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SPE 56518 Efficient Conditioning of 3D Fine-Scale Reservoir Model To Multiphase Production Data Using Streamline-Based Coarse-Scale Inversion and Geostatistical Downscaling Thomas T. Tran, SPE, Xian-Huan Wen, SPE, Ronald A. Behrens, SPE, Chevron Petroleum Technology Company

Copyright 1999, Society of Petroleum Engineers Inc. This paper was prepared for presentation at the 1999 SPE Annual Technical Conference and Exhibition held in Houston, Texas, 3–6 October 1999. 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 In addition to seismic and well constraints, production data must be integrated into geostatistical reservoir models for reliable reservoir performance predictions. An iterative inversion algorithm is required for such integration and is usually computationally intensive since forward flow simulation must be performed at each iteration. This paper presents an efficient approach for generating fine-scale three dimensional (3D) reservoir models that are conditioned to multiphase production data by combining a recently developed streamline-based inversion technique with a geostatistical downscaling algorithm. Production data can not reveal fine scale details of reservoir heterogeneity. By solving the streamline pressure solution at a coarse scale consistent with the production data we are able to invert numerous geostatistical realizations. Additionally, the streamline method allows fine resolution along the 1D streamlines independent of the coarse grid pressure solution so we do not need to explicitly address multiphase scale-up. Multiple geostatistical fine scale models are up-scaled to a coarse scale used in the inversion process. After inversion, the models are each geostatistically downscaled to multiple fine scale realizations. These fine scale models are now preconditioned to the production data and can be up-scaled to any scale for final flow simulation. A 3D extension of the prior 2D sequential-self calibration method (SSC) is developed for the inversion step. This method updates the coarse models to match production data while preserving as much of geostatistical constraint as possible. A new geostatistical algorithm is developed for the downscaling step. We use Sequential Gaussian Simulation

with either block kriging or Bayesian updating to “downscale” the history-matched coarse scale models to fine-scale models honoring fine-scale spatial statistics. Combining these two developments we are able to efficiently generate multiple fine scale geostatistical models constrained to well and production data. Introduction The reliability of geostatistical models increases as more data is included in their construction. Historically only hard data conditioned the models. Now, soft data such as geologic maps or seismic data are included routinely. More recently there has been a growing interest and ability to include dynamic data as a constraint during the construction of reservoir models. The concept of incorporating production data into models by inversion is certainly not new and was commonly termed automatic history matching. These early attempts suffered from poor computer power, early algorithms, and an insufficient appreciation for geologic complexity. These problems are rapidly being overcome and we are now starting to precondition geostatistical models with dynamic data before starting rigorous flow simulation. We use the term "precondition" as a euphemism to avoid the now bitter taste of "automatic history matching" and to more accurately reflect the less ambitious goal of merely dramatically improving the initial flow results as an input to formal history matching and not to replace it entirely. Inversion is both CPU intensive and under-determined because we have so many parameters (cells with unknown properties) to set. Furthermore, the resolution represented by this large number of parameters is typically finer than the spatial resolution of the dynamic data available to us in producer watercut, pressure transient analysis, pressure and/or saturation estimates from 4D seismic. All of these reasons suggest that we reduce the number of parameters before inversion. We can easily reduce the number of parameters before inversion by up-scaling but we want our final model to have fine scale details for flow simulation. The up-scaling process that selectively refines the grid is motivated by observation that fine scale high permeability streaks often dominate breakthrough time and strongly affect ultimate recovery. We want our final geostatistical model to again have high resolution after the coarse scale inversion.

