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Transport in Porous Media

, Volume 131, Issue 1, pp 315–332 | Cite as

Modeling and Experimental Validation of Rheological Transition During Foam Flow in Porous Media

  • M. SimjooEmail author
  • P. L. J. Zitha
Article

Abstract

Flow of nitrogen foam stabilized by alpha olefin sulfonate (C14-16 AOS) was studied in a natural sandstone porous media using X-ray Computed Tomography. Foam was generated by a simultaneous injection of gas and surfactant solution into a porous medium initially saturated with the surfactant solution. It was found that the foam undergoes a transition from a weak to a strong state at a characteristic gas saturation of Sgc = 0.75 ± 0.02. This transition coincided with a substantial reduction in foam mobility by a two-order of magnitude and also with a large reduction in overall water saturation to as low as 0.10 ± 0.02. Foam mobility transition was interpreted by the surge of yield stress as gas saturation exceeded the Sgc. We proposed a simple power-law functional relationship between yield stress and gas saturation. The proposed rheological model captured successfully the mobility transition of foams stabilized by different surfactant concentrations and for different core lengths.

Keywords

Foam flow Mobility transition Yield stress Porous media 

1 Introduction

Foam is a dispersion of gas in a continuum liquid phase stabilized by a surfactant (Exerowa and Kruglyakov 1998). Foaming leads to gas mobility decreases significantly (Patzek 1988; Schramm and Wassmuth 1994; Rossen 1996). This has numerous applications in oil and gas recovery operations, including acid diversion during matrix stimulation, (Rossen 1996; Kibodeaux et al. 1994; Behenna 1995) water and gas shutoff, (Hanssen and Dalland 1994; Zhdanov et al. 1996) and mobility control for enhanced oil recovery (EOR) (Turta and Singhal 1998; Simjoo et al. 2012a; Simjoo and Zitha 2018).

In our previous paper (Simjoo et al. 2012c), we investigated in detail foam propagation in porous media where an anomalous transition in foam mobility was observed due to the presence of a backward secondary foam front. A similar behavior of foam mobility was also reported (Apaydin and Kovscek 2001; Nguyen et al. 2003), but was not discussed in detail. The mobility transition is crucial since it determines the mobility reduction factor used as input in modeling and numerical simulations of foam applications. This study attempts to demonstrate that foam mobility transition is a real physical effect and to provide an explanation for this. The premise of this study is that from a macroscopic perspective, foam in porous media is a non-Newtonian fluid (Falls et al. 1988; Zitha and Du 2010), which is characterized by a yield stress: when external forces are smaller than yield stress, foam does not shear, but when external forces are larger than yield stress, then foam shears with a power-law behavior. Since we expect yield stress to increase during foam flow, this rheological model could account for gas trapping and for the observed mobility transition.

In previous studies (Rossen 1988; Falls et al. 1989; Kovscek and Radke 1994; Robert and Mack 1997; Chen et al. 2005), the existence of yield stress during foam flow was described by a threshold pressure gradient. It was found that when the pressure gradient exceeds a certain value, foam generation is triggered due to the mobilization of stationary lamellae, leading to a further reduction in foam mobility (Falls et al. 1989; Rossen and Gauglitz 1990; Tanzil et al. 2002; Kam and Rossen 2003). The threshold pressure gradient may vary from several psi/ft to as low as 1 psi/ft depending on the type of gas, petrophysical properties of porous media, and surfactant type and concentration (Rossen and Gauglitz 1990; Tanzil et al. 2002; Ranshoff and Radke 1988; Yang 1968; Friedmann et al. 1994; Gauglitz et al. 2002). To describe the yield behavior of stationary lamellae on the pore scale level several studies (Falls et al. 1989; Rossen 1990; Cohen et al. 1997; Nguyen et al. 2004) employed the Young–Laplace relation as a balance between the imposed pressure gradient and the equilibrium lamella tension. Others (Rossen and Wang 1999; Balan et al. 2011) described yield stress as a fixed parameter defined by a ratio of surface tension to pore-throat radius, by approximating the porous medium as a bundle of capillary tubes. However, for modeling foam in porous media most of the previous studies (Friedmann et al. 1991; Kovscek et al. 1997; Bertin et al. 1998; Myers and Radke 2000; Kam 2008; Chen et al. 2010; Ashoori et al. 2012) described foam as a simple power-law fluid, thus ignoring the contribution of yield stress. Neglect of yield stress may not always be satisfied during foam flow and using this simplification without caution may lead to miscalculation of foam mobility.

