Copyright 2005, Society of Petroleum Engineers Inc. This paper was prepared for presentation at the 2005 SPE Reservoir Simulation Symposium held in Houston, Texas U.S.A., 31 January 2005 – 2 February 2005. This paper was selected for presentation by an SPE Program Committee following review of information contained in a proposal 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 a proposal of not more than 300 words; illustrations may not be copied. The proposal 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 Response surfaces (RS) are useful and simple proxies to simulators, which relate in a closed form output variables (oil recovery factor, oil rate, etc.) of a simulator to input parameters over the entire parameter space. These proxies can then be used to compute production forecast uncertainty profiles with Monte Carlo calculation, instead of relying on time consuming reservoir simulation.

Typically, design of experiment (DOE) and response surface methodologies (RSM) are used to both identify statistically significant factors and to generate RS. RS are usually low order polynomials, which are generated by using the regression method. These methodologies work well if response surfaces are smooth with weak non-linearity. However, when these polynomials fail to meet reasonable accuracy criteria, adding more points to the design will only marginally improve RS accuracy. Response surface methodology can be adapted for such cases by using interpolation methods (Kriging, Spline, etc.) instead of regression method. But, most interpolation methods tend to smooth out non-linearity. These interpolated proxies can become substantially inaccurate, if: 1. Non-linear effects are very strong; 2. The non-linearity is not uniformly distributed throughout the parameter space.

In this paper, we introduce a novel RSM and its supporting DOE methodology to construct response surfaces as proxies of a simulator when input factors of the simulator such as faults cause strong non-linear effects. These methodologies are used to generate RS of arbitrary shapes by iteratively interpolating on a multilevel grid in the experimental space, which allows the local subdivision of the parameter domain. New partitions and interpolation points are added adaptively in the selected parameters regions, if local errors of the constructed response

surface exceed a pre-selected threshold. The art of these methodologies consists in:

1. Splitting the whole domain into sub-domains of multiple scale levels, where the components of a RS can be accurately modeled with ‘thin plate’ spline interpolants. The resultant RS is obtained by combining its global and local components.

2. Achieving adequate RS accuracy with the minimum number of simulation runs.

We modify the SPE9 Benchmark problem to investigate the method’s validity and efficiency. We compare it with traditional DOE and RSM methodologies. Finally, we perform Monte Carlo simulation using the RS proxies to assess uncertainty of the production forecasts. Results show that the presented methodologies work accurately and efficiently for problems with strongly non-linear effects, especially when the non-linear effects are non-uniformly distributed. Introduction DOE is an experimental strategy1-3 used to efficiently collect experimental/simulation data to construct response surfaces with RSM, which is a collection of mathematical methods and statistical inferences1-2. DOE and RSM were initially and systematically presented by Box and Wilson 4.

The applications of DOE and RSM in petroleum industry began in the early 1990’s 5-7. They have been primarily applied to uncertainty analysis of performance forecasts of reservoirs8-11, history matching12-13, and well scheme optimization11-14. In the early development phase of a reservoir, geological, petrophysical, and engineering data are scarce, because of high acquisition cost. Consequently, there exist large uncertainties in geological/flow models generated from insufficient physical data. This results in large uncertainties for the predicted performance, such as oil recovery factor, field oil production rate, on which the reservoir management decision process is based. Quantifying uncertainty in production forecast helps decision makers assess the potential economic rate of return. Response surfaces can be applied to analyze the uncertainty in performance forecast of reservoirs and to quantify the financial investment risks. They are simple functions, unlike complicated reservoir simulators, which are time consuming non-linear partial differential equation solvers. Hence, they are much easier and faster to use than reservoir simulators for Monte Carlo experiments, to collect data for uncertainty analysis. History matching is an inverse problem, in which an engineer calibrates key geological/reservoir model parameters via

SPE 92853

Novel Multiple Resolutions Design of Experiment/Response Surface Methodology for Uncertainty Analysis of Reservoir Simulation Forecasts B. Li and F. Friedmann, SPE, California Inst. of Technology

2 SPE 92853

fitting simulator’s output to the real reservoir production history. History matching is a tedious process, in which the reservoir engineer’s experience plays a key role. It is subjective, because model parameters are often adjusted using a ‘one factor at a time’ approach. However, history matching can be very time consuming, since many simulation runs may be required. As an alterative, a response surface could be used as proxies to a reservoir simulator to guide automatic history matching. Therefore, engineers could achieve reasonable history match results faster than that with traditional methods. Similarly, RS provide an efficient approach to build up an objective function to be optimized for well scheme optimization. All these applications of DOE/RSM rely on efficiently constructing accurate response surfaces, which can then be used in lieu of the simulator to perform incremental calculations.

Response surfaces are usually constructed with regression method5-12, interpolation method12, 14, and neural network7. The regression method is easy to implement. However, the constructed surfaces are inaccurate unless the problem is weakly non-linear or is confined to small regions of a parameter space2, 7. Interpolation methods, such as Kriging12-14 and ‘thin plate’ spline methods15, have the advantage over the regression method of honoring experimental data. They can also handle scattered data. However, they tend to smooth out rapid changes in response surfaces. Neural networks could provide an accurate proxy to a reservoir simulator. However, the required training and testing data sets would have to be very large. Therefore, its computing cost would be too high, since reservoir simulation is performed to collect these data sets8.