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T. T. TRAN, X.-H. WEN, R. A. BEHRENS

The resolution of the coarse inversion grid is not necessarily the same as that of the flow simulation grid. Flow simulation grid resolution is typically set to achieve one or two history match runs overnight in a full featured flow simulator. Inversion grid resolution should be governed by the information content of the dynamic data used as constraints and by the CPU constraints of the inversion algorithm and embedded forward model. The embedded forward model in our case is a very efficient coarse grid 3Dstreamline method but in principle it could even be a traditional finite-difference method if we were dealing with complicated flows. There is no reason to expect these grids to have similar resolutions so we can't avoid the final downscaling by simply up-scaling once for both flow simulation and inversion. In this paper we extend a prior method for incorporating production data to 3D models having both saturation data from time lapse seismic (4D seismic) and watercut data at producers. The conditioning is done at a coarse scale consistent with the information content of the data, which both reduces CPU demand and improves the inversion result by making it less under-determined. A downscaling process then takes the coarse inversion grid and restores the fine grid resolution but retains the conditioning from the production data done in the coarse grid inversion. As a result, the final 3D fine-scale model honors both the large-scale features derived from production data and the fine-scale heterogeneity observed at the well logs. We will first present the workflow of our approach with an example, which is followed by a detailed Fine scale k from Seism ic or other w ell logs & core soft data for k description of the methodology for both the G eostatistical m ethod coarse grid inversion and the geostatistical M ultiple fine scale downscaling. Some m odels on uniform grid sensitivities for the inversion and the Scale-up a given fine m odel downscaling are to coarse grid and invert w ith SSC using stream lines investigated before we conclude with a Single inversion result for discussion on limitations each coarse grid input and additional work. Basic Workflow Figure 1 shows the basic workflow. We start with any geostatistical technique to incorporate well data and soft spatial data such as from 3D seismic. Commonly multiple realizations are generated consistent with the static data. A fine scale 3D model is upscaled to the coarse grid

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resolution for inversion and then inverted with the available dynamic data. This coarse grid model is finally downscaled with one of the two new methods presented in this paper to create multiple fine scale realizations all consistent with a given coarse grid inversion result. These fine scale realizations form the basis for a traditional simulation workflow of up-scaling and history matching. The reference field A reference true Ln(k) (natural logarithm of permeability) field was first constructed for testing the proposed approach. This reference field was generated using Sequential Gaussian Simulation at a fine grid consisting of 100x100x20 grid cells. The size of each grid cell is 10x10x5 ft. A spherical variogram was used: the areal direction with maximum spatial continuity is SW-NE with a range of 800 ft, the range in the SE-NW

dow nscaling constrained by coarse grid inversion M ultiple fine grid results for each coarse grid inversion result

Figure 1: We create an initial fine grid field unconstrained by production data, coarsen grid, constrain to production data, and finally downscale to fine grid.

Figure 2: Top: the up-scaled reference (20x20x5) field; Middle: one realization of initial model before inversion; Bottom: SSC Inverted model corresponding to the initial model in the middle.

SPE 56518 EFFICIENT CONDITIONING OF 3D FINE-SCALE RESERVOIR MODEL TO MULTIPHASE PRODUCTION DATA USING STREAMLINEBASED COARSE-SCALE INVERSION AND GEOSTATISTICAL DOWNSCALING 3

direction is 200 ft, and the vertical range is 40 ft. The mean and variance of Ln(k) are 5.95 and 2.8, respectively. This reference field was then up-scaled uniformly to a coarse grid consisting of 20x20x5 blocks using an arithmetic average. The top image of Figure 2 shows the up-scaled reference model. Porosity is assumed constant with 0.2 for the entire model. Flow simulation was performed on the fine-scale reference model to generate synthetic production data with the following conditions: • The reservoir is initially saturated with oil. • A nine-well pattern, shown on Figure 2, is used with an injection well at the center and 8 production wells at the edges. All wells are constrained with constant flow rates (3200 RBBL/day for injection well, and 400 RBBL/day for production wells). • No-flow conditions are imposed at the reservoir boundaries. • Standard power-2 relative permeability curves are used. The fractional flow of water at the 8 producers and saturation distribution at 500 days (corresponding to 0.45 PVI) were retained as “truth” production data for inversion. In practical situations, saturation data are usually obtained at a coarser scale. Thus, to make our example more realistic, the fine-scale saturation distribution was up-scaled to the 20x20x5 coarse grid by simple average; this up-scaled saturation distribution was used as input for inversion.

Figure 3: Initial and updated water saturation vs. reference results for each block at 0.45 PVI.

