Technique to Integrate Production and Static Data in a Self-Consistent Way Jorge L. Landa, SPE, Chevron Petroleum Technology Co.

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1 SPE Technique to Integrate Production and Static Data in a Self-Consistent Way Jorge L. Landa, SPE, Chevron Petroleum Technology Co. Copyright 2001, Society of Petroleum Engineers Inc. This paper was prepared for presentation at the 2001 SPE Annual Technical Conference and Exhibition held in New Orleans, Louisiana, 30 September 3 October 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 subect 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 subect 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 , Richardson, TX , U.S.A., fax Abstract This work presents a technique to integrate production data into the reservoir characterization workflow. The application of the technique ensures that the construction of earth models for reservoir simulation will be geologically sound and consistent not only with all the observed static and production data, but also with the earth modeling workflow philosophy, regardless of its complexity. The technique can be considered as an advanced reservoir simulation history matching process with a complex level of parameterization. The observed production data is applied to adust the key parameters in the workflow used to construct the static model. The magnitude of the parameter adustments is dependent on the uncertainty associated with the parameters. The problem is posed as an inverse problem and it requires the computation of the sensitivity coefficients of the field responses (production data) with respect to the parameters for inversion. This paper also introduces the concept of "static" sensitivity coefficients and explores their usefulness. An example is presented where the parameters for inversion included variogram parameters along with the mean and variance of a reference porosity histogram. The inversion was performed by adusting parameters in the fine scale model while the production data was computed in an upscaled model. Introduction Reservoir simulation models are essential in modern reservoir management. Reservoir models are used to predict the future performance of the field under different scenarios with the obective of optimizing the application of resources. The quality and uncertainty in the reservoir models will translate into the quality of the predictions. It has been recognized that high quality models should be geologically sound and be consistent with all the data available. The task of constructing a reservoir model with such characteristics is not a simple one. The difficulty resides in processing substantially different types of data. One subect of a large research effort in the industry is integrating geological, static and dynamic data. In general, the current reservoir modeling approach consists of two steps. The first step is constructing "static" models using static data, such as well logs, cores, geostatistics, seismic, etc. Very complex workflows are used in this step to ensure consistency with all the static data and the geological information. Next, in a separate step, the models generated in the previous step are adusted to honor the production data observed in the field. Current techniques to make the adustments are disconnected from the complex work made in the previous step in the static data side. As a result the final reservoir model may match the production data but may have lost the consistency with the static and geological data. Forecasts performed with these models will be more uncertain than with the models generated while full consistency is preserved. This work addresses this problem and provides one alternative for consideration. The work presented in this paper is an extension of previous work in the area of inversion using obect-modeling techniques combined with reservoir simulation 1,2,3. Description of the Method The application of the method described in this work is conceptually simple, although its implementation is complex. The procedure can be outlined as two steps. 1. Earth scientists construct a first reservoir model by using mainly static data, such as surfaces inferred from seismic information, well logs, core analysis, stratigraphic interpretations, geostistical data analysis, analog studies, outcrop information and geological setting models. For large fields this task usually requires a substantial amount of efforts and time. Integrated earth modeling software, such as GOCAD, is used to implement complex workflows that are specific to each reservoir. By the time the first static model is completed, an agreement and a definition about what is the correct workflow process for the field under study have been reached, and what are the most uncertain and relevant parameters and their

