Simulation of axis-symmetric seismic waves in fluid-filled boreholes in the presence of a drill string

A numerical algorithm for simulation of 2-D (axis-symmetric) wave propagation using a multidomain approach is proposed. The method uses a cylindrical coordinate system, Chebyshev and Fourier differential operators to calculate the spatial derivatives along the radial and vertical direction, respectively, and a Runge–Kutta time-integration scheme. The numerical technique is based on the solution of the equations of momentum conservation combined with the stress–strain relations of the fluid (drilling mud) and isotropic elastic media (drill string and formation). Wave modes and radiated waves are simulated in the borehole-formation system. The algorithm satisfies the reciprocity condition and the results agree with an analytical solution and low-frequency simulation of wave-propagation modes reported in the literature. Examples illustrating the propagation of waves are presented for hard and soft formations. Moreover, the presence of casing, cement, and formation heterogeneity have been considered. Since the algorithm is based on a direct (grid) method, the geometry and the properties defining the media at each grid point, can be general, i.e., there are no limitations such as planar interfaces or uniform (homogeneous) properties for each medium.      

Upscaling Hydraulic Conductivities in Cross-Bedded Formations

Modern geostatistical techniques allow the generation of high-resolution heterogeneous models of hydraulic conductivity containing millions to billions of cells. Selective upscaling is a numerical approach for the change of scale of fine-scale hydraulic conductivity models into coarser scale models that are suitable for numerical simulations of groundwater flow and mass transport. Selective upscaling uses an elastic gridding technique to selectively determine the geometry of the coarse grid by an iterative procedure. The geometry of the coarse grid is built so that the variances of flow velocities within the coarse blocks are minimum. Selective upscaling is able to handle complex geological formations and flow patterns, and provides full hydraulic conductivity tensor for each block. Selective upscaling is applied to a cross-bedded formation in which the fine-scale hydraulic conductivities are full tensors with principal directions not parallel to the statistical anisotropy of their spatial distribution. Mass transport results from three coarse-scale models constructed by different upscaling techniques are compared to the fine-scale results for different flow conditions. Selective upscaling provides coarse grids in which mass transport simulation is in good agreement with the fine-scale simulations, and consistently superior to simulations on traditional regular (equal-sized) grids or elastic grids built without accounting for flow velocities.     

Multiscale Parameterization of Petrophysical Properties for Efficient History-Matching

The prediction of fluid flows within hydrocarbon reservoirs requires the characterization of petrophysical properties. Such characterization is performed on the basis of geostatistics and history-matching; in short, a reservoir model is first randomly drawn, and then sequentially adjusted until it reproduces the available dynamic data. Two main concerns typical of the problem under consideration are the heterogeneity of rocks occurring at all scales and the use of data of distinct resolution levels. Therefore, referring to sequential Gaussian simulation, this paper proposes a new stochastic simulation method able to handle several scales for both continuous or discrete random fields. This method adds flexibility to history-matching as it boils down to the multiscale parameterization of reservoir models. In other words, reservoir models can be updated at either coarse or fine scales, or both. Parameterization adapts to the available data; the coarser the scale targeted, the smaller the number of unknown parameters, and the more efficient the history-matching process. This paper focuses on the use of variational optimization techniques driven by the gradual deformation method to vary reservoir models. Other data assimilation methods and perturbation processes could have been envisioned as well. Last, a numerical application case is presented in order to highlight the advantages of the proposed method for conditioning permeability models to dynamic data. For simplicity, we focus on two-scale processes. The coarse scale describes the variations in the trend while the fine scale characterizes local variations around the trend. The relationships between data resolution and parameterization are investigated.     

Sensitivity Analysis of a Combined Groundwater Flow and Solute Transport Model Using Local-Grid Refinement: A Case Study

