reservoir characterization and inversion uncertainty via a ......reservoir characterization and...

Reservoir characterization and inversion uncertainty via a family of Particle Swarm Optimizers and differential evolution.

Application to the synthetic Stanford VI reservoir.

Juan Luis Fernández Martínez (1,2,3), Tapan Mukerji (1), David Echeverría (1), Esperanza García Gonzalo (2), and Amit Suman (1)

(1) Engineering Energy Resources Department. Stanford University. Palo Alto, California, USA.

(2) Department.of Mathematics, University of Oviedo, Oviedo, Spain. (3) Department of Civil and Environmental Engineering. University of California Berkeley,

Berkeley, USA.

Abstract

History matching provides to the reservoir engineers an improved spatial distribution of physical properties to be used in forecasting the reservoir response for field management. The ill-posed character of the history matching problem yields non-uniqueness and numerical instabilities that increase with the reservoir complexity. These features might cause local optimization methods to provide unpredictable results not being able to discriminate among the multiple models that fit the observed data (production history). In this manuscript the ill-conditioned character of this history matching inverse problem is attenuated by reducing the model complexity using the Spatial Principal Component base and by combining as observables flow production measurements and time lapse seismic cross-well tomographic images. Additionally the inverse problem is solved in a stochastic framework. For this purpose we use a family of Particle Swarm Optimizers that have been deduced from a physical analogy of the swarm system. For the synthetic Stanford VI sand-and-shale reservoir we analyze the performance of the different PSO optimizers, both in terms of exploration and convergence rate for two different reservoir models with different complexity and under the presence of different levels of white Gaussian noise. We show that PSO optimizers have a very good convergence rate and provide in addition, approximate measures of uncertainty around the optimum facies model. Uncertainty estimation makes our algorithms more robust in presence of noise which is always the case for real data. This is an important achievement since in cases where the reservoir exhibits small scale features local methods get trapped and clearly fail to find a good solution.

Finally we briefly introduce differential evolution and we show some preliminary results of its performance on the Stanford VI reservoir, showing that we are able to achieve similar results.

Application of PSO and DE for reservoir characterization and inversion uncertainty

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1. Introduction

Characterizing the spatial distribution of heterogeneous reservoir properties is one of the major challenges in reservoir modeling for optimizing production. Well data together with seismic data are typically used to infer the spatial distribution of properties such as facies, porosity and permeability. The seismic history matching problem consists then in obtaining reservoir models that match production data as well as seismic time lapse data. Different methodologies have been proposed in the literature to approach this problem using local optimization algorithms and regularization techniques, global optimization methods, geostatistical inversion, filtering techniques, etc (e.g. Landa, 1997; Huang et al., 1997; Sarma et al., 2006; Castro, 2007; ;Dadashpour et al., 2007; Oliver et al., 2008, Echeverria and Mukerji, 2009).

Independently of the algorithm that is used in the inversion, the main challenges of this inverse problem are the following:

1. The production data alone does not contain enough information to uniquely constrain the porosity and permeability of the reservoir. In addition, like all data, the production data are noisy, spatially very localized, and describe an integrated behavior of the reservoir. Integrating different types of data with different spatial-temporal scales can help to partially address this problem. Combining flow production measurements with time lapse seismic data has been useful for better constraining the history matching. This idea has been used by several authors, e.g. Huang et al., 1997; Aanonsen et al., 2003; Sena et al., 2009a,b; Echeverría and Mukerji, 2009; Echeverría et al., 2009; Xia and Huang, 2009; and Dadashpour et al., 2009, among others.

2. Furthermore, to run the flow simulator and produce accurate results a detailed description of the reservoir is needed. This is performed by discretizing the reservoir unit into small blocks causing the inversion problem to be highly ill-posed due to its underdetermined character. One solution commonly found in the literature is to use non linear least squares methods with Tikhonov regularization around a reservoir reference model that is constructed using prior geological and geophysical knowledge. In this case the parameterization of the model space is the same for the forward and for the inverse problems and regularization techniques are needed to achieve uniqueness and to stabilize the solution. The L-curve methodology (Hansen, 1998) is used to select the minimum amount of regularization needed to fit the data. The result of this procedure is a unique reservoir model that shows the best trade-off between the data prediction and the model complexity. One of the dangers of this methodology is that no uncertainty estimation is usually performed around this model, due to the large number of parameters involved in the reservoir model and the high computational cost of performing uncertainty by linear analysis techniques. In addition, the validity of the uncertainty analysis is limited by the highly non-linear character of the history matching inverse problem. Moreover, local methods for non-linear problems are highly dependent on the initial guess, a feature that hampers its robustness.


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Stochastic approaches to inverse problems consist in shifting attention to the probability of existence of certain interesting subsurface structures instead of looking for a unique model. Global optimization algorithms include among others, well known techniques such as genetic algorithms (Goldberg, 1989; Holland, 1992), simulated annealing (Kirkpatrick et al., 1983), particle swarm optimization (PSO) (Kennedy and Eberhart, 1995), differential evolution (Storn and Price, 1997), the neighborhood algorithm (Sambridge, 1999), etc.

The use of global optimization algorithms such us genetic algorithms, simulated annealing and neighborhood algorithm is very well documented in geosciences for geophysical inversion (e.g. Sen and Stoffa, 1991; Stoffa and Sen , 1991; Sambridge and Drijkoningen, 1992; Sen and Stoffa, 1995; Ma, 2002; Fernández-Álvarez et al., 2008). In reservoir optimization global optimization algorithms have been used in well placement problems when the search space has a low dimension, but their use is hampered in real history matching problems by the number of parameters needed to accurately describe the reservoir. Global optimization algorithms and derivative-free techniques have been also used in history matching problems combined with parameter reduction techniques (Echeverría and Mukerji, 2009; Echeverría et al., 2009; Sena et al., 2009a,b; Romary, 2009a,b). Sena et al. (2009a) used instead the pilot point parameterization and very fast simulated annealing as an optimization method with exploitative purposes to obtain the optimal solution. Although in their work they use simulated annealing, they do not provide any uncertainty assessment, because the use of simulated annealing as a sampler (Mosegaard and Tarantola, 1995) needs the cooling parameter to be one, highly worsening its convergence rate (Fernández-Alvarez et al., 2008). With respect to particle swarm optimization, although it has been used to solve a variety of optimization and inverse problems in different branches of engineering and technology (Poli, 2008b), 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). In reservoir engineering PSO has been used to determine the optimum well location and type in very heterogeneous reservoirs (Onwunalu and Durlofsky, 2009).

Although the global algorithms can be used in an exploitative form to quickly converge to an optimum, their main advantage is that they can potentially address the inverse problem as a sampling problem. Nevertheless their use as samplers is still limited by two main reasons: 1. they require reasonably fast forward modeling, 2. the sampling procedure is hampered by the curse of dimensionality (Curtis and Lomax, 2001). The first problem can be solved using faster approximations of the forward problem and running the forward simulation in parallel. The second issue is addressed by using parameterizations that reduce the dimensionality of the model space. This reduction also serves to regularize the inverse problem since all dimensions of the full model space are usually not informed by the observed data. In this paper we use the same reduction technique (spatial principal component analysis) that has been used in Echeverría et al. (2009) and Echeverría and Mukerji (2009) but with the novel contribution of applying the PSO family as global optimizer.

