Fractional Flow Analysis of Foam Displacement in Tight Porous Media with Quasi-Static Pore Network Modeling and Core-Flooding Experiments

Abstract

Fractional flow analysis is an efficient tool to evaluate the gas-trapping performance of foam in porous media. The pore-scale simulation study and the core-scale experimental work have been bridged via the fractional flow analysis to distinguish the characteristics of foam displacement inside the tight porous media with varying absolute permeability, injection rate, and foam quality. In this work, the combined investigation suggests that conventional foam-enhancing strategies, pursuing higher foam quality and stronger foam regime, are inefficient and restricted in tight reservoirs that the critical S_w corresponding to the limiting capillary pressure has increased around 37~43%, which indicates severely weakened gas-trapping capacity as permeability reduces one order of magnitude. The moderate mobility adjustment and corresponding optimized fluid injectivity exerting from the “weak foam” flow presents a staged decline feature of decreasing water fractional flow, which implies the existence of the delayed gas-trapping phenomenon when water saturation reduces to 0.5~0.6. The finding has supported the engineering ideal of promoting low-tension gas (LTG) drive processes as a potential solution to assist field gas injection applications suffering from gas channeling. Also, the validation with core-flooding experimental results has revealed several defects of the current pore network model of foam displacement in tight porous media, including exaggerated gas trapping and overestimated confining water saturation. This study has innovatively demonstrated the feasibility and potential of optimizing the foam performance of gas trapping and mobility control in tight reservoirs, which provides a clue that may eventually boost the efficiency of the gas injection process in enhanced oil recovery or CO₂ sequestration projects.

1. Introduction

Gas injection in tight reservoirs is an essential but risky process encountered in multiple carbon capture, utilization and storage (CCUS) applications [1,2]. Such an attempt to enhance oil recovery (EOR) or accomplish higher CO₂ storage capacity is challenging due to the severe gas channeling caused by unfavorable mobility ratio between the defending wetting fluid, either brine or crude oil, and the immiscible non-wetting gaseous phase injected [3,4,5].

As a representative mobility control technique for decades, foam displacement in porous media has been adopted in EOR projects and soil remediation applications, benefitting from its advantages in flexibility, operation cost, and synergistic effect with customized surfactant treatment [2,6,7]. As the core mechanism optimizing the gas injection process, the mobility adjustment of foam displacement is realized by creating clusters of lamellae that intercept the continuous gas flow along preferential paths inside the porous media. Temporary blockage of the immobile foam bank forms and effectively improves the sweep efficiency of a reservoir via converting the subsequently injected gas towards the bypassed region during the preceding continuous gas injection process [6,8]. Because of the nature of foam instability, in situ performance analysis and profile control optimization are long-term subjects of foam studies in porous media.

Quantitative evaluation of foam ability and foam stability in bulk phase is commonly used to distinguish the properties of foaming surfactant, whereas specific variables and standard procedures have been established to quantify the resistance and mobility of foam in porous media, such as resistance factor analysis, mobility reduction factor estimation, fractional flow analysis, etc. [8]. The fractional flow analysis based on core-flooding experimental studies, which was originally developed to efficiently identify key features of the immiscible waterflooding process, has been extended to foam displacement processes to quantify the characteristics of foam state as well as the gas-trapping ability exerted from the immobile foam bank [9]. The fractional flow analysis of foam displacement has also been extended to non-Newtonian fluid flow in porous media for its advantages in upscaling operations among experimental studies and corresponding simulator calibration [10]. The fractional flow method is an efficient tool to predict foam performance, especially its gas-trapping ability via the f_w vs. S_w relation. The critical water saturation corresponding to the “limiting capillary pressure” distinguishes the strong foam injection state with reduced gas mobility and the weak foam or no-foam injection state with higher gas mobility.

Multiscale performance prediction of immiscible displacement processes in tight reservoirs based on numerical simulation is essential to optimize the gas injection process [11,12]. Various mechanistic modeling approaches have been proposed to simulate the dynamics of foam propagation incorporated with continuous foam generation and coalescence, among which the population balance model has shown advantages in coupling with other macro-scale reservoir simulators [13,14]. Among pore network model studies, the simulation of fluid flow within the bond-site lattice can be classified into quasi-static pore network model and dynamic pore network model. In quasi-static pore network models, the multiphase displacement inside the lattice is assumed to be dominated by capillary forces, whereas the viscous forces are neglected. In dynamic pore network models, both capillary forces and viscous forces are incorporated to quantify the flow field and pressure field within the lattice [15]. For the lamellae evolution inside the pore structure, numerous studies on foam displacement have successfully incorporated quasi-static interfacial activities into the foaming gas drainage process [15,16,17]. Characteristic pore-scale events like lamellae formation triggered during the Haines jump across snap-off sites, as well as the stepwise lamellae thinning or rupture due to the capillary suction, have been included in recent work [18]. Thus, like other immiscible multiphase flow simulated with the pore network model, relative permeability profiles of foam displacement processes governed by the invasion percolation with memory (IPM) can be estimated which further forms a feasible upscaling method to fill the gap between the simulation of pore-scale lamellae behaviors and the simulation of core-scale foam propagation.

