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Investigating Supercritical CO2 Foam Propagation Distance: Conversion from Strong Foam to Weak Foam vs. Gravity Segregation

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Abstract

This study investigates how to determine the optimal supercritical CO2 foam injection strategies, in terms of total injection rate (or injection pressure, equivalently) and injection foam quality, to place injected foams deep and far into the reservoir. Two different mechanisms that limit field foam propagation, such as “conversion from strong foam to weak foam” and “gravity segregation,” are examined separately, and the results are combined together. The first is performed by using a mechanistic foam model based on bubble population balance, while the second is conducted by an analytical model (called Stone and Jenkins model) and reservoir simulations with a commercial software (CMG-STARS). Note that the gas-phase mobility, required as a key input parameter for gravity segregation simulations, is calibrated by the mechanistic model, which is a significant advance in this study.

The results from both mechanisms show in general that foam propagation distance increases with increasing injection pressure or rate (which is often limited by the formation fracturing pressure) and increases with decreasing foam quality down to a certain threshold foam quality below which the distance is not sensitive to foam quality any longer. It is found that the mobilization pressure gradient (i.e., the pressure gradient above which foam films are mobilized to create a population of bubbles) plays a key role to determine the distance. Therefore, the injection of supercritical CO2 foams with lower mobilization pressure gradient should be more favored in field applications. As a step prior to real-world reservoir applications, this study deals with a relatively ideal reservoir (i.e., large homogeneous cylindrical reservoir) focusing on the steady state after foam treatment in the absence of oil.

Introduction

EOR Using Gas and Water Injection

Gas injection is one of the most widely used methods in enhanced oil recovery (EOR) processes because of its economic advantages. The gas phase (commonly CO2, N2, produced natural gas, flue gas, or a combination of these) is injected into the reservoir either as miscible or immiscible with reservoir oil. Since these gas phases generally have lower viscosity and density compared to the reservoir fluids, an early breakthrough of the injected gas into the production well typically occurs resulting in poor sweep efficiency. The main underlying mechanism behind it is the instability at the interface between the displacing and displaced phases caused by poor mobility ratio (leading to fingering or channeling) and density contrast (leading to gravity segregation).

Extensive research efforts have been made to predict the volume fraction of a reservoir that can be swept by gas injection at different reservoir and fluid conditions. For example, many laboratory studies in early days (Dyes et al. 1954; Offeringa and Van Der Poel 1954; Blackwell et al. 1959) found out that the low-viscosity and low-density gas tends to channel through and bypass oil in sands with no dip angles. Therefore, in horizontal sands, gas flooding is less efficient as the oil viscosity increases, and the desired oil recovery can be achieved only by injecting a large volume of gas. For dipping sands, there is a competition between gravity segregation and channeling (Lacey et al. 1958). There exists a critical rate (Hill and Inst 1952; Dietz 1953) below which gravity segregation prevails and no channeling occurs, and above which channeling plays a more significant role.

Caudle and Dyes (1957) first suggested the simultaneous injection of water and gas as a method to improve sweep efficiency over gas injection. It was based on the fact that water, if flowing together with gas, decreases gas relative permeability. They attempted to determine the optimal gas–water injection ratio, by using relative permeability curves and fluid viscosities, resulting in the conditions at which gas and water flow at the same velocity.

It is sometimes more convenient in the field tests to inject water alternatively with gas, rather than water and gas co-injection, and this process is called water alternating gas (WAG). Christensen et al. (2001) provide a thorough review of WAG field experiences. The initial design of a WAG process is usually constructed by reservoir simulation studies, and then, the design is optimized, as the field process matures, with recommended gas slug size (i.e., volume of gas to be injected) and WAG ratio (Attanucci et al. 1993). Blackwell et al. (1959) investigated the effect of gravity on WAG process to find that the mobility of gas–water region becomes less of an issue as gas and water segregate more rapidly.

Stone (1982) first investigated the gravity segregation of gas from liquid for water and gas co-injections in a homogenous reservoir once water fractional flow (\( f_{\text{w}} \)) or gas fractional flow (\( f_{\text{g}} \)) is given (note \( f_{\text{w}} + f_{\text{g}} = 1 \)). He developed an analytical equation by applying the Buckley–Leverett (1942) theory to predict the size of the region around the injection well where vertical conformance was good before complete segregation. Jenkins (1984) extended Stone’s equations to obtain a closed-form solution to the equations for estimation of incremental recovery beyond water flooding for homogenous reservoirs. Combining these two together, so-called Stone and Jenkins model, is shown to be also valid in the presence of surfactants (Rossen and Shen 2007), even though it was originally designed for gas–water co-injection. Figure 1 shows three constant regions at the steady state which represent a gas override region [i.e., only gas flowing (\( f_{\text{g}} = 1 \) and \( f_{\text{w}} = 0 \)) at residual water saturation (\( S_{\text{wr}} \))], a water underride region [i.e., only water flowing (\( f_{\text{g}} = 0 \) and \( f_{\text{w}} = 1 \)) at residual gas saturation (\( S_{\text{gr}} \))], and a mixed region in between [i.e., both gas and water flowing at constant water saturation (\( S_{\text{w}} \))]. Note that the prediction of “traveling distance before complete segregation by gravity (\( R_{\text{gs}} \))” is key to successful field implementation during gas–liquid co-injection EOR processes.

Fig. 1
figure1

Three constant-state regions observed at the steady-state gas–liquid co-injection in 2D space predicted by earlier studies (Stone 1982; Jenkins 1984; Rossen and Shen 2007) (gas and liquid phases are assumed to be incompressible)

Gravity Segregation During Foam EOR Process

Foaming gas with surfactant solutions has been suggested to mitigate gravity segregation and improve the mobility ratio within the mixed region (Shi and Rossen 1998). From a series of two-dimensional laboratory experiments, Holt and Vassenden (1997) found reasonably good agreement between the Stone and Jenkins model and their experimental results for the complete gravity segregation distance (i.e., \( R_{\text{gs}} \) in Fig. 1) in gas and water co-injection tests. They observed, however, that when foam is injected, the segregation into gas and liquid is difficult to measure in small-scale experiments because of kinetics involved in foam decay. Rossen and van Duijn (2004) showed that the Stone and Jenkins model is rigorously correct to use for foam if several assumptions are met. Those assumptions include (a) homogenous reservoir, although anisotropic, (b) cylindrical reservoir with open outer boundary, (c) injection well penetrates full reservoir height, (d) steady-state conditions reached during the injection, (e) incompressible phases, (f) no dispersion, and (g) Newtonian rheology for all phases. Analytical modeling (Stone 1982; Jenkins 1984), simulation studies (Shi and Rossen 1998), and experimental results (Holt and Vassenden 1997) indicate that gravity override in foam depends on dimensionless gravity number that is the ratio of gravity force to viscous force.

