Combining finite volume and finite element methods to simulate fluid flow in geologic media (original) (raw)
Geofluids, 2004
The permeability of the Earth's crust commonly varies over many orders of magnitude. Flow velocity can range over several orders of magnitude in structures of interest that vary in scale from centimeters to kilometers. To accurately and efficiently model multiphase flow in geologic media, we introduce a fully conservative node-centered finite volume method coupled with a Galerkin finite element method on an unstructured triangular grid with a complementary finite volume subgrid. The effectiveness of this approach is demonstrated by comparison with traditional solution methods and by multiphase flow simulations for heterogeneous permeability fields including complex geometries that produce transport parameters and lengths scales varying over four orders of magnitude.
A parallel FE–FV scheme to solve fluid flow in complex geologic media
Computers & Geosciences, 2008
Field data-based simulations of geologic systems require much computational time because of their mathematical complexity and the often desired large scales in space and time. To conduct accurate simulations in an acceptable time period, methods to reduce runtime are required. A parallelization approach is attractive because fast multi-processor clusters are nowadays readily available. Here we report on our recent efforts to parallelize our multiphysics code CSMP þ þ (Complex System Modelling Platform). In particular, we describe a parallel finite element-finite volume method for multi-phase fluid flow in heterogeneous porous media. We take a domain partitioning approach where the finite element mesh is partitioned into sub-domains, assigning each of them to a single processor. For each sub-domain a local finite volume mesh is constructed. We can now solve advection-dispersion type equations taking an operator splitting approach: Pressure diffusion is calculated with an implicit finite element method and advection with an implicit or explicit finite volume scheme. We have tested the accuracy, robustness and computational speedup of our new parallel scheme on a Linux cluster by means of three geologic applications. All tests give excellent computational speedup with increasing number of up to 32 processors. These results broaden the range of possible simulations in terms of spatial and temporal scale and resolution as well as numerical accuracy up to two orders of magnitude.
2004
This paper addresses the development of a numerical model for two-phase immiscible displacement in core samples, considering a two-dimensional representation of the flow. The final objective of this development is the application of such simulation tool on a parameter estimation methodology for determining relative permeability curves from experimental data collected in laboratory experiments. All possible influencing factors on the core-scale flow, such as rock heterogeneities, capillary pressure, gravity effects and fluid compressibility, are included in the model in order to cover a wide range of experimental scenarios and to deal with detailed porous media properties measurements now available from imaging methods. The Elementbased Finite Volume Method (EbFVM) is used for discretizing the mass-conservation differential equations, whereas an accelerated version of the IMPES algorithm is proposed for solving them. Since EbFVM is a relatively novel method for simulating flow in porous media, in this paper is given special emphasis to the description of essential details in a systematic way, in order to provide a methodology basis for future developments in the reservoir simulation field.
A Quadrilateral Element-Based Finite-Volume Formulation for the Simulation of Complex Reservoirs
Proceedings of Latin American & Caribbean Petroleum Engineering Conference, 2007
In this work is presented a numerical formulation for reservoir simulation in which the element-based finite-volume method (EbFVM) is applied to the discretization of the differential equations that describe macroscopic multiphase flow in petroleum reservoirs. The spatial discretization is performed by means of quadrilateral unstructured grids, which are adequate for representing two-dimensional domains of any complexity in an accurate and efficient manner. Although mass conservation is enforced over polygonal control volumes constructed in a vertex-centered fashion, media properties are assigned to the primal-grid quadrilateral elements. In this way, non-homogeneous full tensor permeabilities can be handled straightforwardly. Piecewise bilinear shape functions are used for approximating the main variables in the differential equations. The exception is the advection term in the saturation equation, which is approximated by means of a twodimensional positivity-preserving upwind scheme. Numerical results without noticeable grid orientation effects were obtained using this type of approximation, even for the most adverse cases with high mobility ratios and piston-type displacements. Additionally, some simple problems with known analytical solution were solved in order to assess the accuracy of the method. We show that the approximation of the pressure field is second-order even for non-homogeneous anisotropic media. Finally, the ability for solving fluid displacements in faulted reservoirs of complex geometry was tested with a synthetic problem.
1996
The construction of grids that accurately reflect geologic structure and stratigraphy for computational flow and transport models poses a formidable task. Even with a complete understanding of stratigraphy, material properties, boundary and initial conditions, the task of incorporating data into a numerical model can be difficult and time consuming. Furthermore, most tools available for representing complex geologic surfaces and volumes are not designed for producing optimal grids for flow and transport computation. We have developed a modeling tool, GEOMESH, for automating finite element grid generation that maintains the geometric integrity of geologic structure and stratigraphy. The method produces an optimal (Delaunay) tetrahedral grid that can be used for flow and transport computations. The process of developing a flow and transport model can be divided into three parts: (1)Developing accurate conceptual models inclusive of geologic interpretation, material characterization and construction of a stratigraphic and hydrostratigraphic framework model, (2)Building and initializing computational frameworks; grid generation, boundary and initial conditions, (3)Computational physics models of flow and transport. Process (1) and (3) have received considerable attention whereas (2) has not. This work concentrates on grid generation and its connections to geologic characterization and process modeling. Applications of GEOMESH illustrate grid generation for two dimensional cross sections, three dimensional regional models, and adaptive grid refinement in three dimensions. Examples of grid representation of wells and tunnels with GEOMESH can be found in Cherry et al. [1]. The resulting grid can be utilized by unstructured finite element or integrated finite difference models.