We first describe and then apply a family of particle swarm optimizers to two different synthetic sand-and-shale reservoir models under the presence of different levels of noise. We compare the behavior of the different PSO family members to show empirically that particle swarm optimizers have a very good convergence rate. We are also able to produce a measure of uncertainty, even with a small number of


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forward evaluations. This feature makes our methodology robust in the presence of noise in contrast with local methods that might get trapped and fail to find a good solution.

2. Methodology

Let us refer to the model parameters by nRMm ⊂∈ , where M is the set of admissible models formulated in terms of geological consistency. The model m can be a physical property (permeability and/or porosity) or an indicator (facies) associated with every grid block in the reservoir simulator. In the inverse problem we are interested in, the number of parameters, n, is typically of the order of a few thousands. The optimization problem is stated as follows: 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 ,

Here ( )F m represents the predictions made by the reservoir model with respect to the observables

for the model m, and || ||p

⋅ stands for the norm needed to compute the distance between the data and the

predictions. In this case ( )F m symbolizes all the forward models needed to compute the data: a

reservoir flow simulator to predict the production data using the Stanford’s General Purpose Research Simulator (GPRS), (Aziz and Settari, 1979), and the wave propagation and inversion associated with crosswell tomography (Echeverría et al., 2009) to reconstruct the seismic velocity image from the seismic

traces measured at the boreholes. ( )F m also includes the rock physics model to compute porosity and

permeability for a given facies, and the models to compute changes in seismic velocities due to production-related changes in pore fluid saturations. In our conceptual model, porosity field associated to given a facies distribution is computed by regression with respect to the measurements made at the wells. We assume that the porosity and permeability are related by the Kozeny-Carman equation with parameters regressed against well data. The rock physics model takes into account facies-specific relations between porosity, saturations, and elastic velocities (Mavko et al., 2003).

The methodology includes the following steps:

1. Dimensionality reduction and regularization using the spatial principal component base. Further details can be found in (Echeverría and Mukerji, 2009; Echeverría et al., 2009). This reduction allows us to perform sampling on the reduced model space using global optimization algorithms, particle swarm optimizers in this case.

2. Sampling using the different members of the PSO family. Their convergence is related to the first and second order stability of the particle trajectories. In this paper we use a cloud version of the PSO (García-Gonzalo and Fernández-Martínez, 2009, Fernández-Martínez et al., 2010c) that has been extended to the other family members (Fernández-Martínez and García-Gonzalo, 2010). This procedure


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has several advantages: the tuning of the PSO parameters is automatic, it increases the exploration, and at the same time we avoid having to artificially clamp the particle velocities to avoid instabilities. Also all the PSO family members can be used in one-point modality, that is, all the particles on the swarm have the same inertia, local and global acceleration constants.

3. Posterior analysis to estimate the posterior distribution of the model parameters from the samples gathered on the low misfit region. Based on the PSO samples we get an E-type of the facies models and a measure of the uncertainty of these facies in each pixel of the reservoir (the inter-quartile range for example).

3. Parameter reduction and regularization: the PCA spatial base

Principal component analysis (Pearson, 1901) is a well-known mathematical procedure that transforms a number of correlated variables into a smaller number of uncorrelated variables called principal components. The resulting transformation is such that the first principal component accounts for as much of the variability and each succeeding component accounts for as much of the remaining variability as possible (Jolliffe, 2002). Usually PCA is performed in the data space, but in this case it is used to reduce the dimensionality of the model space based on a priori samples obtained from conditional geostatistical realizations that have been constrained to static data. Applied to our context, PCA consists in finding an orthogonal base of the experimental covariance matrix estimated with these prior geological models, and then selecting a subset of the most important eigenvalues and associated eigenvectors that are used as a reduced model space base.

This method has been extensively used in the literature in several fields, such us weather prediction and operational oceanography, fluid dynamics, turbulence, statistics, etc. In history matching this model reduction technique has been used amongst others by Echeverría et al. (2009), Echevarría and Mukerji (2009) and Romary (2009b). Sometimes the method is also known under other terminologies such as Proper Orthogonal Decomposition or Orthogonal Empirical bases. In this paper we use the same methodology and data set as in Echeverría et al. (2009) and Echeverría and Mukerji (2009) but a different algorithm (PSO in this case) as the optimization algorithm and sampler.

The method works as follows:

1. First we generate an ensemble of plausible reservoirs ( )lmmm ,,, 21 K=Χ n

i R∈m

conditioned to borehole data using multipoint geostatistics (Strebelle, 2002). In our case the reservoir models of facies are 3D and are composed of 4000 pixels (ten layers with 400 pixels). We had l = 1000 different reservoir realizations all conditioned to static data.

2. The problem consist in finding a set of patterns ( )1 2, , , dV = v v vK that provide an accurate low

dimensional representation of the original set with d l<< . There exist several methods to accomplish this task. PCA does it by diagonalising the experimental centered covariance. The centered character of the experimental covariance is crucial to maintain consistency with the static data after reconstruction. Ranging the eigenvalues in decreasing order allows us to select a


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certain number of PCA terms ( d N<< ) to match most of the variability in the model ensemble. Then, any reservoir model in the reduced space is represented as a unique linear combination of

the eigenmodels, 1

ˆ ,d

k i i

i

a=

= +∑m µ v where µ is the experimental model mean. Although the

original facies model is discrete in nature (binary), the reconstructed models lie in a continuous space. This is taken into account in our rock physics model through a threshold procedure to determine which facies we should use to compute the rock physics properties in each case.

3. In our case the rank of the covariance experimental matrix is 1000. Nevertheless, we need a fewer number of PCA terms to expand what we think it is the variability on the model space. Figure 1A shows the model variability expansion with increasing number of principal component terms. It can be observed that with 30 principal components we match 50% of the variance, and with 300 we arrive at almost 90% of the total variability. In practice we have adopted 30 principal components. Romary (2009a) investigated different criteria for selecting a reduced basis set, including flow simulation and geometric connectivity criterion. In practice the number of components to be retained can be affected by different parameters, and a preliminary study should be performed to select the level of decomposition in accordance with available computation resources (Romary, 2009a). The reduction of the dimension in the model space is from 4000 model parameters (20x20x10) to 30 coefficients corresponding to the 30 first principal components. This reduction allows global optimization methods to perform sampling in this reduced model space.

Figure 1: A) Percentage of the model variance spanned by PCA. B) Percentage of the

data variance modeled with an increasing number of principal components.


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The orthogonal character of the vectors i

v provides to this base a telescopic (nested) character; that

is, if we add the next eigenvector 1d +v to the base, the vector ˆk

−m µ will be expressed in these two

bases as follows:

( ){ }

( ){ }1 2 1 2 1

1 2 1 2, , , , , , ,ˆ , , , , , , ,0 . (1)

d d dk d da a a a a a

+

− = =v v v v v v v

m µK K

K K

This property allows an easy implementation of a multi-scale inversion approach adding more eigenvalues to match finer scales of heterogeneity as needed. Determining the level of detail is an important question since all the finer scales might not be informed by the observables, that is, they might belong to the null space of our local linear forward operator. By truncating the number of PCA terms that we use in the expansion we are setting these finer scales to zero avoiding also the risk of over fitting the data. In other words, the use of a truncated PCA base provides a natural smoothing of the solution. Figure 1B shows how the data reconstruction error varies with increasing principal components retained in the reconstruction. These curves were computed using the data for the known true model. It can be observed that:

1. The production data in this case is more sensitive than the seismic data to the number of PCA terms. This might depend on the boundary conditions used to run the flow simulator and on the kind of seismic information that is available.