Shi and Rossen (1998) performed several numerical simulations with homogeneous and anisotropic rectangular and radial reservoirs using UTCOMP (University of Texas Compositional Flood Simulation). They found that the Stone and Jenkins model matches remarkably well with simulation results over a wide range of reservoir properties, geometries, flow rates, foam qualities [or, gas fractions (\( f_{\text{g}} \)), equivalently], foam strengths, foam collapse mechanisms, and coarseness of simulation grids. The results also confirmed that a successful gas injection EOR to overcome gravity segregation, with and without foams, requires horizontal pressure gradient outweighs vertical pressure gradient. Performing N2 foam numerical simulation using CMG-STARS, Rossen and Shen (2007) observed that at fixed injection rate, the length of injection interval does not affect the distance for gravity segregation. They proposed a first guess in required injection pressure by providing an explicit relationship between the injection well pressure and distance to the point of segregation.

Conversion of Strong Foam to Weak Foam

In addition to gravity segregation, there is another mechanism that limits foam propagation in field applications, that is, the conversion of strong foam into weak foam, as demonstrated in Fig. 2. There is a threshold distance (\( R_{\text{csw}} \)) beyond which fine-textured strong foam created near the well (often caused by the turbulence in the well) turns into coarsely textured weak foam, as foam moves away from the injection well. Such a concept of three different states of foam when the pressure gradient (\( \nabla P \)) is controlled was first suggested by the experimental study of Gauglitz et al. (2002) and incorporated into the mechanistic foam modeling later (Kam and Rossen 2003; Lee et al. 2016).

Fig. 2
figure2

Reproduced with permission from Gauglitz et al. (2002), Kam and Rossen (2003) and Lee et al. (2016)

Three different foam states and its implication in field-scale applications.

Figure 2 shows more details about what happens when strong foam is injected into a cylindrical reservoir at the total injection flow rate \( q_{\text{t}} = q_{\text{t}}^{\text{in}} \) that corresponds to the total injection velocity \( u_{\text{t}} = u_{\text{t}}^{\text{in}} \). Note that \( q_{\text{t}} \) remains the same at any radial distance (\( r \)) between the wellbore radius (rw) and the radius to the external boundary (re), if gas and liquid are incompressible, while \( u_{\text{t}} \) decreases with \( r \) (i.e., \( u_{\text{t}} = \frac{{q_{\text{t}} }}{2\pi rH} \) at any \( r \), for a cylindrical reservoir with the uniform thickness of H). It is the pressure gradient (\( \nabla P \)) that governs which state of foam is to be formed at given radial distance, because foam texture (\( n_{\text{f}} \)) increases with the pressure gradient (\( \nabla P \)) monotonically (Kam and Rossen 2003). For example, starting from the injection well (\( r = r_{\text{w}} \)), the reservoir is occupied by strong foam up to the distance where the conversion from strong foam to weak foam takes place (\( r = R_{\text{csw}} \)) and then by weak foam for the radial distance beyond (\( r > R_{\text{csw}} \)).

Once strong foam is formed, the rheology follows the two flow regimes of strong-foam state as discovered by Osterloh and Jante (1992) and Alvarez et al. (2001). Foam rheology in the high-quality regime of strong-foam state is governed by bubble stability near the limiting capillary pressure (\( P_{\text{c}}^{*} \)) [or limiting water saturation (\( S_{\text{w}}^{*} \))]. On the contrary, foam rheology in the low-quality regime of strong-foam state is governed by the transport of bubbles at or near the maximum foam texture (\( n_{\text{f}} = n_{{{\text{f}}\,{ \hbox{max} }}} \)).

Bubble population balance modeling, which this study is based on, is a modeling technique that deals with physical phenomena of bubble creation and coalescence, gas trapping, non-Newtonian rheology, and fluid transports in porous medium. It has been widely used in mechanistic foam modeling in the literature (Kovscek et al. 1995; Friedmann et al. 1991; Kam and Rossen 2003; Lee et al. 2016).

Motivations and Objectives

The objective of this study is to predict how far supercritical CO2 foam can propagate based on two different mechanisms: (1) the conversion of strong foam to weak foam (\( R_{\text{csw}} \)) and (2) gravity segregation (\( R_{\text{gs}} \)), in a wide range of injection conditions. This study can be distinguished from other previous studies on similar topics as follows.

  • First, how strong foam would propagate into the reservoir (before turning into weak foam) where the velocity monotonically decreases with radial distance has not been investigated before, especially when supercritical CO2 foam with very low mobilization pressure gradient (\( \nabla P_{\text{o}} \)) is applied;

  • Second, gravity segregation simulations with commercial software have been performed by many previous studies, but none of them employed gas-phase mobility reduction factors actually calibrated from mechanistic foam model (based on true foam physics and whose model parameters determined from actual model fit to laboratory data);

  • Third, none of the previous studies in the literature have put the above two mechanisms together to demonstrate which of the two becomes more influential under what circumstances; and

  • Finally, this study makes a unique contribution by presenting contour maps to show how far foam can travel, before converting to weak foam or gravity segregation, as a function of injection foam quality (\( f_{\text{g}}^{\text{in}} \)) and total injection rate (\( q_{\text{t}}^{\text{in}} \)), [or injection pressure (\( P_{\text{in}} \)), equivalently]. This is especially important to fill the current gap present for the field implementation of supercritical CO2 foams.

Note that a mechanistic model from Izadi and Kam (2018) is used to provide mobility reduction factors (MRFs) as an input parameter for gravity segregation simulations by CMG-STARS. This study deals with a relatively ideal reservoir (i.e., large homogeneous cylindrical reservoir) in the absence of oil, as a first step, prior to the application in the real situations.

Methodology

Population Balance Modeling

A mechanistic modeling approach based on bubble population balance makes it possible to keep track of the population of bubbles (i.e., foam films or lamellae, equivalently), as foam propagates further into the reservoir. The mechanistic modeling approach captures not only the number of bubbles in unit volume of space [i.e., foam texture (\( n_{\text{f}} \))] but also the relationship between foam texture and other variables such as effective gas viscosity (\( \mu_{\text{g}}^{\text{f}} \)), changes in gas relative permeability (\( k_{\text{rg}}^{\text{f}} \)), trapped and flowing gas saturations (\( S_{\text{gt}} \), \( S_{\text{gf}} \)), non-Newtonian flow behavior.

