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Tracking a Foam Front in a 3D, Heterogeneous Porous Medium

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Abstract

Foam is to be used as a blocking agent for confining a pollutant source zone and avoid spreading in an aquifer. To this end, it is necessary to determine where injected foam flows and stays inside a porous medium. This study examines the use of electrical resistivity tomography for this purpose. Foam is injected in a large-scale 3D heterogeneous porous medium (0.84 × 0.84 × 0.84 m). During the injection, electrical resistivity tomography measurements are performed. We show that combining a large number of measurements with inversion techniques allows for the monitoring of a foam front in 3D during the injection process.

Introduction

Foam injection is used in petroleum engineering applications as an enhancing oil recovery method. More conventional gas injection processes suffer from poor volumetric sweep (i.e., the portion of the reservoir that is contacted by the gas is low). This is caused by differences in physical properties between the injected gas and the displaced fluid (i.e., viscosity, density). The injected gas has a very low viscosity which leads to fingering through the resident fluid or channel through high-permeability zones. Also its density is low causing it to flow to the top of the reservoir bypassing a large portion of the fluids. Foam can be used to enhance the oil recovery as it traps the injected gas in small bubbles which are separated by thin liquid films known as lamellae. This trapping means that the gas is no longer free to flow and thus cannot easily bypass the resident fluids. Therefore, foam injection processes can provide a much more stable displacement front than conventional gas injection. Most studies on foam in porous media available in the literature focus on petroleum applications including experimental work (e.g., Chabert et al. 2014; Singh and Mohanty 2016; Batôt et al. 2016) and modeling studies (e.g., Ma et al. 2014; Masoudi et al. 2015).

In more recent years, foam injection has been applied for environmental remediation purposes. It can be used (i) as a selective reductive permeability agent (e.g., Hirasaki et al. 1997; Bertin et al. 2017), (ii) to increase the mobilization of contaminant (Mulligan and Eftekhari 2003; Huang and Chang 2000; Wang and Chen 2014; Longpré-Girard et al. 2016; Maire and Fatin-Rouge 2017), (iii) to increase the contact between remedial agent and contaminant (Rothmel et al. 1998; Shi et al. 2018) or (iv) as a confining agent (Portois et al. 2018). Only few attempts were done in the field (Hirasaki et al. 1997; Maire et al. 2018; Portois et al. 2018), and all authors highlighted the difficulties to predict foam behavior in 3D dimensions. Two main metrics were used to monitor the foam propagation, namely breakthrough of foam in observation wells (Hirasaki et al. 1997; Maire et al. 2018) and the use of modeling to approximate the progressing foam front (Portois et al. 2018). Foam remains an unstable fluid, and its behavior is quite difficult to predict in the field, even more in the case of environmental remediation where optimal conditions such as a high injection pressure cannot be achieved. As a result due to the lower apparent viscosity (compared to oil production), foam can also suffer from gravity override and lead to a relatively poor sweep efficiency of the porous media. Injection pressure is the commonly tracked parameter during an injection. Considering radial flow in a 2D or a 3D medium, pressure decreases drastically with increasing radial distance from the injection well, leading to foam collapse and subsequent gravity override of the free gas. Thus, injection pressure alone does not provide sufficient information of the foam behavior far away from the injection well.

Apart from pressure monitoring, X-ray computed tomography is one of the most metrics used to monitor foam behavior in 1D (Du et al. 2007; Nguyen et al. 2007; Simjoo et al. 2013), but this method has received limited attention for 2D or 3D injection cases. The work of Bertin et al. (1999) did feature 3D cross-flow effects, but the physical scale of the porous medium they used was limited. Tsai et al. (2009) studied the propagation of foam in a laboratory sandbox using iron powder as a tracer. They were able to draw gas saturation contours (identified as foam) right after the foam injection, but this method is highly invasive and requires the collection of several soil samples. Also it does not allow real-time tracking of the foam propagation. Portois et al. (2018) highlighted the use of hydraulic tests (pumping test combined with several observation wells) to monitor a foam front in a real aquifer. As stated by the authors, the extent to which foam propagates horizontally and vertically can only be roughly estimated.

This study focuses on the monitoring of a foam front during an injection process using electrical resistivity tomography, which can provide 3D noninvasive imaging of the foam inside the porous medium during the injection process. We performed a foam injection process in a large-scale 3D heterogeneous porous medium, and the resistivity measurements were carried out at various stages during the injection process to track the position of the foam front.

Background Theory

Electrical resistivity tomography (ERT) is a method for characterizing the spatial distribution of electrical charge carriers, such as ions in mineralized water, clay and metal particles.

