Karen Sophie Schmid

Higher order FE-FV method on unstructured grids for transport and two-phase flow with variable viscosity in heterogeneous porous media

This paper presents higher order methods for the numerical modeling of two-phase flow with simultaneous transport and adsorption of viscosifying species within the individual phases in permeable porous media. The numerical scheme presented addresses the three major challenges in simulating this process. Firstly, the component transport is strongly coupled with the viscous and capillary forces that act on the movement of the carrier phase. The discretization of the capillary parts is especially difficult since its effect on flow yields non-linear parabolic conservation equations. These are amenable to non-linear finite elements (FEs), while the capillary contribution on the component transport is first-order hyperbolic, where classical FEs are unsuitable. We solve this efficiently by a Strang splitting that uses finite volumes (FVs) with explicit time-stepping for the viscous parts and a combined finite element-finite volume (FEFV) scheme with implicit time-stepping for the capillary parts. Secondly, the components undergo hydrodynamic dispersion and discerning between numerical and physical dispersion is essential. We develop higher-order formulations for the phase and component fluxes that keep numerical dispersion low and combine them with implicit FEs such that the non-linearities of the dispersion tensor are fully incorporated. Thirdly, subsurface permeable media show strong spatial heterogeneity, with coefficients varying over many orders of magnitude and geometric complexity that make the use of unstructured grids essential. In this work, we employ node-centered FVs that combine their ability to resolve flow with the flexibility of FEs. Numerical examples of increasing complexity are presented that demonstrate the convergence and robustness of our approach and prove its versatility for highly heterogeneous, and geometrically complex fractured porous media.

Analytical Solutions for Spontaneous Imbibition: Fractional-Flow Theory and Experimental Analysis

We present analytical solutions for capillary-controlled displacement in one dimension by use of fractional-flow theory. We show how to construct solutions with a spreadsheet that can be used for the analysis of experiments as well as matrix-block-scale recovery in field settings. The solutions can be understood as the capillary analog to the classical Buckley-Leverett solution (Buckley and Leverett 1942) for viscous-dominated flow, and are valid for cocurrent and countercurrent spontaneous imbibition (SI), as well as for arbitrary capillary pressure and relative permeability curves. They can be used to study the influence of wettability, predicting saturation profiles and production rates characteristic for water-wet and mixed-wet conditions. We compare our results with in-situ measurements of saturation profiles for SI in a waterwet medium. We show that the characteristic shape of the saturation profile is consistent with the expected form of the relative permeabilities. We discuss how measurements of imbibition profiles, in combination with other measurements, could be used to determine relative permeability and capillary pressure.

Analytical solutions for co- and counter-current imbibition of sorbing, dispersive solutes in immiscible two-phase flow

We derive a set of analytical solutions for the transport of adsorbing solutes in an immiscible, incompressible two-phase system. This work extends recent results for the analytical description for the movement of inert tracers due to capillary and viscous forces and dispersion to the case of adsorbing solutes. We thereby obtain the first known analytical expression for the description of the effect of adsorption, dispersion, capillary forces and viscous forces on solute movement in two-phase flow. For the purely advective transport, we combine a known exact solution for the description of flow with the method of characteristics for the advective transport equations to obtain solutions that describe both co- and spontaneous counter-current imbibition and advective transport in one dimension. We show that for both cases, the solute front can be located graphically by a modified Welge tangent. For the dispersion, we derive approximate analytical solutions by the method of singular perturbation expansion. The solutions reveal that the amount of spreading depends on the flow regime and that adsorption diminishes the spreading behavior of the solute. We give some illustrative examples and compare the analytical solutions with numerical results.

