The microscopic analysis of high forchheimer number flow in porous media

High Forchheimer number flow through a rigid porous medium is numerically analysed by means of the volumetric averaging concept. The microscopic flow mechanisms, which must be known in order to understand the macroscopic flow phenomena, are studied by utilising a periodic diverging-converging representative unit cell (RUC). The detailed information for the microscopic flow field, in association with the locally averaged momentum balance, makes it possible to quantitatively demonstrate that the microscopic inertial phenomenon, which leads to distorted velocity and pressure fields, is the fundamental reason for the onset of nonlinear (non-Darcy) effects as velocity increases. The hydrodynamic definitions for Darcy's law permeabilityk, the inertial coefficient and Forchheimer number Fo are obtained by applying the averaging theorem to the pore level Navier-Stokes equations. Finally, these macroscopic parameters are numerically calculated at various combinations of micro-geometry and flow rate, and graphically correlated with the relevant microscopic parameters.Nomenclature a _i body force acceleration (m/s²) - A viscous integral term defined in (4.6) - A _f area of entrance and exist of RUC (m²) - A _fs interfacial area between the fluid and solid phases (m²) - B pressure integral term defined in (4.4) - d throat diameter of RUC (m) - D pore diameter of RUC (m) - Fo Forchheimer number defined in (4.1) and (4.10) - g gravitational acceleration (m/s²) - i, j microscopic unit vector for RUC - k Darcy's law permeability (m²) - k _v velocity dependent permeability defined in (4.1) (m²) - L length of a unit cell (m) - L _p pore length of RUC (m) - L _t throat length of RUC (m) - n unit outwardly directed vector for the fluid phase - p microscopic fluid pressure (N/m²) - P macroscopic fluid pressure (N/m²) - _en mean pressure at entrance of RUC (N/m²) - _ex mean pressure at exit of RUC (N/m²) - r _i,r coordinate on the macroscopic scale (m) - Re _d Reynolds number defined in (4.5) - u _i,u microscopic velocity (m/s) - specific discharge (m/s) - _d mean velocity at the throat of RUC (m/s) - v microscopic velocity (m/s) - V _b representative elementary volume (REV) (m³) - V _f volume occupied by the fluid within REV (m³) - V _s volume occupied by the solid within REV (m³) - x _i,x coordinate on the microscopic scale (m) - X _i,X coordinate on the macroscopic scale (m) Greek the inertia coefficient (1/m) - viscosity coefficient (Ns/m²) - _i microscopic unit vector - areosity at the entrance and the exit cross-section of RUC - fluid density (kg/m³) - porosity - _f a general property of the fluid phase Symbols ^f intrinsic phase average - the fluctuating part of _f - the mean value of _f - _f ^* the dimensionless value of _f     

Experimental investigation of capillary pressure effects on immiscible displacement in lensed and layered porous media

This paper reports the results of extensive experimental studies of the effects of well-defined heterogeneous porous media on immiscible flooding. The heterogeneities were layers and lenses, with some of the lenses being a wettability contrast. Drainage and imbibition displacements, with and without an initial residual fluid saturation, were carried out at a variety of flow rates on layered and lensed two-dimensional glass beads models of the size of a typical large core test (58×10×0.6 cm). These displacements were followed photographically and the effluent saturation profiles recorded. In most of the experiments the glass beads were water-wet, but in some the lens beads were coated with a water repellent chemical. In all experiments, the displacement fronts became highly irregular due to the different capillary pressures acting in the different areas of the models. In this paper, these displacements are fully reported and their implications for reservoir simulation and for interpretation of laboratory core tests, where the inner heterogeneities are not known, are discussed.     

Review on Scale Dependent Characterization of the Microstructure of Porous Media

The paper discusses local porosity theory and its relation with other geometric characterization methods for porous media such as correlation functions and contact distributions. Special emphasis is placed on the charcterization of geometric observables through Hadwigers theorem in stochastic geometry. The four basic Minkowski functionals are introduced into local porosity theory, and for the first time a relationship is established between the Euler characteristic and the local percolation probabilities. Local porosity distributions and local percolation probabilities provide a scale dependent characterization of the microstructure of porous media that can be used in an effective medium approach to predict transport.     