For uncertainty analysis of reservoir performance forecasts, it is required that a reservoir simulator proxy covers full ranges of the factors of investigation. Furthermore, factors which cause strong non-linear effects on the predicted response (i.e. faults’ transmissibility) will increase the challenge of generating accurate response surfaces. Therefore, a novel RSM and its supporting DOE methodology are needed to accurately and efficiently construct RS with strong non-linear effects, especially when these effects are not uniformly distributed throughout the parameter space.

Here, we introduce a nested DOE methodology to collect simulation data and a novel RSM to iteratively construct response surfaces on a nested grid in an N-dimensional parameter space. We call them multiple resolutions design of experiment (MRDOE) and multiple resolutions response surface methodology (MRRSM). To logically introduce the ideas of the novel methodologies, we present the mathematical model and algorithms of MRRSM first. Then, we present MRDOE. Furthermore, we verify the accuracy of response surfaces constructed with this approach using a 5-factors problem based on the modified SPE9 benchmark problem16. We also compare these novel methodologies with classical methodologies. Finally, we perform uncertainty analysis of reservoir forecasts with the reconstructed RS for the modified SPE9 problem 17.

Multiple Resolution Response Surface Methodology Consider a system, whose output response variable y is a function of multiple input parameters xi, i = 1, 2 … n:

ε+= ),...,,( 21 nxxxfy ……………………………... (1)

Here, ε represents the random error, which has an independent normal distribution with zero expectation and uniform variance σ2. The expected value E(y) = f(x1, x2,…, xn) is called a response surface RS. Constructing response surfaces should meet two requirements:

1. RS must accurately represent the output response variable y;

2. Computational efficiency must be optimized. When the output response variable is experimentally

measured, error is introduced with each sampling. In such cases, constructing a RS with multivariate regression methods is the appropriate choice. This is the conventional RSM method2. However, there is no measurement error in reservoir simulation studies (ignoring numerical round off error). In such cases, interpolation methods (kriging and ‘thin plate’ spline interpolation), which exactly honor experiment data, may be more appropriate than multivariate regression methods to construct RS.

Moreover, modeling non-linear responses could be difficult to handle with conventional regression methods, especially if the parameters space is large. One way to handle such problems is to decompose the large parameter domain into sub-domains. One drawback of this approach is that too many partitions and points could be needed in regions where the RS is highly non-linear. Using interpolation methods (Kriging and ‘thin plate’ spline) can reduce the need for excessive partitioning and increase computational efficiency in many cases. However, even these methods will lose their efficiency when the non-linear effects are very stiff. Keep in mind that they tend to smooth the response.

The MRRSM methodology is novel to construct response surfaces. Both global and local information is included as a way to account for non-uniformly distributed non-linear effects. In this work, we use ‘thin plate’ spline interpolants to analyze global and local information on a nested grid. ‘Thin Plate’ Spline Interpolant Next, we briefly describe the ‘thin plate’ spline interpolant. We have extended it from 2-dimensional space15 to n-dimensional space for this work.

In n-dimensional space, the ‘thin plate’ spline is the fundamental solution to the bi-harmonic equation:

0),...,( 212 =∆ nxxxU ….………………………….(2)

It has the radial functional form:

rrrU ln)( 2= , ..……………………………………….(3)

where 2/1222

21 )...( nxxxr +++= .

For a given interpolation data set, a weighted combination of ‘thin plate’ splines, centered at each interpolation point, computes exactly the interpolant passing through these points while minimizing the so-called ‘bending energy’. A ‘thin plate’ spline interpolant has the vector form:

,),(1

)( Ω∈⋅+⎥⎦

⎤⎢⎣

⎡⋅= xxhf

xax s …………………….(4)

SPE 92853 3

where, ,] [ 21t

n, ..., xx,x=x ,][ 210t

n, ..., a, a, aa=a ,][ 21

tN, ..., f, f f=f ,)]()()()( 21

tN, ..., h, hh xxxxh [=

,ln)( 22iii rrh =x ,])([||)( 2/1

1

2,∑

=

−=−=n

lillii xxr xxx xi is

the ith interpolation point vector in the n-dimensional space Ω, and N denotes the total number of interpolation points. Eq. (4) shows that ‘thin plate’ spline interpolant is suited for scattered data points. It uses radial functions as basis functions and also includes a linear polynomial.

The bending energy for the ‘thin plate’ spline interpolant in the n-dimensional space Ω is defined as the integral of the sum of squares of the second derivatives over Ω:

n

n

i in dxdxdxs

xxxxsI 21

2

111 ])[()],([ ∫∫ ∑

Ω = ∂∂

= ...(5)

The name ‘thin plate’ spline refers to a physical analogy to the bending of a thin plate of metal. The deflection of the ‘thin plate’ is orthogonal to the plate.

