Automatic History Matching by use ofResponse Surfaces and Experimental DesignAlfhild L. Eide1 Lars Holden1 Edel Reiso2Sigurd I. Aanonsen21Norwegian Computing Center, Oslo, Norway2Norsk Hydro a.s., Bergen, NorwayPresented at 4th European Conference on the Mathematics ofOil Recovery, R�ros, Norway, 7-10 June, 1994AbstractHistory matching is the process of using production history to im-prove the estimates of geological and petrophysical parameters in theoil �eld.We estimate response surfaces based on a set of reservoir simulationswith di�erent combinations of the reservoir parameters. A response sur-face is a simpli�ed relation y(x) between reservoir simulator input x andoutput (response) y, and gives a rough overview of the behavior of theresponse in the whole region of interest. These response surfaces aresearched to �nd parts that are close to the historical data. Since a his-tory matching problem may have several solutions, surfaces that giveoverview over the whole region are of interest. History matching is doneby minimizing the distance between the observed values of the responsevariables, and the response predicted from the response surfaces y(x).The process is iterative: make experimental design, run the selectedexperiments, generate response surface, optimize, make new re�ned de-sign, run : : : and so on. One or several iterations may be performedautomatically. The method is demonstrated on a synthetic reservoirsimulation example.1 IntroductionHistory matching is the process of using production history to improve theestimates of geological and petrophysical parameters in the oil �eld. Theseparameters should be modi�ed to induce the reservoir simulator to reproduceproduction history. An indirect, or secondary aim is to predict future reservoirperformance. 1

Black box-- --?x y�, observed historyyFigure 1.1: History matching can be seen as an inverse problem. For a given x, it ispossible to calculate y = f(x). What we are looking for, is an x, or a set of x's that willreproduce the observed y-value y�.Traditionally, history matching has been performed by trial and error. Theexperienced reservoir engineer has changed one or a few variables at a timehoping for an improved match. With the increased computer resources avail-able and increased understanding of the problem, it is possible to improve thisprocess. The most straightforward improvement is to reduce the manual workin the process. There is much manual work in generating input �les, startingof reservoir simulations, reading of output �les and visualization of the results.The more di�cult improvement task is automatic generation of a set of inputdata to test out. This will both reduce the required manpower and make thehistory match more objective. To estimate the uncertainty in the predictionsbased on the history match, objectivity is important.Traditional gradient techniques using numerical derivatives usually requiretoo many simulations to be applicable to automatic history matching. Thiscomputation time may be signi�cant reduced by the use of optimal controltheory methods (see Chavent, Dupuy & Lemonier (1975), Palatnik & Zakirov(1992), e.g.), or by calculating sensitivity coe�cients parallel to the simulations(Anterion, Eymard & Karcher 1989), (Tan & Kalogerakis 1991).Our main idea is to use estimated response surfaces based on a set ofreservoir simulations with di�erent combinations of the reservoir parameters.The response surface is a simpli�ed relation y(x) between reservoir simulatorinput x and output (response) y (Figure 1.1), and gives a rough overview ofthe behavior of the response in the whole region of interest. These responsesurfaces are searched to �nd parts that are close to the historical data. Sincea history matching problem may have several solutions, surfaces that giveoverview over the whole region are of interest. If necessary, some parts of theresponse surfaces can be re�ned by running additional reservoir simulations.History matching is done by minimizing the distance between the observedvalues of the response variables yobs, and the response predicted from the re-sponse surface y(x). The process is iterative: make experimental design, runthe selected experiments, generate response surface, optimize, make new re-�ned design, run : : : and so on.Experimental design and response surface methods are discussed in Sec-tion 2. Section 3 lists the steps in the method, and Section 4 gives an example.2

