Prediction of Oil Production With Confidence Intervals*

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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.

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### Prediction of Oil Production With Confidence Intervals*

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

Application of Prediction Theory
• Reservoir Development Choices, for example
• Sizing of Production Equipment
• Location of Infill Drilling
• During Early Development Stages
• Risk high
• Payoff high
Basic Idea: I
• History match with probability of error
• Observe production rates, etc.
Basic Idea: II
• Multiple simulations from ensemble
• (Re)Assign probabilities based on data, degree of mismatch of simulation to history
Basic Idea: III

Redefine probabilities and ensemble to be consistent with:

(a) data

(b) probable errors in simulation and data

Basic Idea: IV

New ensemble of geologies = Posterior

Prediction = sample from posterior

Confidence intervals come from

- posterior probabilities

- errors in forward simulation

New Idea: Arrival Time Error Models
• Formulate solution error model in terms of arrival times, rather than solution values
• errors are equi-distributed relative to solution gradients, ie relative to changes in solution values:
Arrival Time Model is Simple, Robust

Regard

as the new independent variable

and t as the new dependent variable. Thus

and the error is equidistributed in

units of

This makes the error robust.

We compare arrival time and

solution based error models

New Result:

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

Problem Formulation

Simulation study:

Line drive, 2D reservoir

Random permeability field

log normal, random correlation length

Simple Reservoir Description

in unit square

constant

Ensemble

100 random permeability fields for each correlation length

lnK gaussian, correlation length

Upscaling

Solution from fine grid

100 x 100 grid

Solution by upscaling

20 x 20, 10 x 10, 5 x 5

Upscaled grids

Upscaling References

Upscaling by

Wallstrom, Hou, Christie, Durlofsky, Sharp

1. Computational Geoscience 3:69-87 (1999)

2. SPE 51939

3. Transport in Porous Media (submitted)

Examples of Upscaled, Exact Oil Cut Curves

Scale-up: Black (fine grid) Red (20x20)

Blue (10x10) Green (5x5)

Design of Study

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)

Bayes Theorem

Permeability = geology

Observation = past oil cut

prior

posterior;

Errors and Discrepancies

Fine

Coarse

usually

but

implies

geology

geology

Example

Fig. 1 Typical errors (lower, solid curves) and discrepancies

(upper, dashed curves), plotted vs. PVI. The two families of

curves are clearly distinguishable.

Mean error

Sample covariance

Precision Matrix

Gaussian error model: has covariance C, mean

In Bayes Theorem, assume is exact.

Then, is an error, probability

For arrival time error models, the formulation is identical, except that the independent variables s and t now denote the solution values, and not the time values, while the error e(s) denotes an error in the time of arrival of the solution value s.
Model Reduction:

Limited data on solution errors

Don’t over fit data

Replace by finite matrix

Three Prediction Methods

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.

Comparing Prediction Methods
• Window prediction is best, but not practical
• -uses fine grid solutions for complete ensemble
• -tests for inherent uncertainty
• Prior prediction is worst
• - makes no use of production data.
Error Reduction

Prediction error reduction, as

per cent of prior prediction

choose present time to be oil cut of 0.6

Error Reduction

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%

Confidence Intervals

5% - 95% interval in future oil production

Excludes extreme high-low values with 5%

probability of occurrence

Expressed as a per cent of predicted

production

Confidence Intervals

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

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]

Arrival Time Error Analysis

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.

Histogram of off diagonal

elements of the correlation

matrices, 5x5, 10x10,

20x20 scaleup

Diagonal covariance matrix elements, three levels of scaleup, averaged over all correlation lengths
Error covariance for arrival time

error model is proportional

to the degree of scale up

Diagonal covariance matix elements for 10x10

scaleup, showing general lack of dependence

on correlation length (except for s = 0.2 entry)

Covariance matrix diagonal entries

for arrival time error model are

independent of correlation length,

except for final (s = 0.2) entry.

Arrival time error model vs. solution value model: confidence intervals (%) for s = 0.6 and 10x10 scaleup
Summary and Conclusions
• New method to assess risk in prediction of future oil production
• New methods to assess errors in simulations as probabilities
• New upscaling allows consideration of ensemble of geology scenarios
• Bayesian framework provides formal probabilities for risk and uncertainty
References
• J. Glimm, S. Hou, H. Kim, D. H. Sharp, “A Probability Model for Errors in the Numerical Solutions of a Partial Differential Equation”. Computational Fluid Dynamics Journal, Vol. 9, 485-493 (2001).
• J. Glimm, S. Hou, Y. Lee, D. H. Sharp, “Prediction of Oil Production with Confidence Intervals”, SPE reprint SPE66350 (2001).
• J. Glimm, S. Hou, H. Kim, D. H. Sharp, K. Ye, W. Zhu, “Risk Management for Petroleum Reservoir Production”, J. Comp. Geosciences, to appear.
• J. Glimm, Y. Lee, K. Ye, “A Simple Model for Scale Up Error” Cont. Math. 2002 (to appear).