2 2 JORGE L. LANDA SPE respective ranges of variability have been detected. The workflow must describe the full process of reservoir model construction including upscaling and reservoir simulation. The list of uncertain and relevant parameters may include what is traditionally considered hard data, such as well logs. 2. An inverse problem is set, where the question posed is: what are the values that should be assigned to the uncertain parameters to generate a new reservoir model that will reproduce the historical production data when it is subected to numerical simulation. The key elements in this step are the selection of the parameters for inversion and the computation of the sensitivity coefficients of the production data with respect to the parameters of inversion. The last element is described in detail in the next section (Mathematical Theory). The above process seems similar to a simulation historymatching problem. What distinguishes it from the other methods in the literature is the choice of parameters for inversion. In this work the parameters for inversion are of the "workflow" and "hard-data" types. By doing this kind of selection it is guaranteed to end up with a reservoir model that will honor both the static and dynamic data, be geologically sound and, more importantly, be consistent with the workflow philosophy. The essence of the method is based on two concepts: - What is traditionally considered as "hard" data in model construction is not actually "hard". There is always uncertainty associated with this type of data. Thus, it makes sense to change them during the history matching process. - The parameters used in the reservoir model workflow are the result of subective interpretations, such as data analysis, guess work, analog comparisons, etc. Therefore, it is reasonable to change them in the process of history matching. In order to preserve the soundness of the history matched model, the changes indicated above should be restricted to ranges determined by the level of uncertainty associated with each parameter. The method can be viewed from two perspectives: - From the earth science point of view, as a method to preserve geological information. - From the inversion point of view, as a method to reduce the inversion parameter space by choosing a "natural" parameter set. The example included in this paper demonstrates the whole process. Mathematical Theory The inversion part of the process is based on the Gauss- Newton algorithm for parameter estimation. New features have been added to handle regularization of ill conditioned matrices and to restrict the parameter search space within prescribed constraints in an efficient way. The mathematical theory for this method has been extensively described in the literature. The particularities of the implementation for this work have been presented in previous publications 1,2,3,4 and they will not be repeated here. A critical step in the inversion method used in this work is the efficient computation of the sensitivity coefficients (derivatives) of the calculated production data with respect to unconventional parameters of inversion. In the example presented later the derivatives of the oil/water/gas production and pressures at wells with respect to parameters, such as the azimuth of a geostatistical variogram or the mean of a histogram need to be calculated. In previous work 1,2,3 it has been demonstrated that it is possible to compute the derivatives of the primary variables of an implicit numerical simulator with respect to any kind of parameter by using Eqn. 1. ( k + 1) ( k + 1) ( k + 1) ( k + 1) ( k ) f J X = D X (1) where Y X = (2) ( k + 1) k + 1 k f = f ( Y, Y, K, Φ, m) = 0 (3) ( k + 1) ( k + 1) f J = (4) ( k + 1) Y ( k + 1) ( k + 1) f D = (5) k Y Here X is the vector of sensitivity coefficients of the simulator's primary variables corresponding to parameter α. Y is the vector of the primary variables. For a three-phase, blackoil implicit simulator the primary variables can be defined as the pressure of the oil phase, water and gas saturation (Po, Sw, Sg) in each grid block, plus the bottom hole pressure at each well (Pwb). The index k in Equations (1)-(5) is the time step index in the numerical simulator. f is the vector of the discretized material balance equation solved by the numerical simulator. The material balance equations are functions of the primary variables Y, the permeability and porosity fields, K and Φ respectively, and other parameters m, such as relative permeability, fluid and rock properties, rate constraints, etc. The J matrix is the same Jacobian matrix that the implicit numerical simulator utilizes to solve for Y (k+1). The D matrix is a sort of Jacobian matrix, but the derivatives are computed with respect to primary variables converged in the previous time step. The sensitivity coefficients of the calculated production data Z are obtained by processing X, the sensitivity

3 SPE TECHNIQUE TO INTEGRATE PRODUCTION AND STATIC DATA IN A SELF-CONSISTENT WAY 3 coefficients corresponding to the simulator primary variables. That is: dcalc X Z = (6) Once the sensitivity coefficients Z are calculated, they are used in the minimization algorithm to calculate the changes that need to be introduced in the parameters α to reduce the value of the obective function. Computation of the last term at the right-hand side of ( k+ 1) Eqn.1, f, is not straightforward, since in general, the parameter α may not appear explicitly in the simulator's material balance equations. This apparent difficulty is overcome by applying the chain rule for differentiation, which is the key element of the method described in this paper. For example, in the case that α is a geological or geostatistical parameter this term can be computed by: ( k + 1) nxyz f f ki f φi = + (7) i= 1 ki φi where nxyz is the total number of active cells in the simulation grid. Computation of the derivatives of f with respect to permeability and porosity does not offer maor difficulty as they appear explicitly in the simulator material