Foam Drainage Research Articles

A Review on the Water Invasion Mechanism and Enhanced Gas Recovery Methods in Carbonate Bottom-Water Gas Reservoirs

Carbonate gas reservoirs are crucial in gas field development, with carbonate bottom-water gas reservoirs being a significant subset. However, the development of these reservoirs often faces challenges such as water invasion, leading to a low gas recovery rate. Enhancing gas recovery is a primary goal for researchers in this field. This study provides a systematic review of the mechanisms, identification, and dynamic prediction of water invasion in these gas reservoirs. The technical adaptability and application range of different enhanced recovery methods are summarized, and their application effects are evaluated. The results indicate that carbonate gas reservoirs have diverse types of storage and permeability spaces, with a wide distribution of pore size scales, leading to various types of enclosed gas caused by water invasion. The prediction accuracy of water invasion models for bottom-water gas reservoirs with fractures and vugs is relatively low. Therefore, numerical simulation research on the basis of fine reservoir characterization is the key technology. The control of bottom-water invasion and the rescue measures after the bottom-water invasion are the keys to improving gas recovery, which can be divided into four types: drainage gas recovery, water control production, active drainage, and injection medium. Gas production by drainage is the main technology for improving gas recovery, among which foam drainage is the most widespread. The optimization of development parameters in production by water control has a good effect in the early stages of development. The active drainage technology on the water invasion channel is the bottom-up technology for the effective development of strong water-flooded gas reservoirs. CO2 injection may have great potential to improve the recovery of bottom-water gas reservoirs, which is one of the important research directions under the background of “carbon peaking and carbon neutrality”. The research provides theoretical and technical reference significance for enhanced recovery of carbonate bottom-water gas reservoirs.

Preparation and properties of a fluorine-free aerogel foam extinguishing agent

Research on fluorine-free firefighting foams is urgently needed due to the environmental hazards of conventional aqueous film-forming foams. Functional particles have significant research potential in the development of fluorine-free foam extinguishing agents. In this study, hydrophobic aerogel particles were prepared using a template method, and a fluorine-free aerogel foam extinguishing agent was developed by electrostatic and steric stabilization. The research investigated the impact of aerogel particles and polymers (xanthan gum and polyethylene glycol) on foam characteristics, fire extinguishing performance, as well as their synergistic effect. The results indicated that polymers and aerogel particles reduced the average foam radius and enhanced the foam liquid film strength, though they could not inhibit foam coarsening. Although accelerating foam drainage, aerogel particles effectively enhanced the fire extinguishing performance of the foam. With the addition of only 0.4wt.% aerogel particles, the fire extinguishing time of the foam was reduced by 21.7%, while 100% burnback time increased by 45%. Furthermore, the polymer enhanced the burnback performance of the foam in synergy with aerogel particles by improving foam stability on the hot heptane surface.

The series solutions of fractional foam drainage and fractional modified regularized long wave problems

In general, integer order differential equations fall short of adequately describing a range of phenomena in many disciplines of science and engineering when compared to fractional order differential equations. In the context of the Caputo fractional derivative, this study examines the nonlinear fractional order foam drainage problem as well as the fractional order modified regularized long wave equation. In this work, we offer two novel techniques by combining the Elzaki transform with the Adomian decomposition method (ADM) and the homotopy perturbation method (HPM). We first translate the problem into its differential partner using the Elzaki transform, and then we use He’s polynomials and the Adomian polynomials that are derived from HPM and ADM, respectively. These are two very powerful supports for nonlinear problems. The efficacy and precision of the recommended techniques are illustrated through the use of numerical and graphical findings. The numerical simulations of the derived solutions make use of a number of relevant values of the order of fractional parameter λ. It is important to keep in mind that the recommended techniques can achieve higher overall performance since they need less computational effort than alternative strategies while yet retaining a high degree of numerical precision. The solutions acquired through the suggested techniques have been compared with the classical solutions and the solution obtained by ADM, new iterative method (NIM). The considerable results show that these approaches minimize the heavy calculation without restricting variables while requiring no assumptions. The present study can confirm the applicability of the new generalized Caputo type fractional operator to mathematical real-world problems. This approach shows to be one of the most efficient methods to solve fractional order nonlinear differential equations that arise in related fields of engineering and science. These methods also make clear how to use fractal calculus in real life. Furthermore, the results of this study support the value and significance of fractional operators in real-world applications.

