Modelling Imbibition Processes in Heterogeneous Porous Media

Transport in Porous Media - Imbibition is a commonly encountered multiphase problem in various fields, and exact prediction of imbibition processes is a key issue for better understanding capillary...     

Foam Drainage in Porous Media

In this paper we present a simple analysis of liquid drainage in foams confined in porous media. First we derive the equation for the evolution of the liquid saturation using general mass and momentum conservation arguments and phenomenological relations between the transport parameters and liquid saturation. We find an unusual foam drainage equation in which the determinant terms express the competition between the external force field, represented here by the gravity field, and capillary pressure gradient. We present analytical solutions of the drainage equation in three cases: (a) gravity forces are dominant over capillary forces, (b) capillary forces are dominant over gravity forces, and (c) capillary and gravity forces are comparable in order of magnitude.     

Effects of Snap-Off in Imbibition in Porous Media with Different Spatial Correlations

Quasi-static imbibition was simulated using random and correlated stochastic network models. Using the snap-off pore-scale displacement observed by Lernormand et al. (1983) the effects of many parameters on relative permeabilities and residual saturation reported in the literature were reproduced and explained. Increased relative permeabilities and decreased residual non-wetting phase saturation were the results of an increased contact angle (Li and Wardlaw, 1986b; Gauglitz and Radke, 1990; Blunt et al., 1992; Mogensen and Stenby, 1998) a decreased pore–throat aspect ratio, the presence of long-range pore-pore size correlations (Iaonnidis and Chatzis, 1993; Blunt, 1997a), or local pore–throat correlations (Jerauld and Salter, 1990; Iaonnidis and Chatzis, 1993). By modifying the level of snap-off, or its spatial distribution, these parameters varied the efficiency of the displacement patterns and ultimately affect relative permeabilities and residual saturations. Mani and Mohanty (1999) performed simulations on networks with infinite-ranged fractional Brownian motion (fBm) correlations and reported trends of relative permeabilities and residual saturations that were opposite to others’ results (Ioannidis and Chatzis, 1993; Blunt, 1997a). Applying a cut-off length to the fBm correlations reversed Mani and Mohanty’s trends to conform with the common observations.     

A New Mathematical Model for Force Gravity Drainage in Fractured Porous Media

In force gas/oil gravity drainage process in fractured porous media, gas is flowing in both matrix and fractures leading to produce a finite gas pressure gradient. Consequently, viscous force plays an important role for displacing matrix oil toward fractures in addition to gravity force that is required to be modeled appropriately. A new analytical model for estimation of steady state oil saturation distribution with assumption of fixed gas pressure gradient throughout the matrix is presented. Moreover, based on some results of this analytical model a different numerical formulation is developed to predict the performance of oil production process. Comparison of the results obtained from this numerical model with the results of a conventional simulator demonstrates that the newly developed model can be applied with satisfactory accuracy. Numerical simulations show that the viscous displacement in fractured porous media can reduce the capillary threshold height, and thus it suggests the force gravity drainage as a favorable production mechanism when the matrix length is close to the threshold height.     

Effect of Wetting on Gravity Drainage in Porous Media

A series of benchmark experiments on the effect of the wetting state on the flow properties in porous media were performed, allowing us to relate the wetting properties at the pore scale to the macroscale hydrodynamics. Drainage of n-alkanes (oils) displaced by air in a model porous medium consisting of water-wet sand was studied using gamma-ray densitometry and weight measurements. The enormous advantage of our system is that we know and control the wetting properties perfectly: we can tune the wetting properties by changing the salinity of the water. This allows us to perform porous medium flow experiments for the different wetting states without changing the transport properties (viscosity and density) of the oil. Drainage is found to be more efficient, and consequently oil recovery more important for partial wetting.     