Because this study investigates supercritical CO2 foam placement in a homogeneous reservoir, a population balance model based on the mobilization and division mechanism is applied. More details of such a model are available in the literature (Kam and Rossen 2003; Kam 2008; Lee et al. 2016; Ortiz Maestre 2017), and all relevant equations are tabulated in Table 1 following the study of Izadi and Kam (2018). Note that the minimum mobilization pressure gradient (\( \nabla P_{\text{o}} \)) is defined as

$$ \nabla P_{\text{o}} = 4 \left( {{\raise0.7ex\hbox{$\sigma $} \!\mathord{\left/ {\vphantom {\sigma {R_{\text{t}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${R_{\text{t}} }$}}} \right), $$
(1)

for a foam film to be mobilized out of pore throat with its radius \( R_{\text{t}} \), if the interfacial tension between gas and liquid is given by \( \sigma \). The minimum mobilization pressure gradient (\( \nabla P_{\text{o}} \)) for supercritical CO2 foam can range less than 1.0 psi/ft easily (i.e., 0.05–5 psi/ft) with effective foamers, while it ranges around 10–30 psi/ft at a typical reservoir permeability (50–500 md) (Gauglitz et al. 2002; Georgiadis et al. 2010). The coefficient in Eq. (1) is 4, rather than 2 (typically shown in Laplace equation), because a foam film consists of two gas–liquid interfaces with almost identical curvatures.

Table 1 A summary of bubble population balance model used in this study.

Conversion from Strong-Foam to Weak-Foam States

Details on the model fit to CO2 foam core flood experiments are shown in Izadi and Kam (2018). An example of foam model parameters from the study to match three different foam states (weak-foam, intermediate, and strong-foam states) and two flow regimes of strong-foam state (high-quality regime and low-quality regime) is presented in Table 2. These input parameters are used as a basis for the prediction of foam propagation distance before strong foam turns into weak foam (\( R_{\text{csw}} \)) in this study, which is demonstrated in Fig. 2.

Table 2 Mechanistic model parameters for supercritical CO2 foam at different mobilization pressure values (\( \nabla P_{\text{o}} \)) fitting three different foam states and two flow regimes of strong-foam state (see Table 1 for equations)

In the case of gas–liquid co-injection EOR (at fixed \( f_{\text{g}} \) condition), the pre-specified total injection rate (\( q_{\text{t}} \)) corresponds to the total superficial velocity (\( u_{\text{t}} \)) that changes as a function of radial distance (note that gas and liquid compressibility can be reasonably assumed to be negligible at the field pressure condition). For a homogeneous reservoir with constant thickness (H), the total velocity (\( u_{\text{t}} \)) at any given radial distance (r) is given by

$$ u_{\text{t}} = \frac{{q_{\text{t}}^{\text{in}} }}{2\pi rH}\quad {\text{for}}\;\;r_{\text{w}} \le r \le r_{\text{e}} $$
(2)

As described earlier, the region with \( u_{\text{t}} > u_{\text{t}}^{\text{csw}} \) (or \( r < R_{\text{csw}} \)) is occupied by strong foam, while the region beyond with \( u_{\text{t}} > u_{\text{t}}^{\text{csw}} \) (or \( r < R_{\text{csw}} \)) is occupied by weak foam as depicted in Fig. 2. Note that \( u_{\text{t}} \) decreases monotonically with \( r \), even though the total rate (\( q_{\text{t}} \)) remains the same (i.e., \( q_{\text{t}} = q_{\text{t}}^{\text{inj}} \)) at any \( r \). Therefore, \( r = r_{\text{w}} \) corresponds to the highest \( u_{\text{t}} \), and \( r = r_{\text{e}} \) corresponds to the lowest \( u_{\text{t}} \). The mechanistic foam model allows velocity-dependent foam rheology to be calculated at three different foam states.

Gravity Segregation of Foam Into Gas and Liquid

The models presented by Stone (1982) and Jenkins (1984) can be combined together and formulated for foam flow by introducing mobility reduction factor (MRF) for gas mobility. Then, the distance for foam to travel before gravity segregation (\( R_{\text{gs}} \)) becomes

$$ R_{\text{gs}} = \sqrt {\frac{{q_{\text{t}} }}{{\pi k_{\text{z}} \left( {\rho_{\text{w}} - \rho_{\text{g}} } \right)g\left( {\frac{{k_{\text{rg}} }}{{\mu_{\text{g}} }} \times \frac{1}{\text{MRF}} + \frac{{k_{\text{rw}} }}{{\mu_{\text{w}} }}} \right)}}} . $$
(3)

or,

$$ R_{\text{gs}} = \sqrt {\frac{{q_{\text{t}} }}{{\pi k_{\text{z}} \left( {\rho_{\text{w}} - \rho_{\text{g}} } \right)g\left[ {B\left( {\frac{{1 - S_{\text{w}} - S_{\text{gr}} }}{{1 - S_{\text{wc}} - S_{\text{gr}} }}} \right)^{{m_{2} }} \times \frac{1}{{{\text{MRF}}\, \times \,\mu_{\text{g}} }} + A\left( {\frac{{S_{\text{w}} - S_{\text{wc}} }}{{1 - S_{\text{wc}} - S_{\text{gr}} }}} \right)^{{m_{1} }} \times \frac{1}{{\mu_{\text{w}} }}} \right]}}} $$
(4)

in full equation. Note that \( q_{\text{t}} \) is total injection rate, \( k_{\text{z}} \) vertical absolute permeability, \( \rho_{\text{w}} \) and \( \rho_{\text{g}} \) water and gas densities, \( g \) gravitational acceleration, \( S_{\text{w}} \) average water saturation in the mixed foam region, \( S_{\text{wc}} \) and \( S_{\text{gr}} \) connate water saturation and residual gas saturation, respectively, \( {\text{MRF}} \) is mobility reduction factor, \( \mu_{\text{w}} \) and \( \mu_{\text{g}} \) water and gas viscosities, A and \( m_{1} \) coefficient and exponent for Corey-type water relative permeability, and B and \( m_{2} \) coefficient and exponent for Corey-type gas relative permeability.

If rock and fluid properties are available at given total injection rate, calculation of \( R_{\text{gs}} \) requires two main input parameters such as \( S_{\text{w}} \) and \( {\text{MRF}} \) in the mixed region where foam is present (see in Fig. 1). These two parameters are constant values in the original studies of Stone’s and Jenkins’s for gas–water co-injection, while they are variables for foam applications as shown by the mechanistic foam model. Because multi-dimensional foam simulation in CMG-STARS also assumes constant \( S_{\text{w}} \) and \( {\text{MRF}} \) values, the space-averaged \( S_{\text{w}} \) and \( {\text{MRF}} \) values calculated from mechanistic foam model are used as input parameters for CMG-STARS simulations. More details on this follow below. How CMG-STARS performs foam simulation can be found in the manual (CMG 2016).

