Topics in Reservoir Engineering (6).pdf

(1)

Analytical Solution of

1D Single Phase Flow Equation

[ ]

Hoonyoung Jeong

Department of Energy Resources Engineering Seoul National University

[email protected] SNU ERE Hoonyoung Jeong

(2)

Assumptions in Analytical Solution of 1D Flow Equation

• Homogeneous rock

✓Constant porosity and permeability

✓How different from isotropy?

• Constant fluid viscosity Things that are needed to Derive a Flow Equation (for Reservoir Simulation)

• Reservoir structure

✓1-dimension

• Fluid properties

✓Single phase

✓Fluid compressibility (slightly compressible fluid)

• Rock properties

✓Porosity

✓Permeability

✓Rock compressibility (slightly compressible rock)

• Properties of rock and fluid

✓Capillary pressure

✓Relative permeability

• Governing equation

✓Continuity equation

✓Constitutive equation

✓Darcy’s law

• Boundary and initial conditions

(3)

1D Flow Equation

• ^𝜕

𝜕𝑥

𝑘𝜌 𝜇

𝜕𝑃

𝜕𝑥 + ¹

𝜕𝑥𝜕𝑦𝜕𝑧

𝜕 𝜌𝑉_𝑒𝑥𝑡

𝜕𝑡 = ^{𝜕 𝜙𝜌}

𝜕𝑡

• Use the Dirichlet condition for P

✓A location with 𝑉_𝑒𝑥𝑡 is assumed to have infinite volume

✓𝑉_𝑒𝑥𝑡 does not affect P, so it can be ignored

• ^𝜕

𝜕𝑥

𝑘𝜌 𝜇

𝜕𝑃

𝜕𝑥 = ^{𝜕 𝜙𝜌}

𝜕𝑡

• ^𝑘

𝜇

𝜕𝜌

𝜕𝑥

𝜕𝑃

𝜕𝑥 + 𝜌 ^𝜕²^𝑃

𝜕𝑥² = 𝜌 ^𝜕𝜙

𝜕𝑡 + 𝜙 ^𝜕ρ

[email protected]𝜕𝑡

SNU ERE Hoonyoung Jeong

(4)

Analytical Solution

If 𝑃 𝑥 = 0, 𝑡 = 𝑃₀ 𝑃 𝑥 = 𝐿, 𝑡 = 𝑃_𝐿 𝑃 𝑥, 𝑡 = 0 = 𝑃_𝐿

𝑃 𝑥, 𝑡 = 𝑃₀ + 𝑃_𝐿 − 𝑃₀ 𝑥

𝐿 + 2

𝜋 ෍

𝑛=1

∞ 1

𝑛 exp − 𝑛²𝜋² 𝐿²

𝑘

𝜙𝜇𝐶_𝑡 𝑡 sin 𝑛𝜋𝑥

[email protected]𝐿

SNU ERE Hoonyoung Jeong

(5)

Sensitivity Analysis

• If 𝑃₀ > 𝑃_𝐿

• Guess physically and mathematically

✓ What if k is larger?

✓ What if porosity is larger?

✓ What if Ct is larger?

✓ What if 𝜇 is larger?

𝑃 = 𝑃₀ − 𝑃₀ − 𝑃_𝐿 𝑥 𝐿

𝜕²𝑃

𝜕𝑥² = 𝜙𝜇𝐶_𝑡 𝑘

𝜕𝑃

𝜕𝑡