Dynamic Pore-Scale Modeling of Residual Trapping Following Imbibition in a Rough-walled Fracture

Transport in Porous Media - We present a new, fully dynamic pore-network modeling platform that is employed to conduct a systematic pore-scale study of capillary trapping under various two-phase...     

New Trapping Mechanism in Carbon Sequestration

The modes of geologic storage of CO₂ are usually categorized as structural, dissolution, residual, and mineral trapping. Here we argue that the heterogeneity intrinsic to sedimentary rocks gives rise to a fifth category of storage, which we call local capillary trapping. Local capillary trapping occurs during buoyancy-driven migration of bulk phase CO₂ within a saline aquifer. When the rising CO₂ plume encounters a region (10⁻² to 10⁺¹m) where capillary entry pressure is locally larger than average, CO₂ accumulates beneath the region. This form of storage differs from structural trapping in that much of the accumulated saturation will not escape, should the integrity of the seal overlying the aquifer be compromised. Local capillary trapping differs from residual trapping in that the accumulated saturation can be much larger than the residual saturation for the rock. We examine local capillary trapping in a series of numerical simulations. The essential feature is that the drainage curves (capillary pressure versus saturation for CO₂ displacing brine) are required to be consistent with permeabilities in a heterogeneous domain. In this work, we accomplish this with the Leverett J-function, so that each grid block has its own drainage curve, scaled from a reference curve to the permeability and porosity in that block. We find that capillary heterogeneity controls the path taken by rising CO₂. The displacement front is much more ramified than in a homogeneous domain, or in a heterogeneous domain with a single drainage curve. Consequently, residual trapping is overestimated in simulations that ignore capillary heterogeneity. In the cases studied here, the reduction in residual trapping is compensated by local capillary trapping, which yields larger saturations held in a smaller volume of pore space. Moreover, the amount of CO₂ phase remaining mobile after a leak develops in the caprock is smaller. Therefore, the extent of immobilization in a heterogeneous formation exceeds that reported in previous studies of buoyancy-driven plume movement.     

Microtomography and Pore-Scale Modeling of Two-Phase Fluid Distribution

Synchrotron-based X-ray microtomography (micro CT) at the Advanced Light Source (ALS) line 8.3.2 at the Lawrence Berkeley National Laboratory produces three-dimensional micron-scale-resolution digital images of the pore space of the reservoir rock along with the spacial distribution of the fluids. Pore-scale visualization of carbon dioxide flooding experiments performed at a reservoir pressure demonstrates that the injected gas fills some pores and pore clusters, and entirely bypasses the others. Using 3D digital images of the pore space as input data, the method of maximal inscribed spheres (MIS) predicts two-phase fluid distribution in capillary equilibrium. Verification against the tomography images shows a good agreement between the computed fluid distribution in the pores and the experimental data. The model-predicted capillary pressure curves and tomography-based porosimetry distributions compared favorably with the mercury injection data. Thus, micro CT in combination with modeling based on the MIS is a viable approach to study the pore-scale mechanisms of CO₂ injection into an aquifer, as well as more general multi-phase flows.     

A Model for Spontaneous Imbibition as a Mechanism for Oil Recovery in Fractured Reservoirs

The flow of oil and water in naturally fractured reservoirs (NFR) can be highly complex and a simplified model is presented to illustrate some main features of this flow system. NFRs typically consist of low-permeable matrix rock containing a high-permeable fracture network. The effect of this network is that the advective flow bypasses the main portions of the reservoir where the oil is contained. Instead capillary forces and gravity forces are important for recovering the oil from these sections. We consider a linear fracture which is symmetrically surrounded by porous matrix. Advective flow occurs only along the fracture, while capillary driven flow occurs only along the axis of the matrix normal to the fracture. For a given set of relative permeability and capillary pressure curves, the behavior of the system is completely determined by the choice of two dimensionless parameters: (i) the ratio of time scales for advective flow in fracture to capillary flow in matrix $\alpha =\tau ^f/\tau ^m$ ; (ii) the ratio of pore volumes in matrix and fracture $\beta =V^m/V^f$ . A characteristic property of the flow in the coupled fracture–matrix medium is the linear recovery curve (before water breakthrough) which has been referred to as the “filling fracture” regime Rangel-German and Kovscek (J Pet Sci Eng 36:45–60, 2002), followed by a nonlinear period, referred to as the “instantly filled” regime, where the rate is approximately linear with the square root of time. We derive an analytical solution for the limiting case where the time scale $\tau ^{m}$ of the matrix imbibition becomes small relative to the time scale $\tau ^{f}$ of the fracture flow (i.e., $\alpha \rightarrow \infty $ ), and verify by numerical experiments that the model will converge to this limit as $\alpha $ becomes large. The model provides insight into the role played by parameters like saturation functions, injection rate, volume of fractures versus volume of matrix, different viscosity relations, and strength of capillary forces versus injection rate. Especially, a scaling number $\omega $ is suggested that seems to incorporate variations in these parameters. An interesting observation is that at $\omega =1$ there is little to gain in efficiency by reducing the injection rate. The model can be used as a tool for interpretation of laboratory experiments involving fracture–matrix flow as well as a tool for testing different transfer functions that have been suggested to use in reservoir simulators.     