To check this hypothesis we conducted experiments where nitrogen and a surfactant solution (1.0 wt% AOS in 0.5 M NaCl brine) were co-injected in the Bentheimer core with a diameter of 3.8 ± 0.1 cm and a length of 38.4 ± 0.1 cm. CT scan images were taken during foam flow and sectional pressure drops were measured at eleven points through the core length. We also revisited the results of our previous foam flow experiments performed in a shorter core with the length of 17.0 cm (Simjoo et al. 2012b). The paper proceeds with the experimental description, followed by a presentation and discussion of the results. Finally, the main conclusions of the study are drawn.

2 Experimental Description

2.1 Materials

The surfactant used to perform the experiments was C14-16 alpha olefin sulfonate (AOS, Stepan). It was provided as an aqueous solution containing 40.0 wt% of active material and was used as received without further treatment. The surfactant solution was prepared using 0.5 M sodium chloride (NaCl, Merck) and its critical micelle concentration (c.m.c) was 4.0 × 10−3 wt%. Nitrogen gas with a purity of 99.98% was used to perform the experiments.

The porous medium used was Bentheimer sandstone. This is a quartzitic, quasi-homogeneous and isotropic natural porous media. The core samples with a diameter of 3.8 ± 0.1 cm and length of 38.4 ± 0.1 cm were used in the experiments. The absolute permeability to brine was 2.5 ± 0.1 Darcy. The average porosity estimated from the CT images was 21.0 ± 0.1%. The core samples were encapsulated in a thin layer of low X-ray attenuation Araldite self-hardening glue to avoid possible bypassing along the side of the core. After hardening, the glued core was machined to ensure that the core fits precisely into the core-holder. Several holes were also drilled through the glue layer into the core surface along the core length for pressure measurements.

2.2 Experimental Setup

The setup used to perform core-flooding experiments is shown in Fig. 1. It consists of a core-holder in line with a double effect piston displacement pump (Pharmacia Biotech P-500) parallel with a gas mass flow controller (Bronkhorst) and on the other end a back pressure regulator and a fraction-collector for the produced fluids. The pump was used to inject brine and surfactant solution. Nitrogen gas was supplied by a 200 bar cylinder equipped with a pressure regulator (KHP Series, Swagelok) and connected to the core inlet through a mass flow controller. A data acquisition system (National Instruments) was used to record pressure, liquid production and gas and liquid injection rates. The core-holder was placed horizontally on the couch of the scanner. Eleven differential pressure transducers were used to monitor local pressure drops along the core length. The pressure ports divided the core length into twelve equal sections with a length of 3.2 cm. The pressure drop was measured between the core sections. Another differential pressure transducer was also used to record the overall pressure drop.
Fig. 1

Schematic of the experimental setup used to perform foam flow experiment. The core-holder was placed horizontally on the couch of the CT scanner

2.3 CT Imaging Settings

The CT imaging settings used in the experiments are listed in Table 1. The CT scans were obtained using a third generation SAMATOM Volume Zoom Quad slice scanner. The X-ray tube operated at 140 kV and 250 mA. The thickness of each CT slice was 3 mm and one series of scan included 128 slices. The spiral scan mode was used for image acquisition. This allowed fast and continuous acquisition of the CT scan data from a complete volume and generated images using a standard reconstruction kernel after the data interpolation (Mees et al. 2003). The B40 medium filter was used for the reconstruction of the images. A typical slice image consists of 512 × 512 pixels with the pixel size of 0.3 mm × 0.3 mm. Since noise for CT images typically ranges from 3 to 20 Hounsfield units, accuracy of measured fluid saturations was within ± 2%. Image analysis was performed using a series of numerical codes developed by the authors in MATLAB (The MathWorks) and visualization of the 3D CT images was done with Avizo (Visualization Sciences Group).
Table 1

Settings of CT scan measurements

Specification

Quantity

Tube voltage (kV)

140

Tube current (mA)

250

Slice thickness (mm)

3

Pixel size (mm × mm)

0.3 × 0.3

Image reconstruction kernel

B40 medium

Scan mode

Spiral

To compute water saturation Sw, from the measured attenuation coefficients in Hounsfield units (HU), the following equation was used:
$$ S_{\text{w}} = \frac{{{\text{HU}}_{\text{foam}} - {\text{HU}}_{\text{dry}} }}{{{\text{HU}}_{\text{wet}} - {\text{HU}}_{\text{dry}} }} $$
(1)
where subscripts dry, wet, and foam stand for the dry core, surfactant-saturated core, and foam flow, respectively.