ERT for the application of monitoring fluid flows in contaminated sites has been the subject of several studies (e.g., Wilkinson et al. 2010; Van Dam et al. 2014) due to its noninvasive nature. Electrodes can be placed toward the edge of the porous medium therefore not interfering with the fluid flows inside it.

The method is suitable for monitoring dynamic processes as the measurements can be carried out during very short intervals. This is known as time-lapse ERT and has been used extensively for a number of applications ranging from the monitoring of surface water–groundwater interaction (e.g., Slater et al. 2010) to monitoring the salinization of freshwater (Wagner et al. 2013), water injection processes (Kuras et al. 2009), interactions between surface water and groundwater (Slater et al. 2010) and soil–plant interactions (Cassiani et al. 2015, 2016). Analysis of time-lapse ERT data requires data inversion techniques where the time is included explicitly in the procedure (Kim et al. 2009; Karaoulis et al. 2014) and where there is specific attention paid to the sensitivity changes of the parameters over time (Karaoulis et al. 2014) in order not to misinterpret the results.

Another application which has been studied extensively is the use of ERT for monitoring the presence of dense non-aqueous phase liquids (DNAPLs) in the subsurface (e.g., Naudet et al. 2011; Chambers et al. 2004; Power et al. 2015). Lucius et al. (1992) found that DNAPLs tend to have a highly resistive nature compared to ground water, thus using ERT allows for easy distinction between DNAPLs and the ground water. Revil et al. (2011) determine that DNAPLs could have various electrical signatures ranging from highly resistive for fresh and unalterated DNAPL, to very conductive for alterated DNAPL. Also the use of time-lapse ERT has proven useful in monitoring remediation processes of DNAPL spills (e.g., Power et al. 2014).

Tracking the movement of foams in porous media using ERT has not been studied extensively thus far, but uniformity (Wang and Cilliers 1999) and liquid content (Karapantsios and Kostoglou 2011) of bulk foams have been studied previously with ERT. Furthermore, in recent years the storage of CO2 in geological formations has received extensive interest. Several studies (e.g., Carrigan et al. 2013; Yang et al. 2015; Pezard et al. 2015; Sauer et al. 2014) use ERT to monitor the movement of CO2 inside these formations, indicating the feasibility of tracking gas fronts using this method.

Materials and Methods

Experimental Setup

A wooden structure consisting of 28-mm-thick boards is used to house an unconsolidated sand pack (0.84 × 0.84 × 0.84 m), which functions as the porous medium in these experiments. An impervious plastic liner is added to the inside of the wooden container to prevent leaks. The porous medium is created by adding sand and water into the structure, thereby creating a medium that is initially fully saturated with water. Most of the sand used is fairly coarse sand (MI 0.4–0.9) with a mean grain diameter of 677 μm. However, to determine how the foam is affected by heterogeneity we added a cylindrical lens of fine sand in the center of the porous medium consisting of finer sand (MI 31) with a mean grain diameter of 300 μm. This fine sand lens is 54 cm in diameter and 27 cm in height. These grain sizes result in permeabilities of 70 and 20 D for the coarse and fine sand regions, respectively. The porous medium has a total pore volume of 208 L and a porosity of 0.353.

A vertical injection well (64 mm diameter) is placed in the center of the porous medium, and four production wells (30 mm diameter) are placed vertically in each of the corners of the structure. A clay layer with a thickness of approximately 80 mm is placed on top of the sand pack which prevents leakage of gas from the top of the sand pack. An additional wooden plate is placed on top of the clay layer and is used to compact the entire medium. A global overview of the sand pack inside the structure is given in Fig. 1.

Fig. 1
figure1

Schematic overview showing vertical (left) and horizontal (right) cross sections of the porous medium inside the wooden structure including dimensions given in millimeters. The darker area in the middle of the medium is the heterogeneity that consists of finer sand (k = 20 D), which is surrounded by coarser sand (k = 70 D)

A schematic of the injection system used in these experiments is shown in Fig. 2. A Cole-Parmer Masterflex L/S pump and a SMG gas metering system are installed for the injection of surfactant solution and gas, respectively. These are co-injection experiments, so both fluids are injected simultaneously. A tube runs down the injection well in the center of the porous medium and is connected to a model porous medium which functions as a bubble generator. This ensures that foam is pre-generated within the well itself during the co-injection process before it enters the porous medium, which ensures a strong foam right at the well.

Fig. 2
figure2

Injection system showing liquid and gas injection systems connected to injection well. A bubble generator (model porous medium) at the bottom of the well ensures strong foam generation inside the well during co-injection of gas and liquid

A further close-up schematic of the injection well is shown in Fig. 3. This figure shows that the well is screened over the entire height within the sand pack, but not within the clay layer near the top of the well. This means that fluids can be injected over the entire height of the sand pack. Two inflatable packers are installed above and below the bubble generator that can block parts of the well. This allows for injection into certain segments of the injection well rather than the entire height.

