PeGn624a Compositional Modeling - Application to Enhanced Oil Recovery  Spring 2009

PeGn624 Homework # 7

• Assigned: 2009-01-29

• Due: 2009-02-09

Please derive the effective horizontal stress equation Eq. 1.

σ′_{h} = \frac{ν}{1 − ν} (σ_{v} − α_{P} P − 3β_{T} K(T − T_{R}))  (1)

For both derivations, use the following assumptions, Eqs. 2–4.

σ_{h} = σ_{xx} = σ_{yy}  (2)
σ_{v} = σ_{zz}  (3)
ϵ_{xx} = ϵ_{yy} = 0  (4)

The following relationships may be useful, Eqs. 5–7.

G = \frac{E}{2(1 + ν)}  (5)
λ = \frac{νE}{(1 + ν)(1 − 2ν)}  (6)
K = \frac{E}{3(1 − 2ν)}  (7)

1. Start with Hooke’s Law, Eqs. 8–9.

ϵ_{xx} = \frac{1}{E} (σ_{x} − ν(σ_{y} + σ_{z}))  (8)
ϵ_{yy} = \frac{1}{E} (σ_{y} − ν(σ_{x} + σ_{z}))  (9)

2. Start with the general stress-strain relationship, Eq. 10.

σ_{ij} = 2Gϵ_{ij} + λϵ_{b}δ_{ij} + α_{P} P δ_{ij} + 3β_{T b}K_{b}(T − T_{R})δ_{ij}  (10)

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