Combining groundwater flow models with solute transport models represents a common challenge in groundwater resources assessments and contaminant transport modeling. Groundwater flow models are usually constructed at somewhat larger scales (involving a coarser discretization) to include natural boundary conditions. They are commonly calibrated using observed groundwater levels and flows (if available). The groundwater solute transport models may be constructed at a smaller scale with finer discretization than the flow models in order to accurately delineate the solute source and the modeled target, to capture any heterogeneity that may affect contaminant migration, and to minimize numerical dispersion while still maintaining a reasonable computing time. The solution that is explored here is based on defining a finer grid subdomain within a larger coarser domain. The local-grid refinement (LGR) implemented in the Modular 3D finite-difference ground-water flow model (MODFLOW) code has such a provision to simulate groundwater flow in two nested grids: a higher-resolution sub-grid within a coarse grid. Under the premise that the interface between both models was well defined, a comprehensive sensitivity and uncertainty analysis was performed whereby the effect of a parameter perturbation in a coarser-grid model on transport predictions using a higher-resolution grid was quantified. This approach was tested for a groundwater flow and solute transport analysis in support of a safety evaluation of the future Belgian near-surface radioactive waste disposal facility. Our reference coarse-grid groundwater flow model was coupled with a smaller fine sub-grid model in two different ways. While the reference flow model was calibrated using observed groundwater levels at a scale commensurate with that of the coarse-grid model, the fine sub-grid model was used to run a solute transport simulation quantifying concentrations in a hypothetical well nearby the disposal facility. When LGR coupling was compared to a one-way coupling, LGR was found to provide a smoother flow solution resulting in a more CPU-efficient transport solution. Parameter sensitivities performed with the groundwater flow model resulted in sensitivities at the head observation locations. These sensitivities identified the recharge as the most sensitive parameter, with the hydraulic conductivity of the upper aquifer as the second most sensitive parameter in regard to calculated groundwater heads. Based on one-percent sensitivity maps, the spatial distribution of the observations with the highest sensitivities is slightly different for the upper aquifer hydraulic conductivity than for recharge. Sensitivity analyses were further performed to assess the prediction scaled sensitivities for hypothetical contaminant concentrations using the combined groundwater flow and solute transport models. Including all pertinent parameters into the sensitivity analysis identified the hydraulic conductivity of the upper aquifer as the most sensitive parameter with regard to the prediction of contaminant concentrations.     

Successful application of multiscale methods in a real reservoir simulator environment

For the past 10 years or so, a number of so-called multiscale methods have been developed as an alternative approach to upscaling and to accelerate reservoir simulation. The key idea of all these methods is to construct a set of prolongation operators that map between unknowns associated with cells in a fine grid holding the petrophysical properties of the geological reservoir model and unknowns on a coarser grid used for dynamic simulation. The prolongation operators are computed numerically by solving localized flow problems, much in the same way as for flow-based upscaling methods, and can be used to construct a reduced coarse-scale system of flow equations that describe the macro-scale displacement driven by global forces. Unlike effective parameters, the multiscale basis functions have subscale resolution, which ensures that fine-scale heterogeneity is correctly accounted for in a systematic manner. Among all multiscale formulations discussed in the literature, the multiscale restriction-smoothed basis (MsRSB) method has proved to be particularly promising. This method has been implemented in a commercially available simulator and has three main advantages. First, the input grid and its coarse partition can have general polyhedral geometry and unstructured topology. Secondly, MsRSB is accurate and robust when used as an approximate solver and converges relatively fast when used as an iterative fine-scale solver. Finally, the method is formulated on top of a cell-centered, conservative, finite-volume method and is applicable to any flow model for which one can isolate a pressure equation. We discuss numerical challenges posed by contemporary geomodels and report a number of validation cases showing that the MsRSB method is an efficient, robust, and versatile method for simulating complex models of real reservoirs.     

Implementation of a Locally Conservative Numerical Subgrid Upscaling Scheme for Two-Phase Darcy Flow

We present a locally mass conservative scheme for the approximation of two-phase flow in a porous medium that allows us to obtain detailed fine scale solutions on relatively coarse meshes. The permeability is assumed to be resolvable on a fine numerical grid, but limits on computational power require that computations be performed on a coarse grid. We define a two-scale mixed finite element space and resulting method, and describe in detail the solution algorithm. It involves a coarse scale operator coupled to a subgrid scale operator localized in space to each coarse grid element. An influence function (numerical Greens function) technique allows us to solve these subgrid scale problems independently of the coarse grid approximation. The coarse grid problem is modified to take into account the subgrid scale solution and solved as a large linear system of equations posed over a coarse grid. Finally, the coarse scale solution is corrected on the subgrid scale, providing a fine grid representation of the solution. Numerical examples are presented, which show that near-well behavior and even extremely heterogeneous permeability barriers and streaks are upscaled well by the technique.     