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 H.B.Jayaraman在20世纪80年代推导的悬链线索元有限元 法计算精度高，特别适用于精度要求比较高的大型索结构. 但是， 当索原长Lu的取值与悬索两节点之间的直线长度相近时，迭 代不易收敛，甚至发散. 提出了当该迭代不收敛时，应采用的迭 代策略. 计算结果表明，该方法准确，计算精度高，可供悬索结 构设计、施工时参考.     

Effect of flows between formations and capillary forces on the process of oil displacement in a stratified inhomogeneous bed

The results of a numerical investigation of the process of oil displacement in a stratified inhomogeneous formation on the basis of the two-phase flow model with account for capillary forces are presented. It is shown that in many cases the vertical inhomogeneity of oil reservoirs may not be a cause of nonuniform displacement and the non-recovery of large oil reserves by the time of water breakthrough to the extraction surface. The action of the capillary forces is an additional factor leading to equalization of the water propagation front in the inhomogeneous formation, water breakthrough delay, and intensification of the mass transfer between the layers with different permeabilities. Analysis of the contribution of the interlayer flows to the water flooding of low-permeability formation intervals calls into question the practicability of blocking high-permeability inclusions in the neighborhood of pumping wells.     

Relative permeabilities of a near critical binary fluid

Relative permeabilities were measured at very low interfacial tensions (IFT) for two-phase mixtures of methanol and hexane flowing through Clashach sandstone. These two components pass from a two- to a single-phase system as the temperature is increased above the critical solution temperature (CST). The interfacial tension between the coexisting phases approaches zero as the solution reaches miscibility. The phase behaviour of methanol and hexane mixtures has been well characterised allowing the calculation of relative permeabilities, saturations and capillary numbers. Flow data are reported for four different temperatures in the two-phase region (i.e., four values of IFT and capillary number). The capillary desaturation curve (CDC) for the strongly wetting methanol rich phase is also presented. In addition to the novel technique presented for measurement of relative permeability, the results indicate that relative permeabilities approach straight line functions very near the critical point. Furthermore, desaturation of the wetting phase was found to be dependent on the capillary number which, in turn, depends on the location of the mixture on the fluid phase diagram and the proximity to the critical temperature.     

剧变截面圆管内渗流的数值计算方法

对于剧变截面圆管的渗流问题写出不可压缩渗流的基本方程组,对直接求解原始变量(速度和压力)的数值计算方法作出改进。先由非主流方向的运动方程计算压力,后由主流方向的运动方程计算主流方向的速度分量,再由连续性方程计算非主流方向的速度分量。这样可以避免在一般的求解原始变量方法中由连续性方程计算压力时出现的困难和麻烦。根据本方法和剧变截面圆管的特点,采用半交错不等距非正交贴体混合网格系。本文详细写出差分方程和迭代计算公式,对剧变截面圆管内的渗流算例进行数值计算。本方法的优点是简单和实用,在工程上具有较大的应用价值。     

Thermodynamic Automaton Simulations of Fluid Flow and Diffusion in Porous Media

A thermodynamic automaton model of fluid flow in porous media is presented. The model is a nonrelativistic version of a Lorentz invariant lattice gas model constructed by Udey et al. (1998). In the previous model it was shown that the energy momentum tensor and the relativistic Boltzman equation can be rigorously derived from the collision and propagation rules. In the present paper we demonstrate that this nonrelativistic model can be used to accurately simulate well known results involving single phase flow and diffusion in porous media. The simulation results show that (1) one-phase flow simulations in porous media are consistent with Darcy's law; (2) the apparent diffusion coefficient decreases with a decrease in permeability; (3) small scale heterogeneity does not affect diffusion significantly in the cases considered.     

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.     

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.     