Spontaneous Inertial Imbibition in Porous Media Using a Fractal Representation of Pore Wall Rugosity

Considering the separable phenomena of imbibition in complex fine porous media as a function of timescale, it is noted that there are two discrete imbibition rate regimes when expressed in the Lucas–Washburn (L–W) equation. Commonly, to account for this deviation from the single equivalent hydraulic capillary, experimentalists propose an effective contact angle change. In this work, we consider rather the general term of the Wilhelmy wetting force regarding the wetting line length, and apply a proposed increase in the liquid–solid contact line and wetting force provided by the introduction of surface meso/nanoscale structure to the pore wall roughness. An experimental surface pore wall feature size regarding the rugosity area is determined by means of capillary condensation during nitrogen gas sorption in a ground calcium carbonate tablet compact. On this nano size scale, a fractal structure of pore wall is proposed to characterize for the internal rugosity of the porous medium. Comparative models based on the Lucas–Washburn and Bosanquet inertial absorption equations, respectively, for the short timescale imbibition are constructed by applying the extended wetting line length and wetting force to the equivalent hydraulic capillary observed at the long timescale imbibition. The results comparing the models adopting the fractal structure with experimental imbibition rate suggest that the L–W equation at the short timescale cannot match experiment, but that the inertial plug flow in the Bosanquet equation matches the experimental results very well. If the fractal structure can be supported in nature, then this stresses the role of the inertial term in the initial stage of imbibition. Relaxation to a smooth-walled capillary then takes place over the longer timescale as the surface rugosity wetting is overwhelmed by the pore condensation and film flow of the liquid ahead of the bulk wetting front, and thus to a smooth walled capillary undergoing permeation viscosity-controlled flow.     

A Discussion of the Effect of Tortuosity on the Capillary Imbibition in Porous Media

In the past decades, there was considerable controversy over the Lucas–Washburn (LW) equation widely applied in capillary imbibition kinetics. Many experimental results showed that the time exponent of the LW equation is less than 0.5. Based on the tortuous capillary model and fractal geometry, the effect of tortuosity on the capillary imbibition in wetting porous media is discussed in this article. The average height growth of wetting liquid in porous media driven by capillary force following the [`(L)] _s(t) ~ t^1/2D_T{\overline L _{\rm {s}}(t)\sim t^{1/{2D_{\rm {T}}}}} law is obtained (here D _T is the fractal dimension for tortuosity, which represents the heterogeneity of flow in porous media). The LW law turns out to be the special case when the straight capillary tube (D _T = 1) is assumed. The predictions by the present model for the time exponent for capillary imbibition in porous media are compared with available experimental data, and the present model can reproduce approximately the global trend of variation of the time exponent with porosity changing.     

Effect of Wetting on the Dynamics of Drainage in Porous Media

We examine the consequences of the wettability properties on the dynamics of gravity drainage in porous media. The relation between the wetting properties at the pore scale and the macroscale hydrodynamics is studied. Model porous media consisting of hydrophilic and hydrophobic glass beads or sand with well defined wetting properties, are prepared for this study. Gravity drainage experiments with air displacing water (two-phase flow), are performed for different Bond numbers, and using different techniques such as gamma-ray densitometry, magnetic resonance imaging (MRI) and weight measurements. The dynamics of drainage is found to be different for hydrophilic and hydrophobic porous media in the transition zone (funicular regime). Moreover, for hydrophilic (water-wet) porous media, MRI experiments reveal the importance of drainage through the continuous water film, which leads to an increase of the residual quantity of water in the transition zone with time.     

Accurate and Efficient Implementation of Pore-Morphology-based Drainage Modeling in Two-dimensional Porous Media

We have developed an efficient and accurate numerical implementation for pore-morphological modeling of drainage in two-dimensional, totally wetting porous media. The new numerical method uses level sets to describe the fluid distribution and polygons that can be defined with subgrid scale accuracy for the pore boundaries, while a previously developed approach represents the phases by pixels arranged on a square lattice. We analyze and compare the previous and new method. For both approaches, the simulated fluid saturations are first-order accurate. For the level-set approach, the simulated interfacial lengths converge to the real values, while the pixel approach yields biased results. The level-set method is orders of magnitudes faster than the pixel method.     

Boiling in Porous Media: Model and Simulations

We present a modelization of the heat and mass transfers within a porous medium, which takes into account phase transitions. Classical equations are derived for the mass conservation equation, whereas the equation of energy relies on an entropy balance adapted to the case of a rigid porous medium. The approximation of the solution is obtained using a finite volume scheme coupled with the management of phase transitions. This model is shown to apply in the case of an experiment of heat generation in a porous medium. The vapor phase appearance is well reproduced by the simulations, and the size of the two-phase region is correctly predicted. A result of this study is the evidence of the discrepancy between the air – water capillary and relative permeability curves and water – water vapor ones.     