A multiscale method for distributed parameter estimation with application to reservoir history matching

A method for multiscale parameter estimation with application to reservoir history matching is presented. Starting from a given fine-scale model, coarser models are generated using a global upscaling technique where the coarse models are tuned to match the solution of the fine model. Conditioning to dynamic data is done by history-matching the coarse model. Using consistently the same resolution both for the forward and inverse problems, this model is successively refined using a combination of downscaling and history matching until model-matching dynamic data are obtained at the finest scale. Large-scale corrections are obtained using fast models, which, combined with a downscaling procedure, provide a better initial model for the final adjustment on the fine scale. The result is thus a series of models with different resolution, all matching history as good as possible with this grid. Numerical examples show that this method may significantly reduce the computational effort and/or improve the quality of the solution when achieving a fine-scale match as compared to history-matching directly on the fine scale.     

A new grid-cell-based method for error evaluation of vector-to-raster conversion

Error evaluation of rasterization of vector data is one of the most important research topics in the field of geographical information systems. Current methods for evaluating rasterization errors are far from perfect and need further improvement. The objective of this study is to introduce a new error evaluation method that is based on grid cells (EEM-BGC). The EEM-BGC follows four steps. First, the area of each land category inside a square is represented in a vector format. The size and location of the square are exactly the same as those of a grid cell that is to be generated by rasterization. Second, the area is treated as the attribute of the grid cell. Vector data are rasterized into n grids, where n is the number of land categories. Then, the relative area error resulting from rasterization for each land category in the grid cell is calculated in raster format. Lastly, the average of the relative area error for all land categories in the grid cell is computed with the area of a land category as weight. As a case study, the EEM-BGC is applied for evaluating the rasterization error of the land cover data of Beijing at a scale of 1 to 250,000. It is found that the error derived from a conventional method (denoted as y) is significantly underestimated in comparison with that derived from the new method (denoted as x), with y = 0.0014x ^2.6667. The EEM-BGC is effective in capturing not only the spatial distribution of rasterization errors at the grid-cell level but also the numerical distribution range of the errors. The EEM-BGC is more objective and accurate than any conventional method that is used for evaluating rasterization errors.     

Integration of local–global upscaling and grid adaptivity for simulation of subsurface flow in heterogeneous formations

We propose a methodology, called multilevel local–global (MLLG) upscaling, for generating accurate upscaled models of permeabilities or transmissibilities for flow simulation on adapted grids in heterogeneous subsurface formations. The method generates an initial adapted grid based on the given fine-scale reservoir heterogeneity and potential flow paths. It then applies local–global (LG) upscaling for permeability or transmissibility [7], along with adaptivity, in an iterative manner. In each iteration of MLLG, the grid can be adapted where needed to reduce flow solver and upscaling errors. The adaptivity is controlled with a flow-based indicator. The iterative process is continued until consistency between the global solve on the adapted grid and the local solves is obtained. While each application of LG upscaling is also an iterative process, this inner iteration generally takes only one or two iterations to converge. Furthermore, the number of outer iterations is bounded above, and hence, the computational costs of this approach are low. We design a new flow-based weighting of transmissibility values in LG upscaling that significantly improves the accuracy of LG and MLLG over traditional local transmissibility calculations. For highly heterogeneous (e.g., channelized) systems, the integration of grid adaptivity and LG upscaling is shown to consistently provide more accurate coarse-scale models for global flow, relative to reference fine-scale results, than do existing upscaling techniques applied to uniform grids of similar densities. Another attractive property of the integration of upscaling and adaptivity is that process dependency is strongly reduced, that is, the approach computes accurate global flow results also for flows driven by boundary conditions different from the generic boundary conditions used to compute the upscaled parameters. The method is demonstrated on Cartesian cell-based anisotropic refinement (CCAR) grids, but it can be applied to other adaptation strategies for structured grids and extended to unstructured grids.     

Coarse-scale data assimilation as a generic alternative to localization

Coarse-scale data assimilation (DA) with large ensemble size is proposed as a robust alternative to standard DA with localization for reservoir history matching problems. With coarse-scale DA, the unknown property function associated with each ensemble member is upscaled to a grid significantly coarser than the original reservoir simulator grid. The grid coarsening is automatic, ensemble-specific and non-uniform. The selection of regions where the grid can be coarsened without introducing too large modelling errors is performed using a second-generation wavelet transform allowing for seamless handling of non-dyadic grids and inactive grid cells. An inexpensive local-local upscaling is performed on each ensemble member. A DA algorithm that restarts from initial time is utilized, which avoids the need for downscaling. Since the DA computational cost roughly equals the number of ensemble members times the cost of a single forward simulation, coarse-scale DA allows for a significant increase in the number of ensemble members at the same computational cost as standard DA with localization. Fixing the computational cost for both approaches, the quality of coarse-scale DA is compared to that of standard DA with localization (using state-of-the-art localization techniques) on examples spanning a large degree of variability. It is found that coarse-scale DA is more robust with respect to variation in example type than each of the localization techniques considered with standard DA. Although the paper is concerned with two spatial dimensions, coarse-scale DA is easily extendible to three spatial dimensions, where it is expected that its advantage with respect to standard DA with localization will increase.     