Massively Parallel Sector Scale Discrete Fracture and Matrix Simulations

We have been able to solve a reservoir simulation problem which was previously thought of as intractable: We simulated multiphase displacement, including viscous, capillary, and gravitational forces, for highly resolved and geologically realistic models of naturally fractured reservoirs (NFR) at the sector, i.e. kilometre, scale with very reasonable runtime. This has been possible because we used massive parallelisation and hierarchical solvers in conjunction with a new discrete fracture and matrix modelling (DFM) technique that is based on mixed-dimensional unstructured hybrid-element discretisations. High-resolution DFM simulations are important to resolve the non-linear coupling of small scale capillary - viscous and large scale gravitational - viscous processes adequately for sector scale NFR. Cross-scale process coupling in NFR controls oil recovery and NFR often exhibit power-law fracture length distributions, i.e. they do not possess an REV, and highly permeable fractures can extend over the full hydrocarbon column height. As a consequence, emergent displacement patterns have been observed which are difficult to quantify using traditional means of upscaling. However, such patterns could now be used as benchmarks to reach a better consensus on the correctness of promising new upscaling techniques. The parallel DFM technologies presented here allow us to to obtain these results much more efficiently and hence explore the parameter space in greater detail. We observed a linear scaling behaviour for up to 64 processes and a significant decrease in runtime when applying our parallel DFM approach to three highly refined NFR simulations. These contain thousands of fractures, up to 5 million elements, and have local grid-refinements below 1 m for model dimensions between I and 10 kilometres. We achieved this excellent speedup because we reduced inter-processor communication by minimising the overlap between individual domains and decreased idle time of individual processors by distributing the number of unknowns equally among the processors.

Chemical osmosis in two‐phase flow and salinity‐dependent capillary pressures in rocks with microporosity

The situation of multiphase flow with varying salinity through rock formations with microporosity plays a key role in many applications. Experimental data for single-phase flow through rocks with microporosity show that chemical osmosis can lead to osmotic pressures in the range of several megapascal, but the effect of osmosis on multiphase flow so far has received little attention. Pore networks can be used to investigate these effects, but crucially depend on expressions for capillary entry pressures. Here we extend the classical theory for capillary entry pressures to the case where chemical potentials play a role. The inclusion of osmosis results into a “capillary-osmotic” pressure that also depends on salinity differences and temperature. Consequently, also the pore-scale events of piston-like displacement and snap-off depend on salinity contrasts and temperature and not only on pore geometry as has been assumed so far. Examples show that even small salinity differences can lead to significantly different entry pressures and changed pore invasion sequences compared to if osmosis is absent. We show that the ensemble behavior with osmosis often is identical to the case where the medium partly has become more water wet, which implies that osmosis might have a strong impact on laboratory-scale quantities, but also that their detection in experiments will be challenging. Hence, chemical gradients could be important drivers of multiphase flow in rocks with microporosity and should be included into flow models, which currently is not the case.

Analytical Solutions for Co- and Countercurrent Imbibition of Sorbing – Dispersive Solutes in Immiscible Two-phase Flow

We derive a set of analytical solutions for the transport of adsorbing solutes in an immiscible, incompressible two-phase system. This work extends recent results for the analytical description for the movement of inert tracers due to capillary and viscous forces and dispersion to the case of adsorbing solutes. We thereby obtain the first known analytical expression for the description of the effect of adsorption, dispersion, capillary forces and viscous forces on solute movement in two-phase flow. For the purely advective transport, we combine a known exact solution for the description of flow with the method of characteristics for the advective transport equations to obtain solutions that describe both co- and spontaneous counter-current imbibition and advective transport in one dimension. We show that for both cases, the solute front can be located graphically by a modified Welge tangent. For the dispersion, we derive approximate analytical solutions by the method of singular perturbation expansion. The solutions reveal that the amount of spreading depends on the flow regime and that adsorption diminishes the spreading behavior of the solute. We give some illustrative examples and compare the analytical solutions with numerical results.

A Combined Numerical-experimental Approach to Study Sub-grid Transport Processes in Real 3D Carbonate Rocks

We have developed a new finite element – finite volume based simulation approach to study flow and transport processes at sub-grid scales, i.e. at scales below the typical size of a reservoir simulation grid block, using real 3D geometries in carbonate reservoirs. We complement the simulations by high-resolution X-Ray CT experiments which provide us with the 3D structures, allow us to visualise flow processes at the core-scale in real time, and help us to compare the observed processes to numerical simulations to validate and verify the latter. We use this combined numerical-experimental approach to analyse the fundamental processes controlling fluid flow in carbonates at sub-grid scales. Results can be incorporated in existing reservoir simulation workflows to increase the confidence in reservoir performance forecasting. We show applications related to fractured carbonate reservoirs and enhanced oil recovery processes due to injection of low-salinity fluids and hot water.