Estimating Fluid Saturations from Capillary Pressure and Relative Permeability Simulations Using Digital Rock


Direct numerical simulations of fluid flow on three-dimensional pore-scale microstructures derived from images promise more, cheaper, and faster, special core analysis for estimating volumetric and transport properties of rocks. However, the micron-scale X-ray computer tomography images generated by the present imaging technology are limited in resolution, and thus, a significant portion of rock pore volume can remain unresolved. The missing pore volume is not accessible to direct numerical simulations which limits applicability of digital rock physics to infer true residual saturation of reservoir fluids. To derive meaningful results from direct simulations, at minimum, raw fluid saturation inferred from simulations on micron-scale images must be corrected for the missing pore volume. We use concepts of capillary physics in rocks to quantify the impact of image resolution on image-derived fluid saturation and develop novel transforms that compensate for this effect on estimates of fluid saturation from multiphase simulations without the need for higher-resolution imaging. We find that image resolution constraints provide quality indicators when comparing digital rock-derived fluid saturations (e.g., connate water saturation) with those measured in a laboratory.

Article Highlights

  • Theoretical framework that defines the impact of image voxel size on the quality of fluid saturation derived from digital rock simulations

  • Novel transforms to estimate true fluid saturation from direct flow simulations on image-derived microstructures that contain sub-resolution pore volume without the need of laboratory measurements or higher-resolution images

  • Method applicable to other natural and synthetic porous media

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We thank Bochao Zhao, Robert Walsh, Steffen Berg, and Shehadah Masalmeh for their useful discussions and suggestions. We are grateful to Shell International Exploration & Production for permission to publish this paper. We also thank three anonymous reviewers who provided excellent feedback and helped improve the manuscript.

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Correspondence to Nishank Saxena.

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Appendix A: MICP Simulations and Estimation of Transform Parameters

The workflow starts with successful scanning of a rock at an appropriate resolution (voxel size of 1–2 microns) on the Zeiss Versa 520 scanner. The obtained 3D micro-CT image is processed through a non-local means filtering tool. The filtered image is then cropped down to the desired dimensions to eliminate edge artifacts created by the reconstruction and the filtering tool. The cropped volume is then viewed by an expert in order to collect important cluster centers that are utilized by an in-house developed segmentation tool that generates a bimodal. The bimodal segmentation is then loaded into a commercially available software GeoDict and then the pore space and impermeable phases are assigned the material default settings of air (fluid) and minerals (solid), respectively. Then a drainage flow simulation with mercury invading air-saturated pores is run using the inputs listed in Table

Table 2 Settings for drainage simulation using GeoDict

2. The simulation yields the non-wetting liquid capillary pressure curve from the segmented three-dimensional image of the rock. A Thomeer model curve is subsequently fit to obtain the transformed porosity \( \phi_{\infty } \), pore geometric factor \( G \), and entry pressure of mercury \( P_{{{\rm D}}} \) using the non-wetting liquid capillary pressure curve. Pore throat resolution factor \( N \) is then estimated using Eq. 4 with \( D\left( P \right) \) = \( P_{{{\rm D}}} \). The parameter \( \alpha_{{{\rm R}}} \) is estimated using Eq. 10. The parameters \( \alpha \left( \xi \right) \) are obtained utilizing the pore geometric factor and the pore throat resolution factor using Eq. 5. Another parameter, \( V_{{{{\rm closure}}}} \), volume of mercury associated with closure correction is estimated by estimating volume occupied by mercury invaded for capillary pressure lower than \( P_{{{\rm D}}} \) (Table 2).

Appendix B: Direct Pore-Scale Flow Simulator

We use a direct pore-scale visco-capillary flow simulator that is based on the numerical solution of the Helmholtz free energy. The numerical approach is based on the lattice Boltzmann method (LBM) for efficiently simulating two-phase pore-scale flow directly on large 3D images of real rocks obtained from micro-computed tomography (micro-CT) scanning. We refer to the energy-based phase-field model solved by the use of LBM as eLBM. The implementation is performed in CUDA programming language to take maximum advantage of accelerated computing multinode general-purpose graphics processing units (GPGPUs). eLBM’s momentum-balance solver is based on the multiple relaxation time (MRT) model. The Boltzmann equation is discretized in space, velocity (momentum), and time coordinates using a 3D 19-velocity grid (D3Q19 scheme), which provides the best compromise between accuracy and computational efficiency. The Cahn–Hilliard equation that governs the order-parameter distribution is fully integrated into the LBM framework that accelerates the pore-scale simulation on real systems significantly.