balance equations. The derivatives of the permeability and porosity fields with respect to α can be viewed as sort of "static" sensitivity coefficients of the reservoir model. The transfer function between α and permeability and porosity will be a function of the earth model workflow, thus these "static" derivatives are calculated outside the numerical reservoir simulator. In some special and simple cases these static derivatives can be calculated analytically, but in more general cases they need to be estimated by numerical perturbation. The CPU time to compute the "static" derivatives by numerical perturbation of the earth model workflow is relatively small when compared to the CPU time of solving for the production data sensitivity coefficients by Eqn. 1 Implementation The technique described in this work was implemented in a research code that consisted of the following components: a) Earth Science module: customized GOCAD software. Including geostatiscal algorithms and flow based upscaling capabilities. b) Reservoir Simulation module: modified version of Chevron s in-house reservoir simulator with the capability of computing sensitivity coefficients for black-oil models using Eqn. 1. c) Inverse module: composed by a constrained minimization algorithm that performs the parameter inversion. This algorithm is based on the Gauss-Newton algorithm. Details concerning the implementation were reported in previous work 1,2,3. d) Interfaces for the above modules. Example A reservoir model was constructed to test the applicability of the method. The reservoir model corresponded to a sector of the field model described by Cook, Chawathe, Larue, Legarre, and Aayi 5. Ten wells, 6 producers, 2 water inectors and 2 inactive wells, were within the boundaries of the model. The earth model was constructed using actual well data. The fluid and rock properties were the same as in the full-scale field model simulation 5. The workflow used to construct the earth model for this test was similar to one of the workflows that was used to study the full-scale field model, and it can be described as follows: 1) A stratigraphic fine scale grid was laid over the area of the reservoir to be modeled. The grid dimensions were 20 x 42 x 21 in the X, Y and Z directions, respectively. The grid dimensions in the X-Y direction were uniform. The grid dimensions in the Z-direction were constrained to the stratigraphic surfaces via well markers (see paper SPE for more detail). 2) Porosity at each cell of the model was assigned by Sequential Gaussian Simulation (SGS) geostatistical algorithm. The porosity field was constrained to the porosity logs at each well. An exponential variogram with an azimuth of 135 degrees and ranges of 1600 m and 1100 m in the main and secondary axis, respectively, were used as input. The parameters for the variogram were determined from the experimental variogram analysis of the entire well population in the field and by the geological insight on the sedimentological depositional model. 3) Porosity populated in the previous step was adusted to honor an external histogram. The external histogram was constructed using the entire well population in the field. 4) Vshale property was assigned to each cell of the model by using the collocated simulation geostatistical algorithm. The vshale property was constrained to the vshale calculated at the well locations. The porosity calculated in the previous step was used as soft data for the collocated simulation. The correlation coefficient vshale-porosity was set to be equal to -0.8 to The variogram model was the same one used to populate porosity. 5) Vshale populated in the previous step was adusted to honor an external histogram. The external histogram was constructed using the entire well population in the field. The procedure is similar to step 3. 6) Permeability in the X direction was calculated using an experimental function determined by core analysis. The function had the following form: log(k) = k1 + k2*vshale + k3*vshale*vshale (8) where k1, k2, and k3 are constants.

4 4 JORGE L. LANDA SPE ) The permeability in the Y direction was assigned to be equal to the permeability in X direction. The permeability in the Z direction was assigned to be 0.10 times the permeability in the X direction. 8) The permeability and porosity fields were upscaled using the flow based upscaling method developed by Durlofsky et. al 6. The dimensions of the upscaled grid were 20 x 21 x 13. The resulting upscaled grid in the Y and Z directions were not uniform. 9) The effect of an aquifer was modeled in the North and East boundaries by using pore volume multipliers. 10) Predetermined permeability and porosity cut-offs were applied to inactivate cells in the upscaled model that had low porosity and/or low permeability. 