In this work, comprehensive fractional flow analysis based on experimental investigation of foam displacement inside the tight core samples is performed, and the impacts of the absolute permeability, foam injection rate, and foam quality are studied to evaluate the potential and feasibility of further optimization in the parameter design of gas injection processes. The experimental results are compared with the fractional flow profiles simulated with a quasi-static pore network model constructed from modified invasion percolation with memory, to look for the calculable and efficient upscaling approach extending the foam performance prediction into the foam displacement in tight reservoirs.

2. Modeling Foam Propagation in Porous Media

2.1. Modeling Scenarios and Fundamental Assumptions

In the modeling section of this work, foam propagation is realized by injecting CO₂ into a pore network filled with surfactant solution. The 3D pore network is generated by connecting sites with bonds at a coordination number of 6. The site of the network represents the pore body attached with six pore throats at the maximum, whereas the bond represents the pore throat connecting two adjacent pore bodies. A truncated log-normal distribution shown in Equation (1) is adopted to prepare the pore network for this work,

$$f\left(R,\sigma \right)={\displaystyle \frac{\sqrt{2}\exp [-\frac{1}{2}(\frac{ln\frac{R}{R}}{\sigma }{)}^2]}{\overline{R}\sqrt{\pi {\sigma }^2}[\mathrm{erf}\left(\frac{ln\frac{R_{max}}{\overline{R}}}{\sqrt{2{\sigma }^2}}\right)-\mathrm{erf}\left(\frac{ln\frac{R_{min}}{\overline{R}}}{\sqrt{2{\sigma }^2}}\right)]}}$$

(1)

where $\overline{R}$ is the mean radius, R_max is the maximum radius, and R_min is the minimum radius. Both the radii of sites and the radii of bonds are determined based on the size distribution shown in Equation (1). σ is the heterogeneity factor describing the size variation in pore bodies and pore throats. In this work, assigning key geometric properties with the proposed distribution allows more accurate control of the range of pore size. Combining this feature with varying heterogeneity factors, the variation in frontal threshold exerting from capillary forces can be controlled indirectly.

The pore network is generated as shown in Figure 1, which is initially saturated with surfactant solution functioning as a foaming agent. CO₂ is injected via the inlet of the pore network at the exact pressure gradient to mobilize one minimum threshold path of the system [16]. The outlet of the pore network is defined at the constant pressure of 0.1 MPa, and the remaining surfaces are defined as a closed boundary. Figure 1a highlights the biphasic fluids distribution inside the network during the foam propagation, whereas Figure 1b presents the mobilized foam flowing path within the corresponding invasion step.

A pseudo-static algorithmic model based on modified invasion percolation with memory is adopted in this work to simulate foam propagation inside the pore network, which has been developed for decades and incorporated with major mechanisms involved during foam generation, coalescence, and propagation [16,17,19]. Several key assumptions are used in this type of topological foam flow model, including:

(1)

The invading gas phase is injected at the pressure gradient exactly sufficient to activate one foam flowing path at each invasion step.

(2)

The compressibility of both invading and defending phases is negligible, whereas the defending phase becomes trapping once having been disconnected from the outlet of the network.

(3)

A foam lamella is generated when the continuously invading gas phase enters a constricting part fully saturated with the defending liquid phase, and the foaming condition is fulfilled. When the invading gas phase has an active lamella within, the displacement through the snap-off sites will not create more lamellae by duplicating the existing one.

2.2. Foam Flow in Porous Media with Lamellae Interaction

Snap-off and leave-behind are two main mechanisms used in this work to create foam lamellae. Initially, each pore throat was assigned a random foaming value p ranging from 0 to 1, and the critical value p_SO is a model input parameter used to define the activeness of foaming events. A snap-off event occurs when the continuous invading gas phase enters a constricting part, which is the pore throat in this work, fully saturated with the defending liquid phase of the pore network, and a foaming value p lower than critical p_SO has been assigned. Figure 2 presents a representative distribution of snap-off sites at specific p_SO, ranging from 0.1 to 0.9. The higher the p_SO is, the more active the lamellae generation will be.

When an invasion event disconnects the defending phase within a single pore throat away from the main cluster of the defending fluid which stays connected with the outlet of the network, the liquid within can be considered as a lamella according to the interfacial configuration. This lamella generation scenario is equivalent to the classic description of leave-behind as one of the major foam generation mechanisms as shown in Figure 3, that both pore bodies of the same pore throat have been displaced in parallelly. According to the assumption of trapping formation, such an interfacial configuration pattern would result in “bond trapping”. Thus, the leave-behind lamellae do not make an additional contribution to the accumulation of foam flowing threshold in the invasion percolation-based algorithm [20].

