Results 1 - 10 of 125
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[en] In this paper, we describe a single-relaxation-time (SRT) lattice Boltzmann formulation, which can be effectively applied to anisotropic advection-dispersion equations (AADE). The formulation can be applied to space and time variable anisotropic hydrodynamic dispersion tensor. The approach utilizes diffusion velocity lattice Boltzmann formulation which in the case of AADE can represent anisotropic diagonal and off-diagonal elements of the dispersion matrix by the coupling of advective and diffusive fluxes in equilibrium function. With this approach, AADE can be applied to the SRT lattice Boltzmann formulation using the same equilibrium function and without any changes to collision step nor in the application of boundary conditions. The approach shows good stability even for highly anisotropic dispersion tensor and is tested on selected illustrative examples which demonstrate the accuracy and applicability of the proposed method.
[en] This paper presents a novel methodology to model semi-steady state horizontal well flow performance in an anisotropic reservoir taking into account flow in the near-well region for an arbitrary well trajectory. It is based on an analytical productivity model describing coupled axial reservoir flow and radial well inflow. In order to apply this model in an anisotropic reservoir, the permeability field relative to the radial direction perpendicular to the well trajectory and the axial direction along the well trajectory must first be determined. A classical space transformation is used in concert with rotational transforms to obtain a virtual isotropic model. The transformation preserves the volumes and pressures. It is not a novel concept, but different from previous approaches in the sense that it is only applied in the near-well domain to formulate an equally isotropic media. As a result, the use of this virtual isotropic model requires the Dietz shape factor for an ellipse, transformed from the original cylindrical near-well domain. The Dietz shape factors are determined numerically in this research. The semi-steady state well/near-well model is implemented in a numerical simulator incorporating formation anisotropy and wellbore hydraulics. The specific productivity index along the well trajectory is generated using the virtual configuration. Numerical results for different anisotropy ratios and also incorporating frictional losses in the well are presented. Furthermore, the well/near-well model is applied in coupling with streamline reservoir model for a water flooding case. This appears to be the first coupling of a well hydraulics model and a streamline simulator. It presents the application of the well/near-well model in integrated reservoir simulation in an efficient and accurate manner. The results demonstrate that the coupling approach with a streamline reservoir model and the well/near-well is of great potential for advanced well simulation efficiently.
[en] A numerical method is formulated for the solution of the advective Cahn–Hilliard (CH) equation with constant and degenerate mobility in three-dimensional porous media with non-vanishing velocity on the exterior boundary. The CH equation describes phase separation of an immiscible binary mixture at constant temperature in the presence of a conservation constraint and dissipation of free energy. Porous media / pore-scale problems specifically entail images of rocks in which the solid matrix and pore spaces are fully resolved. The interior penalty discontinuous Galerkin method is used for the spatial discretization of the CH equation in mixed form, while a semi-implicit convex–concave splitting is utilized for temporal discretization. The spatial approximation order is arbitrary, while it reduces to a finite volume scheme for the choice of element-wise constants. The resulting nonlinear systems of equations are reduced using the Schur complement and solved via inexact Newton’s method. The numerical scheme is first validated using numerical convergence tests and then applied to a number of fundamental problems for validation and numerical experimentation purposes including the case of degenerate mobility. First-order physical applicability and robustness of the numerical method are shown in a breakthrough scenario on a voxel set obtained from a micro-CT scan of a real sandstone rock sample.
[en] 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.
[en] 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.
[en] A three-dimensional (3D) nuclear magnetic resonance (NMR) spectrum can simultaneously provide distributions of longitudinal relaxation time (T1), transverse relaxation time (T2), and diffusivity (D); thus, it greatly improves the capacity of fluid identification, typing, and quantitative evaluations. However, several challenges that significantly hinder the widespread application of this technique remain. The primary challenges are the high time and memory costs associated with the current 3D NMR inversion algorithms. In addition, an activation sequence optimization method for 3D NMR inversions has not been developed. In this paper, a novel inversion method for 3D NMR spectra and a detailed optimization method for activation sequences and acquisition parameters were proposed. The novel method, namely randomized singular value decomposition (RSVD) inversion algorithm, can reduce memory requirements and ensure computational efficiency and accuracy. Window averaging (WA) technique was also adopted in this study to further increase computational speed. The optimized method for pulse sequences is mainly based on projections of the 3D NMR spectra in the two-dimensional (2D) and one-dimensional (1D) domains. These projections can identify missing NMR properties of different fluids. Because of the efficiency and stability of this novel algorithm and the optimized strategy, the proposed methods presented in this paper could further promote the widespread application of 3D NMR.
[en] We perform a convergence analysis of the fixed stress split iterative scheme for the Biot system modeling coupled flow and deformation in anisotropic poroelastic media with tensor Biot parameter. The fixed stress split iterative scheme solves the flow subproblem with all components of the stress tensor frozen using a multipoint flux mixed finite element method, followed by the poromechanics subproblem using a conforming Galerkin method in every coupling iteration at each time step. The coupling iterations are repeated until convergence and Backward Euler is employed for time marching. The convergence analysis is based on studying the equations satisfied by the difference of iterates to show that the fixed stress split iterative scheme for anisotropic poroelasticity with Biot tensor is contractive. We also demonstrate that the scheme is numerically convergent using the classical Mandel’s problem solution for transverse isotropy.
[en] This article presents a new positivity-preserving finite-volume scheme with a nonlinear two-point flux approximation, which uses optimization techniques for the face stencil calculation. The gradient is reconstructed using harmonic averaging points with the constraint that the sum of the coefficients included in the face stencils must be positive. We compare the proposed scheme to a nonlinear two-point scheme available in literature and a few linear schemes. Using two test cases, taken from the FVCA6 benchmarks, the accuracy of the scheme is investigated. Furthermore, it is shown that the scheme is linearity-preserving on highly complex corner-point grids. Moreover, a two-phase flow problem on the Norne formation, a geological formation in the Norwegian Sea, is simulated. It is demonstrated that the proposed scheme is consistent in contrast to the linear Two-Point Flux Approximation scheme, which is industry standard for simulating subsurface flow on corner-point grids.
[en] We develop a Riemann solver for transport problems including geochemistry related to oil recovery. The example considered here concerns one-dimensional incompressible flow in porous media and the transport for several chemical components, namely H2O, H+, OH−, CO2, , , and decane; they are in chemical equilibrium in the aqueous and oleic phases, leading to mass transfer of CO2 between the oleic and aqueous phases. In our ionic model, we employ equations with zero diffusion coefficients. We do so because it is well known that for upscaled equations, the convection terms dominate the diffusion terms. The Riemann solution for this model can therefore be applied for upscaled transport processes in enhanced oil recovery involving geochemical aspects. In our example, we formulate the conservation equations of hydrogen, oxygen, hydrogen, and decane, in which we substitute regression expressions that are obtained by geochemical software. This can be readily done because Gibbs phase rule together with charge balance shows that all compositions can be rewritten in terms of a single composition, which we choose to be the hydrogen ion concentration (pH). In our example, we use the initial and boundary conditions for the carbonated aqueous phase injection in an oil reservoir containing connate water with some carbon dioxide. We compare the Riemann solution with a numerical solution, which includes capillary and diffusion effects. The significant new contribution is the effective Riemann solver we developed to obtain solutions for oil recovery problems including geochemistry and a variable total Darcy velocity, a situation in which fractional flow theory does not readily apply. We thus obtain an accurate solution for a carbonated waterflood, which elucidates some mechanisms of low salinity carbonated waterflooding.
[en] 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 . 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.