Using X-ray computed microtomography, we have visualized and quantified the in situ structure of a trapped nonwetting phase (oil) in a highly heterogeneous carbonate rock after injecting a wetting phase (brine) at low and high capillary numbers. We imaged the process of capillary desaturation in 3D and demonstrated its impacts on the trapped nonwetting phase cluster size distribution. We have identified a previously unidentified pore-scale event during capillary desaturation. This pore-scale event, described as droplet fragmentation of the nonwetting phase, occurs in larger pores. It increases volumetric production of the nonwetting phase after capillary trapping and enlarges the fluid-fluid interface, which can enhance mass transfer between the phases. Droplet fragmentation therefore has implications for a range of multiphase flow processes in natural and engineered porous media with complex heterogeneous pore spaces.

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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.@en

The physics of multi-phase displacement processes in the individual pores of a connected pore-network of a rock ultimately controls how oil, gas and water move in reservoir rocks and how readily they can be produced. These pore scale processes, including piston-like displacement, snap off, film-flow and fluid redistribution have been studied traditionally in pore-network simulations as well as in 2D micro-model experiments. However, recent advances in X-ray computed micro-tomography (CT) techniques now enable us to visualize and monitor these processes in 3D during in-situ core flooding experiments at pore-scale resolution. This provides new information on the spatial and temporal evolution of oil and water phase clusters and films. In this paper, we present results of a suite of two-phase fluid displacement experiments performed on a dolomite core plug. The experiments consist of a series of fluid injections and in-situ CT scans of the core in certain time steps during the drainage and imbibition displacement processes. The fluid phases are brine and a mineral oil. A simple, low-cost and highly X-ray transparent design for core flooding cells is introduced. Our experiments and CT images allow us to visualize the 3D fluid structures of each phase during fluid displacements in carbonate rocks with excellent clarity. Piston-like displacement and snap off mechanisms have been captured clearly in 3D. In addition, the formation, collapse and reorganisation of brine films surrounding oil blobs in individual pores were clearly visualised. However, the formation of oil films, which could provide connectivity for the hydrocarbon phase at low saturations, could not be observed in these experiments. The observed displacement processes and the particular oil-water/rock configurations seen in the displacements suggest the rock is preferentially water wet.@en

We present a finite element - finite volume simulation method for modelling fluid flow and solute transport accompanied by chemical reactions in experimentally obtained 3D pore geometries. The advantage of the proposed methodology with respect to other pore-scale modelling approaches is that no simplifications regarding the geometry of the porous space are required and no approximations to the flow equations are introduced. We apply this method in a proof-of-concept study of a digitised Fontainebleau sandstone sample. We use the calculated velocity profile with the finite volume procedure to simulate pore-scale transport and diffusion of the adsorbing solute. We also demonstrate how analysis of the pore geometry can be used to identify the locations of oil during the two-phase flow and couple this with the reactive transport modeling to show how this procedure can be used to estimate the potential of the enhanced oil recovery techniques.

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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.

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We derive a set of semianalytical solutions for the movement of solutes in immiscible two-phase flow. Our solutions are new in two ways: First, we fully account for the effects of capillary and viscous forces on the transport for arbitrary capillary-hydraulic properties. Second, we fully take hydrodynamic dispersion for the variable two-phase flow field into account. The understanding of immiscible two-phase flow and the simultaneous miscible displacement and mixing of components within a phase is important for many applications, including the location of nonaqueous phase liquids in the subsurface, the design of contaminant cleanup procedures, the sequestration of carbon dioxide, and enhanced oil-recovery techniques. For purely advective transport we combine a known exact solution for the description of immiscible two-phase flow with the method of characteristics for the advective transport equations to obtain solutions that describe cocurrent flow and countercurrent spontaneous 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 advective-dispersive solute transport, we derive approximate analytical solutions by the method of singular perturbation expansion. On the basis of this, we obtain analytical expressions for the growth of the dispersive zone for the case with and without the influence of capillary pressure. We show that for the case of spontaneous countercurrent imbibition the order of magnitude of the growth rate is far smaller than that for the viscous limit. We give some illustrative examples and compare the analytical expressions with numerical reference solutions.@en

Upscaling pore-scale processes into macroscopic quantities such as hydrodynamic dispersion is still not a straightforward matter for porous media with complex pore space geometries. Recently it has become possible to obtain very realistic 3D geometries for the pore system of real rocks using either numerical reconstruction or micro-CT measurements. In this work, we present a finite element-finite volume simulation method for modeling single-phase fluid flow and solute transport in experimentally obtained 3D pore geometries. Algebraic multigrid techniques and parallelization allow us to solve the Stokes and advection-diffusion equations on large meshes with several millions of elements. We apply this method in a proof-of-concept study of a digitized Fontainebleau sandstone sample. We use the calculated velocity to simulate pore-scale solute transport and diffusion. From this, we are able to calculate the a priori emergent macroscopic hydrodynamic dispersion coefficient of the porous medium for a given molecular diffusion D_m of the solute species. By performing this calculation at a range of flow rates, we can correctly predict all of the observed flow regimes from diffusion dominated to convection dominated.

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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.@en

We derive a set of analytical solutions for the transport of adsorbing solutes in an immiscible, incompressible two-phase system. Our analytical solutions are new in two ways: First, we fully account for the effects of both capillary and viscous forces on the transport for arbitrary capillary-hydraulic properties. Second, we fully take hydrodynamic dispersion for the variable two-phase flow field and adsorption effects of the solutes into account. All previously obtained results for component transport in immiscible two-phase systems account for changes in the flow field due to the components presence but they ignore dispersive effects. This is surprising given the enormous practical importance of diffusive effects both in hydrological settings and for situations that arise in the context of oil production for chemical flooding, polymer flooding and the transport of wettability altering agents. In both cases, one often faces the situation of fractured media or thin-layer structures where the diffussive transport processes of components dominate over viscous forces. We consider a situation where the components do not affect the flow field and focus on dispersive effects. 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 countercurrent flow and advective transport in one dimension. We show that for both cases the solute front can be located graphically by a modified Welge tangent and that the mathematically obtained solutions correspond to the physical notion that the solute concentrations are functions of saturation only. For the advection-dispersion case, we derive approximate analytical solutions by the method of singular perturbation expansion. We give some illustrative examples and compare the analytical solutions with numerical results.

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We present a finite element-finite volume simulation method for modelling fluid flow and solute transport accompanied by chemical reactions in experimentally obtained 3D pore geometries. We solve the stationary Stokes equation on the computational domain with the FE method using the same set of nodes and the same order of basis functions for both velocity and pressure. The resulting linear system is solved by employing the algebraic multigrid library SAMG. To simulate large 3D samples we partition them into subdomains and treat each separately on a different computing node. This approach allows us to use meshes with millions of elements as input geometries without facing limitations in computer resources. We apply this method in a proof-of-concept study of a digitized Fontainebleau sandstone sample. We use the calculated velocity profile with the finite volume procedure to simulate pore-scale solute transport and diffusion. This allows us to demonstrate the correct emerging behaviour of sample s hydrodynamic dispersivity. Finally, we model the transport of an adsorbing solute and the surface coverage dynamics is demonstrated. This information can be used to estimate the local change of a sample wettability state and the ensuing changes of the two-phase flow characteristics.@en