure differences on the SAGD steam circulation and production stage in the above case are investigated through analytical model and numerical simulation. Also, to increase the accuracy of the simulation, a sensitivity analysis of the temperature prediction model for the startup stage is performed using Monte Carlo method and the finest possible grid is used in numerical simulation. A method without the addition of new ground equipment is proposed for achieving effective and uniform preheating of oil sands reservoir.

Theoretical analysis

Physical model

The temperature distribution between the two wells and the temperature at the midpoint are used as the reference indexes for the preheating effect under different simulation parameters to compare and analyze the effects of different injection pressure differences under the same preheating time. Determining whether the SAGD steam circulation is up to standard is also site-specific due to reservoir characteristics and corresponding SAGD operational parameters. For example, the Athabasca field requires the minimum temperature between two wells to reach 70–90 °C, and the fluid flow in the reservoir between the wells is considered to meet the expectation (Yuan and Mcfarlane 2011), and select 80 °C as the initial temperature for the production stage of SAGD here.

The physical model is shown in Fig. 1 and consists of the pair horizontal wells set in which the injector is located above the producer and the distance between the two boreholes is d. During the preheating process of the SAGD steam circulation, steam is circulated between the two boreholes at different saturated steam temperatures, i.e., there are two different conduction heating sources. The cross-section of the bi-horizontal well is shown in Fig. 1, and the temperature at any point in the formation is affected by both heat sources.

Fig. 1
figure 1

Dual heat source steam circulation for the pair horizontal wells

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Mathematic models

Mathematically, the one-dimensional radial heat flux equation is defined as follows:

$$\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial T}{{\partial r}}\right) = \frac{1}{\alpha}\frac{\partial T}{{\partial r}}\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial T}{{\partial r}}\right) = \frac{1}{\alpha}\frac{\partial T}{{\partial r}}$$
(1)

where \(\alpha\) is the thermal diffusion coefficient. The boundary conditions and initial conditions for a one-dimensional single heat source are as follows (Duong 2008):

$$T = T_{{\text{i}}} - \frac{q}{4\pi k}E_{{\text{i}}} \left( { - \frac{\gamma }{t}} \right)T = T_{{\text{i}}} - \frac{q}{4\pi k}E_{{\text{i}}} \left( { - \frac{\gamma }{t}} \right)$$
(2)

Among them,

$$\gamma { = }r^{2} /(4\lambda \alpha )\gamma { = }r^{2} /(4\lambda \alpha )$$
(3)
$$\lambda = 8.64 \times 10^{4} (1{\text{day}} = 86400{\text{s}})\lambda = 8.64 \times 10^{4} (1{\text{day}} = 86400{\text{s}})$$
(4)

\({\text{E}}_{\text{i}}\) is an exponential integral function defined as follows:

$$E_{{\text{i}}} ( - x) = - \int\limits_{x}^{\infty } {\frac{{e^{ - u} }}{u}} duE_{{\text{i}}} ( - x) = - \int\limits_{x}^{\infty } {\frac{{e^{ - u} }}{u}} du$$
(5)

where u is the dummy variable of integration; Here let,

$$x = - \frac{\gamma }{t} = - \frac{{r^{2} }}{4\lambda \alpha t}x = - \frac{\gamma }{t} = - \frac{{r^{2} }}{4\lambda \alpha t}$$
(6)

This exponential integral function can be changed to a simpler form by a series of transformations:

$$E_{{\text{i}}} ( - x) = \ln (1.781x) + \sum\limits_{n = 1}^{\infty } {\frac{{( - x)^{n} }}{nn!}}E_{{\text{i}}} ( - x) = \ln (1.781x) + \sum\limits_{n = 1}^{\infty } {\frac{{( - x)^{n} }}{nn!}}$$
(7)

When x < 0.01, this reduces to:

$$E_{{\text{i}}} ( - x) = \ln (1.781x)E_{{\text{i}}} ( - x) = \ln (1.781x)$$
(8)