Fractal Prediction Model of Spontaneous Imbibition Rate

By utilizing fractal dimension as one of the parameters to characterize rocks, a mathematical model was derived to predict the production rate by spontaneous imbibition. This fractal production model predicts a power law relationship between spontaneous imbibition rate and time. Fractal dimension can be estimated from the fractal production model using the experimental data of spontaneous imbibition in porous media. The experimental data of recovery in gas-water-rock and oil–water–rock systems were used to test the fractal production model. The rocks (Berea sandstone, chalk, and The Geysers graywacke) in which the spontaneous water imbibition experiments were conducted had different permeabilities ranging from 0.5 to over 1000 md. The results demonstrate that the fractal production model can match the experimental data satisfactorily in the cases studied. The fractal dimension data inferred from the model match were approximately equal to the values of fractal dimension measured using a different technique (mercury-intrusion capillary pressure) in Berea sandstone.     

Extraction of Effective Cement Paste Diffusivities from X-ray Microtomography Scans

The problem of extracting effective diffusivities of cement pastes on the basis of X-ray microtomography images is considered. A general computational homogenization framework is developed and applied to a variety of cement pastes whose microstructure has been digitized to a resolution of 1 μm. At this resolution, important submicron features are not resolved. Consequently, we propose a methodology whereby the pore space is ascribed a diffusivity less than the free diffusivity. For this purpose, a simple rule that incorporates microtomography data is proposed and shown to yield satisfactory results.     

On Barenblatt's Model of Spontaneous Countercurrent Imbibition

Water imbibition is a critical mechanism of secondary oil recovery from fractured reservoirs. Spontaneous imbibition also plays a significant role in storage of liquid waste by controlling the extent of rock invasion. In the present paper, we extend a model of countercurrent imbibition based on Barenblatt's theory of non-equilibrium two-phase flow by allowing the model's relaxation time to be a function of the wetting fluid saturation. We obtain two asymptotic self-similar solutions, valid at early and late times, respectively. At a very early stage, the time scale characterizing the cumulative volume of imbibed (and expelled) fluid is a power function with exponent between 1.5 and 1. At a later stage, the time scaling for this volume approaches asymptotically classical square root of time, whereas the saturation profile asymptotically converges to Ryzhik's self-similar solution. Our conclusions are verified against experiments. By fitting the laboratory data, we estimate the characteristic relaxation times for different pairs of liquids.     

Capillary Imbibition and Pore Characterisation in Cement Pastes

The kinetics of capillary imbibition in ordinary Portland cement pastes has been studied experimentally and theoretically. Nuclear magnetic resonance stray field imaging (STRAFI) has been used to record water concentration profiles for various ingress times. The profiles follow a t law and thus a master curve can be formed using the Boltzmann transformation. The distribution of pore sizes within the sample as measured by NMR cryoporometry shows a prominent peak at 100Å. A computer model of the pore structure was developed consisting of a lattice of interconnecting pores with a size distribution consistent with the cryoporometry results. The Hagen–Poiseuille law was used to describe the kinetics of the water in this pore structure. The best agreement between the computer simulations and the experimental master curve was obtained by using a narrower range of pore sizes than indicated by the cryoporometry results.     

