Experimental Study and Fractal Analysis of Heterogeneity in Naturally Fractured Rocks

Capillary pressure curves of six low porosity and low permeability core samples from The Geysers geothermal field were measured using the mercury-intrusion approach to characterize the heterogeneity of rock. One high permeability Berea sandstone core sample was analyzed similarly, for comparison. The maximum pressure of mercury intruded into the rock was about 200 MPa to reach the extremely small pores. Experimental data showed that the capillary pressure curves of The Geysers rock are very different from that of the Berea sandstone. It was found that the frequently used capillary pressure models could not be used to represent the data from The Geysers rock samples. This might be because of the fractures in the rock. To this end, a fractal technique was proposed to model the features of the capillary pressure curves and to characterize the difference in heterogeneity between The Geysers rock and Berea sandstone. The results demonstrated that the rock from The Geysers geothermal field was fractal over a scaling range of about five orders of magnitude. The values of the fractal dimension of all the core samples (six from The Geysers and one Berea sandstone) calculated using the proposed approach were in the range from 2 to 3. The results showed that The Geysers rock with a high density of fractures had a greater fractal dimension than Berea sandstone which is almost without fractures. This shows that The Geysers rock has greater heterogeneity, as expected.     

A Semi-Analytical Model for Two Phase Immiscible Flow in Porous Media Honouring Capillary Pressure

This article describes a semi-analytical model for two-phase immiscible flow in porous media. The model incorporates the effect of capillary pressure gradient on fluid displacement. It also includes a correction to the capillarity-free Buckley–Leverett saturation profile for the stabilized-zone around the displacement front and the end-effects near the core outlet. The model is valid for both drainage and imbibition oil–water displacements in porous media with different wettability conditions. A stepwise procedure is presented to derive relative permeabilities from coreflood displacements using the proposed semi-analytical model. The procedure can be utilized for both before and after breakthrough data and hence is capable to generate a continuous relative permeability curve unlike other analytical/semi-analytical approaches. The model predictions are compared with numerical simulations and laboratory experiments. The comparison shows that the model predictions for drainage process agree well with the numerical simulations for different capillary numbers, whereas there is mismatch between the relative permeability derived using the Johnson–Bossler–Naumann (JBN) method and the simulations. The coreflood experiments carried out on a Berea sandstone core suggest that the proposed model works better than the JBN method for a drainage process in strongly wet rocks. Both methods give similar results for imbibition processes.     

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.     

Analysis of capillary interaction and oil recovery under ultrasonic waves

Although, the effects of ultrasonic irradiation on multiphase flow through porous media have been studied in the past few decades, the physics of the acoustic interaction between fluid and rock is not yet well understood. Various mechanisms may be responsible for enhancing the flow of oil through porous media in the presence of an acoustic field. Capillary related mechanisms are peristaltic transport due to mechanical deformation of the pore walls, reduction of capillary forces due to the destruction of surface films generated across pore boundaries, coalescence of oil drops due to Bjerknes forces, oscillation and excitation of capillary trapped oil drops, forces generated by cavitating bubbles, and sonocapillary effects. Insight into the physical principles governing the mobilization of oil by ultrasonic waves is vital for developing and implementing novel techniques of oil extraction. This paper aims at identifying and analyzing the influence of high-frequency, high-intensity ultrasonic radiation on capillary imbibition. Laboratory experiments were performed using cylindrical Berea sandstone and Indiana limestone samples with all sides (quasi-co-current imbibition), and only one side (counter-current imbibition) contacting with the aqueous phase. The oil saturated cores were placed in an ultrasonic bath, and brought into contact with the aqueous phase. The recovery rate due to capillary imbibition was monitored against time. Air–water, mineral oil–brine, mineral oil–surfactant solution and mineral oil-polymer solution experiments were run each exploring a separate physical process governing acoustic stimulation. Water–air imbibition tests isolate the effect of ultrasound on wettability, capillarity and density, while oil–brine imbibition experiments help outline the ultrasonic effect on viscosity and interfacial interaction between oil, rock and aqueous phase. We find that ultrasonic irradiation enhances capillary imbibition recovery of oil for various fluid pairs, and that such process is dependent on the interfacial tension and density of the fluids. Although more evidence is needed, some runs hint that wettability was not altered substantially under ultrasound. Preliminary analysis of the imbibition recoveries also suggests that ultrasound enhances surfactant solubility and reduce surfactant adsorption onto the rock matrix. Additionally, counter-current experiments involving kerosene and brine in epoxy coated Berea sandstone showed a dramatic decline in recovery. Therefore, the effectiveness of any ultrasonic application may strongly depend on the nature of interaction type, i.e., co- or counter-current flow. A modified form of an exponential model was employed to fit the recovery curves in an attempt to quantify the factors causing the incremental recovery by ultrasonic waves for different fluid pairs and rock types.     

