Counter-current Imbibition Research Articles

Analytical Time-Dependent Shape Factor for Counter-Current Imbibition in Fractal Fractured Reservoirs

Summary The fluid exchange behavior for counter-current imbibition in fractured reservoirs can be quantitatively characterized by the transfer function in numerical simulation. The time-dependent shape factor (TDSF) in the transfer function is one of the main factors controlling fluid transport, which directly affects the result of ultimate oil recovery prediction. In practice, fractured reservoirs with different microscopic pore structures often have varied flow laws under the same flow conditions. However, the current TDSFs proposed for counter-current imbibition assume that the microscopic pore structure has no impact on the fluid inter-porosity flow behavior, which is inconsistent with the actual situation. In this work, the fractal theory is used to establish the TDSF of counter-current imbibition, which is related to the microscopic pore structure. First, the analytical solutions of average water saturation and imbibition rate are obtained under different conditions related to the maximum pore diameter and tortuosity fractal dimension of the matrix. The validity of the new analytical solution for strong water-wet and moderate water-wet reservoirs is ascertained by a single-porosity model and experimental data. Subsequently, the proposed analytical solution is applied to the two-phase transfer function to develop the new TDSF for counter-current imbibition, and the sensitivity analysis is carried out. The results demonstrate that the unsteady-state duration of the TDSF is proportional to the characteristic length and tortuosity fractal dimension of the matrix, and it is negatively proportional to the maximum pore diameter of the matrix. The influence of the characteristic length, tortuosity fractal dimension, and maximum pore diameter of the matrix on a constant shape factor (SF) under quasi-steady-state is exactly the opposite. This work provides an enhanced clarification of the fluid exchange behavior of counter-current imbibition in strong water-wet and moderate water-wet fractured reservoirs.

Numerical treatment of temporal-fractional porous medium model occurring in fractured media

This paper proposes a temporal-fractional porous medium model (T-FPMM) for describing the co-current and counter-current imbibition, which arises in a water-wet fractured porous media. The correlation between the co-current and counter-current imbibition for the fractures and porous matrix are examined to determine the saturation and recovery rate of the reservoir. For different fractional orders in both porous matrix and fractured porous media, the homotopy analysis technique and its stability analysis are used to explore the parametric behavior of the saturation and recovery rates. Finally, the effects of wettability and inclination on the recovery rate and saturation are studied for distinct fractional values.

Experiments and phase-field simulation of counter-current imbibition in porous media with different pore structure

Spontaneous imbibition is relevant to many processes in nature and industry. In this work, the effect of pore structure of porous media on counter-current imbibition is studied. The one-dimensional counter-current imbibition experiments were conducted in packed porous media with different pore structure. Experimental results showed that counter-current imbibition can only proceed for a short time in homogeneous porous media for all fluids. In heterogeneous porous media, counter-current imbibition can only proceed a short time for water–air experiments as well. However, imbibition can proceed continuously until the imbibition front reaches the end of packed porous media for water–oil experiments and the mass of water infiltration is proportional to the square root of time. The efficiency of water infiltration behind the imbibition front decreases with increase in viscosity of non-wetting phase fluid. In addition, the numerical simulation of counter-current imbibition in porous media with different pore structure were conducted using phase-field method. Simulation results showed that very little oil was produced in homogeneous porous media and more oil was produced in heterogeneous porous media. It is indicated that the difference between capillary driving pressure and capillary back pressure, which are closely related to pore structure and pore size distribution of porous media, has important influence on counter-current imbibition. Furthermore, the snap-off of oil or gas production channels in narrow pore throats may lead to termination of imbibition when the water saturation behind the imbibition front is higher.

