Analytical and numerical investigation of multicomponent multiphase WAG displacements

In this paper the ability of analytical solutions for four-component three-phase flow to predict displacement efficiency in water alternating gas (WAG) injection processes is studied. First analytical solutions for Riemann problems with injection compositions that are the average water and gas mixture for various WAG injection schemes are presented. These solutions are compared to numerical calculations with variable slug sizes and used to explore the effect of slug size, injecting water vs gas first, and the average injection composition on displacement efficiency in compositional WAG schemes. The example model is partially miscible WAG injection of water and CO₂ into an oil reservoir containing C₁₀ and CH₄ with and without a mobile aqueous phase present initially. The trailing end of the water and gas profiles are sensitive to whether water or gas is injected first, but the magnitude of the oil bank and the breakthrough time of the injected fluids are accurately predicted by the analytical solutions, even for displacements where large water and gas slugs are injected. Fluctuations in the saturation and composition profiles resulting from the alternating injection sequence in the WAG simulations appear as super-imposed on top of the sequence of rarefaction and shock waves predicted by analytical solutions. As the number of slugs increases, the effect of alternating boundary conditions diminishes and the displacements predicted by numerical calculations converge to the analytical solutions.     

The effect of viscosity ratio on the dispersal of fracturing fluids into groundwater system

In the development of oil and gas reservoirs, the transport of miscible fluids in porous rocks is a key issue for oil and gas recovery. The simplified unidirectional flow model is employed to investigate the effects of the viscosity ratio on dispersion in semi-infinite homogenous media. The viscosity is supposed to be unsteady due to changing component concentration over time. In cases of both a viscosity ratio larger and less than 1, the pollutant concentration and flow velocity are computed at different initial conditions and viscosity ratios. The analytical solutions are then developed by introducing new variables and transforming the equation governing advection–diffusion equation in semi-infinite homogeneous media with a continuous source. A comparison of the numerical solution with the analytical solution revealed a similarity over 98%, highlighting the usability of the analytical solution. If the viscosity ratio is larger than 1, flow velocity declines exponentially and concentration attenuates with transporting time. In addition, the timescale plays a significant role and the effects become more prominent in long-term transport. In the case of a viscosity ratio less than 1, both the timescale and viscosity ratio variables have little influence on the changing speed of the concentration profile. This work helps to predict the position and the time required to reach the harmless pollutant concentration when monitoring fracturing fluids transportation into groundwater system and would be especially useful in designing and interpreting laboratory experiments studying the miscible flow.     

Analytical solution of polymer slug injection with viscous fingering

We present an analytical solution to estimate the minimum polymer slug size needed to ensure that viscous fingering of chase water does not cause its breakdown during secondary oil recovery. Polymer flooding is typically used to improve oil recovery from more viscous oil reservoirs. The polymer is injected as a slug followed by chase water to reduce costs; however, the water is less viscous than the oil. This can result in miscible viscous fingering of the water into the polymer, breaking down the slug and reducing recovery. The solution assumes that the average effect of fingering can be represented by the empirical Todd and Longstaff model. The analytical calculation of minimum slug size is compared against numerical solutions using the Todd and Longstaff model as well as high resolution first contact miscible simulation of the fingering. The ability to rapidly determine the minimum polymer slug size is potentially very useful during enhanced oil recovery (EOR) screening studies.     

Adjoint formulation and constraint handling for gradient-based optimization of compositional reservoir flow

An adjoint formulation for the gradient-based optimization of oil–gas compositional reservoir simulation problems is presented. The method is implemented within an automatic differentiation-based compositional flow simulator (Stanford’s Automatic Differentiation-based General Purpose Research Simulator, AD-GPRS). The development of adjoint procedures for general compositional problems is much more challenging than for oil–water problems due to the increased complexity of the code and the underlying physics. The treatment of nonlinear constraints, an example of which is a maximum gas rate specification in injection or production wells, when the control variables are well bottom-hole pressures, poses a particular challenge. Two approaches for handling these constraints are presented—a formal treatment within the optimizer and a simpler heuristic treatment in the forward model. The relationship between discrete and continuous adjoint formulations is also elucidated. Results for four example cases of increasing complexity are presented. Improvements in the objective function (cumulative oil produced) relative to reference solutions range from 4.2 to 11.6 %. The heuristic treatment of nonlinear constraints is shown to offer a cost-effective means for obtaining feasible solutions, which are, in some cases, better than those obtained using the formal constraint handling procedure.     