SPE-163669: Multiscale method for simulating two and three-phase flow in porous media
Multiscale methods developed to solve coupled flow equations for reservoir simulation are based on a hierarchical strategy in which the pressure equation is solved on a coarsened grid and transport equation is solved on the fine grid as a decoupled system. The multiscale mixed finite-element (MsMFE) method attempts to capture sub-grid geological heterogeneity directly into the coarse-scale via mathematical basis functions. These basis functions are able to capture important multiscale information and are coupled through a global formulation to provide good approximation of the subsurface flow solution. In the literature, the general formulation of the MsMFE method for incompressible two-phase and compressible three-phase flow has mainly addressed problems with idealized flow physics. In this paper, we present a new formulation that accounts for compressibility, gravity, and spatially-dependent capillary and relative-permeability effects. We evaluate the computational efficiency and accuracy of the method by reporting the result of series of representative benchmark tests that have a high degree of realism with respect to flow physics, heterogeneity in petrophysical model, and geometry/topology of the corner-point grids. In particular, the MsMFE method is validated and compared against Shell's in-house simulator MoReS. The fine-scale flux, pressure, and saturation fields computed by the multiscale simulation show a noteworthy improvement in resolution and accuracy compared with coarse-scale models.
The Multiscale Finite Volume Method on Unstructured Grids
SPE Reservoir Simulation Symposium, 2013
Finding a pressure solution for large-scale reservoirs that takes into account fine-scale heterogeneities can be very computationally intensive. One way of reducing the workload is to employ multiscale methods that capture local geological variations using a set of reusable basis functions. One of these methods, the multiscale finite-volume (MsFV) method is well studied for 2D Cartesian grids, but has not been implemented for stratigraphic and unstructured grids with faults in 3D. With reservoirs and other geological structures spanning several kilometers, running simulations on the meter scale can be prohibitively expensive in terms of time and hardware requirements. Multiscale methods are a possible solution to this problem, and extending the MsFV method to realistic grids is a step on the way towards fast and accurate solutions for large-scale reservoirs.
Finite Volume Method for Modelling Gas Flow in Shale
ECMOR XIV - 14th European conference on the mathematics of oil recovery, 2014
Gas flow in shale is a complex phenomenon and is currently being investigated using a variety of modelling and experimental approaches. A range of flow mechanisms need to be taken into account when describing gas flow in shale including continuum, slip, transitional flow and Knudsen diffusion. A finite volume method (FVM) is presented to mathematically model these flow mechanisms. The approach incorporates the Knudsen number as well as the gas adsorption isotherm, allowing different flow mechanisms to be taken into account as well as methane sorption on organic matter. The approach is applicable to non-linear diffusion problems, in which the permeability and fluid density both depend on the scalar variable, the pressure. The FVM is fully conservative, as it obeys exact conservation laws in a discrete sense integrated over finite volumes. The method is validated first on unsteady-state problems for which analytical or numerical solutions are available. The approach is then applied for solving pressurepulse decay tests and a comparison with an alternative finite element numerical solution is made. Results for practical laboratory pressure-pulse decay tests of samples with very low permeability are also presented.
Transport in Porous Media, 2010
Discrete-fracture and rock matrix (DFM) modelling necessitates a physically realistic discretisation of the large aspect ratio fractures and the dissected material domains. Using unstructured spatially adaptively refined finite-element meshes, we find that the fastest flow often occurs in the smallest elements. Flow velocity and element size vary over many orders of magnitude, disqualifying global Courant number (CFL)-dependent transport schemes because too many time steps would be necessary to investigate displacements of interest. Here, we present a higher-order accurate implicit pressure-(semi)-implicit transport scheme for the advection-diffusion equation that overcomes this CFL limitation for DFM models. Using operator splitting, we solve the pressure and the transport equations on finite-element, node-centred finite-volume meshes, respectively, using algebraic multigrid methods. We apply this approach to field data-based DFM models where the fracture flow velocity and mesh refinement is 2-4 orders of magnitude greater than that of the matrix. For a global CFL of ≤10,000, this implies sub-CFL, second-order accurate behaviour in the matrix, and super-CFL, at least first-order accurate, transports in fast-flowing fractures. Their greater refinement, however, largely offsets this numerical dispersion, promoting a