Upscaling equivalent hydrofacies simulation based on borehole data generalization and transition probability geostatistics

The heterogeneity of hydrofacies is represented as spatial variability on different scales, and it has a significant impact on the behavior of groundwater flow and pollutant transport. However, effectively characterizing hydrofacies heterogeneity on different scales remains one of the most challenging problems in hydrogeology. In this study, an upscaling hydrofacies simulation (UHS) framework is proposed by integrating the upscaling borehole generalization (UBG) approach and transition probability geostatistics (TPG). A new UBG approach for generating virtual boreholes with equivalent hydrofacies information based on relatively high-density borehole lithological data is proposed, and the TPG is used to delineate the multiscale facies distribution. The results show that the UBG approach can significantly reduce borehole data volume while retaining the key equivalent hydrofacies information on a coarser scale. The UHS method can well characterize the overall distribution of equivalent hydrofacies on coarser scales, with the minor-component hydrofacies underestimated and the major-component hydrofacies overestimated to a lesser extent, and more equivalent facies appearing in strong heterogeneous areas. These results demonstrate that the UHS method can provide valuable capacity insights and advantages in characterizing hydrofacies heterogeneity on different scales using such high-density borehole lithological data.

     

A genetic algorithm approach for assessing soil liquefaction potential based on reliability method

Deterministic approaches are unable to account for the variations in soil’s strength properties, earthquake loads, as well as source of errors in evaluations of liquefaction potential in sandy soils which make them questionable against other reliability concepts. Furthermore, deterministic approaches are incapable of precisely relating the probability of liquefaction and the factor of safety (FS). Therefore, the use of probabilistic approaches and especially, reliability analysis is considered since a complementary solution is needed to reach better engineering decisions. In this study, Advanced First-Order Second-Moment (AFOSM) technique associated with genetic algorithm (GA) and its corresponding sophisticated optimization techniques have been used to calculate the reliability index and the probability of liquefaction. The use of GA provides a reliable mechanism suitable for computer programming and fast convergence. A new relation is developed here, by which the liquefaction potential can be directly calculated based on the estimated probability of liquefaction (P _L ), cyclic stress ratio (CSR) and normalized standard penetration test (SPT) blow counts while containing a mean error of less than 10% from the observational data. The validity of the proposed concept is examined through comparison of the results obtained by the new relation and those predicted by other investigators. A further advantage of the proposed relation is that it relates P _L and FS and hence it provides possibility of decision making based on the liquefaction risk and the use of deterministic approaches. This could be beneficial to geotechnical engineers who use the common methods of FS for evaluation of liquefaction. As an application, the city of Babolsar which is located on the southern coasts of Caspian Sea is investigated for liquefaction potential. The investigation is based primarily on in situ tests in which the results of SPT are analysed.     

The methodological basis for fine‐resolution,multi‐proxy reconstructions of ombrotrophic peat bog surface wetness

Amesbury, M. J., Barber, K. E. & Hughes, P. D. M. 2010: The methodological basis for fine‐resolution, multi‐proxy reconstructions of ombrotrophic peat bog surface wetness. Boreas, 10.1111/j.1502‐3885.2010.00152.x. ISSN 0300‐9483. The need for Holocene peat‐based palaeoclimatic records of increased temporal resolution has been widely identified in recent research. The often rapid growth rates of ombrotrophic bogs, when combined with fine‐resolution (i.e. millimetre‐scale) sampling, provide an as yet largely unexploited potential to derive sub‐decadal palaeoclimatic data from this proxy‐archive. However, multi‐proxy, fine‐resolution analyses require changes to standard methodologies, and the application of sampling techniques that are new to peat‐based palaeoclimate research. A peat sampler was custom‐built to allow precise and replicable millimetre‐scale subsampling. Subsequent methodological testing revealed that, irrespective of sample thickness (i.e. resolution), halving the standard sample volume used for plant macrofossil (from 4 cm³ to 2 cm³) and testate amoebae (from 2 cm³ to 1 cm³) analyses and the sample weight used for peat humification analysis (from 0.2 g to 0.1 g dried peat) did not affect the interpretation of the results. A contiguous 1‐mm sampling resolution for plant macrofossil analysis was also tested, but it was found that contiguous 5‐mm samples provided a more reliable background record to fine‐resolution testate amoebae and peat humification analyses. Based on these findings, a standardized and systematic methodological approach was developed, using the custom‐built peat slicer to take millimetre‐scale samples that provide enough sample material for both testate amoebae and peat humification analyses to be performed at 1‐mm resolution. This approach will facilitate the testing of the palaeoclimatic reliability of multi‐proxy, fine‐resolution peat‐based records.     