2.4 Experimental Procedure

The basic sequence used to conduct core-flooding experiments is given in Table 2. After flushing the core with CO2, it was saturated by injecting brine for at least 10 pore volumes (PV) while maintaining a back pressure of 25 bar. This was done to dissolve any CO2 present in the core and thus to ensure complete core saturation with brine. Then, three pore volumes of the surfactant solution were injected into the core to satisfy its adsorption capacity. Next, N2 and surfactant solution were co-injected to generate foam in the porous medium at a fixed superficial velocity of 4.58 ft/day and with foam quality of 91% at a back pressure of 20 bar and ambient temperature (21 ± 1 °C).
Table 2

Basic sequence used to perform core-flooding experiments

Step

Description

Back pressure (bar)

Surfactant concentration (wt%)

Flow rate (cm3/min)

1

CO2 flushing

5.0

2

Core saturation

25

1.0

3

Surfactant pre-flush

20

1.0

1.0

4

Co-injection of N2 and surfactant

20

1.0

1.0 (gas) + 0.1 (surfactant)

The foam flow experiments were analyzed in terms of CT images, water saturation profiles obtained from these images and reduction in foam mobility. In this study, the foam mobility reduction factor (MRF) was defined as a ratio of measured pressure drop for foam flow to the corresponding pressure drop for the flow of water at the same superficial velocity:
$$ {\text{MRF}} = \left. {\frac{{\Delta P_{\text{foam}} }}{{\Delta P_{\text{water}} }}} \right|_{u} $$
(2)

3 Results and Discussion

In the following section we will first elucidate the dynamics of foam flow for 1.0 wt% AOS foam in a core with the length of 38.4 cm (long core). A detail description of AOS foam flow using a shorter core length (17.0 cm) at different surfactant concentrations was described earlier by Simjoo et al (2012b). Then, we will examine the relation between foam mobility and gas saturation for different AOS concentrations and core lengths.

3.1 Long Core Foam Flow

Figure 2 shows a series of 3D CT images taken at different injected pore volumes during 1.0 wt% AOS foam in the long core. As foam generation proceeds, first a blue front appears in the left side (inlet) of the core after 0.09 PV. This gives a first qualitative impression about the change in water saturation. The frontal region is sharp, indicating that foam displaces the surfactant solution in a characteristic front-like manner. The foam front breaks through the core outlet at around 0.67 PV. Slightly after breakthrough time, a secondary foam front emerges at the outlet region, which propagates backward. The presence of a secondary front is evident by a progressive darkening and spreading of the green zone into the core (see the image at 2.0 PV). The change in color from blue to green means that the secondary front is stronger leading to more liquid desaturation from the previously foam-filled porous medium.
Fig. 2

3D CT images obtained during 1.0 wt% AOS foam in the Bentheimer sandstone core with the length of 38.4 cm. First, a forward foam front (blue colored zone) propagates throughout the core and breaks through at around 0.67 ± 0.02 PV. Slightly after breakthrough time, a secondary foam front (green color) emerges at the outlet region and propagates backward. This leads to further liquid desaturation from the previously foam-filled porous medium. Note that the change in color from blue to green corresponds to lower water saturation

For further analysis of foam flow, water saturation (Sw) profiles at different injected pore volumes are shown in Fig. 3. For each profile, before foam breakthrough, Sw increases from 0.35 ± 0.02 to unity through a sharp transition zone. This behavior of the Sw profile supports the stable displacement of the surfactant solution by foam as shown in the CT images. After foam breakthrough the remaining Sw is distributed uniformly through the core length and exhibits an average value of 0.35 ± 0.02 (see the Sw profile at 1.5 PV). Thereafter, a secondary desaturation front emerges in the outlet region, leading to a substantial reduction in Sw to as low as 0.10 ± 0.02. The secondary foam front propagates with a slower rate compared to the initial forward foam front. Let us consider the Sw profile at 18.7 PV. As the secondary front crosses the downstream core sections, the average Sw diminishes from 0.35 ± 0.02 to 0.10 ± 0.02. This part is followed by a long transition zone extending from 23.0 ± 0.1 to 10.0 ± 0.1 cm toward the core inlet. Through this length Sw increases gradually from 0.10 ± 0.02 to 0.35 ± 0.02.
Fig. 3

Water saturation profiles obtained from the CT images shown in Fig. 2. Closed symbols correspond to the forward foam front until breakthrough time. Open symbols correspond to foam flow after breakthrough time. The presence of secondary foam front is more clearly evident after 1.5 PV: it leads to a significant reduction in the average Sw from 0.35 ± 0.2 to 0.10 ± 0.02