A Dynamic Two-Phase Pore-Scale Model of Imbibition

We present a dynamic pore-scale network model of imbibition, capable of calculating residual oil saturation for any given capillary number, viscosity ratio, contact angle, and aspect ratio. Our goal is not to predict the outcome of core floods, but rather to perform a sensitivity analysis of the above-mentioned parameters, except from the viscosity ratio. We find that contact angle, aspect ratio, and capillary number all have a significant influence on the competition between piston-like advance, leading to high recovery, and snap-off, causing oil entrapment. Due to significant CPU-time requirements we did not incorporate long-range correlations among pore and throat sizes in our network, but were limited to small-range correlations. Consequently, the gradual suppression of snap-off occurs within one order of magnitude of the capillary number. At capillary numbers around 10⁸ - 10⁷ snap-off has been entirely inhibited, in agreement with results obtained by Blunt (1997) who used a quasi-static model. For higher aspect ratios, the effect of rate and contact angle is more pronounced.     

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.     

Pore Scale Modeling of Rate Effects in Imbibition

We use pore scale network modeling to study the effects of flow rate and contact angle on imbibition relative permeabilities. The model accounts for flow in wetting layers that occupy roughness or crevices in the pore space. Viscous forces are accounted for by solving for the wetting phase pressure and assuming a fixed conductance in wetting layers. Three-dimensional simulations model granular media, whereas two-dimensional runs represent fracture flow.We identify five generic types of displacement pattern as we vary capillary number, contact angle, and initial wetting phase saturation: flat frontal advance, dendritic frontal advance, bond percolation, compact cluster growth, and ramified cluster growth. Using phase diagrams we quantify the range of physical properties under which each regime is observed. The work explains apparently inconsistent experimental measurements of relative permeability in granular media and fractures.     

An In-Situ X-Ray Microtomography Study of Split Cylinder Fracture in Cement-Based Materials

In an effort to quantify microstructure-property relationships, three dimensional imaging experiments were conducted on small cylinder specimens subjected to split cylinder fracture. 3-D images were made using synchrotron-based x-ray microtomography, and the experiments were conducted with an in-situ frame such that a specimen could be examined while under load at varying degrees of damage. The specimens were made of fine-grained portland cement mortar and 0.5 mm glass beads, which served as aggregates. The diameter of the specimens was 5 mm. 3-D image analysis routines were developed or adapted to characterize microstructure and internal damage, which could then be related to bulk splitting strength and fracture energy. For fracture energy calculation, crack surface area could be measured in a way that accounted for roughness, branching, and fragmentation. Results showed that, for the specimens tested, aggregate surface roughness had little effect on strength but significant effect on fracture energy. Split cylinder strength showed correlation with specimen porosity, although there was considerable scatter. Strength did not correlate with maximum flaw size, although flaw location was not evaluated.     

Diffraction-Assisted Light Field Microscopy for Microtomography and Digital Volume Correlation with Improved Spatial Resolution

Experimental Mechanics - Light field microscopy (LFM) is capable of ultrafast tomographic reconstruction using a light field image acquired in a single snapshot. However, the axial resolution of...     

Estimation of Dynamic Relative Permeability and Capillary Pressure from Countercurrent Imbibition Experiments

Direct laboratory measurements of in situ water-phase saturation history are used to estimate relative permeability and capillary pressure functions. The magnitude of so-called nonequilibrium effects during spontaneous imbibition is quantified and, if significant, these effects are incorporated within the estimation technique. The primary constraint employed is that curves must increase or decrease monotonically; otherwise, no predetermined functionality is assumed. The technique is demonstrated using water saturation profile histories obtained for diatomite (a low-permeability and high-porosity rock). Results indicate that nonequilibrium effects detected at laboratory scale in low-permeability rocks influence the estimation of unsteady-state relative permeability and capillary pressure.     