Study on the Liquid‐Holding Rate Law of Gas–Water–Foam Three‐Phase Flow in a Wavy Pipe

ABSTRACTAs a common method for liquid drainage in wave‐shaped pipelines, foam drainage has been used in gas‐gathering pipelines in various oil and gas fields. One of the common problems encountered in wave‐shaped foam drainage pipelines is the inaccurate prediction of the liquid retention rate. This issue makes it difficult to predict liquid accumulation points, which affects gas output efficiency, causes pipeline corrosion, and generates natural gas hydrates. To clarify the liquid‐holding rate law of wavy foam drainage pipelines is studied. In this study, through experiments to verify the accuracy of the numerical simulation method, we investigate the easiest liquid accumulation points and the liquid‐holding rate of wave‐shaped foam drainage pipelines, with factors such as inlet gas velocity, inlet liquid velocity, import and export differential pressure, undulation angle, and inlet temperature change rule. Among them, the error between the numerical simulation results and the experimental results is 4.68%. Finally, after considering the influence of the aforementioned factors on the liquid retention rate of wave‐shaped pipelines, a new liquid retention rate calculation model for wave‐shaped pipelines with foam drainage is established by introducing dimensionless numbers, such as gas‐phase velocity coefficient, liquid‐phase velocity coefficient, and angle correction coefficient. The method is compared with the results of previous research, and the error is within 15%. The model has a simple form and high calculation accuracy, providing a theoretical basis for pipeline inspectors to predict liquid accumulation conditions and reasonably adjust production schedules.

Investigating the impact of iron oxide nanoparticles on the stability of class A foam for wildfire suppression

This study investigates the effect of Iron Oxide Nanoparticles (IONPs) on the stability of a class A foam. IONPs are synthesised via two different pathways, namely the reflux and the hydrothermal methods. The synthesised nanoparticles are then added to the foam solution at different concentrations, applying various techniques to test how they affect properties and processes such as particle size distribution, surface tension, and coalescence. Results show that (i) the addition of IONPs improves the foam stability by reducing the bubble coarsening and disproportionation; (ii) the IONPs assemble at the plateau borders and nodes of the bubbles creating a protective layer on the gas–liquid interface, which delays foam drainage hence improving foam stability; and (iii) stability is also improved by the increase in the foam half-life due to the accumulation of nanoparticles on the surface of the bubble. IONPs obtained using the reflux method are shown to affect the firefighting foam more positively.

Novel foam‐draining and anti‐polymerization integrated agent for gas well application

AbstractIn this article, surfactants such as cetyl trimethyl ammonium chloride (CTAC) and coconut oil amide propyl betaine (CAB) were compounded (TCJ) and used as a double function with foam drainage and hydrate anti‐polymerization. According to the result it was found that 0.05% of CTAC and 0.7% of CAB exhibited both excellent foam property and hydrate anti‐polymerization property. For further clarifying the stability characteristic of foam, the foam half‐life, drainage process, microcosmic liquid film structure, and stability mechanism were investigated in this article. The high initial foam volume of 415 mL with relatively long half‐life time of 5.5 min under high‐speed agitation at 50°C after 25 min indicated a good temperature resistance. The excellent foam properties still remained even amount of methanol and NaCl was added, which indicated its well stability and salt resistance. Furthermore, it was found that TCJ exhibited high liquid carrying rate of 62.5% at 65°C and good compatibility with poly vinyl pyrrolidone (PVP), which played great potential to the application in oil field. By the results of DSC curves, it can be found that the phase transformation point of hydrate decreases after addition of TCJ, which indicates the inhibition to the formation of hydrate as thermodynamic inhibitors.