The foam generation mechanism of division, which duplicates active lamella at specific pore structures, is incorporated in this study because the division sites are not assigned to the ideal bond-site pore network.

The pressure threshold required to mobilize an entire foam flowing path (Δp_f in Equation (2)) includes two terms: one is the frontal capillary term Δp_c defined in Equation (3) as the red dashed line in Figure 4 and the other is the train-like sum term Δp_L of lamellae highlighted in red dashed frame in Figure 4 and Equation (4).

$$∆p_f=∆p_c+∆p_L$$

(2)

$$∆p_c={\displaystyle \frac{2\gamma }{R_t}}$$

(3)

$$∆p_L=\sum _n\left[{\displaystyle \frac{4\gamma }{R_{t,i}}}\right]$$

(4)

where γ is the interfacial tension and R_t is the radius of the pore throat. Because the foam displacement scenario studied in this work is a drainage process, the capillary force exerted on the invasion front is the minimum resistance of gas invasion. Since the foam lamella has a double-layered interfacial structure, the pressure drop required to mobilize it is doubled at a quasi-static state. At each step, the invasion candidate with the minimized Δp_f would become the minimum threshold path (MTP) of the foam propagation and complete the invasion as shown in Figure 4b.

Except for the fresh foam lamellae generated during the invasion along the MTP, the number and thickness of active lamellae should be updated between two consecutive invasion steps, as the results of defoaming activities due to capillary suction. Recent studies on modeling foam propagation with the IPM approach have provided a systematic method of incorporating lamellae thinning and rupture via stepwise estimation of film thickness, which is dependent on the equilibrium between capillary pressure and disjoining pressure [19]. Also, additional lamellae rupture due to periodical squeezing and stretching of liquid films inside porous media can be equivalently considered based on the estimation of the limiting capillary pressure [21]. Schematics of two consecutive invasion steps are shown in Figure 4, where the non-wetting gas phase invades site E from site D, whereas two lamellae are mobilized forward and one more new lamella is generated within the bond AB. Also, two invasion candidates at site F and site G are removed due to the formation of two new leave-behind lamellae inside bond EF and EG as stated in assumption (2). As assumed by the invasion percolation algorithm, such a configuration of biphasic fluid will be treated as immobile trapping in the remaining modeling process.

2.3. Involving Post-Breakthrough Viscous Impacts in Flow Field Calculations

Before the breakthrough of invading foam, the increased pressure gradient will directly contribute to building up and satisfying the pressure threshold required from the MTP. When the invading front of foaming gas reaches the outlet of the lattice, a stable connection of the foam flowing path from the inlet is established, so a fraction of the further increased pressure gradient will be consumed by the quasi-static foam flow along this already opened path (AOP). As a result, the pressure threshold required to invade more non-swept regions becomes higher.

Previous work on fluid flow with yield stress in pore networks has given out the quasi-static model estimating the pressure thresholds required to extend non-swept regions after foam breakthrough [17]. After foam breakthrough, the pressure threshold required to mobilize the foam lamella inside any pore throat along the candidate foam flowing path should be updated as

$$p_{LD}={\displaystyle \frac{1}{r_{tD}}}+{\displaystyle \frac{q_{tD}}{r_{tD}^4}}$$

(5)

where the dimensionless flow rate along the throat is

$$\left|q_{tD}\right|=r_{tD}^4\left[1-{\displaystyle \frac{1}{\left|{∆p}_{tD}\right|r_{tD}}}\right]\left|{∆p}_{tD}\right|, \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} \left|{∆p}_{tD}\right|>{\displaystyle \frac{1}{r_{tD}}}$$

(6)

$$0, when \left|{∆p}_{tD}\right|\le {\displaystyle \frac{1}{r_{tD}}}$$

(7)

The dimensionless pressure p_LD is $\overline{R}$P_L/4γ, the dimensionless radius of pore throat r_tD is defined as R_t/$\overline{R}$, and the dimensionless flow rate q_tD is 2μ_iQ_i/πγ$\overline{R^2}$, where μ_i and Q_i are the viscosity and flow rate of the non-wetting invading gas phase, respectively. For a lamella in a pore throat that belongs to a branched foam flowing path yet to reach the outlet of lattice, the pressure drop required to mobilize it is still 1/r_tD by combining Equations (5) and (7). Otherwise, if the pore throat with the active lamella belongs to an AOP, even if it is only a part of it, then an additional rate-relevant pressure drop should be added while estimating the stepwise pressure threshold of the entire candidate foam flowing path.

With these algorithmic invasion rules, the quasi-static foam displacement proceeds by updating fluid configuration and the distribution of foam lamellae. Figure 5 presents a representative fluid distribution during foam propagation concluding all four typical element configurations. If both adjacent pore bodies and the intermediate pore throat are filled with the defending liquid phase, which is assumed as a Newtonian fluid in this work like CD in Figure 5, the flow rate can be quantified with Poiseuille’s law

$$q_{tD}={∆p}_{tD}^{ab}{r}_{tD}^4{=r}_{tD}^4\left(p_{dD}^a-p_{dD}^b\right)$$

(8)

where d indicates the defending phase, whereas the dimensionless flow rate q_tD is 2μ_dQ_d/πγ$\overline{R^2}$.