We use a process-based approach to numerically compute relative permeability in eLBM. This approach accounts for the drainage–imbibition hysteresis by the use of direct simulations. This method represents a steady-state type of displacement approach to relative permeability computation. A forced-drainage process is simulated first. Here, the in situ wetting phase fully saturating the pore space is displaced by an injected non-wetting phase. The rock model is fully saturated by the wetting phase. A buffer layer saturated with oil is placed to the inlet, and one saturated with water is placed to the outlet. Oil is injected from the inlet buffer layer into the pore space with a constant prescribed velocity until the average water saturation in the pore space does not exhibit an appreciable change (convergence), at which time the drainage calculations are terminated. Note that the water saturation achieved in this way may not necessarily correspond to the irreducible or connate water saturation of a representative elementary volume, as the viscous pressure drop achieved in a rather small computational domain size (much smaller than a core plug) is significantly less than capillary pressure established in the porous plate or centrifuge method corresponding to the saturation height in an oil reservoir. Snapshots of the fluid configuration at different average water saturation values are used for the drainage relative permeability calculations. The post-drainage fluid configuration is then used as the initial state for the forced-imbibition simulation where the non-wetting phase is displaced by the wetting phase. The fluids in the inlet and outlet buffers are interchanged prior to the forced-imbibition calculations. The forced-imbibition calculations are continued again until average water saturation in the porous domain converges. It is important to note that the visco-capillary model formulation does not entirely capture the physics of spontaneous imbibition, which is a solely capillary dominated process. For stability reasons, there is a (problem dependent) minimum limit to the inlet velocity that can be imposed to a forced-imbibition simulation. Therefore, the spontaneous imbibition is approximated by using a small constant velocity boundary condition that can be stably simulated by the use of eLBM. Depending on the accuracy of this approximation, the oil saturation remaining at the end of the forced-imbibition process may or may not necessarily correspond to the “ground truth” residual oil saturation due to a potentially limited viscous pressure drop. Again, the fluid-configuration snapshots at different average water saturation values acquired during forced-imbibition simulations are then used for relative permeability computations. In order to ensure steady-state conditions in the relative permeability computation, for a given saturation snapshot, calculations are performed using the loop-type periodic boundary conditions imposed on all faces of the domain. The full mirroring of the porous domain in the main flow direction is applied such that the periodic boundary condition is fully honored on the outlet face of the domain. The system is then driven to steady state, which closely mimics the steady-state relative permeability experiment. Upon convergence of the system to steady state, effective wetting and non-wetting phase permeabilities computed by eLBM are normalized by the absolute permeability computed with an accurate and efficient MRT-LBM code (Alpak et al. 2018b) prior to the relative permeability simulation. The fluxes used for computing effective wetting and non-wetting phase permeabilities are derived by monitoring the average velocities of wetting and non-wetting phases, respectively, over the part of the domain that corresponds to the porous rock. The process is then repeated for all saturation snapshots that cover the saturation range observed during the forced-drainage or forced-imbibition process.

Details of eLBM and its validation are documented in Alpak et al. (2018a) and Alpak et al. (2019). We refer the reader to Alpak et al. (2019) for the description and validation of eLBM’s forced-drainage and forced-imbibition modules. Description and validation of the relative permeability computation module can be found in Alpak et al. (2018a).

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Saxena, N., Alpak, F.O., Hows, A. et al. Estimating Fluid Saturations from Capillary Pressure and Relative Permeability Simulations Using Digital Rock. Transp Porous Med 136, 863–878 (2021).

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  • Digital rock
  • Fluid saturation
  • Multiphase flow
  • Micro-CT images
  • Sub-resolution porosity