Counter-current Spontaneous Imbibition Research Articles

Improved Amott Method to Determine Oil Recovery Dynamics from Water-Wet Limestone Using GEV Statistics

Counter-current spontaneous imbibition of water is a critical oil recovery mechanism. In the laboratory, the Amott test is a commonly used method to assess the efficacy of brine imbibition into oil-saturated core plugs. The classic Amott-cell experiment estimates ultimate oil recovery, but not the recovery dynamics that hold fundamental information about the imbibition mechanisms. Retention of oil droplets at the outer core surface and initial production delay are the two key artifacts of the classic Amott experiment. This retention, referred to here as the “external-surface oil holdup effect” or simply “oil holdup effect”, often results in stepwise recovery curves that obscure the true dynamics of spontaneous imbibition. To address these holdup drawbacks of the classic Amott method, we modified the Amott cell and experimental procedure. For the first time, using water-wet Indiana limestone cores saturated with brine and mineral oil, we showed that our improvements of the Amott method enabled accurate and reproducible measurements of oil recovery dynamics. Also for the first time, we used the generalized extreme value (GEV) statistics to describe oil production histories from water-wet heterogeneous limestone cores with finite initial water saturations. We demonstrated that our four-parameter GEV model accurately described the recovery dynamics, and that optimal GEV parameter values systematically reflected the key characteristics of the oil–rock system, such as oil viscosity and rock permeability. These findings gave us a more fundamental understanding of spontaneous, counter-current imbibition mechanisms and insights into what constitutes a predictive model of counter-current water imbibition into oil-saturated rocks with finite initial water saturation.

The Role of Core Sample Geometry on Countercurrent Spontaneous Imbibition: Mathematical Foundation, Examples, and Experiments Accounting for Realistic Geometries

Summary Countercurrent spontaneous imbibition (SI) experiments are among the most common multiphase experiments performed on porous rock samples. Although the samples rarely are designed to give linear flow, they are often modeled and interpreted using mathematical descriptions assuming linear flow. In this work, the goal is to derive general understanding of how imbibition into different sample geometries behaves compared with linear (1D) imbibition. Using the mathematical theory of N-volume spheres (N being the dimension), we consider core samples as N-spherical and quantify their geometry by a dimension N and length scale L. For the special cases, N=1,2,3, we obtain linear, radial, and spherical flow; however, we treat the dimension as an arbitrary real number for cases not adhering to either of these regimes. Particularly, for rectangular or cylindrical core plugs, a continuous range of dimensions is produced. Theoretical calculations of N and L of arbitrary sample shapes are derived based on relations with area per volume and derivative of area with respect to volume. They correctly produce limit cases and physically meaningful values for symmetrical, cylindrical, and rectangular geometries. The differential equation for countercurrent imbibition in N-dimensions is derived and solved with numerical examples. Also, a simplified analytical solution assuming piston-like displacement is derived to get illustrative relations between system parameters (including dimension) and recovery and front position. Predicted recovery profiles of the N-dimensional (N-D) solution overlap consistently with numerical simulations (by an alternative simulator) into cylindrical geometries with a wide range of height/diameter ratios and viscosity ratios. At early time, the saturation profiles are self-similar (look the same plotted against position divided by square root of time) and identical regardless of dimension. As long as the profiles are self-similar near the open boundary, recovery is proportional to the square root of time. For N=1, this lasts long after the front has reached the closed boundary, while for N >1, it can happen long before the boundary has been reached. The same time scale was applicable for all geometries, stating especially that doubling the length scale increased the time of the entire recovery profile by a factor 4. As long as recovery is proportional to square root of time, at a given time, it is also proportional to dimension over length, NL, and the time needed to reach same level of recovery is proportional to LN2. Permeability anisotropy (lower vertical than horizontal permeability) could effectively be modeled using an effective increased height, which further was captured by the dimension and length scale. Literature and in-house experimental data were matched by the model and used to validate model predictions such as the variation in time, shape of recovery curves with changes in dimension, and the importance of accounting for the dimension (geometry) during data interpretation. The model is valid for all wetting states but assumes negligible gravity and compressibility effects.

