On persistent primary variables for numerical modeling of gas migration in a nuclear waste repository

Numerical simulation of gas migration driven by compressible two-phase partially miscible flow in porous media is of major importance for safety assessment of deep geological repositories for long-lived high-level nuclear waste. We present modeling of compositional liquid and gas flow for numerical simulations of hydrogen migration in deep geological radioactive waste repository based on persistent primary variables. Two-phase flow is considered, with incompressible liquid and compressible gas, which includes capillary effects, gas dissolution, and diffusivity. After discussing briefly the existing approaches to deal with phase appearance and disappearance problem, including a persistent set of variables already considered in a previous paper (Bourgeat et al., Comput Geosci 13(1):29–42, 2009), we focus on a new variant of the primary variables: dissolved hydrogen mass concentration and liquid pressure. This choice leads to a unique and consistent formulation in liquid saturated and unsaturated regions, which is well adapted to heterogeneous media. We use this new set of variable for numerical simulations and show computational evidences of its adequacy to simulate gas phase appearance and disappearance in different but typical situations for gas migration in an underground radioactive waste repository.     

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.     

Numerical modeling of the flow and transport of radionuclides in heterogeneous porous media

This paper is concerned with numerical methods for the modeling of flow and transport of contaminant in porous media. The numerical methods feature the mixed finite element method over triangles as a solver to the Darcy flow equation and a conservative finite volume scheme for the concentration equation. The convective term is approximated with a Godunov scheme over the dual finite volume mesh, whereas the diffusion–dispersion term is discretized by piecewise linear conforming triangular finite elements. It is shown that the scheme satisfies a discrete maximum principle. Numerical examples demonstrate the effectiveness of the methodology for a coupled system that includes an elliptic equation and a diffusion–convection–reaction equation arising when modeling flow and transport in heterogeneous porous media. The proposed scheme is robust, conservative, efficient, and stable, as confirmed by numerical simulations.      

Two-phase, partially miscible flow and transport modeling in porous media; application to gas migration in a nuclear waste repository

We derive a compositional compressible two-phase, liquid and gas, flow model for numerical simulations of hydrogen migration in deep geological repository for radioactive waste. This model includes capillary effects and the gas high diffusivity. Moreover, it is written in variables (total hydrogen mass density and liquid pressure) chosen in order to be consistent with gas appearance or disappearance. We discuss the well possedness of this model and give some computational evidences of its adequacy to simulate gas generation in a water-saturated repository.     

Mathematical and numerical filtration–advection–dispersion model of miscible grout propagation in saturated porous media

The development of a predictive model of behaviour of porous media during injection of miscible grout, taking into account convection, dilution and filtration of grout solution with interstitial water, as well as consolidation aspects, is presented. Model assumptions are reviewed and discussed first. During the establishment of the model, we insist on surface terms and their physical relevance in expressing adsorption effects. Constitutive laws such as Fick's law for diffusive mass transport, hydrodynamic dispersion tensor dealing with miscibility, are modified by taking into account filtration effects. A new surface term appears in mass balance equations as a consequence of filtration. According to the filtration laws used, an initial filtration rate is estimated on the basis of a one‐dimensional experimental campaign. The field equations are discretized by using Galerkin finite element and θ‐scheme standard method. For transport equation, Streamline Upwind Petrov Galerkin method is employed to prevent numerical oscillations. Lastly, confrontation of numerical results with laboratory experiments constitutes a first step to validate the model on a realistic basis. Copyright © 2001 John Wiley & Sons, Ltd.     

Vertex-centred discretization of multiphase compositional Darcy flows on general meshes

This paper concerns the discretisation on general 3D meshes of multiphase compositional Darcy flows in heterogeneous anisotropic porous media. Extending Coats’ formulation [15] to an arbitrary number of phases, the model accounts for the coupling of the mass balance of each component with the pore volume conservation and the thermodynamical equilibrium and dynamically manages phase appearance and disappearance. The spatial discretisation of the multiphase compositional Darcy flows is based on a generalisation of the Vertex Approximate Gradient scheme, already introduced for single-phase diffusive problems in [24]. It leads to an unconditionally coercive scheme for arbitrary meshes and permeability tensors. The stencil of this vertex-centred scheme typically comprises 27 points on topologically Cartesian meshes, and the number of unknowns on tetrahedral meshes is considerably reduced, compared with the usual cell-centred approaches. The efficiency of our approach is exhibited on several examples, including the nearwell injection of miscible CO₂ in a saline aquifer taking into account the vaporisation of H₂O in the gas phase as well as the precipitation of salt.     