SSC Inversion SSC inversion was performed on the 20x20x5 coarse grid. Multiple initial coarse-scale Ln(k) realizations were built using Sequential Gaussian Simulation conditional to data from the nine wells. Since (1) it is difficult to compute a horizontal variogram from only 9 wells, and (2) previous studies by Wen et al.1, 2 have shown that SSC inversion results are not very sensitive to input variogram model, a variogram model was assumed. This variogram is the same as that used to generate the reference model, except that it has a smaller areal anisotropy ratio of 600:300 ft. The histogram from the upscaled well data was used. These initial permeability models were modified by SSC to match fine-scale fractional flow rates and the coarse-scale saturation distribution. The middle and bottom images of Figure 2 show one realization of the initial SGS-generated coarse-scale Ln(k) and the corresponding SSC-updated model. Next, flow simulation was performed on each initial coarse-grid model and its SSC-updated version with the previously defined initial and boundary conditions. Figure 3 shows the cell-by-cell cross-plots of simulated saturation for the reference vs. the initial and SSC-updated models respectively. Note the significant improvement of saturation matching after inversion. The mismatch at those cells with low saturation is expected because these cells contribute very little

Figure 4: Water fractional flows of 10 realizations before (top) and after (bottom) SSC inversion on a coarse scale grid.

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mobile water over the entire lower saturation range and hence contribute little to the watercut at the wells. The inversion was weighted more to watercut data than saturation data because of greater uncertainty of saturation data from, e.g., time-lapse (4D) seismic. Comparison between simulated and reference fractional flow at producer 1 (located at top-left corner) for the first 10 initial and SSC-updated realizations is given in Figure 4. Note the improvement of matching and reduction of uncertainty in inverted models. The inversion also improves recovery at this particular well by correctly placing a permeability baffle between it and the injector thus delaying water breakthrough. This water fractional flow delay is seen in the shift of the curves comparing the top and bottom halves of Figure 4. Downscaling The multiple coarse scale Ln(k) permeability models inverted

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by the SSC method were then downscaled back to the fine scale grid using a geostatistical downscaling method (which will be described in detail in a later section). The downscaled permeability fields honor the well data, variogram and histogram, yet the average of fine scale Ln(k) within a coarse block is preserved as the value inverted by SSC. Multiple downscaled realizations can be generated for each coarse scale model. Figure 5 shows three realizations of Ln(k) downscaled with the Bayesian method from one coarse scale model shown at the bottom of Figure 2. The overall structure of this coarse scale model can be seen in each of the realizations. Only small-scale variations exist between these where the flow response between various realizations is more similar than between unconstrained realizations as expected. The high quality of the fractional flow match to the reference solution of these downscaled models is similar to that of the inverted coarse grid models. In summary, we have demonstrated an efficient workflow to generate highly detailed 3D reservoir models conditioned to (static) geologic data and (dynamic) multiphase production data by an example using a synthetic data set. This workflow consists of a streamline-based coarse-scale inversion and a geostatistically-based downscaling. Detailed description of these two components is given next. SSC Inversion Method The Sequential Self-Calibration (SSC) method, originally proposed by Gomez-Hernandez et al.3, is an inverse technique that iteratively modifies multiple initial geostatistical reservoir models to match dynamic production data, yet preserves the geostatistical features in the initial models. It utilizes the master point concept coupled with a kriging process to propagate perturbations at master point locations to the entire field. It is computationally efficient and robust. The SSC method has been used previously for integrating single-phase pressure data 1, 2, 4, and two-phase fractional flow rate data coupled with streamline simulation and fast semi-analytical sensitivity coefficient calculation1,5. In this work, the SSC method is extended to 3-D and includes the inversion of spatially distributed saturation data. The objective function used in this study is: nw nt ) O = α f ∑ ∑ W f ( w, t ) f ( w, t ) − f ( w , t ) w =1 t =1

) +α s ∑ Ws (x) s (x) − s(x)

2

(1)

2

x ∈D

) where f ( w, t ) and f ( w, t ) are the observed and simulated ) fractional flow rates at well w and time t, s ( x) and s( x) are observed and simulated saturation at location x for a given