Fig. 3
figure3

Cross-sectional schematic view of the injection well showing screened well within the sand pack. By placing two inflatable packers inside the injection well, the injection region can be controlled

During the injection process, liquid that is initially in place is produced from the four production wells that are placed vertically in all four corners of the setup. The Cole-Palmer pump that is used for the injection of liquid is also used for the liquid production by installing a second pump head. This head is connected to four tubes which are placed inside the production wells and continuously sucks out fluid from the wells. The bottom of these tubes is aligned with the top of the sand pack, so the water level in the production wells can never exceed the top of the sand pack which prevents flooding of water from the top of the wells. This system is shown schematically in Fig. 4.

Fig. 4
figure4

Aspiration system drawing liquid from the vertical production wells in all four corners of the porous medium

Resistivity Measurement System and Electrode Layout

The ERT measurements carried out to track the foam front inside the porous medium use a series of electrodes placed inside and along the sides of the sand pack, which are used for the current injection and potential measurement. In total, 128 electrodes are placed inside the porous medium. Sixty-four of these are positioned along vertical lines in the corners of the sand pack, approximately 10 cm away from the production wells. The other 64 electrodes are placed inside the porous medium on horizontal and vertical lines. These electrodes are spread out to all sides of the pilot to ensure that combinations of the electrodes can cover the entire sand pack.

The electrodes used in this experiment are brass rods approximately 1 cm in length and 3 mm in diameter. The electrodes on the inside of the pilot may influence the flow somewhat, but due to their small physical size, their influence is limited. Good contact between electrodes and the medium is ensured by the fully saturated medium. Figure 5 shows the position of the electrodes inside the pilot. The blue cube represents the porous medium, and the small spheres represent the electrodes. The figure shows that the electrodes are positioned on several lines along the sides and inside the pilot. Each line has 16 electrodes on it. Since the sand pack has a thickness of 84 cm, the chosen distance between the electrodes is 5 cm to cover the entire sand pack.

Fig. 5
figure5

Positions of electrodes within the sand pack indicated by colored dots. Different colors represent different lines of electrodes. White dots are electrodes placed along vertical lines near the corners of the porous medium, 10 cm away from the production wells, black dots are electrodes placed on horizontal, diagonal lines at heights of 15, 40 and 65 cm and red dots form a vertical line of electrodes closer to the well to provide better resolution near the center of the sand pack

In order to allow for 3D imaging of the fluids inside the porous medium, a large number of resistivity measurements are required combined with inversion techniques. A single measurement uses four electrodes (quadripole) two of which are used for the current injection and the other two for the simultaneous measurement of potential difference that is caused by the injected current through the medium. The measurement of DC resistivity ρ is based on Ohm’s law, with the resistivity being equal to the ratio of the measured potential U (V) to the product of the injected current I (A) and the geometrical factor K (m). The latter is based on the positions of the electrodes used in the measurements and the boundary conditions of the overall geometry of the porous medium. In order to reach a high sensitivity all over the medium, we have to combine a high number of electrodes well distributed all over the medium, as well as a high number of quadripoles also well distributed. Once all the quadripole combinations (called a sequence) are acquired, the resistivity distribution can be estimated through inversion of the combined measurements. In total, the first measurement sequence contains more than 5000 measurement quadripoles. This includes reciprocal measurements (i.e., quadripoles where the current injection and potential measurement electrodes are inversed) which are used to check the quality of the measured data. Inversion techniques are required to turn the measured results into a 3D representation of the resistivity distribution. In this study, the software R3t (Binley 2009) is used to invert the resistivity data. This software allows for inversion on bounded, unstructured, 3D geometries such as the porous medium used here. The software uses a 3D finite element model of the porous medium. The model was created using Gmsh (Geuzaine and Remacle 2009) and uses a fully unstructured tetrahedral mesh. Electrode positions are integrated into the model and appear as nodes between certain elements. Here, we use null flux conditions on the outer bounds of the geometry of the porous medium to serve as boundary conditions which are used in the inversion process. The simulation uses an initial estimate of the porous medium as a starting point for the inversion. Here, we assume that the initial state of the porous medium is homogeneous with a resistivity of 100 Ωm, which is a representative value for a porous medium that is fully saturated with water.

