Introduction

Steam flooding is a major EOR process applied to heavy oil reservoirs (Alajmi 2011). As the world is moving toward a depletion era, EOR is required in every field. The demand for energy has been increasing day by day, while the conventional oil reserves are shrinking speedily. In this scenario, heavy oil reserves have become a good source of energy and can fulfill oil demand for a long time in the future. All over the world, billions of barrels of heavy oil and tar sand which cannot be produced by conventional techniques have been reserved. To recover heavy oil, many EOR techniques (Thermal Method, Chemical flooding etc.) are used, of which the thermal method is widely used presently.

Three processes have been evolved in the thermal method: cyclic steam stimulation, steam flooding and in situ combustion. In cyclic steam stimulation, steam is injected into a production well for a period of 2–4 weeks. The well is shut and allowed to soak before re-starting production. The initial production is high because of reduced viscosity at the increased temperature. In steam flooding, steam is injected from the injection well and simultaneously production takes place from the production well. In situ combustion is a displacement process in which an oxygen-containing gas is injected into a reservoir where it reacts with the crude oil to create a high-temperature combustion front that is propagated through the reservoir. In most cases, the injected gas is air.

The present study is concerned with steam flooding. In steam flooding, as the steam zone grows, more oil moves from the steam zone to the unheated zone ahead of the steam front. The oil gets accumulated to form an oil bank. The condensed hot water also moves across the steam front, heating and displacing the accumulated oil. The heated oil with reduced viscosity moves toward the producing well and is produced usually by artificial lifting. There are many factors that promote high displacement efficiency of a steam drive. Firstly, there is expansion of oil and reduction in oil viscosity with temperature (Willman et al. 1961). Secondly, steam distillation of oil left behind the hot water flood increases the overall oil recovery. When a vapor phase (steam) is in the presence of two immiscible liquids (oil and water), each liquid phase exerts its own vapor phase at the temperature of the system. Distillation begins when the sum of the vapor phase (oil and water) equals the total pressure on the system. As a result, oil starts distilling at a temperature much lower than the normal boiling point of its constituents. Thirdly, miscible displacement of residual oil takes place due to the solvent extraction process (Volek and Pryor 1972): Steam distillation strips the more volatile component from residual oil. Steam, enriched with these hydrocarbons, flows through the steam zone and gets condensed at the condensation front. The condensed hydrocarbon contains lighter constituents than the oil resident in that region and can displace some of the oil miscibly. Finally, steam drive is inherently more stable than hot water floods (Prats 1982).

A steam flood project typically proceeds through four phases of development: (1) reservoir screening; (2) pilot tests; (3) field-wide implementation; and (4) reservoir management. Performance prediction is essential to provide information for proper execution of each of these development phases. Three different mathematical models (statistical, numerical and analytical models) are commonly used to predict steam flood performance. The analytical models (volumetric) generally require the entering of few, but critical data. Since, it is much faster to obtain results from analytical models than from simulation, analytical model are still useful tools for preliminary forecasting purposes and sensitivity studies. In addition, the models provide a better insight than simulation into the physics of the thermal process. The economic feasibility of any steam flooding project depends on the accuracy of the predicted production data proposed by the model. One of the most widely used analytical models is the Jeff Jones model. Later Chandra and Mamora (2005) tried to improve this aspect and presented a new analytical model. The objective of this study is focused on improving this aspect of the Jeff Jones model and Chandra and Mamora model. The results of the modified model are tested against results based on field performance for steam drive to verify its accuracy and validity. A more accurate steam flood model will provide engineers with an improved and useful tool for prediction of steam flood production performance. This new model will help to decide the economical feasibility of the project and will help to estimate the total production throughout its economic life.

Theory

Jeff Jones model

Jones (1981) developed an analytical model to calculate the oil production rate during steam flooding, based on the works of Myhill and Stegemeier (1978) and Lookeren (1977). Jeff Jones’ steam flood model is divided into two different parts. The first part of the model calculates an optimal steam injection rate (Eq. 1). The second part uses the related data calculated in the first part in conjunction with additional inputs to predict the oil production history (Eq. 4). Jeff Jones modified Myhill–Stegemeier’s (Myhill and Stegemeier 1978) oil displacement rate and converted it into oil production rate based on correlation with 14 different steam flood projects.

