A fast marching method for reservoir simulation

We present a fast marching level set method for reservoir simulation based on a fractional flow formulation of two-phase, incompressible, immiscible flow in two or three space dimensions. The method uses a fast marching approach and is therefore considerably faster than conventional finite difference methods. The fast marching approach compares favorably with a front tracking method as regards both efficiency and accuracy. In addition, it maintains the advantage of being able to handle changing topologies of the front structure. This revised version was published online in July 2006 with corrections to the Cover Date.     

浅海海底表层速度建模技术及其应用研究

浅海海底起伏和速度变化对OBC资料成像质量产生较大影响。建立浅海海底表层速度模型不仅能够解决OBC资料的静校正问题,也可用于海底反射系数计算、双检资料合并、多次波压制等,但是,目前针对浅海海底表层速度建模的研究还不多。文中提出了针对浅海地区OBC资料的海底表层速度建模的三维初至走时反演技术,主要包括:(1)震源位置校正技术。根据地震波在海水中传播特征,把在海水中激发的震源位置校正至海底,使震源和接收点都位于海底,利于初至走时反演计算;(2)快速三维初至走时反演方法。利用回折波走时和射线方程,形成了高效率初至走时反演方法。将该技术应用于胜利油田莱州湾浅海区海底OBC资料的处理中,建立了三维海底表层速度模型,用此速度模型进行静校正,取得了良好的应用效果。     

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.     

层状非均质结构包气带入渗过程单相流与两相流数值模拟对比研究

包气带水分入渗过程受多种因素的影响。定量研究层状非均质岩性结构和入渗速率对其影响,有助于解决根据不同条件选择单相流模型或水气二相流模型模拟包气带水分入渗过程的问题。结合填埋场等场地地层条件及污废水入渗特征,分别建立了“上细下粗”和“上粗下细”包气带层状非均质岩性结构水分入渗单相流和水气二相流模型,探讨不同层状非均质岩性结构条件下模型的适用性。在“上粗下细”岩性结构模型基础上,进一步探究入渗速率对水气两相运移结果的影响。基于论文模型研究表明:（1）在包气带岩性结构为“上细下粗”的条件下,气相的影响基本可以忽略,可直接采用单相流模型对包气带水分运移进行模拟;在“上粗下细”岩性结构和本次模型设定的底部压力保持不变及污废水泄漏前场地未接受降水入渗补给等条件下,当包气带上下层介质渗透率比值大于16倍时,气相会对水相运移产生明显影响,且下层介质渗透率越小、上下层介质渗透率比值越大,单相流与两相流的运移结果差别越大,需要采用水气二相流模型模拟包气带水分运移。（2）在包气带“上粗下细”岩性结构条件下,入渗速率越大,气相对水流入渗的阻滞作用越明显,此时包气带水分运移模拟应采用水气二相流模型。     

Streamline method for resolving sharp fronts for complex two-phase flow in porous media

In this paper, we present a fast streamline-based numerical method for the two-phase flow equations in high-rate flooding scenarios for incompressible fluids in heterogeneous and anisotropic porous media. A fractional flow formulation is adopted and a discontinuous Galerkin method (DG) is employed to solve the pressure equation. Capillary effects can be neglected in high-rate flooding scenarios. This allows us to present an improved streamline approach in combination with the one-dimensional front tracking method to solve the transport equation. To handle the high computational costs of the DG approximation, domain decomposition is applied combined with an algebraic multigrid preconditioner to solve the linear system. Special care at the interior interfaces is required and the streamline tracer has to include a dynamic communication strategy. The method is validated in various two- and three-dimensional tests, where comparisons of the solutions in terms of approximation of flow front propagation with standard fully implicit finite-volume methods are provided.     

