Authors:Dong-Sin Shih; Gour-Tsyh Yeh Abstract: The theoretical behavior of a one-dimensional (1-D) open-channel flow is embedded in the Saint-Venant equation, which is derived from the Navier–Stokes equations. The flow motion is described by the momentum equations, in which the terms for the inertia, pressure, gravity, and friction loss are retained while all other terms are discarded. Although the problem is valid for most channel-flow scenarios, it is numerically challenging to solve because robust, accurate, and efficient algorithms are critical for models to field applications. The method of characteristics (MOC) is applied to solve the diagonalized Saint-Venant equations. Most importantly, the boundary conditions can be naturally implemented based on the wave directions. This is considered more closely related to realistic flow conditions and sufficiently flexible to handle mixed sub- and supercritical fluid flows in natural rivers. A computer model, WASH1DF, derived from the proposed numerical method, and which differs from other commercial software packages such as HEC-RAS and SOBEK, was developed. To test the accuracy of the proposed method, four benchmark problems were examined. Analytical solutions to these benchmark problems, covering a wide range of cases, were provided by MacDonald et al. (J. Hydrol. Eng. ASCE 123(11), 1041–1045, 1997). The simulations indicate that the proposed method provides accurate results for all benchmark cases, which are valid for all transient flow scenarios. Comparisons of WASH1DF with other commercially available software packages were also conducted under the same simulation conditions. The results indicate that our proposed model demonstrates high accuracy for all problems and achieves the highest simulation precision among all packages tested. PubDate: 2017-11-10 DOI: 10.1007/s10596-017-9703-7

Authors:Mohamed Hayek; Anis Younes; Jabran Zouali; Noura Fajraoui; Marwan Fahs Abstract: A new analytical solution is developed for interference hydraulic pumping tests in fractal fractured porous media using the dual-porosity concept. Heterogeneous fractured reservoirs are considered with hydrodynamic parameters assumed to follow power-law functions in radial distance. The developed analytical solution is verified by comparison against a finite volume numerical solution. The comparison shows that the numerical solution converges toward the analytical one when the size of the time step decreases. The applicability of the fractal dual-porosity model is then assessed by investigating the identifiability of the parameters from a synthetic interference pumping test with a set of noisy data using Bayesian parameter inference. The results show that if the storage coefficient in the matrix is fixed, the rest of the parameters can be appropriately inferred; otherwise, the identification of the parameters is faced with convergence problems because of equifinality issues. PubDate: 2017-11-03 DOI: 10.1007/s10596-017-9701-9

Authors:X. Tang; L. A. James; T. E. Johansen Abstract: Streamline simulation is a powerful tool that can be used for full field forecasting, history matching, flood optimization, and displacement visualization. This paper presents the development and the application of a new semi-analytical streamline simulation method in the near-wellbore region in polar/cylindrical coordinate systems. The main objective of this paper is to study the effects of the permeability heterogeneity and well completion details in the near-wellbore region. These effects dictate the streamline geometries, which in turn influence well productivity. Previous streamline applications used a constant flow rate for each stream tube. In this paper, streamline simulation is performed under the assumption of constant pressure boundaries, which is a novel and non-trivial extension of streamline simulation. Solutions are constructed by treating each stream tube as a flow unit by invoking analytical solutions for such geometries. In addition, visualization experiments are conducted to investigate the effect of the heterogeneity. Two-dimensional waterflooding visualization experiments in radial porous media are performed with constant pressure boundaries. The streamline simulator is applied to history match the relative permeabilities using these experiments, thereby validating the new near-well streamline method. PubDate: 2017-10-18 DOI: 10.1007/s10596-017-9697-1

Authors:Fayadhoi Ibrahima; Hamdi A. Tchelepi; Daniel W. Meyer Abstract: In the context of stochastic two-phase flow in porous media, we introduce a novel and efficient method to estimate the probability distribution of the wetting saturation field under uncertain rock properties in highly heterogeneous porous systems, where streamline patterns are dominated by permeability heterogeneity, and for slow displacement processes (viscosity ratio close to unity). Our method, referred to as the frozen streamline distribution method (FROST), is based on a physical understanding of the stochastic problem. Indeed, we identify key random fields that guide the wetting saturation variability, namely fluid particle times of flight and injection times. By comparing saturation statistics against full-physics Monte Carlo simulations, we illustrate how this simple, yet accurate FROST method performs under the preliminary approximation of frozen streamlines. Further, we inspect the performance of an accelerated FROST variant that relies on a simplification about injection time statistics. Finally, we introduce how quantiles of saturation can be efficiently computed within the FROST framework, hence leading to robust uncertainty assessment. PubDate: 2017-10-14 DOI: 10.1007/s10596-017-9698-0

