Numerical Simulation of Single-Phase and Multiphase Non-Darcy Flow in Porous and Fractured Reservoirs

A numerical method as well as a theoretical study of non-Darcy fluid flow through porous and fractured reservoirs is described. The non-Darcy behavior is handled in a three-dimensional, multiphase flow reservoir simulator, while the model formulation incorporates the Forchheimer equation for describing single-phase or multiphase non-Darcy flow and displacement. The non-Darcy flow through a fractured reservoir is handled using a general dual-continuum approach. The numerical scheme has been verified by comparing its results against those of analytical methods. Numerical solutions are used to obtain some insight into the physics of non-Darcy flow and displacement in reservoirs. In addition, several type curves are provided for well-test analyses of non-Darcy flow to demonstrate a methodology for modeling this type of flow in porous and fractured rocks, including flow in petroleum and geothermal reservoirs.     

Non-Darcy effects on convective boundary layer flow past a semi-infinite vertical plate in saturated porous media

The composite effects of viscosity, porosity, buoyancy parameter, thermal conductivity ratio and non-Darcy effects of Brinkman friction and Forscheimmer quadratic drag on the mixed convection boundary layer flow past a semi-infinite plate in a fully-saturated porous regime are theoretically and numerically investigated using Keller’s implicit finite-difference technique and a double-shooting Runge-Kutta method. The Brinkman Forcheimer-extended Darcy model is implemented in the hydrodynamic boundary layer equation. The effects of the various non-dimensional thermofluid parameters, viz Grashof number, Darcy number, and Forchheimer number, and also porosity, thermal conductivity and viscosity parameters on the velocity and temperature fields are discussed. Computations for both numerical schemes are made where possible and found to be in excellent agreement.     

Network Modeling of Non-Darcy Flow Through Porous Media

Darcy's law is inadequate for describing high-velocity gas flow in porous media, which occurs in the near well-bore region of high capacity gas and condensate reservoirs. This study is directed at understanding the non-Darcy flow behavior. A pore-level network model has been developed to describe high velocity flow. The inputs to the model are pore size distributions and network coordination numbers. The outputs are permeability, non-Darcy coefficient, tortuousity and porosity. The additional pressure gradient term is found to be proportional to the square of the velocity in accordance with the Forchheimer's equation. The correlation between the non-Darcy coefficient and other flow properties (the permeability, the porosity and the tortuousity) is found to depend on the morphological parameters being changed. General correlations are derived between these flow properties.     

Effects of Chemical Reaction and Double Dispersion on Non-Darcy Free Convection Heat and Mass Transfer

In this article, the effects of chemical reaction and double dispersion on non-Darcy free convection heat and mass transfer from semi-infinite, impermeable vertical wall in a fluid saturated porous medium are investigated. The Forchheimer extension (non-Darcy term) is considered in the flow equations, while the chemical reaction power–law term is considered in the concentration equation. The first order chemical reaction (n = 1) was used as an example of calculations. The Darcy and non-Darcy flow, temperature and concentration fields in this study are observed to be governed by complex interactions among dispersion and natural convection mechanisms. The governing set of partial differential equations were non-dimensionalized and reduced to a set of ordinary differential equations for which Runge–Kutta-based numerical technique were implemented. Numerical results for the detail of the velocity, temperature, and concentration profiles as well as heat transfer rates (Nusselt number) and mass transfer rates (Sherwood number) are presented in graphs.     

Two-equation (k, ϵ) turbulence computations by the use of a finite element model

A finite element technique is presented and applied to some one- and two-dimensional turbulent flow problems. The basic equations are the Reynolds averaged momentum equations in conjunction with a two-equation (k, ?) turbulence model. The equations are written in time-dependent form and stationary problems are solved by a time iteration procedure. The advection parts of the equations are treated by the use of a method of characteristics, while the continuity requirement is satisfied by a penalty function approach. The general numerical formulation is based on Galerkin's method. Computational results are presented for one-dimensional steady-state and oscillatory channel flow problems and for steady-state flow over a two-dimensional backward-facing step.     

