Primary Drainage Curve Research Articles

Construction of pore network models for Berea and Fontainebleau sandstones using non-linear programing and optimization techniques

Pore network models (PNM) of Berea and Fontainebleau sandstones were constructed using nonlinear programming (NLP) and optimization methods. The constructed PNMs are considered as a digital representation of the rock samples which were based on matching the macroscopic properties of the porous media and used to conduct fluid transport simulations including single and two-phase flow. The PNMs consisted of cubic networks of randomly distributed pores and throats sizes and with various connectivity levels. The networks were optimized such that the upper and lower bounds of the pore sizes are determined using the capillary tube bundle model and the Nelder–Mead method instead of guessing them, which reduces the optimization computational time significantly. An open-source PNM framework was employed to conduct transport and percolation simulations such as invasion percolation and Darcian flow. The PNM model was subsequently used to compute the macroscopic properties; porosity, absolute permeability, specific surface area, breakthrough capillary pressure, and primary drainage curve. The pore networks were optimized to allow for the simulation results of the macroscopic properties to be in excellent agreement with the experimental measurements. This study demonstrates that non-linear programming and optimization methods provide a promising method for pore network modeling when computed tomography imaging may not be readily available.

Hysteretic trapping and relative permeability of CO2 in sandstone at reservoir conditions

Hysteretic nonwetting phase trapping behavior is predicted to be a significant factor determining long term immobilization of injected CO2. This paper presents results from an experimental investigation of hysteresis in residual trapping and relative permeability of CO2 in a CO2/water system at 50°C and 9MPa in a Berea sandstone core. Carbon dioxide saturation data were collected using the steady-state method in a horizontal core-flooding apparatus with X-ray computed tomography. Three flooding cycles were completed at a constant total volumetric flow rate by incrementally increasing and decreasing the fractional flow rates of supercritical CO2 and water. The fractional flow rates were chosen such that the maximum CO2 saturation of each cycle was greater than that of the previous cycle. During the injection stage of a cycle, CO2 displaced water in a wetting-phase drainage process (water saturation decrease). Following injection, water displaced CO2 in a wetting-phase imbibition process (water saturation increase). The fractional flow of CO2 was then increased during the next cycle. Results showed the CO2 saturations trapped during wetting-phase imbibition increased with the maximum CO2 saturations reached during each cycle. The residually trapped CO2 saturation is a hysteretic function, depending on the maximum saturation reached by the core before imbibition. A linear model with coefficient 0.5 describes the nonwetting trapping relationship. The experimental relative permeability data for CO2 fit a single bounding primary drainage curve while CO2 saturations increased, but followed three different scanning curves during wetting-phase imbibition. The CO2 relative permeability data can be fit by making a minor modification in the Van Genuchten–Burdine relative permeability data to account for hysteresis.

Modeling the Primary Drainage Curve of Prefractal Porous Media

Fractal models for the soil water retention curve have largely ignored hysteresis. Such models generally require that all pores be connected to the atmosphere via a network of similar‐sized or larger pores. In random mass fractals and natural porous media, however, not all pores of a given size class empty during drainage due to incomplete pore connectivity and/or the presence of surrounding smaller pores. A modification of Rieu and Sposito's (1991a) prefractal water retention equation is proposed to accommodate this hysteresis during monotonic drying. The resulting expression is identical in form to the empirical Brooks and Corey (1964) model and contains three physically‐based parameters: the proportion of nondraining pores (Pd), the scale factor (b), and the apparent fractal dimension of the primary drainage curve (Dd), which is related to the underlying mass fractal dimension (D) of the porous medium by Dd = D + [log(Pd)]/[log(b)]. Model testing consisted of fitting the modified equation to previously published water retention data for 30 randomized Sierpinski carpets and seven soils. Error sums of squares ranged from 0.001 to 0.093 for the carpets, and from 0.015 to 0.047 for the soils. In contrast, the unmodified Rieu and Sposito (1991a) equation performed very poorly. Best fit estimates of Dd ranged from 0.295… to 1.640… for the two‐dimensional carpets, and from 2.467… to 2.902… for the three‐dimensional soils. Prediction of D, based on estimates of Dd derived from the primary drainage curve, will require additional research on how to obtain Pd and b. Based on the analytical approach outlined here, it should also be possible to model the primary wetting curve, thus providing a more complete fractal description of soil water hysteresis.

Formulation of Capillary Force Barriers in Moderately Oil-Wet Systems and Their Application to Reservoir Simulation

SummaryAs a common production aspect of the Thamama formation (a carbonate reservoir) in both onshore and offshore Abu Dhabi fields, unexpected early-water breakthrough through specific high-permeability layers without a clearly impermeable layer underneath has been observed in several water-injection schemes. Observed field data such as pulsed neutron capture (PNC) logs indicate the absence of injected water slumping away from wellbores. The concept of capillary force barriers was introduced a decade ago to resolve this issue, in which the role of capillary pressure forces on crossflow in stratified layers is modeled.1–3This paper tries to revisit and fine-tune the concept of capillary force barrier and model hysteresis expected in a moderately oil-wet system. First, some measurements of special core analysis and related interpretations are presented in which the results are analytically formulated by a published methodology to generate saturation functions consistent with hysteresis using an assumption of wettability.4An application of the formulation to numerical reservoir simulation was carried out in a systematic manner because the reservoir-rock-type (RRT) scheme of the model was based on primary-drainage curves that can be fully linked with the generated saturation functions. It is demonstrated on cross sections how small differences in imbibition capillary pressures can affect the water movement across contrasting RRT boundaries in a moderately oil-wet system.The proposed formulation is an effective tool for generalizing saturation functions related to matrix properties in a consistent manner, and it systematically incorporates hysteresis and wettability into the numerical reservoir-simulation model.