A multi-scale particle-tracking framework for dispersive solute transport modeling

Particle-tracking simulation offers a fast and robust alternative to conventional numerical discretization techniques for modeling solute transport in subsurface formations. A common challenge is that the modeling scale is typically much larger than the volume scale over which measurements of rock properties are made, and the scale-up of measurements have to be made accounting for the pattern of spatial heterogeneity exhibited at different scales. In this paper, a statistical scale-up procedure developed in our previous work is adopted to estimate coarse-scale (effective) transition time functions for transport modeling, while two significant improvements are proposed: considering the effects of non-stationarity (trend), as well as unresolved (residual) heterogeneity below the fine-scale model. Rock property is modeled as a multivariate random function, which is decomposed into the sum of a trend (which is defined at the same resolution of the transport modeling scale) and a residual (representing all heterogeneities below the transport modeling scale). To construct realizations of a given rock property at the transport modeling scale, multiple realizations of the residual components are sampled. Next, a flow-based technique is adopted to compute the effective transport parameters: firstly, it is assumed that additional unresolved heterogeneities occurring below the fine scale can be described by a probabilistic transit time distribution; secondly, multiple realizations of the rock property, with the same physical size as the transport modeling scale, are generated; thirdly, each realization is subjected to particle-tracking simulation; finally, probability distributions of effective transition time function are estimated by matching the corresponding effluent history for each realization with an equivalent medium consisting of averaged homogeneous rock properties and aggregating results from all realizations. The proposed method is flexible that it does not invoke any explicit assumption regarding the multivariate distribution of the heterogeneity.     

An interactive global optimization algorithm for geological problems

A number of problems in geology can be formulated so that they consist of optimizing a real-valued function (termed the objective function) on some interval or over some region. Many methods are available for solution if the function is unimodal within the domain of interest. Direct methods, involving only function evaluations, are particularly useful in geological problems where the objective function may be strongly nonlinear and constructed from sampled data. In practical problems, the objective function often is not unimodal. Standard optimization routines are not capable of distinguishing between local extrema or of locating the global extremum, which is the point of interest in most cases. The usual approach—trying several different starting points in the hope that the best local extremum found is the global extremum—is inefficient and unreliable. An ancillary algorithm has been developed which avoids these problems and which couples with a variety of local optimization routines. The algorithm first constructs a grid of objective function values over some feasible region. The region dimensions and grid spacings are based on specific problem considerations. First differences are then calculated for successive points along each grid line and monitored in sign only, which rapidly locates extrema. User interaction determines how many of these extrema will undergo further investigation, which is carried out by passing locations to a local optimization subroutine. The algorithm has proved successful on a number of problems. A geological example—determination of benthic mixing parameters in deep-sea sediments via minimization of stratigraphic offset between ¹⁸ O signals from two different species of planktonic foraminifera—is given. FORTRAN code is provided for the global optimization routine, a golden section search subroutine for one-dimensional objective functions, and a simplex subroutine for multidimensional problems.     

Heterogeneity preserving upscaling for heat transport in fractured geothermal reservoirs

In simulation of fluid injection in fractured geothermal reservoirs, the characteristics of the physical processes are severely affected by the local occurence of connected fractures. To resolve these structurally dominated processes, there is a need to develop discretization strategies that also limit computational effort. In this paper, we present an upscaling methodology for geothermal heat transport with fractures represented explicitly in the computational grid. The heat transport is modeled by an advection-conduction equation for the temperature, and solved on a highly irregular coarse grid that preserves the fracture heterogeneity. The upscaling is based on different strategies for the advective term and the conductive term. The coarse scale advective term is constructed from sums of fine scale fluxes, whereas the coarse scale conductive term is constructed based on numerically computed basis functions. The method naturally incorporates the coupling between solution variables in the matrix and in the fractures, respectively, via the discretization. In this way, explicit transfer terms that couple fracture and matrix solution variables are avoided. Numerical results show that the upscaling methodology performs well, in particular for large upscaling ratios, and that it is applicable also to highly complex fracture networks.     

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.