J. Geophys. Eng. 5 (2008) 54–66 doi:10.1088/1742-2132/5/1/006
Nonlinear inversion for estimating
reservoir parameters from time-lapse
seismic data
Mohsen Dadashpour, Martin Landrø and Jon Kleppe
Department of Petroleum Engineering and Applied Geophysics, NTNU, 7491, Trondheim, Norway E-mail:dadashpo@stud.ntnu.no,martin.landro@ntnu.noandjon.kleppe@ntnu.no
Received 3 October 2006
Accepted for publication 18 September 2007 Published 28 November 2007
Online atstacks.iop.org/JGE/5/54 Abstract
Saturation and pore pressure changes within a reservoir can be estimated by a history matching process based on production data. If time-lapse seismic data are available, the same
parameters might be estimated directly from the seismic data as well. There are several ways to combine these data sources for estimating these reservoir parameters. In this work, we formulate a nonlinear inversion scheme to estimate pressure and saturation changes from time-lapse seismic data. We believe that such a formulation will enable us to include seismic data in the reservoir simulator in an efficient way, by including a second term in the least-squares objective function. A nonlinear Gauss–Newton inversion method is tested on a 2D synthetic dataset inspired by a field offshore from Norway. A conventional reservoir simulator has been used to produce saturation and pore pressure changes as a function of production time. A rock physics model converts these data into synthetic time-lapse seismic data. Finally, the synthetic time-lapse data are used to test the derived inversion algorithm. We find that the inversion results are strongly dependent on the input model, and this is expected since we are dealing with an ill-posed inversion problem. Since we estimate pressure and saturation change for each grid cell in the reservoir model, the number of model parameters is high, and
therefore the problem is undetermined. From testing, using this particular dataset, we assume neither pressure nor saturation changes for the initial model. Although uncertainties associated with the proposed method are high, we think this might be a useful tool, since there are ways to reduce the number of model parameters and constrain the objective function by including production data and reservoir simulation data into this algorithm.
Keywords: saturation and pressure changes, time-lapse seismic, parameter estimation, inversion, reservoir simulation
Nomenclature
H Hessian matrix
c average number of contacts per grain
d production quantities
F objective function
f data residual vector
Error normalized error
J Jacobian matrix
keff effective bulk modulus
kfr bulk modulus of solid framework
kg gas bulk modulus
kL bulk modulus of the liquids
kma bulk modulus for solid
ko oil bulk modulus
kw water bulk modulus
MERR minimum error
MAXIT maximum iteration number
N number of parameters
NRMS normalized RMS
P reservoir pore pressure
Peff effective pressure
Pext external pressure
RMS root mean square
Sg gas saturation
So oil saturation
Sor residual oil saturation
Sw water saturation
Swc irreducible water saturation
Vp P-wave velocity
Vs S-wave velocity
V Poisson ratio
µeff effective shear modulus
α model parameter vector
δα error vector
α amount of perturbation of the function.
M shear modulus of solid framework
ρ bulk densities
ρf fluid densities
ρma matrix densities
ϕ porosity
Subscripts and superscripts
nobs number of observation
npar number of parameter
obs observed
cal calculated
nc number of cells
i data space (indicator from 1 to npar)
j model space (indicator from 1 to nobs) ˆ scaled (Hessian, gradient, . . . )
1. Introduction
An ultimate goal in reservoir engineering is to build a model that is as close as possible to nature. This model should be able to predict all crucial data acquired during field production, and at the same time access predictive features suitable for reservoir management. History matching is the method used to meet this goal. History matching involves a construction of an initial model with an initial approximation of the reservoir parameters, followed by a systematic process to minimize an objective function that represents the mismatch between observed and simulated responses. This optimization process is for instance achieved by perturbing the relevant parameters, and determining updates that reduce the objective function. Parameter estimation or determination of reservoir properties is one useful application of the history matching process, which is a well-established method for reservoir property estimation.