A discussion on analytical and numerical modelling of the land subsidence induced by coal seam gas extraction

Coal seam gas (CSG) is an increasingly important source of natural gas all over the world. Although the influence of conventional oil and gas extraction on surface subsidence has been widely recognized and studied, few studies are carried out on the surface subsidence in coal seam gas fields and its impact on surface infrastructure and the environment. This paper discusses modelling of the surface subsidence associated with coal seam gas production by applying both analytical and numerical methods. By comparison of results from the numerical model and two analytical models, i.e. the disc-shaped reservoir model and the uniaxial compaction model, the analytical solutions cannot describe the complex process of water and gas extraction and have the limitations to predict the surface subsidence, while the numerical model can be better used in prediction of subsidence. After applying the numerical model in numerical analysis, the deformation characteristics of coupled fluid flow, and the effects of permeability change of coal seam, associated overlying and underlying layers, and depressurization rates on surface subsidence are investigated. The results demonstrate that the proposed model can simulate the production of water and gas from coal seams and the associated surface subsidence.     

大庆长垣外围特低渗透扶杨油层CO_2非混相驱油试验研究

大庆长垣外围有3×108t特低渗透的扶杨油层储量在注水开发条件下难以有效动用,为提高这部分储量的有效动用率,开展了CO2非混相驱油试验研究。PVT测试结果表明,扶杨油层原油较稠,体积系数小,膨胀性和收缩性小,溶解系数较低;CO2驱油细管实验最小混相压力为29 MPa,比原始地层压力高8.6 MPa,现场试验为非混相驱;长岩心实验CO2驱油采收率比注水高4~6个百分点。综合室内可行性评价实验结果看,大庆长垣外围扶杨油层开展CO2驱油试验是可行的,并于2003年初在宋芳屯油田南部开辟了注气试验区,有注气井1口,采出井5口。矿场试验结果表明,CO2驱油能够使特低渗透、裂缝不发育的扶杨油层建立起有效驱动体系,从根本上改善其开发效果,该技术有望成为特低渗透扶杨油层有效动用的可行技术。     

Analytical 3D transient elastodynamic fundamental solution of unsaturated soils

Unsaturated soils are considered as porous continua, composed of porous skeleton with its pores filled by water and air. The governing partial differential equations (PDE) are derived based on the mechanics for isothermal and infinitesimal evolution of unsaturated porous media in terms of skeleton displacement vector, liquid, and gas scalar pressures. Meanwhile, isotropic linear elastic behavior and liquid retention curve are presented in terms of net stress and capillary pressure as constitutive relations. Later, an explicit 3D Laplace transform domain fundamental solution is obtained for governing PDE and then closed‐form analytical transient 3D fundamental solution is presented by means of analytical inverse Laplace transform technique. Finally, a numerical example is presented to validate the assumptions used to derive the analytical solution by comparing them with the numerically inverted ones. The transient fundamental solutions represent important features of the elastic wave propagation theory in the unsaturated soils. Copyright © 2010 John Wiley & Sons, Ltd.     

High order discontinuous Galerkin method for simulating miscible flooding in porous media

We present a high-order method for miscible displacement simulation in porous media. The method is based on discontinuous Galerkin discretization with weighted average stabilization technique and flux reconstruction post processing. The mathematical model is decoupled and solved sequentially. We apply domain decomposition and algebraic multigrid preconditioner for the linear system resulting from the high-order discretization. The accuracy and robustness of the method are demonstrated in the convergence study with analytical solutions and heterogeneous porous media, respectively. We also investigate the effect of grid orientation and anisotropic permeability using high-order discontinuous Galerkin method in contrast with cell-centered finite volume method. The study of the parallel implementation shows the scalability and efficiency of the method on parallel architecture. We also verify the simulation result on highly heterogeneous permeability field from the SPE10 model.     