Enhancing melt foam stability, form-filling process, and pore structure evolution of shaped aluminum foam

This study explores the stability of Al–Si–Ca–Mg-Sc melt foams and investigates the melt foam filling process and the dynamic evolution of pore structures in the fabrication of foam aluminum profiles using the melt transfer foaming technique. The results show that Al–Si–Ca–Mg-Sc melt foam maintains high porosity and a stable pore structure even after 30 min of foaming, without any signs of boundary rupture or destabilization. This stability is attributed to the increased viscosity from finely dispersed second-phase particles, which hinder the growth, coalescence, and drainage of the melt foam. Regarding melt foam filling, enhanced filling capability is achieved by elevating the foaming temperature, extending the foaming time, and increasing the TiH2 addition. The melt foam exhibits a smooth flow from the lower part of the mold to the corners, completing the filling process with well-defined edges in the profile. During lateral movement for filling, a gradual increase in porosity, a decrease in pore number, and a slow enlargement of pore size and standard deviation along the lateral (x-axis) are observed. This is due to the slower liquid phase flow compared to the gas phase, resulting in bubble coalescence. After filling is complete, this phenomenon improves, contributing to a slow and stable evolution of the pore structure.Under optimized conditions of 1.4 wt% TiH2 addition at 600°Cfor 9 min of foaming with Al–Si–Ca–Mg-Sc alloy melting, we successfully fabricated foam aluminum profiles with complete filling, a porosity of 81.5%, an average pore size of 2–3 mm, and a uniform pore structure.

Tuning thermal stability of fluorine-free foam by nano-MDH inorganic flame retardant

Foams enhanced by nanoparticles (NPs) have become an important way to develop new type of fluorine-free firefighting foams. This study focuses on thermal stability of foams generated by the mixed dispersion of CoatOsil-77 silicone and BS-12 hydrocarbon surfactants as foaming agent and nano-magnesium hydroxide (nano-MDH) as foam stabilizer. The interaction between nano-MDHs and surfactants were studied. Thermal stability of nano-MDH enhanced foam was explored by investigating drainage and volume attenuation of foam and heat transfer in vertical foam blanket. Stabilization mechanism of nano-MDH enhanced foam under heat was clarified. Results show that nano-MDHs have a weak interaction with the BS-12 molecules and almost has no interaction with the CoatOsil-77 molecules. Initial foam height of mixed dispersion increases at the presence of nano-MDHs, but further increase in the concentration of nano-MDH decreases its foaming ability. At room temperature, foam drainage and coarsening are accelerated by low concentration nano-MDH (1 wt%) and delayed by high concentration nano-MDH (>1 wt%). Meanwhile, foam was strengthened by rising nano-MDH content after exceeding 1 wt%. Drainage and volume attenuation of foam and heat transfer in vertical foam blanket were significantly delayed and the corresponding thermal stability was strengthened by nano-MDHs. This study can offer a possibility for tuning thermal stability of fluorine-free foams based on nano-MDHs.

Effect of gas type on properties of anionic surfactant foam system

The foam drainage technique is considered to be a promising measure for solving the problem of liquid accumulation in natural gas wells and gathering pipelines. However, differences in gases can greatly affect the performance of surfactant foam, thus limiting the application of this technology. To elucidate the mechanism of natural gas components influencing foam properties, the interfacial properties of hydrocarbon gas-surfactant-water system were analyzed thermodynamically and kinetically by combining molecular dynamics simulations and foaming experiments. The results show that with the increasing alkane chain length, the foaming volume and half-life of the foam are gradually rising, while the gas–liquid interfacial tension is decreasing, which is consistent with the results of molecular dynamics simulations. Compared with air, hydrocarbon gases are more beneficial for foam generation and stabilization. The interfacial adsorption of alkane molecules and the interaction between alkanes and surfactants are the main factors affecting the interfacial tension and film stability of different systems.