Authors:Shahid Manzoor; Michael G Edwards; Ali H Dogru; Tareq M Al-Shaalan Abstract: Grid generation for reservoir simulation must honor classical key constraints and be boundary aligned such that control-volume boundaries are aligned with geological features such as layers, shale barriers, fractures, faults, pinch-outs, and multilateral wells. An unstructured grid generation procedure is proposed that automates control-volume and/or control point boundary alignment and yields a PEBI-mesh both with respect to primal and dual (essentially PEBI) cells. In order to honor geological features in the primal configuration, we introduce the idea of protection circles, and to generate a dual-cell feature based grid, we construct halos around key geological features. The grids generated are employed to study comparative performance of cell-centred versus cell-vertex control-volume distributed multi-point flux approximation (CVD-MPFA) finite-volume formulations using equivalent degrees of freedom. The formulation of CVD-MPFA schemes in cell-centred and cell-vertex modes is analogous and requires switching control volume from primal to dual or vice versa together with appropriate data structures and boundary conditions. The relative benefits of both types of approximation, i.e., cell-centred versus vertex-centred, are made clear in terms of flow resolution and degrees of freedom required. PubDate: 2017-10-03 DOI: 10.1007/s10596-017-9686-4

Authors:Carsten Burstedde; Jose A. Fonseca; Stefan Kollet Abstract: Regional hydrology studies are often supported by high-resolution simulations of subsurface flow that require expensive and extensive computations. Efficient usage of the latest high performance parallel computing systems becomes a necessity. The simulation software ParFlow has been demonstrated to meet this requirement and shown to have excellent solver scalability for up to 16,384 processes. In the present work, we show that the code requires further enhancements in order to fully take advantage of current petascale machines. We identify ParFlow’s way of parallelization of the computational mesh as a central bottleneck. We propose to reorganize this subsystem using fast mesh partition algorithms provided by the parallel adaptive mesh refinement library p4est. We realize this in a minimally invasive manner by modifying selected parts of the code to reinterpret the existing mesh data structures. We evaluate the scaling performance of the modified version of ParFlow, demonstrating good weak and strong scaling up to 458k cores of the Juqueen supercomputer, and test an example application at large scale. PubDate: 2017-09-30 DOI: 10.1007/s10596-017-9696-2

Authors:Zhe Liu; Fahim Forouzanfar Abstract: In the development of naturally fractured reservoirs (NFRs), the existence of natural fractures induces severe fingering and breakthrough. To manage the flooding process and improve the ultimate recovery, we propose a numerical workflow to generate optimal production schedules for smart wells, in which the inflow control valve (ICV) settings can be controlled individually. To properly consider the uncertainty introduced by randomly distributed natural fractures, the robust optimization would require a large ensemble size and it would be computationally demanding. In this work, a hierarchical clustering method is proposed to select representative models for the robust optimization in order to avoid redundant simulation runs and improve the efficiency of the robust optimization. By reducing the full ensemble of models into a small subset ensemble, the efficiency of the robust optimization algorithm is significantly improved. The robust optimization is performed using the StoSAG scheme to find the optimal well controls that maximize the net-present-value (NPV) of the NFR’s development. Due to the discrete property of a natural fracture field, traditional feature extraction methods such as model-parameter-based clustering may not be directly applicable. Therefore, two different kinds of clustering-based optimization methods, a state-based (e.g., s w profiles) clustering and a response-based (e.g., production rates) clustering, are proposed and compared. The computational results show that the robust clustering optimization could increase the computational efficiency significantly without sacrificing much expected NPV of the robust optimization. Moreover, the performance of different clustering algorithms varies widely in correspondence to different selections of clustering features. By properly extracting model features, the clustered subset could adequately represent the uncertainty of the full ensemble. PubDate: 2017-09-18 DOI: 10.1007/s10596-017-9689-1