Figure 5: Three Bayesian downscaling realizations from single coarse grid inversion.

time, nw and nt are the number of production wells and the number of the time for fractional flow data measurements, D is the reservoir domain, Wf and Ws are weights assigned to fractional flow and saturation data reflecting the quality of


α f and α s are relative weights for fractional flow and

∂s(x) nl ( x ) ∂τ l = ∑ sl′(τ ) βl ∂∆k j ∂∆k j l =1

0.5 0.4 0.3 0.2 0.1 1E1

1E2

1E3 1E4 Cells in 3D model

1E5

(2)

∆k j is the perturbation of absolute permeability at master point j, τ l is the time-of-flight of streamline l at x , sl′(τ ) is the derivative of 1-D saturation solution along streamline l at time of saturation measurement, ∂τ l / ∂∆k j is where

the sensitivity coefficient of time-of-flight on the permeability perturbation which is given in Wen et al.5, and β l is the weight assigned to streamline l when computing saturation, which is usually proportional to its time-of-flight at that cell6. It is important to note that the computation of sensitivity coefficients has taken into account the spatial correlation of permeability in the model. Selecting coarse grid scale for inversion The coarse grid scale for inversion should reflect the information content of production data. This information content varies spatially but unless unstructured grids are employed a compromise is reached with a structured grid. In this work a series of coarse grids at different resolutions were inverted and the solution spectra evaluated against the reference solution. The finer the grid used in inversion, the closer we can match the data, but with more CPU time and higher uncertainty. The coarsest grid giving an acceptable fit to the constraining production data was chosen to continue in the workflow. Figure 6 shows the average objective function for 30 realizations on four different coarse grids: they are 5x5x2, 10x10x4, 20x20x5, and 50x50x10. The point of diminishing returns is chosen at 20x20x5 so this coarse grid is retained for the remaining steps. The choice of coarse grid size is also somewhat pragmatic. There is a CPU cost tradeoff between inversion done on a finer scale and downscaling done at a coarser scale. As the coarse blocks encompass more fine cells the downscaling method must solve larger matrices and thus takes more CPU. Streamline simulation modified in well cells

0.6 Objective function

saturation components. This objective function is minimized in SSC using an optimization method to find an optimal set of perturbation values at selected master locations. These perturbations are then propagated to the entire model using kriging and the prescribed variogram model. The sensitivity coefficients of fractional flow and saturation are required to minimize the objective function. Detailed discussion on the computation of the sensitivity coefficient of fractional flow rate using semianalytical streamline solution is given in Wen et al.5. Using the same method, the sensitivity coefficient of saturation at a given time can be obtained as:

Figure 6: Objective function is reduced with increasing resolution until a point of diminishing returns. Although inversion is done on a coarse grid, the streamline simulator embedded in the SSC method tracks the streamlines in the fine scale within the blocks containing wells. The modification better tracks streamlines within the coarse blocks and avoids the ambiguity of launching streamlines at different scales. The idea is the following: (1) solve pressure field on coarse model; (2) solve flow for each coarse well block by refining the well block to the base fine grid using fixed flow rate boundary conditions computed from coarse model solution; (3) track each streamline from the borders of the coarse well block to the wellbore location. Figure 7 shows the comparison of water fractional flow curves at a well computed from a coarse grid using this method and from the corresponding fine grid model where a 1.0 Fractional Flow Water

data,

0.8 0.6 Fine scale reference

0.4 Coarse scale with well cell refinement

0.2 0.0 0.0

0.5

1.0 PVI

1.5

2.0

Figure 7: Single well water fractional flow for fine scale and coarse scale grid with fine scale resolution at well cells.

