PeGn624a  Compositional Modeling - Application to Enhanced Oil Recovery  Spring 2009

PeGn624 Homework # 18

• Assigned: Wednesday, March 25, 2009

• Due: Thursday, April 2, 2009 in class

1 Numerical Values

Calculate the numerical values of the matrix equations at the initial conditions for the project.

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 of 5

1.1 Part (a): Initial Form, T < T_{s}

Calculate the numerical values for A_{11}, A_{12}, R_{11}, R_{12}, R_{13}, R_{14}, and R_{15} defined in Eq. 1.1. Use the initial conditions of P_{o} = 500psia and T_{R} = 100^{∘}F.

\overbrace{\begin{matrix}W : & X & 0 & 0 & 0 & X & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ O : & X & 0 & 0 & 0 & X & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ E : & X & X & 0 & 0 & X & X & 0 & 0 & 0 & 0 & 0 & 0 \\ G : & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ W : & X & 0 & 0 & 0 & X & 0 & 0 & 0 & X & 0 & 0 & 0 \\ O : & X & 0 & 0 & 0 & X & 0 & 0 & 0 & X & 0 & 0 & 0 \\ E : & X & X & 0 & 0 & X & X & 0 & 0 & X & X & 0 & 0 \\ G : & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ W : & 0 & 0 & 0 & 0 & X & 0 & 0 & 0 & X & 0 & 0 & 0 \\ O : & 0 & 0 & 0 & 0 & X & 0 & 0 & 0 & X & 0 & 0 & 0 \\ E : & 0 & 0 & 0 & 0 & X & X & 0 & 0 & X & X & 0 & 0 \\ G : & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}}^{A_{11}=LHS} × \overbrace{\begin{matrix}δP_{o} \\ δT \\ δS_{o} \\ δT_{s} \\ δP_{o} \\ δT \\ δS_{o} \\ δT_{s} \\ δP_{o} \\ δT \\ δS_{o} \\ δT_{s} \end{matrix}}^{δ} − \overbrace{\begin{matrix}W : & X & X & X & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ O : & X & X & X & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ E : & X & X & X & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ G : & X & 0 & X & X & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ W : & 0 & 0 & 0 & 0 & X & X & X & 0 & 0 & 0 & 0 & 0 \\ O : & 0 & 0 & 0 & 0 & X & X & X & 0 & 0 & 0 & 0 & 0 \\ E : & 0 & 0 & 0 & 0 & X & X & X & 0 & 0 & 0 & 0 & 0 \\ G : & 0 & 0 & 0 & 0 & X & 0 & X & X & 0 & 0 & 0 & 0 \\ W : & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & X & X & X & 0 \\ O : & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & X & X & X & 0 \\ E : & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & X & X & X & 0 \\ G : & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & X & 0 & X & X \end{matrix}}^{A_{12}=RHS} × \overbrace{\begin{matrix}δP_{o} \\ δT \\ δS_{o} \\ δT_{s} \\ δP_{o} \\ δT \\ δS_{o} \\ δT_{s} \\ δP_{o} \\ δT \\ δS_{o} \\ δT_{s} \end{matrix}}^{δ} =

\overbrace{\begin{matrix}X \\ X \\ X \\ 0 \\ X \\ X \\ X \\ 0 \\ X \\ X \\ X \\ 0 \end{matrix}}^{R_{11}=LHS_{Pc}} + \overbrace{\begin{matrix}X \\ X \\ X \\ 0 \\ X \\ X \\ X \\ 0 \\ X \\ X \\ X \\ 0 \end{matrix}}^{R_{12}=LHS_{D}} + \overbrace{\begin{matrix}X \\ X \\ X \\ 0 \\ X \\ X \\ X \\ 0 \\ X \\ X \\ X \\ 0 \end{matrix}}^{R_{13}=LHS_{other}} + \overbrace{\begin{matrix}X \\ X \\ X \\ 0 \\ X \\ X \\ X \\ 0 \\ X \\ X \\ X \\ 0 \end{matrix}}^{R_{14}=RHS^{n}} + \overbrace{\begin{matrix}X \\ X \\ X \\ X \\ X \\ X \\ X \\ X \\ X \\ X \\ X \\ X \end{matrix}}^{R_{15}=RHS^{ℓ}} (1.1)

2 of 5

1.2 Part (b): Revised Form, T < T_{s}

Eliminate zeros in A_{12} and adjust all terms in the appropriate rows. Calculate the numerical values for A_{31}, and R_{31} defined in Eq. 1.2. The X^{⋆} represents adjusted values based on eliminating zeros. The 0^{⋆} represents the values which have been eliminated. Use the initial conditions of P_{o} = 500psia and T_{R} = 100^{∘}F.