Flow behaviour in the presence of wettability heterogeneities

The physical processes occurring during fluid flow and displacement within porous media having wettability heterogeneities have been investigated in specially designed heterogeneous visual models. The models were packed with glass beads, areas of which were treated with a water repellent to create wettability variations. Immiscible displacement experiments show visually the effect of wettability heterogeneities on the formation of residual oil and recovery due to capillary trapping. This work demonstrates by experiment the importance of incorporating reservoir heterogeneity into pore displacement analysis, essential for the correct interpretation of core data and for directing the route for scale-up of the processes to reservoir scale.     

A Scanning Electron Microscopy study of water in soil

Cold stage Scanning Electron Microscopy (SEM) with a rapid cooling technique makes it possible to investigate the water phase within unsaturated porous media. It is thought that this technique preserves the main features of the micromorphology of the water menisci as it exists in the liquid phase in soils. Saddle-shaped elements, as well as pendular rings of water, were observed with concave and convex curvatures of the water-air interface. The hydraulic conductivity of an unsaturated soil may be inferred from SEM photographs. Observations of isolated water menisci indicate the existence of an immobile water domain. The surface geometry of the water menisci was analyzed quantitatively and surface tension and capillary pressure were determined.     

Pore Scale Modeling of Rate Effects in Imbibition

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

Two-phase flow in heterogeneous porous media: The method of large-scale averaging

The analysis of two-phase flow in porous media begins with the Stokes equations and an appropriate set of boundary conditions. Local volume averaging can then be used to produce the well known extension of Darcy's law for two-phase flow. In addition, a method of closure exists that can be used to predict the individual permeability tensors for each phase. For a heterogeneous porous medium, the local volume average closure problem becomes exceedingly complex and an alternate theoretical resolution of the problem is necessary. This is provided by the method of large-scale averaging which is used to average the Darcy-scale equations over a region that is large compared to the length scale of the heterogeneities. In this paper we present the derivation of the large-scale averaged continuity and momentum equations, and we develop a method of closure that can be used to predict the large-scale permeability tensors and the large-scale capillary pressure. The closure problem is limited by the principle of local mechanical equilibrium. This means that the local fluid distribution is determined by capillary pressure-saturation relations and is not constrained by the solution of an evolutionary transport equation. Special attention is given to the fact that both fluids can be trapped in regions where the saturation is equal to the irreducible saturation, in addition to being trapped in regions where the saturation is greater than the irreducible saturation. Theoretical results are given for stratified porous media and a two-dimensional model for a heterogeneous porous medium.     

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.     

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.     

Transport properties of heterogeneous materials. Combining computerised X-ray micro-tomography and direct numerical simulations

Feasibility of a method for finding flow permeability of porous materials, based on combining computerised X-ray micro-tomography and numerical simulations, is assessed. The permeability is found by solving fluid flow through the complex 3D pore structures obtained by tomography for actual material samples. We estimate overall accuracy of the method and compare numerical and experimental results. Factors contributing to uncertainty of the method include numerical error arising from the finite resolution of tomographic images and the rather small sample size available with the present tomographic techniques. The total uncertainty of computed values of permeability is, however, not essentially larger than that of experimental results. We conclude that the method provides a feasible alternative for finding fluid flow properties of the kind of materials studied. It can be used to estimate all components of permeability tensor and is useful in cases where direct measurements are not achievable. Analogous methods can be applied to other modes of transport, such as diffusion and heat conduction.     

三维非均匀不稳定渗流方程的二维不均匀粗化算法

在无源汇条件下,根据流过某一个横截面的流体流量等于流过这一横截面内所有精细网格的流体流量之和这一特点提出了粗化网格等效渗透率的计算方法。在粗化区内,利用直接解法求解二维渗流方程,再用这些解合成粗化网格的三维合成解,并由合成解计算粗化网格的等效渗透率。根据精度的要求采用了不均匀网格粗化,在流体流速大的区域采用精细网格。利用所得等效渗透率计算了粗化网格的某三维非均匀不稳定渗流场的压降解,结果表明三维非均匀不稳定渗流方程的二维不均匀粗化解非常逼近采用精细网格的解,但计算的速度比采用精细网格提高了80倍。