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constant permeability value is assigned to all fine cells within each coarse block. The acceptable agreement between the two curves indicates the validity of this method. There is less dispersion with the coarse-grid model than the fine-grid model as expected. With this refinement at the well blocks we have a much faster computation and yet retain most of the benefit of the fine scale model. Geostatistical Downscaling Methods The models generated by any inversion are necessarily coarse since production data is a crude non-linear response that integrates and convolutes fine-scale reservoir details. Moreover, limited computer resources prevent practical inversion of fine-scale models. Thus, to generate fine-scale geostatistical models constrained to inversion generated coarse-block average constraints, a downscaling process is required. Previous work by Wen et al.2 used a simulated annealing technique to downscale a coarse scale model inverted by SSC. One shortcoming of the annealing technique is that it is CPU intensive, yet the resulting downscaled model has typical annealing type of fuzzy features (i.e., higher nugget effect than desired). In this work, we employ two methods: (1) a Bayesian approach originally proposed by Doyen et al.11 and later extended by Behrens and Tran7 to integrate seismic map constraints into 3D reservoir models and (2) a block kriging approach10. These downscaling methods are depicted in Figure 9.

Figure 9: The downscaling method replaces each coarse block with multiple fine blocks whose aggregate properties are equivalent to the coarse block. n

represent the linear block average

N e w "S G S B a ye s" P os te rio r P D F

c

p zo {zSGS }, zV

SGS L o ca l P rio r P D F

L ike lih o o d P D F

h pcz {z }h gcz z ,{z }h o

~

SGS

V

o

SGS

*

Figure 8: The local posterior distribution of zo is the SGS-generated conditional distribution updated by the likelihood function from the coarse grid constraint.

where

i =1

n

∑a

i

Downscaling using Sequential Gaussian Simulation with Bayesian Updating The following derivation depicted in Figure 8 is similar to those shown in the above references. Let Z be a multivariateGaussian random function with zero mean and unit variance. In a discretized 3D geologic model, let V be a coarse block that comprises n point values z ( u i ) , i = 1,..., n . As an example, if each coarse block V is a rectangular volume consisting of 5 rows, 5 columns, and 4 layers of fine-scale grid cells, as shown in Figure 9, then n = (5)(5)(4) = 100. Let zV

zV = ∑ ai z (ui )

=1

(3)

i =1

The weights ai ’s could be constant or spatially varying; they are set to 1 n in this paper so each fine-scale grid cell contributes equally to the linear coarse-block average. Note that z ( u i ) ’s can be considered to have quasi-point support, whereas zV informs the volume V and hence is of block support. The objective is to geostatistically simulate a 3D realization of Z that would honor the above block-average constraint, given by Equation 1, in addition to prescribed histogram and spatial covariance models and any conditioning well data. We will populate the 3D grid with a modified form of Sequential Gaussian Simulation8 (SGS) which visits the grid cells via a random path. Let zo be the value to be simulated at the current grid cell, {zSGS } be the set of previously simulated values, and {zc } be the set of previously simulated values of only those grid cells that are contained within the current coarse block. Note that {zc } is a subset of {zSGS } . A value for zo can be simulated by random sampling of the local posterior probability distribution p( zo {zSGS }, zV ) , which is the probability distribution of zo given previously simulated values {zSGS } and the coarseblock linear average constraint zV . Application of Bayes theorem yields

p( zo {zSGS }, zV ) ∝ p( zo {zSGS }) g ( zV zo ,{zSGS })

(4)

8

4.

5.


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previously simulated data, those that are nearest to the current cell, are retained for kriging to keep CPU cost low. Solve an additional kriging system, Equation 9, to obtain the kriging weights λ i , for i ∈ c + o . The size of this kriging system varies from 1 × 1 (when the coarse block corresponding to the current cell does not contain any previously simulated values) to n × n (when the current cell is the last one visited in its corresponding coarse 2 block). Calculate σ g using Equation 8. Calculate mo and σ o using Equations 12 and 13; this involves subtracting the block average, zV , from the weighted sum of previously simulated values in the 2

Figure 11: Bayesian downscaling using the correct variogram (on left) and 90-degree incorrect variogram (on right). The variogram reproductions are given below which show that the method still honors the input inversion data even if it conflicts with the specified variogram. models. In this work, we focus on sensitivity in downscaling process on the selection of variogram model. The variogram model used in the SSC inversion step, γ SSC , will be reflected in the spatial distribution of the coarsegrid permeability fields. The geostatistical downscaling algorithm requires as input a variogram model, γ DS , and will

Figure 10: Layer 2 of coarse grid permeability from inversion used as constraint in downscaling. Note the SW-NE trend.

