Copyright 2007, Society of Petroleum Engineers This paper was prepared for presentation at the 15th SPE Middle East Oil & Gas Show and Conference held in Bahrain International Exhibition Centre, Kingdom of Bahrain, 11–14 March 2007. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

Abstract A method based on the Gauss-Newton optimization technique for continuous model updating with respect to 4D seismic data is presented. The study uses a commercial finite difference black oil reservoir simulator and a standard rock physics model to predict seismic amplitudes as a function of porosity and permeabilities. The main objective of the study is to test the feasibility of using 4D seismic data as input to reservoir parameter estimation problems.

The algorithm written for this study, which was initially developed for the estimation of saturation and pressure changes from time-lapse seismic data, consists of three parts: the reservoir simulator, the rock physics petro-elastic model, and the optimization algorithm. The time-lapse seismic data are used for observation purposes. In our example, a simulation model generated the seismic data, then the model was modified after this the algorithm was used to fit the data generated in the previous step.

History matching of reservoir behavior is difficult because of the problem is not unique. More than one solution exists that matches the available data. Therefore, empirical knowledge about rock types from laboratory measurements are used to constraint the inversion process.

The Gauss-Newton inversion reduces the misfit between observed and calculated time-lapse seismic amplitudes. With this method, it is possible to estimate porosity and permeability distributions from time-lapse data. Since these parameters are estimated for every single grid cell in the reservoir model, the number of model parameters is high, and therefore the problem will be underdetermined. Therefore, a good fit with the observation data is not necessary for a good estimation of the unknown reservoir properties. The methods for reducing the number of unknown parameters and the associated uncertainties is discussed.

Introduction Parameter estimation using new geophysical knowledge like time-lapse seismic data is a new and probably underdeveloped method so far. The main challenges are to develop a method for estimation of key reservoir parameters with the lowest possible estimation error. Parameter estimation is an iterative trial-and-error process that estimates uncertain parameters by perturbing these parameters in order to make the model fit observed data. History matching is difficult since it requires a lot of experience and knowledge of the field. In addition, the process is inherently non-unique. History matching is not new, and various optimization techniques have been suggested1-5. This process is non-unique and this is the main reason for using time-lapse information in addition to other information to limit the solution space. Time-lapse seismic data are time dependent dynamic measurements6-10, aiming at determining the reservoir changes that have occurred in the intervening time. Most of the reservoir parameters, which are used in reservoir simulators, come from laboratory measurements that are not representative for the entire reservoir and therefore there is a need for correlation. The main advantage of using time-lapse seismic data is that they do not need any correlation, which means they are representative for the entire reservoir. However, the results are associated with errors and uncertainties, which are related to the repeatability of data acquisition, data processing sequences, lack of understanding of rock physics and error in up-scaling and cross-scaling seismic and simulation data. Time-lapse seismic technology was first introduced in the early 1980s and many works have been published since that time in this area 11-17. Porosity and permeability are two of the most important parameters in each reservoir simulation model and they have the largest impact on reserves, production forecasts and economics of the reservoir. These two parameters are most difficult to estimate. The main reasons for these difficulties in the estimation of permeability and porosity are 18:

• Spatial variability of permeability and porosity • Very few sampling locations compared to the extent

of the reservoir. • Using different technologies for obtaining

measurements • Complexity of the mathematical model of the

reservoir

Some previous examples estimating of these parameters

SPE 104519

Porosity and Permeability Estimation by Gradient-Based History Matching using Time-Lapse Seismic Data M. Dadashpour and J. Kleppe, SPE, NTNU; M. Landrø, NTNU

2 SPE 104519

are discussed by Jacquard, and Jain20, Chu et al.21, and Oliver22. Landa et al.18 integrates well-test data, reservoir performance history and time-lapse seismic information for this purpose. Vasco et al.19 introduced a new trajectory-based approach for time-lapse reservoir characterization.

The purpose of this work is to develop an efficient procedure for the estimation of porosity and permeability distributions in the reservoir by using time-lapse seismic data. This work is an extension of our previous work about the estimation of saturation and pressure changes using time-lapse seismic data23. In this paper, we present a nonlinear inversion based on the Gauss-Newton optimization technique. This procedure is used for joint estimation of permeability and porosity distributions during depletion and water injection of a 2D synthetic case, which is generated by using field data from a reservoir in the Haltenbanken area offshore Norway. Reservoir Flow Simulation The first step in forward modeling is flow simulation. The simulation obtains fluid saturations and pore pressures in the entire reservoir. An efficient simulator is critical for the entire process since it must be repeated many times.

