Pore Network Modelling of Electrical Resistivity and Capillary Pressure Characteristics

In order to model petrophysical properties of hydrocarbon reservoir rocks, the underlying physics occurring in realistic rock pore structures must be captured. Experimental evidence showing variations of wetting occurring within a pore, and existence of the so-called 'non-Archie' behaviour, has led to numerical models using pore shapes with crevices (for example, square, elliptic, star-like shapes, etc.). This paper presents theoretical derivations and simulation results of a new pore space network model for the prediction of petrophysical properties of reservoir rocks. The effects of key pore geometrical factors such as pore shape, pore size distribution and pore co-ordination number (pore connectivity) have been incorporated into the theoretical model. In particular, the model is used to investigate the effects of wettability and saturation history on electrical resistivity and capillary pressure characteristics. The petrophysical characteristics were simulated for reservoir rock samples. The use of the more realistic grain boundary pore (GBP) shape allows simulation of the generic behaviour of sandstone rocks, with various wetting scenarios. The predictions are in close agreement with electrical resistivity and capillary pressure characteristics observed in experiments.     

Pore Network Analysis of Resistivity Index for Water-Wet Porous Media

Pore network analysis is used to investigate the effects of microscopic parameters of the pore structure such as pore geometry, pore-size distribution, pore space topology and fractal roughness porosity on resistivity index curves of strongly water-wet porous media. The pore structure is represented by a three-dimensional network of lamellar capillary tubes with fractal roughness features along their pore-walls. Oil-water drainage (conventional porous plate method) is simulated with a bond percolation-and-fractal roughness model without trapping of wetting fluid. The resistivity index, saturation exponent and capillary pressure are expressed as approximate functions of the pore network parameters by adopting some simplifying assumptions and using effective medium approximation, universal scaling laws of percolation theory and fractal geometry. Some new phenomenological models of resistivity index curves of porous media are derived. Finally, the eventual changes of resistivity index caused by the permanent entrapment of wetting fluid in the pore network are also studied.Resistivity index and saturation exponent are decreasing functions of the degree of correlation between pore volume and pore size as well as the width of the pore size distribution, whereas they are independent on the mean pore size. At low water saturations, the saturation exponent decreases or increases for pore systems of low or high fractal roughness porosity respectively, and obtains finite values only when the wetting fluid is not trapped in the pore network. The dependence of saturation exponent on water saturation weakens for strong correlation between pore volume and pore size, high network connectivity, medium pore-wall roughness porosity and medium width of the pore size distribution. The resistivity index can be described succesfully by generalized 3-parameter power functions of water saturation where the parameter values are related closely with the geometrical, topological and fractal properties of the pore structure.     

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.     

Three-Phase Capillary Pressure and Relative Permeability Relationships in Mixed-Wet Systems

A simple process-based model of three-phase displacement cycles for both spreading and non-spreading oils in a mixed-wet capillary bundle model is presented. All possible pore filling sequences are determined analytically and it is found that the number of pore occupancies that are permitted on physical grounds is actually quite restricted. For typical non-spreading gas/oil/water systems, only two important cases need to be considered to see all types of allowed qualitative behaviour for non-spreading oils. These two cases correspond to whether water or gas is the intermediate-wetting phase in oil-wet pores as determined by the corresponding contact angles, that is, cos ^o _gw > 0 or cos ^o _gw < 0, respectively. Analysis of the derived pore occupancies leads to the establishment of a number of relationships showing the phase dependencies of three-phase capillary pressures and relative permeabilities in mixed-wet systems. It is shown that different relationships hold in different regions of the ternary diagram and the morphology of these regions is discussed in terms of various rock/fluid properties. Up to three distinct phase-dependency regions may appear for a non-spreading oil and this reduces to two for a spreading oil. In each region, we find that only one phase may be specified as being the intermediate-wetting phase and it is only the relative permeability of this phase and the capillary pressure between the two remaining phases that depend upon more than one saturation. Given the simplicity of the model, a remarkable variety of behaviour is predicted. Moreover, the emergent three-phase saturation-dependency regions developed in this paper should prove useful in: (a) guiding improved empirical approaches of how two-phase data should be combined to obtain the corresponding three-phase capillary pressures and relative permeabilities; and (b) determining particular displacement sequences that require additional investigation using a more complete process-based 3D pore-scale network model.     