Accuracy and efficiency of stencils for the eikonal equation in earth modelling

Motivated by the needs for creating fast and accurate models of complex geological scenarios, accuracy and efficiency of three stencils for the isotropic eikonal equation on rectangular grids are evaluated using a fast marching implementation. The stencils are derived by direct modelling of the wave front, resulting in new and valuable insight in terms of improved upwind and causality conditions. After introducing a method for generalising first-order upwind stencils to higher order, a new second-order diagonal stencil is presented. Similarly to the multistencil fast marching approach, the diagonal stencil makes use of nodes in the diagonal directions, whereas the traditional Godunov stencil uses solely edge-connected neighbours. The diagonal stencil uses nodes close to each other, reaching upwind, to get a more accurate estimate of the angle of incidence of the arriving wave front. Although the stencils are evaluated in a fast marching setting, they can be adapted to other efficient eikonal solvers. All first- and second-order stencils are evaluated in a range of tests. The first test case models a folded structure from the Zagros fold belt in Iran. The other test cases are constructed to investigate specific properties of the examined stencils. The numerical investigation considers convergence rates and CPU times for non-constant and constant speed first-arrival computations. In conclusion, the diagonal stencil is the most efficient and accurate of the three alternatives.     

Immiscible two-phase Darcy flow model accounting for vanishing and discontinuous capillary pressures: application to the flow in fractured porous media

Fully implicit time-space discretizations applied to the two-phase Darcy flow problem leads to the systems of nonlinear equations, which are traditionally solved by some variant of Newton’s method. The efficiency of the resulting algorithms heavily depends on the choice of the primary unknowns since Newton’s method is not invariant with respect to a nonlinear change of variable. In this regard, the role of capillary pressure/saturation relation is paramount because the choice of primary unknowns is restricted by its shape. We propose an elegant mathematical framework for two-phase flow in heterogeneous porous media resulting in a family of formulations, which apply to general monotone capillary pressure/saturation relations and handle the saturation jumps at rocktype interfaces. The presented approach is applied to the hybrid dimensional model of two-phase water-gas Darcy flow in fractured porous media for which the fractures are modelled as interfaces of co-dimension one. The problem is discretized using an extension of vertex approximate gradient scheme. As for the phase pressure formulation, the discrete model requires only two unknowns by degree of freedom.     

Distance parameterization for efficient seismic history matching with the ensemble Kalman Filter

The availability of multiple history matched models is essential for proper handling of uncertainty in determining the optimal development of producing hydrocarbon fields. The ensemble Kalman Filter in particular is becoming recognized as an efficient method for quantitative conditioning of multiple models to history data. It is known, however, that the ensemble Kalman Filter (EnKF) may have problems with finding solutions in history matching cases that are highly nonlinear and involve very large numbers of data, such is typical when time-lapse seismic surveys are available. Recently, a parameterization of seismic anomalies due to saturation effects was proposed in terms of arrival times of fronts that reduces both nonlinearity and the effective number of data. A disadvantage of the parameterization in terms of arrival times is that it requires simulation of models beyond the update time. An alternative distance parameterization is proposed here for flood fronts, or more generally, for isolines of arbitrary seismic attributes representing a front that removes the need for additional simulation time. An accurate fast marching method for solution of the Eikonal equation in Cartesian grids is used to calculate distances between observed and simulated fronts, which are used as innovations in the EnKF. Experiments are presented that demonstrate the functioning of the method in synthetic 2D and realistic 3D cases. Results are compared with those resulting from use of saturation data, as they could potentially be inverted from seismic data, with and without localization. The proposed algorithm significantly reduces the number of data while still capturing the essential information. It furthermore removes the need for seismic inversion when the oil-water front is only identified, and it produces a more favorable distribution of simulated data, leading to a very efficient and improved functioning of the EnKF.     

Development of efficient and robust Eikonal solver variants for first-arrival seismic modeling

Computational Geosciences - We have developed and tested a new Eikonal first-arrival forward model scheme by combining a fast marching method (FMM) algorithm, an upwind Eikonal solver scheme first...     

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.     