Results

For field-scale supercritical CO2 foam propagation prediction, this study uses a hypothetical cylindrical reservoir with an injection well at the center, penetrating the entire reservoir thickness. Reservoir and operational conditions are selected similar to the Rangely Weber Sand Unit, CO (Jonas et al. 1990) where supercritical CO2 and surfactant solutions are co-injected during field EOR tests (as shown in Table 3). This particular field is chosen because it is relatively homogeneous with a good reservoir thickness (H = 275 ft). Table 4 shows a brief summary of operational conditions in foam field EOR applications available in the literature.

Table 3 Rock and fluid properties of a cylindrical reservoir of interest to be tested in this study
Table 4 Examples of foam field EOR tests in the literature with operation conditions

Propagation Distance Based on Conversion from Strong-Foam to Weak-Foam State (\( R_{\text{csw}} \))

How far the fine-textured strong foam injected at the center propagates into the reservoir before it converts into weak foam is evaluated at three different values of mobilization pressure gradient (\( \nabla P_{\text{o}} \) = 1.0, 5.0, and 30.0 psi/ft).

The first step is to make a fit to experimental data showing three foam states and two flow regimes of strong-foam state [not shown; see Izadi and Kam (2018) for more]. Figure 3 shows the results of pressure gradient (\( \nabla P \)) as a function of total velocity (\( u_{\text{t}} \)) at \( \nabla P_{\text{o}} \) = 1.0, 5.0, and 30.0 psi/ft at \( f_{\text{g}}^{\text{in}} = \;70\% \). For \( \nabla P_{\text{o}} \) = 5.0, and 30.0 psi/ft, the S-shaped curve folding back and forth showing three foam states is shown clearly (e.g., \( \nabla P \) < 25.0 psi/ft for weak-foam state, 25.0 psi/ft < \( \nabla P \) < 30.0 psi/ft for intermediate state, and \(\nabla {\text{P}} \) > 30.0 psi/ft for strong-foam state for \( \nabla P_{\text{o}} \) = 30.0 psi/ft). The fact that the curve does not fold back and forth at low \( \nabla P_{\text{o}} \) looks interesting. In such a case, there is a smooth transition from weak-foam to strong-foam state without intermediate state as shown in the case of \( \nabla P_{\text{o}} \) = 1.0 psi/ft.

Fig. 3
figure3

Foam flow characteristics showing three foam states (strong-foam, weak-foam, and intermediate states) at the mobilization pressure gradient (\( \nabla P_{\text{o}} \)) of 1.0, 5.0, and 30.0 psi/ft (injection foam quality (\( f_{\text{g}}^{\text{in}} \)) = 70%)

By using the results in Fig. 3 and reservoir properties in Table 3 for a hypothetical cylindrical reservoir, Figs. 4 through 6 show how \( {\text{MRF}} \) and \( S_{\text{w}} \) values are distributed as a function of radial distance (r) for \( \nabla P_{\text{o}} \) = 30.0, 5.0, and 1.0 psi/ft, respectively. These figures show the steady-state results when foam is injected into the cylindrical reservoir at \( q_{\text{t}}^{\text{in}} \) = 17,970 ft3/day at a pre-specified \( f_{\text{g}}^{\text{in}} \), ranging from 60% (i.e., wet foam) to 99% (i.e., dry foam). Note that \( q_{\text{t}} \) is identical at any \( r \) (i.e., \( q_{\text{t}} = q_{\text{t}}^{\text{inj}} \)) due to incompressible gas and liquid phases, and thus, \( f_{\text{g}} \) is assumed to be identical at any radial and vertical locations (i.e., \( f_{\text{g}} = f_{\text{g}}^{\text{in}} \)). The threshold foam quality separating the high-quality regime from the low-quality regime (\( f_{\text{g}}^{*} \)) is slightly greater than 70%.

Fig. 4
figure4

Results showing foam propagation distance for strong foam to convert into weak foam (\( R_{\text{csw}} \)) (\( q_{\text{t}}^{\text{in}} \) = 17,970 ft3/day in a range of \( f_{\text{g}}^{\text{in}} \)) at the mobilization pressure gradient (\( \nabla P_{\text{o}} \)) of 30.0 psi/ft: a MRF versus radial distance and bSw versus radial distance

Figure 4a, b shows the steady-state response of mobility reduction factor (\( {\text{MRF}} \)) and water saturation (\( S_{\text{w}} \)) as a function of radial distance (\( r \)) when \( \nabla P_{\text{o}} \) = 30.0 psi/ft. The results show folding curves that are consistent with three foam states. Strong foam that is injected at the center of the cylindrical reservoir propagates further out up to the point where the curves fold back, beyond which the strong foam turns into weak foam. This point, as described in Fig. 2, is called \( R_{\text{csw}} \). For example, for \( f_{\text{g}} \) = 60%, \( {\text{MRF}} \) remains high (between 200 and 300) and \( S_{\text{w}} \) remains low (between 43 and 44%), which is a typical response for strong foam, until \( r = R_{\text{csw}} \) (about 5.5 ft). For \( r > R_{\text{csw}} \), \( {\text{MRF}} \) remains low and \( S_{\text{w}} \) remains high, which is a typical response for weak foam. The portion of the curves folding back (representing the intermediate state) and the weak-foam portion of the curves for \( r < R_{\text{csw}} \) do not appear explicitly, because they are hidden solutions (Gauglitz et al. 2002). Similar aspects are shown in Fig. 5a, b when \( \nabla P_{\text{o}} \) = 5.0 psi/ft with \( R_{\text{csw}} \) about 39.7 ft.

Fig. 5
figure5

Results showing foam propagation distance for strong foam to convert into weak foam (\( R_{\text{csw}} \)) (\( q_{\text{t}}^{\text{in}} \) = 17,970 ft3/day in a range of \( f_{\text{g}}^{\text{in}} \)) at the mobilization pressure gradient (\( \nabla P_{\text{o}} \)) of 5.0 psi/ft: a MRF versus radial distance and bSw versus radial distance

A couple of interesting observations can be made in Figs. 4 and 5 that investigate a range of \( f_{\text{g}} \) at \( q_{\text{t}}^{\text{in}} \) = 17,970 ft3/day and H = 275 ft. First, the cases of \( \nabla P_{\text{o}} \) = 30.0 and 5 psi/ft allow foam propagation distance of only about 5.5 and 39.7 ft, respectively, which seems to be unacceptable in most EOR field applications. Second, thinking of the fact that \( q_{\text{t}}^{\text{in}} \) is proportional to “H × \( R_{\text{csw}} \),” this could be translated into 55 and 397 ft (or, \( R_{\text{csw}} \) 10 times higher) if the reservoir thickness was 27.5 ft (or, H 10 times lower), which then becomes quite acceptable. Last, for strong foams in the low-quality regime (\( f_{\text{g}} \) = 60 and 70%) at \( r < R_{\text{csw}} \), \( {\text{MRF}} \) values are comparable and \( R_{\text{csw}} \) values are almost the same. On the contrary, for strong foams in the high-quality regime (\( f_{\text{g}} \) = 80, 90, 95, and 99%) at \( r < R_{\text{csw}} \), both \( {\text{MRF}} \) and \( R_{\text{csw}} \) values decrease sensitively as foam becomes drier. This demonstrates the importance of injection foam quality: (1) propagation of dry foam becomes increasingly more difficult with increasing foam quality and (2) even when relatively wet foam is required for propagation of stable foams, there is not much benefit of going below \( f_{\text{g}}^{*} \). The former is because of foam instability at high foam quality, and the latter is because of foam texture near its maximum as long as the condition falls in the low-quality regime.