Effective Relative Permeabilities and Capillary Pressure for One-Dimensional Heterogeneous Media

The paper presents an analytical construction of effective two-phase parameters for one-dimensional heterogeneous porous media, and studies their properties. We base the computation of effective parameters on analytical solutions for steady-state saturation distributions. Special care has to be taken with respect to saturation and pressure discontinuities at the interface between different rocks. The ensuing effective relative permeabilities and effective capillary pressure will be functions of rate, flow direction, fluid viscosities, and spatial scale of the heterogeneities.The applicability of the effective parameters in dynamic displacement situations is studied by comparing fine-gridded simulations in heterogeneous media with simulations in their homogeneous (effective) counterparts. Performance is quite satisfactory, even with strong fronts present. Also, we report computations studying the applicability of capillary limit parameters outside the strict limit.     

Immiscible Displacement in the Interacting Capillary Bundle Model Part I. Development of Interacting Capillary Bundle Model

An interacting capillary bundle model is developed for analysing immiscible displacement processes in porous media. In this model, pressure equilibration among the capillaries is stipulated and capillary forces are included. This feature makes the model entirely different from the traditional tube bundle models in which fluids in different capillaries are independent of each other. In this work, displacements of a non-wetting phase by a wetting phase at different injection rates were analysed using the interacting capillary bundle model. The predicted evolutions of saturation profiles were consistent with both numerical simulation and experimental results for porous media reported in literature which cannot be re-produced with the non-interacting tube bundle models.     

An Integral Method to Calculate Water Saturation in a Centrifuge Experiment to Determine Capillary Pressure

The centrifuge method is commonly used to determine the capillary pressure of a porous medium, and the original approximating method for data analysis developed by Hassler and Brunner is still being used. Its limitations are, however, not well understood. Application to analyze experiments where one of the assumptions was obviously violated had been given in the literature. While the result appeared to be quite reasonable, it was not clear how close was it to reality. One of the objectives of this paper is to review the assumptions that is required to develop this method, so that the experimental condition in which it is applicable can be established. The other objective is to derive a completely different solution technique to this problem. There is no need to assume that the ratio of the inlet radius to the exit radius of the core to the center of the centrifuge be close to 1. With the freedom from this limitation it is, therefore, possible to construct machines at lower cost and to improve on the data quality by allowing longer cores.     

A Markov Chain Model for Effective Relative Permeabilities and Capillary Pressure

We present a new method for calculating the effective two-phase parameters of one-dimensional randomly heterogeneous porous media, which avoids the timeconsuming use of simulations on explicit realizations. The procedure is based on the steady state saturation distribution. The idea is to model the local variation of saturation and saturation dependent parameters as Markov chains, in such a way that the effective parameters are given by the asymptotic expectations of the chains. We derive the exact asymptotic moment equations and solve them numerically, based on their second order approximation. The method determines the effective parameters to a high degree of accuracy, even with large variations in rock properties. In particular, the capillary limit and viscous limit effective parameters are recovered exactly. The applicability of the effective parameters in the unsteady state case is studied by comparing the displacement production profiles in heterogeneous media and their homogenized counterpart.     

Relative Permeability Analysis of Tube Bundle Models,Including Capillary Pressure

The analytical equations for calculating two-phase flow, including local capillary pressures, are developed for the bundle of parallel capillary tubes model. The flow equations that are derived were used to calculate dynamic immiscible displacements of oil by water under the constraint of a constant overall pressure drop across the tube bundle. Expressions for averaged fluid pressure gradients and total flow rates are developed, and relative permeabilities are calculated directly from the two-phase form of Darcy's law. The effects of pressure drop and viscosity ratio on the relative permeabilities are discussed. Capillary pressure as a function of water saturation was delineated for several cases and compared to a steady-state mercury-injection drainage type of capillary pressure profile. The bundle of serial tubes model (a model containing tubes whose diameters change randomly at periodic intervals along the direction of flow), including local Young-Laplace capillary pressures, was analyzed with respect to obtaining relative permeabilities and macroscopic capillary pressures. Relative permeabilities for the bundle of parallel tubes model were seen to be significantly affected by altering the overall pressure drop and the viscosity ratio; relative permeabilities for the bundle of serial tubes were seen to be relatively insensitive to viscosity ratio and pressure, and were consistently X-like in profile. This work also considers the standard Leverett (1941) type of capillary pressure versus saturation profile, where drainage of a wetting phase is completed in a step-wise steady fashion; it was delineated for both tube bundle models. Although the expected increase in capillary pressure at low wetting-phase saturation was produced, comparison of the primary-drainage capillary pressure curves with the pseudo-capillary pressure profiles, that are computed directly using the averaged pressures during the displacements, revealed inconsistencies between the two definitions of capillary pressure.     