Mixed Systems of Quaternary Ammonium Foam Drainage Agent with Carbon Quantum Dots and Silica Nanoparticles for Improved Gas Field Performance.

Foam drainage agents enhance gas production by removing wellbore liquids. However, due to the ultra-high salinity environments of the Hechuan gas field (salinity up to 32.5 × 104 mg/L), no foam drainage agent is suitable for this gas field. To address this challenge, we developed a novel nanocomposite foam drainage system composed of quaternary ammonium and two types of nanoparticles. This work describes the design and synthesis of a quaternary ammonium foam drainage agent and nano-engineered stabilizers. Nonylphenol polyoxyethylene ether sulfosuccinate quaternary ammonium foam drainage agent was synthesized using maleic anhydride, sodium chloroacetate, N,N-dimethylpropylenediamine, etc., as precursors. We employed the Stöber method to create hydrophobic silica nanoparticles. Carbon quantum dots were then prepared and functionalized with dodecylamine. Finally, carbon quantum dots were incorporated into the mesopores of silica nanoparticles to enhance stability. Through optimization, the best performance was achieved with a (quaternary ammonium foam drainage agents)-(carbon quantum dots/silica nanoparticles) ratio of 5:1 and a total dosage of 1.1%. Under harsh conditions (salinity 35 × 104 mg/L, condensate oil 250 cm3/m3, temperature 80 °C), the system exhibited excellent stability with an initial foam height of 160 mm, remaining at 110 mm after 5 min. Additionally, it displayed good liquid-carrying capacity (160 mL), low surface tension (27.91 mN/m), and a long half-life (659 s). These results suggest the effectiveness of nanoparticle-enhanced foam drainage systems in overcoming high-salinity challenges. Previous foam drainage agents typically exhibited a salinity resistance of no more than 25 × 104 mg/L. In contrast, this innovative system demonstrates a superior salinity tolerance of up to 35 × 104 mg/L, addressing a significant gap in available agents for high-salinity gas fields. This paves the way for future development of advanced foam systems for gas well applications with high salinity.

Experimental study of foam stability and interfacial behaviour of cellulose nanocrystals-enhanced C22-tailed zwitterionic betaine surfactant

The intrinsic thermodynamic instability of foam systems significantly constrains their effectiveness in enhancing oil and gas recovery from unconventional reservoirs. This study investigated the foam properties and interfacial rheological characteristics of the EDAB/CNC/APG system, analyzing the influence of various factors including temperature, pH, and inorganic salts. The results indicated that the optimal concentration ratio for CNC-surfactant system consists of 0.3 % EDAB, 1.5 % CNC and 0.5 % APG by mass fraction. In this system, the addition of CNC forms a three-dimensional network structure at the gas–liquid continuous interface, significantly extending the drainage half-life of the system. At elevated temperatures, the intense molecular thermal motion disrupts the interactions between CNC and EDAB. This disruption leads to a decrease in surface tension and the viscoelastic modulus of interfacial expansion, resulting in an increase of foam volume but a decline in foam drainage half-life. Foam performance and interfacial rheological characteristics are optimal under neutral pH conditions and manifest higher stability in alkaline conditions compared to acidic conditions. In highly mineralized reservoirs (≥5.0 % salt concentration), inorganic salt ions shield the attraction between CNC and oppositely charged surfactants. This shielding weakens the cross-linking within CNC-surfactant systems, altering the structure of the liquid film interface and causing a sharp decrease in foam half-life. The C22-tailed zwitterionic betaine surfactant EDAB, enhanced by CNC, provides a novel, environmentally friendly, efficient, and safe approach to the exploitation of complex, low-permeability oil and gas reservoirs.

