Reservoir characterization by integration of production and time lapse seismic data – An

application to the Norne field

Amit Suman1, Juan Luis Fernández Martínez1,2, Tapan Mukerji1 and Esperanza García-Gonzalo2

(1) Department of Energy Resources Engineering Stanford University

(2) Department of Mathematics.

University of Oviedo, Spain.

Abstract

Time lapse seismic data has evolved as an important diagnostic tool in efficient reservoir characterization and monitoring. In combination with geological and flow modeling as a part of history matching process it can provide better description of the reservoir and thus better reservoir forecasting. However joint inversion of seismic and flow data for reservoir parameter is highly non-linear and complex. Stochastic optimization based inversion has shown very good results in integration of time-lapse seismic and production data in reservoir history matching. In this paper we have used a family of particle swarm optimizers for inversion of semi-synthetic Norne field data set. We analyze the performance of the different PSO optimizers, both in terms of exploration and convergence rate. Finally we also show some promising and preliminary results of the application of differential evolution.

1. Introduction Time lapse seismic data has begun to play an important role in reservoir characterization, management and monitoring. It can provide information on the dynamics of fluids in the reservoir based on the relation between variations of seismic signals and movement of hydrocarbons and changes in formation pressure. Movement of fluids and changes in pore pressure depends on the petrophysical properties of the reservoir rock. Thus reservoir monitoring by repeated seismic or time lapse surveys can help in reducing the uncertainties attached to reservoir models. Reservoir models, optimally constrained to seismic response as well as flow response can provide a better description of the reservoir and thus more reliable forecast. Huang et al., (1997, 1998) formulated the simultaneous matching of production and seismic data as an optimization problem, with updating of model parameters such as porosity. Walker and Lane (2007) presented a case study that included time-lapse seismic data as a part of the production history matching process, and show how the use of seismic monitoring can improve

reservoir prediction. It is difficult to obtain a global optimum match of production as well as seismic data using conventional gradient based optimization methods (Sen and Stoffa, 1996). Stochastic optimization based inversion methods have shown advantages in integration of production and time-lapse seismic data in reservoir history matching (Jin et al, 2007, 2008). Particle swarm optimization has been used in a variety of optimization and inverse problems in different branches of engineering and technology (Poli, 2008b), but its use in geosciences still remains restrained (Shaw and Srivastava, 2007; Fernández-Martínez et al., 2008b; Naudet et al., 2008; Yuan et al., 2009; Fernández-Martínez et al., 2009). Recently it has been used to optimize the well types and locations in a reservoir (Onwunalu and Durlofsky, 2009). Fernández Martínez et al (2010) has used particle swarm optimizers (GPSO, CC-PSO and CP-PSO) to invert production data and time lapse tomographic data for the synthetic Stanford VI reservoir, In this paper we use the PSO family and two other novel variants RR-PSO and PP-PSO (García Gonzalo and Fernández Martínez, 2010) as global optimizers for integration of production and time-lapse seismic data in history matching for a data set in the Norne field. We study their convergence rate and compare their behavior in terms of their uncertainty estimations.

2. Norne field and Synthetic Norne The Norne field is located in the blocks 6608/10 and 6508/10 on a horst block in the southern part of the Nordland II area in the Norwegian Sea. It has 29 producer and 10 injector wells. The rocks within the Norne reservoir are of Late Triassic to Middle Jurassic age. The present geological model consists of five reservoir zones. They are Garn, Not, Ile, Tofte and Tilje. Oil is mainly found in the Ile and Tofte Formations, and gas in the Garn formation (Figure 3). The sandstones are buried at a depth of 2500-2700 m. The porosity is in the range of 25-30 % while permeability varies from 20 to 2500 mD (Steffensen and Karstad, 1995; Osdal et al., 2006). In this study we apply the method on a 2-D section of synthetic Norne field. A 2D section of synthetic model is used which is a cross section of a real field in offshore Norway (Norne field) (Figure 1 and 2). The model has 1014 grid block (39 x 1 x 26), with one injector and one producer at column 6 and 36 respectively. The properties of these wells are similar to the well F-4H in the real Norne field. The grid sizes vary from 2-20 m in the vertical direction and from 80-100 m in the horizontal direction. We generate one thousand numbers of realizations of porosity field to assess the uncertainties attached with reservoir characterization. Realizations are generated using SGEMS and conditioned to the well data and variogram. One thousand number of permeability fields are generated based on the relationship between porosity and permeability in different zones. Horizontal permeability is equal to the vertical permeability.

