Micro-scale experimental investigation of the effect of flow rate on trapping in sandstone and carbonate rock samples

We present the results of a pore-scale experimental study of residual trapping in consolidated sandstone and carbonate rock samples under confining stress. We investigate how the changes in wetting phase flow rate impacts pore-scale distribution of fluids during imbibition in natural, water-wet porous media. We systematically study pore-scale trapping of the nonwetting phase as well as size and distribution of its disconnected globules. Seven sets of drainage-imbibition experiments were performed with brine and oil as the wetting and nonwetting phases, respectively. We utilized a two-phase miniature core-flooding apparatus integrated with an X-ray microtomography system to examine pore-scale fluid distributions in small Bentheimer sandstone (D = 4.9 mm and L = 13 mm) and Gambier limestone (D = 4.4 mm and L = 75 mm) core samples. The results show that with increase in capillary number, the residual oil saturation at the end of the imbibition reduces from 0.46 to 0.20 in Bemtheimer sandstone and from 0.46 to 0.28 in Gambier limestone. We use pore-scale displacement mechanisms, in-situ wettability characteristics, and pore size distribution information to explain the observed capillary desaturation trends. The reduction was believed to be caused by alteration of the order in which pore-scale displacements took place during imbibition. Furthermore, increase in capillary number produced significantly different pore-scale fluid distributions during imbibition. We explored the pore fluid occupancies and studied size and distribution of the trapped oil clusters during different imbibition experiments. The results clearly show that as the capillary number increases, imbibition produces smaller trapped oil globules. In other words, the volume of individual trapped oil globules decreased at higher brine flow rates. Finally, we observed that the pore space in the limestone sample was considerably altered through matrix dissolution at extremely high brine flow rates. This increased the sample porosity from 44% to 62% and permeability from 7.3 D to 80 D. Imbibition in the altered pore space produced lower residual oil saturation (from 0.28 to 0.22) and significantly different distribution of trapped oil globules.     

Extended Roof snap-off for a continuous nonwetting fluid and an example case for supercritical CO2

A dominant mechanism for residual trapping of a nonwetting fluid in porous media during imbibition is snap-off or the disconnection of a continuous stream of the nonwetting fluid when it passes through pore constrictions and when a criterion based on capillary pressure imbalance is met. While quasi-static criteria for Roof snap-off have been defined for pores based on the imbalance between capillary pressure across the front/tail meniscus and local capillary pressure at the pore throat, and expressed in terms of pore body to pore throat ratio for simplification, we extended the previous quasi-static snap-off criterion by considering the local capillary pressure imbalance between the pore body and the pore throat for both circular and noncircular pores when the wetting film exists. We then used the criterion to analyze results from computational fluid dynamics (CFD) simulations of multi-phase flow with supercritical CO₂ as the nonwetting fluid and water as the wetting fluid. The extended criterion successfully described most situations we modeled. Furthermore, we compared fluid interface shape for a noncircular 3D pore predicted by the minimum surface energy (MSE) theory against 3D CFD simulations. While the fluid interface shape at the pore throat for 3D simulation was consistent with the shape predicted by MSE theory, the shape could not be successfully predicted by the MSE theory at the upstream and downstream pore body. Moreover, film flow existed for the noncircular pore at the downstream pore body.     

岩石弹性波速度和饱和度、孔隙流体分布的关系

在实验室对6种砂岩进行了连续的进水和失水实验. 测 量了两个过程的纵波速度VP、横波速度VS进水和失水速率随饱和度SW的 变化；分析了低饱和度时流体对岩石弹性性质的影响. 实验表明，进水和失水过程显示不同 的纵、横波速度与饱和度关系，速度不仅与饱和度有关，还与不同饱和阶段的孔隙流体分布 有关，而且也是水和岩石骨架之间的物理及化学作用所致.     