However, when two heat sources exist in the formation, e.g., a steam injection well with a temperature of \({\text{T}}_{\text{s1}}\) and a heating flow rate of \({\text{q}}_{1}\) and a production well with a temperature of \({\text{T}}_{\text{s2}}\) and a heating flow rate of \({\text{q}}_{2}\) at the same time, the change of temperature at any point in the formation is instantaneously altered by the combined effect of these two heat sources. It is known that the temperature change \(\Delta {\text{T}}_{{\text{x}}}\Delta {\text{T}}_{{\text{x}}}\) at any point is:

$$\Delta T_{x} = - \left( {\frac{{q_{1} }}{{4\pi k_{1} }}} \right)E_{i} \left[ { - \frac{{r_{1}^{2} }}{{4\lambda \alpha_{1} t_{1} }}} \right] - \left( {\frac{{q_{2} }}{{4\pi k_{2} }}} \right)E_{i} \left[ { - \frac{{r_{2}^{2} }}{{4\lambda \alpha_{2} t_{2} }}} \right]\Delta T_{x} = - \left( {\frac{{q_{1} }}{{4\pi k_{1} }}} \right)E_{i} \left[ { - \frac{{r_{1}^{2} }}{{4\lambda \alpha_{1} t_{1} }}} \right] - \left( {\frac{{q_{2} }}{{4\pi k_{2} }}} \right)E_{i} \left[ { - \frac{{r_{2}^{2} }}{{4\lambda \alpha_{2} t_{2} }}} \right]$$
(9)

When the temperature change is \(\Delta T_{{{\text{s1}}}}\Delta T_{{{\text{s1}}}}\)、\(\Delta T_{{{\text{s2}}}}\Delta T_{{{\text{s2}}}}\), it is known:

$$\Delta T_{s1} = - \left(\frac{{q_{1} }}{{4\pi k_{1} }}\right)E_{i} \left[ { - \frac{{r_{w1}^{2} }}{{4\lambda \alpha_{1} t_{1} }}} \right] - \left(\frac{{q_{2} }}{{4\pi k_{2} }}\right)E_{i} \left[ { - \frac{{d^{2} }}{{4\lambda \alpha_{2} t_{2} }}} \right]\Delta T_{s1} = - \left(\frac{{q_{1} }}{{4\pi k_{1} }}\right)E_{i} \left[ { - \frac{{r_{w1}^{2} }}{{4\lambda \alpha_{1} t_{1} }}} \right] - \left(\frac{{q_{2} }}{{4\pi k_{2} }}\right)E_{i} \left[ { - \frac{{d^{2} }}{{4\lambda \alpha_{2} t_{2} }}} \right]$$
(10)
$$\Delta T_{s2} = - \left(\frac{{q_{2} }}{{4\pi k_{2} }}\right)E_{i} \left[ { - \frac{{r_{w2}^{2} }}{{4\lambda \alpha_{2} t_{2} }}} \right] - \left(\frac{{q_{1} }}{{4\pi k_{1} }}\right)E_{i} \left[ { - \frac{{d^{2} }}{{4\lambda \alpha_{1} t_{1} }}} \right]\Delta T_{s2} = - \left(\frac{{q_{2} }}{{4\pi k_{2} }}\right)E_{i} \left[ { - \frac{{r_{w2}^{2} }}{{4\lambda \alpha_{2} t_{2} }}} \right] - \left(\frac{{q_{1} }}{{4\pi k_{1} }}\right)E_{i} \left[ { - \frac{{d^{2} }}{{4\lambda \alpha_{1} t_{1} }}} \right]$$
(11)

Since \(q_{1}q_{1}\) and \(q_{2}q_{2}\) in equation cannot be measured, they can only be obtained by mathematically joining the three equations above and eliminating \(q_{1}q_{1}\) and \(q_{2}q_{2}\):

