Nishank Saxena

Digital rock physics benchmarks-part II: Computing effective properties

This is the second and final part of our digital rock physics (DRP) benchmarking study. We use segmented 3-D images (one for Fontainebleau, three for Berea, three for a carbonate, and one for a sphere pack) to directly compute the absolute permeability, the electrical resistivity, and elastic moduli. The numerical methods tested include a finite-element solver (elastic moduli and electrical conductivity), two finite-difference solvers (elastic moduli and electrical conductivity), a Fourier-based Lippmann-Schwinger solver (elastic moduli), a lattice-Boltzmann solver (hydraulic permeability), and the explicit-jump method (hydraulic permeability and electrical conductivity). The set-ups for these numerical experiments, including the boundary conditions and the total model size, varied as well. The results thus produced vary from each other. For example, the highest computed permeability value may differ from the lowest one by a factor of 1.5. Nevertheless, all these results fall within the ranges consistent with the relevant laboratory data. Our analysis provides the DRP community with a range of possible outcomes which can be expected depending on the solver and its setup.

Estimating elastic moduli of rocks from thin sections

Estimation of elastic rock moduli using 2D plane strain computations from thin sections has several numerical and analytical advantages over using 3D rock images, including faster computation, smaller memory requirements, and the availability of cheap thin sections. These advantages, however, must be weighed against the estimation accuracy of 3D rock properties from thin sections. We present a new method for predicting elastic properties of natural rocks using thin sections. Our method is based on a simple power-law transform that correlates computed 2D thin section moduli and the corresponding 3D rock moduli. The validity of this transform is established using a dataset comprised of FEM-computed elastic moduli of rock samples from various geologic formations, including Fontainebleau sandstone, Berea sandstone, Bituminous sand, and Grossmont carbonate. We note that using the power-law transform with a power-law coefficient between 0.4-0.6 contains 2D moduli to 3D moduli transformations for all rocks that are considered in this study. We also find that reliable estimates of P-wave (Vp) and S-wave velocity (Vs) trends can be obtained using 2D thin sections. New method for estimating elastic moduli of rocks using 2D thin sections.Method is based on a simple power-law 2D-3D transform.Power-law coefficient between 0.4-0.6 contains 2D-3D moduli transformations.Rock thin sections yield accurate estimates of P-wave and S-wave velocity trends.

Estimating permeability from thin sections without reconstruction: Digital rock study of 3D properties from 2D images

We present a new approach for predicting permeability of natural rocks using thin sections. Our approach involves two steps: (1) computing permeability of the thin sections for flow normal to the face, and (2) application of new robust 2D-3D transforms that relate thin section permeability to 3D rock permeability using calibration parameters. We perform step 1 using Lattice-Boltzmann and finite difference schemes, which are memory efficient. We discuss two models to perform step 2. Our two-step approach is fast and efficient, since it does not require reconstruction of the unknown 3D rock using 2D thin section information. We establish the applicability of this new approach using a dataset comprised of LBM-computed permeability of rock samples from various geologic formations, including Fontainebleau sandstone, Berea sandstone, Bituminous sand, and Grosmont carbonate. We find that for sandstones our approach predicts fairly accurate permeability with little calibration. Predicting permeability of carbonates from thin sections is more challenging due to microstructural complexity thus model parameters require more calibration. For general workflow, we propose to first calibrate the proposed models using the available 3D information on the rock microstructure (from microCT, SEM, etc.) and then predict the permeability for rocks from the same geological formation for which only 2D thin sections are available.

A dual-porosity scheme for fluid/solid substitution

Estimating the impact of solid pore fill on effective elastic properties of rocks is important for a number of applications such as seismic monitoring of production of heavy oil or gas hydrates. We develop a simple model relating effective seismic properties of a rock saturated with a liquid, solid, or viscoelastic pore fill, which is assumed to be much softer than the constituent minerals. A key feature of the model is division of porosity into stiff matrix pores and compliant crack-like pores because the presence of a solid material in thin voids stiffens the rock to a much greater extent than its presence in stiff pores. We approximate a typical compliant pore as a plane circular interlayer surrounded by empty pores. The effect of saturation of the stiff pores is then taken into account using generalized Gassmann’s equations. The proposed model provides a good fit to measurements of the shear stiffness and loss factor of the Uvalde heavy-oil rock at different temperatures and frequencies. When the pore fill is solid, the predictions of the scheme are close to the predictions of the solid squirt model recently proposed by Saxena and Mavko. At the same time, the present scheme also gives a continuous transition to the classic Gassmann’s equations for a liquid pore fill at low frequencies and the squirt theory at high frequencies.