Tight dual models of pore spaces

The pore throats in a porous medium control permeability, drainage, and straining through their pore scale geometry and through the way they are connected via pore bodies on the macroscale. Likewise, imbibition is controlled through the geometry of the pore bodies (pore scale) and through the way the pore bodies are connected via pore throats on the macroscale. In an effort to account for both scales at the same time we recently introduced an image-based model for pore spaces that consists of two parts related by duality: (1) a decomposition of a polyhedral pore space into polyhedral pore bodies separated by polygonal pore throats and (2) a polygonal pore network that is homotopy equivalent to the pore space. In this paper we stick to the dual concept while amending the definition of the pore throats and, as a consequence, the other elements of the dual model. Formerly, the pore throats consisted of single two-dimensional Delaunay cells, while they now usually consist of more than one two-dimensional Delaunay cell and extend all the way into the narrowing ends of the pore channel cross sections. This is the first reason for naming the amended dual model “tight”. The second reason is that the formation of the pore throats is now guided by an objective function that always attains its global optimum (tight optimization). At the end of the paper we report on simulations of drainage performed on tight dual models derived from simulated sphere packings and 3D gray-level images. The C-code for the generation of the tight dual model and the simulation of drainage is publicly available at https://jshare.johnshopkins.edu/mhilper1/public_html/tdm.html.     

A THEORETICAL TREATMENT OF THE EFFECT OF MICROSCOPIC FLUID DISTRIBUTION ON THE DIELECTRIC PROPERTIES OF PARTIALLY SATURATED ROCKS1

Microscopic fluid distribution can have a significant effect on the dielectric properties of partially saturated rocks. Evidence of this effect is found in the laboratory data presented by Knight and Nur in which different methods for controlling saturation produced very different results for the dependence of the dielectric response on water saturation. In this study, previously derived models for the dielectric response of a heterogeneous medium are generalized and the case of a pore space occupied by multiple pore fluids is considered. By using various geometrical distributions of water and gas, it is observed that both the pore geometry in which saturation conditions are changing and the gas–water geometry within a given pore space are critical factors in determining the effective dielectric response of a partially saturated rock. As an example, data for a tight gas sandstone undergoing a cycle of imbibition and drying are analysed. Previous research has demonstrated that significantly different microscopic fluid distributions result from the application of these two techniques to control the level of water saturation. By approximating these microscopic fluid distributions using simple geometrical models, good agreement is found between experimental data and calculated dielectric properties.     

Petrophysical characterization of porous media starting from micro-tomographic images

A quasi-static scheme based on pore space spatial statistics is presented to simulate pore-scale two-phase capillary-dominant displacement processes. The algorithm is coupled with computational fluid dynamics in order to evaluate saturation functions. Wettability heterogeneity in partial and fractional/mixed-wet media is implemented using a contact angle map. The simulation process is pixel-wised and performed directly on binary images. Bypassing and snap-off are tackled as non-wetting phase trapping mechanisms. Post-processing results include residual saturations, effective permeability and capillary pressure curves for drainage and imbibition scenarios. The primary advantages of the proposed workflow are eliminating pore space skeletisation/ discretization, superior time efficiency and minimal numerical drawbacks when compared to other direct or network-based simulation techniques.     

A THEORETICAL STUDY ON THE REDUCTION OF STATISTICAL PORE SYSTEM PARAMETERS TO MEASURABLE QUANTITIES*

As shown previously [50], the relations between formation factor, permeability, and porosity can be expressed in statistical parameters of the microscopic pore system. A number of other macroscopically measurable quantities depend on those same parameters. Thus, by proper combination of these quantities, it should be possible to solve for those parameters, that cannot be retrieved by direct methods. In this article, the dependencies on these statistical parameters of the capillarity with imbibition and drainage, the irreducible water saturation, the flow dispersion with miscible displacement, and quantities that are directly observable at ground surfaces or thin sections, are preliminarily studied as a preparation for possible practical measurements.     