For the invasion front where the interface stands, the capillary forces are the resistance against the propagation since a drainage process is assumed. Thus, for the invasion front like EF in Figure 5, the impact of capillary pressure should be taken into consideration for the flow rate

$$q_{tD}=r_{tD}^4\left(p_{dD}^a-p_{dD}^b-p_c\right)$$

(9)

For the non-frontal displaced pore-body–throat set without any active foam lamella like AB in Figure 5, the flow rate of the invading gas phase is estimated by

$$q_{tD}{=r}_{tD}^4\left(p_{iD}^a-p_{iD}^b\right)$$

(10)

For the non-frontal displaced pore-body–throat set without the active foam lamella, the flow rate can be quantified by Equation (6) once the pressure threshold of mobilization is achieved.

Thus, at each step after the foam breakthrough, the overall outlet flow rates of both phases can be solved by these pore-body–throat sets and the mass conservation shown in Equation (11).

$$\sum _{a=1}^n{q}_{ab}=0$$

(11)

where n is the coordination number, which is 4 or 6, and in ordinary 2D or 3D cases, respectively.

The matrix of biphasic fluid flow in the pore network can be solved with successive over-relaxation and conjugate gradient approaches. After solving the flow field of the foam propagation with decreasing water saturation, the absolute permeability and relative permeability of biphasic fluids can be calculated with Equations (12)–(14) below:

$$k=-{\displaystyle \frac{q_d{\mu }_d}{\nabla p_d}} \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}{ S}_w=1$$

(12)

$$k_{ri}=-{\displaystyle \frac{q_i{\mu }_i}{k\nabla p_i}} \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} { S}_g>0$$

(13)

$$k_{rd}=-{\displaystyle \frac{q_d{\mu }_d}{k\nabla p_d}} \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} { S}_w<1$$

(14)

where q_d and q_i are the fluid flux per unit area of the defending phase and invading phase, respectively. Then, the fractional flow f_w can be given out with Equation (15),

$$f_w={\displaystyle \frac{1}{1+\frac{k_{ri}{\mu }_d}{k_{rd}{\mu }_i}}}$$

(15)

in which the viscosities of the defending phase and invading phase are defined as 1 cP and 0.018 cP, respectively.

The complete procedures of modeling foam propagation and corresponding fractional flow analysis in the pore network are summarized in the flow chart of Figure 6. The entire process can be divided into two stages that are distinguished by the breakthrough of the invading phase. Before the gas breakthrough, the biphasic fluid distribution and corresponding local flow resistance are estimated with modified invasion percolation with a memory approach [19]. Although the invading phase has not reached the outlet of the network, the gas saturation has been affected already due to the local flow resistance contributed by active foam lamellae within the network. After the breakthrough of invading gas, relative permeability and fractional flow of non-wetting gas and wetting liquid can be quantified accordingly with quasi-static fluid distribution inside the pore network during the stepwise foam displacement process. The invasion sequence of foaming gas and corresponding biphasic fluid distribution is estimated inside the pore network within the main loop of the process shown in Figure 6. Combined with them, the stepwise flow field and pressure field of the network can be calculated accordingly, when the entire displacement is complete. Then, the corresponding relative permeability curves and fractional flow profile can be plotted as a function of declining water saturation.

3. Core-Flooding Investigation on Foam Propagation in Porous Media

3.1. Materials and Preparation

The core-flooding test is adopted to evaluate foam propagation in porous media at the mesoscale, which is an effective tool to validate the proposed method and results from microscale pore network modeling and optimize parameter design for field application of foam-related processes such as foam-involved EOR projects and foam-assisted CO₂ sequestration.

The experimental apparatus of the core-flooding test is shown in Figure 7. The core sample was installed inside the core holder (RCHR-1.5, Core Lab, Amsterdam, The Netherlands) and sealed with the constant confining pressure sustained by a hydraulic pump (250D, Teledyne ISCO, Lincoln, NE, USA). The stable foam flow was realized by co-injecting the foaming surfactant solution and CO₂ at the constant flow rate, respectively. To ensure the fixed foam quality during the process, CO₂ was injected at a constant flow rate controlled by a mass flow controller (SLA 5850, Brooks, Seattle, WA, USA) and the foaming surfactant solution was injected with a hydraulic pump (500D, Teledyne ISCO). The backpressure regulator (BPR) was installed to simulate the reservoir pressure and reduce the impact of gas compressibility. An actuator was connected to the outlet of the core holder to switch the liquid sampler periodically, and the gas flow rate was measured with a gas flow meter (MilliGascounter, Ritter, Germany), which was installed with the liquid sampler.