A Criterion for Non-Darcy Flow in Porous Media

Non-Darcy behavior is important for describing fluid flow in porous media in situations where high velocity occurs. A criterion to identify the beginning of non-Darcy flow is needed. Two types of criteria, the Reynolds number and the Forchheimer number, have been used in the past for identifying the beginning of non-Darcy flow. Because each of these criteria has different versions of definitions, consistent results cannot be achieved. Based on a review of previous work, the Forchheimer number is revised and recommended here as a criterion for identifying non-Darcy flow in porous media. Physically, this revised Forchheimer number has the advantage of clear meaning and wide applicability. It equals the ratio of pressure drop caused by liquid–solid interactions to that by viscous resistance. It is directly related to the non-Darcy effect. Forchheimer numbers are experimentally determined for nitrogen flow in Dakota sandstone, Indiana limestone and Berea sandstone at flowrates varying four orders of magnitude. These results indicate that superficial velocity in the rocks increases non-linearly with the Forchheimer number. The critical Forchheimer number for non-Darcy flow is expressed in terms of the critical non-Darcy effect. Considering a 10% non-Darcy effect, the critical Forchheimer number would be 0.11.     

A hybrid method for computing the flow of viscoelastic fluids

A hybrid method for computing the flow of viscoelastic and second-order fluids is presented. It combines the features of the finite difference technique and the shooting method. The method is accurate because it uses central differences. Its convergence is at least superlinear. The method is applied to obtain the solutions to three problems of flow of Walters' B' fluid: (a) flow near a stagnation point, (b) flow over a stretching sheet and (c) flow near a rotating disk. Numerical results reveal some new characteristics of flows which are not easy to demonstrate using the perturbation technique.     

Prediction of Non-Darcy Coefficients for Inertial Flows Through the Castlegate Sandstone Using Image-Based Modeling

Near wellbore flow in high rate gas wells shows the deviation from Darcy??s law that is typical for high Reynolds number flows, and prediction requires an accurate estimate of the non-Darcy coefficient (?? factor). This numerical investigation addresses the issues of predicting non-Darcy coefficients for a realistic porous media. A CT-image of real porous medium (Castlegate Sandstone) was obtained at a resolution of 7.57???m. The segmented image provides a voxel map of pore-grain space that is used as the computational domain for the lattice Boltzmann method (LBM) based flow simulations. Results are obtained for pressure-driven flow in the above-mentioned porous media in all directions at increasing Reynolds number to capture the transition from the Darcy regime as well as quantitatively predict the macroscopic parameters such as absolute permeability and ?? factor (Forchheimer coefficient). Comparison of numerical results against experimental data and other existing correlations is also presented. It is inferred that for a well-resolved realistic porous media images, LBM can be a useful computational tool for predicting macroscopic porous media properties such as permeability and ?? factor.     

A finite element method for analysis of fluid flow,heat transfer and free interfaces in Czochralski crystal growth

A finite element algorithm is presented for simultaneous calculation of the steady state, axisymmetric flows and the crystal, melt/crystal and melt/ambient interface shapes in the Czochralski technique for crystal growth from the melt. The analysis is based on mixed Lagrangian finite element approximations to the velocity, temperature and pressure fields and isoparametric approximations to the interface shape. Galerkin's method is used to reduce the problem to a non-linear algebraic set, which is solved by Newton's method. Sample solutions are reported for the thermophysical properties appropriate for silicon, a low-Prandtl-number semiconductor, and for GGG, a high–Prandtl–number oxide material. The algorithm is capable of computing solutions for both materials at realistic values of the Grashof number, and the calculations are convergent with mesh refinement. Flow transitions and interface shapes are calculated as a function of increasing flow intensity and compared for the two material systems. The flow pattern near the melt/gas/crystal tri-junction has the asymptotic form predicted by an inertialess analysis assuming the meniscus and solidification interfaces are fixed.     