This interpolant has the following constrains: • Nodal values

,21,)( , ..., N, iss ii == x …………………………... (6)

where si is the exact value of s(x) at interpolation point xi; • Zero total ‘force’

01

=∑=

N

iif ; …...………………………………………. (7)

• Zero total ‘force moment’

0x =∑=

N

iiif

1

. ………………………………………… (8)

By using these constraints, we define a linear equation system for solving constant parameter vectors a and f. Mathematical Model and Algorithm MRRSM is a response surface methodology which iteratively constructs the components of a RS on a hierarchical grid in its parameter space. The reconstructed response surface incorporates both global and local information in order to efficiently handle non-linear features.

Its mathematical model is defined as:

,),()( , Ω∈= ∑ ∑ xxx j k

kjs φ ……………………… (9)

where, x is a point vector in the n-dimensional parameters space Ω, )(, xkjφ is a non-zero scaling function in the kth sub-domain Dj,k at scale level j (resolution level j or grid level j). We define it as:

⎩⎨⎧

−Ω∈∈

=; 0,

kj

kjkjkj

s

,

,,,

;),()(

DD

xxx

xφ ………………………... (10)

where, s j, k(x) is a ‘thin plate’ spline interpolant. We select the ‘thin plate’ spline interpolant as a scaling

function for the following reasons. First, it is a rational basis function combined with a linear polynomial, which can be easily extended to accommodate multi-dimensional parameters space. Second, it can handle scattered data, which

provides flexibility for constructing response surfaces. Third, it is not like a Kriging interpolant, the variogram of which needs to be assumed or estimated17. There are no extra parameters to be estimated. Fourth, ‘thin plate’ spline interpolant can adequately match a response surface in sub-domains where the response is non-linear.

The first scaling function )(1,1 xφ in Eq. (9) is constructed from an interpolation data set on the whole domain D1,1, i.e. Ω (scale level 1). It is the first approximation )(1 xs of the RS s(x). We have:

1,11,11 ),()( D∈= xxx φs . …………………………….(11)

In order to estimate the accuracy of )(1 xs , we compute the residual error of )(1 xs at a set of verification points xp:

1,11 ),()()( D∈−= pppp ssr xxxx . ………………...(12)

We define a norm, such as maximum relative error or average relative error, to measure the residual error r(x) of a reconstructed RS. We require that a reconstructed RS should be accurate enough, so that it is a ‘true’ approximation to the real RS. If a norm ||)(|| xr of the residual error of the initial approximation )(1 xs of RS is smaller than a pre-selected threshold εr, we exit the modeling process at scale level 1. Else, we perform domain decomposition on the whole domain D1,1 to generate sub-domains at scale level 2 in regions where the norm of residual error )(xr exceeds εr. In these sub-domains, we generate the approximation )(1 xr to the residual error surface with computed residual error at points xp,

1,1D∈px , p = 1, 2, … These data points px belong to either the interpolation data set or the verification data set at scale level 1. The second scaling function k2, ),(xφ k,2D∈x , k = 1, 2, …, N2 is approximated by )(1 xr . We check the accuracy of the second reconstructed RS )( )()( ,2112 xxx ks φφ += using artfully selected verification points on sub-domains at scale level 2. We exit the modeling process, if the reconstructed RS

)(2 xs is accurate enough. Else, we continue space partitioning and residual surface constructing, until the convergence is achieved.

The approximation )(xMs of RS )(xs with M analysis scale levels can be written as:

)()()()()( 1211 xxxxx −++++= MM rrrss . …………(13)

The number of error components )(xjr in Eq. (13) is an indicator of the convergence rate to achieve an accurately estimated RS of the ‘true’ surface )(xs .

The modeling process of MRRSM is iterative. Essentially, the RS model defined in Eq. (9) is a functional expansion. Reconstructing RS with MRRSM consists in estimating the expansion terms. This is done iteratively until the residual error is below a pre-selected threshold.

The following features differentiate MRRSM from conventional response surface methodology: • The response surface is decomposed into multiple

components to better handle non-linear responses.

4 SPE 92853

• Components of a response surface are analyzed at multiple scale levels.

• ‘Thin plate’ spline interpolation is iteratively used to improve the accuracy of reconstructed surfaces.

• Each new scale level is adaptively introduced based on accuracy testing results of the reconstructed RS at the previous scale level.

• The resulting RS incorporates both global and local information of the non-linear system.

• No assumptions about the error term in the RS model are required.

• Accuracy checking of a reconstructed RS is performed with verification points, (different from interpolation data points).

The art of this method consists in: 1. Selecting the nested grid in the parameter space,

in which RS is reconstructed. 2. Efficiently selecting both interpolation and

verification data sets at each grid level to minimize computational cost for reconstructing RS.

We present a supporting DOE methodology for MRRSM, which is used to generate the nested grid, interpolation and verification data sets in next section.

Multiple Resolution Design of Experiment Methodology 2-level and 3-level factorial designs are standard DOE methodologies2. Often, a two-level full or fractional factorial design is conducted first as screening design to reduce dimension of the parameter space2,8-11. Then, three-level designs3 can be conducted on the reduced parameter space for quadratic response surface modeling. An efficient 3-level design is the central composite design2 (CCD). When the number of factors to be considered is large, this kind of design can significantly reduce the number of experimental runs. All these designs have a common feature, i.e., they are designs used for RS modeling with the regression method. They cannot be directly used for MRRSM.