Counter-current spontaneous imbibition dynamics: Combined effects of wettability, fracture flow, and pore structure

Spontaneous imbibition plays a crucial role in various engineering and industrial applications, with its efficiency significantly influenced by a range of factors. To unravel the intricate mechanisms behind these factors, our study employs pore-scale numerical simulations. Utilizing a color gradient model within the framework of the lattice Boltzmann method, we delve into how pore structure, wettability, and flow velocity within fractures collectively impact spontaneous imbibition. Our findings reveal that the dynamics of drainage and imbibition interfaces during countercurrent spontaneous imbibition are key determinants of imbibition efficiency. Specifically, the synergy between wettability and pore structure markedly affects the penetration depth and distribution characteristics of the imbibition interface, which, in turn, influences the imbibition's speed and duration. Moreover, the interaction between the flow velocity inside fractures and the configuration of adjacent pore structures significantly shapes the evolution of the drainage interface. This interplay is crucial as it can either enhance or hinder countercurrent spontaneous imbibition. These insights deepen our understanding of the pore-scale processes governing countercurrent spontaneous imbibition, laying a solid theoretical foundation for optimizing its application in engineering and industrial settings.

Application of Physics-Informed Neural Networks for Estimation of Saturation Functions from Countercurrent Spontaneous Imbibition Tests

Summary In this work, physics-informed neural networks (PINNs) are used for history matching data from core-scale countercurrent spontaneous imbibition (COUCSI) tests. To our knowledge, this is the first work exploring the variation in saturation function solutions from COUCSI tests. 1D flow was considered, in which two phases flow in opposite directions driven by capillary forces with one boundary open to flow. The partial differential equation (PDE) depends only on a saturation-dependent capillary diffusion coefficient (CDC). Static properties such as porosity, permeability, interfacial tension, and fluid viscosities are considered known. In contrast, the CDC or its components [relative permeability (RP) and capillary pressure (PC)], are considered unknown. We investigate the range of functions (CDCs or RP/PC combinations) that explain different (synthetic or real) experimental COUCSI data: recovery from varying extents of early-time and late-time periods, pressure transducers, and in-situ saturation profiles. History matching was performed by training a PINN to minimize a loss function based on observational data and terms related to the PDE, boundary, and initial conditions. The PINN model was generated with feedforward neural networks, Fourier/inverse-Fourier transformation, and an adaptive tanh activation function, and trained using full batching. The trainable parameters of both the neural networks and saturation functions (parameters in RP and PC correlations) were initialized randomly. The PINN method successfully matched the observed data and returned a range of possible saturation function solutions. When a full observed recovery curve was provided (recovery data reaching close to its final value), unique and correct CDC functions and correct spatial saturation profiles were obtained. However, different RP/PC combinations composing the CDC were calculated. For limited amounts of recovery data, different CDCs matched the observations equally well but predicted different recovery behavior beyond the collected data period. With limited recovery data, when all points were still following a square root of time trend, a CDC with a low magnitude and peak shifted to high saturations gave the same match as a CDC with a high magnitude and peak shifted to low saturations. Recovery data with sufficient points not being proportional to the square root of time strongly constrained how future recovery would behave and thus which CDCs could explain the results. Limited recovery data combined with an observed in-situ profile of saturations allowed for accurate determination of CDC and prediction of future recovery, suggesting in-situ data allowed for shortened experiments. With full recovery data, in-situ PC data calibrated the PC toward unique solutions matching the input. The RPs were determined, where their phase had much lower mobility than the others. The CDC is virtually independent of the highest fluid mobility, and RPs could not be matched at their high values. Adding artificial noise in the recovery data increased the variation of the estimated CDCs.

Effective Relative Permeabilities Based on Momentum Equations with Brinkman Terms and Viscous Coupling