Reservoir fluid saturation and pore pressure changes have direct relationships with the drainage and recovery of the reservoir. Robust estimation of changes in these parameters during the life of a hydrocarbon field will lead to well founded future predictions and decisions in reservoir management, and thus will increase the ultimate recovery from the reservoir and reduce the production costs. Furthermore, increased knowledge about these crucial reservoir parameters
will ensure safer and more environmental friendly production of hydrocarbons.
There are many methods for estimation of these parameters. Traditionally, reservoir fluid flow simulators are used for this purpose. These methods estimate reservoir parameters primarily based on traditionally known observation data such as bottom hole pressure (Slater and Durrer 1971, Thomas et al1971), oil, water and gas production data (Tan
1995), well test shut-in pressure data (Landa et al1996) and some combinations of these data (Schulze-Riegert et al2001, Aanonsen et al2003). However, traditional history matching is associated with several uncertainties such as the following. • Parameters from individual wells are not representing all reservoir compartments and zones (up-scaling problem). • Fluid flow formulae are associated with some
uncertainties.
• The geological description of the reservoir model is always a simplification of nature, and we must therefore expect that some production related effects cannot be fully explained by the idealized reservoir model.
• In most cases, consideration of all production processes in reservoir simulation is both numerically expensive and extremely difficult to simulate correctly.
Because of these uncertainties in the fluid flow simulators, there might be a significant difference between historical and numerically modelled data. Calibration or conditioning of the reservoir simulation models to fit the historical production data is history matching. Usually, history matching requires numerous iterations that are time consuming and costly, and it is also a non-unique problem.
Domenico (1974) and Nur (1989) investigated the effect of saturation and pressure changes on seismic parameters. They have concluded that changing these parameters has a detectable effect in seismic attributes. Landrø (2001) tried to map the reservoir pressure and saturation changes using seismic methods. Landrø (2001) tried to represent an efficient method to discriminate between pressure and saturation effects from time-lapse seismic data. Recently, time-lapse, or 4D, seismic data have become available as an additional set of dynamic data. Time-lapse seismic is now a useful tool for monitoring thermal processes (Eastwood et al 1994), CO2
flooding (Lazaratos and Marion1997) as well as gas and water injection (Landrø et al 1999, Burkhart et al 2000, Behrens
et al2002).
Today, 4D seismic data are used to monitor fluid movements within the reservoir. This is a new dimension for history matching since 4D seismic contains information about fluid movement and pressure changes between and beyond the wells. Most reservoir parameters which are being used in reservoir simulators are estimated from laboratory measurements that are not representative of the entire reservoir. By integrating time-lapse data with the production data in a history matching process, the uniqueness of the results improves. Many authors try to update the model by using this additional information (Landa and Horne 1997, Van Ditzhuijzen et al 2001, Mantica et al 2001, Kretz et al
assumed time-lapse seismic is already converted to saturation changes and in addition they assumed seismic variations were only a result of saturation changes and they ignored pressure effects. This research addresses this missing part.
Estimation of saturation and pressure changes using time-lapse seismic data is a new challenge in the petroleum industry. This is the first step in history matching using both reservoir simulation and time-lapse seismic data. In this paper we present a nonlinear inversion method for a joint estimation of pressure and saturation changes during depletion and water injection of a 2D synthetic case, which is generated by using field data from a reservoir offshore from Norway.
2. Forward seismic modelling
Key reservoir parameters, such as saturations and pressures, are changing during production and this might cause significant changes in the seismic responses. Highly porous sand reservoirs show significant 4D seismic changes, while low porosity carbonate reservoirs show less 4D changes. Time-lapse seismic data are time-dependent measurements (dynamic measurements), acquired from the reservoir using the same acquisition parameters for the various time-lapse seismic surveys. Converting saturations and pressure changes into seismic properties such as P-wave velocity, S-wave velocity and density changes requires information about the rock properties. The dependence between fluid saturation changes and pore pressure changes and the seismic parameters are described by rock physics models. Once these rock physics relationships are established for a given reservoir rock, the seismic forward modelling can be done. That means converting a given pressure and fluid saturation state for a given reservoir rock into a seismic section. Moreover, this procedure acts as a bridge that relates seismic parameter changes to reservoir parameter changes and vice versa. Thus, understanding this part of the inversion loop is vital for doing good history matching. This means that rock physics is a key element in such a process. Uncertainties and systematic errors in the rock physics model will be of crucial importance to the final result. The main limitation of the calibrated rock physics models is that they cannot capture variations in rock properties between wells.