2 Experimental design and response surfacemethodsReservoir simulations can be time consuming and expensive. It is important toget maximum information from a relatively small number of reservoir simula-tions. An experimental design is a plan describing the di�erent choices of eachof a number of input variables in a series of simulation runs. Experimentaldesign provides alternatives to the traditional \vary one at a time" strategy.Among other things, it is possible to estimate the joint e�ect of changing twoparameters simultaneously (Box, Hunter & Hunter 1978).The theory of experimental design was developed and applied in agriculturein the 1920's. Since the mid 1980's, experimental design has also been stud-ied and used for `computer experiments', see for instance Welch, Buck, Sacks,Wynn, Mitchell & Morris (1992), Sacks, Welch, Mitchell & Wynn (1989), Mor-ris, Mitchell & Ylvisaker (1993). Experimental design of reservoir simulationsis demonstrated by Damsleth, Hage & Volden (1992) and has been followedup by Egeland, Hatlebakk, Holden & Larsen (1992).Like Damsleth et al. (1992) and Egeland et al. (1992) we use D-optimaldesigns. D-optimality is a mathematical procedure to select the optimal runsfrom a (large) set of possible runs (the candidate set). Data from previousruns are utilized.Based on the input:� a set of candidate experiments,� the number of experiments to be selected,� an a priori regression equation describing the relations between inputand response variables,the output is:� the optimal design w.r.t. obtaining optimally precise estimates of thecoe�cients in the equation given.The regression equation may be written:y = F (x)b+ �;(2.1)where y is a vector of response values, b is a vector of coe�cients to be esti-mated, and � is a vector of independent random variables, each with expecta-tion 0 and variance �2. The vector function F may be a general function ofx, but in this paper we limit F to be second order polynomials in x. Let thenumber of coe�cients bi be n. For a given set of m experiments, F (x) becomesan m� n matrix; the design matrix. 3

The least squares estimate for b is given byb = (F TF )�1F Ty(2.2)while the covariance matrix isCov b = �2(F TF )�1(2.3)It can be shown that the optimal precise estimates of b is obtained whenthe determinant of F TF is maximized over the region of interest, and this iswhy the method is called D-optimality (Determinant-optimality) (See St. John(1971) or Fedorov (1972) for a complete discussion.)2.1 Response surfacesA response surface is a simpli�ed relation between the simulator input and out-put. We consider the reservoir simulator as a \black box" (Figure 1.1). Basedon the results from the selected simulations, a response surface is generatedwhich can be used to predict simulator output for other input values.The most standard way of estimating a response surface is regression. Fordeterministic simulations, interpolating surfaces, for instance kriging surfaces,is an interesting alternative. Our practical experience with these responsesurfaces suggests that in an early stage, when there are few data points, onemight as well use regression surfaces, but as more data points are included,kriging gives better results. Addition of an extra data point does not changethe regression surface much, but when kriging is used, the response surface willalways be changed to interpolate the new data point. This is an advantagewhen new data points are added to improve the quality of the response surfacenear a possible optimum.If gradient information is available, this can also be used in the generationof response surfaces, see Morris et al. (1993).3 Method1. Experimental design: For each input variable to be adjusted throughhistory matching, 3 levels are speci�ed: low, base case and high level.The candidate set contains all combinations of these levels (possibly re-moving unfeasible combinations of input variables). Then a D-optimaldesign is generated (selected from the candidate set), based on the apriori equation and the number of simulations required.2. Reservoir simulations: Run the reservoir simulations in the D-optimaldesign. 4