Summary The relative permeability expresses the mobility reduction factor when a fluid flows through a porous medium in the presence of another fluid and appears in Darcy’s law for multiphase flow. In this work, we replace Darcy’s law with more general momentum equations accounting for fluid-rock interaction (flow resistance), fluid-fluid interaction (drag), and Brinkman terms responding to gradients in fluid interstitial velocities. By coupling the momentum equations with phase transport equations, we study two important flow processes—forced imbibition (coreflooding) and countercurrent spontaneous imbibition. In the former, a constant water injection rate is applied and capillary forces are neglected, while in the latter, capillary forces drive the process and the total flux is zero. Our aim is to understand what relative permeabilities result from these systems and flow configurations. From previous work, when using momentum equations without Brinkman terms, unique saturation-dependent relative permeabilities are obtained for the two flow modes that depend on the flow mode. Now, with Brinkman terms included, the relative permeabilities depend on local spatial derivatives of interstitial velocity and pressure. Local relative permeabilities are calculated for both phases utilizing the ratio of phase Darcy velocity and phase pressure gradient. In addition, we use the Johnson-Bossler-Naumann (JBN) method for forced imbibition (with data simulated under the assumption of negligible capillary end effects) to calculate interpreted relative permeabilities from pressure drop and average saturation. Both flow setups are parameterized with literature data, and sensitivity analysis is performed. During coreflooding, Brinkman terms give a flatter saturation profile and higher front saturation. The saturation profile shape changes with time. Local water relative permeabilities are reduced, while they are slightly raised for oil. The saturation range where relative permeabilities can be evaluated locally is raised and made narrower with increased Brinkman terms. JBN relative permeabilities deviate from the local values: The trends in curves and saturation range are the same but more pronounced as they incorporate average measurements, including the strong impact at the inlet. Brinkman effects vanish after sufficient distance traveled, resulting in the unique saturation functions as a limit. Unsteady state (USS) relative permeabilities (based on transient data from single-phase injection) differ from steady-state (SS) relative permeabilities (based on SS data from coinjection of two fluids) because the Brinkman terms are zero at SS. During spontaneous imbibition, the higher effect from the Brinkman terms caused oil relative permeabilities to decrease at low water saturations and slightly increase at high saturations, while water relative permeability was only slightly reduced. The net effect was a delay in the imbibition profile. Local relative permeabilities approached the unique saturation functions without Brinkman terms deeper in the system because phase velocities (involved in the Brinkman terms) decreased with distance. In both systems, scaling and simulations demonstrate that the relative change in relative permeabilities due to Brinkman terms increases with the Brinkman coefficient, permeability, and inverse squared distance from the inlet.

Simulation and Prediction of Countercurrent Spontaneous Imbibition at Early and Late Time Using Physics-Informed Neural Networks

Simulation and Prediction of Countercurrent Spontaneous Imbibition at Early and Late Time Using Physics-Informed Neural Networks

Early- and Late-Time Prediction of Counter-Current Spontaneous Imbibition, Scaling Analysis and Estimation of the Capillary Diffusion Coefficient

Solutions are investigated for 1D linear counter-current spontaneous imbibition (COUSI). It is shown theoretically that all COUSI scaled solutions depend only on a normalized coefficient {Lambda }_{n}left({S}_{n}right) with mean 1 and no other parameters (regardless of wettability, saturation functions, viscosities, etc.). 5500 realistic functions {Lambda }_{n} were generated using (mixed-wet and strongly water-wet) relative permeabilities, capillary pressure and mobility ratios. The variation in {Lambda }_{n} appears limited, and the generated functions span most/all relevant cases. The scaled diffusion equation was solved for each case, and recovery vs time RF was analyzed. RF could be characterized by two (case specific) parameters RFtr and lr (the correlation overlapped the 5500 curves with mean {R}^{2}=0.9989): Recovery follows exactly mathrm{RF}={T}_{n}^{0.5} before water meets the no-flow boundary (early time) but continues (late time) with marginal error until RFtr (highest recovery reached as {T}_{n}^{0.5}) in an extended early-time regime. Recovery then approaches 1, with lr quantifying the decline in imbibition rate. RFtr was 0.05 to 0.2 higher than recovery when water reached the no-flow boundary (critical time). A new scaled time formulation {T}_{n}=t/tau {T}_{mathrm{ch}} accounts for system length L and magnitude overline{D } of the unscaled diffusion coefficient via tau ={L}^{2}/overline{D }, and {T}_{mathrm{ch}} separately accounts for shape via {Lambda }_{n}. Parameters describing {Lambda }_{n} and recovery were correlated which permitted (1) predicting recovery (without solving the diffusion equation); (2) predicting diffusion coefficients explaining experimental recovery data; (3) explaining the challenging interaction between inputs such as wettability, saturation functions and viscosities with time scales, early- and late-time recovery behavior.