As Romeu et al.24 mentioned, each reservoir flow modeling consists of two complementary components: a functional model and representation model. A set of differential equations and numerical methods for solving these equations are known as the functional model. The representation model mathematically describes a particular reservoir by spatial variable coefficients, external boundary conditions and initial conditions that complete the formulation.

The study uses a commercial finite difference black oil reservoir simulator (Eclipse 100) for this purpose. It gets some rock and fluid properties such as porosities and permeabilities as input and calculates fluid saturations and pore pressures for each cell at desired time steps.

Petro-Elastic Model (PEM) A petro-elastic model is a set of several equations, which relates some reservoir properties such as pore space, pore fluid, fluid saturation, reservoir pressures, and rock composition to seismic elastic parameters such as P- wave and S-wave velocities, seismic acoustic impedance, poisson’s ratio or even seismic amplitudes. A PEM can be used in both inversion and forward seismic modeling and as Falcone et al.25 mention it can be used for seismic modeling, interpretation of seismic data in term of lithology, and history matching.

Time-lapse seismic is defined as repeated 2D or 3D seismic data and these data are a set of dynamic data in the same reservoir under the same condition in different time. Seismic amplitudes are function of variation in the source and acoustic properties of the reservoir. With the assumption of no variation in the source, amplitudes are only function of reservoir acoustic properties in the 4D seismic analysis. Variations in acoustic properties are functions of temperature, compaction, fluid saturation, and reservoir pressure. The effect of temperature and compaction are neglected in this study. The Gassmann equation26 and the Hertz Mindlin27 model are used for estimating seismic parameter changes caused by fluid saturation and reservoir pressure changes respectively.

Conversion of reservoir properties to seismic amplitudes is done in two steps. First reservoir parameters such as pressure or saturation are converted to seismic elastic properties such as P-wave and S-wave velocities by using the petrophysical model (rock-physic relationship) Then, seismic amplitudes are calculated from them based on the matrix propagation method28 (Appendices A and B).

Computer aided history matching Calibration or conditioning of the reservoir simulation models to historical production or survey data is called history matching and it is simply done by reducing the misfit between historical and simulated data. Usually, history matching requires numerous iteration runs that make this procedure very costly. History matching is not only difficult but it is also by its nature non-unique.

One useful application of the history matching process is parameter estimation or determination of reservoir properties. Every method that is in the category of parameter estimation problems consists of three major parts (Appendix C):

• Mathematical Model • Objective Function • Minimization Algorithm

Synthetic Test Case In order to test the efficiency and accuracy of the presented optimization technique a complex, synthetic reservoir has been set up to test the method described in this project. This model was a two dimensional inspired model from a realistic field data from a reservoir in the Haltenbanken area. Some changes have been made to create suitable conditions for a realistic history matching process. Reservoir model description. This reservoir model is subdivided into four different formations from top to base, which correspond to formations A, B, C and D. Hydrocarbons in this reservoir are located in the Lower- to Middle-Jurassic sandstones. Different geological and environments during deposition of sands made nine different rock types with different properties in this reservoir (Figures 1-4).

The reservoir model is two phase (oil and water) two dimensional black oil model with 39*1*26 grid cells containing 733 active cells. Four different faults subdivided this reservoir into four different parts. Two synthetic wells are defined for injection and production purposes. The production well is located in grid block number 6 and perforated in blocks 1-12 and the water injection well is located in block number 36 and perforated in blocks 1-26. Complete connections are chosen between the segments on both sides of the faults. Simulation is done for 20 years (Figure 5).

4D seismic and observed inversion data. 4D seismic or time-lapse data are a new set of time dependent dynamical measurements which are added to inversion process for improving the problem of non-uniqueness of the inversion algorithm. Time-lapse amplitudes vs. time are essential input in the project to feed the inversion loop.