BIOT’S CONSOLIDATION ANALYSIS FOR CYLINDRICAL SOIL-SAMPLE BASED ON HANSBO’S FLOW1)

诸多黏性土渗流试验表明,在低水力梯度下渗流会出现明显偏离Darcy定律的现象. 为了分析渗流的非Darcy特性对固结过程 的影响,引入Hansbo渗流方程描述圆柱土样内的渗流,重新推导轴对称条件下的Biot固结方程,并给出方程的Crank--Nicolson有限 差分格式. 通过与Darcy渗流条件下轴对称Biot固结方程解析解的对比,验证计算方法的有效性. 然后分析Hansbo模型参数对圆柱 土样固结过程的影响. 计算结果表明：与Darcy渗流相比,Hansbo渗流会延缓圆柱土样的固结过程. 随着Hansbo渗流参数m或I¹的增大,在固结前期,Mandel--Cryer效应会更加显著,即孔隙水压力峰值将提高,且达到该峰值的时间 会延迟;在固结中后期,孔压消散滞后的现象也更加明显. 不过,Hansbo渗流对位移的影响很小.     

Viscous Dissipation and Thermal Radiation Effects on Mixed Convection from a Vertical Plate in a Non-Darcy Porous Medium

The paper presents an investigation of the influence of thermal radiation and viscous dissipation on the mixed convective flow due to a vertical plate immersed in a non-Darcy porous medium saturated with a Newtonian fluid. The physical properties of the fluid are assumed to be constant. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. The governing partial differential equations are transformed into a system of ordinary differential equations and solved numerically using a shooting method. The results are analyzed for the effects of various physical parameters such as viscous dissipation, thermal radiation, mixed convection parameters, and the modified Reynolds number on dynamics. The heat transfer coefficient is also tabulated for different values of the physical parameters.     

低渗透多孔介质渗流动边界模型的解析与数值解

考虑启动压力梯度的低渗透多孔介质非达西渗流模型属于强非线性动边界问题, 分别利用相似变量变换方法和基于空间坐标变换的有限差分方法, 对内边界变压力情况下、考虑启动压力梯度的一维低渗透多孔介质非达西渗流动边界模型进行了精确解析与数值求解研究. 研究结果表明:该动边界模型存在唯一的精确解析解, 且所求得的精确解析解可严格验证数值解的正确性;且当启动压力梯度值趋于零时, 非达西渗流动边界模型的精确解析解将退化为达西渗流情况下的精确解析解. 由求解结果作出的非零无因次启动压力梯度下的地层压力分布曲线表现出紧支性特点, 其与达西渗流模型的有显著不同. 因此, 研究低渗透多孔介质中非稳态渗流问题时, 应该考虑动边界的影响. 研究内容完善了低渗透多孔介质的非达西渗流力学理论, 为低渗透油气藏开发的试井解释与油藏数值模拟技术提供了理论基础.      

Analysis of Multiphase Non-Darcy Flow in Porous Media

Recent laboratory studies and analyses (Lai et al. Presented at the 2009 Rocky Mountain Petroleum Technology Conference, 14–16 April, Denver, CO, 2009) have shown that the Barree and Conway model is able to describe the entire range of relationships between flow rate and potential gradient from low- to high-flow rates through porous media. A Buckley and Leverett type analytical solution is derived for non-Darcy displacement of immiscible fluids in porous media, in which non-Darcy flow is described using the Barree and Conway model. The comparison between Forchheimer and Barree and Conway non-Darcy models is discussed. We also present a general mathematical and numerical model for incorporating the Barree and Conway model in a general reservoir simulator to simulate multiphase non-Darcy flow in porous media. As an application example, we use the analytical solution to verify the numerical solution for and to obtain some insight into one-dimensional non-Darcy displacement of two immiscible fluids with the Barree and Conway model. The results show how non-Darcy displacement is controlled not only by relative permeability, but also by non-Darcy coefficients, characteristic length, and injection rates. Overall, this study provides an analysis approach for modeling multiphase non-Darcy flow in reservoirs according to the Barree and Conway model.     