Figure 1: Permeability distribution of 2D synthetic section of Norne field

Figure 2: Porosity distribution of 2D synthetic section of Norne field

Figure 3: Porosity distribution in Real Norne field reservoir model

3. Particle Swarm Optimization

Particle swarm optimization is a stochastic evolutionary computation technique used in optimization, which is inspired by the social behaviour of individuals (called particles) in nature, such as bird flocking and fish schooling (Kennedy and Eberhart, 1995)

Let us consider an inverse problem of the form ( ) =F m d , where Rn∈ ⊂m M are the model

parameters, R s∈d the discrete observed data, and

1 2( ) ( ( ), ( ), , ( ))s

f f f=F m m m mK

is the vector field representing the forward operator and ( )j

f m is the scalar field that accounts

for the j-th data. The "classical" goal of inversion given a particular data set (often affected by

noise), is to find a set a unique set of parameters m, such the data prediction error ( )p

−F m d in

a certain norm p, is minimized. The PSO algorithm to approach this inverse problem is at first glance very easy to understand and implement: 1. A prismatic space of admissible models, M, is defined:

, 1 , 1j ij j size

l m u j n i N≤ ≤ ≤ ≤ ≤ ≤

where ,j j

l u are the lower and upper limits for the j-th coordinate of each particle in the swarm, n

is the number of parameters in the optimization problem and size

N is the swarm size.

2. The misfit for each particle of the swarm is calculated, ( )p

−F m d and the we determine for

each particle its local best position found so far (called ( )i

kl ) and the minimum of all of them is

called the global best ( ( )kg ).

3. At each iteration the algorithm updates the positions ( )i

kx , and velocities ( )i

kv , of each

model in the swarm. The velocity of each particle, i, at each iteration, k, is a function of three major components:

i) The inertia term, which consists of the old velocity of the particle, ( )i

kv weighted by a real

constant, ω , called inertia. ii) The social learning term, which is the difference between the global best position found so far

(called ( )kg ) and the particle's current position ( ( )i

kx ).

iii) The cognitive learning term, which is the difference between the particle's best position

(called ( )i

kl ) and the particle's current position ( ( )i

kx ):

1 1( 1) ( ) ( ( ) ( )) ( ( ) ( ))

( 1) ( ) ( 1)i i i i i

i i i

k k k k k k

k k k

ω φ φ+ = + − + −

+ = + +

v v g x l x

x x v

, ,g l

a aω are the PSO parameters: inertia and local and global acceleration constants;

1 1 2 2,g l

r a r aφ φ= = are the stochastic global and local accelerations, and 1 2,r r are vectors of

random numbers uniformly distributed in ( )0, 1 to weight the global and local acceleration

constants. In the classical PSO algorithm these parameters are the same for all the particles of the swarm. In an inverse problem the position are the coordinates of the model m on the search space and the velocities the perturbations needed to find the low misfit models. The PSO algorithm can be physically interpreted as a particular discretization of a stochastic damped mass-spring system (Fernández Martínez et al., 2008; Fernández Martínez and García Gonzalo, 2008):

1 2 1 0 2 0''( ) (1 ) '( ) ( ) ( ) ( ) ( ).i i i i

t t t t t t tω φ φ φ φ+ − + + = − + −x x x g l (1)