Numerical modelling of coupled flow and deformation in fractured rock specimens

A dual-porosity poroelastic model is extended to represent behaviour in cylindrical co-ordinates for the evaluation of flow-deformation effects in cylindrical laboratory samples incorporating a central wellbore or non-repeating axisymmetric injection on the periphery. Nine-node quadratic elements are used to represent mechanical deformation, while eight-node linear elements are used to interpolate the pressure fields, which offers significant advantages over the behaviour of other non-conforming elements. The model presented is validated against simplified analytical results, and extended to describe the behaviour of homogeneous and heterogeneous laboratory specimens subjected to controlled triaxial state of stress and injection tests. Apparent from the results is the significant influence of stress-deformation effects over system behaviour. Copyright © 1999 John Wiley & Sons, Ltd.     

An analytical solution for modeling thermal energy transfer in a confined aquifer system

A mathematical model is developed for simulating the thermal energy transfer in a confined aquifer with different geological properties in the underlying and overlying rocks. The solutions for temperature distributions in the aquifer, underlying rock, and overlying rock are derived by the Laplace transforms and their corresponding time-domain solutions are evaluated by the modified Crump method. Field data adopted from the literature are used as examples to demonstrate the applicability of the solutions in modeling the heat transfer in an aquifer thermal energy storage (ATES) system. The results show that the aquifer temperature increases with time, injection flow rate, and water temperature. However, the temperature decreases with increasing radial and vertical distances. The heat transfer in the rocks is slow and has an effect on the aquifer temperature only after a long period of injection time. The influence distance depends on the aquifer physical and thermal properties, injection flow rate, and injected water temperature. A larger value of thermal diffusivity or injection flow rate will result in a longer influence distance. The present solution can be used as a tool for designing the heat injection facilities for an ATES system.     

水平受荷长桩弹塑性计算解析解

当考虑桩侧土体非线性本构关系时对水平受荷桩的计算一般需采用数值方法,解析结果相对较少。基于Winkler地基模型和桩侧土体简化的弹塑性本构关系,对均质地基中水平荷载作用下桩头嵌固的长桩进行了解析推导,得到了桩身最大挠度及最大弯矩与荷载关系的统一解析表达式,并采用相同的方法求得高桩情形下桩头挠度的计算式。计算表明,联合荷载作用下桩身泥面处的挠度和转角不等于单个荷载作用时的线性叠加,采用常规的线性叠加法计算将偏于不安全。所求解析式借助计算器即可进行最大挠度和最大弯矩的计算,大大方便了工程的计算应用。     

双误差回归模型在ICP-MS测定饮用水中锶的不确定度评定的简化应用

在计算电感耦合等离子体质谱法测定饮用水中锶的浓度(x)时,由于标准系列配制和仪器检测过程中信号(y)漂移产生的不确定度会传递给最终的计算结果。普通的一次或多次线性拟合结果不能真实地反映对于x、y值都含误差的数据拟合情况。文章对标准曲线进行了双误差回归计算,从双误差回归线性方程推导出校准曲线拟合过程产生不确定度的计算公式,建立了方法检出限与曲线拟合参数(x、y、曲线截距和斜率)及其相关不确定度之间的关系式,依据误差连续传递公式及不确定度分量计算公式简化了合成不确定度的表达式。从合成不确定度计算公式中可得出,水样中锶含量的浓度越低,其对应的不确定度越大;标准系列配制过程不细致、仪器灵敏度低,方法检出限越差,与实际测试情况符合。     

Vertical equilibrium with sub-scale analytical methods for geological CO2 sequestration