3. Analysis: Fit a regression or kriging model to the data. Automaticmodel selection (see Miller (1990) for a review) may be used for selectionof terms in the regression model.4. Optimization using response surface: MinimizeF(y(x)) =Xi wi(yi(x)� y�i )2(3.1)where y(x) is the response surface, y� is the observed history, x is avector of input variables (reservoir parameters) and wi are weights. Weuse a standard multidimensional optimization routine (Powell's methodfrom Press, Flannery, Teukolsky & Vetterling (1988)), but since there areoften several optima, we select a set of di�erent starting points by �rstcalculating the value of the object function on a grid and then selectingthe best points in the grid as starting points. Optimization results arepresented as a list of possible optima.5. Reservoir simulations: Run reservoir simulations close to the opti-mal values. The user may select one or several optima and explore theresponse surface near these by performing additional simulations.6. Iteration: If necessary, go back to step 1 to set up a new design and doadditional experiments. To improve the estimate of the response surfacenear a predicted optimum, a new design can be generated close to thisoptimum. The levels in the new design can be generated as follows:The optimum input values are base case. Calculate the distance to thenearest data point, and let the di�erence between base case and high(low) levels in the new design depend on this distance. Then a new D-optimal design can be generated around the optimum, using runs thathave already been made.With a program system where experimental design and analysis are integratedwith a possibility to start reservoir simulations and read the results back intothe system, this iteration process is easy to run. The Norwegian ComputingCenter has developed the computer program \DECISION" for design and anal-ysis of reservoir simulations. This program has been extended with a historymatching module and it is possible to start the reservoir simulator from within\DECISION". One or several iterations may be performed automatically withdefault options in design and analysis.4 Example4.1 ModelA synthetic, two-dimensional reservoir model was chosen to test the method.The reservoir dips 8 degrees, and has two oil producers and one water injector,5

see Figure 4.1. The reservoir is divided into 3 zones with unknown perme-ability. There is a barrier between the two producers, perpendicular to theoil/water contact, modeled as a row of thin blocks. The two oil zones com-municate through the aquifer, and in addition there is a small opening in thebarrier at the top of the reservoir covering two simulator blocks. The degreeof communication through this opening was also considered as unknown in thehistory matching process (Table 4.1).The following production strategy is chosen: Produce well 1 for 1 year, thenopen well 2 and produce both wells for 1/2 year. Then start water injectionand run production and injection until water breakthrough. As matchingvariables were used bottom hole pressure in producer 2 after 1 and 1.5 years;bottom hole pressure in producer 1 after 1 year; and breakthrough times inthe two producers (Table 4.2). Due to the shape of the water cut curves, itturned out to be di�cult to �nd a consistent way of de�ning the time of waterbreakthrough automatically. In the �nal optimization, water breakthrough wasde�ned as the time when water cut exceeded 0.1. However, other de�nitionswould have given di�erent results (see discussion below).4.2 Experimental design and response surfacesFor each input variable, low, high and base case levels were selected. Theselevels were transformed to a -1,1,0 scale. This makes it possible to get anorthogonal design (i.e. the matrix F in Eq.(2.1) is orthogonal), and it is easierto interpret the regression coe�cients because all variables are on the samescale and symmetric around zero. The levels used, both untransformed andtransformed, can be seen in Table 4.3.The initial design consisted of 8 runs. Automatic model selection was usedto select a regression model for the overall trend. This trend was used in thekriging model for the response surface. Equations 4.1{4.3 give an example ofwhat the trend may look like. The regression equations given are based on the�rst 8 runs.BHP = 214:12 + 6:43 �KWAT� 7:42 �KBAR+ �(x)(4.1) BHP1 = 90:64 + 54:04 �KO1 + 19:36 �KWAT+ �(x)(4.2) BHP2 = 100:45 + 22:51 �KWAT+ 44:95 �KO2 + �(x)(4.3)4.3 OptimaIn history matching, it often happens that more and more data become avail-able as time passes. In this example we �rst used 3 response variables, namelyBHP, BHP1 and BHP2, where history data would have been available after112 year of production. Later we added BT1 and BT2 to see how the additionof more data reduces the uncertainty and the number of possible optima.6

P1

P2

I1

KO2

KO1

KWAT

KBAR

Water/oil contact

Figure 4.1: Example reservoir. Only part of the aquifer is shown. The position of thebarrier is indicated with a horizontal line.KO1 permeability near oil producer 1KO2 permeability near oil producer 2KWAT permeability near injectorKBAR permeability in the two uppermost barrier blocksTable 4.1: Input variables.BHP Bottom hole pressure in producer 2 after 1 yearBHP1 Bottom hole pressure in producer 1 after 1.5 yearsBHP2 Bottom hole pressure in producer 2 after 1.5 yearsBT1 Breakthrough time well 1BT2 Breakthrough time well 2Table 4.2: Response variables.7