A supporting optimal DOE methodology for MRRSM must have the following features2, 7:

1. Supply an appropriate distribution of data points used for constructing response surfaces,

2. Provide model accuracy checking strategy, 3. Allow designs used for improving the accuracy of the

reconstructed RS to be sequentially built up, 4. Use a minimum number of data points to model a

response surface, and 5. Compute model parameters in a simple way. Additionally, MRDOE for MRRSM must have the

following unique features, since MRRSM may require domain partitions and a nested grid in the design space of a response surface:

a. Provide partitions of the global design space only where a reconstructed surface is not accurate,

b. Verify model in both global and local sub-domains, c. Recognize non-uniformly distributed non-linear

effects of a RS, d. Implement adaptive resolution levels of factors.

Non-linear effects caused by factors may not be uniformly distributed on a response surface. It is important to recognize the non-linearity distribution of the surface and adaptively select the scales of each sub-domain, to obtain optimal partition of an experimental.. Similarly, the realization of optimal sampling depends on the identification of the non-linearity distribution of a surface. Consequently, sampling points will be sequentially added only in the sub-domains where factors have non-linear effects on the surface to be reconstructed.

Here, an optimal partition means that the components of a RS constructed with MRRSM are optimally matched with ‘thin plate’ spline interpolants with the minimum number of partitions. Optimal sampling means that a non-linear response surface is accurately constructed in MRRSM with the minimum number of sample points.

Design Criteria To achieve the above features, we develop the MRDOE methodology. It can be used to systematically partition the parameter space into sub-spaces and iteratively use standard DOE methodologies to collect both interpolation and verification data on each sub-space. Then, a response surface is constructed with MRRSM on a hierarchical grid in the design space. The design criteria of the MRDOE methodology are given in the followings: • Standard DOE methodologies are iteratively used in

whole/sub- design space. • At each partition level, the minimum number of sampling

points is used to reconstruct a component of the RS. • The accuracy of the reconstructed RS is checked in each

sub-domain. The non-uniformly distributed non-linearity is localized based on the verification results of the reconstructed RS in sub-domains.

• Space partitioning will continue in sub-spaces where the approximation error of constructed RS exceeds a pre-selected threshold.

• Verification points for error estimation of RS may be reused as interpolation points for constructing improved response surfaces.

Partitioning Design Space We define two types of partition schemes for MRDOE. They are bisection and piecewise partitions. The scheme selection is based on minimizing the required number of partitions for a design space. • bisection partition

For this scheme, the scale of a sub-domain at level n+1 is one half of its parent sub-domain at level n, (see Fig. 1a). The partition lines for sub-domains at level n are their bisection lines. This partition scheme is simple and can easily be implemented as an automatic partition scheme. It is optimal when a surface has uniformly distributed non-linearity in its parameter domain. This partition scheme should be applied if no sensitivity analysis results are available to locate non-linear effects distribution. • piecewise partition

The piecewise partition is recommended when sensitivity analysis results provide a map for the distribution of the non-

SPE 92853 5

linearity of a RS. It is an adaptive partition. This type of partition is used to split the whole domain into sub-domains where the response surface to be reconstructed has weakly non-linearity, (see Fig. 1b).

Kernel Design Space Often, parameter spaces are large for many reservoir performance evaluations. Consequently, very large numbers of simulation runs may be required to generate accurate response surfaces. Introducing a kernel design space to MRDOE can substantially reduce the number of both interpolation and verification points for constructing a RS. We define a kernel design space Dk for the design space Ω composed of n factors, as a sub-space of Ω, which is only composed of the factors having statistically significant effects on the response surface. There are m significant factors for a RS in design space Ω, then the kernel space Dk is the sub-space of Ω where the (n-m) non-significant factors are kept at their median values. Identification of the m significant factors is done with a standard DOE screening design2 performed in design space Ω. For the MRDOE methodology, we determine kernel spaces of all of sub-spaces resulting from the partitions of a global design space. Kernel spaces will improve the computational efficiency if the number of significant factors is significantly different the total number of factors of a design space. Design Algorithm for MRDOE/MRRSM The design algorithm, which is required to implement MRDOE/MRRSM, is presented below: Step 1: Perform screening design (2-level full/fractional factorial design) on the whole design space D1,1 to identify its kernel space Dk

1,1. Step 2: Use sampling points of the screening design to construct the first component )(1,1 xφ of the response surface on D1,1. Take center points of potential sub-domains D2,k at level 2, k = 1, 2, …, N’

2 as verification points for reconstructed RS )()( 1,11 xx φ=s . Calculate residual errors of surface )(1 xs at verification points. If residual errors are below a selected threshold value rε , then )(1 xs is acceptable and exit the iterative modeling process. Otherwise, continue. Step 3: Perform central composite design (CCD) on kernel space Dk