Authors:Maziar Veyskarami; Amir Hossein Hassani; Mohammad Hossein Ghazanfari Abstract: The network modeling approach is applied to provide a new insight into the onset of non-Darcy flow through porous media. The analytical solutions of one-dimensional Navier-Stokes equation in sinusoidal and conical converging/diverging throats are used to calculate the pressure drop/flow rate responses in the capillaries of the network. The analysis of flow in a single pore revealed that there are two different regions for the flow coefficient ratio as a function of the aspect ratio. It is found that the critical Reynolds number strongly depends on the pore geometrical properties including throat length, average aspect ratio, and average coordination number of the porous media, and an estimation of such properties is required to achieve more reliable predictions. New criteria for the onset of non-Darcy flow are also proposed to overcome the lack of geometrical data. Although the average aspect ratio is the main parameter which controls the inertia effects, the effect of tortuosity on the onset of non-Darcy flow increases when the coordination number of media decreases. In addition, the higher non-Darcy coefficient does not essentially accelerate the onset of inertial flow. The results of this work can help to better understand how the onset of inertial flow may be controlled/changed by the pore architecture of porous media. PubDate: 2017-09-14 DOI: 10.1007/s10596-017-9695-3

Abstract: A polynomial chaos (PC) surrogate is proposed to reconstruct seismic time series in one-dimensional (1D) uncertain media. Our approach overcomes the deterioration of the PC convergence rate during long time integration. It is based on a double decomposition of the signal: a damped harmonic decomposition combined with a polynomial chaos expansion of the four coefficients of each harmonic term (amplitude, decay constant, pulsation, and phase). These PC expansions are obtained through the least squares method which requires the solution of nonlinear least squares problems for each sample point of the stochastic domain. The use of the surrogate is illustrated on vertically incident plane waves traveling in 1D layered, vertically stratified, isotropic, viscoelastic soil structure with uncertainties in the geological data (geometry, wave velocities, quality factors). Computational tests show that the stochastic coefficients can be efficiently represented with a low-order PC expansion involving few evaluations of the direct model. For the test cases, a global sensitivity analysis is performed in time and frequency domains to investigate the relative impact of the random parameters. PubDate: 2017-09-07 DOI: 10.1007/s10596-017-9677-5

Authors:Ya-wei Xie; Michael G. Edwards Abstract: Novel cell-centred finite-volume formulations are presented for incompressible and immiscible two-phase flow with both gravity and capillary pressure effects on structured and unstructured grids. The Darcy-flux is approximated by a control-volume distributed multipoint flux approximation (CVD-MPFA) coupled with a higher resolution approximation for convective transport. The CVD-MPFA method is used for Darcy-flux approximation involving pressure, gravity, and capillary pressure flux operators. Two IMPES formulations for coupling the pressure equation with fluid transport are presented. The first is based on the classical total velocity Vt fractional flow (Buckley Leverett) formulation, and the second is based on a more recent Va formulation. The CVD-MPFA method is employed for both Vt and Va formulations. The advantages of both coupled formulations are contrasted. The methods are tested on a range of structured and unstructured quadrilateral and triangular grids. The tests show that the resulting methods are found to be comparable for a number of classical cases, including channel flow problems. However, when gravity is present, flow regimes are identified where the Va formulation becomes locally unstable, in contrast to the total velocity formulation. The test cases also show the advantages of the higher resolution method compared to standard first-order single-point upstream weighting. PubDate: 2017-09-05 DOI: 10.1007/s10596-017-9669-5

Authors:Halvor M. Nilsen; Jan Nordbotten; Xavier Raynaud Abstract: In this paper, we study newly developed methods for linear elasticity on polyhedral meshes. Our emphasis is on applications of the methods to geological models. Models of subsurface, and in particular sedimentary rocks, naturally lead to general polyhedral meshes. Numerical methods which can directly handle such representation are highly desirable. Many of the numerical challenges in simulation of subsurface applications come from the lack of robustness and accuracy of numerical methods in the case of highly distorted grids. In this paper, we investigate and compare the Multi-Point Stress Approximation (MPSA) and the Virtual Element Method (VEM) with regard to grid features that are frequently seen in geological models and likely to lead to a lack of accuracy of the methods. In particular, we look at how the methods perform near the incompressible limit. This work shows that both methods are promising for flexible modeling of subsurface mechanics. PubDate: 2017-09-04 DOI: 10.1007/s10596-017-9687-3