$$\Delta T_{x} = \frac{{E_{i} \left[ { - \frac{{r_{1}^{2} }}{{4\lambda \alpha_{1} t_{1} }}} \right]\left( {\Delta T_{s2} E_{i} \left[ { - \frac{{d^{2} }}{{4\lambda \alpha_{1} t_{1} }}} \right] - \Delta T_{s1} E_{i} \left[ { - \frac{{r_{w2}^{2} }}{{4\lambda \alpha_{2} t_{2} }}} \right]} \right) + E_{i} \left[ { - \frac{{r_{2}^{2} }}{{4\lambda \alpha_{2} t_{2} }}} \right]\left( {\Delta T_{s1} E_{i} \left[ { - \frac{{d^{2} }}{{4\lambda \alpha_{2} t_{2} }}} \right] - \Delta T_{s2} E_{i} \left[ { - \frac{{r_{w1}^{2} }}{{4\lambda \alpha_{1} t_{1} }}} \right]} \right)}}{{E_{i} \left[ { - \frac{{d^{2} }}{{4\lambda \alpha_{1} t_{1} }}} \right] \cdot E_{i} \left[ { - \frac{{d^{2} }}{{4\lambda \alpha_{2} t_{2} }}} \right] - E_{i} \left[ { - \frac{{r_{w1}^{2} }}{{4\lambda \alpha_{1} t_{1} }}} \right] \cdot E_{i} \left[ { - \frac{{r_{w2}^{2} }}{{4\lambda \alpha_{2} t_{2} }}} \right]}}$$
(12)

\(\Delta T_{x}\Delta T_{x}\) can be calculated by substituting the parameters that are known (d, r1,r2, rw1, rw2) or can be estimated (α1, α2) into Eq. 12.

Given specific parameters that are known (d, r1,r2, rw1, rw2) or can be estimated (α1, α2), \(\Delta T_{x}\Delta T_{x}\) can be calculated by Eq. 12. The equation yields the relationship between distance and temperature change, i.e., the distribution of temperature increase between two wells at a specific moment in SAGD steam circulation stage.

The dual heat sources, i.e., the pair horizontal wells parallel to the top and bottom, are injected with steam of the same temperature at the same timein SAGD steam circulation stage, and the spacing between the two wells is kept unchanged, i.e., \(r_{w1} = r_{w2}r_{w1} = r_{w2}\), \(r_{1} = r_{2} = d/2r_{1} = r_{2} = d/2\), \(\alpha_{1} = \alpha_{2}\alpha_{1} = \alpha_{2}\), \(t_{1} = t_{2}t_{1} = t_{2}\), \(\Delta T_{s1} = \Delta T_{s2} { = }\Delta T_{s}\Delta T_{s1} = \Delta T_{s2} { = }\Delta T_{s}\), and it is assumed that:

  1. 1)

    The thermal diffusion coefficient α is a constant value, and the change of the formation's thermal properties is ignored;

  2. 2)

    The inter-floor heat transfer is by heat conduction only, and heat convection is not considered;

  3. 3)

    The heat transfer rate q per unit length is a constant value;

  4. 4)

    There is no interference between adjacent heat sources;

  5. 5)

    The temperature at any point in the borehole is considered uniform and constant \({\text{T}}_{\text{s2}}\).

Temperature change at the mid-point between two wellbores will be (Duong et al. 2008):

$$\Delta T_{{{\text{mid}}}} = \frac{{2\Delta T_{s} E_{i} \left[ { - \frac{{(d/2)^{2} }}{4\lambda \alpha t}} \right]}}{{E_{i} \left[ { - \frac{{d^{2} }}{4\lambda \alpha t}} \right] + E_{i} \left[ { - \frac{{r_{w}^{2} }}{4\lambda \alpha t}} \right]}}\Delta T_{{{\text{mid}}}} = \frac{{2\Delta T_{s} E_{i} \left[ { - \frac{{(d/2)^{2} }}{4\lambda \alpha t}} \right]}}{{E_{i} \left[ { - \frac{{d^{2} }}{4\lambda \alpha t}} \right] + E_{i} \left[ { - \frac{{r_{w}^{2} }}{4\lambda \alpha t}} \right]}}$$
(13)

Sensitivity analysis to mathematic model

However, there are some unavoidable differences among the mathematic model’s predicted value, numerical model simulations and field experiments, because of the existence of a lot of essential model simplifications and hypotheses. Therefore, searching out these differences and interpreting these substantive reasons are supposed to be focused on. For example, the effects of the taking of model parameters on the predicted values or heat convection. They will similarly bring some impact on model validation.