Figure 6a, b shows the steady-state response of mobility reduction factor (\( {\text{MRF}} \)) and water saturation (\( S_{\text{w}} \)) as a function of radial distance (r) when \( \nabla P_{\text{o}} \) = 1.0 psi/ft. The results do not show the intermediate state in this case; rather, in both \( {\text{MRF}} \) and \( S_{\text{w}} \) plots, the transition from the strong foam to weak foam takes place progressively with radial distance.

Fig. 6
figure6

Results showing foam propagation distance for strong foam to convert into weak foam (\( R_{\text{csw}} \)) (\( R_{\text{csw}} \) = 17,970 ft3/day in a range of \( f_{\text{g}}^{\text{in}} \)) at the mobilization pressure gradient (\( \nabla P_{\text{o}} \)) of 1.0 psi/ft: a MRF versus radial distance and bSw versus radial distance

One complication with low \( \nabla P_{\text{o}} \) is that there is no clear cut for \( R_{\text{csw}} \) because the curve does not fold back. Thus, \( R_{\text{csw}} \) is evaluated in two different ways in such a case: (1) \( R_{\text{csw}} \) determined based on the maximum \( {\text{MRF}} \) and (2) \( R_{\text{csw}} \) determined based on a pre-specified MRF value that is still meaningful in the field applications. (MRF = 10 seems to serve as a reasonable target as chosen by this study.) Of course, the former (cutoff based on maximum MRF) provides much more conservative \( R_{\text{csw}} \) values than the latter (cutoff based on MRF = 10). Note that the former (cutoff based on maximum MRF) is in some sense consistent with the earlier examples with folding-back curves (Figs. 4, 5), but the latter (cutoff based on MRF = 10) seems more reasonable way to account for the benefits of lower \( \nabla P_{\text{o}} \). (One may choose MRF value other than 10 such as 20 or 50, but the major findings remain the same.)

An example is shown in Fig. 6a where two horizontal lines determine two different cutoff points and therefore two different \( R_{\text{csw}} \) values. For \( f_{\text{g}} \) = 60%, the cutoff based on maximum MRF gives \( R_{\text{csw}} \) = 92.3 ft, while the cutoff based on MRF = 10 gives \( R_{\text{csw}} \) = 1079.0 ft. This proves the benefit of injecting CO2 with low \( \nabla P_{\text{o}} \) values—that way, supercritical CO2 foam can travel a quite significant distance before turning into weak foam. Except \( f_{\text{g}} \) = 99%, all other foam qualities ranging from 60 to 90% allow stable foam to propagate as much as hundreds or thousands of feet easily, if MRF = 10 is used as a cutoff line.

Once the results similar to Figs. 4 through 6 are constructed, the use of contour plot offers a convenient means to predict how far strong foam propagates before turning into weak foam (\( R_{\text{csw}} \)) as a function of total injection rate and injection foam quality. Such a contour plot, shown in Figs. 7 through 10, is especially helpful to guide field implementation of foam EOR processes. (These contours are constructed based on the calculated values at the positions specified by blue open circles [Figs. 7 through 10).] Note that \( R_{\text{csw}} \) values in these plots are for the reservoir thickness (H) of 275 ft—for other reservoir thickness, the new propagation distance then becomes (H × \( R_{\text{csw}} \))/h, h being the new thickness of interest, at given \( q_{\text{t}}^{\text{in}} \) and \( f_{\text{g}}^{\text{in}} \).

Fig. 7
figure7

Contour plot of strong-foam propagation distance (ft) before turning into weak foam (\( R_{\text{csw}} \)) based on bubble population balance model at the mobilization pressure gradient (\( \nabla P_{\text{o}} \)) of 30.0 psi/ft [reservoir thickness (H) = 275 ft]

Figures 7 and 8 show the contours of propagation distance (\( R_{\text{csw}} \)) at \( \nabla P_{\text{o}} \) = 30.0 and 5.0 psi/ft, respectively. The contour plots show the values of \( R_{\text{csw}} \) in (ft) at different combinations of \( q_{\text{t}}^{\text{in}} \) and \( f_{\text{g}}^{\text{in}} \) (H = 275 ft). The results show that one can make strong foam propagate more, by using higher \( q_{\text{t}}^{\text{in}} \) if \( f_{\text{g}}^{\text{in}} \) is fixed, or lower \( f_{\text{g}}^{\text{in}} \) if \( q_{\text{t}}^{\text{in}} \) is fixed. The sensitivity of \( R_{\text{csw}} \) to \( f_{\text{g}}^{\text{in}} \) at given \( q_{\text{t}}^{\text{in}} \) becomes more significant as foam becomes drier in general, while such a tendency is negligible when foam is wet enough, especially \( f_{\text{g}}^{\text{in}} \) < \( f_{\text{g}}^{*} \) (i.e., foams in the low-quality regime [\( f_{\text{g}}^{*} \) = 70%)].

Fig. 8
figure8

Contour plot of strong-foam propagation distance (ft) before turning into weak foam (\( R_{\text{csw}} \)) based on bubble population balance model at the mobilization pressure gradient (\( \nabla P_{\text{o}} \)) of 5.0 psi/ft [reservoir thickness (H) = 275 ft]

Figures 9 and 10 show the contours of propagation distance (\( R_{\text{csw}} \)) at \( \nabla P_{\text{o}} \) = 1.0 psi/ft, using the cutoff based on maximum MRF and MRF = 10, respectively. The same trend as shown in Figs. 7 and 8 is observed. As discussed earlier, the use of CO2 foams with lower \( \nabla P_{\text{o}} \) [see Eq. (1)] seems much more advantageous when it comes to foam placement deep in the reservoir.