A finite element solutionof three-dimensional mixed convection gas flows in horizontal chnnels using preconditioned iterative metrix methods

An algorithm is presented for the finite element solution of three-dimensional mixed convection gas flows in channels heated from below. The algorithm uses Newton's method and iterative matrix methods. Two iterative solution algorithms, conjugate gradient squared (CGS) and generalized minimal residual (GMERS), are used in conjunction with a preconditioning technique that is simple to implement. The preconditioner is a subset of the full Jacobian matrix centered around the main diagonal but retaining the most fundamental axial coupling of the residual equations. A domain-renumbering scheme that enhances the overall algorithm performance is proposed on the basis of and analysis of the preconditioner. Comparison with the frontal elimination method demonstrates that the iterative method will be faster when the front width exceeds approximately 500. Techniques for the direct assembly f the problem into a compressed sparse row storage format are demonstrated. Elimination of fixed boundary conditions is shown to decrease the size of the matrix problem by up to 30%. Finally, fluid flow solutions obtained with the numerical technique are presented. These solutions reveal complex three-dimensional mixed convection fluid flow phenomena at low Reynolds numbers, including the reversal of the direction of longitudinal rolls in the presence of a strong recirculation in the entrance region of the channel.     

3D pore network modeling and simulation for dynamic displacement of gas and condensate in wellbore region

Condensate dropout in near a wellbore region in gas condensate reservoir is the main reason of low well deliverability. Many researchers have studied gas and condensate relative permeabilities (RPs) in this region to find the condition for better deliverability. It is known that RP is a function of capillary number in low interfacial tension (IFT) systems such as gas-condensate. The positive dependency of RPs to velocity which is referred to as "positive coupling effect" is related to the simultaneous flow of gas and condensate associated with intermittent opening and closure of channels in porous media. The negative dependency of RPs to velocity named "negative inertia" is due to non-Darcy high-velocity flow. In this study, a 3D pore network modeling is developed to investigate fluids distribution in a gas-condense system at a pore scale to find out the effects of IFT and velocity on RPs. A new method is developed that applies a flash calculation in all throats in the network to estimate the amount of accumulated condensate in throats' corners at different time steps. A modified form of Poiseuille's law for polygonal cross-sectional throats is used to find and update pressure field and fluids distribution in the network and to determine the quantity of pushed out condensate from closed throats to neighboring throats. The displacement mechanism is considered to be determined by the volume of displaced phases in throats without applying any correlation. Simulation results indicate that gas and condensate RPs are increased by an increase in velocity. However, RPs sensitivity to velocity is reduced by increasing IFT which is in agreement with previous studies.     

A New Non-Darcy Flow Model for Low-Velocity Multiphase Flow in Tight Reservoirs

The pore and pore-throat sizes of shale and tight rock formations are on the order of tens of nanometers. The fluid flow in such small pores is significantly affected by walls of pores and pore-throats. This boundary layer effect on fluid flow in tight rocks has been investigated through laboratory work on capillary tubes. It is observed that low permeability is associated with large boundary layer effect on fluid flow. The experimental results from a single capillary tube are extended to a bundle of tubes and finally to porous media of tight formations. A physics-based, non-Darcy low-velocity flow equation is derived to account for the boundary layer effect of tight reservoirs by adding a non-Darcy coefficient term. This non-Darcy equation describes the fluid flow more accurately for tight oil reservoir with low production rate and low pressure gradient. Both analytical and numerical solutions are obtained for the new non-Darcy flow model. First, a Buckley–Leverett-type analytical solution is derived with this non-Darcy flow equation. Then, a numerical model has been developed for implementing this non-Darcy flow model for accurate simulation of multidimensional porous and fractured tight oil reservoirs. Finally, the numerical studies on an actual field example in China demonstrate the non-negligible effect of boundary layer on fluid flow in tight formations.     