The description of foam propagation can be elaborated further by plotting the sectional MRF for different injected pore volumes as shown in Fig. 4. The inset shows the MRF profiles until slightly after foam breakthrough. It shows that when the foam front reaches each section, the corresponding MRF first increases steeply and then remains practically constant as the foam front passes through the section. The behavior of the sectional MRF correlates well with the Sw profile. Let us consider the MRF profile at 0.22 PV injected. Over the first 10.0 cm the sectional MRF increases from unity to 25 ± 5 and Sw diminishes to about 0.35 ± 0.02 as a result of foam propagation. Then, MRF decreases steeply and reaches unity for the downstream sections subject to single-phase flow, i.e., Sw= 1.0. After foam breakthrough a constant MRF of about 28 ± 5 is established in all core sections, see for instance the MRF profile at 0.71 PV in Fig. 4. This MRF leads to an average water saturation Sw = 0.35 ± 0.02 over the core length (Fig. 3), except near the outlet region where Sw is slightly higher due to the capillary end effect. As foam flow continues, MRF increases significantly starting from the last section near the core outlet proceeding through consecutive core sections toward the core inlet. For instance the sectional MRF over the last section of the core increases substantially from 28 ± 5 to 500 ± 5 after 3.0 PV of foam injection. This means that foam with a higher strength was developed near the core outlet. The steep increase observed above in the sectional MRF coincides with the development of a backward secondary foam front noted in the CT images. The MRF profile also confirms that the secondary foam front travels at a slower rate compared to the initial forward foam front. For instance, after 3.0 PV the secondary front travels up to 27.2 ± 0.1 cm from the core inlet.
Fig. 4

Sectional mobility reduction factors (MRFs) obtained during 1.0 wt% AOS foam. MRF was defined as a ratio of pressure drop for foam flow to that for single-phase water flow. In the inset figure sharp increase in the sectional MRF corresponds to reach foam front to that section. The presence of secondary foam front leads to a substantial increase in MRF (open symbols) compared to the forward foam front (inset figure)

Figure 4 shows that MRF due to the backward secondary foam front is much higher than that due to the forward foam front. A substantial increase in MRF leads to a steep decrease in remaining water saturation: its average value after the forward foam front was 0.35 ± 0.02, whereas it diminished further to 0.10 ± 0.02 in the region swept by the secondary foam front. From the above observations, it follows that during the forward foam flow a weak foam state is established over the entire core length. Then, when the secondary foam front emerges, the foam undergoes a transition to a strong state: through this transition MRF increases by two orders of magnitude. The occurrence of the secondary foam front is consistent with an increase in lamellae generation induced by capillary snap-off mechanism starting at the outlet boundary. Previous studies found that, as non-wetting phase approaches an abrupt permeability increase, capillary pressure may diminish sufficiently in the low permeable region to cause higher capillary snap-off (Yortsos and Chang 1990; van Lingen 1998). Other works showed that foam generation by snap-off at a sudden permeability increase could significantly reduce gas mobility (Falls et al. 1988; Ranshoff and Radke 1988; Rossen 1999; Tanzil et al. 2000; Shah et al. 2018). Coming back to the core experiment results, one could imagine the core outlet boundary as the extreme case of abrupt permeability increase from a finite value to practically infinity. This is highly favorable for generation of gas bubbles, since capillary pressure is nearly zero near the outlet boundary, and thus it is much smaller than the critical capillary pressure required for snap-off. Such foam bubbles can invade into smaller liquid-filled pores, which were not contacted by gas during forward foam propagation.

3.2 Foam Mobility Transition

We shall now examine more closely the behavior of MRF as a function of Sg for different AOS concentrations and core lengths, as shown in Figs. 5, 6 and 7. Gas saturation was obtained by averaging the CT data over each core section. Let us first consider foam flow in the short core for 0.5 wt% AOS (Fig. 5). For the first section, MRF remains at a low value slightly higher than 150 ± 5 with the Sg of 0.73 ± 0.02. For the second section, MRF is initially very low, but when Sg exceeds a certain critical value, MRF increases steeply, reaching a final value of 988 ± 5 at Sg = 0.86 ± 0.02. MRF behavior is even more striking in the third and fourth sections. MRF remains very low as long as Sg is below 0.70 ± 0.02. However, slightly above this value MRF increases significantly and finally reaches a value nearly equal to 2000 ± 5 at Sg = 0.88 ± 0.02.
Fig. 5

Foam mobility reduction factor versus gas saturation during flow of N2 foam stabilized by 0.5 wt% AOS in the Bentheimer sandstone core with length 17.0 ± 0.1 cm. The length of each section is 4.25 cm. Foam mobility transition occurs at a characteristic gas saturation of Sgc= 0.75 ± 0.02

Fig. 6

Foam mobility reduction factor versus gas saturation during 1.0 wt% AOS foam in the Bentheimer sandstone core with the length of 17.0 ± 0.1 cm. The length of each section is 4.25 cm

Fig. 7

Foam mobility reduction factor versus gas saturation during 1.0 wt% AOS foam flow in the Bentheimer sandstone core with the length of 38.4 ± 0.1 cm. The length of each section is 3.2 cm

Qualitatively, similar MRF behavior was observed for 1.0 wt% AOS foam in the short core as shown in Fig. 6. For the Sg below 0.70 ± 0.02, MRF in all sections exhibits values that are lower than 100 ± 5. However, above a critical gas saturation MRF increases steeply, reaching a final value of 2500 ± 5 at the Sg = 0.90 ± 0.02.