The inversion process will introduce a certain degree of error (defined as root-mean-square (RMS) error between calculated apparent resistivity and measured apparent resistivity) and equivalence. The error is caused both by the finite number of measurements and the finite element model used in the inversion process which is not a perfect realization of the actual porous medium. By equivalence, we mean the uncertainty in resistivity value at each inverted point that is inherent to potential inversion methods due to the nonlinearity of the equation and to ill-posed nature of the problem (less measurements compared to parameters to estimate). Also, the simulation performs iterative calculations until a certain error threshold is reached. R3t allows for setting both offset and relative errors. In these simulations, only a relative error is used of 0.05. An additional series of measurements are performed using electrode quadripoles that are placed in the center of the pilot (i.e., using the electrodes shown as black and red dots in Fig. 5). These measurements allow for directly measuring the resistance at their location. These measurements only provide local information of the medium’s resistance and are combined with the measurements from the larger sequence to increase the accuracy of the inversion process. In total, more than 2000 quadripoles were used for every inversion.

The measurements are performed using an SYSCAL PRO (Iris Instrument) resistivity measurement multielectrodes and multichannels system. This system allows for the simultaneous connection of up to 72 electrodes, meaning that a single sequence can use 72 electrodes allowing the use of all possible quadripole combinations. Since two different sequences are used in these measurements, two different subsets of 72 electrodes out of the total of 128 electrodes that are present in the porous medium are used. Figure 6 shows which electrodes are used during the sequences. Note that some electrodes are used in both sequences. For one identical couple of injection electrodes, this system allows the simultaneous acquisition of 10 quadripoles.

Fig. 6
figure6

Electrode layout within the porous medium. Color of the dots indicates in which sequence the electrodes are used. White is used in the first sequence aimed at providing 3D distribution of resistivity, black is used in the second sequence aimed at reducing uncertainty in the results on the plane of measurement, and red is used in both sequences

Salt Solution Injection

Before the foam injection, injection of salt (NaCl, 0.5 g/L) solution was performed to analyze the porous medium. Adding salt to the water increases the number of ions inside the fluid, thus the amount of electrical charge carriers and thus the conductivity of the fluid, thereby reducing the resistivity of the porous medium. The salt solution used here has a conductivity of 1330 µS/cm versus 420 µS/cm for the fresh water that it replaces, and thus the salt solution is more than three times more conductive than the water that is originally in place. This is expected to be sufficient contrast for the salt solution to be detectable. Running the ERT measurement sequences at various stages during the salt injection process (i.e., the injection is stopped, while the measurements are being performed) allows for determining where the injected fluids flow as the injected salt solution will show up as regions of lower resistivity compared to the water that is initially in place. The effect of the heterogeneity of the porous medium on the flow can be studied in this way. It is also a good method for testing the measurement approach and inversion method as the salt solution can be flushed out of the porous medium afterward to reverse the porous medium back to its initial state, which is not possible after the injection of foam.

A total of volume 150 L (= 0.72 PV) of salt solution was injected into the porous medium at a rate of 600 mL/min. ERT measurements were performed at 5 stages during the injection process, namely after 0, 25, 60, 100 and 150 L (or 0, 0.12, 0.29, 0.48 and 0.72 PV). During the entire salt solution injection process, the injection well is screened over its entire height, so no packers are installed to block off certain sections of the well. After the salt injection, fresh water was injected to remove the salt solution from the porous medium.

Foam Injection Approach

The main experiment is a foam injection process. Initially, 2 PV of surfactant solution (ammonium alcohol ether sulfate, Stepan Petrostep ES-65A) is injected into the porous medium to satisfy the adsorption criterion. The surfactant concentration used throughout this experiment is 0.1 wt% (wt./wt.) active content. This is followed by performing the ERT measurement sequence, which is used as the baseline result (so a porous medium without any gas present in it). Thereafter, the foam injection process is started by co-injecting gas (air from a bottle supplied by Air Liquide) and surfactant solution at a total flow rate of 440 mL/min with foam quality (= gas fractional flow) of 0.9. A preliminary foam injection experiment in a column showed that the decent foaming performance (i.e., mobility reduction) can be expected using this surfactant concentration and foam quality. Both fluids flow combined through the filter at the end of the injection tube which serves as a bubble generator that ensures strong foam is formed inside the injection well.

Results and Discussion

The three-dimensional nature of these experiments means that some concessions need to be made in order to show presentable figures in this manuscript. For clarity reasons, the majority of results are a 2D plane representation of the complete inverted data which are in 3D. The chosen plane is a vertical diagonal plane through the center of the porous medium (so also through the injection well and two out of the four production wells). This is also the plane in which the electrodes on the horizontal diagonal lines (cf. black dots in Fig. 5) are located. This plane was chosen as it allows for observation of the radial and vertical position of the foam front as it progresses from the injection well. Also, since it has the most electrodes directly on the plane, it is likely to have the lowest level of uncertainty within the 3D medium. Figure 7 shows the chosen plane on which most of the results are shown.

