PeGn624a Compositional Modeling - Application to Enhanced Oil Recovery  Spring 2009

PeGn624 Homework # 11

• Assigned: 2009-02-13

• Due: 2009-02-20

1 Question 1: Right-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 for T < T_{s}

Eq. 1 represents the right-hand-side of the water component equation. Expand the finite difference operators in Eq. 1 in terms of the primary variables for a 1D problem. Collect terms for δP_{o}, δS_{o}, δT, and δT_{s}.

RHSW = \frac{VR}{Δt}Δ_{t} (ϕ(S_{w}ξ_{w}X_{w1} + S_{g}ξ_{g}Y_{w1}))  (1)

1.2 Part (b): Oil Equation for T < T_{s}

Eq. 2 represents the right-hand-side of the oil component equation. Expand the finite difference operators in Eq. 2 in terms of the primary variables for a 1D problem. Collect terms for δP_{o}, δS_{o}, δT, and δT_{s}.

RHSO = \frac{VR}{Δt}Δ_{t} (ϕ(S_{o}ξ_{o}X_{o2}))  (2)

1.3 Part (c): Water Equation for T ≥ T_{s}

Eq. 1 represents the right-hand-side of the water component equation. Expand the finite difference operators in Eq. 1 in terms of the primary variables for a 1D problem. Collect terms for δP_{o}, δS_{o}, δT, and δS_{g}.

1.4 Part (d): Oil Equation for T ≥ T_{s}

Eq. 2 represents the right-hand-side of the oil component equation. Expand the finite difference operators in Eq. 1 in terms of the primary variables for a 1D problem. Collect terms for δP_{o}, δS_{o}, δT, and δS_{g}.

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2 Question 2: Alternate Right-Hand-Side Expansion

A = (ϕ(S_{w}ξ_{w}X_{w1} + S_{g}ξ_{g}Y_{w1}))  (1)

A^{ℓ+1} = A^{ℓ} + (\frac{∂A}{∂P_{o}} )^{ℓ}δP_{o} + (\frac{∂A}{∂T})^{ℓ}δT + (\frac{∂A}{∂S_{o}} )^{ℓ}δS_{o} + (\frac{∂A}{∂S_{g}} )^{ℓ}δS_{g}  (2)

2.1 Part (a): Water Equation for T ≥ T_{s}

Expand Eq. 1 using the following approach. Start with Eq. 1. Evaluate A^{ℓ+1} using Eq. 2. Expand the finite difference operators in Eq. 1 in terms of the primary variables for a 1D problem. Collect terms for δP_{o}, δS_{o}, δT, and δS_{g}. Prove that the result is the same as 1(c).

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