Peyman Mostaghimi

Insights into non‐Fickian solute transport in carbonates

[1] We study and explain the origin of early breakthrough and long tailing plume behavior by simulating solute transport through 3-D X-ray images of six different carbonate rock samples, representing geological media with a high degree of pore-scale complexity. A Stokes solver is employed to compute the flow field, and the particles are then transported along streamlines to represent advection, while the random walk method is used to model diffusion. We compute the propagators (concentration versus displacement) for a range of Peclet numbers (Pe) and relate it to the velocity distribution obtained directly on the images. There is a very wide distribution of velocity that quantifies the impact of pore structure on transport. In samples with a relatively narrow spread of velocities, transport is characterized by a small immobile concentration peak, representing essentially stagnant portions of the pore space, and a dominant secondary peak of mobile solute moving at approximately the average flow speed. On the other hand, in carbonates with a wider velocity distribution, there is a significant immobile peak concentration and an elongated tail of moving fluid. An increase in Pe, decreasing the relative impact of diffusion, leads to the faster formation of secondary mobile peak(s). This behavior indicates highly anomalous transport. The implications for modeling field-scale transport are discussed. Citation: Bijeljic, B., P. Mostaghimi, and M. J. Blunt (2013), Insights into non-Fickian solute transport in carbonates, Water Resour. Res., 49, 2714–2728, doi:10.1002/wrcr.20238.

Simulation of Flow and Dispersion on Pore-Space Images

We simulate flow and transport directly on pore-space images obtained from a microcomputed-tomography (micro-CT) scan of rock cores. An efficient Stokes solver is used to simulate lowReynolds-number flows. The flow simulator uses a finite-difference method along with a standard predictor/corrector procedure to decouple pressure and velocity. An algebraic multigrid technique solves the linear systems of equations. We then predict permeability, and the results are compared with lattice-Boltzmannmethod (LBM) numerical results and available experimental data. For solute transport, we apply a streamline-based algorithm that is similar to the Pollock algorithm common in field-scale reservoir simulation, but which uses a novel semianalytic formulation near solid boundaries to capture, with subgrid resolution, the variation in velocity near the grains. A random-walk method accounts for molecular diffusion. The streamline-based algorithm is validated by comparison with published results for Taylor-Aris dispersion in a single capillary with a square cross section. We then predict accurately the available experimental data in the literature for the longitudinal dispersion coefficient for a range of Péclet numbers (10 to 10). We introduce a characteristic length on the basis of the ratio of volume to pore/grain surface area that can be used for consolidated porous media to calculate the Péclet number.

Anisotropic Mesh Adaptivity and Control Volume Finite Element Methods for Numerical Simulation of Multiphase Flow in Porous Media

Numerical simulation of multiphase flow in porous media is of great importance in a wide range of applications in science and engineering. The governing equations are the continuity equation and Darcy’s law. A novel control volume finite element (CVFE) approach is developed to discretize the governing equations in which a node-centered control volume approach is applied for the saturation equation, while a CVFE method is used for discretization of the pressure equation. We embed the discrete continuity equation into the pressure equation and ensure that the continuity equation is exactly enforced. Furthermore, the scheme is equipped with dynamic anisotropic mesh adaptivity which uses a metric tensor field approach, based on the curvature of fields of interest, to control the size and shape of elements in the metric space. This improves the resolution of the mesh in the zones of dynamic interest. Moreover, the mesh adaptivity algorithm employs multi-constraints on element size in different regions of the porous medium to resolve multi-scale transport phenomena. The advantages of mesh adaptivity and the capability of the scheme are demonstrated for simulation of flow in several challenging computational domains. The scheme captures the key features of flow while preserving the initial geometry and can be applied for efficient simulation of flow in heterogeneous porous media and geological formations.