SPE 104519 3

For the real case, these data come directly from the field after some seismic data processing procedures, but for this case, there is no real seismic data available. This project uses 2D synthetic case based on real data because the main objective of this project is to check the accuracy and precision of this method in the estimation of the reservoir parameters. As observation data, synthetic time-lapse seismic data are created based on forward seismic modeling equations. Figure 6 shows the time-lapse amplitudes that are used as observation data in this work. Constraints. Limiting the results is done for validation of estimated porosity and permeabilities in the cells. This is because these data should be accepted as geological concepts from the reservoir point of view. These limits are known as constraints in the inversion process. Two other constraints are used in this work which divided the work into two different cases. Case A. Simple constraint is defined for this case:

• Porosities should be between 15 % and 40%. Because of typical porosity for typical sandstone reservoirs

40.015.0 ≤≤ ϕ …………...……………......…………... (1)

• Permeability should be higher than 0 md and less

than 2000 md.

20000 ≤≤ K ………………………….….………….. (2)

CaseB. In addition to the constraints used in Case A, some additional constraints are used for limiting the solution space. These constraints are based on empirical knowledge about rock types from laboratory measurements which simply shows correlation between porosity and permeability in core measurements. And based on this knowledge nine different rock types are defined from different geological environments during the deposition of sands. (Figures 1-4)

Sensitivity of time-lapse amplitude change to saturation and pressure changes. The effect of saturation and pore pressure variations in time-lapse amplitudes is very critical in the parameter estimation problem. The quantity of estimation is increased with rising sensitivity then, it is crucial to carry out sensitivity analysis before the estimation of any parameters using optimization theory.

Based on our previous work23, the amplitude responses in each column are functions of saturation, pore pressure and porosity. This is based on the assumption that no compaction in the reservoir, effect of porosity is negligible in this case. Sensitivity of water saturations and pressures is found in the sensitivity of the Jacobian matrix. Each column in this matrix represents the derivative of amplitude with respect to one parameter.

Figures 6 and 7 respectively show the amplitude sensitivities with respect to water saturation and pore pressures in each cell in this column. It is clear that seismic amplitudes are more sensitive to saturation changes than pressure changes (at least in this case) and for saturation and

pressure perturbations from 1% to 5%, the overall behavior remains the same. Sensitivity of time-lapse amplitude change to porosity and permeability changes. The study of the effect of reservoir parameter changes such as porosities and permiabilities in seismic amplitudes is vital in the parameter estimation problem. That is because of the dependency of seismic amplitude changes to these parameters. The numerical results of sensitivity analysis are found in following steps:

• Perturbing reservoir parameters (porosity and permeability)

• Run reservoir simulation to compute saturation and pressure at two different times (based and monitor seismic surveys)

• Calculate the amplitudes of a reflection at these two different times by using petro-elastic model (PEM)

• Compute the amplitude changes by subtracting the results of thetwo computations

• The ratio of the change in the differential amplitude to the change in porosity and permeabilities provides the numerical sensitivity of amplitudes

Figures 8 and 9 respectively show the amplitude

sensitivities with respect to porosities and permabilities in each cell in this column. It is clear that seismic amplitudes are more sensitive to porosity than permeability (at least in this case). Porosity and permeability perturbations from 1% to 10% have a significant effect on sensitivity especially for permeability. It will decrease with the increasing perturbation rate from 1% to 10%.

Data Match. Figure 10 shows the amount of mismatch between real and calculated time-lapse seismic data for Cases A and B. In this figure the solid line represents the amount of mismatch between real and calculated time-lapse seismic data in each iteration for Case A and the dashed line represents this error for Case B. This figure shows that in both cases this algorithm reduces the mismatch.

Figures 11 and 12 show initial and final objective functions for Case B. These figures represent time-lapse amplitude differences between observed and calculated data for the initial and final iteration, respectively. NRMS (Normalized RMS - Appendix D) is reduced from 77% to 14%. This algorithm can properly reduce the mismatch between the time-lapse seismic data for the model and the real reservoir.

With this method it is possible to estimate porosity and permeability distributions from time-lapse seismic data if we limit the solution space with geological constraints in addition to simple constraints. Since these parameters are estimated for every single grid cell in the reservoir model, the number of model parameters is high, and therefore the problem will be underdetermined. Therefore, a good fit with the observation data is not necessary for a good estimation of the unknown reservoir properties.

