# Ingrain Digital Rock Physics Lab

# Glossary

^{2}. Common units of permeability are Darcy (1 D = 10

^{-12}m

^{2}). 1 milliDarcy (mD) is 10

^{-3}D = 10

^{-15}m

^{2}. Absolute permeability is always normalized by the viscosity. Absolute permeability may be affected by the type of fluid because certain fluids can react with the mineral matrix, thereby altering the pore geometry. Another reason that measured permeability depends on the fluid type is that some fluids, such as air, do not precisely obey Newtonian viscosity formalism. In digital permeability computations, these deviations are nonexistent simply because we can impose desired viscosity laws as required by the process under examination.

Capillary Force

See Surface Tension

Core

A core is a cylindrical fragment of rock extracted from a well using specialized drilling equipment. The diameter of a core is on the order of several centimeters. The length of a continuous core may reach tens of meters. A sidewall (or percussion) core is taken directly in the well at its wall.

Core Plug

A core plug is a cylindrical rock fragment cut out of a core for laboratory measurements. Its size is on the order of 3-5 centimeters.

CT Scan

Computed tomography (or CT scan) makes a cross-sectional x-ray picture of a "slice" of fragment of rock. The sample rotates and the machine X-rays it from different angles. The resulting 2D images are then processed by a computer to produce a 3D image. In this image, the mineral matrix and pore space are assigned very different shades of gray, respectively. This difference is then used in image processing to subdivide (segment) the image into pores and grains.

Deformation

Deformation (or strain) is the relative change of size (or shape) due to applied force. For example, if a 1-m bar is stretched to 1.01 m, its deformation is 0.01, or 1%.

^{3}or g/cc. 1 g/cc = 1000 kg/m

^{3}. The density of pure water is about 1 g/cc. The density of oil is smaller and may be, e.g., 0.6 g/cc. The density of air at the earth surface is about 0.0012 g/cc. The density of natural gas in the subsurface is larger due to high pressure and may be, e.g., 0.02 g/cc. The density of minerals is much larger and may be 2.65 g/cc for quartz and 2.71 g/cc for calcite. The density of porous rock is smaller than that of pure mineral and larger than that of fluid. A typical (but not general) value may be between 1.9 and 2.5 g/cc.

Drill Cuttings

Drill cuttings are small (just a few mm) fragments of crushed rock washed to the surface by the drilling mud.

Elasticity

Material is called elastic if it recovers its original shape when the loadings causing the deformation are removed. No energy is lost during elastic recovery.

Elastic Moduli

An elastic modulus is a proportionality coefficient between stress and the resulting strain in an elastic body. For example, if strain 0.01 is caused by stress 10

^{8}Pa, the corresponding modulus is 10 GPa. Because the type of deformation depends on the set of stresses applied, relations between stress and strain are often complex. In general, they are governed by 21 elastic stiffnesses (or moduli). However, in an isotropic elastic body, this large number reduces to only two independent moduli (also called elastic constants). Most commonly, we use such pairs as the bulk and shear moduli (related to the changes in the volume and distortion, respectively); Lame’s constants (λ and μ); and Young’s modulus and Poisson’s ratio. Fluids offer no resistance to shear (shape distortion). As a result, there is only one elastic modulus in fluid – the bulk modulus.

Elastic-Wave Velocity

The elastic-wave velocity (or simply velocity) is the speed of propagation of an elastic disturbance (stress and the accompanying strain) in solid or fluid. Because an isotropic solid has only two independent elastic constants, it has two types of velocity – compressional (

*V*) and shear (

_{p}*V*), also called the P- and S-wave velocity, respectively. The former is always larger than the latter. The velocity is linked to the elastic moduli by remarkably simple relations resulting from the wave equation (the equation that governs the propagation of elastic disturbances). The P-wave velocity is the square root of the compressional modulus divided by density while the S-wave velocity is the square root of the shear modulus divided by density. The compressional modulus is the bulk modulus plus 4/3 shear modulus. Because fluids do not resist shearing, the S-wave velocity in fluid is zero. Typical (but not general) values for

_{s}*V*may be 0.34 km/s in air; 1.5 km/s in water; 2.0 to 5.0 km/s in porous rock; and over 7.0 km/s in pure mineral (e.g., dolomite). Typical (but not general) values for

_{p}*V*may be 1.0 to 3.0 km/s in porous rock and over 4.0 km/s in pure mineral (e.g., dolomite).