In order to examine the effect of core length on foam mobility transition, Fig. 7 shows MRF as a function of Sg for 1.0 wt% AOS foam in the long core. The general behavior of mobility transition is similar to that observed in the short cores. The first and second sections are characterized by a very low value of MRF only slightly higher than 20 ± 5 and with the Sg of 0.65 ± 0.02. For the third section, MRF increases slightly to 50 ± 5 as Sg approaches 0.70 ± 0.02. By further increase in Sg, MRF exhibits a modest increase as can be seen for the fourth section. As Sg goes beyond a certain value, MRF increases more steeply and reaches a maximum value nearly equal to 1400 ± 5 at the Sg = 0.90 ± 0.02. From the above results, it is inferred that as gas saturation exceeds a critical value, MRF rises sharply indicating again a transition from a weak to a strong foam state. This mobility transition occurs at a characteristic gas saturation of Sgc= 0.75 ± 0.02.

3.3 Mechanistic Description of Foam Mobility Transition

To interpret foam mobility transition, we provide a mechanistic description of foam rheological behavior within a wide range of gas saturations. To this end, we rely upon the stochastic bubble population model (Zitha and Du 2010; Simjoo and Zitha 2015) where the basic postulate is that bubble generation is a stochastic process. In this model foam flow in porous media is described as a yield stress fluid obeying a Herschel–Bulkley rheological model, which can be expressed as follows:
$$ \left\{ {\begin{array}{*{20}l} {\dot{\gamma } = 0} \hfill & {\tau \le \tau_{\text{y}} } \hfill \\ {\tau = \tau_{\text{y}} + \mu_{\text{p}} \left| {\dot{\gamma }} \right|} \hfill & {\tau > \tau_{\text{y}} } \hfill \\ \end{array} } \right. $$
(3)
where \( \dot{\gamma } \) and \( \mu_{\text{p}} \) are shear rate and plastic viscosity, respectively. The plastic viscosity \( \mu_{\text{p}} \) is defined as follows:
$$ \mu_{\text{p}} = \mu_{\text{g}} + K_{0} \left| {\dot{\gamma }} \right|^{m - 1} $$
(4)
where \( K_{0} \) and \( m \) are, respectively, plasticity coefficient and power-law exponent of foam. Foam viscosity can be described by dividing Eq. (3) by shear rate and then we will have:
$$ \mu_{\text{f}} = \frac{\tau }{{\left| {\dot{\gamma }} \right|}} = \frac{{\tau_{\text{y}} }}{{\left| {\dot{\gamma }} \right|}} + \mu_{\text{p}} $$
(5)
Equation (5) shows that foam rheology exhibits different behaviors depending on shear rate. Foam behaves like a solid material if the shear rate approaches zero, while a gas-like behavior is expected when the shear rate becomes very large. To describe the shear rate we approximate porous medium as a bundle of capillary tubes, thus we have:
$$ \left| {\dot{\gamma }} \right| = c_{0} v_{\text{f}} /r $$
(6)
where \( r = c_{1} (k/\phi )^{1/2} \) is pore radius, \( k \) is absolute permeability, \( \phi \) is porosity, and \( c_{0} \) and \( c_{1} \) are geometric constants. Substituting in Eq. (3) using Eqs. (4), (5) and (6) we obtain:
$$ \left\{ {\begin{array}{*{20}l} {v_{\text{f}} = 0} \hfill & {\tau \le \tau_{\text{y}} } \hfill \\ {\mu_{\text{f}} = \mu_{\text{g}} + K_{1} v_{\text{f}}^{m - 1} + K_{2} \frac{{\tau_{\text{y}} }}{{v_{\text{f}} }}} \hfill & {\tau > \tau_{\text{y}} } \hfill \\ \end{array} } \right. $$
(7)
where \( K_{1} \) and \( K_{2} \) read as follows:
$$ K_{1} = K_{0} \left( {\frac{{c_{0} }}{{c_{1} }}} \right)^{m - 1} \left( {\frac{\phi }{k}} \right)^{{\tfrac{1}{2}\left( {m - 1} \right)}} ; \, \quad K_{2} = \frac{{c_{1} }}{{c_{0} }}\left( {\frac{k}{\phi }} \right)^{{\tfrac{1}{2}}} $$
(8)
If yield stress is left out from Eq. (7), the resulting expression will be similar to the common form of foam viscosity equation, which is as follows:
$$ \mu_{\text{f}} = \mu_{\text{g}} + \frac{\alpha n}{{v_{\text{f}}^{1/3} }} $$
(9)
where \( \alpha \) is a constant parameter, and \( n \) is bubble density. By comparing Eq. (9) with Eq. (7) while leaving yield stress out we will have:
$$ m = 2/3 $$
(10)
$$ K_{1} = \alpha n $$
(11)
Since in the porous medium bubbles are as large as the pores, bubble density could be described as a number of pores occupied by gas bubbles per unit volume. From the Kozeny–Carman relationship the number of pores per unit volume is obtained as follows:
$$ n_{\text{pores}} = \kappa \left[ {\frac{{150(1 - \phi )^{2} }}{{\phi^{3} }}k} \right]^{ - 3/2} $$
(12)
where \( \kappa \) is a coefficient depending on the packing system. Here it is assumed to be equal to unity. Therefore, bubble density reads:
$$ n = S_{\text{g}} \left[ {\frac{{150(1 - \phi )^{2} }}{{\phi^{3} }}k} \right]^{ - 3/2} $$
(13)
by substituting in Eq. (7) using Eqs. (10), (11) and (13) we will have the following expression to describe foam viscosity:
$$ \left\{ {\begin{array}{*{20}l} {v_{\text{f}} = 0} \hfill & {\tau \le \tau_{\text{y}} } \hfill \\ {\mu_{\text{f}} = \mu_{\text{g}} + \alpha S_{\text{g}} \left[ {\frac{{150(1 - \phi )^{2} }}{{\phi^{3} }}k} \right]^{ - 3/2} v_{\text{f}}^{ - 1/3} + K_{2} \frac{{\tau_{\text{y}} }}{{v_{\text{f}} }}} \hfill & {\tau > \tau_{\text{y}} } \hfill \\ \end{array} } \right. $$
(14)
For using Eq. (14), we need to find a way to directly quantify yield stress during foam flow. Moreover, the relevant data to describe the behavior of yield stress are scarce in the literature of foam in porous media. In order to examine how yield stress evolves during foam flow, we used Eq. (14) to calculate yield stress by substituting foam viscosity from Darcy’s equation and using the experimental pressure drop and saturation data. For water and gas relative permeability functions we used the following Corey-type correlations:
$$ k_{\text{rw}} (S_{\text{w}} ) = \left\{ {\begin{array}{*{20}l} 0 \hfill & {0 \le S_{\text{w}} \le S_{\text{wc}} } \hfill \\ {k_{\text{rw}}^{0} \left( {\frac{{S_{\text{w}} - S_{\text{wc}} }}{{1 - S_{\text{wc}} - S_{\text{gr}} }}} \right)^{{n_{\text{w}} }} } \hfill & {S_{\text{wc}} \le S_{\text{w}} \le 1 - S_{\text{gr}} } \hfill \\ {k_{\text{rw}}^{0} + \left( {1 - k_{\text{rw}}^{0} } \right)\frac{{S_{\text{w}} + S_{\text{gr}} - 1}}{{S_{\text{gr}} }}} \hfill & {1 - S_{\text{gr}} \le S_{\text{w}} \le 1} \hfill \\ \end{array} } \right. $$
(15)
where \( S_{\text{wc}} \), \( k_{\text{rw}}^{0} \) and \( n_{\text{w}} \) are connate water saturation, end-point water relative permeability and the relative permeability exponent for water. Similarly \( k_{\text{rg}}^{\text{f}} \left( {S_{\text{w}} } \right) \) is expressed as:
$$ k_{\text{rg}}^{\text{f}} (S_{\text{w}} ) = \left\{ {\begin{array}{*{20}l} {1 - \left( {1 - k_{\text{rg}}^{{{\text{f}},0}} } \right)\frac{{S_{\text{w}} }}{{S_{\text{gr}} }}} \hfill & {0 \le S_{\text{w}} \le S_{\text{wc}} } \hfill \\ {k_{\text{rg}}^{{{\text{f}},0}} \left( {\frac{{1 - S_{\text{w}} - S_{\text{gr}} }}{{1 - S_{\text{wc}} - S_{\text{gr}} }}} \right)^{{n_{f} }} } \hfill & {S_{\text{wc}} \le S_{\text{w}} \le 1 - S_{\text{gr}} } \hfill \\ 0 \hfill & {1 - S_{\text{gr}} \le S_{\text{w}} \le 1} \hfill \\ \end{array} } \right. $$
(16)
where \( k_{\text{rg}}^{{{\text{f}},0}} \) is the end-point foamed-gas relative permeability, and \( n_{f} \) is the corresponding Corey exponent. As Eq. (16) shows, no trapped gas term was used in the gas relative permeability function. The absence of gas trapping indicates that during foam development flow occurs at a finite velocity. This is consistent with the CT images (Fig. 2) and water saturation profiles (Fig. 3), reflecting the fact that both forward and backward foam fronts propagate with finite velocity: 58.0 ± 0.1 cm/PV for forward foam and 0.8 ± 0.1 cm/PV for backward foam. Thus, as a first approximation we can discount the trapping effect and use the gas relative permeability function without modification. Now by using the measured pressure drops and saturation data along with other required parameters, Table 3, yield stress can be calculated during the course of foam flow by using Eq. (14).
Table 3