Immiscible Displacement in the Interacting Capillary Bundle Model Part II. Applications of Model and Comparison of Interacting and Non-Interacting Capillary Bundle Models

In the first part of this work (Dong et al., Transport Porous Media, 59, 1–18, 2005), an interacting capillary bundle model was developed for analysing immiscible displacement processes in porous media. In this paper, the second part of the work, the model is applied to analyse the fluid dynamics of immiscible displacements. The analysis includes: (1) free spontaneous imbibition, (2) the effects of injection rate and oil–water viscosity ratio on the displacement interface profile, and (3) the effect of oil–water viscosity ratio on the relative permeability curves. Analysis of a non-interacting tube bundle model is also presented for comparison. Because pressure equilibration between the capillaries is stipulated in the interacting capillary model, it is able to reproduce the behaviour of immiscible displacement observed in porous media which cannot be modelled by using non-interacting tube bundle models.     

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.     

脑血流与颅内压的仿真模型研究

颅内压（intracranial pressure,ICP）是指颅腔内容物对颅腔壁产生的压力,内外多种原因会引起颅内压变化,同时心搏、呼吸和神经调节等的影响也会使颅内压出现波动。本研究在动物（犬）实验结果的基础上,建立了模拟颅内压动力学变化的集中参数数学模型。模型中整合了颅内压和脑血流以及自调节机制之间的非线性相互作用。模型的仿真结果与实验结果十分接近,较好地模拟了实验中颅内压波形以及颅内代偿功能接近耗尽时产生的高原波。模拟中模型参数的变化和动物实验状况的变化相吻合,这将为临床颅内压监护及诊疗提供一定的参考。     

腰区脑脊液压预估颅内压的数学模拟

颅内压（Intracranial Pressure，ICP）的实时监测临床上有着十分重要的意义。临床测量颅内压的方法是通过穿刺小脑延髓池或侧脑室用压力管测量，然而此方法存在一定的风险。由于正常成人的颅腔和脊髓腔是一个连通的容器，因此腰区脑脊液压（1umbarcerebrospinal fluid pressure，LCP）和颅内压存在一定联系，检测腰区脑脊液压来预估颅内压成为临床的需要。本研究在动物（犬）实验结果的基础上，建立了反映颅内压变化的集中参数模型，模型中整合了腰区脑脊液压与颅内压和脑血流的相互作用，数值模拟了颅内压。结果显示，数值模拟得到的ICP曲线与动物实验检测的ICP曲线经秩和检验无显著性差异，模型参数的变化和动物实验状况的变化相吻合，可以为临床颅内压监测及诊疗提供一定的参考。     

Effects of Organic Base Chemistry on Interfacial Tension,Wettability, and Capillary Pressure in Multiphase Subsurface Waste Systems

The presence of surfactants may have profound effects on the transport of organic contaminants in multiphase systems. It is a common practice, however, to model the subsurface migration of liquids independently of the aqueous phase composition. As such, transport in these systems may not be adequately characterized. This study investigates the impact of pH on interfacial tension, wettability, and the drainage capillary pressure–saturation relationship in air–water–quartz and oxylene–water–quartz systems containing dodecylamine, an organic base. In these systems, three mechanisms, speciation, partitioning, and sorption, are important in determining the interfacial tension and contact angle, and consequently, important in determining the capillary pressure. By adjusting the pH above and below, the pK_a of the base, the relative importance of these mechanisms was altered. Below dodecylamine's pK_a of 10.6, the base was primarily in a cationic form resulting in minimal partitioning into the nonaqueous liquid and greater sorption at the quartz surface. Above the pK_a, the base was primarily in a neutral form which did not sorb to the quartz, and, furthermore, partitioned into the organic liquid phase where its surface activity was minimized. The combination of these processes caused the capillary pressure to change in a manner consistent with porescale theory of capillarity. The utility in this approach lies in the possibility of predicting transport properties in multiphase systems while incorporating the direct effects of solution chemistry.     

Discussion of Techien Chen, 1997, ‘An Integral Method to Calculate Water Saturation in a Centrifuge Experiment to Determine Capillary Pressure’

That discussion recognizes that the above paper deals with one of the main problems for reserves and reservoir evaluation, i.e. the determination of capillary pressure curves using the centrifuge technique. However, it is explained that the subject was not properly addressed and that sections of the paper are out of date, confusing or eventually wrong. For a more relevant overview on the subject, it is recommended to refer to the corresponding survey by the Society of Core Analysts.     