Fig. 7
figure7

Vertical diagonal clipping plane through the center of the porous medium (so also through the injection well). The majority of the resistivity distribution results in the remainder of this text are shown on this plane. The white dots indicate the positions of the electrodes that are inside the pilot (so excluding the ones near the corners). Note that the electrodes on the horizontal lines (which are also shown as black dots in Fig. 5) are on this vertical plane, thereby minimizing the uncertainty within the results shown in the remainder of this text

Injection of Salt Solution

Figure 8 shows the initial distribution of resistivity on the vertical diagonal plane which was obtained by inverting the measured ERT data, so before the injection process has started. Large variations of resistivity are observed at this stage. These may have several reasons. First of all, the region of low resistivity near the top of the figure (blue region) is caused by the presence of the clay layer on top of the sand pack. Note that the simulation model only intended to capture the sand pack and not the clay layer on top, but the clay layer apparently affects the inverted resistivity distribution in the top region of the model. Another possibility is that the clay layer is actually physically thicker than intended and extends further down into the sand pack.

Fig. 8
figure8

Base case resistivity before the salt injection. The low-resistivity region near the top represents the clay layer on top of the porous medium. The remainder of the porous medium shows a significant variation in resistivity

The high-resistivity regions in the top left and right portions of the figures are likely to be caused by a lack of electrodes in those regions. The far-left and far-right portions of this and subsequent figures represent regions that near the corners of the medium, and there are no electrodes present within them (cf. Fig. 5 which shows that electrodes are at least 10 cm away from the corners). Errors in the obtained resistivity distribution are to be expected in these regions, because there are no possible electrode quadripoles that can capture the resistivity here.

The remainder of the figure also shows large variation in resistivity. These may have been caused by dissolution of minerals from the sand during the construction of the sand pack which may have affected the resistivity distribution.

Figure 9a–d shows the resistivity distribution on the vertical plane for various stages during the injection of salt solution. These figures show the advancing salt solution front as it progresses through the medium. As expected, the regions where the salt solution has swept show up as regions with a significantly lower resistivity than the remaining regions. Hand-drawn yellow curves are added to the figures which provide an estimate of the advancing salt front. The clay layer on top makes it hard to judge the position of the salt in the top region of these figures, but the salt can be accurately tracked in the remainder of the medium. It is clear that the coarser sand near the top and bottom of the medium cause preferential flow due to its higher permeability and the salt solution progresses a lot slower through the finer sand in the center. After injection of 0.72 PV of solution, breakthrough of salt solution in the production wells was observed by measuring increased conductivity of the produced water, so the injection process was stopped.

Fig. 9
figure9

ERT measurements on a vertical cross section along the diagonal of the pilot showing resistivity distributions at the indicated points in time in during the salt injection process. Note that the color axis is the same as that shown in Fig. 8 for all of these figures. The yellow lines are hand-drawn and represent an estimate of the advancing salt front in the center portion of the medium. at = 0.12 PV, bt = 0.29 PV, ct = 0.48 PV and dt = 0.72 PV

The salt solution injection process has shown that the utilized ERT measurements are capable of tracking fluids with intrinsically different resistivity values than the fluids that are being displaced. The proposed sequences are sufficient for tracking the front of injected fluid. In the case of the porous medium analyzed here, the heterogeneity formed by the low-permeability lens in the center of the sand pack influences the flow such that there is limited flow going through the center section and most flow is diverted to the coarse sand regions.

The Foam Injection Process

After the salt solution injection process finished, the salt is flushed out of the system by first injecting fresh water and afterward 2 PV of surfactant solution. This provides the initial state for the foam injection process, and its resistivity distribution is shown in Fig. 10. This is once again the resistivity distribution on the vertical diagonal plane that was also used for showing the results of the salt injection along with an identical color axis to allow direct comparison with previous results. Note that the initial state shown here appears far more homogeneous than that found before the salt injection had started (cf. Fig. 8). Some slight variations in resistivity are still observed here, but not nearly as severe as that found before. Those were already implied to be at least partially the result of dissolution of minerals from the sand and clay. The distribution shown in Fig. 10 was taken directly after fully saturating the porous medium with surfactant solution, so the distribution was expected to be homogeneous. Still the contrast between the two figures is very stark, and the complete reason behind the large variations found in Fig. 8 is unknown.