First part

$$ q_{\text{od}} = \frac{{N_{\text{dn}} - N_{{{\text{dn}} - 1}} }}{\Updelta t} $$
(1)
$$ {\text{where}},\quad N_{\text{d}} = F_{\text{os}} V_{\text{s,inj}} $$
(2)
$$ F_{\text{os}} = \frac{{\rho_{\text{w}} C_{\text{w}} h_{\text{n}} }}{{M_{1} h_{\text{t}} }}\Updelta S_{\text{o}} \phi \left( {1 + F_{\text{hd}} } \right)E_{\text{hs}} . $$
(3)

Second part

$$ {\text{Capture}}\,{\text{efficiency}}\, ( {\text{sweep}}\,{\text{efficiency)}} = A_{\text{cD}} \times V_{\text{oD}} \times V_{\text{pD}} $$
(4)

  • AcD = dimensionless area (area sweep efficiency)

  • VoD = volume of displaced oil produced, fraction (volumetric displacement efficiency)

  • VpD = initial pore void filled with steam as water

  • q0 = q0 × capture efficeience.

Chandra and Mamora model

Chandra and Mamora presented an improved model of Jeff Jones seam flood analytical model. AcD (Eqs. 5, 6) was modified to account for the decrease in oil viscosity during steam flood and its dependence on the steam injection rate. VoD (Eqs. 7–10) was modified from its square-root format to an exponential form. The third component, Vpd (Eq. 11), was kept unchanged. The oil production profile was divided into three stages as shown in Fig. 1.

Fig. 1
figure 1

Chandra and Mamora: three stages of oil production profile

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$$ A_{\text{cD}} = \left[ {\frac{{A_{\text{s}} }}{{A\left\{ {\alpha \ln \left( {\mu_{\text{oi}} /100} \right)} \right\}^{1/2} }}} \right]^{2} $$
(5)
$$ \alpha = 0.00015i_{\text{s}} + 0.05 $$
(6)

Limit: 0 ≤ AcD ≤ 1.0 and AcD = 1.0 at μ0 ≤ 100 cp

$$ V_{\text{oD}} = A_{{{\text{cD}}_{\max } }} \left[ {e^{{\left\{ { - \beta \left( {\frac{{N_{\text{d}} - N_{{{\text{d}}_{\max } }} S_{\text{oi}} }}{{N\Updelta S_{\text{o}} }}} \right)} \right\}}} } \right] $$
(7)
$$ \beta = 17.93N_{\text{c}} + 1.3401 $$
(8)
$$ N_{\text{c}} = \frac{{7758Ah\phi \left( {1 - S_{\text{or}} - S_{\text{wc}} } \right)}}{{365i_{\text{s}} t_{\text{c}} }} $$
(9)
$$ t_{\text{c}} = \frac{{t_{\text{cD}} h_{\text{t}}^{2} M_{1}^{2} }}{{35,040K_{\text{h}} M_{2} }} $$
(10)
$$ {\text{Limit: }}A_{{{\text{cD}}_{\max } }} \ge V_{\text{oD}} \ge 0 $$
$$ V_{\text{pD}} = \left[ {\frac{{V_{\text{s,inj}} \times 5.62}}{{43,560A_{\text{s}} h_{\text{n}} \phi S_{\text{g}} }}} \right]^{2} $$
(11)

Limit: 0 ≤ VpD ≤ 1.0 and VpD = 1.0 at Sg = 0.

Modified volumetric model

Steam is lighter than oil, therefore when steam is injected gravity segregation occurs and steam moves vertically upward. This process takes place so rapidly that the reservoir heating occurs mainly due to the vertical expansion of steam zone. Since steam zone thickness varies with time, a new vertical sweep efficiency (Eq. 13) term gets incorporated (ESV) into the capture efficiency. This vertical sweep efficiency has a role when reservoir thickness is more than 200 ft. For reservoir thickness less than 200 ft, it is found that steam zone thickness is equal to sand zone thickness (Green and Willhite 1998). The calculation of average steam thickness is based on energy and mass balance over a thin slice of reservoir in the vertical cross section (Neuman 1985).