Influence of sequential changes in the crude oil-water interfacial tension on spontaneous imbibition in oil-wet sandstone

Crude oil-water interfacial tension in petroleum reservoir is reduced or increased due to surfactant injection or surfactant retention, respectively. Changes in the interfacial tension crucially attribute to a governing capillary pressure and hence an oil displacement in spontaneous imbibition process. The current study attempts to elucidate an influence of such changes on spontaneous imbibition by replacing surfactant concentration consecutively with two approaches: sequential decrease and sequential increase in the interfacial tension. Two fluid flow directions were examined simultaneously: co-current and counter-current flows. With strongly oil-wet wettability, capillarity was a resisting element to oil displacement and therefore controlled by the oil-water interfacial tension. The sequential-reduced interfacial tension was found to weaken such resisting capillary force gradually and resulted in consecutive incremental oil production. On the contrary, the sequential-increased interfacial tension initiated the lowest interfacial tension fluid that produced an immediate large amount of oil, but did not much displace further oil. The current study observed a greater oil recovery obtained from a sequential reduction in the interfacial tension scheme (26.9%) compared to a conventional single reduction scheme (22.4%), with both schemes attaining same interfacial tension at last. In counter-current imbibition, same characteristics of oil displacement were observed as in co-current imbibition, with less oil produced and less sensitive to fluid changes due to negligible gravitational contribution.

Inertial Effect on Oil/Water Countercurrent Imbibition in Porous Media from a Pore-Scale Perspective

Summary The color-gradient lattice Boltzmann (LB) method is used to investigate the inertial effect on oil/water countercurrent imbibition characteristics in a matrix-fracture system. The interplay between capillarity, fluid inertia, and viscous force during the imbibition under different viscosity ratios is delineated. Pore-scale dynamics, the interfacial front morphology, and oil recovery under the influence of fluid inertia are also elucidated. Additionally, we study the energy conversion during the imbibition displacement from the perspective of energy balance. Finally, the application of the theoretical scaling model is discussed based on the simulated data. Results show that the pore-scale events involved mainly consist of cooperative pore filling, oil expelled from large pores, and the motion of jetting-like oil clusters under high viscosity ratios. The curve of pressure difference between the fracture inlet and outlet vs. imbibition time can be regarded as a signal to discern the imbibition regime, which is taken together with the energy conversion analysis could further determine how capillarity, external pressure, and viscous dissipation contribute to water imbibition. Capillary force dominates in the cases of low viscosity ratios, and the majority of the surface energy is dissipated. The external pressure becomes increasingly significant and even governs the countercurrent imbibition as the viscosity ratio increases. Furthermore, the oil recovery, interfacial area, and fractal dimension of the nonwetting phase strongly rely on the Ohnesorge (Oh) number when the viscosity ratio is low. In contrast, the inertial effect can be neglected in the cases of high viscosity ratios. Besides, the relationship between the simulated imbibition recovery and imbibition time follows the theoretical scaling model as the external pressure is trivial. The comparable exponents fitted from different Oh numbers reveal that the inertial effect does not alter the imbibition dynamics. In sum, fluid inertia only affects the local fluid behaviors and thus the imbibition oil recovery when the viscosity ratio is low. These results could provide important implications for a range of energy-related and environmental applications, such as the evaluation of fracturing fluids loss, oil recovery by water huff n puff, microfluidic devices, and hydrological sciences.

Effects of grain shape and packing pattern on spontaneous imbibition under different boundary conditions: Pore-scale simulation

Core-scale spontaneous imbibition experiments and numerical simulations have demonstrated that the macroscopic imbibition performance is significantly different under different boundary conditions. However, the detailed pore-scale flow mechanisms behind these phenomena and the influence of porous media's geometric features on the imbibition behavior under different boundaries have not been addressed in depth. In this work, an optimized color-gradient lattice Boltzmann model is applied to simulate spontaneous imbibition in granular media under four boundary conditions. The influence of grain shape and packing pattern on the two-phase interface evolution and the recovery factor during spontaneous imbibition is investigated while fixing different models' porosity, grain number, and size distribution. It is found that the grain shape has little influence on the drainage interface evolution at the initial stage but it has a more significant influence on the imbibition interface during counter-current spontaneous imbibition. The grain packing pattern influences the evolution of drainage and imbibition interfaces and the recovery factor. Spontaneous imbibition with different boundary conditions has different imbibition rates and ultimate recovery factors. The fastest imbibition rate and highest ultimate recovery are obtained under all faces open and two faces open (free) boundary conditions, respectively. This research provides pore-scale insights into the complex dynamic fluid displacement mechanism between the fracture and matrix of fractured oil and gas reservoirs.