2. Some of these curves are not monotonously decreasing, thus, when adding more detailed components of the model the prediction may sometimes worsen. This can be taken as an illustration of the degree of ill-posedness of the history matching inverse problem.

The total misfit decreases with increasing terms in the PCA base, and for 30-50 PCA terms the reconstruction error remains acceptable (relative error less than 0.1). Although, if the true reservoir model is far from the median behavior described by the PCA base, then the number of principal components needed to describe accurately the observed data will increase and the dimension reduction will be seriously affected. This is expected to happen when the “true” reservoir model exhibit low-scale features that might highly influence the fluid flow. This also illustrates the fact that the model reduction technique, although is a modeling decision, has to be performed adequately.

4. The Particle Swarm Optimizers

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

The algorithm consists of the following:

1. A prismatic space of admissible geophysical models M, is defined:

, 1 , 1 ,j ji j size

l x u j n i n≤ ≤ ≤ ≤ ≤ ≤


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where , ,j jl u are the lower and upper limits for the j-th coordinate for each geophysical model. In

PSO terminology, each reservoir model will be called a particle, which is represented by a vector whose length is the number of (reduced) model parameters of the inverse problem. Each particle has its own position in the search space. The particle velocity represents the parameter perturbations needed for these particles to move around in the search space and explore solutions of the inverse problem.

2. At each iteration the algorithm updates the positions, ( )i kx and velocities, ( )i kv of each

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

a) The inertia term, which consists of the old velocity of the particle, ( )i kv , weighted by a

real constant, ω , called inertia.

b) The so-called social term, which is the difference between the global best position found

thus far in the entire swarm (called ( )kg ) and the particle's current position ( ( )i kx ).

c) The so-called cognitive term, which is the difference between the particle's best position

found so far (called ( )i kl , the local best) and the particle’s current position ( ( )i kx ):

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

1 2

1 1 2 2 1 2

1 ( ) ( )

1 1 ,

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

k

i i i i i

i i i

g l g l

k k k k k

k k k

r a r a r r U a a

ω φ φ

φ φ ω

+ = + − + −

+ = + +

= = ∈ ∈

v v g x l x

x x v

R

1 2,r r are vectors of random numbers uniformly distributed in (0,1), to weight the global and local

acceleration constants, and g l

a a .2

g la a

φ+

= is the total mean acceleration and plays an important role

on determining the algorithm’s stability and convergence.

Traditionally researches have focused on the analysis of the deterministic particle trajectories to understand its convergence properties (Ozcan and Mohan, 1999; Carlisle and Dozier, 2001; Clerc and Kennedy, 2002; Trelea, 2003; Zheng et al., 2003; van den Bergh and Engelbrecht, 2006; Fernández-Martínez et al., 2008a). Nevertheless to properly understand the PSO dynamics the second order and higher moments are needed (Poli, 2008a; Fernández-Martínez and García-Gonzalo, 2008, 2009a).

The PSO algorithm can be physically interpreted as a particular discretization of a stochastic damped mass-spring system (Fernández-Martínez and García-Gonzalo, 2008). Based on analysing this stochastic differential model Fernández-Martínez and García-Gonzalo (2009b) proposed a family of PSO algorithms: the GPSO or centered-regressive PSO (Fernández-Martínez and García-Gonzalo, 2009a), the CC-GPSO or centered PSO, and the CP-PSO or centered-progressive PSO. Although these algorithms are stochastic in nature they are not heuristic, since their convergence can be related to the first and second


9

order stability of the trajectories now represented as stochastic processes (Fernández-Martínez and García-Gonzalo, 2008; Fernández-Martínez and García-Gonzalo, 2009 a,b, 2010). Both stability regions

can be calculated analytically. The stability regions can be defined in the space of ω φ− (inertia-mean

acceleration). Figure 2 shows the first and second order stability regions for all the family members with 1t∆ = . In these three cases the second order stability regions are imbedded within the corresponding first order stability regions. For the CC version both regions are unbounded on the inertia axis. For the CP version both regions of stability are unbounded and are composed of two different disjoint zones. We also show the isolines of the first and second order spectral radii that control the attenuation of the first and second order trajectory moments. Blue areas indicate a very fast attenuation. Choosing the PSO parameters in these areas can cause entrapment in local minima due to the almost instantaneous attenuation of the particles trajectories, that is, the algorithm does not explore enough the search space. The second-order spectral radius of different variants in the algorithm family determine the explorative capabilities of the swarm. CP-GPSO has greater exploration capabilities than the GPSO and CC-PSO (Fernández-Martínez and García-Gonzalo, 2009a). This fact has also been found in practice in the solution and appraisal of inverse problems (Fernández-Martínez et al., 2008b, 2009). Finally, the continuous PSO model does not represent only a physical analogy since there exists a full stochastic convergence between the discrete and the continuous versions (Fernández-Martínez and García-Gonzalo, 2009b, 2010).

Figure 2: First (first row) and second (second row) order stability regions for GPSO, CP-

PSO and CC-PSO, and corresponding spectral radius isolines.


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Although these algorithms are linearly isomorphic in their homogeneous parts (without taking into account the effect of the local and global attractors in the inhomogeneous part) they behave very differently in practice due to the way they update the positions and the velocities of particles: GPSO updates first the velocity and then the trajectory; CC-PSO updates first the trajectory and then the velocity using two consecutive centers of attraction; CP-PSO updates the trajectory and the velocity at the same time.

In all cases the upper boundary of the second order stability region is the line where the action of the cost function attractors is weak, that is, above this line we achieve a very explorative behavior. Also along the median lines of the first order stability regions the temporal covariance between trajectories is close to zero. Thus, each algorithm member has greater exploration capabilities when the inertia and acceleration parameters are selected close to the intersection of these lines (Fernández-Martínez and García-Gonzalo, 2009a). This fact has also been confirmed by numerical experiments using benchmark functions in 30 dimensions, which coincides with the number of PCA terms that we are going to use on our study cases. Figure 3 shows for each algorithm the contour plots of the misfit error (in logarithmic scale) after a certain number of iterations (200) applied to the Rosenbrock function in 30 dimensions. This function has the global minimum located in a very flat valley. This analogy is important since as shown later this topography is also the case for the seismic history matching problem. The numerical analysis shown in

Figure 3 is done for a lattice of ( ,ω φ ) points values located in the corresponding first order stability

regions, over 50 independent simulations. These simulations show that in all the cases (PSO, CC-PSO,

CP-PSO) parameters sets ( , ,g l

a aω ) with lower misfit (and thus, with a higher probability to reach the

global minimum) are close to the upper boundary of the second order stability region. Similar results are achieved for the Griewank function that has a global minimum surrounded by many local minima (Fernández-Martínez and García-Gonzalo, 2009a).

Based on these results we have designed for each PSO version a cloud-PSO algorithm where each

particle of the swarm has associated a different ( , ,g l

a aω ) set located on this region, instead of the more

common algorithm where every particle has the same set of parameters. The cloud PSO algorithm has been tested on different benchmark functions obtaining better results than those published for hard benchmark functions in several dimensions (García-Gonzalo and Fernández-Martínez, 2009). In this work the cloud algorithm is also expanded and tested for other family members (CC-PSO and CP-PSO).