Air and water entrapment in the vicinity of the water table

A laboratory tank was used to study entrapment of water in coarse sand lenses above the water table and of air in coarse sand lenses below the water table. Monitoring of these experiments involved a combination of visual inspection, measurement of moisture content, and measurement of air/water pressure. The medium consisted of coarse sand lenses with various degrees of vertical connectivity embedded within a fine sand matrix. Experiments were performed under conditions of both drainage (from a fully saturated medium) and imbibition. Observations during drainage included: (1) water was trapped in the coarse sand zones above the water table at heights significantly greater than anticipated from consideration of capillary rise in the coarse sand; (2) rapid drainage of these same coarse zones occurred when air penetrated into these zones through the surrounding fine sands; and (3) prior to the time of penetration of the coarse sand by air, water pressure in the coarse zone dropped significantly below atmospheric pressure. Observations during imbibition included: (1) entrapment of air within coarse sands below the water table, (2) the pore fluids in these zones varied spatially from predominantly air to predominantly water, and (3) pressure in the trapped air phase was significantly greater than pressure in the water phase in the surrounding fine sand. Overall, these results demonstrated significant sensitivity to the geometry of the coarse sand inclusions, particularly the vertical connectivity of the coarse sand lens.     

Pore-scale imbibition experiments in dry and prewetted porous media

Two-phase imbibition behavior of immiscible fluids was studied in dry and prewetted porous media using a laser-induced fluorescence technique. Imbibition was first investigated in two-dimensional (2-D) systems under conditions comparable to those for a study of drainage [Ovdat H, Berkowitz B. Pore-scale study of drainage displacement under combined capillary and gravity effects in index-matched porous media. Water Resources Research 2006;42:W06411. doi:10.1029/2005WR004553] in the capillary-dominated regime. The effect of initial wetting saturation (IWS) was then explored in 2-D and 3-D porous media under the combined effect of gravity, capillary and viscous forces, within and outside the capillary-dominated regime. Parameters that describe maximum vertical advance, volumetric fraction, total surface area and specific surface area of the invading fluid were used to quantify the behavior. Comparison of 2-D drainage and imbibition patterns demonstrates significant qualitative differences under analogous viscosity ratio, buoyancy number, and capillary number values. However, quantitative analyses show strong pore-scale similarities between these patterns. Invasion structures in 3-D, prewetted (IWS ≈ 8% of the pore volume) porous media are ramified, with lateral branching and regions containing trapped residual fluid. These structures are qualitatively and quantitatively different from the compact, branchless structures that develop in dry (IWS = 0) porous media.     

Research note: Laboratory study of the influence of changing the injection rate on the geometry of the fluid front and on P‐wave ultrasonic velocities in sandstone

Forced imbibition was performed in reservoir sandstone by injecting water into a dry sample. The injection was monitored with X‐ray computed tomography and acoustic acquisition to simultaneously visualize the displacement of the fluid and quantify its presence by calculating saturation and P‐wave velocities. We observed a strong influence when changing the injection rates on the acoustic response. Upon decreasing the injection rate from 5 mL/h to 0.1 mL/h, P‐wave velocities decreased sharply: 100 m/s in 1 h. This behaviour is related to the partially saturated conditions of the sample (76% of saturation) before decreasing the injection rate. The air that is still trapped is free to move due to a decrease of pore pressure that is no longer forced by the higher injection rate. After 1 hour, P‐wave velocities started increasing with small changes in saturation. Stopping injection for 16 hrs decreased saturation by 8% and P‐wave velocities by 100 m/s. Restarting injection at 5 mL/h increased saturation to 76% while P‐wave velocities fluctuated considerably for 2 hrs until they stabilized at 2253 m/s. Through the computed tomography scans we observed a water front advancing through the sample and how its shape changed from a plane to a curve after decreasing the injection rate.     