4 SPE 104519

For an accuracy check and precision of the estimation of reservoir properties with this method and to check the effect of constraint in the algorithm, two different cases with differents definition of constraints are created.

For porosity we think that volumetric weighted averaging method, gives us a reasonable estimate for the error in the estimated porosities. However since the permeability is a flow parameters, the choice of various averaging method is not straight forward and obvious. We tested three averaging methods for error estimation, volumetric weighted, harmonic, and weighted root mean square (RMS) (Appendix D).

Figures 13 and 17 show amount of average error in the estimation of permeability and porosity for both cases. The solid line in these figures represents amount of average error in the estimation of reservoir parameters in each iteration in Case A and dashed line represents this error in Case B.

Volumetric and normalized root mean square (NRMS) averaging method seems more stable than harmonic average. The NRMS averaging method is defined as the root mean square (RMS) of the difference between two datasets divided by the average of the root mean square (RMS) of each dataset39. It shows how much are datasets identical. It can be the best method error controlling method in parameter estimation problem with many independent parameters.

Based on these figures it is obvious that Case A gave erroneous estimates. Although it reduced the objective function the estimated parameters from this method are far from the real answer even it could not improve this estimation with respect to the initial case. The average error in the estimation of porosity increases from 3.35% to 3.40% (NRMS error from 15.31% to 17.42%) and the reduction of error in permeability estimation is not considerable from 655.3 md to 501.2 md (NRMS error from 85.91% to 79.42 %).

Limiting the solution space with some extra information about the rock types from laboratory measurements increased the accuracy in the estimation of reservoir parameters. Figures 13 - 16 show this fact. Based on these figures, the average error in the estimation of porosity decreases from 3.35% to 1.64% (NRMS error from 15.31% to 8.87%). Moreover, the average error in permeability estimation decreases from 655.3 md to 358.7 md (NRMS error from 85.91% to 66.82%). Initial, real and calculated porosity and permeability distributions are presented in Figures 18 – 23 and Figures 24 - 27 show the amount of actual errors in the estimation of porosity and permeability distributions in the first and last iterations in Case B. Obviously, this is not a perfect answer but at least it could improve the answer with respect to the initial case. The main reason for getting an imperfect answer is the high number of unknowns and the high degree of nonlinearity between parameters especially between porosities and permeabilities. This made this problem underdetermined. It seems that adding extra information in the algorithm for limiting the solution space has a positive effect in increasing the accuracy of the estimation process. Conclusions A systematic approach for using time-lapse seismic data to estimate reservoir parameters such as porosity and permeability was tested. The approach uses a standard Gauss-Newton optimization technique for reducing the mismatch

between modeled and measured 4D seismic data. Analysis of this algorithm for a 2D synthetic reservoir, based on field data from a complex reservoir in the Haltenbanken area offshore Norway indicates that this algorithm has good capability for matching observed and calculated data. With this method it is possible to estimate porosity and permeability distributions from time-lapse data and it enables us to improve the estimation of these parameters with respect to the initial guess. However, this study shows that additional constraints are needed to stabilize the estimates. For instance correlation between porosity and permeability which comes from laboratory measurements could subdivide the reservoir into different rock types. Seismic amplitude changes seem more sensitive to porosity changes than permeability. Porosity and permeability perturbation from 1% to 10% has a significant effect on sensitivity especially for permeability.

Since these parameters are estimated for every single grid cell in the reservoir model, the number of model parameters is high, and therefore the problem will be underdetermined. Therefore, a good fit with the observation data is not necessary for a good estimation of the unknown reservoir properties. Major weaknesses of this algorithm are:

• The numbers of simulation runs are equal to the amount of unknown parameters.

• It is a very time-consuming procedure. • Strong dependency on the initial model • Considerable requirement for the constraint to limit

the solution space. Future steps for this methodology are using AVO gradient together with actual amplitude changes as observation data. With this improvement we hope that we can improve the estimation of reservoir properties with respect to the common method. Acknowledgment The authors want to thank the Norwegian University of Science and Technology (NTNU) for financial support. We also want to thank Alexey Stovas for preparing the seismic forward modeling used in this project.

References

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SPE 104519 5

History Matching”, paper SPE 90058, presented at 2004 SPE Annual Technical Conference and Exhibition, Houston, Texas, September 26-29.