Upscaling LBM-TPM simulation approach of Darcy and non-Darcy fluid flow in deformable,heterogeneous porous media

This paper presents a numerical approach for the simulation of fluid flow through porous media by proposing a theoretical and numerical meso-to-macro multiscale framework, which combines the advantages of the lattice Boltzmann method (LBM) with the continuum Theory of Porous Media (TPM) to efficiently and accurately model fluid transport in heterogeneous porous media. In particular, LBM presents an alternative to experiments by studying the flow from a mesoscopic perspective, which in turn, allows the derivation of the material parameters needed for simulating the flow in the macroscopic TPM model. In this work, a meso-macro hierarchic upscaling scheme is applied to investigate the deformation-dependent intrinsic permeability properties and the Darcy/non-Darcy fluid flow regime. Concerning the mesoscale, the intrinsic permeability of the porous domain is computed by means of the LBM model at the first stage. Subsequently, deformation of the medium takes place in furtherance of determining the relation of the aforementioned deformation dependency. Thereupon, these findings are input into the TPM model in order to compute the primary unknown variables, where special focus is laid on the stability challenges in the compaction and near compaction states. With respect to the criteria of non-Darcy fluid flow, the conditions of its onset, i.e. the induced pressure gradient and mean fluid flow velocity, are computed as well using the LBM solver and conveyed afterwards to the macroscopic TPM model. Herein, the non-Darcy intrinsic permeability has been investigated in the TPM approach based on the Forchheimer equation. Simulations done on a synthetic porous micro-structure show that the combined framework proved to stand well between the two approaches.     

低渗油层压裂水平井两相流研究

依据压裂水平井不同流动区域的流动规律, 将压裂水平井的渗流分为裂缝中的高速非达西流 动区、裂缝控制影响的椭圆渗流区、远离裂缝的基质非达西渗流区, 考虑启动压力梯度的影 响, 对压裂水平井两相渗流进行了分析, 得到了低渗透油层压裂水平井的产量公式. 研究结 果表明, 裂缝的导流能力越大, 压裂水平井的产量越高. 但随着开采时间的增加, 其产量递减幅度越 大; 压裂裂缝长度越小, 压裂水平井的初始产量越高. 但随着生产时间的推移, 压裂裂缝的 长度越大, 产量的递减幅度越小; 中间裂缝长, 两翼裂缝短的情况下, 压裂水平井的产量最 高.     

峰后岩石非Darcy渗流的一维非定常严格简明解析解

煤炭是中国近期的主要能源,仍需要大力研究. 而煤矿采动围岩大多处于峰后应力状态或破碎状态,其渗流一般不符合Darcy定律. 探讨非Darcy渗流系统, 对其研究既有理论创新价值,尤其在煤矿安全中更有重要的实用价值. 用近年发展的求解偏微分方程新的分离变量法------加法分离变量法,得出了Ahmed-Sunada型非Darcy渗流的3套非常简明的一维非定常严格解析解,以发展相应的渗流理论,以及作为标准解推进渗流数值计算的水平.      

Predicting the Onset of Inertial Effects in Sandstone Rocks

This study presents a method to determine the onset of inertial effects at the microscopic level, to distinguish between Darcy and non-Darcy flow regions within porous media at the pore level, and to quantify the effects of retained polymer on gas mobility. Capillary pressure and polymer flood experiments were conducted using Elgin and Okesa sandstone samples. The pore-size distributions were used to study the high-velocity flow effects. A modified capillary-orifice model was used to determine the non-Darcy flow effects at the pore level, with and without residual polymer.The overall flow behavior at any flow rate may be described as the average of all contributions from the Darcy and the non-Darcy terms in all pores. Results of this study suggest that the conventional Reynolds number may lead to incorrect analysis of flow behavior when evaluating non-Darcy flow effects in porous media. The Forchheimer number, defined as the ratio of inertial forces to viscous forces, is found more adequate for analyzing microscopic flow behavior in porous media.