Efficiency of Capillary Imbibition Dominated Displacement of Nonwetting Phase by Wetting Phase in Fractured Porous Media

The critical and optimum injection rates as well as the critical fracture capillary number for an efficient displacement process are determined based on the experimental and numerical modeling of the displacement of nonwetting phase (oil) by wetting phase (water) in fractured porous media. The efficiency of the process is defined in terms of the nonwetting phase displaced from the system per amount of wetting phase injected and per time. Also, the effects of injection rate on capillary imbibition transfer dominated two-phase flow in fractured porous media are clarified by visualizing the experiments. The results reveal that as the injection rate is increased, fracture pattern begins to become an effective parameter on the matrix saturation distribution. As the rate is lowered, however, the system begins to behave like a homogeneous system showing a frontal displacement regardless the fracture configuration.     

An Analytic Solution for the Frontal Flow Period in 1D Counter-Current Spontaneous Imbibition into Fractured Porous Media Including Gravity and Wettability Effects

Including gravity and wettability effects, a full analytical solution for the frontal flow period for 1D counter-current spontaneous imbibition of a wetting phase into a porous medium saturated initially with non-wetting phase at initial wetting phase saturation is presented. The analytical solution applicable for liquid–liquid and liquid–gas systems is essentially valid for the cases when the gravity forces are relatively large and before the wetting phase front hits the no-flow boundary in the capillary-dominated regime. The new analytical solution free of any arbitrary parameters can also be utilized for predicting non-wetting phase recovery by spontaneous imbibition. In addition, a new dimensionless time equation for predicting dimensionless distances travelled by the wetting phase front versus dimensionless time is presented. Dimensionless distance travelled by the waterfront versus time was calculated varying the non-wetting phase viscosity between 1 and 100 mPas. The new dimensionless time expression was able to perfectly scale all these calculated dimensionless distance versus time responses into one single curve confirming the ability for the new scaling equation to properly account for variations in non-wetting phase viscosities. The dimensionless stabilization time, defined as the time at which the capillary forces are balanced by the gravity forces, was calculated to be approximately 0.6. The full analytical solution was finally used to derive a new transfer function with application to dual-porosity simulation.     

Pore-Level Observation of Free Gravity Drainage of Oil in Fractured Porous Media

This work presents results from two sets of experiments conducted to study, in pore level, the role of fracture aperture and tilt angle on the stability of liquid bridges and the shape of a front during free gravity drainage process. Glass micromodels of two different aperture sizes were used to monitor the mechanism of gravity drainage of air?Ccrude oil system, rotating around a bottom corner to create different tilting angles. Oil content within the matrix blocks was determined as a function of time using a series of images obtained during the experiments, from which net drainage rate from the upper and lower matrix blocks is calculated. Liquid bridges are more frequent but less stable at early time of drainage. The liquid bridges, which have widths as thin as 50 ??m, can resist instability to maintain continuity. Liquid bridges formed in stacks with higher tilt angles are more stable, enhancing oil drainage from the upper matrix block and causing higher recoveries. Quantitative analysis of the results shows that a wider fracture aperture increases the oil production rate, but reduces the ultimate recovery. Furthermore, stacks with higher tilt angles present larger ultimate recoveries and smaller production rates. The front geometry in the lower block deviates from linearity due to formation of liquid bridges in the middle fracture. The results of this work can be helpful to better understand the interaction between fractures and matrix blocks.     