11) Numerical reservoir simulation was performed in the upscaled model. Oil production and water inection were constrained to historical data. The entire earth model, including the scale-up, was constructed using GOCAD software. Numerical reservoir simulation was performed using Chevron's in-house simulator. Synthetic production data was generated for the test model by constraining the producing wells to maximum oil production, minimum bottom hole pressure and maximum liquid production. Water inector wells were constrained to maximum bottom hole pressure and to approximate voidage replacement at the producing wells. The production data generated during this step was used as data for the inverse problem. The production data consisted of: bottom hole pressure, oil, water, gas production for the producing wells; and bottom hole pressures, water inection rates for the inectors. The set of reservoir parameters and production data calculated in the previous steps will be referred as the "true" reservoir in the rest of this paper. The "hard" data for this example consisted of the porosity and vshale at the well locations as they were interpreted from the well logs. The workflow parameters included variogram parameters (azimuth and ranges), porosity and vshale histograms, the parameters of the deterministic relationship between permeability and vshale (Eqn. 8), the vshale-porosity correlation index, and the seed numbers for the random number generators in the geostatistical algorithms. To check the applicability of the method, first a "wrong" model was constructed by using the same workflow described above with "wrong" parameters in the workflow and wrong" hard-data at one of the wells. Next the technique introduced in this paper was applied to see whether it was possible to recover the "true" workflow and hard-data type parameters. This process was to mimic the situation of history matching a reservoir simulation model when the simulated production data failed to reproduce the historical production data. The focus of this paper is to perform inversion using nonconventional history matching parameters. The purpose is to explore the possibility of using "workflow" and hard-data parameters for inversion in order to ensure consistency in the final model. For this example, the following seven parameters were chosen for inversion/history matching: - Parameter # 1 (az): Azimuth of the variogram. The azimuth represents the direction of maximum continuity in the reservoir. - Parameter # 2 (r1): Range of the variogram in the direction of the azimuth. - Parameter # 3 (r2): Range of the variogram in the direction 90 of the azimuth. - Parameter # 4 (pw38): Porosity multiplier to the porosity log in well #38. - Parameter # 5 (k1): First parameter in the log(k) vs. vshale relationship (Eqn. 8), the other parameters k2 and k3 were considered as constants. - Parameter # 6 (mpor): The statistical mean of the porosity histogram used to constrain the porosity geostatistically simulated in steps 2 and 3 of the workflow for this example. - Parameter # 7 (sdpor): The statistical standard deviation of the porosity histogram used to constrain the porosity geostatistically simulated in steps 2 and 3 of the workflow. Parameters # 1,2,3,5,6 and 7 are of the workflow type. Parameter # 4 is of the hard-data type. Parameters # 1, 2 and 3 (variogram parameters) are usually ill determined in actual field studies. Sampling points (wells) are scarce when compared to the area to be populated. Well spacing also limits the estimated variogram range. In most of the cases these parameters are set using apriori information such as the sedimentological environment model for the area. Thus, it is natural to use these parameters for inversion. Parameter # 4 is a multiplier in the porosity log of a well. Here the intention is to study how the uncertainty in the well log interpretation can be used as a parameter for inversion. In conventional history matching, permeability and porosity multipliers are frequently used at the well locations and they affect only the grid cells where the wells are completed. In the method subect of this paper the multiplier effect propagates through the entire model via the geostatistical algorithms. Parameter # 5 has a maor influence on permeability estimation. This parameter is calculated from core analysis and may be subect to many interpretations. Parameters # 6 and 7 are the statistical mean and standard deviation of a property distribution used to constrain geostatistical algorithms. Usually these parameters are calculated from interpreted well log information. One of the problems inherent to the last procedure is that the wells do not provide an un-biased sampling of the reservoirs. In many cases a large number of wells are located in the "best" areas of the reservoir. Models constructed using this information may result in being too optimistic. It is then natural to choose this as a parameter to study. The unique characteristics of the set of parameters chosen for this study are:

5 SPE TECHNIQUE TO INTEGRATE PRODUCTION AND STATIC DATA IN A SELF-CONSISTENT WAY 5 a) All the parameters are related to the fine scale model. Their connections to the simulation model are dependent on the upscaling algorithm. In other words: our intent is to modify the fine scale model to obtain an history match in the upscaled model. b) Porosity is indirectly related to permeability. Porosity is used as soft data in the collocated geostatistical simulation of permeability. c) Parameter # 5 has no connection to porosity. d) Parameters # 1, 2 and 3 have direct connections to both permeability and porosity. e) Parameters # 4, 6, and 7 have direct connections to porosity and indirect connections to permeability. Obviously the list of the inversion parameters can be extended to cover other hard-data and workflow type parameters. The obective of this exercise was to test the applicability of the method before it can be applied to the full-scale field model. In a full-scale field other parameters related to the vshale property should be included. Fig.1 shows a three-dimensional view of one layer of the fine scale model (grid 20 x 42 x 21) constructed using the workflow described before. Figures. 