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Hubbert’s Oil Peak Revisited by a Simulation Model
Pierre-Noël Giraud, Aline Sutter, Timothée Denis, Cédric Léonard
To cite this version:
Pierre-Noël Giraud, Aline Sutter, Timothée Denis, Cédric Léonard. Hubbert’s Oil Peak Revisited by a Simulation Model. 2010. �hal-00530077�
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CERNA WORKING PAPER SERIES
Hubbert’s Oil Peak Revisited by a Simulation Model
Pierre-Noël Giraud, Aline Sutter, Timothée Denis & Cédric Léonard
Working Paper 2010-17
Cerna, Centre d’économie industrielle
MINES ParisTech
60, boulevard Saint Michel
75272 Paris Cedex 06 – France
Tél. : 33 (1) 40 51 90 00
October 2010
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1 80 70 3 45 1 330 165 12 2 1
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N i j ˜nij(t) = Nn˜N˜ijsample(t)sample(t) ˜nij(t) i j t N ˜nijsample(t) i j t ˜ Nsample(t) t ˜ nijsample(t) ˜ Nsample(t) ˜ρi(t) = !NC j=1˜nij(t) N
˜ρj(t) = !NS i=1˜nij(t) N ˜ρi(t) i ˜ Nsample(t) ˜ρj(t) j ˜ Nsample(t) NC NS t∈ [1, ..., te] D(t) = NC " i=1 Pj(t) + PS(t) te D(t) t NC Pj(t) j t PS(t) t CS PS(t) > 0
j Vi τ Vi i i NS rPj(t − τ + 1) j t− τ + 1 j t t− τ + 2 t j t! t! ∈ [t − τ + 2, ..., t − 1] t j t rPj(t)+rτ Pj(t!) Pj(t) j t j τ − 1 Pj(t) = t " t!=t−τ+1 rPj(t!) τ
j RNj(t + 1) = RNj(t) + rDj(t) − rPj(t) RNj(t) j rDj(t) j t rPj(t) j t j t + 1 t t rDj(t) t rPj(t) ˜
Nsample(t + 1) = ˜Nsample(t) + ˆndmax(t)
˜ Nsample(t) t ˆndmax(t) ˆnd(t) t ndmax(t) t+1 N˜sample(t+1) t ˜Nsample(t) 1 t− 1 ˆndmax(t) t ndmax(t) nd(t) ∀t ∈ {1, ..., te} ∀j ∈ {1, ..., NC}
RNj(t) 0 nd(t) 0 rDj(t) 0 rPj(t) 0 Pj(t) 0 PS(t) 0 ∀t ∈ {1, ..., te} ˜ Nsample(t) ≤ N nd(t) ≤ ndmax(t) N ndmax(t) j j NC 1 rPj(t) = min # τ # D(t)− j−1 " k=1 rPk(t) τ − PF(t) $ , RNj(t) + rDj(t) $ Pf(t) = !Nk=1C %!tt−1!=t−τ+1 rPk(t!) τ & Pf(t) D(t)− Pf(t) t j 1 NC rPj(t) j j rPj(t) rPj(t) t rDj(t)
t
ˆnd(t)
ndmax(t)
t ˆndmax(t) = max (ˆnd(t), ndmax(t))
j i j NC i NS t ˆnd(t) D(t) nd(t) C˜T(t, nd(t)) ˆnd(t) ˆnd(t) = min nd(t) % ˜ CT (t, nd(t)) & ˜ CT (t, nd(t)) ˜ CT (t, nd(t)) = ˜CP(t, nd(t)) + Cend(t) Ce Cend(t) ˜ CP (t, n d(t)) ˜ CP(t, nd(t)) = NC " i=1 CjP˜j(t, nd(t)) + CSP˜S(t, nd(t))
˜ Pj(t, nd(t)) j nd(t) ˜ PS(t, nd(t)) nd(t) Cj j CS ˜ρi(t) ˜ρj(t) ˜ρi(t) i ˜ρj(t) j nd(t) j ˜rDj(t, nd(t)) = nd(t) ˜Vexpj(t) ˜ Vexpj(t) = ˜ρj(t) !Ni=1S ˜ρi(t)Vi rDj(t) ˜rDj(t, nd(t)) j NC ˜rPj(t, nd(t)) j nd(t) ˜ Pj(t, nd(t)) j NC ˜ PS(t, nd(t)) t C˜P (t, nd(t)) nd(t) %+ %+ ¯n1(t) nd(t) ∈ [0, ¯n1(t)] ˜ CP(t, nd(t)) NC nd(t) ∈ [¯n1(t), +∞[ ˜CP (t, nd(t)) NC = 5 ˜ CT (t, n d(t)) nd(t) %+ %+ NC nd(t) ∈ [¯n1(t), +∞[ C˜T (t, nd(t)) Ce ˜ CT (t, n d(t)) nd(t) ˆnd(t) t ˜ CP(t, nd(t)) Cend(t) Ce ˜ G(t) = ˜CT(t, 0)− ˜ CT(t, nd(t))
ˆnd(t) ˆnd(t) = maxnd(t) % ˜ G(t, nd(t)) & ˆndmax(t) t ˆndmax(t) =
max (ˆnd(t), ndmax(t)) ndmax(t)
• CT(t) ˜ CT(t) • ˜ρj(t) ˜ρi(t) x D(t) !x−1i=0D(t + i)
D(t)− PF(t) PF(t) k k∈ {1, 2} k∈ {1, 2} ˆndk(t) = min ndk(t) % ˜ CkT(t, ndk(t)) & 1 ˆndk(t) k∈ {1, 2} D(t)− PF(t) ˆnd1(t) ˆnd2(t) ¯nd1(t) = ˆnd1(t) ˜ G1(t, ˆnd1(t)) ˜ G1(t, ˆnd1(t)) + ˜G2(t, ˆnd2(t)) ¯nd2(t) = ˆnd2(t) ˜ G2(t, ˆnd2(t)) ˜ G1(t, ˆnd1(t)) + ˜G2(t, ˆnd2(t)) ¯ndk(t) k ˆndk(t) k ˜ Gk(t, ˆndk(t)) k ˜ρik(t) ˜ρjk(t) ˜ρik(t) i k ˜ρjk(t) j k
j
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• 35 45 55
• 25 35 45
• 15 25