\overbrace{\begin{matrix}W : & X^{⋆} & X^{⋆} & 0^{⋆} & 0 & X^{⋆} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ O : & X^{⋆} & X^{⋆} & 0^{⋆} & 0 & X^{⋆} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ E : & X & X & X & 0 & X & X & 0 & 0 & 0 & 0 & 0 & 0 \\ G : & X & 0 & X & X & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ W : & X^{⋆} & 0 & 0 & 0 & X^{⋆} & X^{⋆} & 0^{⋆} & 0 & X^{⋆} & 0 & 0 & 0 \\ O : & X^{⋆} & 0 & 0 & 0 & X^{⋆} & X^{⋆} & 0^{⋆} & 0 & X^{⋆} & 0 & 0 & 0 \\ E : & X & X & 0 & 0 & X & X & X & 0 & X & X & 0 & 0 \\ G : & 0 & 0 & 0 & 0 & X & 0 & X & X & 0 & 0 & 0 & 0 \\ W : & 0 & 0 & 0 & 0 & X^{⋆} & 0 & 0 & 0 & X^{⋆} & X^{⋆} & 0^{⋆} & 0 \\ O : & 0 & 0 & 0 & 0 & X^{⋆} & 0 & 0 & 0 & X^{⋆} & X^{⋆} & 0^{⋆} & 0 \\ E : & 0 & 0 & 0 & 0 & X & X & 0 & 0 & X & X & X & 0 \\ G : & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & X & 0 & X & X \end{matrix}}^{A_{31}} × \overbrace{\begin{matrix}δP_{o} \\ δT \\ δS_{o} \\ δT_{s} \\ δP_{o} \\ δT \\ δS_{o} \\ δT_{s} \\ δP_{o} \\ δT \\ δS_{o} \\ δT_{s} \end{matrix}}^{δ} = \overbrace{\begin{matrix}X^{⋆} \\ X^{⋆} \\ X \\ X \\ X^{⋆} \\ X^{⋆} \\ X \\ X \\ X^{⋆} \\ X^{⋆} \\ X \\ X \end{matrix}}^{R_{31}}  (1.2)

1.3 Part (c): Extract Temperature and Pressure, T < T_{s}

Extract the pressure and temperatures from A_{41}. Calculate the numerical values for A_{41}, and R_{41} defined in Eq. 1.3. Use the initial conditions of P_{o} = 500psia and T_{R} = 100^{∘}F.

\overbrace{\begin{matrix}X & X & X & 0 & 0 & 0 \\ X & X & X & 0 & 0 & 0 \\ X & 0 & X & X & X & 0 \\ X & 0 & X & X & X & 0 \\ 0 & 0 & X & 0 & X & X \\ 0 & 0 & X & 0 & X & X \end{matrix}}^{A_{41}} × \overbrace{\begin{matrix}δP_{o} \\ δT \\ δP_{o} \\ δT \\ δP_{o} \\ δT \end{matrix}}^{δ} = \overbrace{\begin{matrix}X \\ X \\ X \\ X \\ X \\ X \end{matrix}}^{R_{41}}  (1.3)

3 of 5

1.4 Part (d): Initial Form, T ≥ T_{s}

Calculate the numerical values for A_{11}, A_{12}, R_{11}, R_{12}, R_{13}, R_{14}, and R_{15} defined in Eq. 1.4. Use the conditions of P_{o} = 500psia and T = 500^{∘}F.

\overbrace{\begin{matrix}W : & X & 0 & 0 & 0 & X & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ O : & X & 0 & 0 & 0 & X & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ E : & X & X & 0 & 0 & X & X & 0 & 0 & 0 & 0 & 0 & 0 \\ G : & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ W : & X & 0 & 0 & 0 & X & 0 & 0 & 0 & X & 0 & 0 & 0 \\ O : & X & 0 & 0 & 0 & X & 0 & 0 & 0 & X & 0 & 0 & 0 \\ E : & X & X & 0 & 0 & X & X & 0 & 0 & X & X & 0 & 0 \\ G : & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ W : & 0 & 0 & 0 & 0 & X & 0 & 0 & 0 & X & 0 & 0 & 0 \\ O : & 0 & 0 & 0 & 0 & X & 0 & 0 & 0 & X & 0 & 0 & 0 \\ E : & 0 & 0 & 0 & 0 & X & X & 0 & 0 & X & X & 0 & 0 \\ G : & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}}^{A_{11}=LHS} × \overbrace{\begin{matrix}δP_{o} \\ δT \\ δS_{o} \\ δS_{g} \\ δP_{o} \\ δT \\ δS_{o} \\ δS_{g} \\ δP_{o} \\ δT \\ δS_{o} \\ δS_{g} \end{matrix}}^{δ} − \overbrace{\begin{matrix}W : & X & X & X & X & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ O : & X & X & X & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ E : & X & X & X & X & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ G : & X & X & X & X & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ W : & 0 & 0 & 0 & 0 & X & X & X & X & 0 & 0 & 0 & 0 \\ O : & 0 & 0 & 0 & 0 & X & X & X & 0 & 0 & 0 & 0 & 0 \\ E : & 0 & 0 & 0 & 0 & X & X & X & X & 0 & 0 & 0 & 0 \\ G : & 0 & 0 & 0 & 0 & X & X & X & X & 0 & 0 & 0 & 0 \\ W : & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & X & X & X & X \\ O : & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & X & X & X & 0 \\ E : & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & X & X & X & X \\ G : & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & X & X & X & X \end{matrix}}^{A_{12}=RHS} × \overbrace{\begin{matrix}δP_{o} \\ δT \\ δS_{o} \\ δS_{g} \\ δP_{o} \\ δT \\ δS_{o} \\ δS_{g} \\ δP_{o} \\ δT \\ δS_{o} \\ δS_{g} \end{matrix}}^{δ} =

