Numerical Methods in Petroleum Engineering – Vocabulary Review

These vocabulary flashcards cover the critical terms, equations, and concepts introduced in the Numerical Methods in Petroleum Engineering lecture, providing concise definitions for exam preparation.

Numerical Methods

Computer‐based techniques for solving mathematical problems that usually give approximate solutions whose accuracy depends on computational effort.

Scientific Computing

A broad field that covers computer applications such as CAD, root finding, and complex model development to analyze scientific or engineering problems.

Petroleum

A complex mixture of thousands of hydrocarbons composed mainly of hydrogen and carbon.

Programming (in simulation)

The act of converting a mathematical model of hypotheses, data, and assumptions into executable computer code.

Simulation

Modeling a physical system on a computer to study its behavior and obtain predictive results.

Reservoir Simulation

A multidisciplinary tool that combines physics, math, reservoir engineering, and programming to predict reservoir performance under varying conditions.

Geological Model

A static, three‐dimensional representation of the subsurface created from geophysical and geological data before production starts.

Mathematical Model

A set of equations expressing the essential features of a physical system, relating dependent variables to independent variables, forcing functions, and parameters.

Shared Earth Model

An updatable, multidisciplinary knowledge base that integrates subsurface information for consistent decision making.

Finite-Difference Method

A numerical technique that approximates derivatives by discrete differences to solve differential equations on a grid.

Diffusivity Equation

A partial differential equation derived from Darcy’s law and the continuity equation describing pressure changes in porous media; second order in space and first order in time.

Darcy’s Law (1-D, lab units)

v = −(k/μ)(dp/dl); relates flow velocity to permeability, viscosity, and pressure gradient.

Continuity (Mass-Balance) Equation

−∇·(ρv) + ρq = ∂(ρφ)/∂t; states that mass in minus mass out plus sources equals accumulation.

Equation of State (EOS)

A relation that expresses fluid density as a function of pressure and temperature.

Constitutive Equation

An equation, such as Darcy’s law, that links flux to driving forces in a porous medium.

Permeability (k)

A rock property measuring the ability to transmit fluids; typically expressed in Darcy or millidarcy.

Viscosity (μ)

A fluid’s resistance to flow; measured in centipoise (cP) or Pa·s.

Porosity (φ)

The fraction of rock volume that is pore space; dimensionless.

Total Compressibility (ct)

Sum of fluid and rock compressibilities; ct = c + CRφ0⁄φ0.

Formation Volume Factor (FVF)

Ratio of fluid volume at reservoir conditions to volume at standard conditions (Bo, Bw, Bg).

Fluid Compressibility (c)

Fractional volume change of a fluid per unit pressure change; 1/psi.

Solution Gas–Oil Ratio (Rs)

SCF of gas dissolved per STB of oil at reservoir conditions.

Fluid Density (ρ)

Mass per unit volume; lbm/ft³, kg/m³, or g/cc.

Fluid Viscosity

Resistance of fluid to deformation or flow; symbol μ, unit cP.

Fluid Saturation (S)

Fraction of pore volume occupied by a specific fluid (So, Sw, Sg).

Capillary Pressure (Pc)

Pressure difference between non-wetting and wetting phases across an interface; Pc = ρg h.

Relative Permeability (kr)

Effective permeability of a phase relative to absolute permeability; varies with saturation.

Fluid Potential (Φ)

Φ = p + γD; sum of pressure head and elevation head, measured in psi.

Fluid Head (h)

h = p/γ + D; height equivalent of fluid potential, ft.

Fluid Gravity (γ)

Pressure gradient of a fluid column, ψ/ft; γ = ρg/144.

Gradient Operator (∇, nabla)

Vector differential operator ∂/∂x î + ∂/∂y ĵ + ∂/∂z k̂ indicating spatial derivatives.

Dependent Variable

A quantity whose value depends on other variables, e.g., pressure P in P = Patm + ρg h.

Independent Variable

A variable that can be freely changed, e.g., elevation h in pressure-elevation relation.

Dirichlet Boundary

Boundary condition specifying the value of a variable (e.g., constant pressure).

Neumann Boundary

Boundary condition specifying the value of a derivative or flux (e.g., constant rate).

Mixed Boundary

Boundary that combines Dirichlet and Neumann specifications.

Laplace Equation

∇²p = 0; incompressible-fluid flow equation, a boundary-value problem.

Pressure-Squared Diffusivity

Form of gas-flow equation derived for slightly compressible gas: ∇²(p²) = (2μct/k) ∂(p²)/∂t.

Taylor Series Expansion

Mathematical series that expresses a function as sum of derivatives at a point; basis of finite differences.

Forward Difference

Approximation df/dx ≈ [f(x+Δx) − f(x)]/Δx.

Backward Difference

Approximation df/dx ≈ [f(x) − f(x−Δx)]/Δx.

Central Difference

Approximation df/dx ≈ [f(x+Δx) − f(x−Δx)]/(2Δx).

Second Central Difference

Approximation d²f/dx² ≈ [f(x+Δx) − 2f(x) + f(x−Δx)]/Δx².

Tensor (k̿)

Matrix representing permeability in principal directions; accounts for anisotropy.

Vector

Quantity with magnitude and direction (e.g., velocity v).

Scalar

Quantity described by magnitude only (e.g., pressure p).

Transmissibility (T)

Coefficient that couples flow between grid blocks; proportional to area, permeability, and inversely to viscosity and distance.

Accumulation Term

Time derivative part of PDE representing mass stored or depleted: ∂(ρφ)/∂t.

Material Balance Equation

Reservoir statement that cumulative production equals initial volume minus remaining volume plus influx.

Finite Volume (Gridblock Bulk Volume, Vb)

Total volume assigned to a grid cell, used in discretizing flow equations.

No-Flow Boundary

Boundary where normal velocity is zero; ∂p/∂n = 0.

Rate-Specified Boundary

Boundary where inflow or outflow rate is fixed; Neumann type.

Pressure-Specified Boundary

Boundary where pressure is held constant; Dirichlet type.

Incompressible-Flow Equation

∇²p = 0; Laplace form for fluids with zero compressibility.

Slightly Compressible-Flow Equation

∇²p = (μct/k) ∂p/∂t; governs liquids with small compressibility.

Compressible-Flow Equation

General PDE that considers large density changes with pressure, often for gases.

Order of a PDE

Highest derivative present; diffusivity equation is second order in space, first order in time.

Reservoir Boundary Conditions

External constraints (pressure, flux, mixed) applied to solve flow equations.

Series Conductivity (Interface Conductivity)

Combined flow capacity for two adjacent blocks: (A1k1A2k2)/(A1k1L2 + A2k2L1).