Insights from Boltzmann transformation in solving 1D counter-current spontaneous imbibition at early and late time

Counter-current 1D spontaneous imbibition in scaled form is investigated using Boltzmann transform before and after water meets the closed boundary (early and late time). At early time the system is self-similar and only depends on position divided by square root of time. At late time it also depends on the interaction with the no-flow boundary and hence a second variable, which is set as the square root of time. Diffusion coefficients shifted to high saturations result in early time spatial saturation profiles with shorter front distance, higher average saturation within the imbibition profile and larger imbibed amount. Strongly water-wet systems have zero oil mobility at the inlet, while mixed-wet systems have finite non-zero mobility. The imbibition rate is proportional to inlet diffusion coefficient, inlet saturation gradient (regarding position divided by square root of time) and inverse square root of time. Accordingly, the saturation gradient is infinite and finite for strongly water-wet and mixed-wet systems. At early time, the profile does not change, thus recovery is proportional to square root of time. When the front meets the no-flow boundary (critical time), the saturation profile deviates from the early time profile first at the no-flow boundary, then towards the inlet. When the inlet gradient changes, imbibition rate declines faster than inverse square root of time. The interaction at the inlet and not the closed boundary, thus determines when recovery stops being proportional to square root of time and explains why such proportionality after critical time is reported. The findings were confirmed by matching experimental data. Cited as: Andersen, P. Ø. Insights from Boltzmann transformation in solving 1D counter-current spontaneous imbibition at early and late time. Advances in Geo-Energy Research, 2023, 7(3): 164-175. https://doi.org/10.46690/ager.2023.03.03

An analytic solution for Counter-Current spontaneous imbibition in porous media by the perturbation method

Following primary production of hydrocarbon from naturally fractured reservoirs, secondary recovery by waterflooding relies on Spontaneous Imbibition (SI) as the primary mechanism for oil expulsion to the fracture network, where it can be easily transported to neighboring production wells. Thus, a more complete understanding of this phenomenon is warranted and essential for process improvement. The imbibition process, taking advantage of inherent preferential affinity for water contact over oil of high energy solids, is the rate determining step in recovery and can proceed in either co-current or counter-current fashion, depending upon entrance and exit boundary conditions. Due to the prominence of linear flow normal to extensive planar fracture faces, the phenomena can be largely described as one-dimensional displacement. In this study, the nonlinear one-dimensional diffusivity equation, derived by combining Darcy’s law for both wetting and non-wetting phases with the mass conservation equation, is solved for counter-current spontaneous imbibition using a direct regular perturbation scheme. Saturation and phase pressure profiles in time and space were found to agree with literature solutions. In its analytic formulation, the method constitutes a simpler, more accurate, and more direct way to compute spontaneous imbibition performance than previously available. We claim generality with regard to functional forms for relative permeability and capillary pressure and applicability over the entire range of wettability, provided these effects are reflected in the parameterization of capillary pressure and saturation normalization. In particular, the present method avoids the difficulty in solving nonlinear integral equations. Instead, we reduce computations to a set of simple numerical integrations of the diffusion coefficient. Such a streamlined solution method can more easily provide a compute engine for inverse modeling or optimization schemes to recovery rock and fluid properties from displacement experiments. The solution method is benchmarked against oil–water and air–water solutions by alternative techniques and experimental data. The case of spontaneous imbibition into dry diatomite represents extreme contrast in phase mobility and a rigorous test of the perturbation method as an easier and more direct way to model 1D SI. Following method construction and benchmarking of the new streamlined countercurrent spontaneous imbibition solution method, a subsequent parametric study contributes to a greater understanding of controlling factors in fracture-matrix exchange in porous media.

