Traveling waves in a finite condensation rate model for steam injection

Steam drive recovery of oil is an economical way of producing oil even in times of low oil prices and is used worldwide. This paper focuses on the one-dimensional setting, where steam is injected into a core initially containing oil and connate water while oil and water are produced at the other end. A three-phase (oil, water, steam) hot zone develops, which is abruptly separated from the two-phase (oil + water) cold zone by the steam condensation front. The oil, water and energy balance equations (Rankine–Hugoniot conditions) cannot uniquely solve the system of equations at the steam condensation front. In a previous study, we showed that two additional constraints follow from an analysis of the traveling wave equation representing the shock; however, within the shock, we assumed local thermodynamic equilibrium. Here we extend the previous study and include finite condensation rates; using that appropriate scaling requires that the Peclet number and the Damkohler number are of the same order of magnitude. We give a numerical proof, using a color-coding technique, that, given the capillary diffusion behavior and the rate equation, a unique solution can be obtained. It is proven analytically that the solution for large condensation rates tends to the solution obtained assuming local thermodynamic equilibrium. Computations with realistic values to describe the viscous and capillary effects show that the condensation rate can have a significant effect on the global saturation profile, e.g. the oil saturation just upstream of the steam condensation front.     

Numerical methods for flow in a porous medium with internal boundaries

Flow of fluids and transport of solutes in porous media are subjects of wide interest in several fields of applications: reservoir engineering, subsurface hydrology, chemical engineering, etc. In this paper we will study two-phase flow in a model consisting of two different types of sediments. Here, the absolute permeability, the relative permeabilities and the capillary pressure are discontinuous functions in space. This leads to interior boundary value problems at the interface between the sediments. The saturation S_w will be discontinuous or experience large gradients at the interface. A new solution procedure for such problems will be presented. The method combines the modified method of characteristics with a weak formulation where the basis functions are discontinuous at the interior boundary. The modified method of characteristics will provide a good first approximation for the jump in the discontinuous basis functions, which leads to a fast converging iterative solution scheme for the complete problem. The method has been implemented in a two-dimensional simulator, and results from numerical experiments will be presented. This revised version was published online in August 2006 with corrections to the Cover Date.     

Solving the Buckley–Leverett equation with gravity in a heterogeneous porous medium

Immiscible two‐phase flow in porous media can be described by the fractional flow model. If capillary forces are neglected, then the saturation equation is a non‐linear hyperbolic conservation law, known as the Buckley–Leverett equation. This equation can be numerically solved by the method of Godunov, in which the saturation is computed from the solution of Riemann problems at cell interfaces. At a discontinuity of permeability this solution has to be constructed from two flux functions. In order to determine a unique solution an entropy inequality is needed. In this article an entropy inequality is derived from a regularisation procedure, where the physical capillary pressure term is added to the Buckley‐Leverett equation. This entropy inequality determines unique solutions of Riemann problems for all initial conditions. It leads to a simple recipe for the computation of interface fluxes for the method of Godunov. This revised version was published online in July 2006 with corrections to the Cover Date.     

Uniqueness conditions in a hyperbolic model for oil recovery by steamdrive

In this paper we study a one-dimensional model for oil recovery by steamdrive. This model consists of two parts: a (global) interface model and a (local) steam condensation/capillary diffusion model. In the interface model a steam condensation front (SCF) is present as an internal boundary between the hot steam zone (containing water, oil and steam) and the cold liquid zone (containing only water and oil). Disregarding capillary pressure away from the SCF, a 2× 2 hyperbolic system arises for the water and steam saturation. This system cannot be solved uniquely without additional conditions at the SCF. To find such conditions we blow up the SCF and consider a parabolic transition model, including capillary diffusion. We study in detail the existence conditions for traveling wave solutions. These conditions provide the missing matching conditions at the SCF in the hyperbolic limit. We show that different transition models yield different matching conditions, and thus different solutions of the interface model. We also give a relatively straightforward approximation and investigate its validity for certain ranges of model parameters. This revised version was published online in July 2006 with corrections to the Cover Date.     

