Errors in History Matching by - TU Delft
Errors in History Matching by Zohreh Tavassoli Jonathan Carter Peter King Department of Earth Science & Engineering Imperial College, London, UK
Introduction History Matching is used to find parameters in a reservoir model from production data The assumption is that a model which has a good history match gives i) A good estimate of model parameters ii) A good forecast of future performance In this study we show, for a simple model, that this may not be the case
Concept Prior Model P(m)
Data (d)
Likelihood Function P(d|m) Bayes Theorem
History Match (MAP)
Maximise
Posterior Model P(m|d)~P(d|m)P(m)
Reservoir Model • Poor Sand, kp (blue)
• Good Sand, kg (red)
h
• The model has 3 parameters (high and low permeability & fault throw) • Waterflood from left hand side.
Parameters Vary Randomly • 3 unknown parameters are: fault throw – h∈[0,60] high permeability – kg ∈ [100,200] low permeability – kp ∈[0,50] • We choose a point in this parameter space as a base case and run a flow simulation. We then carried out simulations for 159645 other realisations of the reservoir by choosing the parameters randomly from each range. • We define objective functions for the match of production data between the base case and each realisation.
Objective Function, ∆m weighted sum of squares of the difference between the production data of the base case for each realisation
( Qi − Qib ) 1 ∆m = ∑ 2 nh i =1 2σ ib nh
2
nh is the end of the history match (3 years)
Objective Function, ∆f Define ∆f to be a measure of match between the production data from start to the end of prediction
( Qi − Qib ) 1 ∆f = ∑ 2 nf − nh i =nh 2σ ib nf
2
where nf is the end of forecast (4 years)
Oil/ Water Production Rates for the Base Case (blue) and the Best Production Matched Model (red) for 4 years Parameters for Best Production Match
h = 33.1 ft kg = 135.9 md kp = 2.62 md Base Model parameters
h0 = 10.4 ft kg0 = 131.6 md kp0 = 1.3 md
Oil
Objective functions,
∆m = 0.118 ∆P=1066 Good fit to Production rates Bad fit for Parameters
Water
The vertical dotted line is the end of history matching (nh=3 years).
Oil/Water Production Rates for the Base Case (blue) and the Best Parameter Matched Model (red) for 4 years Best Parameters Matched to Base Model
h = 11.15 ft kg = 134.8 md kp = 1.35 md Base Model Parameters
h0 = 10.4 ft kg0 = 131.7 md kp0 = 1.31 md
Oil
Objective function,
∆m = 68.18 ∆p= 1.23 Good fit to parameters Bad fit to Production rates
Water
The vertical dotted line is the end of history matching (nh=3 years).
Water Saturation Maps at the end of History Match Base Case
Best History matched model
Best Model Best Parameter matched model
Changing the Parameters Systematically ∆m and ∆f are functions of Parameters of the Assumed Model, h,kg,kp and of the Base Case, h0,kg0,kp0, Fix kg=131.6 md, kp=1.3 md, vary h between 0 and 60 ft.
Q (h,t) −Q (h ,t)2 Q (h,t) −Q (h ,t)2 w wbase 0 o obase 0 ∆(h| h0 ) = ∑ + ( ) ( ) aQ h , t aQ h , t t w o 0 0 base base ∆(h=h0) = 0
It is frequently assumed that there is a single, simple minimum of ∆m = 0 at the True Model, h=h0.
Cross Sections of Objective Function ∆m versus h for three base cases h0=7.3, h0=10.4 and h0=30
The surfaces are complicated and DON’T have a single simple minimum. For each Base Case there are many local minima at different values of h and a global minimum for which ∆m(h=h0) = 0.
Blue: ∆m = 0 Green: 0 < ∆m
Introduction History Matching is used to find parameters in a reservoir model from production data The assumption is that a model which has a good history match gives i) A good estimate of model parameters ii) A good forecast of future performance In this study we show, for a simple model, that this may not be the case
Concept Prior Model P(m)
Data (d)
Likelihood Function P(d|m) Bayes Theorem
History Match (MAP)
Maximise
Posterior Model P(m|d)~P(d|m)P(m)
Reservoir Model • Poor Sand, kp (blue)
• Good Sand, kg (red)
h
• The model has 3 parameters (high and low permeability & fault throw) • Waterflood from left hand side.
Parameters Vary Randomly • 3 unknown parameters are: fault throw – h∈[0,60] high permeability – kg ∈ [100,200] low permeability – kp ∈[0,50] • We choose a point in this parameter space as a base case and run a flow simulation. We then carried out simulations for 159645 other realisations of the reservoir by choosing the parameters randomly from each range. • We define objective functions for the match of production data between the base case and each realisation.
Objective Function, ∆m weighted sum of squares of the difference between the production data of the base case for each realisation
( Qi − Qib ) 1 ∆m = ∑ 2 nh i =1 2σ ib nh
2
nh is the end of the history match (3 years)
Objective Function, ∆f Define ∆f to be a measure of match between the production data from start to the end of prediction
( Qi − Qib ) 1 ∆f = ∑ 2 nf − nh i =nh 2σ ib nf
2
where nf is the end of forecast (4 years)
Oil/ Water Production Rates for the Base Case (blue) and the Best Production Matched Model (red) for 4 years Parameters for Best Production Match
h = 33.1 ft kg = 135.9 md kp = 2.62 md Base Model parameters
h0 = 10.4 ft kg0 = 131.6 md kp0 = 1.3 md
Oil
Objective functions,
∆m = 0.118 ∆P=1066 Good fit to Production rates Bad fit for Parameters
Water
The vertical dotted line is the end of history matching (nh=3 years).
Oil/Water Production Rates for the Base Case (blue) and the Best Parameter Matched Model (red) for 4 years Best Parameters Matched to Base Model
h = 11.15 ft kg = 134.8 md kp = 1.35 md Base Model Parameters
h0 = 10.4 ft kg0 = 131.7 md kp0 = 1.31 md
Oil
Objective function,
∆m = 68.18 ∆p= 1.23 Good fit to parameters Bad fit to Production rates
Water
The vertical dotted line is the end of history matching (nh=3 years).
Water Saturation Maps at the end of History Match Base Case
Best History matched model
Best Model Best Parameter matched model
Changing the Parameters Systematically ∆m and ∆f are functions of Parameters of the Assumed Model, h,kg,kp and of the Base Case, h0,kg0,kp0, Fix kg=131.6 md, kp=1.3 md, vary h between 0 and 60 ft.
Q (h,t) −Q (h ,t)2 Q (h,t) −Q (h ,t)2 w wbase 0 o obase 0 ∆(h| h0 ) = ∑ + ( ) ( ) aQ h , t aQ h , t t w o 0 0 base base ∆(h=h0) = 0
It is frequently assumed that there is a single, simple minimum of ∆m = 0 at the True Model, h=h0.
Cross Sections of Objective Function ∆m versus h for three base cases h0=7.3, h0=10.4 and h0=30
The surfaces are complicated and DON’T have a single simple minimum. For each Base Case there are many local minima at different values of h and a global minimum for which ∆m(h=h0) = 0.
Blue: ∆m = 0 Green: 0 < ∆m