Direct Observation of Trapped Gas Bubbles by Capillarity in Sandy Porous Media

We investigated the mechanism of residual gas trapping at a microscopic level. We imaged trapped air bubbles in a Berea sandstone chip after spontaneous imbibition at atmospheric pressure. The pore structure and trapped bubbles were observed by microfocused X-ray computed tomography. Distributions of trapped bubbles in Berea and Tako sandstone were imaged in coreflooding at a capillary number of 1.0 × 10⁻⁶. Trapped bubbles are of two types, those occupying the center of the pore with a pore-scale size and others having a pore-network scale size. In low-porosity media such as sandstone, connected bubbles contribute greatly to trapped gas saturation. Effects of capillary number and injected water volume were investigated using a packed bed of glass beads 600μm in diameter, which had high porosity (38%). The trapped N₂ bubbles are stable against the water flow rate corresponding to a capillary number of 1.0 × 10⁻⁴.     

基于核磁共振孔隙划分的致密油藏自发渗吸原油可动性研究

自发渗吸驱油是致密油藏提高采收率的有效手段,但不同的孔隙划分方法会导致原油可动性精细定量表征存在差异性.基于此,以鄂尔多斯盆地延长组致密油藏为研究对象,开展了四种典型致密岩心的自发渗吸驱油实验,利用基于核磁共振分形理论的流体分布孔隙精细划分方法,区分了致密砂岩岩心孔隙类型,明确了不同类型岩心孔隙结构对原油可动性和自发渗吸驱油速率的控制特征.研究结果表明不同类型岩心自发渗吸模拟油动用程度介于22.07%～33.26%,核磁共振T2谱双峰型岩心自发渗吸模拟油动用程度高于单峰型岩心;不同类型致密砂岩岩心中流体分布孔隙可初步划分出P1和P2两种类型, P1类型孔隙则可进一步划分出P1-1, P1-2和P1-3三种亚类型孔隙;致密砂岩岩心中P1和P2类孔隙中模拟油均有不同程度的动用, P1类孔隙作为致密岩心中主要孔隙,尤其是P1类孔隙中P1-2和P1-3类孔隙的数量决定了自发渗吸模拟油动用程度;P1-1, P1-2和P1-3类孔隙结构差异性对自发渗吸模拟油动用程度起决定性作用,较小尺寸孔径孔隙较大的孔隙结构差异性不仅提升了自发渗吸模拟油动用程度,而且提升了自发渗吸驱油速率;流体可动性指数较高的P...     

Effect of dynamic contact angle variation on spontaneous imbibition in porous materials

We investigate the influence of contact angle variations on spontaneous imbibition of moisture in porous materials. While the contact angle is typically assumed constant when modelling the moisture transfer in porous media, experimental findings put this assumption into question. It has been shown that during imbibition the contact angle notably rises with increasing meniscus velocity. This phenomenon resultantly affects the moisture retention curve, the relation linking the local capillary pressure to the local moisture saturation, which in turn impacts the imbibition rate and moisture distribution. This study investigates these dynamic effects via a pore network technique as well as a continuum approach. It is shown that the impacts of pore-scale contact angle variations on the imbibition process can be reproduced at the continuum scale through a modified moisture retention curve including a dynamic term. Complementarily a closed-form equation expressing the dynamic capillary pressure in terms of local saturation and saturation rate is derived. The continuum approach is then finally employed to predict measured moisture saturation profiles for imbibition in Berea sandstone and diatomite found in literature, and a fair agreement between simulated and measured outcomes is observed.