Modeling of Chemical Tracers for Two-Phase Flow in Capillary-Dominated Porous Media

Transport of chemical tracers in porous media is an important physical mechanism for applications in oil reservoir characterization, aquifer remediation, and CO2 storage. Although numerous analytical solutions exist to model the flow of solutes under single-phase flow, the modeling of solute transport in two-phase media driven by imbibition has been insufficiently examined. In tight porous media and the matrix of fractured reservoirs, spontaneous imbibition (SI) represents a key driving mechanism for fluid infiltration because the low permeability in these systems results in negligible transport by advection. Here, we present a new semi-analytical solution to model the transport of chemical tracers under one-dimensional countercurrent SI in oil–water porous media.The model presented is derived from the analysis of fluid imbibition driven by capillarity and numerically solved as a function of water distribution. We model ideal and partitioning tracers to examine the contacted region and estimate the average oil saturation in capillary-dominated media under countercurrent SI. The concentration profiles obtained with the derived model show an excellent agreement against numerical simulation results, verifying that the semi-analytical solution accurately models the mechanisms of partitioning, hydrodynamic dispersion, and adsorption. The concentration profiles exhibit a significant delay in displacement behind the imbibition front when hydrodynamic dispersion is ignored and for high partitioning coefficients, demonstrating the importance of determining these properties under two-phase flow conditions. We consider that our solution derived represents an accurate alternative to time-consuming simulations that can be extended for the analysis of tracers in fractured reservoirs to estimate oil saturation in the matrix medium.

Modeling of counter-current spontaneous imbibition in independent capillaries with unequal diameters

Spontaneous imbibition is a crucial process for oil recovery from fractured and unconventional reservoirs. Herein, with the assumption of capillaries being independent, a new mathematical model for spontaneous imbibition is proposed and solved using a numerical method. The simulated results show that the wetting phase preferentially enters smaller capillaries where the advancement velocity is higher than that in larger ones, while the non-wetting phase can be displaced out in the larger capillaries. In addition, the effect of fluid viscosity ratio on counter-current imbibition is analyzed. The results show that imbibition velocity becomes higher with the increase in the viscosity ratio. When the viscosity of the non-wetting phase is larger than that of the wetting phase, the end pressure gradually increases as the imbibition front advances. In contrast, when the viscosity of the non-wetting phase is less than that of the wetting phase, the end pressure decreases with the infiltration. With a higher viscosity ratio of non-wetting and wetting phase, the heterogeneity of the interface advancement among different capillaries increases. Cited as: Chen, K., Xu, H., Zhang, Z., Meng, Q., Zhang, T. Modeling of counter-current spontaneous imbibition in independent capillaries with unequal diameters. Capillarity, 2022, 5(6): 115-122. https://doi.org/10.46690/capi.2022.06.02

Counter-current imbibition and non-linear diffusion in fractured porous media: Analysis of early- and late-time regimes and application to inter-porosity flux

The displacement of non-aqueous phase liquid (NAPL) in permeable porous rock by water saturating the surrounding fractures is studied. In many situations of practical interest, capillarity is the dominant driving force. Using the global pressure concept, it is shown that the water saturation is driven by a generic non-linear diffusion equation governing the matrix block counter-current spontaneous imbibition. In addition, in most cases, the saturation-dependent diffusion coefficient vanishes at the saturation end points, that renders the driving equation highly singular.In this paper, two exact asymptotic solutions valid for short and long times are presented, under the assumption that the conductivity vanishes as a power-law of both phase saturations at the extreme values of the fluid saturation.Focusing on the late-time domain, the asymptotic solution is derived using an Ansatz that is written under the form of a power-law time decay of the NAPL saturation. In the spatial domain, this solution is an eigenvector of the non-linear diffusion operator driving the saturation, for a problem with Dirichlet boundary conditions. If the diffusion coefficient varies as a power law of the NAPL saturation, the spatial variations of the solution are given analytically for a one-dimensional porous medium corresponding to parallel fracture planes. The analytical solution is in very good agreement with results of numerical simulations involving various realistic sets of input transport parameters.Generalization to the case of two- or three-dimensional matrix blocks of arbitrary shape is proposed using a similar Ansatz, solution of a non-linear eigenvalue problem. A fast converging algorithm based on a fixed-point sequence starting from a suitable first guess was developed. Comparisons with full-time simulations for several typical block geometries show an excellent agreement.These results permit to set-up an analytical formulation generalizing linear single-phase representation of matrix-to-fracture exchange term. It accounts for the non-linearity of the local flow equations using the power-law dependence of the conductivity for low NAPL saturation. The corresponding exponent can be predicted from the input conductivity parameters. Similar findings are also presented and validated numerically for two- or three-dimensional matrix blocks. That original approach paves the way to research leading to a more faithful description of matrix-to-fracture exchanges when considering a realistic fractured medium composed of a population of matrix blocks of various size and shapes.