From the above mathematic model (Eq. 12) while heat convection is neglected, it can be seen that the main factors affecting ΔTx are distance between two wellbores (d), thermal diffusivity (α), steam temperature (Ts), preheating time (t) and other variables, and it is possible to compare the extent of the impact of these parameters on ΔTx through sensitivity analysis, thereby gaining further insight into their significance in the accuracy of model output For parameter sensitivity analysis, a common method is to change the value of each parameter and observe the change of ΔTx, which can only analyze a variable at a point in time and called univariate analysis. The global sensitivity analysis (SA) method is selected for this paper, which can rank the input variables qualitatively or quantitatively with respect to their influences on the model output uncertainty (Saltelli 2002; Chong and Hermann 2019; Zdeněk, 2019). To facilitate the application of SA, many methods have been proposed to estimate sensitivity indices, such as Monte Carlo (MC)method (Saltelli 2002), Polynomial Chaos Expansions(PCA)(Shao 2017), and state-dependent parameter method (Li and Lu 2017). Among them, Monte Carlo approaches can be classified into two categories: event-driven and time-driven (Zhao et al. 2007). The basic distinction is the respective treatment of the relationship between events and time steps. Considering the variable t in the mathematic model, MC method is used here for sensitivity analysis, a statistical approach established by Metropolis and Ulam (Metropolis and Ulam 1949).

The midpoint temperature prediction model (Eq. 13) is used here as an example for the sensitivity analysis. For dynamic models, variance-based SA can be applied to each discrete time output, i.e. the corresponding sensitivity indices are generated at each time instant (Lamboni et al. 2011).The distribution of the parameters needs to be defined first, and the interactions between the factors of the parameters as well as possible nonlinear effects are ignored here. Based on the field data (Duong 2008; (AER), 2017), d, rw, α, and ΔTs are considered as random variables whose distribution parameters are given in Table 1. t represents the time variable varying within (0, 100] which is discretized into 10 time nodes evenly, and other parameters are normally distributed.

Table 1 Distribution parameters of the input
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Table 2 show the results of the first-order sensitivity indices and the total sensitivity indices at various moments. In general, first-order sensitivity index (S1) and total sensitivity index (ST) are used to evaluate the results of SA, which indicate the contributions of individual input parameters to the variability of model output. In the process of mathematical modeling, sensitivity indices is the rate of change of the model output when one of the input parameters of the model is changed slightly. S1 represents the contribution of a single parameter to the output variability, with larger values indicating more significant influence of that parameter value changes on the output. ST represents the comprehensive contribution of a single parameter value changes and its interactions with other parameters to the output variability, with larger values indicating more significant influence of the parameter and its interactions on the output. If the ST of a parameter is significantly greater than its S1, then the interactions of that parameter are relatively significant. By comparing S1 and ST, it is able to analyze which parameter nearly have no impact on the variance of the whole dynamic model output, thus, can be fixed at any value in their uncertainty ranges to simplify models.

Table 2 First-order/total sensitivity indices estimated at various moments
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For a clearer comparison, the generalized first-order sensitivities and total sensitivities at t = 50 day are graphically shown in Fig. 2. It can be seen from Table 2 and Fig. 2 that the first-order sensitivity indices are nearly the same as the total sensitivity indices, this indicates that the interaction of the input variables has little effect on the variance of the model output in this example. Besides, the importance ranking obtained by both the first order and total order effect is ΔTs > d > rw > α. Thus, by reducing the effect or enhancing accuracy of ΔTs, the variability of the dynamic output can be significantly reduced. d, rw and α can be taken as a generic constant value, ignoring the subtle effects of their taken values on the prediction results.

Fig. 2
figure 2

First-order/total sensitivity indices at t = 50 day

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In published articles, there are numerous discussions of each parameter in the process of values being considered as averages or approximations and substituted into similar mathematical models (Duong 2008; Yuan and McFarlane 2011; Yanfang et al 2019). However, based on the results of the sensitivity analysis above, it is recommended to focus on the values of ΔTs taken due to the high variability of steam temperatures in horizontal wells, thus to predict ΔTx for each horizontal section separately. This will serve better the purpose of optimizing, predicting and guiding the oil sands project during the SAGD steam circulation.