Seismic amplitudes change with variation in the source strength and with the angle of incidence. In addition, variations in acoustic properties are the function of reservoir properties such as pressure, fluid saturation, temperature and compaction. Effects of temperature and compaction are not considered in this work.
Forward modelling is done in two main steps. First, reservoir parameters are converted to seismic parameters by using the rock physics models. The Gassmann equation (Gassmann 1951) and the Hertz–Mindlin model (Mindlin
1949) are used to estimate seismic parameter changes caused by fluid saturation and pore pressure changes, respectively. Then, synthetic seismic sections are calculated based on these parameters. For simplicity, a one-dimensional reflectivity-based modelling is used. This scheme includes internal multiple reflections. In this case, the reflection coefficients
are defined by contrasts in acoustic impedance. The two-way travel time is computed from the P-wave velocity. Time-lapse changes in reflection strength are caused by changes in acoustic impedance. 4D time shifts are introduced by changes in P-wave velocity within the reservoir layers. Both impedance and velocity are saturation–pressure dependent (Stovas and Landrø2005).
We study a two-phase system (oil and water) and focus on variation of oil and water saturation, denoted by Soand Sw, respectively.
2.1. The Gassmann equation
Seismic velocities in a porous medium saturated with water depend on three constants, namely the bulk modulus k (we use the lowercase letter to avoid any confusion with the permeability sign), shear modulus µ and density ρ. The bulk modulus or incompressibility of an isotropic rock is defined as the ratio of hydrostatic stress to volumetric strain. In other words, it tells us how difficult it is to compress the rock. The shear modulus or shear stiffness of the rock is defined as the ratio of shear stress to shear strain and in other words, how difficult it is to change the shape of a rock sample. In the long-wavelength limit, the speed of sound is related to the elastic constants of the aggregate:
vp= k+4 3µ ρB , (1) vs= µ ρB , (2)
where ρBdenotes bulk density, i.e. the volume average density of the solid and liquid phases, which is a function of porosity, matrix and fluid densities and can be calculated from the following equation:
ρB= φρf + (1− φ)ρma, (3)
where ρf and ρma are the fluid and matrix densities,
respectively. The downscaling steps allow the calculation of the fluid density:
ρf = ρoSo+ ρwSw, (4) where ρo and ρw are oil and water densities, respectively. The most commonly used theoretical approach for fluid substitution employs the low-frequency Gassmann theory. The Gassmann equations divide the bulk modulus of a fluid saturated rock into three parts:
• the bulk modulus of the mineral matrix, • the bulk modulus of the porous rock frame, • the bulk modulus of the pore-filling fluids.
Gassmann equations can be written as the following formula: ksat= kf r+ (kHM− kf r)2 kHM 1− φ + φkHM kf − kf r kHM , (5)
where kfr is the bulk modulus of the solid framework,
(equation (7)), kfis the bulk modulus of the saturating fluid (water, gas or oil) and φ is the effective porosity of the medium.
kfis the bulk modulus of the pore fluid (water and oil) and is estimated by Wood’s law for oil and water given as (Reuss harmonic average 1929): 1 kf = So ko +Sw kw , (6)
where ko and kw denote oil and water bulk modulus, respectively.
According to Mavko et al (1998) the Gassmann equation is associated with some assumptions:
• Pore fluid is firmly coupled to the pore wall. • Gas and liquid are uniformly distributed in the pores. • Constant shear modulus in dry and saturated rocks. • There is homogeneous mineral modulus and statistical
isotropy.
However, there is no assumption made about the pore geometry. In addition, the Gassmann equation is only valid for low frequencies when the pore pressures are equilibrated through the pore space.