6.

current coarse block, zi for i ∈ c ; see Equation 12. Generate zo by random sampling from the local posterior

20%

65%

probability distribution N ( mo , σ o ) . Add zo to the list of previously simulated values and return to Step 3 until all grid cells have been visited. Back-transform the simulated normal score values to the original data space. 2

7. 8.

Sensitivities in Bayesian Method The sensitivity of SSC inversion results on the use of different initial models, as well as variogram models has been discussed previously by Wen. et al.1,2. It has been shown that SSC method is quite robust and major heterogeneity features can be revealed with different initial models and different variogram

90%

99%

Figure 12: Increasing amounts of nugget (20, 65, 90 and 90%) in the downscaled result.


attempt to honor it. If γ SSC = γ DS , then the spatial structure of the fine-scale permeability fields is consistent with γ SSC for all variogram lags. If γ SSC ≠ γ DS , then the SSC-downscaling workflow can be thought of as a two-scale characterization process where the coarse-scale spatial constraint is imposed in the SSC step and the fine-scale spatial constraint is imposed in the downscaling step. Figure 10 shows an areal slice of an SSC-generated 20x20x5 coarse model of Ln(k). The SW-NE trend, imposed by γ SSC , is clearly apparent. Downscaling is performed twice, once with γ SSC = γ DS , see upper left of Figure 11, and again with γ DS having the same anisotropy ratio as that of γ SSC but with an opposite SE-NW trend, see upper right of Figure 11. In the first case, the downscaled image seems consistent with the input variogram model as proven by the lower left of Figure 11. In the second case, the spatial characteristics of the coarse-grid constraint dominates the appearance of the downscaled image, but small-scale SENW trend could still be seen; the corresponding directional variograms are shown in the lower right of Figure 11. Figure 12 shows another experiment where the “correct” variogram form, i.e., anisotropy ratio and direction of γ DS is consistent with those of γ SSC , was used with increasing amount of relative nugget. As the relative nugget increases, the resemblance of the downscaled image to the coarse-grid constraint slowly diminishes. At 99% relative nugget, there is virtually no visual resemblance and yet, the coarse-grid average constraint is still honored with almost perfect accuracy. Based on our experience with the downscaling algorithm, we have concluded that the spatial structure imposed by γ SSC tends to dictate that of the fine-scale downscaled permeability field. The main utility of γ DS is to control small-scale spatial continuity which in turn “smoothes” the transition between cell values on either side of the coarse-block boundaries.

µ V is the kriging weight given to the coarse-block average constraint zV , µ i is the kriging weight corresponding to the previously simulated value zi , Coi is the (normalized) point-to-point covariance between the values at cell o and cell i , CoV is the point-to-block covariance between the value at cell o and the block value at V , and {SGS} denotes the set of previously simulated (visited)

• • • • •

cells. The above kriging weights are the unknowns in the following system of equations:

∑µ C j

ij

+ µ V CiV = Coi for i ∈{SGS}

j ∈{SGS }

and

∑µ C j

jV

+ µ V CVV = CoV

j ∈{SGS }

where CVV is the block-to-block covariance of volume V with itself. Figure 13 shows two realizations generated by SGSBK using the same coarse-scale constraints and variogram models as those of Figure 11. We have observed that SGSBK requires less CPU time because SGSBK procedure requires only one

Downscaling using Sequential Gaussian Simulation with Block Kriging We also investigated using Sequential Gaussian Simulation with Block Kriging (SGSBK), proposed by Behrens et al.10, for downscaling purpose. During the sequential simulation procedure, the simulated normal-score values at the current grid cell can be randomly drawn from a Gaussian distribution with mean and variance given by

mo , BK =

∑µ z

i i

+ µ V zV

i ∈{SGS }

and

σ o2 , BK = 1 −

∑µ C i

i ∈{ SGS }

where

oi

− µ V CoV

Figure 13: Block kriging downscaling using the correct variogram (on left) and 90-degree incorrect variogram (on right). The variogram reproductions are given below which show that the method still honors the input inversion data even if it conflicts with the specified variogram.