The cloud design allows the different swarm members to find the sets of parameters that are better suited for solving each inverse problem. In the cloud some of the particles will have a more exploratory behavior while others will show a higher exploitative character. This feature helps to avoid two main drawbacks of the standard PSO algorithm: the tuning of the PSO parameters and the artificial clamping of the velocities.

In the algorithm design the time step parameter ( t∆ ) plays a very important role: increasing 1t∆ >

provides more exploratory character, while decreasing 1t∆ < allows for a more detailed local search

around the global best. Switching of t∆ with iterations has been named the lime and sand algorithm (Fernández-Martínez and García-Gonzalo, 2008) and allows escaping entrapment in local minima. In the


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case study the CP version with a variable time step has shown to be a very efficient algorithm.

Figure 3: Numerical results obtained for the Rosenbrock function (flat valley function) in

30 dimensions. Median error contourplot (in logarithmic scale) after 50 different

simulations in a very fine lattice of ( ,w φ ). We also show the cloud of points that are used

for the optimization for different family members: (A) PSO, (B) CC-PSO, (C) CP-PSO.

The clouds might also incorporate other points located on the gray areas.


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5. Uncertainty analysis

Uncertainty analysis in a Bayesian framework (see for instance Scales and Tenorio, 2001) consists of estimating ( / )P m d the posterior distribution of the model parameters m given the observed data d:

( / ) ( / ) ( ).P L P∝m d d m m

( / )L d m is called the likelihood and involves calculation of the misfit error ( )|| ||p−F m d , while

( )P m is the prior probability of the model. In the history matching example the search is done in the

reduced PCA base and ( )P m is considered uniform on the reduced search space (see figure 4). Although

it is a non informative prior on the reduced base, prior knowledge about the reservoir is already incorporated in building the PCA base through an ensemble of 1000 reservoir models generated by multipoint geostatistics conditioned to prior static reservoir data. In figure 4 we show the co-ordinates of the 1000 initial reservoir models projected onto the PCA base formed by the first 30 eigenvectors. As can be observed from Figure 4, the size of the search space can be reduced using upper and lower bounds on the coordinates for each principal component defined by the space where most of the initial reservoir models lie.

Figure 4: Reduced search space on the PCA base with 30 eigenmodels (from Echeverría

et al., 2009). We also show the coordinates of the 1000 prior model realizations projected

on the PCA base. The higher frequency components correspond to higher indexes. Lower

and upper bound in the reduced search space are shown in blue and red respectively.


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Bayesian approaches and Monte Carlo methods (Scales and Tenorio, 2001; Mosegaard and Tarantola, 1995) can be used to solve the inverse problem in a different way accomplishing implicitly the posterior appraisal task. Theoretically to perform a correct posterior sampling Monte Carlo methods are needed. In this case the posterior probability distribution is sampled many times in a random walk, with a bias towards increased sampling of areas of higher probability, thus accurately approximating the whole posterior probability (the so-called importance sampling). The drawback of such approaches is that they are computationally expensive, and they might not even be feasible for inverse problems with costly forward solvers such as in our history matching case. Previous work has shown that, under certain conditions, global algorithms can provide reasonably accurate measures for model uncertainty. Mosegaard and Tarantola (1995) used simulated tempering to perform importance sampling of the model parameters through the Metropolis-Hastings algorithm. In this theoretical development the cooling parameter has to be fixed to one in order to sample by importance. This basically means that any convergence property of simulated annealing is lost and the algorithm amounts to almost a random search. More recently, Fernández-Álvarez et al. (2008) have shown numerically the ability of the binary genetic algorithms to perform an acceptable sampling of the posterior distribution when they are used in an explorative way, i.e., the algorithm cannot be elitist and the mutation probability has to remain high enough. Although genetic algorithm is a population based algorithm, the conclusion reached in Fernández-Álvarez et al. (2008) is similar to the one in Mosegaard and Tarantola (1995), that is, to sample correctly, a high exploratory capacity is needed and this severely worsens the algorithm speed. Finally, the neighborhood algorithm can also be used to accomplish this task (Sambrigde, 1999) but requires a resampling procedure to produce an accurate posterior.

Although so far there do not exist any theoretical results about conditions under which the PSO family members might perform an accurate posterior sampling, it has been shown in practice that if the parameters are chosen close to the region of second order stability (high exploration capabilities), the marginal distributions deduced from the posterior sampling can be interpreted as a proxy for uncertainty or dispersion of the model parameters (Fernández-Martínez et al., 2009; 2010 a,b). In other words, PSO can provide a proxy for the posterior distribution for the model parameters if it is used in its explorative form. PSO also has the advantage of being a very fast sampler showing a good equilibrium between exploration and convergence.

In this manuscript we use particle swarm sampling to provide a measure of uncertainty about the optimum facies updates by seismic history matching. Although these results are empirical, these measures of uncertainty can help as inputs to perform risk analysis. In practice the uncertainty analysis is performed by taking into account the particles that have been collected on a certain region of the low misfit area, with a chosen misfit cut-off. We also keep track of the evolution of the median distance between the global best and the particles of the swarm. When this distance is smaller than a certain percentage of the initial value (5% in this case) 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. The algorithm also has the possibility to increase the exploration by adopting a new center of attraction based on previous statistics


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or by dispersing the swarm introducing repulsive forces by switching to negative sign of the acceleration constants. 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. The posterior marginal distributions show us how any individual PCA term is resolved, and the posterior covariance helps us to understand the existing linear tradeoffs between model parameters. These last measures are related to the topography of the misfit surface near the low misfit region.

6. Synthetic case Study: application to the Stanford VI synthetic

sand and shale reservoir

The case study is based on two synthetic datasets generated by a reservoir model composed of 4000 cells organized in ten layers of 20x20 pixels extracted from the Stanford VI reservoir (Castro et al., 2005). This synthetic reservoir is a good framework for comparing model inversion methodologies since the true facies model is known. We used two different synthetic models to test the PSO performance. In the first case the sand channels exhibit a larger lateral continuity than in the second case where small scale local channel features are present. This makes the optimization problem harder in the second case. In both cases we examine how the solutions are perturbed by the presence of noise (5 and 10% of Gaussian level) added to the data. The noise on data is expected to increase the number of equivalent solutions and to move the perturbed global minimum from its “true position”. Paradoxically we also show that noise makes the convergence of PSO faster to the valley region. The PCA base in both cases is the same that has been used in Echeverría et al. (2009) and Echevarría and Mukerji (2009). It has been generated using an ensemble of 1000 models generated by multipoint geostatitics (Strebelle, 2002) using SGEMS (Remy et al., 2009).

6.1. The topography of the cost function

The misfit error has two terms and is defined as follows:

( ) ( )

( ) ( )

( )

( ) ( )

( )

1 1 2 2

1 2

1 2

2 2* *

2 21 1 22 2

2 2

2 2* *

2 22 3 42 2

2 2

( / ) ( / ) ( / ),

( / ) ,

( / ) .

prod tomo

o o

inj inj oil oil

prodo o

inj oil

o o

T T T T

tomoo o

T T

E E E

W W P PE w w

W P

t t t tE w w

t t

= +

− −= +

− −= +

m d m d m d

m mm d

m m m mm d

m m

,o o

inj oilW P are the injected water and the produced oil during 90 days.