Large-scale implementation of geological CO₂ sequestration requires quantification of risk and leakage potential. One potentially important leakage pathway for the injected CO₂ involves existing oil and gas wells. Wells are particularly important in North America, where more than a century of drilling has created millions of oil and gas wells. Models of CO₂ injection and leakage will involve large uncertainties in parameters associated with wells, and therefore a probabilistic framework is required. These models must be able to capture both the large-scale CO₂ plume associated with the injection and the small-scale leakage problem associated with localized flow along wells. Within a typical simulation domain, many hundreds of wells may exist. One effective modeling strategy combines both numerical and analytical models with a specific set of simplifying assumptions to produce an efficient numerical–analytical hybrid model. The model solves a set of governing equations derived by vertical averaging with assumptions of a macroscopic sharp interface and vertical equilibrium. These equations are solved numerically on a relatively coarse grid, with an analytical model embedded to solve for wellbore flow occurring at the sub-gridblock scale. This vertical equilibrium with sub-scale analytical method (VESA) combines the flexibility of a numerical method, allowing for heterogeneous and geologically complex systems, with the efficiency and accuracy of an analytical method, thereby eliminating expensive grid refinement for sub-scale features. Through a series of benchmark problems, we show that VESA compares well with traditional numerical simulations and to a semi-analytical model which applies to appropriately simple systems. We believe that the VESA model provides the necessary accuracy and efficiency for applications of risk analysis in many CO₂ sequestration problems.     

Numerical modeling of groundwater inflow from a confined aquifer into Sangan open pit mine,northeast Iran

Numerical models are useful in the evaluation of the interaction between groundwater systems and mining activities. They can be successfully used to predict the quantity of inflow into open pits and to design an appropriate dewatering system. In this paper, a two-dimensional axi-symmetric finite element model called SEEP/W has been used to predict the groundwater inflow into Sangan open pit mine (anomaly north C). The Sangan iron mine is located at 280 km south-east of Mashhad, Iran, in arid and warm climate conditions wherein precipitation is generally limited. The water inflow to the pit is mainly from a confined aquifer, mainly by horizontal flow in the upper layers and vertical flow at the pit bottom. The results of the numerical model of the ground water inflow are presented and compared with those obtained from Theis, Cooper-Jacob and Jacob-Lohman analytical solutions. Ground water inflow monitoring was also carried out in a trial excavation at the Sangan mine in order to calibrate the model. The model was then used to predict groundwater inflow into Sangan open pit mine during its advancement. This model provides valuable information for designing an appropriate dewatering system.     

A segregated flow scheme to control numerical dispersion for multi-component flow simulations

One of the challenges for reservoir simulation is numerical dispersion. For waterflooding applications the effect is controlled due to the self-sharpening nature of a Buckley–Leverett shock. However, for multi-component flow simulations, incorrect wavespeeds can develop leading to the excessive smearing of fronts because of the coupling of compositional dispersion with the fractional flow. Rather than implementing a higher-order discretization method, we propose a simple scheme based on segregation-in-flow within a gridblock to control numerical dispersion. We extend the method originally proposed for polymer flooding to augmented waterflooding simulations in general as well as simulations of miscible or near miscible gas injection. For compositional simulations of gas injection, this is done through a coupled limited-flash/upstream-exclusion assumption. To test the scheme, an in-house streamline simulator has been modified and validated for modeling low-salinity floods as well as ternary two-phase displacements. Simulation results presented with and without segregation demonstrate the potential of the approach as a heuristic method to control numerical dispersion in multi-component flow simulations.     

Finite element models for use in mechanics including geological engineering

A number of finite element (FE) models have been developed, starting with a purely magnetic one. The magnetic model was verified against analytical solutions and the commercial FE-programme ANSYS. The magnetic model was then coupled to the wave-equation, resulting in a model governing linear magnetostriction. In the magnetostrictive model a property of two degrees of freedom, that is, displacements, was coupled to a property of one degree of freedom, that is, magnetic potential. To make this possible, non-quadratic coupling matrices had to be developed. This linear model was then verified against analytical solutions, with a good conceptual agreement. In the next step the still linear system of equations was time-propagated, using the Newmark method. This model was verified by comparing the differential step-response of the system calculated with the time propagation method, to the frequency function of the system, calculated with the harmonic model. Further, non-linear and non-linear transient models were developed and discussed. The non-linear transient model was developed for a situation where a Terfenol rod was assumed to collide periodically with a rigid wall, thus indicating a tool for analysing a Terfenol based resonant system, which could be used, for example, for ultrasonic cutting of hard and brittle materials and also for rock-blasting.     