2 and 3 show X-Y and Y-Z cross sections of the upscaled grid respectively (grid 20 x 21 x 13). Note the non-uniform grid in the X-Y section. Table 1 shows the workflow parameters that were used for the test. The true reservoir column contains the parameters that will produce the true reservoir. The initial guess column contains the parameters that will produce the first model for history matching. The last column of the table shows the maximum and minimum values that are allowable for each of the parameters provided by the earth scientist. Any model constructed within the ranges predetermined in this table and matching reasonably the production data will be considered acceptable and consistent with the static and dynamic data and the earth model workflow. Figures 4 and 5 show the bottom hole pressure and water production rate of well #38 calculated by simulation using the true reservoir model and the initial guess. The plots for the other wells are not shown here because of space limitations. The production data used as input for inverse module consisted of: (a) Oil, water and gas production rates for 6 wells. (b) Water inection rates for 2 wells. (c) Bottom hole pressure - either static or flowing pressure (to mimic the shut-in s and permanent down hole pressure gauge information), for all the active wells in the sector of the field under study. The obective function for the minimization algorithm was defined as a general least square type, E(α) = (d cal -d obs ) T C d -1 (d cal -d obs ), where α is the vector of parameters for inversion, d is the vector of measurements (pressure and production/inection rates), the indices calc and obs mean calculated with the reservoir simulator and observed in the field respectively. C d is the covariance matrix of the data and it is assumed to be diagonal. The obective function measures the mismatch between the observed data and the one calculated by the numerical simulator during the history matching process. The constrained optimization in the inverse module applied the Gauss-Newton algorithm with Levenberg- Marquardt method for stabilization. Constrains in the parameter space were defined in column 5 of Table 1. At each iteration of the inversion process, 1) The fine scale reservoir model was constructed using GOCAD. The input consisted of the fine scale grid skeleton, well logs and the workflow scripts prepared to be dependent of the seven inversion parameters. 2) The fine scale model was upscaled. For simplicity the upscaling template was kept unchanged during all the process (the process can include a variable template option). 3) The "static" sensitivity coefficients required for Eqn. 7, which are the derivatives of the permeability and porosity fields in the upscaled model with respect to the inversion parameters were estimated by perturbing the fine scale model and tracking the changes through the upscaling process. 4) The upscaled porosity and permeability fields and the "static" sensitivity coefficients were used as input for the numerical simulator. The simulator provided two important inputs for the inverse module: the calculated production data d calc and the sensitivity coefficients Z of the production data to the inversion parameters. 5) The inverse module calculated a new set of parameters. The process, steps 1 to 5 were repeated in an iterative manner until the obective function E satisfied the convergence criteria or the change in the new parameter was less than a predetermined threshold. If the last situation had arisen and the value of the obective was considered to be high (bad match of production data), then one option was to change the weights of the data and to continue with the process. Figures 6 and 7 show the sensitivity coefficients corresponding to the same calculated production data shown in Figures 4 and 5, that is the bottom hole pressure and water production rate at well # 38 in the "true" reservoir model. Not shown in the paper but calculated in the simulator are the sensitivities of the gas production rate, static pressure and oil production. The sensitivity coefficient of the oil production is zero when the history matching is constrained to oil production and it is not zero when the history matching is constrained to total well fluid production, or when a bottom hole pressure constraint is reached during the simulation. The sensitivity coefficients are time and reservoir model dependent. They are different for each well and the relative importance of each parameter is also different for each well, depending mostly on the relative position of the wells, boundaries, faults etc. Figures 6 and 7 provide a sense of