_{p}Electrical Conductivity

Electrical (or specific) conductivity is the ratio of the electric current density to the electric field strength that is applied to a conductor to generate this electric current. Conductivity is measured in Siemens/m (S/m). Electrical conductivity of gold is 45.106 S/m, that of seawater is 5 S/m, while that of fresh water is on the order of 0.01 S/m. The inverse of electrical conductivity is electrical resistivity, measured in Ohm.m ( .m). Natural rock is often a conductor because it is partly or totally filled with conductive brine. Other elements of rock, such as iron-containing minerals and porous clays are conductors as well.

Finite Element Method

The deformation of a solid or fluid can be described by partial differential equations with certain boundary conditions. In most complex simulations, analytical solutions are nonexistent. Direct finite-difference computations of these equations are often convoluted and practically unattainable. An alternative is to subdivide the material domain under examination into finite elements for which simple analytical solutions are available. Then, finite element method (FEM) is used to “glue” these analytical solutions together to approximate the deformation of the entire system. The same method can be applied not only to describe elastic deformation, but also fluid and electrical flow.

Formation Factor

Electrical formation factor (FF) is defined as the ratio of the resistivity of porous rock to that of the conductive brine that fully saturates the rock. FF does not depend on the saturating fluid and is only affected by the pore-space geometry.

Image Segmentation

In a 3D CT scan image of porous rock, the mineral matrix and pore space are assigned very different shades of gray, respectively. This difference is used in image processing to subdivide the image into pores and grains. This task of separating the pores from grains in such 3D objects is called image segmentation. The main technical challenge in image segmentation is the gradual transition from dark to light shade of gray at the edges of the pore space. A CT scan image may be segmented into more than two entities. Density contrast permitting, segmented image may include three (e.g., mineral, pores with air, and pores with liquid) or more (e.g., light versus dense minerals) phases.

Irreducible Water Saturation

Irreducible water saturation is the minimum possible water saturation in rock. It is almost always larger than zero because water cannot be moved from very small capillaries. Also, very thin layers of water strongly stick to the surface of some minerals and cannot be easily removed.

Lattice-Boltzmann Method (LBM)

LBM is a computational method designed to simulate slow flow of viscous Newtonian fluid. This flow is governed by the Navier-Stokes equations, which are obtained from the law of mass conservation, Newton’s second law, and the linear friction law between layers of viscous fluid. The main problem of computing these equations in a complex pore space is the boundary no-slip conditions at the pore walls. LBM is structured to simulate collisions of virtual particles on a 3D lattice. It can easily handle the no-slip boundary conditions on any surface and, therefore, does not require a replacement of the pore space with idealized geometrical bodies. The particle collision rules in LBM are specified so that viscous flow automatically emerges from the behavior of the ensemble of these virtual particles. LBM can simulate a single-phase flow as well as a multiphase flow, accounting for the viscosities of the phases as well as wettability and surface tension.

Laplace Equation

The Laplace (or elliptical) equation is a partial differential equation that describes a steady-state flow of electricity (or heat exchange) through a conductive medium. One specific application is the resistivity of porous rock filled with conductive fluid. Analytical solutions of this equation for a realistic pore space are nonexistent. One way of computing this equation is FEM.

Micro-core

A micro-core is a small fragment of rock, typically a cylinder of approximately 1 mm in size, used in CT scanning.

Multiphase Flow

Multiphase flow occurs in a porous system where more than one fluid phase is present. In the subsurface, these phases may be water and oil; water and gas; or water, oil, and gas. Multiphase flow and the mobility of each phase in the presence of the others is affected by the viscosities of the phases and their surface tension. In turn, these physical properties affect the relative permeabilities.