( ) ( )1 1

,o o

T Tt tm m are the cross-


15

well tomograms observables in two diagonal sections of a five spot configuration (Echeverría et al., 2009; Echeverría and Mukerji, 2009). In these cases the weights (w1, w2, w3, and w4) have been set to one. For the relative misfit we used the squared Euclidean norm to keep consistency with the error definition that had been used in Echeverría et al. (2009). Nevertheless any other norm (such as the L1 norm) can be used. The use of the norm is also related to the character of the solutions we seek. L1 norm is recognized to provide blocky and sparser solutions (Scales and Gersztenkorn, 1998).

Figure 5: Topography of the cost function for data sets I (A, B) and II (C,D). We show

how the cost function topography deforms in the presence of noise: the flat valley region

increases its size and some singularities (breaks and inflections) develop on the

neighbourhood of the global minimum.


16

It has been shown analytically that for nonlinear inverse problems, equivalent models are located along flat elongated valleys of misfit function landscape (Fernández Martínez et al, 2010d). Figure 5 shows a 2D section of the topography of the cost function when we prescribe the 28 last parameters (PC coefficients) to the true value and we make a finer grid around the two main PCA coefficients for both data sets (cases 1 and 2). We also show how these surfaces deform when 10% of Gaussian noise is added to the observed data. In case 2 the global minimum is located close to a range of higher misfits making the search harder. Also, when noise is added it is possible to observe that the local minimum region becomes broader and its amplitude smaller. This means that unexpectedly the noise has a regularization effect allowing faster location of the low misfit models. This is a dangerous situation when no uncertainty analysis is performed because we might converge to a wrong facies model. It also raises the question whether it is possible in practice to detect the global minimum (which is unknown in real cases) and if the global minimum really has any relevance in real cases; rather one should always consider a whole family of plausible models. Obviously the results shown in figure 5 are partial since we only see a particular cross-section of the cost function topography, the actual topography being much more complicated due to the dimensionality of the problem. Nevertheless this exemplifies perfectly the kind of ill-posedness of the seismic history matching problem.

6.2. First synthetic data set. Convergence rate curves

The first numerical experiment we run is to analyze how the PSO algorithms optimize the history matching problem using a small number of particles. In this case we used a swarm of 20 models for 75 iterations and performance behaviour was averaged over 10 independent simulations for each family member. For this numerical simulation we have used the first data case.

Figure 6 shows the median convergence curve for different family members for case I with 5% Gaussian noise. It can be observed that:

1. Most of the family members reach a zone of low misfit within approximately 30 iterations. Obviously in the different simulations there are runs with lower misfit than the median. It can be shown that as iterations progress both curves (the minimum error and the median) approach each other, that is, with iterations the swarm is composed mainly of low misfit reservoir models. This allows us to perform posterior statistics even with a reduced number of forward runs provided that we perform enough exploration. As explained before, these samples can be derived keeping track of the median distance between the swarm and the global best.

2. The CP-PSO version gives the lower median curve when it is used in the lime and sand modality, that is, with t∆ varying between 0.8 and 1.0 with each of the iterations. With t∆ =1 the CP-PSO version is very explorative and the error does not decrease with the iterations, meaning this that we are exploring the region of medium misfits (0.01).

3. The PSO and CC-PSO versions although they also perform well, they usually converge in an area of higher misfit. Nevertheless we can use the results provided by these algorithms to perform posterior uncertainty analysis. We used for this purpose the region of models with squared


17

relative misfit lower than 0.01. This is the region that has been considered to perform posterior statistics since all the PSO family members are able to sample it quite efficiently within this region from the 10th iteration onwards and these models provide a quite acceptable data misfit.

Figure 6: Stanford VI reservoir-case I with 5% of Gaussian noise. Median convergence

curves for different PSO versions with a swarm size of 20.

Similar convergence curves are obtained (not shown) when the size of the swarm is increased to 50 and 100 members, but in this cases a greater percentage of samples are placed on the low misfit area.

As noted before, the telescopic (nested) nature of the principal component base allows performing multiscale inversion in a simple way using relationship (1). Mutiscale inversion first optimizes the broader scales of heterogeneity until a certain error threshold is reached, and then higher principal components (higher spatial frequencies) are added to decrease the misfit. The transition between both stages can be done by adapting the whole swarm to the new augmented base or by just passing the global best obtained in the first step and generating randomly the rest of the particles in the expanded search space. Multiscale approaches have already been used in reservoir modeling and history marching problems (see for instance Tureyen and Caers, 2005; Aanonsen and Eydinov, 2006). Figure 7 shows the median convergence curve for different family members obtained by mutiscale inversion with 10 and 20


18

PCA terms respectively using a swarm of 20 particles. It can be observed that the convergence rate for all the PSO members is similar compared to those shown in figure 6, though in this case the PSO version achieves the lowest median misfit instead of the CP lime-sand version.. Thus, multiscale inversion can be performed in a natural way in the reduced model space and at each stage we are able to explore different scales, adapting dynamically our model parameterization to the data resolution.

Figure 7: Stanford VI reservoir-case I with 5% of Gaussian noise. Median convergence

curves for different PSO versions using multiscale inversion with a swarm of twenty

particles and ten and twenty PCA terms respectively.

As we have shown before analyzing 2-D cross-sections of the cost function, the optimization problem exhibits flat valley topography. The use of PSO as a proxy uncertainty sampler requires exploring the search space while maintaining a descent convergence rate. The cloud versions of the PSO algorithms

help to perform this task. Since all the particles have different PSO parameters ( , ,g l

w a a ) located close to

the upper boundary of the second order stability region, the attraction from the oscillation center is weak, allowing exploration of the search space. Also, as explained in the uncertainty analysis section once the swarm shrinks towards the oscillation center these particles will not be treated as independent, but as a one particle for posterior uncertainty analysis.


19

Figure 8 shows the optimum facies model found by CP-PSO in the presence of 5% of Gaussian noise compared to the true model. The figure shows the ten horizontal layers of the true model projected on the PCA base. Therefore instead of binary sand-shale model we see a continuous gradation of values. We also show the uncertainty analysis deduced form the samples that can be associated to this “optimum” facies model. Although the true model is binary (sand and shale) the optimum facies model and the interquartile range show a continuous color gradation due to the truncation adopted on the PCA base. It can be observed that the inverted model approaches the true synthetic model, and although they are different, the uncertainty measures in each pixel computed in the region of square relative misfit lower than 0.010 serve to account for the difference between these models. Similar results were achieved with other family members. When the noise level is increased (10% of Gaussian noise) the uncertainty estimates are still very reliable, that is, the interquartile range help us to imagine different plausible scenarios consistent with the true variability (figure 9). This feature provides to our methodology a robust character.

Figure 8: Stanford VI reservoir-case I with 5% of Gaussian noise. Horizontal slices from

the synthetic reservoir showing the inverted solution, true model, and uncertainty on the

model parameters (inter-quartile range) calculated with samples gathered on the low-

misfit region.

Finally figure 10 shows the comparison between the convergence rate curves for 5% and 10% Gaussian noise cases. Although the final misfit is higher for the 10% case, we can still perform uncertainty analysis on the 0.010 misfit region and produce good results as we have shown in figures 8 and 9. Similar results can be achieved for other family members (PSO and CC-PSO).


20

Figure 9: Stanford VI reservoir-case I with 10% Gaussian noise. Inverted model, true

solution and uncertainty on the model parameters (inter-quartile range) for the different

layers (1 to 10) of the 3D reservoir model.