6 6 JORGE L. LANDA SPE importance of each parameter for well #38. To facilitate the comparison the derivatives were scaled to the following units: - azimuth: prod data / 10 degrees - variogram ranges: prod data / 200 meters - parameters related to porosity: prod data /1 % porosity - k1: prod data / 1% change The accuracy of the "dynamic" sensitivity calculation was verified by re-computing them carrying the numerical perturbation to the simulation model. Fig. 8 shows the "static" sensitivity coefficients for layer 4 of the "true" model. They are the derivatives of the porosity and permeability field in the upscaled model with respect to (wrt) the inversion parameters, which in this example are related to fine scale model. The "static" sensitivity coefficients are different for each layer and are dependent on the values of the parameters. Figures 8.a.1 and 8.a.2 show the permeability and porosity fields respectively in a color scale. The index 1 and 2 in Fig.8 mean "related to" permeability and porosity respectively. Figures 8.b - 8.h show the static sensitivity coefficients in "flag" style maps where blue represents positive values, red represents negative values and white represents zero. A more detailed analysis on the information that these plots provide is given in the next section (Observations). After ten iterations the inversion/history-matching process was stopped as the obective function reached a low value. Table 2 shows the evolution of each parameter during the inversion process. The last row of the table shows the true value of the parameters. It can be verified that it was possible to recover all the parameters within a reasonable limit, thus the test can be considered as a success. Observations At the first iteration of the history matching process, the variogram parameters were changed significantly and the maor reduction in the mismatch of the production data was observed in the water production data. In this example, despite that the variogram ranges were of the same order of magnitude as the dimensions of the test model they still had an important effect on the water breakthrough and water cut behavior. It is expected that variogram parameters will have an even more pronounced effect in the full-scale field studies since variogram parameters control the direction and length of the continuity in the reservoir and thus control how the fluids move in the reservoir. Variogram parameters are not considered as parameters during conventional history matching. The mean and standard deviation of the reference histogram have important effects on reservoir behavior. Using them as parameters for inversion resembles to using zonal permeability and porosity multipliers in conventional history matching. The conceptual difference is that in our method the selection of the cells and the modification of the value of the property are not arbitrarily, but in a way that ensures consistency with the workflow. Thus, our method will always provide reservoir models that will be accepted from the geological point of view. The concept of "static" sensitivity coefficients was developed along with the new inversion technique. The static sensitivity maps provide very useful information that can be applied in different ways, for example: a) The maps give "insight" to the earth scientist about how each piece of information (hard data) and the decisions made during the construction of model (workflow parameters) affect the final product, the reservoir model. They indicate what is important and what is not relevant. Obviously the analysis has similar limitation of any linearization methods. Conventional statistical analysis performed on the sensitivity data combined with the qualitative visual analysis improves the evaluation. Table 3 summarizes the main statistical indices for each sensitive coefficient. The sensitivities were scaled in such a way that it was possible to perform comparisons between them. The histograms are illustrated in Figures 9, 10 and 11. These means are valuable tools for evaluation since statistical indices alone are not enough to describe the distributions. b) The sensitivity maps indicate the effect of the algorithms used in the model construction. For example many of the "flag" type maps in Fig. 8 show that positive-value zones balance out with the negative-value zones. This is due to the feature in the algorithms to honor histograms. Thus, increases in one area should be balanced by decreases in another. What is not intuitive is which parts in the reservoir are "blue" zones and which are "red" zones. The sensitivity maps help to locate different color (positivenegative) zones. c) The sensitivity maps give an indication on how the uncertainty in the hard data propagates through the model during its construction. Figures 8.e.1 and 8.e.2 illustrate how an increase in the porosity at well #38 results in a positive increase of permeability and porosity in the surrounding area but also causes a reduction in the rest of the reservoir. d) The maps and associated plots can be used to determine in a simple way whether there is correlation between parameters. These correlations cause stabilization problem in the inversion process and add unnecessary computational work during the calculation of the "dynamic" sensitivity coefficients in the simulator. For example, in Fig. 8 the oint visual comparisons of maps 8.c.1 vs. 8.d.1 and 8.c.2 vs. 8.d.2 indicate a possible correlation between the two main ranges in the variogram (r1, and r2 respectively). A cross plot of the sensitivity of permeability in X-direction with respect to both variogram ranges (Fig. 12) confirms that a mild correlation exists. On the other hand, a cross plot of the sensitivity of permeability with respect to the azimuth and main variogram range (Fig. 13) clearly indicates the independence of these two parameters. Correlation in the "static" sensitivity space implies correlation in the "dynamic data" space. Unfortunately the opposite, independence, does not apply. The procedure helps in choosing the parameters for inversion.