Experimental study on the mechanism of nanoparticles improving the stability of high expansion foam

AbstractHigh expansion (Hi‐Ex) foam is recommended to suppress the leakage and diffusion of cryogenic liquid due to its light weight and large volume. However, the disadvantages of low stability and high break rate under environmental conditions are all limited the further application in vapor mitigation and fire extinguishing. So that, this paper focus on the effect and mechanism of nanoparticles in stabilizing Hi‐Ex foam. Three kinds of nanoparticles with different concentration were selected to evaluate the effect of foam half‐life and the mechanism of particles on improving the foam stability. The results indicated that different particle concentrations can improve the foam stability to a specific extent, and the maximum improving of half‐life can increase by 95.4% in the presence of the hydrophilic SiO2 at .5 wt%. Meanwhile, the hydrophilicity, size, and morphology of the particles have a specific impact on the foam stability. The foam expansion rate first increased and then decreased. From the microscopic point of view, the bubble size gradually increases with time by two processes of ripening and coalescence and satisfied in a logarithmic distribution. While, the liquid film thickness remarkably decreases due to foam drainage without particles and the adsorption and accumulation of nanoparticles on foam lamella can provide a spatial barrier for the film thinning and the inter bubble diffusion. Finally, the microscopic interaction mechanism on improving the foam stability has been further explored and revealed in these two aspects.

Study on Dynamic Liquid-Carrying Process of Foaming Agent and Establishment of Mathematical Model.

The efficiency of foam drainage gas recovery is predominantly dictated by the performance of the foaming agent. To better understand their behavior, a novel testing apparatus was developed to simulate the foam drainage gas recovery process within the wellbore. Through the dynamic liquid-carrying performance tests of four foaming agents under uniform conditions, it was discerned that there existed significant disparities in the liquid-carrying performance and action duration. Further interface performance analysis disclosed that the liquid-carrying capacity and the duration were correlated with their adsorption capacity and interface activity at the gas-liquid interface. Notably, foaming agents with lower adsorption capacity and higher interfacial activity demonstrated superior liquid-carrying performance and longer action duration. By analyzing the consumption of foaming agents during the liquid-carrying process, five dynamic liquid-carrying equations were derived based on first-order reaction kinetics, the Malthusian population model, and the logistic function. The outcomes demonstrated that all these five equations could precisely delineate the dynamic liquid-carrying process of the foaming agent. During the research, we found that the consumption of the foaming agent in the foam drainage gas recovery process is related to its adsorption behavior at the gas-liquid interface, and revealed that the dynamic liquid-carrying process of foaming agent is the increasing process of liquid-carrying capacity under the continuous consumption of limited foaming agent resources. This laid a foundation for the further exploration of the functional mechanism of the foaming agent in the foam drainage gas recovery process.

Classical solvability to the free boundary problem for a foam drainage equation. II. From the interface to the bottom

We study a nonlinear partial differential equation describing the evolution of a foam drainage in one dimensional case which was proposed by Goldfarb et al. in 1988 in order to investigate the flow of liquid through channels (Plateau borders) and nodes (intersections of four channels) between the bubbles, driven by gravity and capillarity. Its mathematical studies so far are mainly restricted within numerical and particular solutions; as for mathematical analysis of it there are only a few. We prove that the free boundary problem for the foam drainage equation in the region below an interface to the bottom in a foam column admits a unique global-in-time classical solution by the standard classical mathematical method, the maximum principle. Moreover, the existence of its steady solution and its stability are shown.