Reservoir simulation

Simulation methods

The effects of heat convection is ignored in the mathematic model derivation, and thus the impacts of injection pressure difference on the startup stage cannot be reflected. For comparison, the existing horizontal well model provided by CMG STARS is used to realize the simulation of the pair horizontal wells, and analyze the influence of the injection pressure difference ΔP (= Pinj-Pprod + pgd) between the two wells on the results of reservoir preheating. It is assumed that the tubing and the tailpipe are coaxial in the model, but in the actual situation of the field, the oil tubing and the tailpipe may have different axes. Where Pprod is the steam injection pressure of the toe end of the horizontal section of the short tubing, and Pinj is the gas injection pressure of the toe end of the horizontal section of the long tubing, and the injection pressure of the perforation location in the horizontal section is controlled to be the same during the process of the SAGD steam circulation. The vertical distance between the two horizontal wells is 49 m (at this time the natural hydraulic pressure is about 49 kPa, i.e., ρgd = 49 kPa). The injection pressure difference should shorten the time to achieve uniform connectivity between wells as much as possible, while ensuring that the injection pressure difference does not lead to fractures between the injector and producer.

Reservoir and fluid properties

Reservoir model parameters are summarized as shown in Table 3. A three-dimensional non-homogeneous grid model with a width of 80.5 m, a height of 34 m, and a well length of 600 m was established for the pair horizontal wells preheated by the steam circulation stage, and a spatially correlated but stochastic distribution of porosity and permeability was set up for each layer to achieve non-homogeneous reservoir, and the average values of the porosity, horizontal permeability, and vertical permeability of the grid area were 0.35, 4 Darcy, and 2 Darcy, respectively, and the distribution of porosity and permeability is shown in Fig. 3. The grid size and Cell with wellbore is 0.2 × 0.2 m and 0.2 m, respectively. The original oil and water saturations are 0.83 and 0.17, respectively, and the initial oil content is about 35,181 Sm3. The distance between horizontal wells (d) is constant.

Table 3 Reservoir model parameters
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Fig. 3
figure 3

Schematic diagram of three-dimensional non-homogeneous model with permeability and porosity distribution in the horizontal plane (Left horizontal permeability, right porosity)

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Model grid sensitivity analysis

In order to reduce the influence of grid delineation on the simulation results, the grids including reservoir and wellbore were analyzed against each other using 0.5 × 0.5 m, 0.2 × 0.2 m, and 0.1 × 0.1 m grids, respectively. At this time, the control injection pressure difference ΔP = 49 kPa (where Pprod = Pinj = 2000 kPa), and the temperature distributions of the grid models of 0.5 × 0.5 m, 0.2 × 0.2 m, and 0.1 × 0.1 m along one perforation-hole cross-section of the pair horizontal wells correspond to Fig. 4a, b, and c, respectively, during the simulation of steam circulation of 90 days.

Fig. 4
figure 4

Temperature distribution along a section of pair horizontal wells with different wellbore/grid (Day 90)

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As observed in Fig. 4, the shapes of temperature distributions under different grids follow roughly the same trend, and when the injection pressure difference between the producer and the injector is 0, the area of the steam chamber in the region near the injector is larger than that of the producer. This is due to the existence of the natural hydraulic pressure between the two wells, so under the differential effect of gravity, the steam flows upward and the mobile condensation flows naturally to the producer. The corresponding midpoint temperatures between the two wells under different grids are 111.3 °C(0.5 × 0.5 m), 71.5 °C(0.2 × 0.2 m) and 56.7 °C(0.1 × 0.1 m), respectively. That is, when the grid size is 0.5 × 0.5 m, the simulation results of the midpoint temperature are about two times higher than that of the grid of 0.1 × 0.1 m. The degree of steam chamber development and the temperature of the area near the wellbore of the former are also significantly larger than that of the latter. It can be seen that the accuracy of grid delineation notably affects the temperature distribution of steam circulation. The simulated temperature values between the two pair horizontal wells decrease as the fineness of the model grid increases.

