PE 437 – Numerical Methods in Petroleum Engineering: Chapter 4 Vocabulary

Vocabulary flashcards covering main concepts, equations, coordinate systems, numerical methods, flow regimes, and key physical properties introduced in Chapter 4 on single-phase flow formulations.

Single-Phase Flow

Movement of a single fluid phase (oil, water, or gas) through a porous medium.

Darcy’s Equation

Empirical law that relates volumetric flow rate of a fluid through porous media to pressure gradient, permeability, fluid viscosity, and formation area.

Continuity (Mass-Balance) Equation

Statement of conservation of mass for flow in a porous medium; equates accumulation to inflow minus outflow plus sources/sinks.

Diffusivity Equation

Partial-differential equation obtained by combining Darcy’s law with the continuity equation; second order in space and first order in time.

Rectangular Coordinates

Linear (x, y, z) coordinate system used to model 1-D, 2-D, or 3-D flow in reservoir simulation.

Cylindrical Coordinates

(r, θ, z) system convenient for radial flow problems such as around a wellbore.

Spherical Coordinates

(r, Θ, θ) system useful for fully radial flow from a point source in three dimensions.

Finite-Difference Method

Numerical technique that solves differential equations by replacing derivatives with difference quotients on a grid.

Taylor Series Truncation

Process of approximating a function by a finite number of terms in its Taylor expansion; basis for finite-difference formulas.

Incompressible Fluid Flow (Laplace Equation)

Form of the diffusivity equation when fluid density is constant; time-independent and lacks porosity term.

Slightly Compressible Fluid Flow

Diffusivity equation that includes porosity and time but assumes small fluid compressibility; typical for liquid reservoirs.

Compressible Fluid Flow

Form of the flow equation where fluid density and viscosity vary significantly with pressure; characteristic of gas reservoirs.

Boundary-Value Problem

Mathematical problem where spatial derivatives exist but time does not; requires specification of conditions only at boundaries.

Initial-Boundary-Value Problem

Problem involving both time and space derivatives; needs an initial condition and boundary conditions for t > 0.

Initial Condition

Value of pressure (and/or saturation) specified at t = 0 throughout the reservoir to start a transient simulation.

Boundary Condition

Constraint applied at reservoir limits or wellbores to obtain a unique solution; may specify pressure, flow rate, or their derivatives.

Neumann Boundary Condition

Boundary specification that sets the spatial derivative (usually rate or flux) at the boundary; ‘no-flow’ or constant-rate condition.

Dirichlet Boundary Condition

Boundary specification that sets the primary variable (usually pressure) at the boundary; constant-pressure condition.

External Boundary

Physical limit of the reservoir model where boundary conditions (e.g., no-flow or constant pressure) are applied.

Internal Boundary (Wellbore)

Interior surface within the reservoir, representing a well, where pressure or rate conditions are enforced.

Mixed Boundary Condition

Situation where different segments of the boundary have different specifications, e.g., part no-flow, part constant pressure.

Porous Medium

Rock framework containing interconnected voids that allow fluid flow.

Permeability (k)

Measure of the ability of porous rock to transmit fluids; expressed in darcies or μm².

Porosity (ϕ)

Fraction of bulk rock volume that is pore space available for fluids.

Viscosity (μ)

Fluid property indicating resistance to flow; appears in Darcy’s law denominator.

Compressibility (c)

Fractional change in volume (or density) per unit change in pressure; important for slightly compressible and gas flow equations.

Formation Volume Factor (Bo, Bg)

Ratio relating reservoir-volume fluid to surface-volume fluid for oil (Bo) or gas (Bg).

Solution Gas-Oil Ratio (Rs)

Amount of gas dissolved in oil at reservoir conditions, expressed as scf/STB.

Homogeneous Isotropic Reservoir

Reservoir with uniform properties (kx = ky = k_z) in all directions.

Gridblock

Small volumetric cell in a numerical model over which average properties are assumed constant.

Transmissibility

Coefficient representing ease of flow between adjacent gridblocks; combines permeability, area, viscosity, and cell dimensions.

Volumetric Velocity (u or q/A)

Darcy (superficial) velocity obtained by dividing volumetric flow rate by cross-sectional area.

Gravitational Acceleration (g)

Acceleration due to gravity (32.174 ft/s² or 9.8066 m/s²) appearing in gravity terms of flow equations.