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Appendices Appendix A. Gassmann Equation. Seismic velocities in a porous medium saturated with water are dependent on two constants, namely the bulk modulus K and the shear modulus µ. Navier’s equation is used to calculate pressure vp and shear wave vs velocities from these two.

ρμ

ρ

μ

=

+=

s

p

V

kV 3

4 …...………………………………………….. (3)

where ρ is bulk density and is volume average density of solid and liquid phase, which is calculated from following equation:

mafB ρφφρρ )1( −+= ………………………………………..(4) where, ρf and ρma are the fluid and matrix densities respectively. The downscaling steps allow the computing fluid density:

ggwwoof SSS ρρρρ ++= ……….………………………….. (5)

where, ρ, P, and S are phase density, pressure, and saturation respectively. Calculation of bulk modulus is done by the well-known theory proposed by Gassmann. This equation expresses bulk modulus as a function of frame and continuous properties.

( )

⎟⎟⎠

⎞⎜⎜⎝

⎛−+−

−+=

ma

fr

f

mama

frmafr

kk

kk

k

kkkk

φφ1

2 ……………………………... (6)

where, k is rock bulk modulus of saturated medium, kfr is bulk modulus of solid framework, μ is shear modulus of solid framework, kma is bulk modulus for solid when assuming zero porosity for the rock, φ is effective porosity of the medium, and kf is bulk modulus of saturating fluid (water, gas or oil) which was computed as follows:

w

w

g

g

o

o

f kS

kS

kS

k++=

1 …………………………………...………(7)

where kw, ko and kg are the moduli for water, oil, and gas respectively. Gassman’s theory assumes that the rock is homogeneous and isotopic and that all pore space is connected. Some other assumptions regarding Mavco et al.29 are:

• Pore fluid is firmly coupled to the pore wall • Gas and liquid are uniformly distributed in the pores • Shear modulus is not affected by the pore fluid • Pore shapes are spherical

However, there is no assumption about the pore geometry. In addition, it is only valid sufficiently in low frequency when pore pressures are equilibrated thought the pore space, and when fluid bearing rock is completely saturated. Hertz mindlin. The main effective stress and pore pressure in the porurs medium is explained by the Hertz-Mindlin model. The effective bulk modulus and shear modulus of a dry random identical sphere packing are given by:

n effeff

n effeff

pc

pck

22

222

22

222

)1(2

)1(3)2(5

45

)1(18

)1(

νπ

μφννμ

νπ

μφ

−

−

−−

=

−

−=

……….……...…………. (8)

where v and μ are the Poisson’s ratio and shear modulus of the solid grains, respectively, c = 9 is the average number of contact per grain and the effective pressure, Peff, used in Hertz-Mindlin theory is taken as the difference between the lithostatic and hydrostatic pressure.

PPP exteff η−= ………………….…………………………… (9) where, η is coefficient of internal deformation. It is common to assume its value is close to one.

In original Hertz-Mindlin theory the degree of the root in Equation 8 was 3. Vidal et al.30 found n=5.6 for the P-wave and n=3.8 for the S-wave in gas sands, while Landrø 31 used n=5 for oil sands. This correction is used for developing forward modeling in this study (n=5 instead n=3).

This method has some limitations, for instance cannot compute seismic parameters properly in very small amounts of effective pressure. Appendix B. Getting seismic amplitude from seismic velocities. Acoustic impedances are defined as the product of bulk density and related velocities.

sBs

pBp

VI

VI

ρ

ρ

=

= …………..………………………..………….. (10)

where, Ip and Is are compressional and shear impedances respectively.

The seismic reflection coefficient at one interface between two layers is defined as:

12

12IIIIR

+

−= …………..………………………………….….. (11)

where I2 is seismic impedance of the layer below the interface and I1 is seismic impedance of the layer above the interface.

In an isotopic medium, the S-wave and P-wave velocities are independent from the direction along the medium. For computing the transmission and reflection response from a stack of plane layers we use the propagator-matrix method32,33

which is modified by Stovas and Arntsen28 for quasi-vertical wave propogation.

SPE 104519 7

Appendix C. Mathematical Model. Every parameter estimation problem needs to compute the reservoir response numerically, and this is done by forward modeling, which as already mentioned consists of fluid simulator and petro-elastic modeling.