Exact Analytical Solutions of Countercurrent Imbibition with Both Capillary and Gravity Effects

Exact Analytical Solutions of Countercurrent Imbibition with Both Capillary and Gravity Effects

Modeling of early- and late-time countercurrent spontaneous imbibition in porous media: A semi-analytical approach

Countercurrent spontaneous imbibition (SI) is an important flow mechanism to recover oil in water-wet porous media through capillarity. Experimental imbibition tests conducted in core samples show that oil recovery as a function of time occurs in a characteristic S-shaped curve, which describes the infinite-acting and the boundary-dominated regime. Although several dimensionless time groups have been proposed to model countercurrent SI in porous media, they fail to properly scale the results, causing inaccurate estimates of oil recovery. In this work, we present a new approach, using a hybrid solution, to model countercurrent SI for oil-water systems dominated by capillary forces. The infinite-acting regime is modeled by an early-time solution using a new dimensionless time group, which enables the correct scaling of imbibition results onto a universal single curve. To model the boundary-dominated regime, we introduce a late-time solution derived as a function of a characteristic distribution of saturation to estimate the flow behavior after the imbibing fluid reaches the no-flow boundary. The novelty of the model is that it enables the accurate estimation of fluid imbibition under the boundary-dominated regime, a flow condition critical to evaluate the true potential of countercurrent SI driven by capillarity for oil recovery in porous media. We verified the model against numerical simulations under a wide range of flow conditions relevant in water-oil systems, and used experimental data reported in the literature to validate our model. The solution presented provides an accurate approach to model countercurrent SI, which could be extended for dynamic conditions and the modeling of the flow of fluids in fractured media.

An experimental and field case study to evaluate the effects of shut-in on well performance

There is still debate on how shutting fractured wells for a period of time can affect the well performance. In this study, we combined two approaches to better understand the effects of shut-in time on well performance. First, we analyzed flowback and post-flowback production data from a horizontal well drilled in the Montney Formation, which was fractured with water containing a microemulsion (ME) additive. After the shut-in time for 7 months, the oil and solution gas rates significantly increased by 750% and 671%, respectively. However, the free gas rate decreased by 95% in 65 days, before it started to build up again to exceed the values at the start of the production. Second, we performed a series of imbibition oil-recovery, dynamic liquid-liquid contact angle, and interfacial tension measurements to investigate how the interplay of (i) capillary suction and (ii) osmotic pressure affects oil production from core plugs during the counter-current imbibition tests. Combined analyses of field and laboratory results suggest that the increase in oil production rate after the shut-in period is due to combined effects of (i) free-gas dissolution into the oil (ii) capillary imbibition of fracturing water containing ME solution into the rock matrix driven by wettability alteration and osmotic pressure, and (iii) reduction in phase trapping near fracture face due to interfacial tension reduction.

Simulation of Counter-Current Spontaneous Imbibition Based on Momentum Equations with Viscous Coupling, Brinkman Terms and Compressible Fluids

A numerical model is investigated representing counter-current spontaneous imbibition of water to displace oil or gas from a core plug. The model is based on mass and momentum conservation equations in the framework of the theory of mixtures. We extend a previous imbibition model that included fluid–rock friction and fluid–fluid drag interaction (viscous coupling) by including fluid compressibility and Brinkman viscous terms. Gas compressibility accelerated recovery due to gas expansion from high initial non-wetting pressure to ambient pressure at typical lab conditions. Gas compressibility gave a recovery profile with two characteristic linear sections against square root of time which could match tight rock literature experiments. Brinkman terms decelerated recovery and delayed onset of imbibition. Experiments where this was prominent were successfully matched. Both compressibility and Brinkman terms caused recovery deviation from classical linearity with the square root of time. Scaling yielded dimensionless numbers when Brinkman term effects were significant.Article HighlightsSpontaneous imbibition with viscous coupling, compressibility and Brinkman terms.Viscous coupling reduces spontaneous imbibition rate by fluid–fluid friction.Brinkman terms delay early recovery and explain seen delayed onset of imbibition.Gas compressibility accelerates recovery and can be significant at lab conditions.Gas compressibility gives recovery with two root of time lines as seen for shale.