In the work, the core samples, including a Bandera Gray core and a Colton core, were ordered from Texas, and the basic properties are listed in

Table 1. (a) Basic properties of core samples. (b) Mineralogy and elemental composition of core samples.

(a)
Type of Core Sample	Bandera Gray	Colton
Length, cm	10.16	10.16
Radius, cm	1.91	1.91
Porosity	0.204	0.143
Pore Volume, cm3	23.75	16.65
Permeability, mD	3.14	0.15
Bulk Density, g/cm3	2.20	2.38
Grain Density, g/cm3	2.71	2.68
Young’s Modulus, 106 psi	2.60	2.80–6.0
Poisson’s Ratio	0.2	0.13–0.16
(b)
Mineral composition/Element, wt%	Bandera Gray	Colton
Quartz	87	87
Kaolinite	6	2
Albite	3	5
Illite	2	2
Na	0.75	0.58
Mg	0.68	0.82
Ca	0.23	0.33
Fe	0.98	0.1071

a, whereas the mineralogy and element composition information are listed in Table 1b. The synthetic brine in this work, either injected for pre-saturation or used for preparation of the foaming agent, was prepared based on the composition listed in

Table 2. Chemical composition of experimental brine.

Ion Type	Salinity, mg/L
Potassium and Sodium (Ka+, Na+)	5112
Calcium (Ca2+)	10,276
Magnesium (Mg2+)	322
Chloridion (Cl−)	26,669
Sulfate (SO42−)	74
Bicarbonate (HCO3−)	445

. Anionic surfactant (Bio-Terge AS-40, Stepan, Northfield, IL, USA) was applied to prepare the foaming agent at 1.0 wt%. The crude oil sample was collected from the Bakken formation in southern Saskatchewan, which was 5.75 ± 0.5 cP in viscosity and 0.849 g/cm³ in density.

3.2. Experimental Procedures

After the installation of the experimental apparatus, the basic properties such as porosity and permeability of the core sample were measured. Darcy’s law was used to estimate the permeability of the core sample at varying liquid injection rates from 0.1 cm³/min to 0.3 cm³/min, and the absolute permeabilities of the Bandera Gray core and Colton core used in this study were 3.14 mD and 0.15 mD, respectively.

Before foam injection, 1 PV of surfactant solution was injected to pre-flush the core sample in case of severe surfactant adsorption. Then, foam injection was realized via co-injecting foaming surfactant solution with a hydraulic pump and CO₂ with a mass flow controller at a constant flow rate, respectively. As listed in

Table 3. Grouping design of core-flooding tests.

Test No.	Core Type	Injection Rate, cm3/min	Foam Quality
1	Bandera Gray	0.05	0.8
2		0.1	0.8
3			0.8 foam free
4			0.5
5			0.5 foam free
6		0.2	0.8
7	Colton	0.05	0.8
8		0.1	0.8
9			0.8 foam free
10			0.5
11			0.5 foam free
12		0.2	0.8

, the permeability of the core sample, total injection rate of the foaming agent and gas, and foam quality are key factors investigated in this study to reveal the foam performance in the tight reservoir and help optimize our simulation results of micro-scale foam propagation. When the steady state of foam flow was achieved, surfactant injection was stopped, and pure CO₂ was injected at the designed total injection rate until no more liquid was produced. The dynamic water saturation was continuously counted based on material balance during both foam and pure CO₂ injection periods.

4. Results and Discussion

4.1. Pore Network Modeling Study: Mobilization and Resistance of Foam Flow in Porous Media

Both the relative permeability curves of the gas–liquid inside the porous media and the corresponding fractional flow relation can be quantified with the proposed algorithmic approach. Figure 8 is the collection of simulation results of relative permeability and corresponding fractional flow of water, based on the modeling method introduced in Section 2. Five representative foaming conditions are discussed varying from extremely weak foam (p_SO = 0.1) to extremely strong foam (p_SO = 0.9). Previous studies on foam flow in pore networks have shown that stronger foaming conditions would lead to more active foam lamellae, which exert higher pressure thresholds in foam mobilization and less free gas flow [19].

In the relative permeability analysis, the decline in gas relative permeability is clear as p_SO increases, indicating more active lamella generation events during the gas invasion process. According to the results of pseudo-static pore network modeling of the foam displacement process, the gas relative permeabilities in the presence of foam have shown linear functional relation corresponding to the increasing water saturation, which differs from the relative permeability curve of typical non-wetting fluids. As stated in the previous studies on modeling foam flow with pore network modeling, a critical snap-off probability (p_SO*), normally from 0.32 to 0.36, is observed that distinguishes foaming status [19]. When geometric properties of the pore network and fluid properties are determined, the simulation with a snap-off probability higher than p_SO* will result in a strong foam state, which does not have any continuous gas-flowing path inside the pore network when the breakthrough occurs. For Figure 8a,b, the relative permeabilities of the gas phase are apparently higher than others, which represents a common weak foam state with limited presence of foam lamellae and reduced continuous gas flow compared with foam-free gas invasion cases. Also, the maximum relative permeability of either the gas phase or aqueous phase, which is achieved at the minimum or maximum end point of relative permeability profiles, reduces as p_SO increases. The semi-logarithmic relative permeability profiles plotted in Figure 9 compare the reduction in gas relative permeability with varying p_SO that shows a clear range of relative permeabilities developing from weak foam to strong foam.