Fig. 10
figure10

Base case resistivity before foam injection. The low-resistivity region near the top represents the clay layer on top of the porous medium. Overall a fairly uniform distribution of resistivity is measured within the porous medium although some regions of higher resistivity can be identified

Figure 11a–h shows the resistivity distribution on the vertical diagonal plane at various stages during the foam injection process. For the initial stages of the foam injection process, two packers were used to block off certain sections of the injection well allowing for targeted injection into specific sections of the porous medium. This was done to determine whether foam can be used to block off certain regions within the porous medium. The idea was to inject into the top and bottom regions first which contain coarse sand and afterward inject over the entire height to see whether the blocking property of the foam in the coarse sand layer is sufficient for diverting flow to the lower-permeability lens in the center of the porous medium. Afterward, foam was injected over the entire height of the injection well for a while, but some additional injection in the top and bottom layers proved necessary to improve the blocking effect. In the end, foam injection over the entire height of the well was carried out for a prolonged period of time to try and sweep the entire porous medium with foam. This injection procedure is outlined in Table 1 which also includes the corresponding figures for each of the injection stages.

Fig. 11
figure11

ERT measurements on a vertical cross section along the diagonal of the pilot showing resistivity distributions at the indicated points in time in during the injection process. Note that the color axis is the same as that for the salt injection shown in Figs. 8 and 9 and the base resistivity shown in Fig. 10 for all of these figures. at = 0.04 PV, bt = 0.08 PV, ct = 0.12 PV, dt = 0.20 PV, et = 0.28 PV, ft = 0.40 PV, gt = 0.58 PV and ht = 0.76 PV

Table 1 Outline of the foam injection procedure

Figure 11a, b shows the result after the initial injection in the top and bottom layers. When comparing these results to the initial state shown in Fig. 10, there are clear regions of increased resistivity which can be observed near the top and the bottom of the medium. This is as expected due to the increased resistivity of the foam swept regions. The next step was injecting over the entire height of the injection well, and its result is shown in Fig. 11c. This figure shows only a very slight increase in resistivity in the fine sand lens in the center of the pilot, but there is a significant increase that is observed near the top of the medium, which implies that gas is flowing there due to gravity override. This means that the foam’s ability to reduce mobility is insufficient and the foam that was already in place in the coarse sand layers was also not enough to prevent the gas flowing to the top.

To improve on this, additional foam is injected in the top and bottom layers. The result of this is shown in Fig. 11d in which an extension of the high-resistivity regions can be observed near the top and bottom of the medium. After this, the injection over the entire height of the well was performed for an extended period of time. The injection was halted at various points in time to perform the ERT measurements. The results of these are shown in Fig. 11e–h. Subsequent figures show an ever growing expansion of the foam swept region, but the gravity override could not be prevented, so in each figure the preferential gas flow to the top of the medium can be clearly observed.

The foam strength was found to be insufficient to provide a stable radial displacement front. During the experiment, the injection pressure was monitored, which serves as an additional measure of the foaming performance. However, since only the injection pressure is monitored and no pressure data are available further away from the injection well, this can only describe the foaming performance in the near-well region. Figure 12 shows the injection pressure during 0.4 and 0.58 PV of injection, which corresponds to the period of time between Fig. 11f, g. This injection timespan was chosen as it represents a pressure profile which is fairly typical for this experiment. During other injection stages within the foam injection process, the resulting pressure profile may vary slightly from this due to the foam front having swept a smaller or larger region, but not to such an extent that it alters the conclusion drawn from the calculations below.

Fig. 12
figure12

Injection pressure profile during one of the foam injection stages

As shown in Fig. 12, the pressure increases gradually during the injection until it reaches a plateau. We will use this plateau value (roughly 5.35 × 104 Pa) to base our calculations on. It should be noted that this elevated injection pressure did not compromise the structure of the apparatus. No deformations in the structure were observed during the experiment. As mentioned above, these experiments were carried out using a fixed total flow rate of 440 mL/min. During the injection stage for which the pressure profile is given in Fig. 12, the injection occurs over the full height of the injection well. This means that injection occurs in both the coarse and fine sand layers, and thus we need to make use of the arithmetic average of the permeability to find an average permeability value. This average value is based on the thickness of the fine and coarse sand layers (see Eq. 1) and amounts to 52 D.

$$ k_{\text{avg}} = \frac{{k_{\text{fine}} \cdot H_{\text{fine}} + k_{\text{coarse}} \cdot H_{\text{coarse}} }}{H} = 52\,{\text{D}} $$
(1)

where kavg is the average permeability of the medium near the well, kfine is the fine sand permeability of 20 D, Hfine is the height of the fine sand layer of 30 cm, kcoarse is the coarse sand permeability of 70 D, Hcoarse is the height of the coarse sand of 54 cm, and H is the overall height of the sand pack of 84 cm. Using this value of permeability, the expected pressure drop is calculated if only water is flowing inside the medium using Darcy’s law in radial form (Eq. 2). This value can then be compared to the measured value of the injection pressure to determine the foam apparent viscosity in the near-well region.