7 SPE TECHNIQUE TO INTEGRATE PRODUCTION AND STATIC DATA IN A SELF-CONSISTENT WAY 7 e) A comparison of the "static" sensitivity coefficients calculated in fine scale with those computed in upscaled coarse model may indicate what information may have been lost in the upscaling process. For example, if the areal coarsening is too large the sensitivity to the variogram ranges may be lost. It is also interesting to notice that in the test example, despite only a "soft" link between porosity and permeability, parameters such as the mean and variance of the reference porosity histogram had a significant effect on the permeability field and on the well responses. The last two rows of Table 3 and Figs. 6 and 7 demonstrate this behavior. Given the complexity of the parameters that are necessary to deal with in the method presented here, the problem of choosing the right parameters for inversion is not trivial. The recommended procedure for choosing parameters is to analyze the "dynamic" sensitivity coefficients using resolution and variance analysis 3,4. The static sensitivity coefficient analysis can be used as a first pass in the parameter selection process. The example emphasized the utilization of workflow type parameters for inversion because it is a new concept. Practical application of the method can include more conventional parameters such as aquifer pore volume multipliers, fault transmissibility, vertical to horizontal permeability ratio, skin / PI multipliers at the wells, etc. The research code developed for this work can accommodate a mix of parameter types. Conclusions A technique has been developed to provide a selfconsistent procedure to integrate production and static data for reservoir model construction. By choosing the key parameters in the workflow for constructing the static model as parameters of inversion one can generate models that: - honor the static data; - honor the dynamic data; - are geological sound; - are consistent with the workflow philosophy. It is possible to perform history matching of reservoir models using non-conventional parameters, such as geostatistical parameters. It is possible to adust the fine scale model while performing history matching in the upscaled model. The concept of static sensitivity coefficients has been introduced, and their usefulness has been explored. Nomenclature az = variogram azimuth, degree. C d = covariance matrix of the production data. d = production data vector. E = obective function. k = permeability, md. k1 = first coefficient in the vshale-permeability correlation. mpor = statistical mean of the reference porosity distribution. r1 = variogram range in the azimuth direction, m. r2 = variogram range 90 o from the azimuth, m. pw38 = porosity multiplier for well # 38. sdpor = statistical standard deviation of reference porosity distribution. wrt = "with respect to". Z = vector of sensitivity coefficients of the production data with respect to the inversion parameters. α = parameter for inversion. φ = porosity. Subscripts i = grid cell index. = parameter index. Superscripts cal = calculated. k = iteration number. obs = observed. Acknowledgements The author thanks the management of Chevron Petroleum Technology Co. for authorizing the presentation of this work. References 1. Horne, R.N.: Reservoir Characterization Constrained to Well Test Data: A Field Example, paper SPE presented at the 1996 SPE Annual Technical Conference and Exhibition, Denver, CO, Oct Landa, J.L.: Reservoir Parameter Estimation Constrained to Pressure Transients, Performance History and Distributed Saturation Data, Ph.D. dissertation, Stanford University, Stanford, CA, Landa, J and Horne, R: A Procedure to Integrate Well Test Data, Reservoir Performance History and 4-D Seismic Information into Reservoir Description, paper SPE presented at the 1997 SPE Annual Technical Conference and Exibition, San Antonio, TX, Oct Anterion, F., Eymard, R., and Karcher, B.: Use of Parameter Gradients for Reservoir History Matching, paper SPE presented at the 1989 SPE Symposium on Reservoir Simulation, Houston, TX, Feb Cook, G., Chawathe, A., Larue, D. Legarre, H., and Aayi, E.: Incorporating Sequence Stratigraphy in Reservoir Simulation: An Integrated Study of the MerenE-01/MR-05 Sands in the Niger Delta, paper SPE presented at the 1999 SPE Reservoir Simulation Symposium, Houston, TX, Feb Durlofsky, L. J., Beherens, R. A, Jones R. C., and Bernath, A.: Scale Up of Heterogeneous Three Dimensional Reservoir Derscriptions, paper SPE presented at the 1995 SPE Annual Technical Conference and Exhibition, Denver, Oct SI Metric Conversion Factors bbl x E-01 = m 3 psia x E+00 = kpa

8 8 JORGE L. LANDA SPE Table 1. Parameter definition for the example problem Parameter # 1 # 2 #3 #4 #5 #6 #7 Parameter type "True" value Initial guess Range Variogram azimuth (degree) Variogarm range 1 (m) Variogram range 2 (m) Porosity well # Permeability -Vshale correlation Mean porosity reference histogram Standard deviation porosity reference histogram Fig. 1 3-D view of the structure and porosity distribution in layer 9 of the fine scale model (20 x 42 x 21). Table 2. Evolution of the parameters during the inversion process Iteration # 1 ( az ) 1 (Initial guess) # 2 ( r1 ) # 3 ( r2 ) Parameter # 4 (pw38) #5 ( k1 ) # 6 (mpor) # 7 (sdpor) True value Fig. 2 X-Y view of upscaled grid (20 x 21 x 13). Table 3. Statistics of the static sensitivity coefficients computed for true reservoir model unit max min mean std-dev δφ / δaz % / 10 degrees δφ / δr1 % / 200 m δφ / δr2 % / 200 m δφ / δpw38 % / 1% δφ / δk1 % / 1% δφ / δmpor % / 1% δφ / δsdpor % / 1% δkx / δaz md / 10 degrees δkx / δr1 md / 200 m δkx / δr2 md / 200 m δkx / δpw38 md / 1% δkx / δk1 md / 1% δkx / δmpor md / 1% δkx / δsdpor md / 1% Fig. 3 Y-Z view of the upscaled grid (20 x 21 x 13).