4. Results and discussion

Model validation

It is impossible to directly measure the midpoint temperature in inter-well area by underground measurement experiments. At present, temperature falloff test has been extensively employed for an in-situ heating process to determine the heating characteristics along the horizontal wellbores (Duong 2008; Zhu 2012; Yanfang et al 2019). Duong (Duong 2008) conducted a comprehensive evaluation of radial conductive heating models for a SAGD well pair and developed a temperature falloff test methodology, which is the first known application of conductive heating analysis. Zhu et al. (Zhu 2012) used transient temperature analysis to evaluate steam circulation in the SAGD startup process; Yanfang et al. (Yanfang et al 2019) considered the latent heat effect caused by the phase change of bitumen was involved for thermo-phase change coupled analyses during SAGD startup process in his modeling, and they all conducted an analysis for the temperature falloff after steam chamber forms, so as to find out whether there is a good match between his model and Duong’s. In the case of SAGD circulation modeling studies above, temperature falloff test are generally performed after steam injection or circulation of the wellbore for a long period of time, and the focus is on the wellbore rather than the midpoint between wells.

In this study, the temperature falloff simulation by CMG was conducted. For example, at Pinj-Pprod = 100 kPa and after 180 days of steam circulation, I and P wells were shut in for 100 days. The temperature evolution at the inter-well midpoint and wellbore with shut-in time are shown in Fig. 5. The semi-log coordinate was used to obtain an approximate linear relation in Fig. 5, which is also used to determine the zero-intercept and cooling time. During the shut-in period, the temperature falloff curves with the shut-in time were observed for analyses. The dimensionless time is equal to (Ts—T)/(Ts—Ti), where T is the reservoir temperature. The results revealed that the temperature simulated by the model has an approximate linear relationship with the model proposed by Duong, Zhu and Yanfang et al. (Duong 2008; Zhu 2012; Yanfang et al 2019).

Fig. 5
figure 5

Semi-log plot for temperature falloff test conducted on the wellbores

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Meanwhile, the mathematic model derived in the front has been verified by formers and applied for decades, which provided a strong credibility. Here, the parameters used in numerical models are also substituted into the mathematical model which neglects thermal convection and the results are compared for validation and analysis. It can be seen from Fig. 6 that the midpoint temperature curves predicted by the numerical simulation have similar trends and values to those predicted by the mathematic model when t < 80 days. The R2 correlation coefficient between the two curves and the data of mathematic model is greater than 0.97 when t < 80 day, which indicates that numerical simulation models have credibility in their predictions at the early stage of the steam circulation. But then the trend of the numerical simulation model prediction curve when Pprod-Pinj = 100 kPa significantly diverges from the other curves at around t > 80 days. That is attributed to the increasing temperature between the wells to over 80 °C, the oil sand bitumen between the two wells gradually beginning to mobility, which results in the intensification of the heat convection between steam and oil sand bitumen (or the formation fluid). In the mathematica model derivation process, the possible heat convection is ignored, so the subsequent prediction value has a large deviation. Besides, possibly influenced by factors such as gravitational divergence, the midpoint temperature is higher when pressure difference is Pinj-Pprod = 100 kPa than the midpoint temperature when pressure difference is Pprod-Pinj = 100 kPa, and the subsequent heating rate is significantly faster than in other cases. As a whole, Fig. 6 shows that the midpoint temperature predicated by the numerical model has a good agreement with mathematical model, especially at the early stage of the steam circulation.

Fig. 6
figure 6

Comparison of Midpoint Temperature calculated in various method

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Combining the results of the above two validation methods, the comparison results indicate that the numerical model can provide a reasonable prediction in the SAGD start-up process. However, due to numerous essential model simplifications and assumptions, there are still unavoidable discrepancies between the predicted model and validation method. It is essential to Identify and interpret these differences.

Pressure difference effects on preheating

The preheating effects of different pressure difference strategies is compared by observing the temperature distribution near the wellbore. The steam injection pressure set in the model, the time to reach full connectivity between wells and the midpoint temperature at day 60 are shown in Table 4, Other fixed inputs of reservoir parameters not specifically stated are shown in Table 3. The wellbore size is set to 0.2 m, the grid size is 0.1 × 0.1 m (including production wells and injection wells), the injection pressure is kept the same at the injection wells and the horizontal well injection holes, and the simulation time of the steam circulation time is 180 days, in order to investigate the effects of the pressure difference in the injector and producer and effects of the value of pressure difference. The steam injection rate is 75m3/day during circulation. Comparison of wellbore cross section temperature profiles at day 60 of model preheating and when full inter-well connectivity is reached for each group is shown in Fig. 7.