Image analysis algorithms for estimating porous media multiphase flow variables from computed microtomography data: a validation study

Image analysis of three-dimensional microtomographic image data has become an integral component of pore scale investigations of multiphase flow through porous media. This study focuses on the validation of image analysis algorithms for identifying phases and estimating porosity, saturation, solid surface area, and interfacial area between fluid phases from gray-scale X-ray microtomographic image data. The data used in this study consisted of (1) a two-phase high precision bead pack from which porosity and solid surface area estimates were obtained and (2) three-phase cylindrical capillary tubes of three different radii, each containing an air–water interface, from which interfacial area was estimated. The image analysis algorithm employed here combines an anisotropic diffusion filter to remove noise from the original gray-scale image data, a k-means cluster analysis to obtain segmented data, and the construction of isosurfaces to estimate solid surface area and interfacial area. Our method was compared with laboratory measurements, as well as estimates obtained from a number of other image analysis algorithms presented in the literature. Porosity estimates for the two-phase bead pack were within 1.5% error of laboratory measurements and agreed well with estimates obtained using an indicator kriging segmentation algorithm. Additionally, our method estimated the solid surface area of the high precision beads within 10% of the laboratory measurements, whereas solid surface area estimates obtained from voxel counting and two-point correlation functions overestimated the surface area by 20–40%. Interfacial area estimates for the air–water menisci contained within the capillary tubes were obtained using our image analysis algorithm, and using other image analysis algorithms, including voxel counting, two-point correlation functions, and the porous media marching cubes. Our image analysis algorithm, and other algorithms based on marching cubes, resulted in errors ranging from 1% to 20% of the analytical interfacial area estimates, whereas voxel counting and two-point correlation functions overestimated the analytical interfacial area by 20–40%. In addition, the sensitivity of the image analysis algorithms on the resolution of the microtomographic image data was investigated, and the results indicated that there was little or no improvement in the comparison with laboratory estimates for the resolutions and conditions tested.     

A two level scaling-up method for multiphase flow in porous media; numerical validation and comparison with other methods

We present a robust and accurate strategy for upscaling two-phase flow in heterogeneous porous media composed of different rock-types. The method is tested by means of numerical simulations and compared with other upscaling methods.     

A Methodology for Establishing a Data Reliability Measure for Value of Spatial Information Problems

We propose a value of information (VOI) methodology for spatial Earth problems. VOI is a tool to determine whether purchasing a new information source would improve a decision-makers’ chances of taking the optimal action. A prior uncertainty assessment of key geologic parameters and a reliability of the data to resolve them are necessary to make a VOI assessment. Both of these elements are challenging to obtain, as this assessment is made before the information is acquired. We present a flexible prior geologic uncertainty modeling scheme that allows for the inclusion of many types of spatial parameter. Next, we describe how to obtain a physics-based reliability measure by simulating the geophysical measurement on the generated prior models and interpreting the simulated data. Repeating this simulation and interpretation for all datasets, a frequency table can be obtained that describes how many times a correct or false interpretation was made by comparing them to their respective original model. This frequency table is the reliability measure and allows a more realistic VOI calculation. An example VOI calculation is demonstrated for a spatial decision related to aquifer recharge where two geophysical techniques are considered for their ability to resolve channel orientations. As necessitated by spatial problems, this methodology preserves the structure, influence and dependence of spatial variables through the prior geological modeling and the explicit geophysical simulation and interpretations.     

基于二维浅水方程的滑坡体兴波数值模型

基于有限体积方法和结构化网格,建立了海底滑坡引起的波浪传播数值模型。模型控制方程为考虑了海床随时间变化的二维浅水方程。采用中心迎风格式计算控制体界面数值通量,采用线性重构技术、局部海床高程处理技术和全隐式离散底摩阻项,保证了格式的和谐性、守恒性和水深非负性,有效处理了海岸动边界问题。时间积分采用具有强稳定性质的二阶龙格-库塔方法（ssp RK）。针对滑坡体兴波经典算例开展数值模拟,将计算结果与解析解、实验结果及其他模型计算结果进行比较和分析。结果表明,对于所考虑的计算工况,模型能较合理地模拟滑坡兴波的产生、传播和爬高过程。     