Authors:Kai Zhang; Xiaoming Zhang; Liming Zhang; Lixin Li; Hai Sun; Zhaoqin Huang; Jun Yao Abstract: Accurate prediction of fracture distribution in fractured reservoirs is important in the development process. Considering that assisted history matching technology is an effective method for the inversion of reservoir parameters, the technology can also be applied for the inversion of fractures. Because applying assisted history matching technology for the inversion of fractures has an inherent defect of multiplicity of solution, it is therefore necessary to alleviate the multiplicity for the success of inversion. Although there are many factors affecting the multiplicity, the paper focuses on the study of the inversion results of different combinations of inversion parameters which are all representative parameters of fractures and determine the distribution of fractures. Firstly, we simulate the flow behavior in fractured media based on the discrete fracture matrix (DFM) module of Matlab Reservoir Simulation Toolbox (MRST) to explicitly describe the effect of fractures on flow behavior. Secondly, history matching objective function is established based on Bayesian theory and different kinds of representative parameters of fractures are chosen as inversion parameters. Thirdly, simultaneous perturbation stochastic approximation (SPSA) algorithm is adopted to minimize the objective function to achieve the inversion of fractures corresponding to different inversion parameters. Finally, theoretical cases verify that the inversion method is effective for the accurate prediction of fracture distribution and proper inversion parameters are crucial to the success of fracture inversion. PubDate: 2017-08-26 DOI: 10.1007/s10596-017-9690-8

Authors:Flávio Dickstein; Paulo Goldfeld; Gustavo T. Pfeiffer; Renan V. Pinto Abstract: We propose a new algorithm for solving the history matching problem in reservoir simulation, truncated conjugate gradient (TCG), which involves a model reparameterization based on the factorization of the prior covariance matrix, C M = L L T . We also revisit the LBFGS algorithm, framing it into the same reparametrization, introducing M-LBFGS. We present numerical evidence that this reparameterization has an important regularizing impact on the solution process. We show how TCG and M-LBFGS, as well as TSVD, can be implemented without the need of actually computing the factor L. Our numerical experiments, including the PUNQ-S3 and the Brugge cases, indicate that TCG and M-LBFGS are effective schemes for history matching. PubDate: 2017-08-23 DOI: 10.1007/s10596-017-9694-4

Authors:Arild Lohne; Oddbjørn Nødland; Arne Stavland; Aksel Hiorth Abstract: Polymeric liquids are of great practical importance for porous media flow as they can be used to improve the sweep of water in the reservoir and therefore improve the recovery of oil. Due to the non-Newtonian behavior of these liquids, they are extremely challenging to model. In this paper, we present a model that is capable of describing the most commonly observed flow regimes in porous media: (i) Newtonian, (ii) Shear thinning, (iii) Shear thickening, and (iv) Mechanical degradation. The novel feature of our model is that the time constants for the shear thinning and shear thickening behavior are related to variations in reservoir properties and conditions, thus making it possible to translate lab results to larger scale without introducing new fitting parameters. Furthermore, we present a way to estimate polymer mechanical degradation in porous media. In our model, the polymer degradation rate is linked to the effective pore radius (using a Kozeny-Carman type equation), shear stress, and polymer molecular weight, M w . The degradation results in a lower M w , while the polymer volumetric concentration is unaffected. The model is applied to a series of laboratory core flood experiments conducted with partially hydrolyzed polyacrylamide, HPAM, of different initial M w ranging from 5 to 20 MDa in seawater, and core permeability varied from 137 to 2019 mD. The flow rate is varied approximately three orders of magnitude and covers the shear thinning, shear thickening, and degradation flow regimes. We show that our model is able to reproduce experimental rate-dependent flow resistance, as well as viscosity of effluent samples. An important aspect supporting the use of the model as a predictive tool is that all the simulations with a given brine have made use of a single set of input parameters to describe the observed shear thickening and degradation behavior. Simulation of a second experimental series using low salinity brine required a separate set of input parameters for the shear thickening and shear degradation. The onset of shear thickening was not affected while shear thickening was reduced and degradation appeared to be slower. PubDate: 2017-08-18 DOI: 10.1007/s10596-017-9692-6

Authors:Calogero B. Rizzo; Felipe P. J. de Barros; Simona Perotto; Luca Oldani; Alberto Guadagnini Abstract: We study the applicability of a model order reduction technique to the solution of transport of passive scalars in homogeneous and heterogeneous porous media. Transport dynamics are modeled through the advection-dispersion equation (ADE) and we employ Proper Orthogonal Decomposition (POD) as a strategy to reduce the computational burden associated with the numerical solution of the ADE. Our application of POD relies on solving the governing ADE for selected times, termed snapshots. The latter are then employed to achieve the desired model order reduction. We introduce a new technique, termed Snapshot Splitting Technique (SST), which allows enriching the dimension of the POD subspace and damping the temporal increase of the modeling error. Coupling SST with a modeling strategy based on alternating over diverse time scales the solution of the full numerical transport model to its reduced counterpart allows extending the benefit of POD over a prolonged temporal window so that the salient features of the process can be captured at a reduced computational cost. The selection of the time scales across which the solution of the full and reduced model are alternated is linked to the Péclet number (P e), representing the interplay between advective and dispersive processes taking place in the system. Thus, the method is adaptive in space and time across the heterogenous structure of the domain through the combined use of POD and SST and by way of alternating the solution of the full and reduced models. We find that the width of the time scale within which the POD-based reduced model solution provides accurate results tends to increase with decreasing P e. This suggests that the effects of local-scale dispersive processes facilitate the POD method to capture the salient features of the system dynamics embedded in the selected snapshots. Since the dimension of the reduced model is much lower than that of the full numerical model, the methodology we propose enables one to accurately simulate transport at a markedly reduced computational cost. PubDate: 2017-08-17 DOI: 10.1007/s10596-017-9693-5