The effect of wettability on capillary trapping in carbonates

We use an organic acid (cyclohexanepentanoic acid) to alter the wettability of three carbonates: Estaillades, Ketton and Portland limestones, and observe the relationship between the initial oil saturation and the residual saturation. We take cores containing oil and a specified initial water saturation and waterflood until 10 pore volumes have been injected. We record the remaining oil saturation as a function of the amount of water injected. In the water-wet case, with no wettability alteration, we observe, as expected, a monotonic increase in the remaining oil saturation with initial saturation. However, when the wettability is altered, we observe an increase, then a decrease, and finally an increase in the trapping curve for Estaillades limestone with a small, but continued, decrease in the remaining saturation as more water is injected. This behavior is indicative of mixed-wet or intermediate-wet conditions, as there is no spontaneous imbibition of oil and water. In contrast, Ketton did not show indications of a significant wettability alteration with a similar observed trapping profile to that observed in the water-wet case. Portland limestone also showed a monotonic increasing trend in remaining saturation with initial saturation but with a higher recovery, and less trapping, than the water-wet case. Again, this is intermediate-wet behavior with no spontaneous imbibition of either oil or water, and slow production of oil after water breakthrough. Finally, we repeat the same experiments but instead we age the three carbonates with a high asphaltenic content and high viscosity crude oil at 70 °C mimicking reservoir conditions. The results show a monotonic increase in residual saturation as a function of initial saturation but with higher recovery than the water-wet cases for Estaillades and Portland, with again no indication of wettability alteration for Ketton. We discuss the results in terms of pore-scale recovery process and contact angle hysteresis. In these experiments, water-saturated micro-porosity appears to protect the solid surfaces from a strong wettability alteration, particularly in Ketton.     

Capillary pressure for the sand–CO2–water system under various pressure conditions. Application to CO2 sequestration

Accurate modeling of storage of carbon dioxide (CO₂) in heterogeneous aquifers requires experiments of the capillary pressure as function of temperature and pressure. We present a method with which static drainage and imbibition capillary pressures can be measured continuously as a function of saturation at various temperature (T) and pressure (P) conditions. The measurements are carried out at (T, P) conditions of practical interest. Static conditions can be assumed as small injection rates are applied. The capillary pressure curves are obtained for the unconsolidated sand–distilled water–CO₂ system. The experimental results show a decrease of drainage and imbibition capillary pressure for increasing CO₂ pressures and pronounced dissolution rate effects for gaseous CO₂. Significant capillary pressure fluctuations and negative values during imbibition are observed at near critical conditions. The measurement procedure is validated by a numerical model that simulates the experiments.     

Forced imbibition into a limestone: measuring P‐wave velocity and water saturation dependence on injection rate

Quantitative interpretation of time‐lapse seismic data requires knowledge of the relationship between elastic wave velocities and fluid saturation. This relationship is not unique but depends on the spatial distribution of the fluid in the pore‐space of the rock. In turn, the fluid distribution depends on the injection rate. To study this dependency, forced imbibition experiments with variable injection rates have been performed on an air‐dry limestone sample. Water was injected into a cylindrical sample and was monitored by X‐Ray Computed Tomography and ultrasonic time‐of‐flight measurements across the sample. The measurements show that the P‐wave velocity decreases well before the saturation front approaches the ultrasonic raypath. This decrease is followed by an increase as the saturation front crosses the raypath. The observed patterns of the acoustic response and water saturation as functions of the injection rate are consistent with previous observations on sandstone. The results confirm that the injection rate has significant influence on fluid distribution and the corresponding acoustic response. The complexity of the acoustic response —‐ that is not monotonic with changes in saturation, and which at the same saturation varies between hydrostatic conditions and states of dynamic fluid flow – may have implications for the interpretation of time‐lapse seismic responses.     

Co-sequestration of SO2 with supercritical CO2 in carbonates: An experimental study of capillary trapping,relative permeability,and capillary pressure