The fluid simulator consists of a set of differential equations and numerical methods for solving these equations accompanied by some spatial variable coefficients, external boundary conditions and initial conditions that complete the formulation. The synthetic pressure, production history and fluid saturations are obtained from this simulator. Objective Function. The amount of discrepancy between observation data such as seismic survey production and pressure historical data and the simulator response for a given set of parameters is known as the objective function.

The main objective of every parameter estimation problem is to minimize the function F, which is a function of the permeability and porosity in this study. Many definitions are available for the objective function and the application of each of them depends on the type of available information.

Two of the most common formulas for objective functions are known as the weighted least square (Equation 12) and the generalized least square34,35 (Equation 13). Based on the type of information available the weighted least square formula is used in this study.

( ) ( )calcobsTcalobs ddddF −−= ………………..……….....….(12)

( ) ( ) ( ) ( )⎥⎦⎤

⎢⎣⎡ −−+−−= −− calcobsTcalobscalcobs

dTcalobs ddCddddCddF 11

21

α... (13)

where, w is a diagonal matrix that assigns individual weights to each measurement and Cd is the covariance matrix of the data, Cd provides information about the correlation among the data. Cα is the covariance matrix of the parameters of the mathematical model. Minimization Algorithm. One of the characteristics of reservoir parameter estimation is the nonlinearity of the objective function and the best way for solving these kinds of problems is using an iterative algorithm that will calculate the unknown parameters by successive approximations. The Initial guess of body parameters is a starting point for this algorithm and the process is iteratively advanced until the best fit is obtained between calculated and observed data. The common algorithm for inversion process is:

• Input initial guess of body parameters • Compute the response of the system through forward

modeling • Compute the objective function • Update parameters by minimizing the objective

function by using some minimization algorithms • If the value of objective function is not certifiable

return to step 3

Several optimization algorithms are developed for this purpose. Some of these algorithms such as Gauss-Newton, Singular Value Decomposition, Conjugate Gradient, Steepest

Descent6,36 require both or at least one of the first (Jacobian) or second (Hessian) derivatives of time dependent the reservoir properties with respect to static reservoir properties. However, some of them such as the genetic algorithm, response surfaces, experimental design methods and Monte Carlo simulation37 do not need to compute the gradient for optimization purpose. The gradient of objective function is defined as:

TFF ⎟⎠⎞

⎜⎝⎛∂∂

=∇α

……...………………………………………. (14)

This is a nonlinear Gauss-Newton optimization technique, which is classified as a gradient method to determine target parameters in this study23. Because of the particular form of the objective function the calculation of the mathematical model gradient with respect to the parameters such as porosities and permeabilites is required. These gradients are known as sensitivity coefficient.

i

calc

i

i

calc

K

AS

KAS

i

φφ ∂∂

=

∂∂

= ………………..…………………………… (15)

At each iteration of the Gauss-Newton algorithm, a linear system of equation is solved.

GHGN −=δα. ……………………………………………… (16)

JJH T= …………………………..……………………….. (17)

FJG T= …………….……………………………………. (18) where, Β is positive defining Hessian matrix38 and HGN is positive defining Hessian matrix (Gauss-Newton approximation to Hessian matrix), and G is the Gradian vector and referred to as sensitivity coefficients. The parameter estimation process continues until an error between observed and calculated data is less than a certain error number by controlling either the RMS value or the normalized value. Appendix D. Normalized RMS (NRMS). Normalized RMS (NRMS) is defined as the RMS of the difference between two datasets divided by the average of the RMS of each dataset.

)()()(200(%)ii

iibRMSaRMS

baRMSNRMS+

−= …………………...……………(19)

where RMS is defined as:

∑∑=

i

iii

xxRMS

ωω )(

)(2 ……………….…………………….. (20)

where w is weighting factor. NRMS is expressed as a percentage and ranges from 0% to

200%.