用油水相对渗透率确定镇泾油田长6储层的产液情况

油水相对渗透率可以精确刻划出油、水二相流体在孔喉中的流动情况,通过阿尔奇方程准确求取储层的含水饱和度,建立适合镇泾油田的束缚水饱和度计算模型,并运用"岩心刻度测井"的方法,通过回归法,用实测油、水相对渗透率曲线求取琼斯方程中的区域参数,最终建立适合本区的油水相对渗透率经验公式。结果表明,用测井数据可以进行油、水相对渗透率的计算,并且完全满足评价储集层的产液情况。由此建立的油水相对渗透率解释模型,可以进行镇泾油田长6储层的油、水层的划分,并对油、水分异不彻底的储层进行测井评价,有着重要的参考价值。     

Effect of connectivity and wettability on the relative permeability of NAPLs

The characteristic relationships among relative permeability (K _r ), saturation (S) and capillary pressure (P) of NAPLs are the important constitutive laws to simulate the NAPLs flow in the subsurface. In this study, a micro model was used to obtain the values of permeability, saturation for the multi-phase flow of five fluid-pairs. The perspective micro model allows one to clearly observe the multiphase flow and allow this study to precisely measure the fluid saturation by digital image analysis. The experimental results showed hysteresis phenomenon of relative permeability versus saturation and that was not interpreted by previous studies. By carefully examining the recorded images, this study found that the degree of the connectivity for the micro channel occupied by wetting phase fluid could influence the relative permeability. Therefore, for the same saturation, the relative permeability in the imbibition is higher than that in the drainage. The results of the K _r –S experiments for five fluid-pairs also showed that the residual saturation of wetting phase fluid decreased with the wettability increasing but increased with the interfacial tension increasing. Those interpretations and experimental results are valuable references for groundwater remediation and oil reservoir development.     

碳酸盐岩储层参数对微观渗流的影响

针对碳酸盐岩油藏孔隙大小的双峰分布特征,首先利用计算机模拟建立了描述不同孔隙特征的大孔隙网络模型和微孔隙网络模型,在此基础上提出一种耦合算法构建出的同时描述大孔隙和微孔隙特征的碳酸盐岩双孔隙网络模型;然后,基于侵入-逾渗理论,模拟双孔隙网络模型中油水两相流体的一次驱替和二次吸吮过程,并建立了毛细管压力和相对渗透率的求解模型;最后,通过调整双孔隙网络结构参数,模拟水湿油藏条件下碳酸盐岩储层参数对相对渗透率曲线的影响. 结果表明,随着微孔隙比例因子和平均配位数的增加,油相相对渗透率曲线升高;随着双孔隙半径比的增加,油相和水相相对渗透率曲线下降,这对碳酸盐岩油藏渗流机理研究有着重要的指导意义.      

Adjoint analysis of Buckley-Leverett and two-phase flow equations

This paper analyzes the adjoint equations and boundary conditions for porous media flow models, specifically the Buckley-Leverett equation, and the compressible two-phase flow equations in mass conservation form. An adjoint analysis of a general scalar hyperbolic conservation law whose primal solutions include a shock jump is initially presented, and the results are later specialized to the Buckley-Leverett equation. The non-convexity of the Buckley-Leverett flux function results in adjoint characteristics that are parallel to the shock front upstream of the shock and emerge from the shock front downstream of the shock. Thus, in contrast to the behavior of Burgers’ equation where the adjoint is continuous at a shock, the Buckley-Leverett adjoint, in general, contains a discontinuous jump across the shock. Discrete adjoint solutions from space-time discontinuous Galerkin finite element approximations of the Buckley-Leverett equation are shown to be consistent with the derived closed-form analytical solutions. Furthermore, a general result relating the adjoint equations for different (though equivalent) primal equations is used to relate the two-phase flow adjoints to the Buckley-Leverett adjoint. Adjoint solutions from space-time discontinuous Galerkin finite element approximations of the two-phase flow equations are observed to obey this relationship.     