Numerical Simulation of Reactive Transport on Micro-CT Images

Reactive transport is a key issue in hydrocarbon reservoirs, hydrogeological and environmental applications. A numerical model is presented to predict alteration of porous medium structure due to the dissolution mechanisms. The model includes the coupling of mass transport, chemical reactions and solid modification. It is validated by comparing reactive flow in a fracture geometry with previously published results and analytical expressions. Flow, transport, and chemical reaction are simulated directly on three-dimensional micro computed tomography images of rocks with increasing degree of heterogeneity: a sand pack, Berea sandstone, and limestone carbonate. Different regimes of transport and reaction are characterised by the dimensionless Péclet and Damköhler numbers. Dissolution patterns and geometrical evolutions of the solid phases obtained from imaging are investigated for different Péclet and Damköhler numbers and the effects of heterogeneity are included. The porosity profiles are presented in different classes of porous media after reaction. The results demonstrate different mechanisms such as uniform and face dissolution at transport- and reactive-limited regimes. The relationships between permeability and porosity are also explored. At high Péclet and Damköhler numbers, high-permeability channels are uniformly dissolved leading to significant increases in permeability. The largest changes in permeability are observed for the most heterogeneous sample, the carbonate, in all Péclet and Damköhler regimes. For low Damköhler but high Péclet numbers, a uniform dissolution occurs over the entire porous medium. The complex correlation for permeability in different porous structures is explained based on connectivity and morphological properties of the porous media obtained from porosity profiles and dissolution patterns. The exponent n in the power-law correlation between permeability and porosity is measured in different samples and our findings are consistent with experimental observations. This study helps improve the understanding of reactive flow at pore scale in porous media highlighting the interplay of Péclet and Damköhler numbers as well as the rock heterogeneity.

Impact of mineralogical heterogeneity on reactive transport modelling

Impact of mineralogical heterogeneity of rocks in reactive modelling is investigated by applying a pore scale model based on the Lattice Boltzmann and Finite Volume Methods. Mass transport, chemical reaction and solid structure modification are included in the model. A two-dimensional mineral map of a sandstone rock is acquired using the imaging technique of QEMSCAN SEM with Energy-Dispersive X-ray Spectroscopy (EDS). The mineralogical heterogeneity is explored by conducting multi-mineral reaction simulations on images containing various minerals. The results are then compared with the prediction of single mineral dissolution modelling. Dissolution patterns and permeability variations of multi-mineral and single mineral reactions are presented. The errors of single mineral reaction modelling are also estimated. Numerical results show that mineralogical heterogeneity can cause significant errors in permeability prediction, if a uniform mineral distribution is assumed. The errors are smaller in high Pclet regimes than in low Pclet regimes in this sample. Effect of mineralogical heterogeneity on reactive transport was investigated.Mineral map of sandstones was acquired using QEMSCAN SEM-EDS.Lattice Boltzmann and Finite Volume Methods were applied to model reactive transport.Pore-scale simulations of multi-mineral reaction were performed directly on rock images.

Reservoir Modeling for Flow Simulation Using Surfaces, Adaptive Unstructured Meshes, and Control-Volume-Finite-Element Methods

We present new approaches to reservoir modeling and flow simulation that dispose of the pillar-grid concept that has persisted since reservoir simulation began. This results in significant improvements to the representation of multi-scale geological heterogeneity and the prediction of flow through that heterogeneity. The research builds on 20+ years of development of innovative numerical methods in geophysical fluid mechanics, refined and modified to deal with the unique challenges associated with reservoir simulation. Geological heterogeneities, whether structural, stratigraphic, sedimentologic or diagenetic in origin, are represented as discrete volumes bounded by surfaces, without reference to a pre-defined grid. Petrophysical properties are uniform within the geologically-defined rock volumes, rather than within grid-cells. The resulting model is discretized for flow simulation using an unstructured, tetrahedral mesh that honors the architecture of the surfaces. This approach allows heterogeneity over multiple length-scales to be explicitly captured using fewer cells than conventional corner-point or unstructured grids. Multiphase flow is simulated using a novel mixed finite element formulation centered on a new family of tetrahedral element types, PN(DG)-PN+1, which has a discontinuous N-order polynomial representation for velocity and a continuous (order N+1) representation for pressure. This method exactly represents Darcy force balances on unstructured meshes and thus accurately calculates pressure, velocity and saturation fields throughout the domain. Computational costs are reduced through (i) automatic mesh adaptivity in time and space and (ii) efficient parallelization. Within each rock volume, the mesh coarsens and refines to capture key flow processes, whilst preserving the surface-based representation of geological heterogeneity. Computational effort is thus focused on regions of the model where it is most required. Having validated the approach against a set of benchmark problems, we demonstrate its capabilities using a number of test models which capture aspects of geological heterogeneity that are difficult or impossible to simulate conventionally, without introducing unacceptably large numbers of cells or highly non-orthogonal grids with associated numerical errors. Our approach preserves key flow features associated with realistic geological features that are typically lost. The approach may also be used to capture near wellbore flow features such as coning, changes in surface geometry across multiple stochastic realizations and, in future applications, geomechanical models with fracture propagation, opening and closing. Introduction Reservoir modelling and flow simulation have become ubiquitous in the hydrocarbon industry over the past 20 years and the development of flow simulation models now follows a widely accepted workflow that is surprisingly similar across companies and academic institutions, regardless of the software tools used (e.g. Bryant and Flint, 1993): 1. The reservoir volume is defined by surfaces representing the top and base of the reservoir and surfaces representing key reservoir bounding faults. 2. Additional faults within the reservoir are represented by additional surfaces, across which the top and base surfaces may be offset. 3. The reservoir is subdivided into geologically defined zones by one or more surfaces, which may be offset across the fault surfaces. These surfaces may be interpreted from seismic data, or correlated between wells, in which case the topography of the surfaces may be dictated by the top and/or base reservoir surfaces. Conventional reservoir models may contain 10s to 100s of these surfaces.