$$ {\text{Capture}}\,{\text{efficiency}} = A_{\text{cD}} \times V_{\text{oD}} \times V_{\text{pD}} \times E_{\text{sv}} $$
(12)

where Esv is assumed to be the ratio of steam zone thickness to that of net pay zone thickness:

$$ E_{\text{sv}} = \frac{{h_{\text{st}} }}{{h_{\text{t}} }} = \frac{{C_{\text{w}} \Updelta T_{\text{s}} }}{{h_{\text{fg}} h_{\text{t}} }}\left( {\frac{{\pi tK_{\text{h}} }}{{M_{2} }}} \right)^{0.5} $$
(13)
$$ {\text{where}},\quad h_{\text{st}} = \frac{{V_{\text{s}} }}{{A_{\text{s}} }} $$
(14)
$$ A_{\text{s}} = \left( {\frac{{h_{\text{fg}} f_{\text{s}} i_{\text{s}} }}{{K_{\text{h}} \Updelta T_{\text{s}} }}} \right)\left( {\frac{{K_{\text{h}} t}}{{M_{2} \pi }}} \right)^{0.5} $$
(15)
$$ V_{\text{s}} = \frac{{C_{\text{w}} Q_{\text{s}} t}}{{M_{2} \left( {h_{\text{fg}} + C_{\text{w}} \Updelta T_{\text{s}} } \right)}}. $$
(16)

Jones and Chandra and Mamora both calculated dimensionless steam zone size in their analysis. In the calculation of AcD, they both took Marx and Langenheim (frontal advance model) (Marx and Langenheim 1959) as a reference to calculate steam zone size, As.

$$ A_{\text{s}} = \frac{{Q_{\text{inj}} h_{\text{n}} M_{1} }}{{4K_{\text{h}} \left( {t_{\text{s}} - t_{\text{f}} } \right)M_{2} \times 43,560}}\left( {e^{{t_{\text{D}} }} {\text{erfc}}\sqrt {t_{\text{D}} } + \sqrt {{{t_{\text{D}} } \mathord{\left/ {\vphantom {{t_{\text{D}} } {\pi - 1}}} \right. \kern-\nulldelimiterspace} {\pi - 1}}} } \right). $$
(17)

But in the reservoir, due to gravity segregation, steam zone size differs from that calculated by the Marx and Langenheim model. Neuman developed a new model to calculate steam zone size for the gravity override model (Neuman 1975). According to Neuman model, steam zone size (Eq. 18) is given by:

$$ A_{\text{s}} = \left( {\frac{{Q_{\text{s}} }}{{K_{\text{h}} \Updelta T_{\text{s}} }}} \right)\left( {\frac{{h_{\text{fg}} }}{{h_{\text{fg}} + C_{\text{w}} \Updelta T_{\text{s}} }}} \right)\left( {\frac{Dt}{\pi }} \right)^{0.5} $$
(18)
$$ {\text{where}}\quad Q_{\text{s}} = m_{\text{s}} f_{\text{s}} \left( {h_{\text{fg}} + C_{\text{w}} \Updelta T_{\text{s}} } \right). $$
(19)
$$ A_{\text{s}} = \left( {\frac{{h_{\text{fg}} f_{\text{s}} i_{\text{s}} }}{{K_{\text{h}} \Updelta T_{\text{s}} }}} \right)\left( {\frac{{K_{\text{h}} t}}{{M_{2} \pi }}} \right)^{0.5} \left( {\frac{{350 \times \sqrt {24} }}{43,560 \times 24}} \right). $$
(20)

Case study: field location and setting

Case 1: Schoonebeek oil field

The Schoonebeek oil field is situated in the eastern part of the Netherlands at the German border. The oil is produced from the Bentheim barrier sands of Neocom/Valanginian age. A completely sealing fault divides the accumulation into two separate reservoirs. The southwestern part produces by solution-gas drive and the remainder of the field by edge-water drive. In the greater part of the field, the reservoir consists of a single, unconsolidated sand body. Schoonebeek pilot project is an inverted five-spot pattern. Due to steam drive, oil recovery from this field increased from 5 to 38 %.