Fig. 9
figure9

Contour plot of strong-foam propagation distance (ft) before turning into weak foam (\( R_{\text{csw}} \)) based on bubble population balance model at the mobilization pressure gradient (\( \nabla P_{\text{o}} \)) of 1.0 psi/ft [reservoir thickness (H) = 275 ft]: cutoff based on maximum MRF

Fig. 10
figure10

Contour plot of strong-foam propagation distance (ft) before turning into weak foam (\( R_{\text{csw}} \)) based on bubble population balance model at the mobilization pressure gradient (\( \nabla P_{\text{o}} \)) of 1.0 psi/ft [reservoir thickness (H) = 275 ft]: cutoff based on MRF = 10

Propagation Distance Based on Gravity Segregation (\( R_{\text{gs}} \))

In addition to the conversion to weak foam, foam propagation is also limited by gravity segregation. This section deals with how to determine the distance before foam segregates into gas and liquid (\( R_{\text{gs}} \)) in two different methods: Stone and Jenkins model and CMG-STARS simulation. The former is a simplified approach assuming fixed values of reservoir and fluid properties, and the latter is more realistic, but complicated, approach accounting for those properties as a function of pressure, temperature, and radial and vertical locations. To evaluate \( R_{\text{gs}} \), the same cylindrical reservoir is selected as shown in the previous section (Table 3). Because both methods assume a fixed and constant value of MRF in the mixed region (even though it is not true physically as shown in Figs. 4 through 6), the results from the mechanistic foam modeling in the previous section are used as input parameters. More specifically, for the mixed region properties, the Stone and Jenkins model uses the maximum MRF value (see Figs. 4a through 6a) and its corresponding \( S_{\text{w}} \) value (see Figs. 4b through 6b) to determine \( R_{\text{gs}} \). This means, for example, MRF = 240 for \( \nabla P_{\text{o}} \) = 30.0 psi/ft, MRF = 278 for \( \nabla P_{\text{o}} \) = 5.0 psi/ft, and MRF = 303 for \( \nabla P_{\text{o}} \) = 1.0 psi/ft, when \( f_{\text{g}}^{\text{in}} \) = 70%, while MRF = 149 for \( \nabla P_{\text{o}} \) = 30.0 psi/ft, MRF = 158 for \( \nabla P_{\text{o}} \) = 5.0 psi/ft, and MRF = 165 for \( \nabla P_{\text{o}} \) = 1.0 psi/ft, when \( f_{\text{g}}^{\text{in}} \) = 90% (\( q_{\text{t}}^{\text{in}} \) remains the same at 17,970 ft3/day). In CMG-STARS simulations, \( R_{\text{gs}} \) is determined by using these MRF values, but letting \( S_{\text{w}} \) values be calculated by the simulator. These \( S_{\text{w}} \) values calculated by the simulator are essentially the same as those \( S_{\text{w}} \) values used for input in the Stone and Jenkins model in Figs. 4b through 6b.

Figures 11 through 13 show the results of CMG-STARS simulations to evaluate the cases of \( \nabla P_{\text{o}} \) = 30.0, 5.0, and 1.0 psi/ft, respectively, at \( f_{\text{g}}^{\text{in}} \) = 70 and 90% (\( q_{\text{t}}^{\text{in}} \) = 17,970 ft3/day). In all cases, the reservoir has wellbore radius (\( r_{\text{w}} \)) of 0.42 ft, radial distance to the reservoir boundary (\( r_{\text{e}} \)) of 1000.0 ft, and reservoir thickness of 275.0 ft. Gas and surfactant solutions, which create strong foam inside the well, are co-injected at the total injection rate (\( q_{\text{t}}^{\text{in}} \)) of 17,970 ft3/day into the reservoir initially saturated with water. The two injection foam qualities (\( f_{\text{g}}^{\text{in}} \)), 70 and 90%, are chosen to represent wet-foam and dry-foam scenarios (or, foams in the low-quality regime and in the high-quality regime), respectively. These results are based on 4000 days of foam injection, which is shown to be (near) steady-state results after some trial-and-error simulations. Additional input parameters specific to CMG-STARS for this simulation task are shown in Table 5 [see Computer Modeling Group (CMG) (2016) for more details on these parameters].

Fig. 11
figure11

Simulation results showing foam propagation distance before gravity segregation (\( R_{\text{gs}} \)) when the mobilization pressure gradient (\( \nabla P_{\text{o}} \)) is 30.0 psi/ft: a\( f_{\text{g}}^{\text{in}} \) = 70% at MRF = 240 and b\( f_{\text{g}}^{\text{in}} \) = 90% at MRF = 149 (Stone and Jenkins model predicts \( R_{\text{gs}} \) = 801 and 701 ft, respectively)

Table 5 Additional foam simulation parameters required by CMG-STARS for gravity segregation simulation.

Figure 11 shows simulation results in terms of the steady-state water saturation for \( \nabla P_{\text{o}} \) = 30.0 psi/ft. When the injection foam qualities (\( f_{\text{g}}^{\text{in}} \)) are 70% (Fig. 11a) and 90% (Fig. 11b), the corresponding MRF values are about 240 and 149 (see Fig. 4a), with \( R_{\text{gs}} \) from simulations leading to 700 and 530 ft, respectively. Drawing horizontal lines from the injection well to the contact point of the three regions, the water saturations in the mixed region are 0.430 and 0.426 in Fig. 11a, b, respectively. For the same case, the Stone and Jenkins model predicts \( R_{\text{gs}} \) values of 801 and 701 ft for \( f_{\text{g}}^{\text{in}} \) = 70 and 90%, respectively. Although there is some difference, the results are comparable showing the same trend. It is believed that the difference is caused by multiple aspects including changes in fluid properties (density, viscosity, compressibility, etc.) as well as simulation artifacts at the injection and production wells (fluid redistribution at the inlet face, capillary end effect, etc.), and as a result the simulation slightly underpredicts \( R_{\text{gs}} \) compared to the Stone and Jenkins model.

Figure 12 shows similar simulation results for \( \nabla P_{\text{o}} \) = 5.0 psi/ft. For the injection foam qualities (\( f_{\text{g}}^{\text{in}} \)) of 70% (Fig. 12a) and 90% (Fig. 12b), the corresponding MRF values are 278 and 158 (see Fig. 5a), the \( S_{\text{w}} \) values are 0.430 and 0.426, and the \( R_{\text{gs}} \) values are 720 and 540 ft, respectively. The Stone and Jenkins model predicts \( R_{\text{gs}} \) of 858 and 720 ft. Once again, the trend is well captured, and the simulation predicts somewhat lower \( R_{\text{gs}} \) values compared to the Stone and Jenkins model.