Figure 10: Stanford VI reservoir, case I. Comparison between the convergence rates for

CP-PSO with 5% and 10% Gaussian noise.


21

6.3. Second data set: reservoir model with small scale features

Finally, we show some results for the second synthetic data set for different levels of noise. Figures 11, 12, 13 show the inverted model and the uncertainty analysis for the noise free, 5% and 10% of Gaussian noise.

Figure 11: Stanford VI reservoir, case II. Noise free case. Inverted solution true model

and uncertainty on the model parameters (inter-quartile range) for the different layers (1

to 10) of the 3D reservoir model.


22

Figure 12: Stanford VI reservoir, case II. True model, inverted solution and uncertainty

on the model parameters (inter-quartile range) for 5% of Gaussian noise.

Figure 13: Stanford VI reservoir, case II. True model, inverted solution and uncertainty

on the model parameters (inter-quartile range) for 10% of Gaussian noise.

It can be observed that although in presence of noise the found median solution is not far from the


23

true model, nevertheless the uncertainty measure (interquartile range) does not catch for some layers some variability that is present in the true model but not in the inverted model as shown for layers 6 and 7 of the 3D reservoir model in Figure 13. Thus, in the presence of high level of data noise we might need more samples in the posterior and more exploration, that is, more forward evaluations.

Figure 14: Stanford VI reservoir, case II. Comparison between the different PSO versions

and different levels of noise.

Figure 14 shows the convergence rate curves for different PSO members for different levels of noise (5% and 10%). In this case the CC-PSO is used in the one point modality, adopting

( ) ( ), , 0.5,2.13,2.13g l

w a a = . CC-PSO gives better performance on the one point modality. This might

be due to the fact that is using two consecutive centers of attraction and when adopting two different sets of PSO parameters the algorithm seems to loose its convergence properties and becomes a more explorative version. All other PSO versions show also a very good performance. In case I CP-PSO with lime-sand modality gave the best median convergence; for the sequential multiscale algorithm, PSO showed best convergence, in both cases with a cloud of acceleration and inertia parameters, while for case II it is CC-PSO with single set of acceleration and inertia parameters. This feature highlights the need of having at disposal a complete set of optimizers and not relying only in one version.

0 10 20 30 40 50 60 70 800.001

0.003

0.005

0.006

0.01

0.015

0.02

iterations

Stanford VI reservoir-Case II

Iterations

Re

lative

mis

fit

CP 0%

CP 5%

CP 10%

PSO 10%

PSO 0%

PSO 5%

CC 0%

CC 10%

CC 5%


24

6.4. Prediction of production and seismic data

To illustrate how the algorithm performs, figures 15 and 16 show for case I (5% of Gaussian noise) the production fit with the initial population and the match that is achieved with the models having the lower misfit and the models that have sampled on the region of misfit less than 0.015. It is possible to observe the big dispersion on the prediction at the first iteration and how the predictions from the samples collected by PSO on the region of lower misfits imbed the observed production data. For the seismic tomography sections (figure 16) we show the median sections in the first iterations and for the models on the 0.015 misfit region. Although the initial guess is far from the observed seismic data, the median tomographic sections of the low misfit models match quite acceptably the real tomographic sections.

Figure 15: Stanford VI reservoir, case I-5% of Gaussian noise. Prediction with the initial

swarm: (A) cumulative oil (B) injected water. Prediction with the samples located on the

0.015 squared misfit region: (C) cumulated oil (D) injected water.


25

Figure 16: Stanford VI reservoir, case I-5% of Gaussian noise. Observed data (A)

Tomographic section 1; (B) Tomographic section 2. Prediction with the initial swarm:

(C) Tomographic section 1; (D) Tomographic section 2. Prediction with the samples

located on the 0.015 squared misfit region: (E) Tomographic section 1; (F) Tomographic

section 2.


26

Finally figures 17 and 18 show the same kind of results for the Stanford VI-case II with 10% of Gaussian noise with similar conclusions.

To conclude, it is important to bear in mind that the main source of errors in our inversion procedure are the following:

1. Noise in data, production and time lapse seismic.

2. The model reduction technique that limits the variability to the most important eigenvectors. Also in the reconstruction after truncation, the facies variable which is by definition discrete, becomes a continuous variable.

3. The forward models that includes the rock physics to assign porosities and permeabilities, to the facies models.

Although all these sources of problems are present in our case, we have shown that the posterior reconstruction match quite considerably well the true synthetic reservoir models.

Figure 17: Stanford VI reservoir, case II-10% of Gaussian noise. Prediction with the

initial swarm: (A) cumulated oil (B) Injected water. Prediction with the samples located

on the 0.015 squared misfit region: (C) cumulated oil (D) injected water.


27

Figure 18: Stanford VI reservoir, case II-10% of Gaussian noise. Observed data (A)

Tomographic diction 1; (B) Tomographic section 2. Prediction with the samples located

on the 0.015 squared misfit region: (C) Tomographic section 1; (D) Tomographic section

2. The initial predictions are not shown but they are very far from the true solution.

7. 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:


28

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

k +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. 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 i

c are then projected over the search space if they are out

of bounds.

3. Selection: Fitness of ( 1)k +i

c is calculated. The selection procedure will make this individual to


29

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 exist to

analyse how to tune the differential evolution parameters. Usually it is recommended 1 2 0.5F F= = and

a crossover probability 0.8r

C = . Also the scheme “target to the best” is usually used for mutation.

Figure 19 shows the logarithmic misfit for different benchmark functions as a function of

1 2F F F= = and .r

C In these simulations we have adopted target to the best as mutation scheme. It is

possible to observe that the good points ar almost parallel to the .r

C 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 (0.1 to 0.1).

Figure 19: Logarithmic misfit error as a function of the mutation scale factor and

crossover parameter for different type of benchmark functions in 50 dimensions.


30

Figure 20: Stanford VI-Case I. A) Convergence error curves for different DE mutation

mechanisms compared to CP-PSO with variable time step. B) Median distance between

the particles of the swarm as a function of the iterations.


31

Figure 20 shows for Stanford VI-Case I. the convergence error curves for different DE mutation mechanisms compared to CP-PSO with variable time step. The rate is very similar in both cases for the mechanisms that include the global best in the mutation. The random mutation mechanisms show a slightly higher misfit, as expected. The evolution of the median distance between the particles of the swarm as a function of the iterations shows that the random mechanisms are much explorative and thus better suited for uncertainty estimation as long as there are enough samples on the low misfit region.

Finally 21 shows the solution and the uncertainty estimation for the Stanford VI reservoir-case I with 5% of Gaussian noise, using the rand-1 mechanism (blue curve in figure 20). These results have to be compared to those shown in figure 8. The uncertainty measures (iqr) explain also quite the deviations observed from the true model. Similar results can be achieved using the other mutation mechanisms.

Figure 21: Stanford VI reservoir-case I with 5% of Gaussian noise. Horizontal slices from

the synthetic reservoir showing the inverted solution, true model, and uncertainty on the

model parameters (inter-quartile range) calculated with samples gathered on the low-

misfit region.