The role of geological heterogeneity and variability in water infiltration on non-aqueous phase liquid migration

This study investigates the influence of two factors—geological heterogeneity and variability in water infiltration—on non-aqueous phase liquid (NAPL) migration in the unsaturated zone. NAPL migration under three-phase flow conditions resulting from a ground surface spill is modeled for multiple heterogeneous realizations of a porous medium with various water infiltration scenarios. Increased water infiltration before the spill has two counteracting effects: NAPL relative permeability (k _rn) increases with increasing water saturation (S _w) for a given NAPL saturation, while higher S _w in the soil near the NAPL source zone leads to less NAPL mass infiltration. It is found that the former effect is overwhelmed by the latter effect, the net effect being that with longer infiltration durations before the spill, both the infiltrated NAPL mass and the depth of the front migration decrease. Simulation results also show strong effect of the medium heterogeneity. Results suggest that total infiltrated mass, front depth and plume spread increase with an increasing standard deviation of log-permeability. Also variability in modeling results among realizations is largely impacted by the log-permeability standard deviation. Spatial correlation in permeability also strongly influences NAPL infiltration. An increase in the isotropic correlation length from 0.75 to 1.5 m leads to a decrease in total infiltrated mass, plume migration depth as well as vertical spread. Lateral spread in this case is not shown to be affected by the correlation length.     

Capillary hysteresis and gravity segregation in two phase flow through porous media

We study the gravity driven flow of two fluid phases in a one dimensional homogeneous porous column when history dependence of the pressure difference between the phases (capillary pressure) is taken into account. In the hyperbolic limit, solutions of such systems satisfy the Buckley-Leverett equation with a non-monotone flux function. However, solutions for the hysteretic case do not converge to the classical solutions in the hyperbolic limit in a wide range of situations. In particular, with Riemann data as initial condition, stationary shocks become possible in addition to classical components such as shocks, rarefaction waves and constant states. We derive an admissibility criterion for the stationary shocks and outline all admissible shocks. Depending on the capillary pressure functions, flux function and the Riemann data, two cases are identified a priori for which the solution consists of a stationary shock. In the first case, the shock remains at the point where the initial condition is discontinuous. In the second case, the solution is frozen in time in at least one semi-infinite half. The predictions are verified using numerical results.

     

吐哈盆地巴喀气田八道湾组致密砂岩气藏气水分布特征

基于核磁共振、压汞测试、物性分析、相渗曲线及生产动态等多种资料,对吐哈盆地巴喀气田八道湾组致密砂岩储层的物性特征、孔喉分布、束缚水含量、气柱高度、"甜点"发育的主控因素进行了分析,总结了研究区气水分布特征和气水分布模式。研究结果表明:八道湾组发育典型的致密砂岩储层,其孔隙度总体小于7%,渗透率小于0.2×10-3μm2;喉道半径较小,主喉道半径的分布范围为0.01～1μm;束缚水饱和度较高,且同孔隙度呈负相关关系。研究区气水过渡带的范围约为640 m,研究区储层总体处于气水过渡带之内;处于过渡带不同构造位置处不同物性的储层具有不同的含气特征和气水生产特征,位于气水过渡带较高部位的物性较好的区域和裂缝发育区是"甜点"的主要发育区。     

Modelling Two-Phase Incompressible Flow in Porous Media Using Mixed Hybrid and Discontinuous Finite Elements

In this paper, we present a numerical model for simulating two-phase (oil–water and air–water) incompressible and immiscible flow in porous media. The mathematical model which is based on a fractional flow formulation is formed of two nonlinear partial differential equations: a mean pressure equation and a water saturation equation. These two equations can be solved in a sequential manner. Two numerical methods are used to discretize the equations of the two-phase flow model: mixed hybrid finite elements are used to treat the pressure equation, h-based Richards' equation and the diffusion term in the saturation equation, the advection term in the saturation equation is treated with the discontinuous finite elements. We propose a better way to calculate the nonlinear coefficients contained in our equations on each element of the discretized domain. In heterogeneous porous media, the saturation becomes discontinuous at the interface between two porous media. We show in this paper how to use the capillary pressure–saturation relationship in order to handle the saturation jump in the mixed hybrid finite element method. The two-phase flow simulator is verified against analytical solutions for some flow problems treated by other authors.     

