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) History Match (MAP) Maximise Bayes Theorem Posterior Model P(m d)~p(d m)p(m)
Reservoir Model Poor Sand k p (blue) Good Sand k g (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 [6] high permeability k g [12] low permeability k p [5] 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 m = ( Q Q ) 1 nh i ib nh 2 i= 1 2σ ib 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 f = nf 1 nh nf i= nh ( Q Q ) i 2 σ ib 2 ib 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 k g = 135.9 md k p = 2.62 md Base Model parameters h = 1.4 ft k g = 131.6 md k p = 1.3 md Objective functions m =.118 P =166 Good fit to Production rates Bad fit for Parameters Oil 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 k g = 134.8 md k p = 1.35 md Base Model Parameters h = 1.4 ft k g = 131.7 md k p = 1.31 md Objective function m = 68.18 p = 1.23 Good fit to parameters Bad fit to Production rates Oil 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 hk g k p and of the Base Case h k g k p Fix k g =131.6 md k p =1.3 md vary h between and 6 ft. ( h h ) = t Q w ( h t) Q ( h t) aq w base w base ( h t) 2 ( ) Q ( h t) Qo h t + aqo base o base ( h t) 2 (h=h ) =
It is frequently assumed that there is a single simple minimum of m = at the True Model h=h.
Cross Sections of Objective Function m versus h for three base cases h =7.3 h =1.4 and h =3 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=h ) =.
Blue: m = Green: < m <1/nh Red: 1/nh < m <1/nh Cyan: 1 3 /nh < m <1 4 /nh Black: m > 1 4 /nh
Water Production Rate versus the Fault Throw h at different time steps For each h the production increases with time. The area of h within the parabolic shapes of production correspond to the green lines with approximately 45º to horizontal on the h-h graph. Regions with low water production correspond to the area with high values of m (black lines on the h-h graph).
Transmisibility in X Direction versus h Regions with low and high transmisibilities correspond to the area with low and high water production rates respectively. They are related to the cases when respectively either two good sand layers or a high and a low permeability layers in the Reservoir model face each other as the two structures slide against each other.
We fix h=h =37 ft and k g =k g =131.6 md Blue: m = Green: < m <1/nh Red: 1/nh < m <1/nh Cyan: 1 3 /nh < m <1 4 /nh Black: m > 1 4 /nh
Base Case and the Assumed model have k g =k g but k p k p ( ) ( ) ( ) ( ) ( ) ( ) ( ) + = t p o p o p o p w p w p w p p t k h aq t k h Q t Q hk t k h aq t k h Q t Q hk k h hk base base base base 2 2 m (h=h ) There is no guarantee that the global minimum is at h=h Base Cases and Assumed Models are not from the same data set
k p =1.3 k p =2.62 for h =7.3 (left) h =3 (right) Global minimum is at h=6.6 ft for the left graph and is at h=24.9 ft for the right graph.
k p =1.3 Base Case and k p =2.62 Assumed Model Blue: m = Green: < m <1/nh Red: 1/nh < m <1/nh Cyan: 1 3 /nh < m <1 4 /nh Black: m > 1 4 /nh
There are many reasonably good solutions whose statistics are robust Prior Model P(m) Data (d) Likelihood Function P(d m) History Match History Match History Match History Match Sample (MCMC) Bayes Theorem Posterior Model P(m d)~p(d m)p(m)
Forecasting f versus h for h =7.3 (left) h =1.4 (middle) and h =3 (right)
m (blue) and f (red) versus h for h =7.3 ft Little correspondence between the local minima of m and f
f m
f against m
Sensitivity to initial conditions t
Conclusions A model with the best history match DOES NOT necessarily give a good estimate of parameters or give a good forecast. The true model IS NOT necessarily the most likely to be obtained using the commonly search algorithms. The best history match can be very sensitive to errors in the model The statistics of good history matches are robust