PeGn624a Compositional Modeling - Application to Enhanced Oil Recovery  Spring 2009

PeGn624 Homework # 13

• Assigned: Monday, February 23, 2009

• Due: Monday, March 2, 2009

1 Question 1: Left-Hand-Side Expansion

The primary variables are P_{o}, S_{o}, T. When T < T_{s}, S_{g} = 0 and T_{s} is used as the fourth primary variable. When T ≥ T_{s}, S_{g} is used as the fourth primary variable.

1.1 Part (a): Water Equation

Eq. 1.1 represents the left-hand-side of the water component equation. In homework #10, you expanded this in terms of P^{n+1}. Now complete this expansion in terms of δP_{o} for a 1D problem. Collect terms for δP_{o}.

LHSW = Δ (T_{w}^{n}ζ_{w}^{n}X_{w1}^{n}(ΔP_{w}^{n+1} − γ_{w}^{n}ΔD) + Δ (T_{g}^{n}ζ_{g}^{n}Y_{w1}^{n}(ΔP_{g}^{n+1} − γ_{g}^{n}ΔD) + q_{w}ζ_{w}X_{w1} + q_{g}ζ_{g}Y_{w1} (1.1)

1.2 Part (b): Oil Equation

Eq. 1.2 represents the left-hand-side of the oil component equation. In homework #10, you expanded this in terms of P^{n+1}. Now complete this expansion in terms of δP_{o} for a 1D problem. Collect terms for δP_{o}.

LHSO = Δ (T_{o}^{n}ζ_{o}^{n}X_{w1}^{n}(ΔP_{o}^{n+1} − γ_{o}^{n}ΔD) + q_{o}ζ_{o}X_{o2}  (1.2)

2 Question 2: Energy Equation Expansion

For the energy equation expansions, note that H(T) is a function of T only. U is a function of H(T), P, and ζ(P, T).

2.1 Part (a): Left-Hand-Side for T < T_{s}

Eq. 2.1 represents the left-hand-side of the energy equation. Complete this expansion in terms of the primary variables for a 1D problem. Collect terms for δP_{o}, δT, δS_{o}, and δT_{s}.

1 of 2 LHSE = Δ (T_{w}^{n}ζ_{w}^{n}X_{w1}^{n}H_{w}^{n}(ΔP_{w}^{n+1} − γ_{w}^{n}ΔD) + Δ (T_{g}^{n}ζ_{g}^{n}Y_{w1}^{n}H_{g}^{n}(ΔP_{g}^{n+1} − γ_{g}^{n}ΔD) + Δ (T_{o}^{n}ζ_{o}^{n}X_{w1}^{n}H_{o}^{n}(ΔP_{o}^{n+1} − γ_{o}^{n}ΔD) + Δ (T_{e}^{n}ΔT) + q_{w}ζ_{w}X_{w1}H_{w} + q_{g}ζ_{g}Y_{w1}H_{g} + q_{o}ζ_{o}X_{o2}H_{o} (2.1)

2.2 Part (b): Left-Hand-Side for T ≥ T_{s}

Eq. 2.1 represents the left-hand-side of the energy equation. Complete this expansion in terms of the primary variables for a 1D problem. Collect terms for δP_{o}, δT, δS_{o}, and δS_{g}.

2.3 Part (c): Right-Hand-Side for T < T_{s}

Eq. 2.2 represents the right-hand-side of the energy equation. Complete this expansion in terms of the primary variables for a 1D problem. Collect terms for δP_{o}, δT, δS_{o}, and δT_{s}.

RHSE = \frac{VR}{Δt}Δ_{t} (ϕ(S_{w}ξ_{w}X_{w1}U_{w} + S_{g}ξ_{g}Y_{w1}U_{g} + S_{o}ξ_{o}X_{o2}U_{o})) + \frac{VR}{Δt}Δ_{t} ((1 − ϕ)(ρ_{R}C_{R}T)) (2.2)

2.4 Part (d): Right-Hand-Side for T ≥ T_{s}

Eq. 2.2 represents the right-hand-side of the energy equation. Complete this expansion in terms of the primary variables for a 1D problem. Collect terms for δP_{o}, δT, δS_{o}, and δS_{g}.

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