This model has been addressed as the PSO continuous model since it describes (together with the

initial conditions) the continuous movement of any particle coordinate in the swarm ( )i

tx , where

i stands for the particle index, and ( )tg and ( )i

tl are its local and global attractors. In model (1)

the trajectories are allowed to be delayed a time 0t with respect to the trajectories. Using this

physical analogy we were able to analyze the PSO particle's trajectories (Fernández-Martínez et al., 2008) and to explain the success in achieving convergence of some popular parameters sets found in the literature (Carlisle and Dozier, 2001 , Clerc and Kennedy, 2002), Trelea, 2003). A whole family of PSO algorithms (Fernández-Martínez and García-Gonzalo, 2009, García-Gonzalo and Fernández-Martínez, 2009) have been derived considering different differencing

schemes for ''( )i

tx and '( )i

tx :

1. GPSO or centered-regressive PSO ( 0 0t = ).

( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )( )

( ) ( ) ( )

1 1 ,

.

v t t t v t t g t x t t l t x t

x t t x t v t t t

ω φ φ+ ∆ = − − ∆ + ∆ − + ∆ −

+ ∆ = + + ∆ ∆

₁ ₂

The GPSO algorithm is the generalization of the PSO algorithm for any time step ∆t , (PSO is the particular case for 1t∆ = ). These expressions for the velocity and position are obtained by employing a regressive scheme in velocity and a centred scheme in acceleration.

2. CC-PSO or centred - centred PSO ( 0 0t = ).

( ) ( ) ( ) ( )( ) ( ) ( )( )

( )1

0 2

2 ( 1)( ) ,

2 2 2

( ( ) ( ))2 ( 1)( )

( ( ) ( ))2 (1 ) 2 (1 ) k

t t tx t t x t v t g t x t l t x t t

l t k t x t k tt tv t t v t

g t k t x t k tt t

ωφ φ

φω

φω ω =

+ − ∆ ∆ ∆ + ∆ = + + − + − ∆

+ ∆ − + ∆ + − ∆ ∆+ ∆ = +

+ + ∆ − + ∆+ − ∆ + − ∆ ∑

₁ ₂

₁

3. CP-PSO or centred -progressive PSO ( 0t t= ∆ )

( )( ) ( ) ( ) ( )( ) ( ) ( )( )

( ) ( ) ( )

21,

1 (1 )

.

t v t t g t x t t l t x tv t t

w t

x t t x t v t t

φ φ φ− ∆ + ∆ − + ∆ −+ ∆ =

+ − ∆

+ ∆ = + ∆

₁ ₂

4. PP-PSO or progressive-progressive PSO ( 0 0t = )

( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )( )

( ) ( ) ( )

1 1 ,

.

v t t t v t t g t x t t l t x t

x t t x t v t t

ω φ φ+ ∆ = − − ∆ + ∆ − + ∆ −

+ ∆ = + ∆

₁ ₂

5. RR-PSO or regressive-regressive PSO ( 0t t= ∆ )

( )( ) ( ) ( )( ) ( ) ( )( )

( ) ( ) ( )

2,

1 (1 )

.

v t t g t x t t l t x tv t t

w t t

x t t x t v t t t

φ φ

φ

+ ∆ − + ∆ −+ ∆ =

+ − ∆ + ∆

+ ∆ = + + ∆ ∆

₁ ₂

These versions have different first and second order stability regions. Figure 4 shows for each family member the stability regions and the contour plots of the misfit error (in logarithmic scale) after a certain number of iterations (500) for the Rosenbrock function that has a valley shape.

Figure 4: Logarithmic median misfit errors for the Rosenbrock function in 50 simulations (after

500 iterations) for different family members.

This numerical analysis is done for a lattice of ( , )ω φ points located in the corresponding first

order stability regions over 50 different simulations. For GPSO, CC-PSO and CP-PSO better

parameter sets , ,g l

a aω are located on the first order complex region, close to the upper border

of the second order stability region where the attraction from the particle oscillation center is lost, i.e. the variance becomes unbounded; and around the intersection to the median lines of the first stability regions where the temporal covariance between trajectories is close to zero (Fernández

Martínez and García Gonzalo, 2009). The PP-PSO does not converge for 0ω < , and the good

parameter sets are in the complex region close to the limit of second order stability and to 0φ = .