1,1 and reconstruct RS )(1 xrs to improve the accuracy of )(1 xs using all collected interpolation points in domain D1,1. Use the same verification points as those for )(1 xs to verify )(1 xrs . If residual error of )(1 xrs is acceptable, exit. Else, go to next step. Step 4: Use one of the partition schemes previously described to split domain D1,1 into sub-domain D2,k at level 2, k = 1, 2, …, N2, where residual error of )(1 xrs is greater than the threshold rε , Set resolution level j as 2. Step 5: If the number of the factors to be handled with is still too large to handle with CCD design, perform a screening design on sub-spaces Dj,k, k = 1, 2, …, Nj to determine their kernel spaces Dk

j,k. Step 6: Reconstruct component )(, xkjφ of response surface with residual errors of )(,1 xkj−φ at sampling points in sub-

domains Dj,k, k = 1, 2, …, Nj. Use runs at center points of potential sub-domains Dj+1,k

at level j+1, k = 1, 2, …, N’j+1 as

verification points for )(xjs . Calculate residual errors of response surface )(xjs at verification points. If residual error for response surface )(xjs is acceptable, exit. Else, proceed to next step. Step 7: Perform CCD in sub-domains Dj,k and reconstruct residual )(, xkjφ with residual error )(,1 xkj−φ at interpolation points in sub-space Dj,k, k = 1, 2, …, Nj to obtain improved RS

)(xjrs . Use verification points in sub-domain Dj,k to verify accuracy of )(xjrs . If residual error of )(xjrs is lower than the error threshold rε , exit. Step 8: Split sub-domain Dj,k, where the residual error of

)(xjrs is not acceptable, into sub-domain Dj+1,k, k = 1, 2, …, Nj+1, update scale level j with j ←j+1, and go to Step 5. Validation of MRDOE/MRRSM We use a modified version of the SPE9 benchmark problem to illustrate and validate the MRDOE/MRRSM methodology. In this example, the response to be analyzed is the produced oil rate at 0.60PV of water injection, Qo@0.60PV. Then, we use the reconstructed RS as a proxy to perform Monte Carlo experiment for assessing uncertainties in oil production forecasts. The simulator of choice is CHEARS™, which is the property of the Chevron-Texaco Petroleum Company.

The modified SPE9 problem16 studies a bottom waterflooding in a dipping reservoir with natural water encroachment from an aquifer. A 3-D representation of the reservoir model is shown in Fig. 2. The following modifications were done on the original SPE9 model: • The bubble point pressure is reduced from 3,600 psia to

2,000 psia. • A partially sealing vertical fault is added at x = 3,000ft,

which separates production wells into down-fault and up-fault families.

• We add a high permeability thief zone at layer 9 and 10, (see Fig.3).

• We increase the aquifer size by 2,000 ft in the x- direction.

• The injector located in the aquifer is changed from vertical to horizontal. It is perforated at layer 14. Its water injection rate is set at 30,000 STB/D with maximum bottom-hole pressure of 5200 psia.

• All the vertical producers are completely perforated, except producers 8, 15, 16, 22, and 25, which are perforated at layer 1-5.

• Volumetric production rates of producers are 1200 STB/D. A minimum bottom hole pressure constraint of 1000 psia is set. Over 1.0 pore volume of water is injected into the reservoir.

The following factors are selected to do the uncertainty analysis of reservoir performance forecasts: Rock absolute permeability multiplier, aquifer pore volume multiplier, thief permeability, viscosity multiplier, and x- fault transmissibility multiplier. Their ranges are listed in Table 1. We select oil field production rate when 0.60 pore volume of water is

6 SPE 92853

injected as the response to study. First, we conduct a sensitivity analysis of factor x- fault transmissibility. The result is shown in Fig. 4. This factor causes highly non-linear effect on the response.

We normalize the factors (parameters) to avoid distortion effects on the constructed RS. A logarithmic transform is performed on x- fault transmissibility multiplier, first. Then a linear transform is operated on all factors to map them onto domain [0, 1]5. The linear transform is

,521,minmax

min,', , ..., , l

xxxx

xll

lilil =

−−

= ……………(14)

where xl,i is the value of the lth component of experiment point xi, and xlmin and xlmax are the minimum and maximum of the lth factor, respectively. Constructing RS Using MRDOE/MRRSM Method Following the MRDOE methodology, we first perform a 2-level screening design with 5 factors (25-0runs) on the global design space D1,1. The 3 significant factors are indicated in a Pareto chart, (Fig.5). They are: rock absolute permeability multiplier, oil viscosity multiplier, and x- fault transmissibility multiplier. We have reduced the dimension the global design space D1,1 from 5 to 3 to obtain the kernel sub-space Dk

1,1. Next, we construct the first RS )(1 xs for Qo@0.60PV. We then verify the accuracy of this first estimated RS, using 8 center points of the potential sub-spaces, shown in Fig.6. The acceptable error threshold rε to achieve an accurate response surface is set as 15%. Surface )(1 xs fails to pass the model verification test at 6 out of 8 points. We perform CCD in kernel space Dk

1,1 to collect more interpolation points (6 points) to improve the accuracy of )(1 xs . The residual error of the improved RS )(1 xrs is not below the percent error threshold rε everywhere in the experimental space. Fig.7 shows that the percent error of )(1 xrs is larger than its threshold at 5 out of 8 points.