2.2. The Hertz–Mindlin model
The Hertz–Mindlin model is used to describe seismic parameter changes caused by pressure changes. The effective bulk modulus and shear modulus of a dry random identical sphere packing are given by
kHM= n c2(1− φ c)2µ2peff 18π2(1− ν)2 , (7) µHM= 5− 4ν 5(2− ν) n 3c2(1− φ c)2µ2peff 2π2(1− ν)2 , (8)
where kHMand µHMare the bulk and shear modulus at critical
porosity φc, respectively; Peffis the effective pressure; µ is the
shear modulus of the solid phase; ν is Poisson’s ratio and n is the coordination number. In the original Hertz–Mindlin theory
n is equal to 3 which means velocity varies with Peffraised to
the 1/6th power. Some laboratory measurements of samples gave a larger number for n. Vidal (2000) found n= 5.6 for P-waves and n= 3.8 for S-waves for gas sands, while Landrø (2001) used n= 5 for oil sands. We have used n = 5 in this work. c denotes the average number of contacts per grain, and in this study we use c= 9. The effective pressure, Peff, used
in Hertz–Mindlin theory is taken as the difference between the lithostatic Pextand the hydrostatic pressure P (Christensen and
Wang1985):
Peff= Pext− ηP, (9)
where η is the coefficient of internal deformation which is usually an unknown parameter. Thus, it is a limiting factor for quantitative use of rock physics measurements for pore pressure estimation. This unknown parameter is often assumed to be equal to 1; however, if η is different from 1, uncertainty in estimation in saturation and pressure changes will increase.
3. Inversion method
The inversion model or parameter estimation procedure determines the values of reservoir parameters such as pressure and saturation from indirect measurements. This procedure traditionally starts with an initial guess of body parameters and the process is iteratively advanced until the best fit is obtained between the calculated and observed data. The common algorithm for an inversion process is as follows:
(1) Enter the initial guess for the body parameters.
(2) Compute the response of the system through forward modelling.
(3) Compute the objective function.
(4) Update parameters by minimizing the objective function using some minimization algorithms.
(5) If the value of the objective function is not certifiable, return to step 2.
Several optimization algorithms are developed for this purpose. Some of these, such as Gauss–Newton, singular value decomposition, conjugate gradient and steepest descent (Landa1979, Gill et al1981) require both or at least one of the first (Jacobian) or second (Hessian) derivative of time-dependent reservoir properties with respect to static reservoir properties. However, some of them such as genetic algorithm, response surfaces, experimental design methods and Monte Carlo simulation (Lepine et al 1998) do not require the computation of gradients for optimization purpose.
In this work, minimization of the objective function is solved numerically by the damped Gauss–Newton method (appendix A) which is classified as a gradient-based method to determine the target parameters automatically. The main purpose of each inversion method is to minimize an objective function, which is defined as deviation between the calculated value (dcal) and the observed value (dobs). In this study d is
amplitude differences between the base and monitor survey (time-lapse seismic data):
F = N i=1 (f )2 f = (dobs− dcal), (10)
where N is the number of observation points, f is the data residual vector and F is the least-squares error, which should be minimized.
The inversion procedure is summarized in figure1and consists of the following steps.
(1) Enter initial guess of parameters (αi), observation data dobs
j
, maximum iteration number (MITR) and minimum acceptable error (MERR). Inverted parameters are saturation and pressure changes for each cell and are defined as
αi = [S1. . . Snc, P1. . . Pnc] ,
(11)
Figure 1. Flowchart of the nonlinear inversion process. (2) Compute the calculated data djcal by using forward
seismic modelling.
(3) Calculate the Jacobian matrix which is defined as Jij =
∂di/∂αi where i runs over data space, j runs over model space and α is the model parameter vector (appendix A.2). (4) Calculate the transpose of the Jacobian matrix (JT).
(5) Calculate the objective function (F) from equation (10). (6) If the error is less than the minimum acceptable error or the
iteration numbers are higher than the maximum iteration number, stop and print the iteration number and the new values for the parameters.