\overbrace{\begin{matrix}X \\ X \\ X \\ 0 \\ X \\ X \\ X \\ 0 \\ X \\ X \\ X \\ 0 \end{matrix}}^{R_{11}=LHS_{Pc}} + \overbrace{\begin{matrix}X \\ X \\ X \\ 0 \\ X \\ X \\ X \\ 0 \\ X \\ X \\ X \\ 0 \end{matrix}}^{R_{12}=LHS_{D}} + \overbrace{\begin{matrix}X \\ X \\ X \\ 0 \\ X \\ X \\ X \\ 0 \\ X \\ X \\ X \\ 0 \end{matrix}}^{R_{13}=LHS_{other}} + \overbrace{\begin{matrix}X \\ X \\ X \\ 0 \\ X \\ X \\ X \\ 0 \\ X \\ X \\ X \\ 0 \end{matrix}}^{R_{14}=RHS^{n}} + \overbrace{\begin{matrix}X \\ X \\ X \\ X \\ X \\ X \\ X \\ X \\ X \\ X \\ X \\ X \end{matrix}}^{R_{15}=RHS^{ℓ}} (1.4)

4 of 5

1.5 Part (e): Revised Form, T ≥ T_{s}

Eliminate zeros in A_{12} and adjust all terms in the appropriate rows. Calculate the numerical values for A_{31}, and R_{31} defined in Eq. 1.5. The X^{⋆} represents adjusted values based on eliminating zeros. The 0^{⋆} represents the values which have been eliminated. Use the conditions of P_{o} = 500psia and T = 500^{∘}F.

\overbrace{\begin{matrix}W : & X^{⋆} & X^{⋆} & 0^{⋆} & 0^{⋆} & X^{⋆} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ O : & X^{⋆} & X^{⋆} & 0^{⋆} & 0 & X^{⋆} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ E : & X^{⋆} & X^{⋆} & X^{⋆} & 0^{⋆} & X^{⋆} & X^{⋆} & 0 & 0 & 0 & 0 & 0 & 0 \\ G : & X & X & X & X & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ W : & X^{⋆} & 0 & 0 & 0 & X^{⋆} & X^{⋆} & 0^{⋆} & 0^{⋆} & X^{⋆} & 0 & 0 & 0 \\ O : & X^{⋆} & 0 & 0 & 0 & X^{⋆} & X^{⋆} & 0^{⋆} & 0 & X^{⋆} & 0 & 0 & 0 \\ E : & X^{⋆} & X^{⋆} & 0 & 0 & X^{⋆} & X^{⋆} & X^{⋆} & 0^{⋆} & X^{⋆} & X^{⋆} & 0 & 0 \\ G : & 0 & 0 & 0 & 0 & X & X & X & X & 0 & 0 & 0 & 0 \\ W : & 0 & 0 & 0 & 0 & X^{⋆} & 0 & 0 & 0 & X^{⋆} & X^{⋆} & 0^{⋆} & 0^{⋆} \\ O : & 0 & 0 & 0 & 0 & X^{⋆} & 0 & 0 & 0 & X^{⋆} & X^{⋆} & 0^{⋆} & 0 \\ E : & 0 & 0 & 0 & 0 & X^{⋆} & X^{⋆} & 0 & 0 & X^{⋆} & X^{⋆} & X^{⋆} & 0^{⋆} \\ G : & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & X & X & X & X \end{matrix}}^{A_{31}} × \overbrace{\begin{matrix}δP_{o} \\ δT \\ δS_{o} \\ δS_{g} \\ δP_{o} \\ δT \\ δS_{o} \\ δS_{g} \\ δP_{o} \\ δT \\ δS_{o} \\ δS_{g} \end{matrix}}^{δ} = \overbrace{\begin{matrix}X^{⋆} \\ X^{⋆} \\ X^{⋆} \\ X \\ X^{⋆} \\ X^{⋆} \\ X^{⋆} \\ X \\ X^{⋆} \\ X^{⋆} \\ X^{⋆} \\ X \end{matrix}}^{R_{31}}  (1.5)

1.6 Part (f): Extract Temperature and Pressure, T ≥ T_{s}

Extract the pressure and temperatures from A_{41}. Calculate the numerical values for A_{41}, and R_{41} defined in Eq. 1.3. Use the conditions of P_{o} = 500psia and T = 500^{∘}F.

5 of 5