Since reconstructed surfaces are not acceptable at level 1, we selectively partition the global design space D1,1 into sub-spaces at level 2, where RS )(1 xrs is not accurate enough, using the bisection partition scheme, seen as Fig. 8. We perform 2-level screening designs to identify the kernel spaces of these sub-spaces, so that we can generate verification points for the second reconstructed RS. The residual errors of RS

)(1 xrs at experimental points are used to reconstruct the second component )(,2 xkφ of the RS in each of these sub-spaces. The accuracy of the second reconstructed RS

)()()( ,212 xxx krss φ+= is checked with the set of verification points, shown in Fig. 8. The maximum percent error of RS is 13.1%, which is below its threshold value 15%. We can stop the iterative modeling of MRDOE/MRRSM.

Comparison of Response Surfaces Reconstructed with MRDOE/MRRSM and DOE/Regression Next, we compare MRDOE/MRRSM with DOE/Regression. To construct the RS of Qo@0.60PV with the regression method, we normalize the factors as what we did before. Then

we use the results of the previously performed screening design on the global domain of the parameter space. Next, CCD is carried out in the reduced design space composed of the three significant factors, i.e., permeability multiplier, viscosity multiplier, and x- fault transmissibility multiplier, to collect the data used for fitting the RS. The polynomial regression model used for fitting Qo@0.60PV is:

,),,(33

1

23

10321 j

jiiij

iiii

iii xxaxaxaaxxxf ∑∑∑

<==

+++= ......(15)

Where x1 is permeability multiplier, x2 denotes the viscosity multiplier, and x3 is x- fault transmissibility multiplier. This is a typical regression model7-9, 11. The estimated coefficients in this model are listed in Table 2. The quality of fit is estimated using indices2, 9 R2=0.99303 and R2

adj. = 0.98049. The fit of the response surface RS-poly to the experimental data is good. We use the same set of verification points as the one used for RS )(2 xs to verify the accuracy of RS-poly. The maximum percent error of RS-poly is 22.2%, which is much larger than that for RS )(2 xs . Fig.9 compares the correlations between CHEARS™ simulation values and prediction values of proxies

)(2 xs and RS-poly. The slope and R-squared values of the linear regression line between CHEARS™ simulation and proxy prediction )(2 xs results are 0.993 and 0.9718, respectively. They are close to their ideal value of 1.0. But the slope and R2 values of the regression line for RS-poly are 0.868 and 0.689, respectively. The ideal slope value is 1.0. Hence, RS-poly is not as accurate as RS )(2 xs . We can visualize the accuracy of each reconstructed surface by plotting their profiles and comparing with CHEARS™ results. The CHEARS™ profile was generated by conducting 121 simulations. For illustrative purpose, we plot RS vs. permeability multiplier and x- fault transmissibility multiplier, keeping all other factors at their median values. The profiles of the “true” RS, reconstructed RS )(2 xs and RS-poly are shown in Fig. 10. The error residuals (normalized differences between “true” and proxy surfaces) are shown in Fig. 11. For the profile of RS-poly, its maximum and average relative errors are 27.1% and 12.4%, respectively. Correspondingly, for the profile of surface )(2 xs , its maximum and average relative errors are 16.2% and 5.1%, respectively. We see that the surface reconstructed with DOE/Regression is too smooth to model the non-linear effects of Qo@0.60PV accurately. The surface reconstructed with MRDOE/MRRSM is not as smooth as RS-poly. It matches the “true” surface fairly well, especially its non-linear effects. If we compare the data sets, which are used by MRDOE/MRRSM and DOE/Regression for constructing the two RS, we find that MRDOE/MRRSM uses 54 simulation runs and DOE/Regression takes 39 runs. Fifteen extra points are required using MRDOE/MRRSM for this 5-factor problem. However, the accuracy of the reconstructed RS is significantly improved . Uncertainty Analysis Using Monte Carlo Simulation Furthermore, we use RS )(2 xs and RS-poly as proxies to CHEARS™ to perform Monte Carlo experiments. We want to evaluate the impact of an inaccurate RS model on the uncertainty assessment of Qo@0.60PV. The three significant

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factors (No. 1, 4, 5 in Table 1) are used in the Monte Carlo simulation. Parameters 4 and 5 in Table 1 are sampled using normal probability distribution function (PDF), while the permeability multiplier is sampled using a lognormal PDF. Figs. 12 and 13 show the histograms for Qo@0.60PV obtained from 1,000 sampling data. The means and standard deviations of the histograms are listed in Table 3. There are significant differences in both mean and standard deviation between these two histograms. Hence, it is important to construct an accurate proxy model to the simulator in order to trust the statistics, which are generated with the Monte Carlo simulator.