多孔介质水油两相系统相对渗透率与饱和度关系试验研究

相对渗透率和饱和度定量关系是多孔介质多相流动系统中重要的动力学参数关系。使用设计的试验装置及试验方法测定了一维砂柱中油水两相动态流动系统的相对渗透率及饱和度数据,由试验结果分析表明:所测得的相对渗透率与实际情况吻合较好;细砂的强亲水性对不同先湿条件下的油、水相相对渗透率及饱和度有较大影响,并造成了不同先湿条件下流体间驱替机制的差异;相饱和度是影响相对渗透率的主要因素,孔隙空间中两种流体的分布方式和流体的饱和历史也影响各相相对渗透率。对试验结果用VGM模型(Parker-Lenhard模型)进行拟合所得结果较好;在水先湿条件下,将van公式拟合毛细压力-饱和度数据所得拟合参数用于VGM模型预测相对渗透率-饱和度曲线,所得结果与VGM模型直接拟合所得结果有差异,但两者所得结果均较好。     

Numerical simulation of ultrasonic waves in reservoir rocks with patchy saturation and fractal petrophysical properties

We simulate the propagation of ultrasonic waves in heterogeneous poroviscoelastic media saturated by immiscible fluids. Our model takes into account capillary forces and viscous and mass coupling effects between the fluid phases under variable saturation and pore fluid pressure. The numerical problem is solved in the space–frequency domain using a finite element procedure and the time–domain solution is obtained by a numerical Fourier transform. Heterogeneities due to fluid distribution and rock porosity–permeability are modeled as stochastic fractals, whose spectral densities reproduce saturation an petrophysical variations similar to those observed in reservoir rocks. The numerical experiments are performed at a central frequency of 500 kHz, and show clearly the effects of the different heterogeneities on the amplitudes of shear and compressional waves and the importance of wave mode conversions at the different interfaces.     

Determination of Soil Parameters via the Solution of Inverse Problems in Infiltration

In this paper, we propose an efficient method for the identification of soil parameters in unsaturated porous media, using measurements from infiltration experiments. The infiltration is governed by Richard's nonlinear equation expressed in terms of effective saturation. The soil retention and hydraulic permeability functions are expressed using the Van Genuchten-Mualem ansatz in terms of the soil parameters. The mathematical algorithm is based on a transformation of Richard's equation to a system of ordinary differential equations completed by the governing equation for the movement of the wetness front. This system can be efficiently solved by specialized packages for the solution of stiff systems of ODE. The unknown parameters are determined using the optimization approach of minimizing a cost functional for the discrepancy between the model output and the measurements. The gradient and Hessian of the solution with respect to soil parameter vector are determined using automatic differentiation. Several numerical experiments are included.     

层状非均质多孔介质中LNAPLs污染物运移与分布的锋面扩展模型

通过建立二维物理模型,研究非饱和层状非均质多孔介质中轻非水相液体LNAPLs(Light Non-a-queous Phase Liquids)的入渗机制与变化特征,建立LNAPLs运移与分布的锋面扩展模型,探讨入渗阶段油流锋面扩展速率的变化规律。结果发现:模型中砂土的渗透系数、孔隙度及油的相对渗透系数对LNAPLs运移的锋面扩展速率影响较大,而油的饱和度与压头变化对其影响较小。当LNAPLs由中砂进入细砂与细砂倾斜透镜体时,砂的渗透系数、孔隙度及油的相对渗透系数减小,这种介质结构面的突变改变了LNAPLs锋面扩展速率,其在中砂中锋面扩展速率快而在细砂与细砂倾斜透镜体中锋面扩展速率慢。此外,当LNAPLs进入干湿界面后,由于毛细作用的增强与含水量的加大,其横向比垂向运移的平均锋面扩展速率大。     

Hydro‐mechanical modeling of cohesive crack propagation in multiphase porous media using the extended finite element method

