An HMM--ELLAM scheme on generic polygonal meshes for miscible incompressible flows in porous media

We design a numerical approximation of a system of partial differential equations modelling the miscible displacement of a fluid by another in a porous medium. The advective part of the system is discretised using a characteristic method, and the diffusive parts by a finite volume method. The scheme is applicable on generic (possibly non-conforming) meshes as encountered in applications. The main features of our work are the reconstruction of a Darcy velocity, from the discrete pressure fluxes, that enjoys a local consistency property, an analysis of implementation issues faced when tracking, via the characteristic method, distorted cells, and a new treatment of cells near the injection well that accounts better for the conservativity of the injected fluid

Vanishing artifficial diffusion as a mechanism to accelerate convergence for multiphase porous media flow

Numerical solution of the equations governing multiphase porous media flow is challenging. A common approach to improve the performance of iterative non-linear solvers for these problems is to introduce artificial diffusion. Here, we present a mass conservative artificial diffusion that accelerates the non-linear solver but vanishes when the solution is converged. The vanishing artificial diffusion term is saturation dependent and is larger in regions of the solution domain where there are steep saturation gradients. The non-linear solver converges more slowly in these regions because of the highly non-linear nature of the solution. The new method provides accurate results while significantly reducing the number of iterations required by the non-linear solver. It is particularly valuable in reducing the computational cost of highly challenging numerical simulations, such as those where physical capillary pressure effects are dominant. Moreover, the method allows converged solutions to be obtained for Courant numbers that are at least two orders of magnitude larger than would otherwise be possible

An exploration of the IGA method for efficient reservoir simulation

Novel numerical methods present exciting opportunities to improve the efficiency of reservoir simulators. Because potentially significant gains to computational speed and accuracy may be obtained, it is worthwhile explore alternative computational algorithms for both general and case-by-case application to the discretization of the equations of porous media flow, fluid-structure interaction, and/or production. In the present work, the fairly new concept of isogeometric analysis (IGA) is evaluated for its suitability to reservoir simulation via direct comparison with the industry standard finite difference (FD) method and 1st order standard finite element method (SFEM). To this end, two main studies are carried out to observe IGA’s performance with regards to geometrical modeling and ability to capture steep saturation fronts. The first study explores IGA’s ability to model complex reservoir geometries, observing L2 error convergence rates under a variety of refinement schemes. The numerical experimental setup includes an 'S' shaped line sink of varying curvature from which water is produced in a 2D homogenous domain. The accompanying study simplifies the domain to 1D, but adds in multiphase physics that traditionally introduce difficulties associated with modeling of a moving saturation front. Results overall demonstrate promise for the IGA method to be a particularly effective tool in handling geometrically difficult features while also managing typically challenging numerical phenomena.Petroleum and Geosystems Engineerin

A Darcy-Cahn-Hilliard model of multiphase fluid-driven fracture

A Darcy-Cahn-Hilliard model coupled with damage is developed to describe multiphase-flow and fluid-driven fracturing in porous media. The model is motivated by recent experimental observations in Hele-Shaw cells of the fluid-driven fracturing of a synthetic porous medium with tunable fracture resistance. The model is derived from continuum thermodynamics and employs several simplifying assumptions, such as linear poroelasticity and viscous-dominated flow. Two distinct phase fields are used to regularize the interface between an invading and a defending fluid, as well as the ensuing damage. The damage model is a cohesive version of a phase-field model for fracture, in which model parameters allow for control over both nucleation and crack growth. Model-based simulations with finite elements are then performed to calibrate the model against recent experimental results. In particular, an experimentally-inferred phase diagram differentiating two flow regimes of porous invasion and fracturing is recovered. Finally, the model is employed to explore the parameter space beyond experimental capabilities, giving rise to the construction of an expanded phase diagram that suggests a new flow regime

Effect of Viscosity Contrast and Wetting on Frictional Flow Patterns

Multiphase flows involving two fluids and a granular material occur in such diverse sce-narios as mud and debris flows, methane venting from sediments, degassing of volatiles from magma, and the processing of granular and particulate systems in the food, pharmaceutical, and chemical industries. The presence of the granular material introduces solid friction as a governing force in the dynamics, alongside viscosity and capillarity. This multitude of interacting elements and forces can give rise to instabilities and the emergence of patterns, making these multiphase frictional flows inherently difficult to predict or control. We refer to these granular-fluid-mixtures as frictional fluids.We explore here systematically the competition between frictional, viscous, and capillary forces in frictional fluid flows. Viscously stable (more viscous invading fluid) and unstable (more viscous defending fluid) scenarios are investigated, and we study wetting conditions from drainage (grains wetted by defending fluid), through mixed-wet, to imbibition (grains wetted by invading fluid). The emerging flow patterns are studied using both experiments and simulations. Firstly, the effect of viscous stabilization on frictional finger pattern formation is discovered. When the flow is viscously stable, increasing the viscous force leads to a striking transition from the growth of one solitary finger to the simultaneous growth of multiple, wandering fingers to the axisymmetric growth of a radial spoke pattern as the flow is increasingly viscously stabilised. When the flow is viscously unstable, in contrast, the invasion patterns transition from frictional fingering to classical viscous fingering as viscous force increases beyond a critical fluidisation threshold. Later, the effects of parameters such as plate spacing and its gradient along the cell, and the tilt angle of the cell, on the pattern formation is studied. Furthermore, viscously unstable fracturing in drainage is studied. A small change on the volume fraction of granular material which govern the friction stress in the system, can convert the invasion from bulldozing fractures to pore invasion. At high air pressure, the fractures form a radially symmetric pattern where the fractures also gradually widen over time. Finally, viscously stable displacement from imbibition through mixed-wet to drainage is explored. Here, three types of invasion dynamics happens simultaneously or sequentially: pore invasion, capillary bulldozing and erosion, and five regimes of invasion patterns are identified: (I) pure pore invasion, (II) pure capillary bulldozing, (III) capillary bulldozing followed by pore invasion, (IV) pore invasion followed by erosion and (V) capillary bulldozing followed by pore invasion and erosion. These are caused by the relative importance of capillarity, friction and viscous pressures determined by the experimentally controlled variables