Numerical Homogenization of Two-Phase Flow in Porous Media

Homogenization has proved its effectiveness as a method of upscaling for linear problems, as they occur in single-phase porous media flow for arbitrary heterogeneous rocks. Here we extend the classical homogenization approach to nonlinear problems by considering incompressible, immiscible two-phase porous media flow. The extensions have been based on the principle of preservation of form, stating that the mathematical form of the fine-scale equations should be preserved as much as possible on the coarse scale. This principle leads to the required extensions, while making the physics underlying homogenization transparent. The method is process-independent in a way that coarse-scale results obtained for a particular reservoir can be used in any simulation, irrespective of the scenario that is simulated. Homogenization is based on steady-state flow equations with periodic boundary conditions for the capillary pressure. The resulting equations are solved numerically by two complementary finite element methods. This makes it possible to assess a posteriori error bounds.     

考虑气相影响的降雨入渗过程分析研究

降雨入渗过程是水在下渗的过程中驱替空气的水-气二相流过程,对这一过程的精确模拟一直是渗流计算的难点,目前的处理方法通常是忽略孔隙气压力变化的影响。根据多相流理论,结合质量守恒定律和达西定律,建立了水-气二相流模型,模型的求解采用积分有限差分法和Newton-Raphson迭代方法,通过变换主要变量来表达相态的变化,实现了水相、气相边界条件及降雨入渗边界的精确模拟。利用上述模型对一土柱试验进行模拟,从而验证了模型的正确性,研究了一均质土层的降雨入渗过程,得到了孔隙水压力、孔隙气压力和毛细压力及含水率的变化过程。根据入渗率与地表孔隙气压力的变化关系,验证了孔隙气压力的增大对入渗水流产生阻滞作用。在求解非稳定渗流问题中,考虑空气压力变化的影响是值得研究的。     

Detection of gravitational waves through observations of a group of pulsars

We suggest a new approach to the detection of gravitational waves using observations of a group of millisecond pulsars. In contrast to the usual method, based on increasing the accuracy of the arrival times of pulses by excluding possible distorting factors, our method supposes that the additive phase noise that is inevitably present even in the most accurate observational data has various spectral components, which have characteristic amplitudes and begin to appear on different time scales. We use the “Caterpillar” (Singular Spectral Analysis, SSA) method to decompose the signal into its components. Our initial data are the residuals of the pulse arrival times for six millisecond pulsars. We constructed the angular correlation function for components of the decomposition of a given number, whose theoretical form for the case of an isotropic and homogeneous gravitational-wave background is known. The individual decomposition components show a statistically significant agreement with the theoretical expectations (correlation coefficient ρ = 0.92 ± 0.10).     

库水位下降时的岸坡非稳定渗流问题研究

水位下降时岸坡的渗流是涉及土体由饱和向非饱和状态过渡的水-气二相流过程,目前相关研究成果大都假设孔隙气压力为0,忽略孔隙气的影响。根据水、空气的质量守恒定律和达西定律,结合多相流理论建立水-气二相流模型,采用高效的积分有限差分法求解,通过变换主要变量,实现饱和（单相）与非饱和（二相）的相互转变,并给出各种边界条件下合理的数学处理方法。通过Muskat渗流问题,验证了上述模型的正确性;并对某土质岸坡水位下降时的非稳定渗流问题进行分析,结果表明,岸坡的基质吸力小于浸润线以上的负孔隙水压力,在浸润线以上的很大区域为毛细管水饱和带,其土体饱和且基质吸力为0,这对边坡稳定十分不利,精确分析水位下降的边坡稳定问题时,孔隙气压力变化的影响值得研究。     

模拟多孔介质中生物化学输运的有限颗粒法的一个修正算法

给出了有限颗粒法(FCM) 的一个修正算法, 用来模拟二维多孔介质中复杂的物理、生物化学输运现象.该算法不仅具有与早先的FCM一样的优点, 而且可以在更微观的水平上保证质量守恒, 获得更为准确的颗粒位置, 从而有利于质量交换的高精度计算.计算结果与精确解和早先的FCM的结果做了比较.