$$ \Delta p_{\text{w}} = \frac{{Q\mu_{\text{w}} \ln \left( {{{R_{\text{e}} } \mathord{\left/ {\vphantom {{R_{\text{e}} } {R_{\text{w}} }}} \right. \kern-0pt} {R_{\text{w}} }}} \right)}}{{2\pi Hk_{\text{avg}} }} = 30.4\,{\text{Pa}} $$
(2)

where Δpw is the expected pressure drop for a single-phase water flood, Q is the overall injection flow rate of 440 mL/min, μw is the dynamic viscosity of water of 1 mPa s, Re is the distance between the injection and the production well, and Rw is the injection well radius. This implies that the injection pressure (i.e., the pressure needed to overcome the viscous forces due to the generated foam) is 5.35 × 104/30.4 = 1.76 × 103 times higher than that expected for a single-phase water flood. This means that, at least near the well and since the viscosity of water equals 1 mPa s, the foam apparent viscosity is 1.76 Pa s. So the foam’s ability to reduce mobility near the well should be able to provide a stable displacement front.

Another measure which determines whether the foam front is stable is a comparison between the viscous and gravitational forces. The viscous pressure is equivalent to the injection pressure as it is this pressure that drives the flow of foam. However, in radial flow configurations such as this, the pressure quickly decreases with increasing distance from the injection well. This means that even though foam may initially be stable, further away from the well, gravitational forces may become dominant leading to gravity override and foam collapse. The maximum possible pressure due to gravity is equal to the hydrostatic pressure, which for the injection of gas in a water-saturated medium is given in Eq. 3.

$$ \hbox{max} \left( {\Delta p_{\text{h}} } \right) = \left( {\rho_{\text{w}} - \rho_{\text{g}} } \right)gH = 8.23 \times 10^{3} \;{\text{Pa}} $$
(3)

Thus, near the well, the viscous pressure gradient can be assumed to be higher than the gravitational one, due to the injection pressure being substantially (more than six times) higher than the maximum possible hydrostatic pressure, so the main driving force of the flow is the viscous pressure. However, if purely radial flow of foam is assumed and a constant foam apparent viscosity is assumed, a pressure profile throughout the pilot as a function of its distance from the injection well can also be estimated. The result of this is given in Fig. 13, which is based on Eq. 2, but using the foam apparent viscosity rather than the water viscosity. This figure shows that pressure quickly decreases as a function of radius and at a distance of 0.31 m from the well center, the pressure has decreased so much that the gravitational forces will become dominant and gravity override could be expected for distances larger than this. Note that this is an ideal scenario where the foam strength is constant and no foam collapse occurs, so in reality, probably gravity override can be expected even sooner. This result shows that the gravity override that was found in the experiment is not unexpected.

Fig. 13
figure13

Estimated pressure profile as a function of radial distance from the well assuming purely radial flow and the presence of foam with a foam apparent viscosity of 1.76 Pa s

The resistivity data can also be used to estimate the liquid saturation within the foam swept region by applying Archie’s law (Archie 1942) (Eq. 4).

$$ \rho = \rho_{\text{l}} \cdot \phi^{ - m} \cdot S_{\text{l}}^{ - n} $$
(4)

where ϕ is the porosity, which was measured during the preparation of the sand pack and is equal to 0.35, ρ is the resistivity of the formation, ρl is the resistivity of the liquid, which in this case is the surfactant solution with a conductivity measured at 1330 μS/cm which corresponds to 7.5 Ωm, Sl is the fluid saturation of the medium (fully saturated Sl = 1), m is the cementation factor estimated at 0.6, and n is the saturation exponent estimated at 2.

Thus, for a fully saturated medium the resistivity is about 15 Ωm. As observed in the resistivity figures (Fig. 11), the foam swept region has resistivity values upward of 200 Ωm. When using this value, we find a liquid saturation of Sl = 0.27. Note that the color axis in Fig. 11 is clipped, so higher resistivity values are found within the foam swept region, and thus the liquid saturation value may vary.