8 SPE 104519

If two datasets are identical, then NRMS is 0%. If two datasets are completely different, then NRMS is 200%. If both datasets are random noise, then NRMS is 141% (the square root of 2). NRMS is very sensitive to static, phase or amplitude differences. If one dataset is a phase-shifted version of the other, then the NRMS for a 10-degree phase shift is 17.4%, for a 90-degree phase shift is 141%, and for a 180-degree phase shift is 200%. If one dataset is a scaled version of the other, then the NRMS for a 0.5 scale is 66.7%. Volumetric weighted average. The weighted mean or weighted average, of a non-empty list of data [x1,x2,…,xn], with corresponding non-negative weights [w1,w2,…,wn], at least one of which is positive, is the quantity calculated by

∑∑

=

==ni i

ni ii

w

xwx

1

1 ……………………………………………(21)

So data elements with a high weight contribute more to the

weighted mean than do elements with a low weight. If all the weights are equal, then the weighted mean is the same as the arithmetic mean.

This average method is known as volumetric weighted average if it is weighted by volume. Harmonic average. In statistics, given a set of data, [x1,x2,…,xn], and corresponding weights [w1,w2,…,wn], the weighted harmonic mean is calculated as

∑

∑

=

==ni

i

i

ni i

xww

x

1

1 …………………………………………...…(22)

Note that if all the weights are equal, the weighted

harmonic mean is the same as the harmonic mean.

Nomenclature A = Seismic amplitude C = Average number of contacts per grain Cd = Data covariance matrix Cα = Covariance matrix of priory parameter F = Objective function G = Gradient matrix HGN = GN approximation to Hessian matrix IP = Compressional impedance Is = Shear impedance J = Jacobian matrix K = Permeability k = Bulk modulus keff = Effective bulk modulus kfr = Bulk modulus of solid framework kma = Bulk modulus for solid ko = Oil bulk modulus kw = Water bulk modulus kg = Gas bulk modulus N = Number of parameters Nob = Number of blocks NRMS =Normalized RMS P = Reservoir pore pressure

Po = Oil pressure Pg = Gas pressure Pw = Water pressure Peff = Effective pressure Pext = External pressure R= Reflection coefficent RMS = Root mean square So = Oil saturation Sg = Gas saturation Sw = Water saturation S = Vector of sensitivity coefficient or gradient Vp = P-wave velocity Vs = S-wave velocity ν = Poisoon’s ratio μ = Shear modulus of solid framework μeff = Effective shear modulus ρB = Bulk densities ρf = Fluid densities ρo = Oil densities φ = Effective porosity ρg = Gas densities ρw = Water densities ρma = Matrix densities w = Weighting factor (matrix) α = Reservoir model parameters η = Coefficient of internal deformation obs = Observed value cal = Calculated value i = Indicator from 1 to number of parameters

Figure 1: Formation A- Rock types 1&2.

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Figure 2: Formation B- Rock types 1&2.

Figure 4: Formation D- Rock types 1&2&3.

Figure 6: Amplitude sensitivities for saturations in column 5 of the reservoir model.

Figure 3: Formation C- Rock types 1&2.

Figure 5: Synthetic reservoir simulation model

Figure 7: Amplitude sensitivities for pressure in column 5 of the reservoir model.

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Figure 8: Amplitude sensitivities for porosity in column 5 of the reservoir model.

Figure 10: Amount of mismatch between real and calculated time-lapse seismic data in each iteration for Cases A and B

Figure 12: Objective Function at the end of inversion process (Mismatch between observation data and calculated data)

Figure 9: Amplitude sensitivities for permeability in column 5 of the reservoir model.

Figure 11: Initial Objective Function (Mismatch between observation data and calculated data)

Figure 13: Volumetric average error in the estimation of permeability distribution

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Figure 14: Volumetric average error in the estimation of porosity distribution

Figure 16: NRMS error in the estimation of porosity distribution

Figure 18: Initial permeability distribution In Case B.

Figure 15: NRMS error in the estimation of permeability distribution

Figure 17: Harmonic average error in the estimation of permeability distribution

Figure 19: Initial porosity distribution In Case B.

12 SPE 104519

Figure 20: Real permeability distribution.

Figure 22: Estimated permeability distribution after 20 Iterations in Case B.

Figure 24: Initial error in the estimation of permeability distribution in Case B.

Figure 21: Real porosity distribution.

Figure 23: Estimated porosity distribution after 20 Iterations in Case B

Figure 25: Initial error in the estimation of porosity distribution in Case B.

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Figure 26: Error in estimation of permeability distribution after 20 iterations in Case B.

Figure 27: Error in estimation of porosity distribution after 20 iterations in Case B.