Incorporating sub-metre scale capillary pressure heterogeneity into upscaled numerical models is key to the successful prediction of low flow potential plume migration and trapping at the field scale. At low flow potential, nearing the capillary limit, the upscaled equivalent relative permeability incorporating capillary pressure heterogeneity is far from that derived at the viscous limit , dependent on the heterogeneity structure and flow rate, i.e. dependent on the capillary number .
To derive upscaled multiphase functions for field scale modelling, numerical or analytical methods are generally used either with statistical realisations of the permeability-capillary pressure field  or with simplified layered systems . However, no protocol currently exists to efficiently derive these upscaled functions on heterogeneous rock cores directly from the subsurface, i.e. cores which are truly representative of the system under consideration. Experimental observations must be combined with numerical upscaling so that subsurface rock cores can be directly characterised and used to accurately inform further modelling efforts.
In this work, we present a characterisation approach that incorporates experiments and numerical simulations to accurately characterise drainage and imbibition multiphase flow functions from the mm-m scale on heterogeneous subsurface rocks. We make use of a rich experimental core flood dataset obtained on N$_2$-DI water and CO$_2$-brine systems across multiple fractional flows, total flow rates and drainage-imbibition cycles. The experiments are performed on three distinct sandstones; a relative homogeneous Bentheimer outcrop core, a Bunter core from the Southern North Sea subsurface and a composite Captain core from the Northern North Sea subsurface.
Through characterisation of the mm scale capillary pressure heterogeneity and the viscous limit relative permeabilities we create parametrised 3D numerical rock cores. We build on the recent work of  and  by incorporating imbibition cycles into the characterisation approach, allowing the hysteretic nature of the systems to be analysed. The numerical cores are validated by simulating core flood experiments across a wide range of flow conditions, showing that mm-m scale experimental saturations and equivalent, low flow potential relative permeabilities can be accurately predicted for both drainage and imbibition cycles.
The numerical cores are then used to derive equivalent relative permeabilities across a wide range of conditions representative of flow regimes from the well to the far field in the subsurface, incorporating mm scale capillary pressure heterogeneities and saturation hysteresis. These functions show significant deviation from those derived in the viscous limit (orders of magnitude at some points), and highlight the impact that heterogeneity and hysteresis have on low potential fluid flow. We show that under this new approach, equivalent relative permeabilties can be effectively derived for drainage and imbibition cycles on heterogeneous rock cores, and that a relatively simple entry pressure scaling (or permeability-porosity scaling with the Leverett J-function) can be used to uniquely describe the capillary pressure heterogeneity with hysteresis.
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