(7) Compute the gradient matrices (G), G = −2JTf, and compute the approximate Hessian matrix (H) H= 2JTJ. Based on Helgesen and Landrø (1993) each parameter class is scaled independently in order to speed up the algorithm. The scaling formulae are
Hij = Hij HjjHii Gii= Gii √ Hii . (12) (8) Calculate the normalized Hessian and gradient matrices. (9) Calculate the scaled model parameter:
ˆ H .δp= − ˆG δαˆi = δαi √ Hii . (13) (10) Update the parameter by adding the update vector to the
old value:
αi = αi+ δ ˆαi, (14)
(11) Apply constraints for these values to have reasonable value.
(12) Add one value to the iteration number. (13) Go to step 2.
4. Example
To test the efficiency and accuracy of our method, we designed a 2D synthetic experiment based on a 2D cross section from a North Sea reservoir model. The propagator-matrix method (Haskell1953, Kennett1983) which is modified by Stovas and Arntsen (2006) for a quasi-vertical wave propagation approach was used to compute the transmission and reflection response from a stack of plane layers. Based on the modelled time-lapse seismic differences, changes in fluid saturations and pore pressure for each cell in the reservoir model were estimated.
Water saturations and pore pressures in each cell are considered to be independent reservoir parameters. Initial guesses of these parameters come from the assumption that there is no depletion (fluid flow changes) and no pore pressure changes in the reservoir. Figures 2 and3show the values of these parameters in each cell. The maximum iteration number and minimum normalized error are set to 20 and 0.01 respectively.
4.1. Observation data
In field cases, observation data come directly from seismic traces. However, in this project (synthetic case) these data come from seismic forward modelling. For this purpose, one reservoir model with 31× 1 × 26 grid cells containing 740 active cells and 3 main faults is constructed. The total pore volume for this reservoir is 12 683 178 m3. A commercial three-phase (oil, gas and water), three-dimensional, black oil reservoir simulator (Eclipse 100) is used to create saturation and pressure changes in each cell.
The production well is located in column 1 and perforated in blocks 1 to 3 and 5 to 13 and the water injection well is located in column 31 and perforated in blocks 5 to 22 and 25 to 26. Complete connections are chosen between the segments on both sides of the faults. Simulation is done for 20 years.
Figure 2. Initial guess of reservoir pore pressure at both the
beginning and end of production (MPa).
Figure 3. Initial guess of reservoir water saturation at both the
beginning and end of production.
Water saturation and pressure changes between these two times and forward seismic modelling are used to create seismic time-lapse traces (we assume no change in gas saturation). Figures4
and 5 show seismic sections at the start and end of the simulation, respectively.
The time-lapse seismic difference section (figure 6) (amplitude difference between base and monitor sections) is used as observation data in the inversion loop.
4.2. Constraints
In this case, water saturation and pore pressures are limited by boundaries. From an engineering point of view,
• water saturation cannot be less than the connate water saturation (Swc) and should always be less than one minus
the irreducible oil saturation (Sor),
Swc Sw 1 − Sor; (15)
• summation of liquids in the cell should equal 1,
So+ Sw = 1; (16)
Figure 4. Modelled zero offset seismic section for the start of the
simulation, corresponding to the base survey.
Figure 5. Modelled zero offset seismic section for the end of the
simulation, corresponding to the monitor survey.
Figure 6. Seismic difference section between the monitor and base
survey.
• because the reservoir pore pressure is maintained by water injection, the values for pore pressures should lie between the bubble point (Pbub) and the overburden pressures (Pext),
Pbub P Pext. (17)
In practice there are several examples where the pressure drops below the bubble point, i.e. P < Pbub. However, for
Figure 7. Amplitude sensitivities for saturations in column 5 of the
reservoir model. The vertical scale is the dimensionless ratio of the fractional change in amplitude to the fractional change in saturation.
this study we assume that the pressure is maintained by water injection.