Imbibition of Anionic Surfactant Solution into Oil-Wet Capillary Tubes

The displacement of oil by anionic surfactant solutions in oil-wet horizontal capillary tubes is studied. The position of the oil–water interface is recorded with time. The surfactant solution used is a mixture of several different surfactants and co-solvents tailored to produce ultra-low interfacial tension (IFT) for the specific oil used in the study. The surfactant solution results in ultra-low IFT at optimum salinity and room temperature. Several experimental parameters including the capillary tube radius and surfactant solution viscosity are varied to study their effect on the interface speed. Two different models are used to predict the oil–water interface position with time. In the first model, it is assumed that the IFT is constant and ultra-low throughout the experiments. The second model involves change of wettability and IFT by adsorption of surfactant molecules to the oil–water interface and the solid surface. Comparing the predictions to the experimental results, it is observed that the second model provides a better match, especially for smaller capillary tubes. The model is then used to predict the imbibition rate for very small capillary tubes, which have equivalent permeability close to oil reservoirs. The results show that the oil displacement rate is limited by the rate of diffusion of surfactants to the interface.     

Predicting the Capillary Imbibition of Porous Rocks from Microstructure

The kinetics of capillary imbibition into porous rocks is studied experimentally and theoretically. The Washburn law is modified by introducing various corrections relating to the microstructure of the rocks, such as tortuosity, pore shape (obtained experimentally), and applying the effective medium approximation (EMA) in order to calculate the effective radius that defines the hydraulic conductance and the topology of the capillary imbibition. The application of the EMA shows that capillary imbibition is mainly produced in 1-D, and the pore structure is constituted by different pore throats in series, linked by chamber pores. The capillary process has been discussed as a function of their petrography and pore structure. Our study of the Washburn equation and the addition of correction factors for the pore structure allows a very accurate prediction of the weight rate.     

Evaluation of Matrix-Fracture Transfer Functions for Counter-Current Capillary Imbibition

Different functions describing matrix-fracture transfer were tested for counter-current capillary imbibition interaction. The recovery curves obtained from capillary imbibition experiments were used to fit the transfer functions. The exponential coefficients yielding the best fit to the experimental data were obtained and correlated to the effective parameters such as viscosity, IFT, matrix length and diameter, matrix permeability and porosity, and wettability using multivariable regression analysis. In order to obtain the recovery curves, experiments were conducted on Berea sandstone and Indiana limestone samples. Cylindrical samples with different shape factors were obtained by cutting the plugs 1, 2.5, and 5 cm in diameter and 2.5, 5, and 10 cm in length. All sides were coated with epoxy except one end. More than fifty static imbibition experiments were carried out on vertically and horizontally situated samples where the imbibition took place upward and lateral directions, respectively. Brine–air, brine–kerosene, brine–mineral oil, and surfactant solution–mineral oil pairs were used as fluids. For many matrix shape factors (especially longer and small diameter ones), dividing the recovery curve into three parts were needed as the early, intermediate, and late times, which are typically distinguished by the time required for the imbibition front to reach the closed boundary at the end of the core. Correlations among the exponential coefficients and rock/fluid properties were developed. It was observed that different rock/fluid properties and transfer mechanisms (capillary imbibition and gravity drainage) govern the process for each part. Hence, the analyses done in this study were useful not only for developing explicit transfer functions but also identifying the physics of the counter-current imbibition recovery.     

Pore-Scale Lattice Boltzmann Modeling and 4D X-ray Computed Microtomography Imaging of Fracture-Matrix Fluid Transfer

We present sequential X-ray computed microtomography (CMT) images of matrix drainage in a fractured, sintered glass-granule-pack. Sequential (4D) CMT imaging captured the capillary-dominated displacement of the oil-occupied matrix by the surfactant-brine-occupied fracture at the pore scale. The sintered glass-granule-pack was designed to have minimal pore space beyond the resolution of CMT imaging, ensuring that the pore space of the matrix connected to the fracture could be captured in its entirety. This provided an opportunity to validate the increasingly common lattice Boltzmann modeling technique against experimental images at the pore scale. Although the surfactant was found to alter the wettability of the originally weakly oil-wet glass to water-wet, the fracture-matrix fluid transfer is found to be a drainage process, showing minimal counter-current migration of the initial wetting phase (decane). The LB simulations were found to closely match experimental rates of fracture-matrix fluid transfer, and trends in the saturation profiles, but not the irreducible wetting-phase saturation behind the flooding front. The underestimation of the irreducible wetting phase saturation suggests that finer image and lattice resolutions than those reported here may be required for accurate prediction of some macroscale multiphase flow properties, at a sizable computational cost.