Insight from analytical solutions for improved simulation of miscible WAG flooding in one dimension

In this work, the analytical and numerical solutions for modeling miscible gas and water injection into an oil reservoir are presented. Conservation laws with three levels of complexity are considered. Only the most complex model has the correct phase behavior for the example system, which is a multicontact miscible condensing gas drive with simultaneous water and gas injection. Example displacements in which one or both of the simpler models result in accurate simulations in a fraction of the computation time are presented, along with an example in which neither simplified thermodynamic model achieves a truly satisfactory result. A methodology is presented that can be used to establish the accuracy of simplified models in 1-D simulation based on convergence to analytical solutions for the full three-phase system.     

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 coarse-scale compositional model

In subsurface flow modeling, compositional simulation is often required to model complex recovery processes, such as gas/CO ₂ injection. However, compositional simulation on fine-scale geological models is still computationally expensive and even prohibitive. Most existing upscaling techniques focus on black-oil models. In this paper, we present a general framework to upscale two-phase multicomponent flow in compositional simulation. Unlike previous studies, our approach explicitly considers the upscaling of flow and thermodynamics. In the flow part, we introduce a new set of upscaled flow functions that account for the effects of compressibility. This is often ignored in the upscaling of black-oil models. In the upscaling of thermodynamics, we show that the oil and gas phases within a coarse block are not at chemical equilibrium. This non-equilibrium behavior is modeled by upscaled thermodynamic functions, which measure the difference between component fugacities among the oil and gas phases. We apply the approach to various gas injection problems with different compositional features, permeability heterogeneity, and coarsening ratios. It is shown that the proposed method accurately reproduces the averaged fine-scale solutions, such as component overall compositions, gas saturation, and density solutions in the compositional flow.     

An empirical theory for gravitationally unstable flow in porous media

In this paper, we follow a similar procedure as proposed by Koval (SPE J 3(2):145–154, 1963) to analytically model CO₂ transfer between the overriding carbon dioxide layer and the brine layer below it. We show that a very thin diffusive layer on top separates the interface from a gravitationally unstable convective flow layer below it. Flow in the gravitationally unstable layer is described by the theory of Koval, a theory that is widely used and which describes miscible displacement as a pseudo two-phase flow problem. The pseudo two-phase flow problem provides the average concentration of CO₂ in the brine as a function of distance. We find that downstream of the diffusive layer, the solution of the convective part of the model, is a rarefaction solution that starts at the saturation corresponding to the highest value of the fractional-flow function. The model uses two free parameters, viz., a dilution factor and a gravity fingering index. A comparison of the Koval model with the horizontally averaged concentrations obtained from 2-D numerical simulations provides a correlation for the two parameters with the Rayleigh number. The obtained scaling relations can be used in numerical simulators to account for the density-driven natural convection, which cannot be currently captured because the grid cells are typically orders of magnitude larger than the wavelength of the initial fingers. The method can be applied both for storage of greenhouse gases in aquifers and for EOR processes using carbon dioxide or other solvents.     

Multiphase flow modeling with density functional method

This paper is a review of applications of density functional theory (DFT) in compositional hydrodynamics. The basic idea is representation of the entropy or the Helmholtz energy of the mixture as the functional depending on the molar densities of chemical components (density functional). The hydrodynamics is governed by local conservation laws of chemical components, momentum, and energy, while constitutive relations and boundary conditions are introduced in accordance with the explicit form of the density functional. The general ideas and the history of the DFT in compositional hydrodynamics are discussed. Then the DFT for multiphase multicomponent mixtures is presented including the exposition of the first principles, governing equations and constitutive relations, and explicit expressions of density functional depending on physical situation. The DFT-based numerical simulator is described, and several multiphase simulation results are presented to illustrate the scope and effectiveness of DFT: sessile drop with and without surfactant, droplet breakup in shear flow, and three-phase hydrodynamics with mobile solid phase. Also, two practical scenarios with multiphase simulations in micro-CT porous rock models are presented: two-phase immiscible water-oil flow and three-phase water-gas-condensate flow with phase transitions. All numerical results are obtained by essentially the same code; both the number of chemical components and the Helmholtz energy have been set up in accordance with physical situation.     