Deformable Porous Media Saturated by Three Immiscible Fluids: Constitutive Modelling and Simulations of Injection and Imbibition Tests

A number of environmental and petroleum engineering applications involve the coexistence of three non-miscible fluids. In this work, basic constitutive relations and computational schemes are developed in order to simulate fluid injection and imbibition processes in a deformable rock through the finite element method. For this purpose, the following ingredients are worked out: (i) simple, but general formulas for the effective saturations; (ii) constitutive expressions for the relative permeabilities of water, oil and gas in terms of effective saturations; and (iii) constitutive capillary pressure relationships. These ingredients are introduced in a domestic finite element code where the primary variables are the solid displacement vector and the three fluid pressures. Given the abundance of experimental data in the petroleum engineering field, the whole framework is firstly tested by simulating gas injection into a rock core sample initially saturated by water and oil. Sensitivity analyses are performed upon varying key constitutive, loading and numerical parameters, to assess the physical and computational outputs of the proposed framework. Particular attention is given to the influence on the model predictions of several expressions defining relative permeabilities. Simulations of water-alternated-gas injection and of counter-current water imbibition tests are also performed, to establish the reliability of the proposed constitutive and computational framework.     

Renormalization Group Model of Large-Scale Turbulence in Porous Media

Renormalization group methods are used to develop a macroscopic (large-scale) turbulence model for incompressible flow in porous media. The model accounts for the large-distance and large-time behavior of velocity correlations generated by the momentum equation for a randomly stirred, incompressible flow. Utilizing the renormalization procedure, the transport equations for the large-scale modes and expressions for effective transport coefficients are obtained. Expressions for renormalized turbulent viscosity, which accounts for the ultraviolet subrange of the turbulent kinetic energy spectrum, are also obtained.     

Imbibition into Highly Porous Layers of Aggregated Particles

The co-occurrence of gravity-driven drainage and forced convective drying in a macroporous medium is investigated in this study. The drainage and drying processes of fully saturated porous asphalt (PA) specimens placed in a custom-made mini wind tunnel are documented with neutron radiography (NR). Six PA specimens of dimensions \(180\times 10\times 30\,\hbox {mm}^{3}\) with a maximum aggregate size of 8 or 11 mm are used in the experiments. In the first few minutes of each experiment, there is significant moisture loss in all the specimens due to gravity-driven drainage. Most of the residual water retention is observed at the bottom region of the specimens due to the strong impact of gravity-driven drainage in the upper regions. The specimens are subjected to many hours of airflow at their top surface; however, forced convection from turbulent airflow near the upper part of the specimens is found to have a minor influence on moisture loss when there are no water clusters in the upper regions of the specimens. This points to the strong resistance to evaporation in PA as a result of the large vapor diffusion lengths. By combining neutron radiography and microcomputer tomography (X-ray \(\upmu \)-CT) images, saturated and unsaturated flows in the pores are distinguished. Fluid flow path during air entry and water redistribution is further analyzed by reconstructing the real three-dimensional pore geometry of the specimens from X-ray \(\upmu \)-CT scans.     

Coupling of Foam Drainage and Viscous Fingering in Porous Media Revealed by X-ray Computed Tomography

We investigate the development foam in granular porous media and the subsequent flow of the surfactant solution, where the fluid fraction variations are visualized and quantified using X-ray computed tomography. It is found that foam flows in a front like manner leading to a residual liquid fraction of 0.18±0.01, far from the inlet surface of the porous sample. A desaturation backward wave is also observed during foam development. We provided direct evidence that the flow of surfactant solution in porous media containing foam gives rise to superposition of a drainage wave and a characteristic viscous fingering pattern. In the wave the liquid fraction ranges from the above residual value to nearly 0.25±0.01. The liquid fraction associated with the viscous fingering decays as a function of distance but the inlet value increases up to 0.06±0.01. Certain ideas about the physics of foam flow in porous media are revised in the light of our findings.     

A Spatially Non-Local Model for Flow in Porous Media

A general mathematical model of steady-state transport driven by spatially non-local driving potential differences is proposed. The porous medium is considered to be a network of short-, medium-, and long-range interstitial channels with impermeable walls and at a continuum of length scales, and the flow rate in each channel is assumed to be linear with respect to the pressure difference between its ends. The flow rate in the model is thus a functional of the non-local driving pressure differences. As special cases, the model reduces to familiar forms of transport equations that are commonly used. An important situation arises when the phenomenon is almost, but not quite, locally dependent. The one-dimensional form of the model discussed here can be extended to multiple dimensions, temporal non-locality, and to heat, mass, and momentum transfer.