9 SPE TECHNIQUE TO INTEGRATE PRODUCTION AND STATIC DATA IN A SELF-CONSISTENT WAY 9 Well # 38 - Well Bottom Hole Pressure (WBHP) Sensitivity of Bottom Hole Pressure for Well # 38 Pressure - PSIA WBHP -true model WBHP - initial guess Time - days psia mpor r2 r1 20 k1 0 pw38 sdpor az Time - days Fig. 4 Well # 38 - Bottom hole pressure calculated using the "true" reservoir model and the initial reservoir model. Fig. 6 Well # 38 - Sensitivity coefficients of the bottom hole pressure to the seven parameters of inversion ("true" reservoir model). Water Production Rate - BWPD Well # 38 - Well Water Production Rate (WWPR) WWPR - True Model WWPR - initial guess Time - days bwpd Sensitivity of Water Production Rate for Well # sdpor pw38 0 k1 az -100 r2-200 r mpor Time - days Fig. 5 Well # 38 - Water production calculated using the "true" reservoir model and the initial reservoir model. Fig. 7 Well # 38 - Sensitivity coefficients of the water production rate to the seven parameters of inversion ("true" reservoir model).

10 10 JORGE L. LANDA SPE (a.1) (a.2) (b.1) (b.2) (c.1) (c.2) (d.1) (d.2) (e.1) (e.2) (f.1) (f.2) (g.1) (g.2) (h.1) (h.2) Fig. 8 Static Sensitivity coefficients for layer 4 of the upscaled model. (a1) X-perm field. (a2) Porosity field. The plots in the left column show derivatives of permeability, and the plots in the right column show derivatives of porosity. (b 1&2) Derivative wrt variogram azimuth. (c 1&2) Derivative wrt to variogram range r1. (d 1&2) Derivative wrt to variogram range r2. (e 1&2): Derivative wrt porosity multiplier in well 38, pw38. (f 1&2) Derivative wrt parameter k1 in eqn. 8. (g 1&2) Derivative wrt the mean of the reference porosity distribution, mor. (h 1&2) Derivative wrt the standard deviation of the reference porosity distribution, sdpor.

11 SPE TECHNIQUE TO INTEGRATE PRODUCTION AND STATIC DATA IN A SELF-CONSISTENT WAY 11 Fig. 9 Histogram of the sensitivity coefficient of the porosity in the upscaled model with respect to the azimuth of the porosity variogram in the fine scale model. Unit: porosity fraction / degree. Fig. 12 Cross Plot of the sensitivity coefficients of the permeability in X-direction in the upscaled model with respect to the main and secondary ranges (r1 and r2) in the fine scale model. Index of correlation = Fig. 10 Histogram of the sensitivity coefficient (derivative) of the permeability in the X-direction in the upscaled model with respect to the azimuth of the porosity (vshale) variogram in the fine scale model. Unit: mdarcy / degree. Fig. 11 Histogram of the sensitivity coefficient (derivative) of the porosity in the upscaled model with respect to the standard deviation of the reference porosity histogram in the fine scale model. Unit: porosity fraction / porosity fraction. Fig. 13 Cross Plot of the sensitivity coefficients of the permeability in X-direction in the upscaled model with respect to the variogram azimuth and main range (r1) in the fine scale model. Index of correlation =