The fractional flow curve of foam flow can be constructed based on relative permeability results with Equation (15), which provides an intuitive presentation of steady-state mobility of foam flow and effective gas trapping with decreasing water saturation [22,23]. For the displacement scenario of gas invasion, the fractional flow of water has declined rapidly since the gas breakthrough due to the significant mobility difference between the gas and aqueous phase. Also, the transition indicates the abrupt conversion from a strong foam regime to a no-foam regime corresponding to the limiting capillary pressure [10]. When the strong foam presents, the decline in the water fractional flow is effectively postponed as water saturation reduces, indicating that the gas is trapped inside the porous media due to the involvement of foam. The infection point of the fractional flow curve with decreasing water saturation is one of the indicators of foaming ability. As water saturation reduces, the further such a turning point of water fractional flow is delayed, and the more effective gas trapping is during the foam propagation. Figure 10a shows the comparison between fractional flow curves with varying p_SO that shows the difference in fractional flow as foaming status changes. When weak foam dominates the foaming gas invasion process, the decline in water fractional flow starts at 0.66 and 0.61 in water saturation (S_w) with the p_SO of 0.1 and 0.2, respectively. For the remaining fractional flow curves of water, which all belong to the strong foam scenario, the turning points are delayed until 0.38 of remaining water saturation when p_SO > p_SO*. Although the fractional flow analysis of the foam injection process gives out the critical S_w corresponding to the limiting capillary pressure dominating the transition from strong foam state to weak/no-foam state, this critical value cannot distinguish the effective gas trapping of the weak foam state from the no-foam state, especially for the gas channeling control attempts inside the tight reservoirs. Since S_w and f_w both range from 0 to 1, the area of the region enclosed by a foam fractional flow curve has become a natural dimensionless number that quantitively describes the gas-trapping ability of foam flow in homogeneous and consistent porous media. In this work, the ratio of the area enclosed with a fractional flow curve to the theoretical maximum area of the fractional flow profile is defined as the dimensionless gas-trapping index. Figure 10b shows the results of the gas-trapping index collected from Figure 10a after interpolation and integration, which provides an intuitive comparison of the gas-trapping ability of foam flow in tight porous media, from weak foam state to strong foam state.

For the fractional flow analysis of foam performance in the conventional reservoirs, the abrupt inspection point of the water fractional flow decline in strong foam is on the far left of the profile corresponding to a low water saturation, especially compared with the foam-free water-alternating gas (WAG) process that has been encountered with the abrupt inspection point at quite a high water saturation [10,22]. Among the studies of foam performance in the conventional porous media with relatively higher permeability, the fractional flow curves of weak foam also decline abruptly when the limiting capillary pressure is achieved [10], which leads to one of the most practical understandings that weak foam is inefficient as a mobility control method, either for soil-enhanced oil recovery projects or for improved CO₂ sequestration. As a result, foam-related techniques have not been widely adopted in unconventional oil development due to the difficulties related to injectivity issues or lack of suitable field facilities. However, recent pilots of different foam-related gas injection projects in unconventional reservoirs have reported quite good results and potential in enhanced oil recovery and gas channeling control [22,24,25].

Except for the inspection point corresponding to the rapid reduction in water fractional flow, the absolute derivative |df_w/dS_w| is the other indicator describing the reduced gas-trapping ability of foam flow in porous media as shown in Figure 11. For the gas immiscible displacement process in conventional porous media, the abrupt decline will result in distinguished |df_w/dS_w| separated by the water saturation corresponding to the limiting capillary pressure. In tight porous media, the absolute derivative of fractional flow profiles, for both strong and weak foam states, have presented asymptotic features as water saturation reduces.

In the tight porous media, as presented in Figure 8 and Figure 10, the transition of water fractional flow profiles is not as abrupt as the conventional ones, for either a strong foam state or a weak foam state. On the one hand, the water fractional flow declines earlier than strong foaming scenarios in conventional porous media as the gas invasion continues when p_SO > p_SO*, indicating the restrained gas trapping with a relatively higher water saturation; on the other hand, the gas-trapping performance of weak foam has been enhanced according to the delayed inspection point and less abrupt decline in fractional flow profiles as water saturation decreases. The results suggest that although weak foam is unable to provide as long-term effective gas trapping during the gas invasion process as strong foam, it avoids the abrupt reduction in fractional flow as weak foam in conventional porous media or water-alternating gas (WAG) in tight porous media, indicating that it exerts a certain degree of effective gas trapping.