Classical solvability to the free boundary problem for a foam drainage equation. I. From the top to the interface

We study a nonlinear partial differential equation describing the evolution of a foam drainage in one dimensional case which was proposed by Goldfarb et al. [Izv. Akad. Nauk SSSR Mekh. Ghidk. Gaza 2, 103–108 (1988)] in order to investigate the flow of liquid through channels (Plateau borders) and nodes (intersections of four channels) between the bubbles, driven by gravity and capillarity. Its mathematical studies so far are mainly restricted within numerical and particular solutions; as for mathematical analysis of it there are only a few. We prove that the free boundary problem for the foam drainage equation in the region from the top to the interface in a foam column admits a unique global-in-time classical solution by the standard classical mathematical method, the maximum principle and the comparison theorem. Moreover, the existence of its steady solution and its stability are discussed.

Enhancing bioactive compounds retention in pomegranate juice powder through foam mat drying

AbstractThe effect of addition of whey protein isolate (WPI; 0%, 5%, 7.5% and 10%) and carboxy methyl cellulose (CMC; 0%, 0.25%, 0.5% and 0.75%) was studied for its foaming properties and stability in pomegranate juice (PJ). High‐quality powders can be produced from fruit juices and heat‐sensitive food items using foam mat drying (FMD). The PJ foam with 10% WPI, a foaming agent, and 0.25% CMC, a stabilizer had high foam expansion (FE) (455.28%) and low foam density (FD) (0.2030 g/cc) and foam drainage volume (FDV) (20.10 mL). The PJ foam was subjected to different drying temperatures (50, 60 and 70°C) and thickness (3 and 6 mm). The drying temperature of 60°C and foam thickness of 3 mm were found to be the acceptable drying conditions based on drying time (180 min) and retention of bioactive compounds. At optimum drying conditions, the retention of phenols, anthocyanin, ascorbic acid, and antioxidant capacity was 97.86%, 83.90%, 77.29%, and 87.11%, respectively. The sensory evaluation of the reconstituted foam mat dried pomegranate juice powder (FPJP) showed lower overall acceptability scores than fresh juice (FJ), but the overall acceptability of the powder dried at optimum drying conditions was within liking limits (>6). FPJP has an amorphous nature with a particle size of 3564 nm and a porous and irregular structure. Bioactive compounds in FPJP were studied after 8 months of storage, with higher retention of bioactive compounds in refrigerated conditions.Practical applicationThe studies revealed that foam mat drying is suitable for the pomegranate juice which has high moisture and sugar content. It requires shorter drying time of 180 min. for drying up to moisture content of 4% (db). The foam mat dried juice powder obtained at optimum drying conditions has retained higher bioactive compounds and has higher sensorial acceptability when compared with the other treatment combinations. This drying process requires low initial investment and can easily be scaled up to the commercial level thus making it profitable business venture. Further, higher storage stability up to 8 months, low volume and consequent low transport cost improves the technical and financial viability of the process in industrial application. The Foam mat dried pomegranate juice developed can be used in many food products such as reconstituted beverages, ice creams, confectioneries, etc.

New exact solutions of the conformable space-time two-mode foam drainage equation by two effective methods

In this article, the two-mode foam drainage equation in terms of time and space conformable sense has been investigated. Two effective methods, the generalized exponential rational function method (GERFM) and the improved version of the Bernoulli sub-equation function method (IBSEFM), are used to get new solutions of underlying equation. The fractional travelling wave transformation is applied to convert nonlinear partial differential equations to nonlinear ordinary differential equations. Proposed methods successfully extract trigonometric, hyperbolic and exponential solutions. Some of the obtained solutions are visualized to understand the effect of fractional orders of time and space derivatives on the wave profile and the dynamic behavior of the solutions.