In this paper, a numerical model is developed for the fully coupled hydro‐mechanical analysis of deformable, progressively fracturing porous media interacting with the flow of two immiscible, compressible wetting and non‐wetting pore fluids, in which the coupling between various processes is taken into account. The governing equations involving the coupled solid skeleton deformation and two‐phase fluid flow in partially saturated porous media including cohesive cracks are derived within the framework of the generalized Biot theory. The fluid flow within the crack is simulated using the Darcy law in which the permeability variation with porosity because of the cracking of the solid skeleton is accounted. The cohesive crack model is integrated into the numerical modeling by means of which the nonlinear fracture processes occurring along the fracture process zone are simulated. The solid phase displacement, the wetting phase pressure and the capillary pressure are taken as the primary variables of the three‐phase formulation. The other variables are incorporated into the model via the experimentally determined functions, which specify the relationship between the hydraulic properties of the fracturing porous medium, that is saturation, permeability and capillary pressure. The spatial discretization is implemented by employing the extended finite element method, and the time domain discretization is performed using the generalized Newmark scheme to derive the final system of fully coupled nonlinear equations of the hydro‐mechanical problem. It is illustrated that by allowing for the interaction between various processes, that is the solid skeleton deformation, the wetting and the non‐wetting pore fluid flow and the cohesive crack propagation, the effect of the presence of the geomechanical discontinuity can be completely captured. Copyright © 2012 John Wiley & Sons, Ltd.     

Prediction and analysis of the stimulated reservoir volume for shale gas reservoirs based on rock failure mechanism

The ultra-low-permeability shale gas reservoir has a lot of well-developed natural fractures. It has been proven that hydraulic fracture growth pattern is usually a complex network fracture rather than conventional single planar fractures by micro-seismic monitoring, which can be explained as the shear and tensile failure of natural fractures or creation of new cracks due to the increase in reservoir pore pressure caused by fluid injection during the process of hydraulic fracturing. In order to simulate the network fracture growth, a mathematical model was established based on full tensor permeability, continuum method and fluid mass conservation equation. Firstly, the governing equation of fluid diffusivity based on permeability tensor was solved to obtain the reservoir pressure distribution. Then Mohr–Coulomb shear failure criterion and tensile failure criterion were used to decide whether the rock failed or not in any block on the basis of the calculated reservoir pressure. The grid-block permeability was modified according to the change of fracture aperture once any type of rock failure criterion was met within a grid block. Finally, the stimulated reservoir volume (SRV) zone was represented by an enhancement permeability zone. After calibrating the numerical solution of the model with the field micro-seismic information, a sensitivity study was performed to analyze the effects of some factors including initial reservoir pressure, injection fluid volume, natural fracture azimuth angle and horizontal stress difference on the SRV (shape, size, bandwidth and length). The results show that the SRV size increases with the increasing initial pore reservoir and injection fluid volume, but decreases with the increase in the horizontal principal stress difference and natural fracture azimuth angle. The SRV shape is always similar for different initial pore reservoir and injection fluid volume. The SRV is observed to become shorter in length and wider in bandwidth with the decrease in natural fracture azimuth angle and horizontal principal stress difference.     

Impact of time-lapse seismic data for permeability estimation

We consider the impact of using time-lapse seismic data in addition to production data for permeability estimation in a porous medium with multiphase fluid flows, such as a petroleum reservoir under water-assisted production. Since modeling seismic wave propagation in addition to modeling fluid flows in the reservoir is quite involved, it is assumed that the time-lapse seismic data have already been inverted into fluid saturation differences (pseudoseismic data). Because an inversion process often leads to considerable error growth, we will consider pseudoseismic data with large uncertainties. The impact of pseudoseismic data is assessed through permeability estimation with and without such data and through application of some uncertainty measures for the estimated parameters. A multiscale algorithm is used for the parameter estimations, so that potential differences in attainable permeability resolution will be easily revealed. The numerical examples clearly indicate that the permeability estimation problem is stabilized at a higher level of resolution when pseudoseismic data are applied in addition to production data, even if the pseudoseismic data have large associated uncertainties. Use of the parameter uncertainty measures confirm these results.     

储层毛管压力曲线的拟合及其地质应用

本文以辽河油田新开地区沙一、沙三段砂岩储层为例阐述了如何利用原始压汞数据建立注汞压力与进汞饱和度之间的函数关系，然后再从这些关系式出发导出它们的一些地质应用。研究结果表明:新开地区注汞压力与进汞饱和度之间存在着良好的双曲线函数关系，根据这些关系可以利用孔渗两个参数反求毛管压力和毛管压力曲线，很方便地计算某一样品的渗透率以及任意进汞区间的渗透率贡献值。