A Theoretical Model of Hysteresis and Dynamic Effects in the Capillary Relation for Two-phase Flow in Porous Media

It is well known that the relationship between capillary pressure and saturation, in two-phase flow problems demonstrates memory effects and, in particular, hysteresis. Explicit representation of full hysteresis with a myriad of scanning curves in models of multiphase flow has been a difficult problem. A second complication relates to the fact that P ^c–S relationships, determined under static conditions, are not necessarily valid in dynamics. There exist P ^c–S relationships which take into account dynamic effects. But the combination of hysteretic and dynamic effects in the capillary relationship has not been considered yet. In this paper, we have developed new models of capillary hysteresis which also include dynamic effects. In doing so, thermodynamic considerations are employed to ensure the admissibility of the new relationships. The simplest model is constructed around main imbibition and drainage curves and assumes that all scanning curves are vertical lines. The dynamic effect is taken into account by introducing a damping coefficient in P ^c–S equation. A second-order model of hysteresis with inclined scanning curves is also developed. The simplest version of proposed models is applied to two-phase incompressible flow and an example problem is solved.     

Problem of Capillary Displacement for One Model of Three-Phase Filtration

The class of capillary pressures corresponding to a triangular tensor of capillary diffusion in a three-phase fluid is studied. Filtration with such a tensor is described by a parabolic system of equations degenerating on solutions. This system is integrodifferential because the desired quantities are the total flow rate and the phase-saturation distribution under conditions of a specified pressure drop on the boundaries of the flow region in one of the phases. It is shown that in the problem of capillary displacement, the degenerate system can be studied using a special maximum principle.     

Petrophysical and Capillary Properties of Compacted Salt

This paper reports experimental results that demonstrate petrophysical and capillary characteristics of compacted salt. The measured data include porosity, gas permeability, pore size distribution, specific surface area, and gas-brine breakthrough and capillary pressure. Salt samples employed in the experiments were prepared by compacting sodium chloride granulates at high stresses for several hours. They represent an intermediate consolidation stage of crushed salt under in-situ conditions. The porosity and permeability of compacted salt showed similar trends to those expected in backfilled regions of waste repositories excavated in salt rock. The correlation between the measured porosity and permeability seems to be independent of the compaction parameters for the range examined in this study. The correlation also shows a different behaviour from that of rock salt. The data of all petrophysical properties show that the pore structure of compacted salt can be better characterized by fracture permeability models rather than capillary bundle ones. Simple creep tests, conducted on the fully-brine-saturated compacted salt samples, yielded similar strain rates to those obtained by a steady-state mechanical model developed from the tests on fully brine-saturated granular salt. A modified procedure is proposed for the evaluation of restored-state capillary pressure data influenced by the material creep. The characteristic parameters for the capillary behaviour of compacted salt are determined by matching the Brooks-Corey and van Genuchten models with the measured data. The Leverett functions determined with different methods agree well.     

Insights into the Relationships Among Capillary Pressure,Saturation, Interfacial Area and Relative Permeability Using Pore-Network Modeling

To gain insight in relationships among capillary pressure, interfacial area, saturation, and relative permeability in two-phase flow in porous media, we have developed two types of pore-network models. The first one, called tube model, has only one element type, namely pore throats. The second one is a sphere-and-tube model with both pore bodies and pore throats. We have shown that the two models produce distinctly different curves for capillary pressure and relative permeability. In particular, we find that the tube model cannot reproduce hysteresis. We have investigated some basic issues such as effect of network size, network dimension, and different trapping assumptions in the two networks. We have also obtained curves of fluid–fluid interfacial area versus saturation. We show that the trend of relationship between interfacial area and saturation is largely influenced by trapping assumptions. Through simulating primary and scanning drainage and imbibition cycles, we have generated two surfaces fitted to capillary pressure, saturation, and interfacial area (P _c –S _w –a _nw ) points as well as to relative permeability, saturation, and interfacial area (k _r –S _w –a _nw ) points. The two fitted three-dimensional surfaces show very good correlation with the data points. We have fitted two different surfaces to P _c –S _w –a _nw points for drainage and imbibition separately. The two surfaces do not completely coincide. But, their mean absolute difference decreases with increasing overlap in the statistical distributions of pore bodies and pore throats. We have shown that interfacial area can be considered as an essential variable for diminishing or eliminating the hysteresis observed in capillary pressure–saturation (P _c –S _w ) and the relative permeability–saturation (k _r –S _w ) curves.