MATHEMATICAL SOLUTION OF IMBIBITION PHENOMENON IN VERTICAL HETEROGENEOUS POROUS MEDIUM

The phenomenon of imbibition is of great importance in the petroleum industry. In this analysis, we have studied the Counter-current imbibition through mathematical modelling in a double non-miscible phase in the vertical heterogeneous porous medium, which ensues in the process of oil recovery. The non-linear partial differential equation of second-order is the result of the mathematical formulation. Variational iteration approach is applied to attain approximate analytical solution of the governing equation. The solution's numerical outcomes and graphical representations are identified with the help of MATLAB.

EXPERIMENTAL RESEARCH ON QUANTIFICATION OF COUNTERCURRENT IMBIBITION DISTANCE FOR TIGHT SANDSTONE

China has an unbelievable number of tight oil reserves in storage, but a large majority of the tight oil reservoirs are in low sweep efficiency and in poor depletion development. Countercurrent imbibition is an important recovery mechanism for enhancing oil recovery during water injection development of tight oil reservoirs. At present, a large number of scholars have mainly conducted research on the imbibition recovery of tight oil reservoirs as well as the factors that may have influences on that, but actually there are few research on the imbibition distance that characterizes the range of imbibition effect in tight oil reservoirs. In this paper, the CT online scanning device is employed to establish a quantification method for countercurrent imbibition distance (CID) of tight cores, determining the range of countercurrent imbibition, and it also can make a further study on the influence of fluid pressure, water saturation, core permeability and surfactant on CID. In addition, it can be utilized to determine the relationship between CID and imbibition recovery. As a result, this study also provides theoretical guidance for enhancing oil recovery of tight oil reservoirs. The research results show that the CID scale of tight core with the permeability of about 0.3 mD is only 1.25 ~ 1.625 cm, and CID of the tight core with 0.302 mD under the condition of 5 MPa is 1.375 cm. Under the experimental conditions in this article, fluid pressure and initial water saturation have little effect on the CID of tight cores, while permeability and surfactant have significant effect on the CID of tight cores. What’s more, the CID of the core with 0.784 mD is 2.63 times higher than that of the core with 0.302 mD. In conclusion, the CID is a crucial parameter for the characterization of imbibition recovery, and it determines the range of countercurrent imbibition.

Simulation of imbibition in porous media with a tree-shaped fracture following the level-set method

Imbibition is an important mechanism for enhancing oil recovery in low-permeability reservoirs, such as shale and tight sandstone, and a tree-shaped network has been successfully used to characterize fracturing fracture. Therefore, understanding the imbibition mechanism in porous media with a tree-shaped fracture (TFPM) is important for developing low-permeability reservoirs. In this study, a simulation model for imbibition in TFPM was established based on the level-set method, and the model was verified by comparing it with an analytical solution. The influences of the fracture width, bifurcation angle, tortuosity, and water flow rate on imbibition in TFPM were then discussed. Based on the results, the following points have been established: (1) During the early stage, the imbibition in TFPM included countercurrent and a combined imbibition, and only countercurrent imbibition occurred during the later stage. (2) At a constant fracture width ratio, increasing the primary fracture width could reduce the residual oil in the TFPM, thereby improving the oil recovery factor. (3) At a fracture bifurcation angle ranging from 0° to 45°, the oil recovery factor increased as the bifurcation angle increased. (4) At a fracture tortuosity of 1.0 to 1.24, changes in tortuosity had little effect on the oil recovery factor during the early stage of imbibition, while it significantly affected the distribution of the residual oil. (5) At a water flow rate of 5 mm/s, the simulated oil recovery factor in the TFPM was highest. This investigation can provide a reference for the development of low-permeability reservoirs.