Nanoscale

Nanoscale pertains to sizes in the range between 1 and 100 nm, where 1 nm = 10

^{-9}m.

Navier-Stoke Equations

Navier-Stokes equations define the dynamic movement of viscous fluid (similar to Newton’s second law defining the movement of a point mass). These equations are obtained from a combination of Newton’s second law applied to an element of fluid, relation between the shear traction and velocity gradient, and mass conservation law.

Plasticity

Material is plastic if it is not elastic. This means that it retains residual deformation after the stresses are removed. Typically, part of the energy used to deform a plastic material is lost to heat.

Pixel

Pixel is the smallest piece of information in 2D image (e.g., 1 by 1 micron).

Porosity

Porosity is a measure of the void spaces in rock. It is defined as the ratio of the pore volume to the total volume of a rock fragment. It is measured as a fraction, between zero and 1, or as a percentage between zero and 100%.

Pressure (as related to stress)

Stress in fluid is called pressure. 10 m of water creates pressure about 1 bar = 0.1 MPa.

Relative Permeability

Relative permeability is a dimensionless measure of the permeability of a fluid phase as it flows through porous rock in the presence of another fluid phase. The apparent (or phase) permeability of a fluid phase in the presence of another fluid phase is measured exactly as its absolute permeability (of course, it will be different because part of the pore space is now occupied by another fluid). Then, the corresponding relative permeability is the ratio of this apparent permeability to the absolute permeability. Therefore, it is dimensionless. It is natural to expect that the relative permeability varies between zero and one.

Residual Oil Saturation

During hydrocarbon production, only a fraction of oil present in the reservoir can be delivered to the well and lifted to the surface. Much oil is trapped in small and disconnected pores. This remaining oil volume normalized by the total pore volume is the residual oil saturation.

Saturation

Saturation is the ratio of the pore volume occupied by a fluid phase to the total pore volume. According to this definition, water, oil, and gas saturations in a given volume of rock should add up to precisely one.

Stress

Stress is defined as the ratio of the total force to the area it is acting upon. Stress is measured in Newtons per m

^{2}(N/m

^{2}). 1 Pa = 1 N/ m

^{2}; 1 MPa = 10

^{6}Pa; 1 GPa = 10

^{9}Pa. Stress in fluid is called pressure. 10 m of water creates pressure about 1 bar = 0.1 MPa.

Strain

See Deformation

Surface Tension and Capillary Force

A surface tension exists at the interface between two immiscible fluids or between a fluid and a solid. The surface tension acts tangentially to the interface surface. Surface tension is responsible for retaining liquid in a thin capillary open on both sides. This is why it is called the capillary force.

Viscosity

Viscosity is a measure of the resistance of fluid to the rate of shear and hence to flow. Dynamic viscosity is the ratio between the shear stress acting along any plane between adjacent fluid elements and the velocity gradient in the direction normal to the shear stress. Kinematic viscosity is the ratio of dynamic viscosity and density. Dynamic viscosity is measured in Pascal times second. The viscosity of water at surface conditions is about 0.001 Pa s = 1 cPs (centiPoise).

Voxel

A voxel is the smallest piece of information in 3D image (e.g., 1 by 1 by 1 micron).

vRock® digital reservoir rock

vRock is a digital object that represents a fragment of real rock in 3D. A vRock is obtained from the CT scan of this fragment where each voxel is assigned a number corresponding to a shade of gray. This spectrum is segmented to single out the pore space and mineral matrix, which, in turn, can be segmented into several mineral types. Mathematically, a vRock is a 3D matrix with each element being 0 or 1 (if the segmentation singles out only two components), or 0, 1, or 2, for 3-component segmentation, and so on. A vRock is a shared digital object used to computationally simulate physical processes in real rock (e.g., viscous flow for permeability).

Wettability

Wettability is the measure of adherence or repellence of fluid and a solid surface. A material is water-wet if a drop of water placed upon its surface spreads along this surface. Wettability is quantified by the contact angle between a drop of fluid and solid surface. The fluid is wetting if this angle is larger than 90 degrees and vice versa.

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