Parameters used in the numerical computations

Parameter

Value

k (m2)

2.5 × 10−12

\( \phi \)

0.21

Swc

0.10

Sgr

0

krw0

0.75

krgf,0

1.0

nw

5

nf

3.4

c1/c0

0.71

α (Pa s2/3 m10/3)

5.8 × 10−16

σgw (N/m)

30.0 × 10−3

uw (m/s)

1.47 × 10−6

ug (m/s)

1.47 × 10−5

µg (Pa s)

1.8 × 10−5

Figure 8 shows the calculated yield stress τy as a function of gas saturation Sg for 1.0 wt% AOS foam in the long core. It shows that τy is small for low Sg, but it rises sharply above a certain gas saturation, Sgc = 0.75 ± 0.020. For the Sg smaller than 0.75 ± 0.02, τy hardly exceeds 1.0 ± 0.1 Pa. However, when Sg is larger than 0.75 ± 0.02, τy rises sharply and finally reaches a maximum value of 43.0 ± 0.1 Pa at Sg = 0.90 ± 0.02.
Fig. 8

Yield stress versus gas saturation for 1.0 wt% AOS foam flow in the Bentheimer sandstone core with the length of 38.4 ± 0.1 cm. A sharp rise of the yield stress occurs above a characteristic gas saturation of Sgc = 0.75 ± 0.02

The dependence of τy on Sg reflects itself in the behavior of MRF as a function of Sg (Figs. 5, 6, 7). When the forward foam front breaks through the core, the average gas saturation along the core is about 0.65 ± 0.02, which is below the Sgc. This results in a very small yield stress according to Fig. 8. Therefore, foam viscosity remains low, leading to a low MRF observed during forward foam flow (see inset in Fig. 4). After foam breakthrough, however, more liquid desaturation occurs in the core due to the presence of the secondary foam front. This leads to a large increase in the average gas saturation, and also, so does yield stress. Thus, a higher foam viscosity is obtained during the secondary foam flow, leading to a large reduction in foam mobility. From the above discussion, it can be inferred that the change in MRF from a low to a high value is essentially due to the build-up of yield stress during foam flow.

3.4 Dependence of Yield Stress on Saturation

The dependence of foam mobility transition on gas saturation has not been discussed in the literature. In order to develop a mechanistic model, we inspired ourselves to bulk foam studies, recognizing that foam in porous media is fundamentally different from bulk foam. The dependence of yield stress on gas volume fraction is a commonly observed feature of bulk foams and concentrated emulsions (Khan et al. 1988; Princen and Kiss 1989; Gardiner et al. 1998; Pal 1999; Quintero et al. 2008): the yield stress rises sharply when the gas or oil volume fraction is higher than a critical value. Several authors (Mason et al. 1996; Saint-Jalmes and Durian 1999; Rouyer et al. 2005) proposed the following power-law relationship to describe the yield stress of bulk foams:
$$ \tau_{\text{y}} = a\frac{\sigma }{r}(S_{\text{g}} - S_{\text{gc}} )^{b} $$
(17)
where \( \sigma \) is surface tension, \( r \) is bubble radius, \( S_{\text{gc}} \) is critical gas volume fraction, \( a \) and \( b \) are model parameters. The behavior of yield stress described by Eq. (17) is qualitatively consistent with our observation for the relation between yield stress and gas saturation shown in Fig. 8. Thus, as a first approximation we adopt this power-law relationship to describe yield stress during foam flow in porous media. We note that the origin of yield stress in bulk foam is from the energy dissipation among the gas bubbles, while in porous media the dissipation mainly results from the interaction of bubbles with the pore walls. To find the model parameters (\( a \) and \( b \)) we fitted the yields stress data obtained from the experiments to Eq. (17). The results are given in Table 4 for foam with different surfactant concentrations and core lengths. The yield stress data are well fitted by the power-law model, in which the coefficient \( a \) ranges from 17 to 52, and the exponent \( b \) ranges from 3.78 to 3.91.
Table 4