Application of spectral element method in simulation of fractured porous media domains modeled by fracture-only technique

In this paper, a new application of the spectral element method (SEM) is proposed, which is combined by the features of the fracture-only reservoir simulation technique. This method has been examined with the numerical simulation of waterflooding recovery mechanism. The effect of co-current imbibition is exhibited by a fracture flow analysis with an impermeable porous media. This analysis has been then repeated to study counter-current imbibition effect considering a homogeneous porous media. The capability of the fracture-only technique has been demonstrated more by analyzing fracture flow with heterogeneous porous media. All the presented SEM's results have shown a closeness agreement with the valid results obtained by analytical and streamline-based formulations. The SEM presented 36% higher accuracy when compared to the solution obtained by the finite element method (FEM).

Numerical investigation of single- and two-phase flow in porous media with a bifurcated fracture

Considering that fractures arising from fracturing systems have bifurcation characteristics, mathematical models for single- and two-phase flow in porous media with a bifurcated fracture (BFPM) were established. The phase-field method was adopted to trace the oil–water interface, and the influence of fracture morphology, boundary conditions, gravity, and wettability on imbibition in BFPM was discussed. The results are as follows: (1) during single-phase flow in BFPM, the velocity in the bifurcated fracture was several orders of magnitude greater than that in the matrix. (2) Imbibition in BFPM includes countercurrent imbibition and the combination of co-current and countercurrent imbibition. The bifurcated fracture produces an increase in matrix pores that participate in imbibition, thus increasing the oil recovery factor by approximately 30% during simulation. (3) Closing an outlet of the bifurcated fracture has minimal effect on imbibition during the early stage (approximately the first 6 s) and an inhibitory effect during the middle and late stages (after 6 s), which leads to a decrease in oil recovery factor. (4) When the surface tension value is low (0.1 and 1 mN/m), disregarding the effect of gravity during imbibition results in an overestimated oil recovery factor value, while a large surface tension value (20 and 25 mN/m) produces the opposite effect. (5) Similar to the laws followed in porous media with pure pores and a single fracture, improving BFPM wettability increases the oil recovery factor value. This study can provide guidance for the production of low-permeability reservoirs.

A Semi-Analytical Solution for Countercurrent Spontaneous Imbibition in Water-Wet Fractured Reservoirs

Countercurrent spontaneous imbibition (SI) is an important flow mechanism for oil recovery in fractured reservoirs during waterflooding. SI plays a key role in the mobilization of oil in the matrix because it facilitates water infiltration by capillarity even when the matrix permeability is low, which limits fluid transport by advection. However, the modeling of SI in fractured media under dynamic conditions has been insufficiently studied. Most imbibition models assume conventional exponential functions and empirical constants based on experimental results of oil recovery under static conditions. Thus, the modeling of water distribution in the fracture and the matrix has been ignored which may lead to incorrect estimates of the efficiency of countercurrent SI to recover oil. Using the classic fractional flow equation, we present a semi-analytical solution to model countercurrent SI in a water-wet fractured medium by including a transfer function to account for continuous fluid exchange between the fracture and the matrix. The model is numerically solved using finite differences by including an effective imbibition time as a function of water advance in the fracture, which overcomes the difficulty of solving iterative numerical summations as shown in other imbibition models. The novelty of the presented solution is that it enables the modeling of water infiltration in the matrix under dynamic conditions. We verified the semi-analytical model against 2D numerical simulations and validated against experimental data to demonstrate that the model accurately predicts oil recovery and water infiltration in the matrix.