     

A Hierarchical Sampling for Capturing Permeability Trend in Rock Physics

Among all properties of reservoir rocks, permeability is an important parameter which is typically measured from core samples in the laboratory. Due to limitations of core drilling all over a reservoir, simulation of rock porous media is demanded to explore more scenarios not seen in the available data. One of the most accurate methods is cross correlation based simulation (CCSIM) which recently has broadly applied in geoscience and porous media. The purpose of this study is producing realizations with the same permeability trend to a real sample. Berea sandstone sample is selected for this aim. Permeability results, extracted from smaller sub-samples of the original sample, showed that classic Kozeny–Carman permeability trend is not suitable for this sample. One reason can be due to lack of including geometrical and fractal properties of pore-space distribution in this equation. Thus, a general trend based on fractal dimensions of pore-space and tortuosity of the Berea sample is applied in this paper. Results show that direct 3D stochastic modeling of porous media preserves porous structure and fractal behavior of rock. On the other hand, using only 2D images for constructing the 3D pore structures does not reproduce the measured experimental permeability. For this aim, a hierarchical sampling is implemented in two and three steps using both 2D and 3D stochastic modeling. Results showed that two-step sampling is not suitable enough, while the utilized three-step sampling occurs to be show excellent performance by which different models of porous media with the same permeability trend as the Berea sandstone sample can be generated.     

Capillary Pressure at the Imbibition Front During Water–Oil Counter-Current Spontaneous Imbibition

Counter-current spontaneous imbibition (COUCSI) in porous media is driven by capillary forces. Capillary action results in a high capillary imbibition pressure at the imbibition front and a low capillary drainage pressure at the outlet face. It is the difference between these two pressures that draws in the wetting phase and pushes out the non-wetting phase. A technique for measuring the capillary pressure at an imbibition front under restricted flow conditions has been developed and applied to Berea sandstone with a range of permeabilities. In the experiments, brine was the wetting phase and refined oil was the non-wetting phase. One end face of a sandstone core was butted to a short section of a finer-pored rock. The composite core surface was then sealed, apart from the end face of the low-permeability segment. A connection to a pressure transducer was set in the opposite end face of the core. Initially, the main core segment was filled with oil. In most cases, the finer-pored segment was filled with brine. Imbibition was started by immersing the core in brine. The purpose of the finer-pored segment was to prevent the escape of non-wetting phase from the open face. For some tests there was an initial period of co-current spontaneous imbibition (COCSI) created by allowing production of non-wetting phase through an outlet tapping in the sealed end face. The outlet was then connected to the transducer and the imbibition changed to COUCSI. There followed an increase in the monitored end pressure to a maximum as fluid redistributed within the core. For the tests in which the fine-pored segments were pre-saturated with brine, even without an initial period of co-current imbibition, limited invasion of the main core segment by brine resulted in an asymptotic rise of the end pressure to a maximum as the imbibition front dispersed. To confirm that the dispersing front did not reach the dead end of the core, the distance of advance of the wetting liquid was detected by a series of electrodes. The maximum value of the end pressure provides an estimate of the capillary pressure at an imbibition front for COUCSI. The maximum capillary pressure generated by the invading fluids ranged from 6.6 kPa to 42 kPa for sandstone with permeabilities between 1.050 (μm)² and 0.06 (μm)².     

Permanent Alteration of Porous Media Wettability from Liquid-Wetting to Intermediate Gas-Wetting

Wettability of Berea and low permeability reservoir rocks are permanently altered from liquid-wetting to intermediate gas-wetting. We use water and decane as model liquid, and air and nitrogen as model gas in the experiments. New chemicals with various functional groups are used in the wettability alteration. We perform compositional analyses of the treated chemical solutions extracted from rock treatment by gas chromatography–mass spectrometry (GCMS) and by inductively coupled plasma-mass spectrometry (ICPMS). The analyses demonstrate reaction between the chemicals and the rock substrate. There is no measurable change in permeability from the chemical reaction for the low molecular weight chemicals. The results reveal the permanent alteration of wettability. Tests are conducted to measure contact angle, spontaneous imbibition, and flow to assess the effect of wettability alteration on flow performance as a function of chemical concentration and functionality. For Berea, the contact angle for the water–air–rock is altered from 0° to ~150° depending on the chemical concentration. For the reservoir rock, the contact angle is altered from ~70° to ~130°. As a result of the treatment, the water flow rate may increase two and a half times for a given pressure drop in the Berea. The permanent alteration of wettability with the new chemicals is intended for prevention of water blocking in gas production from tight reservoirs. Instead of hydraulic fracturing when water is introduced in formations with most of the water retained by the water-wet rocks, one may use the new chemical surfactants in fracturing to avoid water retention for high gas well productivity.     