4.3. Sensitivity of the time-lapse amplitude change to saturation and pressure changes
Before estimation of saturation and pressure changes using optimization theory, we have to understand the effect of perturbing these parameters with respect to amplitude responses. In other words, it is interesting to analyse how much influence each parameter has on the seismogram.
According to our petrophysical model, the amplitude response in each column is a function of saturation, pressure and porosity. However, the porosity term will cancel out because we assume no compaction.
This sensitivity information (water saturation and pressure) is available in the sensitivity or Jacobian matrix. Each column of this matrix represents the derivative of amplitude with respect to each parameter, and might therefore be used directly to study the sensitivity variations. We select one column (column 5) as a representative element for all reservoirs. Figures 7 and 8 respectively show amplitude sensitivities with respect to water saturation and pore pressure changes for each cell in this column. It is clear that seismic amplitudes are more sensitive to saturation changes than pressure changes (in this case). For saturation and pressure perturbations from 1% to 5%, the overall behaviour remains the same. For other scenarios, this might be different of course; however, most 4D field examples show saturation effects that are more pronounced than the pressure effects.
5. Results and discussion
The method presented enables us to automatically estimate reservoir parameters from seismic data. It successfully reduces the initial error in the objective function. Figure9shows initial and final amplitude differences between observation data and calculated data in the first and best iteration respectively (initial
Figure 8. Amplitude sensitivities for pressure in column 5 of the
reservoir model. The vertical scale is the dimensionless ratio of the fractional change in amplitude to the fractional change in pore pressure.
Figure 9. Amplitude differences between observation data and
calculated data for the first and the best iteration, respectively.
and final objective function). Calculated NRMS (normalized RMS) between observation data and calculated data is only 4%, which shows the strength of this method to reduce the objective function (see appendix B for more information about NRMS). NRMS is used to clarify differences between calculated and real data and it is not used as repeatability metric in the present study.
The reservoir was divided into four segments, which are denoted respectively from left to right A, B, C and D (figure2). Segments C and D initially are in the water zone, which means that there are only pore pressure changes in these segments.
After 20 iterations, comparison between calculated and observed data shows that in the best iteration (lowest objective function), all segments show improvement compared with
Figure 10. Comparison between observed data (grey) and calculated data in the lowest objective function (black) in column 5 segment A,
column 14 segment B, column 21 segment C and column 29 segment D.
Figure 11. Normalized error and the difference between average
real and calculated data versus iteration number in column 5 segment A.
the initial guess and they all show an excellent match with observation data (figure10).
We consider columns 5, 14, 21 and 29 to represent the behaviour of segments A, B, C and D. Figures11–14show the normalized error (top) and the difference between average real and calculated saturations and pressures (bottom) with respect to iteration numbers. It is obvious from the figures that in all cases it is possible to find a situation with a normalized error less than the initial one.
From figure11 we see that the pressure error increases after two iterations. This might be caused by the constraints we impose (equation (17)), but we are not sure about this.
Figure 12. Normalized error and the difference between average
real and calculated data versus iteration number in column 14 segment B.
For this example the inversion algorithm successfully estimates both saturation and pressure changes between the base and monitor surveys. Estimated water saturation and pore pressure changes after 20 iterations are 5.7± 8.8% and 2.9± 3.1 MPa respectively, which are improvements in the estimation of both saturation and pressure changes compared with initial guess (0 and 0 MPa). These numbers represent average values for all cells in the model, and their associated standard deviations. Using average numbers as a diagnostic tool for evaluating the performance of the inversion algorithm might be misleading, so these numbers should be interpreted with care. Therefore, it is important to study the detailed differences between estimated and real changes, as shown
Figure 13. Normalized error and the difference between average
real and calculated data versus iteration number in column 21 segment C.
Figure 14. Normalized error and the difference between average
real and calculated data versus iteration number in column 29 segment D.
in figures 15 and 16. We notice that the pressure change estimates for segments C and D are very good and agree very well with the correct pressure changes. This is probably caused by the fact that there are no saturation changes within these two segments (as shown in figure16). For segments A and B, however, the deviations between estimated and correct pressure changes are significant, especially for the middle layers in segment A. Also the estimated saturation changes
Figure 15. Real and calculated changes in reservoir pore pressure
(MPa).