32

8. Conclusions

We have shown, using a synthetic case, that the joint use of particle swarm algorithms and model reduction techniques such as principal component analysis allows solving stochastically the seismic history matching problem in reservoir optimization. The space of solutions is partially constrained by the use of time lapse seismic, cross-well tomography in this case. We have analyzed two different reservoir model subsets extracted from the synthetic Stanford VI reservoir. We have shown that the misfit function has a valley shape and sometimes the global minimum is close to ranges of high misfits making the search harder. We have analyzed the performance of different members of the PSO family. Although all of them can be used to perform this task the CP-PSO version with variable time step seems to be the best, both in terms of convergence rate and exploration capabilities. Using the samples gathered in each case from the low misfit region, we use the inter quartile range as a measure of uncertainty for facies prediction. This feature provides robustness to the PSO algorithms in presence of different levels of Gaussian noise. The uncertainty analysis, although approximate, sheds light to establish probability maps of facies. Finally we present some preliminary results on differential evolution showing that it is possible to achieve similar conclusions.

9. Acknowledgments

Authors acknowledge the financial support of the Smart Fields Consortium and Stanford Center for Reservoir Forecasting (Energy Resources Department, Stanford University). This work also benefited from a one-year sabbatical grant (2008-2009) at the University of California Berkeley (Department of Civil and Environmental Engineering) given by the University of Oviedo (Spain) and by the “Secretaría de Estado de Universidades y de Investigación” of the Spanish Ministry of Science and Innovation. We also acknowledge Eduardo Santos (Stanford University) for the diffraction tomography program used to solve the seismic forward problem, and particularly David Echeverría (Stanford University, Smart Field Consortium) for allowing us to use the forward simulation models to match the Stanford VI synthetic reservoir using the PSO optimizers, and for plenty of fruitful and passionate discussions that helped us to improve the quality of this manuscript. Finally we acknowledge the Center for Computational Earth and Environmental Science at Stanford University for providing the computational resources.

10. References

Aanonsen, S., Aavatsmark, I., Barkve, T., Cominelli, A., Gonard, R., Gosselin, O., Kolasinski, M. and Reme, H., 2003. Effect of Scale Dependent Data Correlations in an Integrated History Matching Loop Combining Production Data and 4D Seismic Data. In: Proceedings SPE Reservoir Simulation Symposium, Houston, TX. Doi: 10.2118/79665-MS.

Aanonsen, S., and Eydinov, D., 2006., A multiscale method for distributed parameter estimation with application to reservoir history matching., Computational Geosciences 10, 97-117.


33

Aziz, K. and Settari, A., 1979. Petroleum Reservoir Simulation, Applied Science Publishers, 476pp.

van den Bergh, F., and Engelbrecht, A., 2006. A study of particle swarm optimization particle trajectories. Information Sciences 176, 937-971.

Carlisle, A., and Dozier, G., 2001. An off-the-shelf PSO. In proceedings of The Workshop On particle Swarm Optimization, Indianapolis, IN, 2001, 1-6.

Castro, S.A. 2007, A probabilistic approach to jointly integrate 3D/4D seismic, production data and geological information for building reservoir models, Ph.D. Thesis, Stanford University.

Castro, S., Caers, J. and Mukerji, T., 2005. The Stanford VI Reservoir. 18th Annual Report. Stanford Center for Reservoir Forecasting (SCRF). Stanford University.

Clerc, M., and Kennedy, J., 2002. The particle swarm--explosion, stability, and convergence in a multidimensional complex space, IEEE Transactions on Evolutionary Computation 6, 58-73.

Curtis, A., and Lomax, A., 2001. Prior information, sampling distributions and the curse of dimensionality. Geophysics 66, 372-378.

Dadashpour, M., Kleppe, J., and Landrø, M., 2007. Porosity and Permeability Estimation by Gradient-Based History Matching using Time- Lapse Seismic Data, SPE 104519, 15th SPE Middle East Oil & Gas Show and Conference, Bahrain, March 11-14.

Dadashpour, M., Echeverría-Ciaurri, D., Kleppe, J., and Landrø, M., 2009. Porosity and permeability estimation by integration of production and time-lapse near and far offset seismic data. Journal of Geophysics and Engineering 6, 325-344.

Echeverría, D., and Mukerji, T., 2009. A robust scheme for spatio-temporal inverse modeling of oil reservoirs. In Anderssen, R.S., R.D. Braddock and L.T.H. Newham (eds) 18th World IMACS Congress and MODSIM09 International Congress on Modelling and Simulation. Modelling and Simulation Society of Australia and New Zealand and International Association for Mathematics and Computers in Simulation, Cairns, Australia, July 2009, pp. 4206-4212 www.mssanz.org.au/modsim09/J1/echeverria.pdf

Echeverría, D., Mukerji, T., and Santos, E. T. F., 2009. Robust scheme for inversion of seismic and production data for reservoir facies modeling, SEG Expanded Abstracts 28, 2432-2436.

Fernández-Álvarez, J. P., Fernández-Martínez, J. L., and Menéndez-Pérez, C. O., 2008. Feasibility analysis of the use of binary genetic algorithms as importance samplers application to a geoelectrical VES inverse problem: Mathematical Geosciences 40, 375-408.

Fernández-Martínez, J. L., García-Gonzalo, E., and Fernández-Alvarez, J. P., 2008a. Theoretical analysis of Particle Swarm trajectories through a mechanical analogy: International Journal of Computational Intelligence Research 4, 93-104.


34

Fernández-Martínez, J. L., Fernández-Álvarez, J. P., García-Gonzalo, E., Menéndez-Pérez, C. O., and Kuzma, H. A., 2008b. Particle Swarm Optimization (PSO): a simple and powerful algorithm family for geophysical inversion. SEG Expanded Abstracts 27, 3568-3571.

Fernández-Martínez, J. L., and García-Gonzalo, E., 2008. The generalized PSO: a new door to PSO evolution. Journal of Artificial Evolution and Applications, 2008, 861275, 15pp. doi:10.1155/2008/861275

Fernández-Martínez, J. L., and García-Gonzalo, E., 2009. The PSO family: deduction, stochastic analysis and comparison. Swarm Intelligence 3, 245-273.

Fernández-Martínez, J. L., and García-Gonzalo, E., 2009b. Stochastic stability analysis of the linear continuous and discrete PSO models. Research report. University of Oviedo, Spain. Submitted to the Transactions on Evolutionary Computing, IEEE.

Fernández-Martínez, J. L., and García-Gonzalo, E., 2010. What makes particle swarm optimization a very interesting and powerful algorithm. Handbook of Swarm Intelligence – Concepts, Principles and Applications Series on Adaptation, Learning, and Optimization. Springer. To appear.

Fernández-Martínez, J. L., H. Kuzma, E. García-Gonzalo, J. M. Fernández-Díaz, J. P. Fernández- Álvarez, and Menéndez-Pérez, C. O., 2009. Application of Global Optimization Algorithms to a Salt Water Intrusion Problem. Symposium on the Application of Geophysics to Engineering and Environmental Problems, SAGEEP 22, 252-260.

Fernández-Martínez, J. L., García-Gonzalo, E., Álvarez, J. P. F., Kuzma, H. A. and Menéndez-Pérez, C. O, 2010a: PSO: A Powerful Algorithm to Solve Geophysical Inverse Problems. Application to a 1D-DC Resistivity Case. Journal of Applied Geophysics. In Press. doi:10.1016/j.jappgeo.2010.02.001

Fernández-Martínez, J. L., García-Gonzalo, E. and Naudet, V. 2010b. Particle Swarm Optimization applied to the solving and appraisal of the Streaming Potential inverse problem. Geophysics. Hydrogeophysics Special Issue. Accepted for publication.