In addition to the results on the vertical diagonal plane, an attempt has been made to visualize the results in three dimensions by means of iso-resistivity contour plots. Three figures showing the last three stages of the foam injection process are shown in Fig. 14a–c. These are 3D representations of the advancing foam front, and they correspond to Fig. 11f–h, respectively, which show results at the same time, but only the in-plane results. The red contours shown in Fig. 14a–c are iso-resistivity contours with a value of 200 Ωm. This value was empirically chosen as it is significantly higher than the resistivity values obtained for a medium fully saturated with water. Therefore, it is a qualitative indication of the region where the foam has swept. These figures allow determining whether there are preferential flow paths in radial direction. Preferential flow appears to be the case in Fig. 14a which shows the foam front after 0.40 PV of injection. The front has already progressed to the top left corner of the medium at this stage, but has not quite reached the other corners. This might be the result of preferential flow, but could also be that the resistivity on the right side of the figure is just slightly below the contour value of 200 Ωm. The latter appears to be the case when comparing the results from the Fig. 14b, c which shows improved axial symmetry, thus not indicating preferential flow toward a particular corner of the medium. Also, these latter two figures are directly comparable to the 2D distributions shown in Fig. 13g, h which implies that for axisymmetric flow the 2D representation provides sufficient information to characterize the advancing foam front inside the medium. Another explanation for the observed asymmetry in Fig. 14a is the constant rate from the production wells. If the permeability of the porous medium was not fully axisymmetric, this would lead to preferential flow toward one of the production well and the constant rate production would mean that more water was pulled out of one of the production wells. Changing to a constant pressure rather than constant rate production could prevent such preferential flow.

Fig. 14
figure14

3D iso-resistivity contours for the indicated injection time. The red contours are iso-contours with a resistivity of 200 Ωm. at = 0.40 PV, bt = 0.58 PV and ct = 0.76 PV

One further observation from Fig. 14c is that the foam swept region adopts an almost spherical shape. Kovscek et al. (1997) found that this is the expected shape for a region that is swept with strong foam. It appears that in this case a region of strong foam does form near the well, but further away from the well the foam strength is reduced which leads to foam collapse and the subsequent flow of free gas to the top of the medium as a result of gravity override.

Conclusions

Foam flow can be accurately monitored inside large-scale porous media using ERT due to the large contrast of resistivity between liquid- and foam-saturated porous media. A large number of measurement quadripoles are required to generate the inverted image showing the resistivity distribution in 3D. This puts some constraints on the experimental process. The injection has to be stopped during the measurement sequence to minimize changing conditions during the entire sequence. Still, foam may flow even when the injection is halted due to the higher pressure near the point of injection and possible gravity override from the gas. This need for intermittent injection may negatively impact the foaming performance inside the porous medium. In the current experiment, the time needed to perform the measurements was similar to the duration of injection and thus the injection was halted about half the time. Both the foam flow and the ERT results are affected by this. Internal electrodes appeared as anomalies on the inverted ERT results. These are local anomalies though and the global flow of foam (or salt solution) can still be tracked. However, the result is qualitative, meaning that results are limited to tracking the foam front due to the large contrast in resistivity between water- and foam-saturated regions. Basic calculations, such as the ones presented here based on injection pressure and resistivity values, can provide initial estimates into more quantitative data, like foam apparent viscosity and liquid saturation, that can help explain the foam performance. For further, more accurate calculations of these data, additional measurements of properties that influence the results are required. These properties include, but are not limited to, the following: salt concentration of the liquid, transfer of water to the gas phase and adsorption of surfactant.

The foam injection performed in this study was not entirely successful. The foam front proved to be unstable and suffered from severe gravity override. This implies that the mobility reduction was not sufficient enough and the volumetric sweep of the medium was therefore limited. Overall, it can be stated that the foam injection process needs to be improved upon in order to prevent the gravity override. However, when generating a stronger foam in the injection well which can be better suited to reduce overall mobility of the fluids, the injection pressure will have to increase as well or the flow rate needs to be lowered. Operating pressure is a major constraint for aquifer application of foam, and thus lower flow rates with a stronger, more effective foam, may be the preferred choice. Another method of injection may be to use a fixed pressure rather than a fixed injection rate. In this way, we can ensure that allowable operating pressure is not exceeded. This method has been used in previous studies by Hirasaki et al. (1997) and Maire and Fatin-Rouge (2017). Furthermore, Shan and Rossen (2004) found that, for oil reservoir conditions, maintaining maximum allowable injection pressure minimizes foam segregation and gravity. Other injection schemes such as surfactant-alternating-gas (SAG) injection could also be considered for this purpose as it would increase injectivity compared to co-injection methods.

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Boeije, C.S., Portois, C., Schmutz, M. et al. Tracking a Foam Front in a 3D, Heterogeneous Porous Medium. Transp Porous Med 131, 23–42 (2020). https://doi.org/10.1007/s11242-018-1185-0

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Keywords

  • Foam
  • Porous media
  • Environmental remediation
  • Electrical resistivity tomography