The good parameters sets for the RR-PSO are concentrated around the line

_

14 / 3( 1)φ ω= − , mainly for inertia values greater than two. This line is located in a zone of medium attenuation and high frequency of trajectories (García Gonzalo and Fernández Martínez, 2009). These results are used to optimize using the cloud versions (García Gonzalo and Fernández Martínez, 2009; Fernández Martínez and García Gonzalo, 2010) where each particle in the swarm has different inertia (damping) and acceleration (rigidity) constants, in contrast to the conventional algorithm where every particle has the same inertia and acceleration parameters. The results obtained for very hard benchmark functions in several dimensions using the PSO-cloud algorithm are comparable to the reference misfits published in the literature. This design avoids two main drawbacks of the PSO algorithm: the tuning of the PSO parameters and the clamping of the velocities. Finally, in this paper we are applying RR-PSO, CP-PSO, CC-PSO, PP-PSO and GPSO methods for joint inversion of production and time-lapse seismic data.

4. Differential Evolution

Differential Evolution is a stochastic evolutionary computation technique (Storn and Price, 1997) consisting in the replacement of a current population of individuals by a better fit new population using 3 different mechanisms:

1. Mutation : for each particle of the population, i, a mutation vector ( 1)

ik +m is generated

( ) ( )1 2( 1) ( ) ( ) ( ) ( ) ,

( 1) ( ) ( 1),i l n r s

i j i

k F k k F k k

k k k

+ = − + −

+ = + +

v x x x x

m x v

where l ,n, r, s are particles indexes randomly chosen on the population. The parameters 1F , 2F

are called mutation scale factors and are usually chosen on [ ]0,2 .

There are several mutation strategies:

• Rand-1: ( )1( 1) ( ) ( ) ( ) ,i j l nk k F k k+ = + −m x x x

• Best-1: ( )1( 1) ( ) ( ) ( ) ,i l nk k F k k+ = + −m g x x

• Target to the best: ( ) ( )1 1( 1) ( ) ( ) ( ) ( ) ( ) ,i i n l nk k F k k F k k+ = + − + −m x g x x x

• Rand-2: ( ) ( )1 2( 1) ( ) ( ) ( ) ( ) ( ) ,i j l n r sk k F k k F k k+ = + − + −m x x x x x

• Best-2: ( ) ( )1 2( 1) ( ) ( ) ( ) ( ) ( ) .i l n r sk k F k k F k k+ = + − + −m g x x x x

2. Crossover: after mutation, the following updating rule is applied:

( 1) (0,1) ,( 1)

( ) .r

k if rand Ck

k otherwise

+ ≤+ =

i

i

i

mc

x

Crossover parameter, r

C , controls the probability of creating new parameters into the population

from the mutant vector. Parameters of the trial vector ic are then projected over the search space if they are out of bounds. 3. Selection: Fitness of ( 1)

ik +c is calculated. The selection procedure will make this

individual to survive if the value of the cost function has decreased, i.e.:

( ) ( )( 1) ( 1) ( ) ,( 1)

( ) .

k if f k f kk

k otherwise

+ + ≤+ =

i i i

i

i

c c xx

x

There is not a proof of convergence for differential evolution and no theoretical analysis exists to analyse how to tune the differential evolution parameters. Usually it is recommended to take

1 2 0.5F F= = and a crossover probability of 0.8r

C = . Also the scheme “target to the best” is

usually used for mutation. In a companion paper (Fernández Martínez et al, 2010) we have shown the location of the optimum parameters F and

rC for different kind of benchmark functions in 50

dimensions (figure 5). It is possible to observe that the good points are almost parallel to the .

rC axis, that is, this parameter seems not to play a crucial role. For the valley shape

function (Rosenbrock) the good values of F seems to be bounded by 0.5. For the Griewank function (multiple minima) or when the complexity of the optimization decreases, the optimum value for F seems to be lower. We have also shown for the Stanford VI synthetic reservoir that although with all mutation mechanisms we obtain very good results, the lower misfit rates are obtained for the mechanisms that include the global best in the mutation strategy. We are using 1 2 0.3F F= = and 0.8.

rC = for Norne

synthetic. We also show some preliminary results obtained with the target-to best and

best-1 mutation mechanisms.