Fig. 12
figure12

Simulation results showing foam propagation distance before gravity segregation (\( R_{\text{gs}} \)) when the mobilization pressure gradient (\( \nabla P_{\text{o}} \)) is 5.0 psi/ft: a\( f_{\text{g}}^{\text{in}} \) = 70% at MRF = 278 and b\( f_{\text{g}}^{\text{in}} \) = 90% at MRF = 158 (Stone and Jenkins model predicts \( R_{\text{gs}} \) = 858 and 720 ft, respectively)

Figure 13 shows the steady-state simulation results for \( \nabla P_{\text{o}} \) = 1.0 psi/ft with the cutoff based on maximum MRF. For the injection foam qualities (\( f_{\text{g}}^{\text{in}} \)) of 70% (Fig. 13a) and 90% (Fig. 13b), the corresponding MRF values are 303 and 165 (see Fig. 6a), the \( S_{\text{w}} \) values are 0.429 and 0.426, and the \( R_{\text{gs}} \) values are 780 and 540 ft, respectively. The Stone and Jenkins model predicts \( R_{\text{gs}} \) of 884 and 734 ft. The results are consistent with other cases.

Fig. 13
figure13

Simulation results showing foam propagation distance before gravity segregation (\( R_{\text{gs}} \)) when the mobilization pressure gradient (\( \nabla P_{\text{o}} \)) is 1.0 psi/ft (cutoff based on maximum MRF): a\( f_{\text{g}}^{\text{in}} \) = 70% at MRF = 303 and b\( f_{\text{g}}^{\text{in}} \) = 90% at MRF = 165 (Stone and Jenkins model predicts \( R_{\text{gs}} \) = 884 and 734 ft, respectively)

Figure 14 shows simulation results for \( \nabla P_{\text{o}} \) = 1.0 psi/ft with the cutoff based on MRF = 10. This situation is somewhat tricky, because the mechanistic modeling results in Fig. 6a show that the MRF values are mostly much greater than 10 for the region occupied by strong foams. Even so, it is believed to provide a useful insight when compared with Fig. 13. For the injection foam qualities (\( f_{\text{g}}^{\text{in}} \)) of 70% (Fig. 14a) and 90% (Fig. 14b) both with MRF = 10, the simulation shows the \( S_{\text{w}} \) values of 0.494 and 0.456, and the \( R_{\text{gs}} \) values of 185 and 175 ft, respectively. The Stone and Jenkins model predicts \( R_{\text{gs}} \) of 211 and 204 ft. These \( R_{\text{gs}} \) values in Fig. 14 are less than those in Fig. 13, because the use of smaller MRF (i.e., MRF = 10 in Fig. 14) provides lower pressure gradient (\( \nabla P \)), resulting in lower \( R_{\text{gs}} \).

Fig. 14
figure14

Simulation results showing foam propagation distance before gravity segregation (\( R_{\text{gs}} \)) when the mobilization pressure gradient (\( \nabla P_{\text{o}} \)) is 1.0 psi/ft (cutoff based on MRF = 10): a\( f_{\text{g}}^{\text{in}} \) = 70% at MRF = 10 and b\( f_{\text{g}}^{\text{in}} \) = 90% at MRF = 10 (Stone and Jenkins model predicts \( R_{\text{gs}} \) = 211 and 204 ft, respectively)

In order to verify the assumption of constant water saturation in the three constant regions in the Stone and Jenkins model, scanning the saturation map in vertical and horizontal directions works as a convenient means. As shown in each of Fig. 13a, b, one horizontal line through the contact point of three regions (not shown, but the same as the horizontal arrows shown) and the other vertical line somewhat before the contact point, where r < \( R_{\text{gs}} \), are selected as an example.

Figure 15 shows the change in water saturation along the vertical and horizontal scanning lines (shown in Fig. 13) from the simulations. It clearly shows three different constant-state regions—the gas override region where \( S_{\text{w}} \) is near \( S_{\text{wc}} \), the water underride region where \( S_{\text{w}} \) is near \( 1 - S_{\text{gr}} \), and the mixed region in between with foams at its steady state \( S_{\text{w}} \) that matches with MRF values from mechanistic model.

Fig. 15
figure15

Steady-state water saturation profiles along the scanning lines [horizontal (a), vertical (b)] in Fig. 13 from gravity segregation simulations showing three constant regions as approximated by Stone (1982) and Jenkins (1984) model

Figure 16 shows the pressure profiles for the same scanning lines as shown in Fig. 13. The pressure decreases sharply along the horizontal scanning line up to r = \( R_{\text{gs}} \) because of high MRF value in the presence of foams, followed by mild change to the outlet because of single-phase flow of water. The pressure profile along the vertical direction follows hydrostatic pressure gradient concept (higher hydrostatic pressure gradient in the underride region, and lower hydrostatic pressure gradient in the override and mixed foam regions).

Fig. 16
figure16

Steady-state pressure profiles along the scanning lines [horizontal (a), vertical (b)] in Fig. 13 from gravity segregation simulations showing three constant regions as approximated by Stone (1982) and Jenkins (1984) model

Figure 17 shows how the bottom-hole injection pressure changes for those examples shown in Figs. 15 and 16, until it reaches 4000 days of foam injection that is believed to be at, or close to, the steady state after some trial-and-error simulations. In both cases, the injection pressure rapidly increases with time in the beginning as strong foam enters and then levels off gradually as the system approaches the steady state. The cases with higher MRF have higher injection pressures. Note that the outlet back pressure is 1555.0 psia.

Fig. 17
figure17

Change in bottom-hole injection pressure with time simulated by CMG-STARS to reach (close to) the steady state at 4000 days of foam injection

Similar to \( R_{\text{csw}} \) contours in Figs. 7 through 10, the results from the Stone and Jenkins model for \( R_{\text{gs}} \) can be plotted as a function \( q_{\text{t}}^{\text{in}} \) and \( f_{\text{g}}^{\text{in}} \) as well. Figures 18 through 21 show how far foam propagates before gravity segregation (\( R_{\text{gs}} \)) when the MRF values are borrowed from the mechanistic foam model for \( \nabla P_{\text{o}} \) = 30.0, 5.0, and 1.0 (cutoff based on maximum MRF and cutoff based on MRF = 10) psi/ft (Figs. 7 through 10), respectively. It is interesting to find that \( R_{\text{gs}} \) is also very sensitive to \( f_{\text{g}}^{\text{in}} \), i.e., it is becoming increasingly difficult to make drier foams propagate deep into the reservoir, while such a sensitivity is much less for relatively wet foams. Similar to \( R_{\text{csw}} \) contours, \( R_{\text{gs}} \) contours also show longer propagation distance at high injection rate (or higher injection pressure, equivalently) (Fig. 19).