Table 4 Different pressure difference scenarios corresponding midpoint temperatures and the date of reaching full connectivity between wells
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Fig. 7
figure 7figure 7

Temperature distribution of the wellbore cross-section at the corresponding date for each modeling

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When there is no pressure difference (No.3, \(\Delta P=0\)) between the two wells, the midpoint temperature is 50.9 °C at 60 days of steam circulation, and the wells are not connected within 180 days of the simulation. Comparing Models No. 1 and 5 as an example, the former is for the conventional strategy for SAGD startup stage where the injection pressure of injector is larger than that of the producer, while the opposite is true for the latter. As can be seen in Fig. 7, the former is fully connected between the two wells of this cross-section at about 120 days, while the latter takes about 90 days only. And the midpoint temperatures of No.1 and No.5 are 52.6 °C and 69.2 °C at day 60, respectively. Variation of midpoint temperature with time for different injection pressure is shown in Fig. 8. Combined with other subsequent model preheating results in Table 4, Fig. 7 and Fig. 8, it can be found that at the same moment, the midpoint temperature rises rapidly with the increase of the pressure difference, and the preheating results will be more pronounced and time-saving when the injection pressure of the producer is larger than that of the injector and the pressure difference is increasing (No.4–8).

Fig. 8
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Temperature variation with time at middle between injector and producer

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This phenomenon in preheating stages when ΔP < 0 is due to the fact that in the area near the production well, the steam is influenced by gravity separation, and it is always the superheated steam which contains a large amount of enthalpy that moves upward to first contact the cold oil sands between the two wells. Then, the mobile condensation flows downward to area near the production well, exacerbating heat convection at the edge of steam chamber between the two wells. This means that most of the heat convection occurs between the two wells when ΔP < 0. Compared to ΔP > 0 (conventional preheating method), the superheated steam transfers most of the heat upward to the region above the injection wells, while the region between the wells is heated by the mobile condensation flows, deriving less heat convention and heat transfer than in the ΔP < 0 preheating strategy.

It is also worth noting from Fig. 7 and Fig. 8 that when ΔP changes from 0 to -200 kPa (No.2–5), the steam chamber area and overall temperature between the two wells increase significantly. For example, the midpoint temperature at day 60 increases from 50.9 to 69.2 ℃, i.e., the increase of producer injection pressure by 10% leads to the increase of the midpoint absolute temperature by 35.95%. When the ΔP increases from − 200 to − 500 kPa (No.5–8), the area and overall temperature also increase, but more significantly than before. The midpoint temperature increases from 69.2 °C to 161.7 °C, and the injection pressure of producer increases by 13.6% leads to the increase of the midpoint absolute temperature by 133.67%. It suggests that a further increase in pressure difference could further shorten the time to reach full connectivity between the two wells. The temperature distribution of the whole horizontal well section is shown in Fig. 9. The regions of higher temperature also correspond to the distribution of steam chamber. The area of high temperature distribution is the volume of steam chamber. In the case of the traditional SAGD without steam circulation, there are only two areas where the steam chamber is obviously developed, which is much smaller than the other case. The preheating effect when Pprod-Pinj = 200 kPa is less than that of when Pprod-Pinj = 400 kPa. At the meanwhile, the total volume of steam chamber when Pinj-Pprod = 400 kPa is larger than that when Pprod-Pinj = 400 kPa. However, it is obvious that there are multiple steam chamber breakthroughs along the entire horizontal segment at around 90 ~ 120 days due to the reservoir non-homogeneity, which will result in the undeveloped areas of steam chambers being difficult to drainage during production stage.

Fig. 9
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Preheating effect of the steam circulation in different case

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In conclusion, when ΔP < 0, the heat convection between the two wells can be enhanced by increasing the absolute value of P during the preheating stage, which could further shorten the preheating time and enable more uniform preheating. But considering the non-homogeneous of reservoir, too high a pressure difference may lead to localized steam chamber breakthrough. Therefore, the appropriate injection pressure difference should be considered according to the specific reservoir characteristics.