Analysis of Computation Results Finally, we conduct a comparative analysis of both the MRDOE/MRRSM and DOE/Regression approaches. The standard CCD is often used for collecting data for modeling a RS with a quadratic polynomial. This method works well as long as the response surface has no highly non-linear effects. Otherwise, more complicated models are needed. Observing the profile of the “true” RS of Qo@0.60PV, (Fig.10a), we see that the non-linear effects are distributed in the region where the normalized x- fault transmissibility multiplier is higher than 0.5 and non-linearity increases with permeability multiplier. The points, which are sampled with CCD, are uniformly distributed in the global parameter space and each parameter has only three sampling values. Therefore, the polynomial model of the RS cannot include terms with power higher than 2. Therefore, the resulting RS reconstructed with regression method is too smooth to fit the RS in the non-linear region. Additionally, the regression method estimates2, 9 the covariance of the coefficients of the terms in the regression model of a RS with the assumption that the error term has an independent normal distribution with zero expectation and uniform variance σ2. The accuracy checking of the polynomial surface is flawed unless the assumption is valid. MRDOE/MRRSM makes no assumption about the error of a reconstructed RS. MRDOE/MRRSM uses independent verification points to check the accuracy of reconstructed surfaces at all iterations of its modeling process. Its data sampling is guided by the accuracy checking results of the reconstructed surfaces. This adaptive sampling will result in high sampling frequency in the non-linear regions, seen as Fig. 8. It is very good for obtaining a high data collecting efficiency to accurately reconstruct a RS. Conclusions We have introduced a novel RSM and its supporting DOE methodology (MRDOE/MRRSM), which is used to iteratively collect experimental data to reconstruct components of a RS on a hierarchical grid in the parameters space. The key feature of this approach is to split the whole domain of a non-linear system into sub-domains where components of the non-linear RS have weak non-linear effects, which can be efficiently modeled with ‘thin plate’ spline interpolant.

From the numerical testing results, we can draw the following conclusions: 1) MRDOE/MRRSM is an efficient method to accurately construct a RS having non-linear effects caused by factors, 2) MRDOE/MRRSM is better than the conventional DOE/Regression method to construct an accurate non-linear RS on a global parameters domain. It is particularly

true for response surfaces with non-uniformly distributed non-linearity. 3) MRDOE/MRRSM is a practical methodology, which can be used to construct proxies of a reservoir simulator with an affordable number of simulation runs. Nomenclature ai = coefficient of the ith linear term in the regression model ai,i = coefficient of the ith quadratic term in the regression

model ai,j = coefficient of the two-way interaction term xixj in the

regression model D k = kernel design space of design space Ω Dj,k = the kth sub-domain at scale level j D k

j,k = kernel space of sub-domain Dj,k, ε = the random error term in the response surface model f(x1, x2,…, xn) = response surface

)(, xkjφ = a non-zero scaling function in sub-domain Dj,k

=)],([ 11 nxxxsI bending energy of a ‘thin plate’ spline interpolant

N = the total number of interpolation points Qo@0.60PV = oil production rate when 0.60PV of water is

injected, )(xjr = the residual error of the jth approximation of

response surface )(xs R2 = coefficient of determination of the regression model R2

adj = adjusted coefficient of determination of the regression model

)(xjs = the jth approximation of response surface )(xs

)(xjrs = the jth improved reconstructed response surface )(xs σ = error standard deviation x = a point vector in the n-dimensional parameters space Ω xi = the ith interpolation point xi = the ith factor xl,i = the value of the lth component of experiment point xi xlmin = minimum value of the lth factor xlmax = maximum value of the lth factor U(r) = rational basis function Acknowledgements This work was conducted at Calfornia Institute of Technology and financially supported by ChevronTexaco Petroleum Company. The authors thank ChevronTexaco for permissions to publish the results.

References

1. Box, G.E.P., and Wilson, K. G.: Empirical Model Building and Response Surfaces, Wiley, New York, 1987.

2. Montgomery, D.C.: Design and Analysis of Experiments, 5th Edition, John Wiley & Sons, INC., 2001.

3. Box, G.E.P., and Hunter, J.S.: “Multifactor Experimental Designs for Exploring Response Surfaces,” Annals of Mathematical Statistics, Vol. 28 (1957), pp. 195-242.

4. Box, G.E.P., and Wilson, K. G.: “On the Experimental Attainment of Optimum Conditions,” Journal of Royal Statistical Society, B., Vol.13 (1951), pp. 1-45.

5. Chu, C.: “Prediction of Steamflood Performance in Heavy-Oil Reservoirs Using Correlations Developed by Factorial Design Method,” paper SPE 20020 presented at the 1990 SPE California Regional Meeting, Ventura, April 4-6.

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6. Egeland, T. et al.: “Designing Better Decisions,” paper SPE 24275 presented at the 1992 SPE European Petroleum Computer Conference, Stavanger, May 25-27.

7. Damsleth, E., Hage, A., and Volden, R.: “Maximum Information at Minimum Cost: A North Sea Field Development Study With an Experimental Design,” J. Pet. Techn. pp. 1350-1356.

8. Friedmann, F., Chawathé, A., and Larue, D. K.: “Assessing Uncertainty in Channelized Reservoir Using Experimental Designs,” SPE Reservoir Evaluation & Engineering, August 2003, p. 264-274.

9. Dejean, J.-P., and Blance, G.: “Managing Uncertainties on Productions Using Integrated Statistical Methods,” paper SPE 56696, presented at the 1999 SPE Annual Technical Conference and Exhibition held in Houston, Texas, 3-6 October.

10. Corre, B., Thore, P., de Feraudy, V., and Vincent, G.: “Integrated Uncertainty Assessment For Project Evaluation and Risk Analysis,” paper SPE 65205, presented at the SPE European Petroleum Conference held in Paris, France, 24-25 October 2000.