Method for Calculating Productivity of Water Imbibition Based on Volume Fracturing Stimulations of Low Permeability Reservoirs

Spontaneous water imbibition is an important mechanism in water-wet fractured reservoirs. For volume-fractured reservoirs, to evaluate the oil productivity and oil recovery through water counter-current imbibition, we propose an analytical method for optimizing the reservoir volume fracturing scheme. Based on the two-phase fluid flow differential equation for capillary force, a three-dimensional water imbibition productivity equation is derived analytically. The equation for the water imbibition productivity considering the fracture network is obtained. A numerical model is constructed to verify the validity of the average capillary diffusivity coefficient and the results of the analytical model. By applying this method to a low permeability reservoir, after volume fracturing and waterflooding huff and puff, the relationship between the tenth year’s oil recovery and oil production rate and the length, width, and density of the fracture network is predicted, which gives an optimization of the field fracturing construction scale. The results show that the length and width of the fracture network should be no less than 50% of the well spacing and row spacing to obtain a reasonable production. Considering the fracturing technique and economic feasibility, the higher the density of the fracture network, the better the production obtained. Through hydraulic volume fracturing and waterflooding huff and puff, water imbibition is brought into full play and the 10 year oil recovery is increased by 6%–8% in this area.

Insights into the solution of counter-current spontaneous imbibition for tight unconventional reservoirs

Oil production from fractured reservoirs is challenging due to tendency of water to flow through the fractures causing reservoir oil to be bypassed and captured in the matrix. The spontaneous imbibition is a key mechanism to recover these captured reservoir hydrocarbons. This study brings pressure loss calculations to be an addition to the semi-analytical solution of counter-current imbibition. The magnitude of pressure loss was shown to be quite considerable for tight unconventional reservoirs contrary to its insignificancy for conventional reservoirs. The validity of the semi-analytical solution for unconventional reservoirs is discussed in this respect. A material balance check methodology is introduced and implemented to check the accuracy of the calculation by means of semi-analytical method. Comparison of the results of semi-analytical and numerical solutions reveals that the efficiency of the numerical solution is poor in respect to requirement of model partition and computation time. In this connection, error analysis is conducted for both numerical and semi-analytical solution to clarify the reasons for this observation. Invariance of the minimum and maximum phase pressures, the circumstance of pressure losses to be higher in the oil phase, extended variations in phase pressures due to greater capillary pressures are the considerable outcomes of the solution of the counter-current spontaneous imbibition.

Effect of viscosity on oil recovery by spontaneous imbibition from partially water-covered matrix blocks

Spontaneous imbibition is an important mechanism of oil recovery from fractured reservoirs and unconventional reservoirs. Oil is produced by combining co- and counter-current imbibition when the matrix blocks was partially covered by water. In this paper, we focused on the effect of viscosity ratios on oil production by spontaneous imbibition and established the numerical model for one-dimensional linear imbibition with TEO-OW boundary conditions, which was validated by the experimental data. The effect of viscosity ratio on co- and counter-current imbibition is investigated and scaling result of the imbibition recovery curve for wide range of viscosity ratio using the conventional scaling equation was tested, which indicates that the close correlation was achieved only when oil-water viscosity ratios are higher. Then, a modified scaling equation was developed based on the piston-like assumption for one-dimensional co-current imbibition and close correlation of imbibition recovery curves was achieved when viscosity ratios are lower. Finally, correlation of imbibition recovery curves was improved for wide range of viscosity ratios by combining conventional and modified scaling equation. Results show that since the shape of imbibition recovery curves is not similar for different viscosity ratios, it is difficult to obtain the perfect correlation using the constant viscosity term.

Numerical modeling of water oil two-phase flow during counter-current spontaneous imbibition in porous media at pore-scale

In this work, phase-field method is used to develop a numerical model to simulate two-phase flow through a heterogeneous fractured porous medium. Various sensitivity analyses were performed to assess the impact of wettability, fracture aperture, interfacial tension, and water injection velocity on the displacement process. It was observed that the water mass imbibed into the matrix block varies linearly with time before the water front meets the outlet, known as “filling fracture” regime, which is captured for the first time in a numerical study. It is revealed that increasing the fracture aperture reduces water breakthrough time and oil recovery.