On the time scales of nonlinear instability in miscible displacement porous media flow

In this paper, we analyze the time scales associated with instable fingering induced by density contrasts in miscible displacement porous media flow. We perform numerical simulations of a two-dimensional domain with boundaries that are closed to flow and identify the three regimes of the dynamics, namely the development of a stable diffusive boundary layer, the onset and growth of instabilities, and the fully nonlinear dynamics. Special focus is given to the onset of the fully nonlinear regime. The results are generic in the sense that there are no parameters in the non-dimensional model problem. Large ensembles are studied and an error estimate is given based on the combined effect of numerical errors and sampling errors. The nonlinear time scales show a dependence on the size of initial perturbations. We estimate this size for three formations used for CO₂ storage and find that the onset of enhanced convective mixing is considerably delayed compared with the linear onset time.     

Real‐scale miscible grout injection experiment and performance of advection–dispersion–filtration model

A model was developed, to describe miscible grout propagation in a saturated deformable porous medium, based on Bear's statistical model with spatial volume averaging. In a previous paper, the model was first successfully confronted to one‐dimensional laboratory experiments. In the present paper, the numerical model is used to simulate practical grouting operation in a cylindrical injection model. The cylindrical injection model lends itself to study main flow and propagation character istics for a dispersed suspension‐type grout, under axisymmetric conditions close to real scale conditions. Comparison between numerical solutions and experimental results is essential to confirm the validity and accuracy of the proposed model from a phenomenological standpoint. The numerical model performances show that the underlying mathematical model constitutes a realistic predictive model reproducing most prominent features during injection of a suspension‐type grout into a deformable porous medium. The basic mechanism by which injected miscible grout permeates a soil mass is discussed in detail. Such a tool leads to quality control criteria for grouting on a theoretical basis, which complements existing criteria acquired through engineering practice. Copyright © 2001 John Wiley & Sons, Ltd.     

Method of negative saturations for flow with variable number of phases in porous media: extension to three-phase multi-component case

Various EOR methods lead to the appearance of specific macroscopic surfaces called interfaces of phase transition (IPT) such that the number of phases on two sides of an IPT is different, and fluids separated by an IPT are in non-equilibrium. Therefore, the flow equations are also different on two sides of an IPT and cannot be deduced from each other by a continuous degeneration, which imposes difficulties in numerical modelling. To describe such systems, we developed a new conceptual mathematical method based on the replacement of the real single-phase fluid by an imaginary multi-phase multi-component continuum having fictitious properties. As the result, the fluid over all zones becomes multi-phase and can be described by uniform multi-phase hydro- and thermodynamic equations, which allows applying the direct numerical simulation. The equivalence principle determines the physical properties of the fictitious multi-phase fluid, as well as the structure of the uniform multi-phase equations. It also proves that the saturation of each phase becomes an extended function negative or higher than unity in non-equilibrium zones, which becomes the efficient method of tracking the interfaces, the number of phases at any point, and their degree of disequilibrium. The method was developed in [1, 2] for the two-phase case. In the present paper, the new version of the method is developed for the three-phase case with gravity, diffusion, and capillarity. We have obtained the new equivalent uniform multi-phase equations which contain additional non-classical terms responsible for the diffusion and gravity across an IPT. The comparison with classical method is presented. The presentation is illustrated by several examples of simulation by means of the code developed by the research group; their concern: EOR by miscible methods and CO ₂ bubble raising in aquifer.     

A coupled model for solid deformation and gas leak flow

From the viewpoint of interaction mechanics of solid and gas, a coupled mathematical model is presented for solid coal/rock‐mass deformation and gas leak flow in parallel deformable coal seams. Numerical solutions using the strong implicit procedure (SIP) method to the coupled mathematical model for double parallel coal seams are also developed in detail. Numerical simulations for the prediction of safety range using protection layer mining are performed with experimental data from a mine with potential danger of coal/gas outbursts. Analyses show that the numerical simulation results are consistent with the measured data on the spot. The coupled model shows a positive future for applications in a wide range of gas‐leak‐flow‐related problems in mining engineering, gas drainage engineering and mining safety engineering. Copyright © 2004 John Wiley & Sons, Ltd.     

A locally conservative Eulerian–Lagrangian numerical method and its application to nonlinear transport in porous media

Eulerian-Lagrangian and Modified Method of Characteristics (MMOC) procedures provide computationally efficient techniques for approximating the solutions of transport-dominated diffusive systems. The original MMOC fails to preserve certain integral identities satisfied by the solution of the differential system; the recently introduced variant, called the MMOCAA, preserves the global form of the identity associated with conservation of mass in petroleum reservoir simulations, but it does not preserve a localized form of this identity. Here, we introduce an Eulerian-Lagrangian method related to these MMOC procedures that guarantees conservation of mass locally for the problem of two-phase, immiscible, incompressible flow in porous media. The computational efficiencies of the older procedures are maintained. Both the original MMOC and the MMOCAA procedures for this problem are derived from a nondivergence form of the saturation equation; the new method is based on the divergence form of the equation. A reasonably extensive set of computational experiments are presented to validate the new method and to show that it produces a more detailed picture of the local behavior in waterflooding a fractally heterogeneous medium. A brief discussion of the application of the new method to miscible flow in porous media is included. This revised version was published online in July 2006 with corrections to the Cover Date.     