11. Manceau, E., Meaghani, M., Zabalza-Mezghani, I., and Roggero, F.: “Combination of Experimental Design and Joint Modeling Methods for Quantifying the Risk Associated With Deterministic and Stochastic Uncertainties – An Integrated Test Study,” paper SPE 71620 presented at the 2001 SPE Annual Technical Conference and Exhibition held in New Orleans, Louisiana, 30 September – 3 October 2001.

12. Eide, A. L., Holden, L., Reiso, E., and Aanaonsen, S. I.: “Automatic History Matching by Use of Response Surfaces and Experimental Design,” presented at 4th European Conference on the Mathematics of Oil Recovery, Røros, Norway, 7-10 June, 1994.

13. Landa, J.L. and Güyagüler, B.: “A Methodology for History Matching and the Assessment of Uncertainties Associated with Flow Prediction,” paper SPE 84465, presented at the SPE Annual Technical Conference and Exhibition held in Denver, Colorado, USA, 5-8 October 2003.

14. Güyagüler, B. and Horne, R. N.: “Uncertainty Assessment of Well-Placement Optimization,” paper SPE 71625, presented at the 2001 SPE Annual Technical Conference and Exhibition held in New Orleans, Louisiana, 30 September – 3 October 2001.

15. Li, B., A Control Volume Function Approximation Method for Reservoir Simulation Using Unstructured Grids, dissertation, Southern Methodist University, 2003.

16. Killough, J. E.: “Ninth SPE Comparative Solution Project: A Examination of Black-Oil Simulation,” paper SPE 29910, presented at the 13th SPE Symposium on Reservoir Simulation held in San Antonio, TX, 12-15 February, 1995.

17. Deutsch, C.V., Geostatistical Reservoir Modeling, Oxford Univ. Press, 2002.

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Table 1 Uncertain Factors and Their Ranges

No. Factor Low (-1) Median (0) High(+1) 1 Absolute Permeability Multiplier 1.0 1.5 2.0 2 Aquifer Pore Volume Multiplier 1.0 5.5 10.0 3 Thief Permeability (md) 500. 2750. 5000. 4 Viscosity Multiplier 1.0 2.3 3.6 5 X- Fault Transmissibility Multiplier 10-6 10-3 1.0

Table 2 Coefficients of Reconstructed Surface RS-poly

Coefficient Value Coefficient Value Coefficient Value a0 1241.26 a11 -1158.59 a13 1338.74 a1 1521.80 a22 26.21 a23 -517.51 a2 -173.95 a33 1943.01 a3 -392.81 a12 -184.89

Table 3 Statistics Results of Monte Carlo Experiments Performed with Proxies

Reconstruction Approach Mean (STB/D) Standard Deviation (STB/D) MRDOE/MRRSM 3081.5 323.3

DOE/PROXY 2747.9 282.8

0

0.2

0.4

0.6

0.8

1

00.1

0.20.3

0.40.5

0.60.7

0.80.910

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2000

2500

f

y

x

(a) bi-section partition (b) piecewise partition

Fig. 1 Partition Schemes for MRDOE

Level n+1

Level n

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Fig. 2 Reservoir Model of Modified SPE9 Fig. 3 Thief Zone Added in Reservoir

1000

1500

2000

2500

3000

3500

4000

1.00E-06 1.00E-01 2.00E-01 3.00E-01 4.00E-01 5.00E-01 6.00E-01 7.00E-01 8.00E-01 9.00E-01 1.00E+00

x- Fault Trans. Multi.

Qo@

0.60

PV (

STB

/D)

Qo@0.60PV

Fig. 4 Sensitivity Analysis of X- Fault Transmissibility Multiplier

-.3494

1.262191

-3.30809

6.312868

12.5824

p=.05

(2)Aqui. Multi.

(3)Thief Perm.

(4)Vis. Multi.

(1)Perm. Multi.

(5)Fault Trans. Multi.

Fig. 5 Pareto Chart of Qo@0.60PV for Screening Design on Global Design Space

X

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SOIL on day 0

d-1.grid

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Fig. 6 Interpolation and Verification Data Sets in Kernel Sub-Space Dk

1,1 at Level 1

Fig. 7 Percent Error of RS )(1 xrs Reconstructed with MRMRDOE/MRRSM

interpolation points verification points

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Fig. 8 Domain Partition, Hierarchical Grid, and Data Sets at Scale Level 2

y = 0.8678xR2 = 0.6887

y = 0.9931xR2 = 0.9718

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Fig. 9 Verification Results of RS Reconstructed with MRMRDOE/MRRSM and DOE/Regression

5

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8 interpolation points level 1

interpolation points level 2 verification points level 2

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(a) Qo@0.60PV-profile (CHEARS)

(b) Qo@0.60PV-profile (MRDOE/MRRSM)

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(c) Qo@0.60PV-profile (DOE/Regression)

Fig. 10 True and Reconstructed Profiles of RS of Qo@0.60PV

(a) Percent Error of Reconstructed Surface )(2 xs

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(b) Percent Error of Reconstructed Surface RS-poly

Fig. 11 Residual Error of Reconstructed Response Surfaces

Fig. 12 Histogram for Qo@0.60PV, Using RS Reconstructed with MRDOE/MRRSM

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Fig. 13 Histogram for Qo@0.60PV, Using RS Reconstructed with DOE/Regression