In this study we performed three categories of steady- and unsteady-state core-flooding experiments to investigate capillary trapping, relative permeability, and capillary pressure, in a scCO₂ + SO₂/brine/limestone system at elevated temperature and pressure conditions, i.e., 60 °C and 19.16 MPa. We used a Madison limestone core sample acquired from the Rock Springs Uplift in southwest Wyoming. We carried out two sets of steady-state drainage-imbibition relative permeability experiments with different initial brine saturations to study hysteresis. We found that the final scCO₂ + SO₂ drainage relative permeability was very low, i.e., 0.04. We also observed a rapid reduction in the scCO₂-rich phase imbibition relative permeability curve, which resulted in a high residual trapping. The results showed that between 62.8% and more than 76% of the initial scCO₂ + SO₂ at the end of drainage was trapped by capillary trapping mechanism (trapping efficiency). We found that at higher initial brine saturations, the trapping efficiency was higher. The maximum initial and residual scCO₂-rich phase saturations at the end of primary drainage and imbibition were 0.525 and 0.329, respectively. Each drainage-imbibition cycle was followed by a dissolution process to re-establish S_w = 1. The dissolution brine relative permeabilities for both cycles were also obtained. We characterized the scCO₂ + SO₂/brine capillary pressure hysteresis behavior through unsteady-state primary drainage, imbibition, and secondary drainage experiments. We observed negative imbibition capillary pressure curve indicative of possible wettability alteration throughout the experiments due to contact with scCO₂ + SO₂/brine fluid system. The trapping results were compared to those reported in literature for other carbonate core samples. We noticed slightly more residual trapping in our sample, which might be attributed to heterogeneity, different viscosity ratio, and pore-space topologies. The impact of dynamic effects, i.e., high brine flow rate imbibition tests, on trapping of the scCO₂-rich phase was also explored. We performed two imbibition experiments with relatively high brine flow rates. The residual scCO₂ saturation dropped to 0.291 and 0.262 at the end of the first and second imbibition tests, i.e., 11.5% and 20.4%, respectively, compared to 0.329 under capillary-dominated regime.     

Solute dilution under imbibition and drainage conditions in a heterogeneous structure: Modeling of a sand tank experiment

This study aims at modeling the transport of a conservative tracer in two dimensions, as experimentally observed in a strongly heterogeneous medium under conditions of variable water saturation during drainage and imbibition. Solute transport experiments were conducted in a sand tank containing an artificial packing of three quartz sands of different particle sizes. The packing was characterized by the presence of numerous homogeneous layers (0.5 × 5 × 5 cm) inclined at 45° and randomly distributed in a tank. Six different stationary flow conditions were sequentially established during imbibition and drainage. When a stationary flow regime was reached, several solute pulses were applied at different positions at the upper surface of the sand structure. The transport regime was studied by monitoring the tracer plumes injected as point-like pulses at the surface, as they travelled through the sand bedding.     

Rapid imbibition of water in fractures within unsaturated sedimentary rock

The spontaneous imbibition of water and other liquids into gas-filled fractures in variably-saturated porous media is important in a variety of engineering and geological contexts. However, surprisingly few studies have investigated this phenomenon. We present a theoretical framework for predicting the 1-dimensional movement of water into air-filled fractures within a porous medium based on early-time capillary dynamics and spreading over the rough surfaces of fracture faces. The theory permits estimation of sorptivity values for the matrix and fracture zone, as well as a dispersion parameter which quantifies the extent of spreading of the wetting front. Quantitative data on spontaneous imbibition of water in unsaturated Berea sandstone cores were acquired to evaluate the proposed model. The cores with different permeability classes ranging from 50 to 500 mD and were fractured using the Brazilian method. Spontaneous imbibition in the fractured cores was measured by dynamic neutron radiography at the Neutron Imaging Prototype Facility (beam line CG-1D, HFIR), Oak Ridge National Laboratory. Water uptake into both the matrix and the fracture zone exhibited square-root-of-time behavior. The matrix sorptivities ranged from 2.9 to 4.6 mm s^−0.5, and increased linearly as the permeability class increased. The sorptivities of the fracture zones ranged from 17.9 to 27.1 mm s^−0.5, and increased linearly with increasing fracture aperture width. The dispersion coefficients ranged from 23.7 to 66.7 mm² s⁻¹ and increased linearly with increasing fracture aperture width and damage zone width. Both theory and observations indicate that fractures can significantly increase spontaneous imbibition in unsaturated sedimentary rock by capillary action and surface spreading on rough fracture faces. Fractures also increase the dispersion of the wetting front. Further research is needed to investigate this phenomenon in other natural and engineered porous media.     