5. Methodology

The methodology does not only consists in looking for the model of minimum misfit, but also to

find the family of models belonging to M that fit the observed data m∈d R (comprising all the

observables, e.g. production and/or seismic data) within the data error tolerance tol;

( )|| ||p tol− <F m d. This family will be used to deduce uncertainty estimates on the porosity and

the permeability of the reservoir as it has been shown in Fernández Martínez et al (2010), for the Stanford VI shale and sand reservoir.

Figure 5: Logarithmic misfit error as a function of the mutation scale factor and crossover

parameter for different type of benchmark functions in 50 dimensions.

In our case the forward model ( )F m has multiple components: a reservoir flow simulator to

predict the production data; a forward seismic modeling (Dadashpour et al. 2008); a geostatistical model to constrain the spatial structure of the reservoir, and finally a rock physics model that takes into account the facies-specific relations between porosity, permeability, saturations and elastic velocities. In this case Dadashpour (2009) used AVO (amplitude versus offset) time lapse-data to constrain the results of the inversion, that is, the use of zero-offset amplitude and AVO gradient differences in conjunction with production data. This method is easy to implement and has not heavy pre-processing requirements, as pointed by Dadashpour (2009). In his PhD work he compared several optimization methods:

1. Two gradient-based optimization techniques: SNOPT (Gill et al, 2005) and Gauss-Newton method.

2. Two direct search methods: Hooke and Jeeves (1961) and the Generalized Pattern Search method (Kolda et al, 2004)

3. Two global optimization algorithms: Genetic Algorithms and Particle Swarm.

To perform optimization using derivative-free and global optimization methods, dimensionality reduction using the spatial principal component base has been performed; this method has also been used for reservoir optimization in Echeverría and Mukerji (2009); Echeverría et al. (2009) and Fernández Martínez et al. (2010). This reduction allows us to perform sampling on the reduced model space using global optimization algorithms. This reduction is also aimed at regularizing the inverse problem since the high frequencies on the model might not be informed by the observables. Although the best results in Dadashpour work were obtained using local optimization algorithms and pattern search methods, it is important to note that in his work he did not use the state of the art available at this time for Particle Swarm. We also provide some preliminary results about the use of Differential Evolution to solve this history matching problem (Storn and Price, 1997).

6. Some preliminary results using a family of Particle

Swarm optimizers

The semi-synthetic Norne field

Particle swarm optimization method has been used for estimation of porosity and permeability of 2D semi-synthetic Norne field (Dadashpour, 2009). It has been shown that PSO has the capability for good convergence of the optimal value. The model porosity and permeability inversion gives acceptable results. The flow simulator model has one injector and one producer which are situated on the right and left side respectively of the section. Water injection starts at the same time with production. Production measurements are the bottom hole pressure for both wells, water production and oil production rates during 20 years. Time-lapse seismic includes zero offset amplitudes and AVO gradients (Dadashpour, 2009). The first task is to determine the number of PCA terms that are required to match adequately the observables. Dadashpour et al (2009) used 100 PCA terms for the porosity and 100 PCA terms for the permeability. Using this reduced PCA base he was able to find a model with minimum misfit 0.075 using a direct search method (Hooke and Jeeves).Figure 6 shows the evolution of the misfit when we consider an increasing number of PCA terms using the model provided by Hooke and Jeeves. It is possible to observe that with 60 PCA terms it is possible to achieve very low misfits. This is the number of terms we will adopt in our simulations. Figure 7 shows the comparison between all the PSO family members. It can be observed that in this case all the family members provide a better result than PSO. Nevertheless we think that the misfit for PSO can be improved adopting a less explorative PSO. In all the cases we have adopted a swarm size of 60 particles.

Figure 6: Evolution of the misfit as a function of the number of PCA terms

Figure 7: Comparison of the convergence curves for different PSO optimizers.

