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Prediction of Oil Production With Confidence Intervals*. James Glimm 1,2 , Shuling Hou 3 , Yoon-ha Lee 1 , David H. Sharp 3 , Kenny Ye 1 1. SUNY at Stony Brook 2. Brookhaven National Laboratory 3. Los Alamos National Laboratory.
James Glimm1,2, Shuling Hou3, Yoon-ha Lee1,
David H. Sharp3, Kenny Ye1
1. SUNY at Stony Brook
2. Brookhaven National Laboratory
3. Los Alamos National Laboratory
*Supported in part by the Department of Energy and National Science Foundation
Redefine probabilities and ensemble to be consistent with:
(b) probable errors in simulation and data
New ensemble of geologies = Posterior
Prediction = sample from posterior
Confidence intervals come from
- posterior probabilities
- errors in forward simulation
as the new independent variable
and t as the new dependent variable. Thus
and the error is equidistributed in
This makes the error robust.
We compare arrival time and
solution based error models
Predict outcomes and risk
Risk is predicted quantitatively
Risk prediction is based on
- formal probabilities of errors
in data and simulation
- methods for simulation error analysis
- Rapid simulation (upscale) allowing
exploration of many scenarios
Line drive, 2D reservoir
Random permeability field
log normal, random correlation length
in unit square
100 random permeability fields for each correlation length
lnK gaussian, correlation length
Solution from fine grid
100 x 100 grid
Solution by upscaling
20 x 20, 10 x 10, 5 x 5
Wallstrom, Hou, Christie, Durlofsky, Sharp
1. Computational Geoscience 3:69-87 (1999)
2. SPE 51939
3. Transport in Porous Media (submitted)
Scale-up: Black (fine grid) Red (20x20)
Blue (10x10) Green (5x5)
Select one geology as exact.
Observe production for
Assign revised probabilities to all
500 geologies in ensemble based on:
(a) coarse grid upscaled solutions
(b) probabilities for coarse grid errors.
Compared to data (from “exact” geology)
Permeability = geology
Observation = past oil cut
Fig. 1 Typical errors (lower, solid curves) and discrepancies
(upper, dashed curves), plotted vs. PVI. The two families of
curves are clearly distinguishable.
Gaussian error model: has covariance C, mean
Then, is an error, probability
Limited data on solution errors
Don’t over fit data
Replace by finite matrix
Prediction based on
(a) Geostatistics only, no history match (prior).
Average over full ensemble
(b) History match with upscaled solutions (posterior). Bayesian weighted average over ensemble.
(c) Window: select all fine grid solutions “close” to exact over past history.
Average over restricted ensemble.
Prediction error reduction, as
per cent of prior prediction
choose present time to be oil cut of 0.6
Window based error reduction: 50%
(fine grid: 100 x 100)
Upscaled error reduction:
5 x 5 23%
10 x 10 32%
20 x 20 36%
5% - 95% interval in future oil production
Excludes extreme high-low values with 5%
probability of occurrence
Expressed as a per cent of predicted
s0 = oil cut at present time.
t0 = present time.
Compute 5%--95% confidence intervals for future oil production, based on posterior and forward prediction using upscaled simulation.
Result is a random variable. We express confidence intervals as a percent of predicted production, and take mean of this statistic.
Confidence intervals in percent for three values of present oil cut s0 and three levels of scaleup with fine grid values included.
s0 100x100 20x20 10x10 5x5
0.8 [-13,22] [-21,36] [-24,35] [-27,34]
0.6 [-14,20] [-18,20] [-22,22] [-29,25]
0.4 [-14,17] [-18,18] [-24,21] [-33,23]
Error Model defined by 5 solution values:
s = 1- (Breakthrough), 0.8, 0.6, 0.4, 0.2.
Covariance is a 5 x 5 matrix, diagonally
dominant, and neglecting diagonal terms,
thus has 5 degrees of freedom. Thus it is simple.
Covariance is basically independent of the
geology correlation length. Thus it is robust.
elements of the correlation
matrices, 5x5, 10x10,
error model is proportional
to the degree of scale up
scaleup, showing general lack of dependence
on correlation length (except for s = 0.2 entry)
for arrival time error model are
independent of correlation length,
except for final (s = 0.2) entry.