A distributed parallel multiple-relaxation-time lattice Boltzmann method on general-purpose graphics processing units for the rapid and scalable computation of absolute permeability from high-resolution 3D micro-CT images

Digital rock physics (DRP) is a rapidly evolving technology targeting fast turnaround times for repeatable core analysis and multi-physics simulation of rock properties. We develop and validate a rapid and scalable distributed-parallel single-phase pore-scale flow simulator for permeability estimation on real 3D pore-scale micro-CT images using a novel variant of the lattice Boltzmann method (LBM). The LBM code implementation is designed to take maximum advantage of distributed computing on multiple general-purpose graphics processing units (GPGPUs). We describe and extensively test the distributed parallel implementation of an innovative LBM algorithm for simulating flow in pore-scale media based on the multiple-relaxation-time (MRT) model that utilizes a precise treatment of body force. While the individual components of the resulting simulator can be separately found in various references, our novel contributions are (1) the integration of all of the mathematical and high-performance computing components together with a highly optimized code implementation and (2) the delivery of quantitative results with the simulator in terms of robustness, accuracy, and computational efficiency for a variety of flow geometries including various types of real rock images. We report on extensive validations of the simulator in terms of accuracy and provide near-ideal distributed parallel scalability results on large pore-scale image volumes that were largely computationally inaccessible prior to our implementation. We validate the accuracy of the MRT-LBM simulator on model geometries with analytical solutions. Permeability estimation results are then provided on large 3D binary microstructures including a sphere pack and rocks from various sandstone and carbonate formations. We quantify the scalability behavior of the distributed parallel implementation of MRT-LBM as a function of model type/size and the number of utilized GPGPUs for a panoply of permeability estimation problems.

Rock physics model for seismic velocity & fluid substitution in sub-resolution interbedded sand-shale sequences: Rock physics model for seismic velocity

The measured geophysical response of sand–shale sequences is an average over multiple layers when the tool resolution (seismic or well log) is coarser than the scale of sand–shale mixing. Shale can be found within sand–shale sequences as laminations, dispersed in sand pores, as well as load bearing clasts. We present a rock physics framework to model seismic/sonic properties of sub-resolution interbedded shaly sands using the so-called solid and mineral substitution models. This modelling approach stays consistent with the conceptual model of the Thomas–Stieber approach for estimating volumetric properties of shaly sands; thus, this work connects established well log data-based petrophysical workflows with quantitative interpretation of seismic data for modelling hydrocarbon signature in sand–shale sequences. We present applications of the new model to infer thickness of sand–shale lamination (i.e., net to gross) and other volumetric properties using seismic data. Another application of the new approach is fluid substitution in sub-resolution interbedded sand–shale sequences that operate directly at the measurement scale without the need to downscale; such a procedure has many practical advantages over the approach of “first-downscale-and-then-upscale” as it is not very sensitive to errors in estimated sand fraction and end member sand/shale properties and remains stable at small sand/shale fractions.

CHANGE IN EFFECTIVE ELASTIC PROPERTIES OF SOLID-FILLED POROUS MEDIA UPON SUBSTITUTION

Biot-Gassmann relations that link the change in effective elastic properties of fluid-filled porous media to the change in pore fluid properties are not applicable for solid-filled porous media. Such problems arise when concerned with pore-filling materials with non-negligible shear modulus; examples of such materials include molten lava, cold bitumen, kerogen, clay, etc. In this paper, we discuss how effective solid-filled bulk and shear moduli of a solid-filled porous medium (or composite) are explicitly related to other effective stiffnesses and pore-filling solid elastic properties. Such relations are known as substitution equations and are particularly useful in exploration geophysics, for example in reservoir monitoring. We present new equations which improve on previously suggested solid substitution relations by Ciz and Shapiro.