Computational Modeling of Fluid Flow through a Fracture in Permeable Rock

Laminar, single-phase, finite-volume solutions to the Navier–Stokes equations of fluid flow through a fracture within permeable media have been obtained. The fracture geometry was acquired from computed tomography scans of a fracture in Berea sandstone, capturing the small-scale roughness of these natural fluid conduits. First, the roughness of the two-dimensional fracture profiles was analyzed and shown to be similar to Brownian fractal structures. The permeability and tortuosity of each fracture profile was determined from simulations of fluid flow through these geometries with impermeable fracture walls. A surrounding permeable medium, assumed to obey Darcy’s Law with permeabilities from 0.2 to 2,000 millidarcies, was then included in the analysis. A series of simulations for flows in fractured permeable rocks was performed, and the results were used to develop a relationship between the flow rate and pressure loss for fractures in porous rocks. The resulting friction-factor, which accounts for the fracture geometric properties, is similar to the cubic law; it has the potential to be of use in discrete fracture reservoir-scale simulations of fluid flow through highly fractured geologic formations with appreciable matrix permeability. The observed fluid flow from the surrounding permeable medium to the fracture was significant when the resistance within the fracture and the medium were of the same order. An increase in the volumetric flow rate within the fracture profile increased by more than 5% was observed for flows within high permeability-fractured porous media.     

A New Method for Calculating Two-Phase Relative Permeability from Resistivity Data in Porous Media

Many resistivity data from laboratory measurements and well logging are available. Papers on the relationship between resistivity and relative permeability have been few. To this end, a new method was developed to infer two-phase relative permeability from the resistivity data in a consolidated porous medium. It was found that the wetting phase relative permeability is inversely proportional to the resistivity index of a porous medium. The proposed model was verified using the experimental data in different rocks (Berea, Boise sandstone, and limestone) at different temperatures up to 300°F. The results demonstrated that the oil and water relative permeabilities calculated from the experimental resistivity data by using the model proposed in this article were close to those calculated from the capillary pressure data in the rock samples with different porosities and permeabilities. The results demonstrated that the proposed approach to calculating two-phase relative permeability from resistivity data works satisfactorily in the cases studied.     

Correction to: Time and Space Fractional Diffusion in Finite Systems

Additive manufacturing technology, or 3D printing, with silica sand has enabled the manufacture of porous rock analogues for the use in experimental studies of geomechanical properties of reservoir rocks. The accurate modelling of the fluid flow phenomena within a reservoir and improving the performance of hydrocarbon recovery require an understanding of physical and chemical interactions of the reservoir fluids and the rock matrix. Therefore, for the 3D printed samples to serve as rock analogues, flow properties have to be equivalent to the petrophysical properties of their natural counterparts, such as Berea sandstone. In this study, sandstones that were 3D printed with silica sand and Poly-Furfuryl alcohol (PFA) binder, were used to investigate interactions between porous media with different fluids. Wettability preference of 3D printed samples was characterized through contact angle measurements, as well as co-current and counter-current spontaneous imbibition experiments. Results indicated that 3D printed sandstones had mixed-wet characteristics due to the high preference of silica grains for polar fluids and the affinity PFA binder to the oleic phase. Printing configurations including binder saturation were found to greatly influence the wettability preference of the 3D printed analogue rocks as higher PFA concentrations resulted in more strongly oil-wet preferences. Efforts to optimize the printing process and challenges to control the wettability preferences of the 3D printed samples are also highlighted.

     

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.     

Efficiency of diffusion controlled miscible displacement in fractured porous media

Experiments were performed to study the diffusion process between matrix and fracture while there is flow in fracture. 2-inch diameter and 6-inch length Berea sandstone and Indiana limestone samples were cut cylindrically. An artificial fracture spanning between injection and production ends was created and the sample was coated with heat-shrinkable teflon tube. A miscible solvent (heptane) was injected from one end of the core saturated with oil at a constant rate. The effects of (a) oil type (mineral oil and kerosene), (b) injection rates, (c) orientation of the core, (d) matrix wettability, (e) core type (a sandstone and a limestone), and (f) amount of water in matrix on the oil recovery performance were examined. The process efficiency in terms of the time required for the recovery as well as the amount of solvent injected was also investigated. It is expected that the experimental results will be useful in deriving the matrix–fracture transfer function by diffusion that is controlled by the flow rate, matrix and fluid properties.     