Bulk moduli and seismic attenuation in partially saturated rocks: hysteresis of liquid bridges effect

A key task of exploration geophysics is to find relationships between seismic attributes (velocities and attenuation) and fluid properties (saturation and pore pressure). Experimental data suggest that at least three different factors affect these relationships, which are not well explained by classical Gassmann, Biot, squirt-flow, mesoscopic-flow and gas dissolution/exsolution models. Some of these additional factors include (i) effect of wettability and surface tension between immiscible fluids, (ii) saturation history effects (drainage versus imbibition) and (iii) effects of wave amplitude and effective stress. We apply a new rock physics model to explain the role of all these additional factors on seismic properties of a partially saturated rock. The model is based on a well-known effect in surface chemistry: hysteresis of liquid bridges. This effect is taking place in cracks, which are partially saturated with two immiscible fluids. Using our model, we investigated (i) physical factors affecting empirical Brie correlation for effective bulk modulus of fluid, (ii) the role of liquids on seismic attenuation in the low frequency (static) limit, (iii) water-weakening effects and (iv) saturation history effects. Our model is applicable in the low frequency limit (seismic frequencies) when capillary forces dominate over viscous forces during wave-induced two-phase fluid flow. The model is relevant for the seismic characterization of immiscible fluids with high contrast in compressibilities, that is, for shallow gas exploration and CO₂ monitoring.     

Dual models of pore spaces

We present a model for pore spaces that consists of two parts related by duality: (1) a decomposition of an open polyhedral pore space into open contractible pore bodies separated by relatively open interfaces and (2) a pore network that is homotopy equivalent to the pore space. The dual model is unique and free of parameters, but it relies on regularity conditions for the pore space. We show how to approximate any pore space by the interior of a polyhedral complex such that the regularity conditions are fulfilled. Thus, we are able to calculate the dual model from synthetic porous media and images of real porous media. The pore bodies are unions of relatively open Delaunay cells with respect to the corners of the pore boundary, and the pore network consists of certain at most two-dimensional (2D) Voronoi cells with respect to the corners of the pore boundary. The pore network describes the neighborhood relations between the pore bodies. In particular, any relatively open 2D Delaunay face f separating two pore bodies has a unique (relatively open) dual network edge. In our model, f is a pore throat only if it is hit by its dual network edge. Thus, as opposed to widespread intuition, any pore throat is convex, and adjacent pore bodies are not necessarily separated by pore throats. Due to the duality between the pore network and the decomposition of the pore space into pore bodies it is straightforward to store the geometrical properties of the pore bodies [pore throats] as attributes of the dual network vertices [edges]. Such an attributed network is used to perform 2D drainage simulations. The results agree very well with those from a pore-morphology based modeling approach performed directly on the digital image of a porous medium. Contractibility of the pore bodies and homotopy equivalence of the pore space and the pore network is proven using discrete Morse theory and the nerve theorem from combinatorial topology.     

油水双相饱和孔隙模型核磁特性理论研究

利用特征函数展开方法，给出了基于扩散效应的核磁共振Bloch控制方程的弛豫模式解表达式, 在此基础上分别给出了水饱和与油水双相饱和孔隙模型的核磁共振特性模拟结果. 结果表明，孔隙流体的核磁弛豫与孔隙大小、孔隙表面弛豫率、孔隙流体的扩散能力以及含油饱和度等有密切的关系. 对于水饱和孔隙，弛豫时间主要由孔隙大小控制. 当孔隙较大时，即使均匀大小孔隙，孔隙流体的弛豫也会表现为多指数弛豫. 而且最小模式弛豫时间与孔隙大小为非线性关系. 对于油水双相饱和孔隙，在孔隙较小时，含油饱和度对弛豫的影响主要表现在弛豫时间随含油饱和度的增加而线性减小；但在孔隙较大时，含油饱和度的增加对弛豫影响表现在两个方面，其一，孔隙水弛豫由多个弛豫模式控制逐渐转变为由最小弛豫模式控制；其二，孔隙水弛豫时间与含油饱和度表现为非线性关系. 对由实际岩芯抽象出的孔隙模型，采用本文获得的理论公式，在油水双相饱和时进行了正反演模拟. 计算结果与已有的实验结果较为一致.