Pressure difference effects on production

In order to further investigate the influence of different steam circulation strategies on subsequent formal production, the subsequent stage is simulated by CMG. In the model, steam circulation is carried out under different strategies for 120 days, and the steam circulation rate was 75 m3/d, then turn to formal production. The temperature distribution, oil distribution and chamber profile at different moments of the model are shown in Figs. 10, 11 and 12, respectively. It can be seen that only the oil sand bitumen in the area affected by the steam has the mobility due to the high temperature, and therefore the oil saturation distribution and temperature distribution are basically consistent with the chamber profile by comparing Figs. 10, 11 and 12. Impacts of different steam circulation strategies on oil production rate, cumulative oil production, oil recovery and so on are shown in Figs. 13, 14, 15, and 16, which shows that all production indexes of traditional SAGD process is lower than that of SAGD with enhanced circulation such as oil production rate or oil recovery. Whereas the emphasis of this paper is to analyze the effects of steam circulation at various injection pressure difference on formal production, the results of the traditional SAGD method will not be overanalyzed extensively and only be used as a benchmark parameter for comparison.

Fig. 10
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Temperature distribution under formal production

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Fig. 11
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Oil saturation distribution under formal production

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Fig. 12
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Chamber profile under formal production with different pressure difference

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Fig. 13
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Impact of different steam circulation strategies on oil rate

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Fig. 14
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Impact of different steam circulation strategies on steam-oil rate

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Fig. 15
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Impact of different steam circulation strategies on cumulative oil

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Fig. 16
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Impact of different steam circulation strategies on oil recovery

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The steam temperature is determined by the saturated steam pressure. And the viscosity of oil sand bitumen is very sensitive to temperature. Thus, the steam chamber pressure has a great influence on the SAGD performance. According to Fig. 13, due to the premature breakthrough of localized steam chambers and the high extent of localized growth, oil production rate when Pinj-Pprod = 400 kPa is higher during the initial phase (0–120 day) than that of the case when Pprod-Pinj = 400 kPa. But in the formal production stage during 120–4015 day, the latter's (Pprod-Pinj = 400) oil production rate is greater than the former's (Pinj-Pprod = 400 kPa) at the same time, due to the more uniform development of steam chambers, which allows for a larger conformance. Comparing the cases of Pprod-Pinj = 200 kPa and Pprod-Pinj = 400 kPa, the oil production rate at low negative pressure difference is slightly lower, suggesting that increasing the value of the suitable pressure difference can accelerate the oil rate and shorten the production period. With the subsequent production turn to the end stage, the temperature, oil and steam chamber distributions corresponding to each model from Figs. 10, 11 and 12 are becoming consistent, and the curves for production indexes in Figs. 13, 14, 15, and 16 also show similar trends at the end of production. At the same time, it is shown in Fig. 14 that the negative injection pressure difference during steam circulation can also reduce steam usage in the first and middle production stage.

To summarize, SAGD process with negative injection pressure difference of steam circulation allows for a more uniform and rapid growth of steam chamber, and better production indexes after transitioning to formal production. In the models with different injection pressure difference scenarios listed in Table 4, the best thermal recovery effectiveness for subsequent production stage is achieved when Pprod-Pinj is in the range of 400–500 kPa, with injection pressure difference being as high as possible to shorten the preheating time. In the numerical model of this paper, certain non-homogeneous coefficient is considered. And in the actual application of the field, for most of the relatively homogeneous oil sand reservoirs, the injection pressure of the pair horizontal wells can be controlled by conventional surface equipments, which can bring about the improvement of economic benefits.

Conclusions

The study could be concluded in the following points:

  1. (1)

    The effects of different injection pressure difference on steam circulation and production stage are investigated by analytical model and numerical simulation, and reliable results are found regarding midpoint temperature by temperature falloff test.

  2. (2)

    A sensitivity analysis of the temperature prediction model for the start-up stage is performed using Monte Carlo method and the finest possible mesh is used during numerical simulations to minimize the factors affecting the simulation accuracy.

  3. (3)

    The preheating results are faster and more uniform than the conventional preheating method when steam injection pressure of producer is greater than that of injector, and the subsequent production indexes are also superior to those of the conventional preheating method.

  4. (4)

    To our knowledge, except for conventional parameter optimization studies, no similar SAGD steam circulation method has been established in terms of maximizing efficiency and uniformity without adding new surface or downhole equipment.