Design Charts for Selecting Minimum Setback Distance from Side Slope to Horizontal Trench System in Bioreactor Landfills

The primary objective of bioreactor landfill is to achieve adequate and rapid distribution of moisture in landfilled municipal solid waste (MSW) to accelerate the anaerobic biodegradation of the organic fraction within MSW. A horizontal trench system (HT) is commonly adopted for leachate distribution in MSW under pressurized conditions. However, this approach should be implemented carefully due to the potential instability of landfill slopes that comes from the generation and distribution of excessive pore fluid pressures. In this study, HT design charts are presented that determine the optimal location of horizontal trench systems from the side slope (i.e., minimum lateral setback distance) under continuous leachate addition with maximum applied injection pressures, for which the landfill slopes remain stable [factor of safety (FOS) where FOS ≥ 1.5]. Use of any higher injection pressure and/or shorter lateral setback distance of HT than the one presented in the design charts would result in an unacceptable design of the bioreactor side slope (FOS < 1.5). The design chart was developed based on a parametric study that used a numerical two-phase flow model that involved different slope configurations and landfill waste depths. MSW heterogeneity and anisotropy, as well as unsaturated hydraulic properties, were taken into consideration in these simulations. Transient changes in pore water and gas pressures due to leachate recirculation were accounted for dually in the slope stability computations. The importance of these design charts is illustrated using a practical example. Site-specific conditions and the expertise and prior experience of a designer or operator must also be adequately considered and utilized with the design charts presented here for the safe design of a horizontal trench system in a bioreactor landfill.     

Hydrogen penetration in water through porous medium: application to a radioactive waste storage site

Hydrogen penetration in water through porous medium was analyzed in the paper. A two-phase compositional model approach was considered. The first part of the work deals with the thermodynamic analysis of the hydrogen–water system. The thermodynamic model was calibrated using the experimental data of hydrogen solubility in water. The phase densities, viscosities and phase concentrations were presented in an analytical form. Moreover, the domain of validity of analytical laws—such as Henry’s, Raoult’s and Kelvin’s laws—for the estimation of phase properties was presented for the analyzed system. The second part deals with two-phase hydrodynamic behaviors. An analytical solution for the non-compressible flow was constructed. In general case, the influence of relative permeabilities on the flow regimes was analyzed numerically. The notion pseudo-saturation was introduced to define phase appearance. Actually, mobile gas created a time displaced front relatively slower than mobile gas flow. Diffusion becomes really important for low mobile gas case as the penetration accelerates for the large range of saturation. In contrast, the mass exchange phenomena have a small influence on the flow type. Thus, the regimes of hydrogen penetration in liquid were shown really sensitive to the relative permeability form.     

大型灌区陆地水循环模式的参数化方案:LWCMPS_ID

地表水系统、地下水系统和土壤植物大气连续体的强烈耦合作用是大型灌区水文过程的基本特点,这导致对大型灌区的陆地水循环和水资源进行评价必须采取综合的方法,然而目前还缺少适用的模拟工具。农业灌区的水文特征与天然流域存在显著差别,常规流域水文模型和陆面过程参数化方案用于大型灌区陆地水循环的分析还存在较大的困难。LWCMPS_ID是本文提出的一个适用于大型灌区陆地水循环模式的参数化方案,采用分块集中参数模型简明地实现了地表水、地下水和土壤水的动力学耦合分析,并且包含了一个土壤水冻结融化的简化模型。对处在黄河流域的内蒙古河套灌区,用LWCMPS_ID进行20 年水文动态的模拟,取得了较好的效果。