Pore network modeling of foam displacement in tight porous media provides an efficient tool for upscaling pore-scale foam development to mesoscale foam performance evaluation with reasonable simplification. Later, a series of core-flooding tests are carried out to validate current simulation results and analyze the impacts of foaming parameters.

4.2. Core-Flooding Experimental Investigation: Gas Trapping During Foam Propagation

Fractional flow profiles are efficient and effective in the rapid evaluation of foam performance in porous media. The onset of the inspection point of water fractional flow decline, the slope of fractional flow decline (or rate of decline), and the confining water saturation of the end point are crucial in the fractional flow analysis of foam propagation.

Commonly, foam injection rate, foam quality during the injection, and the permeability of porous media are considered three major factors of foam performance that are independent of the properties of foaming surfactant.

In the conventional porous media, the gas-trapping ability of foam is positively correlated with its injection rate. For either the initial establishment of the foam bank or continuous lamellae regeneration inside deeper regions of interest, a higher injection rate would create lamellae more frequently that directly boost the foam bank with better gas-trapping ability. In Figure 12, six foam displacement core-flooding tests are plotted with two absolute permeabilities and three typical injection rates, respectively. According to the experimental results within the tight core samples, compared with the injection rate, the absolute permeability is still the dominating factor affecting the foam performance in tight porous media, especially when the difference raised by permeability has reduced as the core sample becomes tighter from 3.14 to 0.15 mD. As learned from foam performance studies in conventional porous media, increasing the foam injection rate is an efficient approach to improve the mobility control ability of foam that makes it easier to achieve a strong foam regime at various water saturation and fractional flow conditions. However, the experimental results revealed in this work, that foam performance becomes less sensitive to its injection rate, implies a hint in the application-designing perspective of foam-involved processes in tight reservoirs, like foam flooding EOR or foam-assisted CO₂ sequestration project, that is relieved from chasing stronger foam regime via increasing its injection rate blindly, which is difficult and risky for these applications in tight reservoirs due to the limitation in fluid injectivity.

Foam quality is another key factor directly affecting foam performance in porous media. Based on earlier experimental studies reported in decades, maintaining higher foam quality is considered a necessary condition to achieve a stronger foam regime that presents more effective and stable gas trapping in conventional porous media. However, as a response to develop efficient and reliable enhanced tight oil recovery techniques, a novel foaming gas injection method is proposed and named low-tension gas (LTG) flooding that pursues desired rheological features for the stable foam displacement of light hydrocarbon resources in tight reservoirs, which is poorly suitable for high-foam-quality injection strategy [26,27,28]. Impacts of foam quality on foam fractional flow profiles in the Bandera Gray core are compared in Figure 13. Two typical foam qualities are chosen, including 0.8 and 0.5, respectively corresponding to a strong foam regime and weak foam regime according to previous foam displacement studies in porous media. Fractional flow profiles of foam-free co-injection cases are matched as well.

According to Figure 13, the presence of both strong foam and weak foam has shown gas trapping that adjusts fractional flow collected at the outlet of core samples. Although the decline in weak foam fractional flow curve is observed at higher water saturation as foam propagation continues, indicating less effective gas trapping at a relatively higher water saturation (or earlier displacement stage), the fractional flow curve of weak foam soon merges with the strong one as the displacement proceeds and reaches the zero water fractional flow at similar water saturation, which implies the gas-trapping performance of weak foam approaches the strong one at later foam invasion stage at relative lower water saturation. A similar trend has been observed within the Colton core sample with an order of magnitude less in absolute permeability as plotted in Figure 14. Unlike the results of the foam-free control group in the Bandera Gray core, a further right water fractional flow with a 0.8 gas fraction indicates more severe gas channeling than the test with a 0.5 gas fraction, because tighter porous media exaggerates the negative impact of unfavorable mobility ratio during the immiscible gas drainage process. When the foaming surfactant presents, the gas-trapping ability of both displacement scenarios is greatly improved. A weak foam regime with lower foam quality results in earlier water fractional flow decline and delayed confining water saturation. The distinguished phenomena imply that weak foam with lower foam quality in extremely tight porous media (k < 0.2 mD) would result in reduced peak gas-trapping capacity and longer effective gas-trapping period with extended range of water saturation. The experimental results suggest that one of the key mechanisms of the LTG flooding process in tight reservoirs is to efficiently convert the peak gas-trapping capacity into the extended effective gas-trapping window of varying water saturation. Such a mechanism can effectively guide the application and parameter design of LTG flooding by optimizing the moderate but continuous gas trapping and mobility adjustment during the foam displacement.