Authors:T. Almani; K. Kumar; M. F. Wheeler Abstract: In this paper, we consider an iterative coupling scheme for solving a fully discretized Biot system based on the fixed-stress split coupling algorithm. Specifically, we derive a priori error estimates for quantifying the error between the solution obtained at any iterate and the true solution. Our approach is based on studying the equations satisfied by the difference of iterates and utilizing a Banach contraction argument to show that the corresponding scheme is a fixed point iteration. Obtained contraction results are then used to derive theoretical convergence error estimates for the single rate iterative coupling scheme. We compare our numerical computations against the theoretically derived contraction estimates and show a good agreement with theory. PubDate: 2017-08-16 DOI: 10.1007/s10596-017-9691-7

Authors:A. V. Novikov; V. S. Posvyanskii; D. V. Posvyanskii Abstract: Well modeling plays an important role in numerical reservoir simulation. The main difficulty in well modeling is the difference in scale between the wellbore radius and well gridblock dimension used in the simulation. The Peaceman equation is widely used in reservoir simulation to match gridblock pressure to the local solution of the diffusivity equation describing the flow near the well. However, this approach was developed under the assumption of radial flow. At the same time, the well inflow equation can be solved within the Green’s function (GF) formalism which allows the solution to be obtained without the assumption of radial flow. The GF solution can be presented as a series over the eigenvalues of the Laplace differential operator. However, this series converges conditionally and its direct summation is time-consuming. In Posvyanskii et al. (2008), a method for fast summation of such a series was proposed and successfully applied for analyzing the pressure build up curves. In this paper, we adopt the same technique for calculating the well indices for horizontal, slanted and partially penetrated wells. Additionally, the role of different boundary conditions is considered. The semi-analytical expressions for well indices are obtained and compared to the solution of the Peaceman equation. It is shown that in some cases, the difference between these solutions can be significant. The use of the obtained expression in numerical flow simulation allows well inflow to be modeled with high accuracy even on a coarse grid. PubDate: 2017-08-10 DOI: 10.1007/s10596-017-9684-6

Authors:Niloofar Misaghian; Mehdi Assareh; MohammadTaqi Sadeghi Abstract: The upscaling process of a high-resolution geostatistical reservoir model to a dynamic simulation grid model plays an important role in a reservoir study. Several upscaling methods have been proposed in order to create balance between the result accuracy and computation speed. Usually, a high-resolution grid model is upscaled according to the heterogeneities assuming single phase flow. However, during injection processes, the relative permeability adjustment is required. The so-called pseudo-relative permeability curves are accepted, if their corresponding coarse model is a good representation of the fine-grid model. In this study, an upscaling method based on discrete wavelet transform (WT) is developed for single-phase upscaling based on the multi-resolution analysis (MRA) concepts. Afterwards, an automated optimization method is used in which evolutionary genetic algorithm is applied to estimate the pseudo-relative permeability curves described with B-spline formulation. In this regard, the formulation of B-spline is modified in order to describe the relative permeability curves. The proposed procedure is evaluated in the gas injection case study from the SPE 10th comparative solution project’s data set which provides a benchmark for upscaling problems [1]. The comparisons of the wavelet-based upscaled model to the high-resolution model and uniformly coarsened model show considerable speedup relative to the fine-grid model and better accuracy relative to the uniformly coarsened model. In addition, the run time of the wavelet-based coarsened model is comparable with the run time of the uniformly upscaled model. The optimized coarse models increase the speed of simulation up to 90% while presenting similar results as fine-grid models. Besides, using two different production/injection scenarios, the superiority of WT upscaling plus relative permeability adjustment over uniform upscaling and relative permeability adjustment is presented. This study demonstrates the proposed upscaling workflow as an effective tool for a reservoir simulation study to reduce the required computational time. PubDate: 2017-08-04 DOI: 10.1007/s10596-017-9688-2