Fig. 18
figure18

Contour plot of foam propagation distance (ft) before gravity segregation (\( R_{\text{gs}} \)) based on Stone and Jenkins model [MRF taken from mechanistic foam model at the mobilization pressure gradient (\( \nabla P_{\text{o}} \)) of 30.0 psi/ft]

Fig. 19
figure19

Contour plot of foam propagation distance (ft) before gravity segregation (\( R_{\text{gs}} \)) based on Stone and Jenkins model [MRF taken from mechanistic foam model at the mobilization pressure gradient (\( \nabla P_{\text{o}} \)) of 5.0 psi/ft]

Combined Results and Discussion

The two mechanisms that limit foam propagation in the field-scale foam EOR can be analyzed together based on the results given in the previous sections. Figures 22 and 23 show such results at the injection foam qualities (\( f_{\text{g}}^{\text{in}} \)) of 70 and 90%, respectively, for \( \nabla P_{\text{o}} \) = 30.0, 5.0, and 1.0 as well as \( \nabla P_{\text{o}} \) = 0.1 psi/ft. Note that when \( \nabla P_{\text{o}} \) is low (1.0 and 0.1 psi/ft), both results from the cutoff line based on maximum MRF as well as MRF = 10 are used. Note in such cases that the results at MRF = 10 overpredicts \( R_{\text{csw}} \) compared to maximum MRF (see Fig. 6), while the results at MRF = 10 underpredicts \( R_{\text{gs}} \) compared to maximum MRF because of lower lateral pressure gradient (see Figs. 20, 21).

Fig. 20
figure20

Contour plot of foam propagation distance (ft) before gravity segregation (\( R_{\text{gs}} \)) based on Stone and Jenkins model [MRF taken from mechanistic foam model at the mobilization pressure gradient (\( \nabla P_{\text{o}} \)) of 1.0 psi/ft (cutoff based on maximum MRF)]

Fig. 21
figure21

Contour plot of foam propagation distance (ft) before gravity segregation (\( R_{\text{gs}} \)) based on Stone and Jenkins model [MRF taken from mechanistic foam model at the mobilization pressure gradient (\( \nabla P_{\text{o}} \)) of 1.0 psi/ft (cutoff based on MRF = 10)]

Both figures show that there is a threshold value (or, range) of \( \nabla P_{\text{o}} \), below which foam propagation is limited by gravity segregation (\( R_{\text{gs}} \)) and above which foam propagation is limited by the conversion of strong foam to weak foam (\( R_{\text{csw}} \)). Because of relatively steep slope in \( R_{\text{csw}} \) curve, foam injection conditions at lower \( \nabla P_{\text{o}} \) (i.e., left-hand side of the figure) have advantages in placing foams deep into the reservoir. Foam propagation distance is less sensitive to \( \nabla P_{\text{o}} \) at lower \( \nabla P_{\text{o}} \), while foam propagation distance can still be improved significantly by making \( \nabla P_{\text{o}} \) lower at higher \( \nabla P_{\text{o}} \). Comparing Figs. 22 and 23, it also shows that foams in the high-quality regime are more difficult to be placed deep in the reservoir than foams in the low-quality regime. Note from Eq. (1) that lower \( \nabla P_{\text{o}} \) translates lower interfacial tension and higher pore throat size, which can be achieved more easily at higher pressure, with better foamer, and at higher absolute permeability.

Fig. 22
figure22

Prediction of propagation distance (ft) of 70% quality foams by combining both mechanisms (conversion to weak foam vs. gravity segregation)

Fig. 23
figure23

Prediction of propagation distance (ft) of 90% quality foams by combining both mechanisms (conversion to weak foam vs. gravity segregation)

This study greatly improves the prediction of \( R_{\text{gs}} \), by using representative MRF values from mechanistic foam model. At the same time, it should be pointed out that there still is room to improve the prediction further, because both Stone and Jenkins model and CMG simulation assume constant values of MRF as an input parameter [see Eqs. (3), (4)]. When it comes to possible errors associated with a constant MRF assumption, the case with lower \( \nabla P_{\text{o}} \) (Fig. 6) would show more errors than the case with higher \( \nabla P_{\text{o}} \) (Figs. 4 or 5), because it presents a continuous and progressive change from strong-foam to weak-foam state without showing hysteretic behaviors (i.e., multi-valued foam rheology surface that folds back and forth). A three-dimensional reservoir simulation with mechanistic modeling capability is believed to reduce the gap essentially.

Even though this study shows how foam propagates in a large system, the results are limited to homogeneous cylindrical reservoirs at the moment. In order to take the results to the real-world field cases, there are challenges to overcome, including (but not limited to) heterogeneity of the system and interaction between foams and reservoir oils. The major finding of this study, however, still holds true—foams with lower \( \nabla P_{\text{o}} \) (e.g., supercritical CO2 foams) are more advantageous over other gaseous foams with higher \( \nabla P_{\text{o}} \) (e.g., foams with gas CO2, gas N2, hydrocarbon gas, flue gas, etc.). It should be noted that the importance of small laboratory-scale coreflood experiments based on the field rock and fluid samples and selected foaming agents cannot be underestimated, because they allow mechanistic model fits to recommend appropriate MRF values at different injection scenarios.

Conclusions

Foam propagation is limited in field EOR processes by two main mechanisms as investigated in this study—the first, conversion from strong-foam to weak-foam state, and the second, gravity segregation of foam into gas and liquid. Dealing with an ideal (large homogeneous cylindrical) reservoir, the results of this study can be summarized as follows:

  • The population-balance foam model shows that the propagation distance before strong foam converts to weak foam (\( R_{\text{csw}} \)) primarily depends on the mobilization pressure gradient (\( \nabla P_{\text{o}} \)). This explains why foams with lower \( \nabla P_{\text{o}} \) (e.g., supercritical CO2 foams) can propagate much further than other gaseous foams with higher \( \nabla P_{\text{o}} \). The results also show theoretically why wetter foams can propagate further than drier foams, and why higher injection rates help longer propagation distance.

  • CMG-STARS simulation and the Stone and Jenkins model confirm that gravity segregation also limits foam propagation distance. Foam propagation distance before gravity segregation (\( R_{\text{gs}} \)) primarily depends on the mobility reduction factor (MRF) that is calibrated by mechanistic model based on fundamental foam physics in this study.

  • Combining both mechanisms together, the results show that foams with lower \( \nabla P_{\text{o}} \) tend to have gravity segregation more dominating factor for foam propagation. On the contrary, foams with higher \( \nabla P_{\text{o}} \) tend to have the conversion to weak foam more dominating factor.

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Acknowledgements

The authors would like to acknowledge Computer Modeling Group (CMG) for generous donation of CMG-STARS for this study.

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Correspondence to S. I. Kam.

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Izadi, M., Kam, S.I. Investigating Supercritical CO2 Foam Propagation Distance: Conversion from Strong Foam to Weak Foam vs. Gravity Segregation. Transp Porous Med 131, 223–250 (2020) doi:10.1007/s11242-018-1125-z

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Keywords

  • CO2 foam
  • Population balance model
  • Foam propagation
  • Gravity segregation