The impact of permeability is distinct during the various foam displacement processes. In Figure 15, four representative fractional flow profiles are shown to compare the gas-trapping ability. As permeability reduces, not only the foam performance of gas trapping has been weakened but more severe gas channeling is observed in the foam-free cases. It supports the necessity of introducing additional mobility control into the immiscible gas injection process among the tight reservoirs. In other words, although the hampered gas-trapping ability of foam becomes less effective inside the tight reservoirs, it will still distinctly boost either the oil recovery factor or the CO₂ sequestration efficiency by providing a degree of mobility adjustment inside the tight porous media to greatly relieve current difficulties in those field applications due to the severe gas channeling.

4.3. Forward of Foam Displacement Simulation Based on the Experimental Investigation

Simulation of foam displacement in porous media is complicated, whereas the foam propagation model constructed based on invasion percolation with memory (IPM) has successfully filled the gap between the simulation of molecular-scale interfacial evolution with the MD approach and macroscopic simulation with the population balance model or empirical method. As discussed in Section 4.1, both relative permeability and fractional flow profile describing foam displacement inside the pore network can be plotted by simplifying and simulating the foam-involved gas invasion as a quasistatic immiscible process. Recent developments have added more realistic mechanisms into the algorithmic model like yield stress and viscous effect along the preferential foam flow paths, the dependence of mobilized foam fraction on pressure distribution, the transition between the immobile foam bank and intermittent gas flow at designated pressure gradient, etc. [17,18,21]. Experimental validation has been carried out on the foam performance within the conventional porous media, which has a good match, especially among the relative permeability of the gas phase [29].

However, for the foam displacement in tight porous media with enhanced capillary forces and accelerated lamella thinning, current pore-scale simulation cannot fully capture the characteristics among these peculiar interfacial activities. The numerical prediction results of limiting capillary pressure corresponding to the rapid decline in fractional flow curved reported in previous studies have been marked as dashed red line in Figure 16, which have shown significant increments of critical water saturation as the gas-trapping ability of foam fades away when the absolute permeability reduces from 1.30 mD to 0.13 mD [10]. Compared with our experimental study, the results predicted by the traditional method have clearly underestimated the gas-trapping ability of foam in tight porous media.

In this work, as for the results compared in Figure 16, simulation results of the pore network model exaggerate the gas-trapping ability of foam in both the Bandera Gray core (~1 mD) and the Colton core (~0.1 mD). From the topological perspective view of foam fractional flow profiles, the curve describing the gas-trapping ability of foam can be simplified into three components, including the onset of the inspection point as the indicator of gas breakthrough, the end point corresponding to the confining water saturation that can no longer be displaced by the specific configuration of foaming gas, and the development of slope connecting these two characteristic points. In Figure 16, the simulation of weak foam displacement within the tight porous media has matched the inspection point and its corresponding water saturation, but the simulated decline in water fractional flow of foaming gas invasion is slower than the experimental observation, indicating the overestimation of gas trapping at lower water saturation. Also, the current algorithm cannot fully capture the late stage of water fractional flow decline due to the present static assumption on the formation of a trapped wetting phase, which results in the lack of details in the fractional flow profiles among low water saturation regions. Thus, adding mechanistic description on weakened gas-trapping ability and continuous reduction in wetting phase clusters among low S_w region can greatly improve the simulation accuracy of foam displacement in tight porous media.

5. Conclusions and Recommendations

Fractional flow analysis is an effective and efficient tool to evaluate foam performance in porous media. A series of core-flooding experiments are performed to compare the gas-trapping ability in tight core samples with fractional analysis, further evaluating the feasibility of LTG processes. The fractional flow profiles of foam flow in tight reservoirs with the quasi-static pore network model (PNM) method based on invasion percolation with the memory algorithm validate the prediction with experimental results, which revealed several unreported shortages in the current model and implied the optimization of it. Combined with a series of core-flooding experiments and quasi-static pore network modeling investigations, this study analyzed and discussed the mobilization and resistance of foam flow in tight porous media, as well as the effectiveness of gas trapping during foam displacement. The key findings include the following:

(1)

Pore network simulation and core-flooding experiments have both identified the advantages of introducing weak foam in tight porous media that balances mobility adjustment and fluid injectivity.

(2)

As absolute permeability reduces, the foam performance in tight porous media becomes less sensitive to the variation in foam injection rate and foam quality, which supports the core idea of a low-tension gas (LTG) flooding process with more flexible foam injection strategies.

(3)

While being applied in tight reservoirs, some defects of the current pore-scale foam model based on the IPM algorithm have been revealed after the validation with various experimental results, which can be optimized via introducing mechanistic description on weakened gas-trapping ability and extended reduction in wetting phase clusters among low Sw regions during the foam displacement.

The foam performance studied in this work, either the core-flooding experimental investigation or the pore network model study, focuses on the fractional analysis of foam flow inside the matrix of tight reservoirs, which can quantify the gas-trapping ability inside homogeneous and consistent porous media affected by absolute permeability, foam quality, and foam injection rate. To thoroughly investigate the foam behavior in tight reservoirs, additional foam displacement tests across the dual-permeability systems are required such as the resistance factor and mobility reduction factor of foam flow in fractured core samples.