Interfacial area measurements for unsaturated flow through a porous medium

Multiphase flow and contaminant transport in porous media are strongly influenced by the presence of fluid‐fluid interfaces. Recent theoretical work based on conservation laws and the second law of thermodynamics has demonstrated the need for quantitative interfacial area information to be incorporated into multiphase flow models. We have used synchrotron based X‐ray microtomography to investigate unsaturated flow through a glass bead column. Fully three‐dimensional images were collected at points on the primary drainage curve and on the secondary imbibition and drainage loops. Analysis of the high‐resolution images (17 micron voxels) allows for computation of interfacial areas and saturation. Corresponding pressure measurements are made during the course of the experiments. Results show the fluid‐fluid interfacial area increasing as saturation decreases, reaching a maximum at saturations ranging from 20 to 35% and then decreasing as the saturation continues to zero. The findings support results of numerical studies reported in the literature.

Computation of the interfacial area for two-fluid porous medium systems

We develop a method to compute interfacial areas from three-dimensional digital representations of multiphase systems. We approximate the interfaces with the isosurface generated by the standard marching-cube algorithm from the discrete phase distribution. We apply this approach to two-fluid pore-scale simulations by (1) simulating a random packing of spheres that obeys the grain-size distribution and porosity of an experimental porous medium system, and (2) using a previously developed pore-morphology-based model in order to predict the phase distribution for a water-wet porous medium that undergoes primary drainage. The predicted primary drainage curve and interfacial areas are in good agreement with the experimental values reported in the literature, where interfacial areas were measured using interfacial tracers. The energy dissipation during Haines jumps is significant: thus, the mechanical work done on the system is not completely converted into surface energy, and interfacial areas may not be deduced from the primary drainage curve.

Measurement and modelling of diffusion, porosity and other pore level characteristics of sandstones

The diffusion coefficients of methane, iso-butane and n-butane through dry and water-saturated sandstones and their dependence on relative molar mass are discussed. The diffusion of iso-butane is anomalously fast in dry sandstones and that of methane is anomalously low in water-saturated sandstones. A three-dimensional computer model of the pores and throats in the sandstone has been developed. Simulations have been made of the sandstone's porosity, surface area, pore and throat size distribution, tortuosity, mercury porosimetry primary drainage curve, average pore connectivity and diffusion coefficient for mixed gaseous alkanes.

Centrifuge Capillary Pressure Curves

Abstract This paper presents a comparison of six methods jar calculating capillary pressure curves from centrifuge data: Hassler and Brunner's iterative method, Hoffman's second method, Ruth and Wong's method (LIM), a new integral method (ElM), and the two parameter estimation techniques proposed by Bentsen and co-workers. Two comparison techniques are used; reinterpretation of synthetic data, using Bentsen's first model as the standard and interpretation of experimental data. The results for the first three methods are shown to be essentially the same, with a marginal advantage shown by the LIM. The EIM is shown to give the best results in some cases, but to completely fail in others. For experimental data, it is shown that Bentsen's second model is preferred over Bentsen's first model. Variations between the models suggest the simultaneous use of more than one method ill order to ensure that data interpretation is correct. Introduction This paper represents a continuation and extension of previous work (Ruth and Wong(1)). The previously reported study examined the accuracy of the common methods used to calculate capillary pressure curves from centrifuge data. The classic method, due to Hassler and Brunner(2), was shown to be inadequate in many instances, while the parameter estimation techniques proposed by Bentsen and co-workers(3,4) were shown to be reliable. A new technique, referred to as the linear interpolation method (LIM), based on approximating the capillary pressure by straight line segments and evaluating the correction integral proposed by Melrose(5) was also shown to give good results. In Reference 1, the iterative and integral methods that exist in the literature were not considered. The primary examples of these methods are the iterative technique proposed by Hassler and Brunner(2) to improve their solutions, and the second method offered by Hoffman (presented in an author's response to comments made by Luffel(6)). The present paper treats these two methods, as well as a new method that is an extension of the LIM described in Reference 1. One of the techniques used to compare the various data analysis methods is to specify type-curves based on the Bentsen equation: Equation (Available In Full Paper) where Pc is the capillary pressure, S is the water saturation, Srw is the irreducible water saturation, and Sro is the residual oil saturation. The b is a curve-shape factor, while for a primary drainage curve, the a represents the threshold capillary pressure. In the current work, this equation is made dimensionless by dividing through with the threshold capillary pressure, and writing it in the form Equation (Available In Full Paper) where the prime denotes a dimensionless capillary pressure, and Be is a "Bentsen type-number", given by b/a. This treatment of the equation provides a convenient method of distinguishing, and studying, different types of capillary pressure curves. Obviously, Be must relate to other properties of the rock, such as pore throat size distribution. Further study of this parameter may lead to new methods of classifying the behaviour of various types of porous material.