Pore scale numerical investigation of counter-current spontaneous imbibition in multi-scaled pore networks

The multi-scaled pore networks of shale or tight reservoirs are considerably different from the conventional sandstone reservoirs. After hydraulic fracturing treatment, the spontaneous imbibition process plays an important role in the productivity of the horizontal wells. Applying the color-gradient model of Lattice Boltzmann Method (LBM) accelerated with parallel computing, we studied the countercurrent spontaneous imbibition process in two kinds of pore structures with different interlacing distributions of large and small pores. The effect of geometry configuration of pore arrays with different pore-scale and the capillary number Ca on the mechanism of counter-current spontaneous imbibition as well as the corresponding oil recovery factor are studied. We found that the wetting phase tends to invade the small pore array under small Ca in both types of geometry configurations of different pore arrays of four pore arrays zones. The wetting phase also tends to invade the pore array near the inlet for injecting the wetting phase no matter if it is a large pore array or small pore array except for the situation when the Ca is large to a certain value. In this situation, the small pore arrays show resistance to the wetting phase, so the wetting phase doesn't invade the small pore near the inlet, but invades the large pore preferentially. Both the geometry configurations of different pore arrays and Ca have a significant effect on the oil recovery factor. This work will help to solve the doubt about the selectivity of the multi-scaled pores of the wetting phase and the role of pores with different sizes in imbibition and oil draining in countercurrent spontaneous imbibition processes.

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.

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.

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.

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.

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.

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.

Simulation study of wettability alteration enhanced oil recovery during co-current spontaneous imbibition

Naturally fractured reservoirs are highly dependent on capillary forces to recover hydrocarbons during water injection. Water can spontaneously imbibe and expel oil if positive capillary forces exist; purely counter-current if all sides of the matrix blocks are exposed to water; and predominantly co-current with some counter-current production if the blocks are exposed to water and oil simultaneously. The latter is referred to as a co-current spontaneous imbibition (SI) setup. Wettability alteration (WA) has been identified as a key mechanism to improve oil recovery from naturally fractured reservoirs, however almost all experimental and modeling studies on WA during SI have focused on counter-current SI. Our review indicates limited systematic experimental work on co-current SI using nonzero initial saturation, mixed wettability or WA processes and This modeling study will investigate enhanced oil recovery by WA during co-current SI where a brine with a general WA component imbibes and causes the system to become more water-wet. We model a 1D oil-saturated core exposed to water at one end (inlet) and oil at the other end (outlet), thus facilitating co-current SI. The core is initially preferentially (not strongly) oil-wet with low SI potential. The component is both transported by the imbibing brine and diffuses towards the imbibition front. Adsorption of the component is assumed to improve the water-wetness of the porous medium and hence the SI potential. The model is parameterized using consistent capillary pressure and relative permeabilities from previous history matching of brine-dependent porous disc experiments. The behavior of co-current SI at mixed-wet state is examined and compared to that of literature strongly water-wet behavior. Both secondary and tertiary enhanced recovery by SI with WA component is then considered in the simulations. Important parameters such as mobility ratio (as via oil viscosity), capillary back pressure, WA component concentration, adsorption and time of WA component exposure are investigated. Under mixed-wet conditions, favorable and unfavorable mobility ratios do not limit oil production as can be the case for strongly wetted media at unfavorable mobility ratio. This is due to oil preserving mobility at all obtained saturations. A third or more of the total production was counter-current, which is high compared to strongly wetted media. It was shown that half the oil could be produced counter-currently as an upper limit. High oil mobility is preserved in the twophase region near the inlet and was found to ensure a high minimum fraction of counter-current production. Twice as much of the incremental oil from WA was produced counter-currently as co-currently, explained by increased oil relative permeability in the WA affected inlet region. Sensitivity analysis revealed that an opposite shift would reduce the incremental counter-current production despite raised local capillary forces. Capillary back pressure resists oil production at the inlet without limiting water from imbibing. As a result, capillary back pressure had significant impact on co-current SI simulations with fixed and changing wettability. The trends discovered in this study, both for mixed-wet and wettability alternating systems, are hoped to inspire future experimental measurements.