Numerical Methods in Petroleum Engineering – Review 1

Vocabulary flashcards summarizing fundamental terms, equations, and concepts from the Numerical Methods in Petroleum Engineering lecture notes.

Numerical Methods

Techniques for solving mathematical problems on a computer that yield approximate solutions whose accuracy depends on computational effort.

Scientific Computing

A broad field of computer applications that includes computer-aided design, root-finding, and building complex numerical models.

Petroleum

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

Programming (in simulation)

Creating a mathematical description of a system, then coding algorithms to solve or exercise the model and obtain results.

Simulation

The computer-based modeling of a system using hypotheses, data, and equations to predict behavior under various conditions.

Analytical Solution

An exact, closed-form mathematical solution obtainable without numerical approximation.

Finite-Difference Method

A numerical technique that approximates derivatives with differences at discrete grid points, often derived from Taylor series.

Darcy’s Law (basic)

v = −(k/μ) ∇p Describes single-phase fluid velocity through porous media proportional to permeability and pressure gradient.

Continuity (Mass-Balance) Equation

−∇·(ρ v) + ρ q = ∂(ρ φ)/∂t Balances mass entering, exiting, accumulating, or added by sources/sinks in a control volume.

Diffusivity Equation

∇·(k ∇p) = (φ μ ct / k) ∂p/∂t Second-order in space, first-order in time; governs pressure diffusion in porous media.

Gradient Operator (∇, Nabla)

Vector differential operator (∂/∂x, ∂/∂y, ∂/∂z) indicating spatial rate of change of a scalar field.

Equation of State (EOS)

Relationship giving fluid density as a function of temperature and pressure, e.g., ρ = ρsc[1 + (p − Psc) cT − (T − Tsc) cT].

Total Compressibility (ct)

Sum of fluid compressibility and rock compressibility: ct = cf + φ CR/φ₀.

Formation Volume Factor (FVF)

Ratio of fluid volume at reservoir conditions to volume at standard conditions; Bo, Bw (bbl/STB) or Bg (ft³/SCF).

Fluid Potential (Φ)

Φ = p + γ D Sum of pressure and elevation head expressed in psi.

Fluid Head (h)

h = p/γ + D Elevation equivalent of fluid potential, measured in feet.

Fluid Gravity (γ)

Weight density; γ = ρ g /144 in psi/ft for oilfield units.

Porosity (φ)

Fraction of bulk rock volume that is pore space; dimensionless.

Permeability (k)

Measure of rock’s ability to transmit fluids; darcy (D) or millidarcy (mD).

Viscosity (μ)

Fluid resistance to flow or deformation; measured in centipoise (cP).

Fluid Compressibility (cf)

Relative change in fluid volume per unit pressure change: cf = −(1/V)(∂V/∂p).

Solution Gas–Oil Ratio (Rs)

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

Fluid Density (ρ)

Mass per unit volume of a fluid; lbm/ft³ or kg/m³.

Capillary Pressure (Pc)

Pressure difference between non-wetting and wetting phases across an interface: Pc = pnw − pw.

Relative Permeability (kr)

Effective permeability of a phase divided by absolute permeability; function of saturation.

Fluid Saturation (Sϕ)

Volume fraction of a specific fluid in pore space (So, Sw, Sg).

Dirichlet Boundary Condition

Boundary with specified pressure (constant-pressure boundary).

Neumann Boundary Condition

Boundary with specified flux/gradient (constant-rate boundary).

Mixed Boundary Condition

Boundary combining specified pressure and flux terms.

No-Flow Boundary

External boundary where normal flux equals zero; simulates impermeable rock or symmetry.

Initial Condition

Specification of pressure (or other variable) at t = 0 throughout the domain.

Tensor

Mathematical object with multiple components; e.g., permeability tensor k_ij expressing directional permeability.

Vector

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

Scalar

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

Transmissibility (T)

Coefficient linking pressures to flow between grid blocks; incorporates k, A, μ, Δx, and conversion factors.

Reservoir Simulation

Multidisciplinary process using physics, math, and programming to predict reservoir performance under various operating scenarios.

Geological Model

Static 3-D representation of reservoir structure and properties produced by geologists/geophysicists.

Mathematical Model

Set of equations capturing essential behavior of a physical system; may include material balance, decline curves, or analytical solutions.

Compressible-Flow Equation (Pressure-Squared Form)

For ideal gas: ∂²(p²)/∂x² = (2 φ μ ct / k) ∂p/∂t used in low-pressure gas reservoirs.

Incompressible Flow Equation

Laplace equation: ∇²p = 0 Describes steady single-phase flow with constant density and no sources.

Slightly Compressible Flow Equation

∇²p = (φ μ ct / k) ∂p/∂t Applicable when density changes with pressure are small.

Finite Volume (Gridblock Bulk Volume, Vb)

Rock volume represented by a computational cell; used in accumulation term of simulators.

Taylor Series Expansion

Series representation of a function used to derive finite-difference approximations.

Forward Difference

Numerical derivative: (f(x+Δx) − f(x)) / Δx.

Backward Difference

Numerical derivative: (f(x) − f(x−Δx)) / Δx.

Central Difference

Numerical derivative: (f(x+Δx) − f(x−Δx)) / (2Δx).

Pressure–Elevation Relationship

P = Patm + ρ g h Links pressure to fluid column height.

Root-Finding

Numerical process of solving f(x) = 0 for x (e.g., Newton-Raphson).

Approximate Solution

Numerical answer that approaches the exact solution as grid is refined or iterations increase.

Source/Sink Term (q)

Mass or volume rate added to or removed from a control volume via wells.

Total System Conductivity (Interface)

Equivalent conductivity for series flow between two blocks: (A1 k1 A2 k2)/(A1 k1 L2 + A2 k2 L1).

Oil FVF Above Bubble-Point

Bo = Bob [1 − co (p − pb)] Linear approximation using oil compressibility.

Reduced Gas Compressibility (c_r)

c_r = (1/Ppr)(∂Z/∂Ppr)Tpr Compressibility expressed in reduced coordinates.

Gamma (γ) Conversion Factor

γ = ρ g /144 Converts weight density to pressure gradient in psi/ft.

Taylor Law of Exponents

a^m a^n = a^{m+n}; used in series expansions for e^x.

Ideal Gas Law (Reservoir Units)

ρg = (M p)/(Z R T) Relates density, pressure, temperature, and compressibility factor.

Material Balance (Rate Form)

Rate in − Rate out + Rate injected = Rate of accumulation within a control volume.

Well-Posed Problem

Mathematical problem possessing existence, uniqueness, and stability of solution with given boundary conditions.

Total Accumulation Term (Mass Conservative)

∂(φ ρ)/∂t Ensures mass conservation when porosity or density are pressure-dependent.