Foam drainage agent with enhanced foam stability through modified Fe3O4 nanoparticles

A nanofoam drainage agent was developed to address the challenges of inadequate foam stability and foaming efficiency in the gas recovery process of foam drainage. The formulation of the primary foam drainage agent was obtained by screening and combining conventional anionic and amphoteric surfactants. Experimental results revealed a synergistic interaction between α-olefin sulfonate (AOS) and lauramidopropyl betaine (LAB-35). Specifically, when a mixture of 0.4 wt.% AOS and 0.1 wt.% LAB-35 was used, the surface tension of the primary foam drainage agent was measured to be 23.32 mN/m. Furthermore, modification of Fe3O4 nanoparticles with the silane coupling agent γ-(methacryloxy)propyl trimethoxysilane (KH-570) was performed to improve the dispersibility in aqueous solutions and enhance foam stability. The results of infrared spectroscopy, x-ray diffraction, thermogravimetric analysis, and contact angle measurements revealed the effective bonding of KH-570 to the surface of Fe3O4 nanoparticles. Additionally, the particle size distribution in the aqueous solution showed the excellent dispersion of the modified Fe3O4 nanoparticles. Through the incorporation of the modified Fe3O4 nanoparticles into the primary foam drainage agent, a nanofoam drainage agent with a half-life of 989 ± 6.5 s was developed. Microscopic observations of foam coarsening behavior revealed the mechanism behind foam stability. Performance testing of the foam drainage agent demonstrated the agent’s outstanding resistance against salinity, temperature, and oil and its remarkable liquid-carrying capacity.

Study on Foaming Agent Foam Composite Index (FCI) Correlation with High Temperature and High Pressure for Unconventional Oil and Gas Reservoirs

In the process of unconventional oil and gas reservoir exploitation, it is difficult to reduce drilling fluid lost in natural fractures, enhance the CO2 displacement effect and reduce foam drainage gas recovery costs. In most cases, foaming agents can solve these problems in a low-cost way in a short period of time. Foaming agent screening and evaluation is the key to this technology. However, there are few experimental tests used in the evaluation of foaming agent properties that match the actual unconventional oil or gas well conditions of high temperature and high pressure. Using the actual temperature and pressure conditions of a wellbore, the foaming capacity and half-life of two common foaming agents were systematically evaluated by using the high-temperature and high-pressure visual foam properties evaluation device (UPMX-500), in which the foaming agent’s volume concentration was 3‰ in a simulated formation water with a pH of 6 and salinity of 9 × 104 mg/L. The high-temperature (40 °C, 60 °C, 80 °C, 100 °C) and high-pressure (0.1 MPa, 6.0 MPa, 8.0 MPa, 10.0 MPa) effect on the foaming capacity and half-life was analyzed. Binary linear regression of pressure and temperature was carried out, taking the foam composite index as the target and using a formula with high correlation. The results showed that the foam composite index (FCI) of the two foaming agents was positively correlated with pressure and temperature. The correlation of UT-7 was FCI = 64.1196T + 735.713p − 2066.2, the correlation of HY-3K was FCI = 62.5523T + 7220.391p − 2415.6, and the coefficients of determination were 0.9799 and 0.9895, respectively, with an error of less than 10%. This correlation equation can provide a reference for accurately predicting the foaming capacity of foaming agents under high-temperature and high-pressure conditions and can also be used to optimize foaming agents or to qualitatively evaluate results for the efficient exploitation of unconventional oil and gas reservoirs.

Solution of the foam-drainage equation with cubic B-spline hybrid approach

This work presents a robust and efficient numerical stratagem for the study of integer and fractional order non-linear Foam-Drainage (FD) model. The scheme first uses, usual forward difference and the L 1 formula, in integer and fractional cases, respectively. Then, the collocation approach together with cubic B-splines (CBS) basis are employed to estimate the unknown solution and its derivatives. With the help of these discretizations and Quasi-linearization, solving non-linear FD model transforms to the system of linear algebraic equations. The solution of the linear system approximates the CBS coefficients which further leads to the numerical solutions. Moreover, by Von Neumann stability it is proved that the proposed scheme is unconditionally stable. To evaluate the performance and accuracy of the technique, absolute error (AE), L 2, and L ∞ norms are presented. The obtained outcomes are also matched with some existing results in literature. It is noted from simulations that the proposed method gives quite accurate solutions.