A Dynamic Mesh Approach for Simulation of Immiscible Viscous Fingering

Viscous fingering is a major concern in the waterflooding of heavy oil reservoirs. Traditional reservoir simulators employ low-order finite volume/difference methods on structured grids to resolve this phenomenon. However, their approach suffers from a significant numerical dispersion error due to insufficient mesh resolution which smears out some important features of the flow. We simulate immiscible incompressible two phase displacements and propose use of unstructured control volume finite element (CVFE) methods for capturing viscous fingering in porous media. Our approach uses anisotropic mesh adaptation where the mesh resolution is optimized based on the evolving flow features. The adaptive algorithm uses a metric tensor field based on solution interpolation error estimates to locally control the size and shape of elements in the metric. We resolve the viscous fingering patterns accurately and reduce the numerical dispersion error significantly. The mesh optimization, generates an unstructured coarse mesh in other regions of the computational domain where a high resolution is not required. We analyze the computational cost of mesh adaptivity on unstructured mesh and compare its results with those obtained by a commercial reservoir simulator based on the finite volume methods.

Adaptive Mesh Optimization for Simulation of Immiscible Viscous Fingering

Viscous fingering can be a major concern when waterflooding heavy-oil reservoirs. Most commercial reservoir simulators use low-order finite-volume/-difference methods on structured grids to resolve this phenomenon. However, this approach suffers from a significant numerical-dispersion error because of insufficient mesh resolution, which smears out some important features of the flow. We simulate immiscible incompressible two-phase displacements and propose the use of unstructured control-volume finite-element (CVFE) methods for capturing viscous fingering in porous media. Our approach uses anisotropic mesh adaptation where the mesh resolution is optimized on the basis of the evolving features of flow. The adaptive algorithm uses a metric tensor field dependent on solution-interpolation-error estimates to locally control the size and shape of elements in the metric. The mesh optimization generates an unstructured finer mesh in areas of the domain where flow properties change more quickly and a coarser mesh in other regions where properties do not vary so rapidly. We analyze the computational cost of mesh adaptivity on unstructured mesh and compare its results with those obtained by a commercial reservoir simulator on the basis of the finite-volume methods.

Hydrodynamics of fingering instability in the presence of a magnetic field

The hydrodynamics of two immiscible fluids in a rectangular Hele-Shaw cell under the influence of a magnetic field is studied, both theoretically and numerically. A linear stability analysis is conducted to determine the effect of magnetic fields on the formation of viscous fingers. As a result, an analytical solution is found to calculate the growth rate of perturbations. For numerical simulation of the two-phase flow, the interfacial tension is treated as a body force using the continuum surface force model and the interface tracking is performed by the volume of fluid method. The variations of the width and growth rate of fingers in an unstable displacement versus Hartmann number, a dimensionless number characterizing the strength of the applied magnetic field, are investigated. By varying the value of Hartmann number systematically, a suppressing effect on the formation of viscous fingers is observed. Consequently, it is detected that there exists a minimum Hartmann number preventing the formation of viscous fingers and ensuring a stable displacement. Our numerical simulations are in agreement with the results of the linear stability analysis and quantify the effect of magnetic fields in mitigating viscous fingering effects and improving the efficiency of the fluid displacement.

Determination of Local Diffusion Coefficients and Directional Anisotropy in Shale From Dynamic Micro-CT Imaging

This research/project was undertaken with the assistance of resources and services from the National Computational Infrastructure (NCI), which is supported by the Australian Government. We acknowledge funding from the member companies of the ANU/UNSW Digicore research consortium.