2.

3.

the various geostatistical and deterministic approaches. The current SSC algorithm would have to be modified to include an additional component in the objective function, Equation 1, to minimize the difference between the initial scaled-up model and the final history-matched model. This is probably the preferred method since seismic data, if used properly, could improve the initial model which would in turn speed up the history matching process. During SSC: Seismic data could also be included as a soft constraint in the SSC algorithm using a probability field perturbation technique as recently proposed by Capilla et al.15. It is particularly appealing to include 4D seismic data (giving estimated saturation and/or pressure changes) as a soft constraint in the SSC. The uncertainty in the porosity and/or pressure changes derived from 4D seismic data varies greatly throughout a reservoir; this variable uncertainty can be handled in the SSC algorithm. During final downscaling: Seismic data could be treated as just another linear constraint, with the appropriate averaging volume and averaging weights, in the Bayesian framework. The danger with this approach is that the SSC and seismic constraints may be conflicting which could lead to neither constraint being adequately honored.

Non-uniform grid Our current implementation assumes that the 3D model is uniformly gridded. For non-uniform grids, the SSC method needs to propagate the master points via kriging which right now is easy with the same block sizes everywhere. Alternatively these could be calculated for the different block sizes or an approximate perturbation could be propagated. On the downscaling side, the derivation remains valid as long as the constitutive fine cells are of equal support. However, the coarse-scale averages have to be properly transformed to the normal score domain to reflect the different volume supports. Relative Permeability The fine scale relative permeability curves were used throughout in coarse scale SSC inversion, i.e., no pseudo relative permeability curves (pseudos) were considered even for numerical dispersion correction at the different scales. Our ultimate goal will iteratively link the SSC inversion process with non-uniform up-scale process, including the up-scaling of relative permeability curves. Summary We have presented a workflow that takes a fine scale geostatistical 3D model conditioned to static data, coarsens it and constrains it to dynamic data, and downscales the coarse model back to a fine scale. New contributions include: 1. Coarse scale 3D streamline simulation with well block refinement, 2. Calculation of sensitivity coefficients for saturation using a semi-analytical streamline method, which allows fast inversion of 3D permeability fields

3.

conditioned to time lapse seismically derived saturations, Two new geostatistical downscaling methods using a Bayesian updating and a block kriging approach accounting for different support volumes and intraand inter-block spatial correlation.

Nomenclature C = covariance function C = point-to-block covariance function C = block-to-block covariance function D = simulation domain N = normal distribution Z = random function W =weight for production data a = weight for block averaging f = fractional flow rate g = likelihood function of k = absolute permeability m = mean of a normal distribution n = number of grid cells comprising a coarse block p = probability distribution function p( | ) = conditional probability s = saturation z = property being simulated zV = linear block average for volume V α = relative weight for objective component β = weight assigned to streamline λ = kriging weight µ = kriging weight γ = variogram model σ = standard deviation of a normal distribution τ = time-of-flight associated to streamline ω = power law average exponent ∈ = belongs to ∉ = does not belong to ∀ = for all ∝ = proportional by a constant factor {} = set of values Subscripts BK = block kriging SGS = sequential Gaussian simulation previously simulated cells SK = simple kriging l = streamline o = current cell to be simulated c = previously simulated cells in current coarse block c + o = union of c and o V = coarse block volume


xxxii, Aug 1996. 14 Xu, W., Tran, T.T., Srivastava, R.M., and Journel, A.G.: “Integrating Seismic Data in Reservoir Modeling: the Collocated Cokriging Alternative,” paper SPE 24742, presented at the 1992 SPE Annual Technical Conference and Exhibition, Washington D.C., October 4-7. 15 Capilla, J.E., Rodrigo, J, and Gomez-Hernandez, J.J.: “Geostatistical Structure of Simulated Transmissivity Fields That Honor Piezometric Data,” presented at the Second European Conference on Geostatistics for Environmental Applications, Nov 18-20, 1998.