Relative Permeability Calculations from Two-Phase Flow Simulations Directly on Digital Images of Porous Rocks

We present results from a systematic study of relative permeability functions derived from two-phase lattice Boltzmann (LB) simulations on X-ray microtomography pore space images of Bentheimer and Berea sandstone. The simulations mimic both unsteady- and steady-state experiments for measuring relative permeability. For steady-state flow, we reproduce drainage and imbibition relative permeability curves that are in good agreement with available experimental steady-state data. Relative permeabilities from unsteady-state displacements are derived by explicit calculations using the Johnson, Bossler and Naumann method with input from simulated production and pressure profiles. We find that the nonwetting phase relative permeability for drainage is over-predicted compared to the steady-state data. This is due to transient dynamic effects causing viscous instabilities. Thus, the calculated unsteady-state relative permeabilities for the drainage is fundamentally different from the steady-state situation where transient effects have vanished. These effects have a larger impact on the invading nonwetting fluid than the defending wetting fluid. Unsteady-state imbibition relative permeabilities are comparable to the steady-state ones. However, the appearance of a piston-like front disguises most of the displacement and data can only be determined for a restricted range of saturations. Relative permeabilities derived from unsteady-state displacements exhibit clear rate effects, and residual saturations depend strongly on the capillary number. We conclude that the LB method can provide a versatile tool to compute multiphase flow properties from pore space images and to explore the effects of imposed flow and fluid conditions on these properties. Also, dynamic effects are properly captured by the method, giving the opportunity to examine differences between steady and unsteady-state setups.     

多孔介质自发渗吸研究进展

自发渗吸是发生在多孔介质里的一种常见自然现象, 存在于众多工程应用和自然科学领域, 多孔介质 中自发渗吸的基本静力学和动力学问题已成为当前国际研究的热点课题之一. 本文综述了传统理论研究中的 Lucas{Washburn(LW) 模型, Terzaghi 模型, Handy 模型, Mattax 和Kyte 无因次时间标度模型, Aronofsky 归 一化采收率标度模型以及近十年最新研究进展, 分析了渗吸机理判别参数研究, 简述了数值模拟研究及渗吸率 影响机理的实验研究现状, 总结了基于分形理论研究多孔介质自发渗吸的最新进展, 并展望了多孔介质以及裂 缝性双重多孔介质中牛顿流体和非牛顿流体自发渗吸研究的方向和课题.      

Modeling the elasto-acoustic hysteretic nonlinearity of dry Berea sandstone

By means of the dynamic acousto-elastic technique the instantaneous state of a material is probed by a high-frequency pulse while an external, low-frequency dynamic source modulates it in time. When it is applied to rocks, a remarkable variety of responses in the form of hysteresis of the material’s modulus defect is obtained. Among them, the response by Berea sandstone is perhaps the most investigated. Hysteresis of the modulus defect in dry Berea sandstone shows intriguing features which have not yet been interpreted in terms of specific mechanisms occurring at atomic or meso-scale. In the present communication, three rheological models providing simplified representations of realistic mechanisms are considered. They involve dislocations interacting with point defects, and microcracks with finite stiffness in compression. Constitutive relations linking macroscopic anelastic strain and stress are derived from them together with the modulus defect associated to each model. These relationships are employed to construct a composite model capable of reproducing several of the main features observed in the experimental data. The limitations of the present approach, the validity of its physical interpretation, and, possibly, of the current implementation of the dynamic acousto-elastic technique are also discussed.     

Nonequilibrium Effects During Spontaneous Imbibition

Accurate models of multiphase flow in porous media and predictions of oil recovery require a thorough understanding of the physics of fluid flow. Current simulators assume, generally, that local capillary equilibrium is reached instantaneously during any flow mode. Consequently, capillary pressure and relative permeability curves are functions solely of water saturation. In the case of imbibition, the assumption of instantaneous local capillary equilibrium allows the balance equations to be cast in the form of a self-similar, diffusion-like problem. Li et al. [J. Petrol. Sci. Eng. 39(3) (2003), 309–326] analyzed oil production data from spontaneous countercurrent imbibition experiments and inferred that they observed the self-similar behavior expected from the mathematical equations. Others (Barenblatt et al. [Soc. Petrol. Eng. J. 8(4) (2002), 409–416]; Silin and Patzek [Transport in Porous Media 54 (2004), 297–322]) assert that local equilibirum is not reached in porous media during spontaneous imbibition and nonequilibirium effects should be taken into account. Simulations and definitive experiments are conducted at core scale in this work to reveal unequivocally nonequilbirium effects. Experimental in-situ saturation data obtained with a computerized tomography scanner illustrate significant deviation from the numerical local-equilibrium based results. The data indicates: (i) capillary imbibition is an inherently nonequilibrium process and (ii) the traditional, multi-phase, reservoir simulation equations may not well represent the true physics of the process.