Figure 16. Real and calculated changes in reservoir water saturation
(fractions).
show significant deviations compared to the correct values, especially for segment B we observe an underestimation of the saturation changes close to the oil–water contact.
This degree of errors varies from segment to segment. Segments A and B show a good improvement in the estimation of water saturation changes with respect to the initial guess (15.5 ± 19.9% and 8.1 ± 12.2% for segments A and B respectively). All segments except segment A show a good
Figure 17. Difference between real and estimated pressure changes
for the first (bottom) and last (top) iteration.
Table 1. Saturation and pressure estimation in each segment.
Real Calculated
Segment P (MPa) Sw(%) P(MPa) Sw(%)
A 3.29± 1.18 28.6 ± 22.2 −0.64 ± 4.05 15.46 ± 19.8 B 4.10± 0.96 31.1 ± 25.5 1.66 ± 3.74 8.07± 12.20 C 6.42± 0.22 0.0 6.14± 1.01 0.0
D 5.89± 2.33 0.0 5.25± 3.73 0.0
improvement in the estimation of pore pressure changes with respect to the initial guess (−0.6 ± 4.0 Mpa, 1.7 ± 3.7 MPa, 6.1± 1.0 MPa and 5.3 ± 3.7 MPa for segments A, B, C and D respectively). As mentioned previously, segments C and D are initially in the water zone and there is no fluid saturation change in these segments (table1).
Figures17and18show the differences between estimated and real pressure and saturation changes, respectively. These figures show that in the water saturation case, this program enables us to reduce the amount of deviation between the calculated and real data with respect to the initial case. However, in the pore pressure estimation case it shows acceptable improvement only in segments C and D, where there is only aquifer. The reason for this can be the high number of estimated parameters, more sensitivity in the forward simulation model to saturation than pore pressure and the type or amount of applied constraints.
Figure19 shows the range of P-wave velocity changes versus effective pressure change in grid cell C55(column 5
and layer 5) and C58(column 5 and layer 8). Based on this
figure low effective pressure gives higher sensitivity to pressure changes. And in this case we are not in a good range for pore pressure estimation. This can be one of the reasons to get unsatisfactory answers for pore pressure estimation.
Figure 18. Difference between real and estimated water saturation
changes for the first (bottom) and last (top) iteration.
Figure 19. Range of P-wave velocity changes versus effective
pressure change in C55and C58.
This method will work better if we reduce the amount of parameters to be estimated and apply some other reservoir parameter constraints.
6. Conclusions
A nonlinear Gauss–Newton optimization technique is used to estimate water saturation and pore pressure changes from time-lapse seismic data. Analysis of this algorithm for a 2D synthetic reservoir, based on field data from a complex reservoir, indicates that this algorithm enables us to match observed and calculated data. This procedure provides reasonable estimates of saturation and pressure changes between the base and monitor surveys. We find that the inversion procedure improves the estimation of saturation and
pressure changes. For our test example, we find that the fluid sensitivity is somewhat larger than the sensitivity for pore pressure changes.
Major weaknesses of the developed algorithm are the strong dependence on the initial model (common for most inversion techniques) and the strong need for constraints to limit the solution space.
Another future step for this methodology will be to include the fluid simulation loop into the algorithm. With this extension, we will estimate reservoir properties such as permeability and porosity distributions from time-lapse seismic data.
Acknowledgments
We thank the Norwegian University of Science and Technology (NTNU), The Research Council of Norway and TOTAL for financial support. We also thank Alexey Stovas for preparing the seismic forward modelling. We would like to extend our gratitude to Jan Ivar Jensen from NTNU for his help during this study. We are grateful to STATOIL and specially Magne Lygren for making it possible to use data from a real reservoir. We are also grateful to Schlumberger-GeoQuest for the use of the Eclipse simulator. One anonymous reviewer is acknowledged for constructive comments and numerous suggestions that improved the paper.