Table 1 shows the minimum misfit reached by the different PSO optimizers and the inter-quartile range of all the visited models after the 20th iteration. It is possible observe that RR-PSO has the lower misfit but also the lower IQR. PP-PSO shows also a similar behavior. This means that both algorithms are able to locate a great percentage of samples in the low misfit area.

Algorithm Minimum Misfit IQR(iter>20)

CC-PSO 0.1169 0.22 CP-PSO 0.1448 0.13 PP-PSO 0.1209 0.077

PSO 0.2383 1.13 RR-PSO 0.0933 0.065

DE target to best 0.2226 DE best-1 0.3616

Table 1: Minimum misfit for different PSO optimizers

7. Uncertainty assessment using different versions All of the versions of PSO provide an acceptable match with the original synthetic model. The advantage of using global optimization method is that uncertainty can be assessed near the optimum point. To assess uncertainty near the optimum point we plot the best, E-type and IQR (Inter quartile range) of porosity and permeability for each version of PSO. To compute uncertainty measures we also keep track of the evolution of the median distance between the global best in each of the iterations and the particles of the swarm. When this distance is smaller than a certain percentage of the initial value this means that the swarm has collapsed towards the global best. Once this happens we can either stop the algorithm, or continue iterating, but in the posterior analysis we count all the particles in this collapsed swarm as one. Taking them into account individually has the effect of overestimating the probability on the low misfit area due to oversampling. Finally, based on the selected samples it is possible to produce averages (E-types) over the samples and interquartile range maps that help us to establish facies probabilities. We can also scan possible existing tradeoffs between the parameters. Figure 8 shows the results obtained for the CC_PSO. The porosity seems to be higher towards the top of the reservoir and exhibit a good lateral continuity. It can be observed in all the maps a zone of medium porosity (0.20) close to the bottom of the reservoir that travels along the section and seems to vanish on the right part of the section. The uncertainty of the porosity is low close to the wells and increases towards the center of the section. This behavior is almost the same (with some local differences) for all the members of the family (figures 9 to 12). In conclusion it seems that the porosity is quite well resolved. The permeability shows a peculiar behavior depending on the runs. For instance in the case of CC-PSO very low permeabilities are present towards the middle of the section, while in the other cases it seems that the permeability increases towards the bottom of the reservoir. In all these simulations the zones of low permeability are highly variable in location and extent (compare the results of CC-PSO and RR-PSO). Thus, it seems that the low permeability zones are not very well informed by the observables. Nevertheless they seem to be associated mainly to the bottom of the third block. Based on the ensemble it is possible to establish the probability of having a very low permeability.

CC-PSO

Figure 8: Centered-Centered PSO (CC-PSO). Best model, Etype and IQR for porosity and permeability. Samples are taken on the 0.2 misfit region.

Best

E-type

IQR

CP-PSO

Figure 9: Centered-Progressive PSO (CP-PSO). Best model, Etype and IQR for porosity and permeability. Samples are taken on the 0.2 misfit region.

Best

E-type

IQR

PP-PSO

Figure 10: Progressive-Progressive PSO (PP-PSO). Best model, Etype and IQR for porosity and permeability. Samples are taken on the 0.20 misfit region.

Best

E-type

IQR

PSO

Figure 11: PSO. Best model, Etype and IQR for porosity and permeability. Samples are taken on the 0.3 misfit region.

Best

E-type

IQR

RR-PSO

Figure 12: Regressive-Regressive PSO (RR-PSO) Best model, Etype and IQR for porosity and permeability. Samples are taken on the 0.2 misfit region.

Best

E-type

IQR

8. Conclusions and Future work

All of the PSO algorithms provide an acceptable match to porosity and permeability of the synthetic Norne. Using PSO for inversion also helps in assessing the uncertainty attached to estimation of porosity and permeability of the reservoir. This is the advantage of using global method over local methods of optimization. We observed that RR-PSO provided minimum misfit as compared to other members of PSO. All of the above indicates PSO family has strong potential as an optimizer in joint inversion of time-lapse seismic data and production data. Our next step is application of these algorithms on real Norne data set. In future we will apply these algorithms on segment E of Norne field.

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