Fernández-Martínez, J. L., García-Gonzalo, E., and Mukerji T., 2010c. How to design a powerful family of Particle Swarm Optimizers for inverse modelling. Transactions of the Institute of Measurement and Control. New trends in bio-inspired computation. To be published.

Fernández-Martínez, J L, Fernández-Muñiz, Z., Tompkins, M., and Mukerji, T , 2010d. The topography of the cost function in inverse problems. Research Report. Stanford Center for Reservoir Forecasting. Stanford University.

García-Gonzalo, E., and Fernández Martínez, J. L., 2009. Design of a simple and powerful Particle Swarm optimizer: Proceedings of the International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE2009, Gijón, Spain.


35

Goldberg, D.E., 1989. Genetic algorithms in search, optimization and machine learning. Addison-Wesley, 412pp

Hansen, C., 1998, Rank deficient and discrete ill-posed problems. Numerical aspects of linear inversion. Society for Industrial Mathematics (SIAM), 147pp.

Holland, J. H., 1992, Adaptation in natural and artificial systems, MIT Press, 228pp.

Huang, X., Meister, L. and Workman, R., 1997. Reservoir Characterization by Integration of Time-Lapse Seismic and Production Data. In: Proceedings, SPE Annual Technical Conference and Exhibition, San Antonio, TX. doi: 10.2118/38695-MS

Jolliffe, I. T., 2002, Principal Component Analysis, Springer-Verlag, 487pp.

Kennedy, J. and Eberhart, R.C., 1995. Particle swarm optimization. In: Proceedings IEEE International Conference on Neural Networks 4, 1942-1948.

Kirkpatrick, S., Gelatt, C. D., and Vecchi, M. P., 1983, Optimization by Simulated Annealing. Science 220, 671-680.

Landa, J.L. 1997. Reservoir parameter estimation constrained to pressure transients, performance history and distributed saturation data, Ph.D. dissertation, Stanford University.

Ma, X.-Q, 2002, Simultaneous inversion of prestack seismic data for rock properties using simulated annealing. Geophysics 67, 1877-1885.

Mavko, G., Mukerji, T. and Dvorkin, J., 2003. The Rock Physics Handbook, Cambridge, 340pp.

Mosegaard, K., and Tarantola, A., 1995. Monte Carlo sampling of solutions to inverse problems. Journal of Geophysical Research 100, 12431–12447.

Naudet, V., Fernández-Martínez, J. L., García-Gonzalo, E. and Álvarez-Fernández, J. P.,2008. Estimation of water table from self-potential data using particle swarm optimization (PSO). SEG Expanded Abstracts 27, 1203-1207.

Oliver, D. S., Reynolds, A. C., and Liu, N., 2008, Inverse Theory for Petroleum Reservoir Characterization and History Matching, Cambridge University Press, 308pp.

Onwunalu, J.E., and Durlofsky, L.J., 2009. Application of a particle swarm optimization algorithm for determining optimum well location and type. Computational Geosciences 14, 183-198.

Ozcan, E. and Mohan, C. K.,1999, Particle swarm optimization: surfing the waves. In: Proceedings of the IEEE Congress on Evolutionary Computation (CEC '99), Washington DC 3, 1939-1944.

Pearson, K., 1901. Principal components analysis. The London, Edinburgh and Dublin Philosophical Magazine and Journal 6, 566.


36

Poli, R., 2008a. Dynamics and Stability of the Sampling Distribution of Particle Swarm Optimisers via Moment Analysis. Journal of Artificial Evolution and Applications, 2008, 761459, 10 pp. doi: 10.1155/2008/761459

Poli, R., 2008b Analysis of the Publications on the Applications of Particle Swarm Optimisation,.Journal of Artificial Evolution and Applications, 2008, 685175, 10 pp. doi: 10.1155/2008/685175

Remy, N., Boucher, A., and Wu, J., 2009. Applied Geostatistics with SGeMS: A User's Guide. Cambridge University Press, 284 pp.

Romary, T., 2009a. Integrating production data under uncertainty by parallel interacting Markov chains on a reduced dimensional space, Computational Geosciences, 13, 103-122.

Romary, T., 2009b. History matching of approximated lithofacies models under uncertainty, Computational Geosciences. 14, 343-355.

Sambridge, M., 1999, Geophysical Inversion with a Neighbourhood Algorithm -II. Appraising the ensemble. Geophysical Journal International 138, 727-746.

Sambridge, M., and Drijkoningen, G., 1992. Genetic algorithms in seismic waveform inversion. Geophysical Journal International 109, 323-342.

Sarma, P., Durlofsky, L.J., Aziz, K. and Chen, W.H., 2006. Efficient real-time reservoir management using adjoint-based optimal control and model updating, Computational Geosciences 10, 3–36.

Scales, J.A, and Gersztenkorn, A., 1988. Robust methods in inverse theory. Inverse problems 4, 1071-1091.

Scales, J. and Tenorio, L., 2001. Prior information and uncertainty in inverse problems. Geophysics 66, 389-397.

Sen, M. K., and Stoffa, P. L., 1991. Nonlinear one-dimensional seismic waveform inversion using simulated annealing. Geophysics 56, 1624-1638.

Sen, M., and Stoffa, P. L., 1995, Global optimization methods in geophysical inversion, Elsevier, 281pp.

Sena, A., Sen, M., Stoffa, P., Seif, R., and Jin L., 2009a. Joint inversion of time-lapse seismic and production data using VFSA with Local Thermal Regulation and pilot point parameterization. SEG Expanded Abstracts 28, 1810-1814.

Sena, A., Stoffa, P., Sen, M., and Seif, R., 2009b. Assessing the value of time-lapse seismic data in joint inversion for reservoir parameter estimation in an oil reservoir subjected to water flooding recovery: a synthetic example. SEG Expanded Abstracts 28, 3904-3908.

Shaw, R., and Srivastava, S., 2007. Particle swarm optimization: A new tool to invert geophysical data. Geophysics 72, F75-F83.


37

Stoffa, P. L., and Sen, M. K., 1991. Nonlinear multiparameter optimization using genetic algorithms: Inversion of plane-wave seismograms. Geophysics 56, 1794-1810.

Storn, R. and Price, K., 1997. Differential Evolution - A Simple and Efficient Heuristic for global Optimization over Continuous Spaces. Journal of Global Optimization 11, 341-359.

Strebelle, S., 2002. Conditional simulation of complex geological structures using multi-point statistics, Mathematical Geology 34, 1–21.

Trelea, I., 2003. The particle swarm optimization algorithm: convergence analysis and parameter selection. Information Processing Letters 85, 317--325.

Tureyen, O. and Caers, J., 2005. A parallel, multiscale approach to reservoir modeling'. Computational Geosciences 9, 75-98.

Xia, O. and Huang, X., 2009. Reservoir Model Updating in the History Matching Process Constrained by 3D Seismic Data. SEG Expanded Abstracts 28, 1865-1869.

Yuan, S., Wang, S., and Tian, N., 2009. Swarm intelligence optimization and its application in geophysical data inversion. Applied Geophysics 6, 166-174.

Zheng, Y.-L., Ma, L.-H., Zhang, L.-Y. and Qian, J.-X., 2003. On the convergence analysis and parameter selection in particle swarm optimization. In: International Conference on Machine Learning and Cybernetics (ICMLC '03) 3, 1802-1807.

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