SEAWAT 2000: modelling unstable flow and sensitivity to discretization levels and numerical schemes

A systematic analysis shows how results from the finite difference code SEAWAT are sensitive to choice of grid dimension, time step, and numerical scheme for unstable flow problems. Guidelines to assist in selecting appropriate combinations of these factors are suggested. While the SEAWAT code has been tested for a wide range of problems, the sensitivity of results to spatial and temporal discretization levels and numerical schemes has not been studied in detail for unstable flow problems. Here, the Elder-Voss-Souza benchmark problem has been used to systematically explore the sensitivity of SEAWAT output to spatio-temporal resolution and numerical solver choice. A grid size of 0.38 and 0.60% of the total domain length and depth respectively is found to be fine enough to deliver results with acceptable accuracy for most of the numerical schemes when Courant number (Cr) is 0.1. All numerical solvers produced similar results for extremely fine meshes; however, some schemes converged faster than others. For instance, the 3rd-order total variation-diminishing method (TVD3) scheme converged at a much coarser mesh than the standard finite difference methods (SFDM) upstream weighting (UW) scheme. The sensitivity of the results to Cr number depends on the numerical scheme as expected.     

煤层气羽状水平井数值模拟模型的交替隐式算法

煤层气开采过程中通常采用数值模拟方法进行生产指标预测,其中的数值模型普遍采用三维全隐式差分法求解,模型涉及的参数变量多,求解难度大,计算复杂。根据煤层气的储气特征、吸附特征和解吸机理建立了反映煤层气解吸、扩散及渗流过程的气、水两相流动煤层气羽状水平井三维数学模型,模型考虑了煤层非均质各向异性的影响,运用块中心差分格式,对气、水相偏微分方程进行空间差分和时间差分,采用交替隐式(ADI)法求解,实现了煤层气羽状水平井开发数值模拟计算。交替方向格式可以把二维问题转化成一维问题,对x,y两个方向的迭代矩阵均为三对角矩阵,结构相同,易于编程计算。所建立的差分格式具有计算量小,稳定性好等优点,数值试验的结果表明效果良好。     

Using vorticity to quantify the relative importance of heterogeneity, viscosity ratio, gravity and diffusion on oil recovery

The vorticity of the displacement velocity is used to derive dimensionless numbers that can be used to quantify the relative importance of viscosity ratio, gravity, diffusion/dispersion and permeability heterogeneity on secondary hydrocarbon recovery. Using this approach, a new objective measure of the impact of permeability and porosity heterogeneity on reservoir performance is obtained. This is used, in conjunction with other dimensionless numbers, to analyse the relative impact of heterogeneity, buoyancy effects, mobility ratio and dispersion on breakthrough time and recovery at 1 pore volume injected during first contact miscible gas injection. This is achieved using results obtained from detailed simulation of miscible displacements through a range of geologically realistic reservoir models. This study goes some way towards developing a unified mathematical framework to determine under which flow conditions reservoir heterogeneity becomes more important than other physical processes. We propose that comparison of these dimensionless numbers can be used to identify the key factors controlling recovery and thus assist the engineer in determining appropriate enhanced oil recovery techniques to improve recovery.     

基于流网单元的污染物优势运移数值模型

垃圾填埋场滤出液、入侵海水、核废物及生产生活废水等污染物随地下水的迁移威胁人类生存。地下水渗流的随机性导致溶质运移问题更加复杂。根据流网特点,利用流线与水头等势线对求解域进行离散,基于质量守恒原理建立流网单元内溶质浓度求解的隐式有限体积差分格式。基于多孔介质孔隙流速分布规律,利用蒙特卡洛法建立流管单元随机流速场进行溶质运移过程的数值模拟,最后根据数值模拟和模型试验的结果对比,验证了数值模拟方法的准确性。基于流网单元的数值模型中沿流线方向的物质交换由对流和扩散共同作用,而流管间的物质交换只有扩散作用,因此,可在不使用弥散系数下进行污染物运移的模拟。引入随机方法确定流管内流速为研究非均匀流场中染污物的优势迁移提供了新的思路。