Model parameters obtained by fitting Eq. (17) to the yields stress data

Sample

a

b

R2

0.5 wt% AOS foam, short core

52

3.91

0.96

1.0 wt% AOS foam, short core

26

3.84

0.95

1.0 wt% AOS foam, long core

17

3.78

0.96

The Sgc is equal to 0.75±0.02 for all the foam flow experiments

For the sake of simplicity we set the exponent \( b \) at four, and thus yield stress as a function of gas saturation reads as follows:
$$ \tau_{\text{y}} = a\frac{\sigma }{r}\left( {S_{\text{g}} - S_{\text{gc}} } \right)^{4} \, $$
(18)
Thus, when gas saturation is below the Sgc, yield stress is zero and above that it is described by a power-law behavior. The proposed rheological model, with the coefficient \( a \) as the only fitting parameter, was used to calculate foam mobility. This is shown by the solid blue line in Figs. 9, 10 and 11. We also provided the model results (solid red line) for the case where yield stress was left out of the foam viscosity equation, see Eq. (9). The figures show that the ‘model without yield stress’ does not capture the observed MRF behavior during mobility transition above the Sgc. However, by including yield stress into the foam viscosity equation, Eq. (14), the model results successfully captured the foam mobility transition. The good match of the model and the measured MRF supports the above idea that yield stress grows during the secondary foam front.
Fig. 9

Foam mobility reduction factor (MRF) versus gas saturation during 0.5 wt% AOS foam in the Bentheimer sandstone core with the length of 17.0 ± 0.1 cm. The solid lines represent the model results and the open symbols are experimental data

Fig. 10

Foam mobility reduction factor (MRF) versus gas saturation during 1.0 wt% AOS foam in the Bentheimer sandstone core with the length of 17.0 ± 0.1 cm

Fig. 11

Foam mobility reduction factor (MRF) versus gas saturation during 1.0 wt% AOS foam in the Bentheimer sandstone core with the length of 38.4 ± 0.1 cm

We showed that the anomalous transition in foam mobility is a real physical effect and could be interpreted by the surge of the yield stress. We proposed a power-law model to describe the yield stress of foam as a function of gas saturation. In the frame of this model, foam flow in porous media is a yield stress fluid obeying a Herschel–Bulkley rheological model. It is based on the premise that the generated bubbles are as large as the pores and thus bubble density could be described as a number of pores occupied by gas bubbles per unit volume. This model also discounted the effect of gas trapping, reflecting that foam moves with a finite velocity consisting with the CT images and water saturation profiles (Figs. 2 and 3). We found that the proposed model is a valid heuristic representation of the foam mobility transition. However, further work is still needed to refine the model for foams at different experimental conditions. Also, the secondary desaturation effect and its connection to the mobility transition need to be elucidated further by the future studies.

4 Conclusions

Foam propagation in Bentheimer sandstone cores generated by the co-injection of nitrogen and alpha olefin sulfonate (C14-16 AOS) surfactant in 0.5 M NaCl brine was studied. The displacement of the foam fronts was visualized with the aid of a CT scanner. The CT data were examined to map fluid saturations at different times. The sectional pressure drops were measured to determine the foam mobility reduction factor (MRF). The following conclusions can be drawn from this study:
  • The 3D CT images revealed two foam propagation fronts: (a) the forward primary foam front, characterized by MRF = 28 ± 5 and an overall water saturation of 0.35 ± 0.02 and (b) the backward secondary foam front which appears after foam breakthrough, characterized by MRF = 1500 ± 5, and an overall water saturation of 0.10 ± 0.02.

  • The transition of foam mobility from a weak (low MRF) to a strong (high MRF) state occurs at a characteristic gas saturation of Sgc = 0.75 ± 0.02.

  • Foam mobility transition was successfully interpreted in terms of the surge of yield stress during foam flow. It was found that when gas saturation is below the Sgc, yield stress is nearly equal to zero coinciding with weak foam. However, when gas saturation is larger than the Sgc, yield stress increases significantly leading to the observed strong foam.

  • A functional relationship between yield stress and gas saturation was proposed to describe foam mobility transition, which is \( \tau_{\text{y}} = a\left( {{\sigma \mathord{\left/ {\vphantom {\sigma r}} \right. \kern-0pt} r}} \right)\left( {S_{\text{g}} - S_{\text{gc}} } \right)^{4} \). When gas saturation is below the Sgc, yield stress falls to zero, and above that it is described by a power-law behavior.

  • Foam mobility transition was successfully described by the stochastic bubble population model where foam rheology was molded as a power-law fluid with a yield stress term. This combined rheological model which capture foam mobility transition for different surfactant concentrations (0.5 and 1.0 wt%) and core lengths (17.0 ± 0.1 and 38.4 ± 0.1 cm).

Notes

Acknowledgements

We thank H. van Asten, J. Etienne, and M. Friebel of TU Delft for their technical support. M. Simjoo acknowledges the financial support of Iran Ministry of Science, Research, and Technology for this study.

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Faculty of Petroleum and Natural Gas EngineeringSahand University of TechnologyTabrizIran
  2. 2.Department of Geoscience and EngineeringDelft University of TechnologyDelftThe Netherlands

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