Case 2: San Ardo field

The San Ardo is a large oil field in Monterey County, California, in the USA. It is in the upper Salinas Valley and has an anticline structure. The productive units are the Aurignac sands, which are a portion of the huge Monterey Formation. Above the Aurignac sands are the thinner but also productive Lombardi sands. All of the productive units are of Miocene age with the underlying basement rocks dated to the Jurassic period

Case 3: Hamaca field

The Hamaca field is located in Venezuela’s Orinoco heavy oil belt and is a huge stratigraphic trap on the southern flank of the Oriente basin. The Hamaca concession area, which covers 160,000 acres, contains oil trapped in shallow fluvial–deltaic reservoirs of the Oficina Formation (Miocene age). Sandstone reservoirs of the Oficina Formation at Hamaca were generally deposited in a bedload-dominated, fluvial–deltaic environment. Hamaca crude is considered “foamy” and is generally saturated with gas at reservoir conditions.

Reservoir characteristic and operating condition

At maximum production rate, the vertical sweep efficiency is unity because the steam zone thickness is equal to the net pay zone thickness. The various reservoir parameters and characteristics data which have been used to predict the performance of steam flooding are shown in Table 1.

Table 1 Reservoir characteristic and operating conditions of Schoonebeek, San Ardo and Hamaca fields
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Results and discussion

The results obtained using the modified volumetric model agrees with the actual field data in comparison to those obtained from Jeff Jones model and Chandra and Mamora model. Three fields were analyzed for this using macro programming in Excel. The barrels of oil produced per day were plotted against the days of production using the Jeff Jones model as shown in Figs. 2a, 3a, 4a. In contrast to this, the same was plotted using the modified volumetric model as shown in Figs. 2b, 3b, 4b. The graphs clearly show the consistency and accuracy of the new model. Moreover, to further prove the validity of this model, the results of the prediction of the model were compared with that of actual field data. The oil peak rate calculated by the modified volumetric model is close to the actual data, as compared to that calculated using Jeff Jones model and Chandra and Mamora model as shown in Table 2. Chandra and Mamora model’s prediction of cumulative oil production data is very large as compared to the actual field data as shown in Table 3. In this table, it can be clearly seen that the new model gives the best prediction. Similarly, maximum recovery prediction for the three fields shows that modified the volumetric prediction is better off than the other widely accepted models as shown in Table 4.

Fig. 2
figure 2

Oil production profile (barrels of oil produced per day vs. days of production) of the Schoonebeek field. a Jeff Jones Model, b Modified volumetric model

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Fig. 3
figure 3

Oil production profile (barrels of oil produced per day vs. days of production) of the San Ardo field. a Jeff Jones Model, b Modified volumetric model

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Fig. 4
figure 4

Oil production profile (barrels of oil produced per day vs. days of production) of the Hamaca field. a Jeff Jones Model, b Modified volumetric model

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Table 2 Comparison of maximum oil production rate as predicted by different models
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Table 3 Comparison of Cumulative oil production as predicted by different models
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Table 4 Comparison of maximum recovery as predicted by different models
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Conclusion

  1. 1.

    Analytical models (volumetric) are used to predict steam flood performance (preliminary forecasting purposes and sensitivity studies) because these require few data, give quick results and provide better insight, than simulation, into the physics of the thermal process.

  2. 2.

    The most widely used analytical model given by Jeff Jones and then modified by Chandra and Mamora has the limitation of the frontal advance model, and hence give unsatisfactory prediction of oil production.

  3. 3.

    The new modified volumetric model, as presented in this paper, removes the above limitation by considering gravity segregation in the development of the model.

  4. 4.

    Results based on the modified model agree with field results for three different sets of reservoir and fluid properties: Schoonebeek field, San Ardo field and Hamaca field.

  5. 5.

    Engineers will find the modified model an improved and useful tool for prediction of steam flood production performance. In this study, it has been proved that the new modified model can predict more accurately than the earlier existing models.

  6. 6.

    Engineers can predict the economic feasibility and optimum injection rate with the new volumetric model. Performance prediction is essential to provide information for proper execution of each of these development phases. Therefore, this model is very useful for engineers in the decision-making process.