Appendix A. Gauss–Newton optimization techniques
A.1. Theory of the Gauss–Newton algorithm
The theoretical anomaly expression of equations, which is mentioned in forward modelling, is nonlinear with respect to the target parameters; therefore, the problem can be solved by the least-squares inversion method.
The main assumptions in these kinds of algorithms are as follows:
• The objective function is smooth. • It is possible to compute ∇F .
The condition for optimization is
F (α0+ α) < F (α0), (A.1)
α= ρ p, (A.2)
where α is the amount of perturbation of the function F(α), p is the unit vector that provides a direction for descent and ρ is a positive scalar that provides the step size.
It is always possible to find a positive value of ρ that results in reduction of the value of F provided thatp satisfies the following condition:
∇F p < 0. (A.3)
wherep is defined as the sufficient descent direction and when p satisfies the condition in equation (A.3).
The structure of the gradient algorithm synthesizes as follows:
• Compute a sufficient descent direction (p).
• Compute an adequate step size (ρ).
At each iteration of the Gauss–Newton algorithm, a linear system of equations is solved:
H α= −∇F, (A.4)
where ∇F is the objective function and H is a symmetric and non-singular matrix. In the Gauss–Newton optimization technique, it is considered as an approximation to the exact Hessian matrix.
When equation (10) is used∇F and H for Gauss–Newton optimization techniques will be calculated from the following:
∇F = −2JT(dobs− dcal), (A.5)
H = 2JTJ, (A.6)
where J is the Jacobian matrix which is the first derivative of time dependent reservoir properties to static reservoir properties, which is also referred to as sensitivity coefficient and it will be explained further in appendix A.2.
The inversion process continues until the error between the observed and the calculated data is less than a certain error number by controlling either the RMS value or the normalized value.
A.2. Jacobian matrix
A Jacobian matrix is a first-order partial derivative of time-dependent reservoir properties (seismic amplitudes in this case) with respect to static reservoir parameters (saturation and pressure in this case). It is used to find the linear approximation of a system of differential equations. The expressions for partial derivatives are usually obtained by analytical differentiation in the inversion process.
By definition the partial derivative is the derivative with respect to a single variable of a function of two or more variables, regarding other variables as constants. In the present study, the numerical Jacobian matrix is made by the calculation of the partial derivative with respect to saturation (α1) and
pressure (α2) in each cell. In the general form of first-order
finite differences, the derivation of any function f (α) at a point
αi(i= 1, . . . , N) is given by the following equation:
∂f (α) ∂αi
= f (αi+ α)− f (α)
α . (A.7)
In this case, the partial derivative of amplitudes that is calculated from forward modelling with respect to the reservoir parameters (saturation and pressure in each cell), can be calculated as ∂dcal j ∂α1 = dcal j (αi+ α)− d(α) α , (A.8) dfcal j ∂αnop = dcal j (αi+ α)− d(α) α . (A.9)
The final Jacobian matrix will be J ≡ ∂ d cal ∂α = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ∂d1cal ∂α1 ∂d1cal ∂α2 . . ∂d cal 1 ∂αnpar ∂dcal 2 ∂α1 . . . . . . . . . . . . . . ∂dcal nobs ∂α1 . . . ∂d cal nobs ∂αnpar ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . (A.10)
For computing partial derivatives for each iteration, the perturbation values for each parameter are taken as 1% of that value (αi= 0.01αi,i= 1, . . . , nop).
Appendix B. 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 (Kragh and Christie2002):
NRMS(%)= 200RMS(ai− bi) RMS(ai)+ RMS(bi) , (B.1) where RMS is defined as RMS(xi)= x2 i N . (B.2)
N is the number of samples in the summation interval. NRMS
is expressed as a percentage and ranges from 0% to 200%. 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 NRMS for a 10◦phase shift is 17.4%, for a 90◦phase shift is 141% and for a 180◦phase shift is 200%. If one dataset is a scaled version of the other, then NRMS for a 0.5 scale is 66.7%.
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