Variable name scale low level base case high levelcoded -1 0 1KO1, KO2, KWAT uncoded log 4.38 5.19 6.00uncoded real 79.84 179.5 403.4coded -1 0 1KBAR uncoded log -2.3 1.15 4.6uncoded real 0.10 3.16 99.5Table 4.3: Levels in the initial design for the 4 input variables. Variables are transformedfrom the real scale by �rst taking logarithms, then coding so that zero becomes the basecase level.Optimization with 8 runs and 3 response variables gave the optima shownin Table 4.4. The simulation runs in these 3 optima were then used to improvethe response surface, which now exhibited 2 distinct optima (runs 12 and 13 ofTable 4.4). However, the match itself is not signi�cantly improved. Figure 4.2shows the response surfaces for BHP after 8 runs and after 11 runs.With 4 input variables and only 3 measurements, we can not expect to�nd a unique optimum. Now imagine that more data becomes available, waterbreakthrough occurs and BT1 and BT2 are now available. We expect that thiswill reduce the number of possible optima.We re-run optimization based on the 11 �rst runs, but using 5 responsevariables instead of 3. To compensate for the di�erence in absolute values be-tween the di�erent variables, we used relative di�erences in the object function,that is F(y(x)) =Xi yi(x)� y�iy�i !2 :(4.4)From the list of possible optima, we selected 3 di�erent points for newsimulations. Results are shown in Table 4.5. The results are now improved,but the match is still not satisfactory, and 3 additional iterations with runsin the optimum and in a small design around the optimum were performed.Figure 4.3 show bottom hole pressures and water cuts for run 5, 15, and 25compared with the history run. For the pressures, the match was signi�cantlyimproved, while the results are less satisfactory for water cut. As mentioned,breakthrough was de�ned as the time when water cut exceeded 0.1. FromFigure 4.3 we see that even if the match is reasonably good at one point intime, the trend is not well reproduced in any of the runs. This shows thatto obtain a better match of the water cut over time|and a better prediction,more than one measurement is needed.8

- 1. 00

- 0. 600

- 0. 200

0. 200

0. 600

1. 00 - 1. 00

- 0. 600

- 0. 200

0. 200

0. 600

1. 00 200. 205. 210. 214. 219. 223.

KB

ARK

WA

TBHP

BHP(0, 0, KWAT, KBAR)

- 1. 00

- 0. 600

- 0. 200

0. 200

0. 600

1. 00 - 1. 00

- 0. 600

- 0. 200

0. 200

0. 600

1. 00 193. 200. 206. 212. 219. 225.

KB

ARK

WA

TBHP

BHP(0, 0, KWAT, KBAR)

Figure 4.2: BHP kriging response surfaces, with 8 runs (above) and 11 runs (below).9

Years

BH

P W

1 (B

ar)

0 1 2 3 4 5 6

100

150

200

250

0 1 2 3 4 5 6

100

150

200

250

0 1 2 3 4 5 6

100

150

200

250

0 1 2 3 4 5 6

100

150

200

250

historyrun 5run 15run 25

Years

BH

P W

2 (B

ar)

0 1 2 3 4 5 6

100

150

200

250

0 1 2 3 4 5 6

100

150

200

250

0 1 2 3 4 5 6

100

150

200

250

0 1 2 3 4 5 6

100

150

200

250

historyrun 5run 15run 25

YearsW

ater

Cut

W1

2 3 4 5 60.

00.

10.

20.

30.

40.

52 3 4 5 6

0.0

0.1

0.2

0.3

0.4

0.5

2 3 4 5 60.

00.

10.

20.

30.

40.

52 3 4 5 6

0.0

0.1

0.2

0.3

0.4

0.5

historyrun 5run 15run 25

Years

Wat

er C

ut W

2

3 4 5 6 7

0.0

0.1

0.2

0.3

0.4

0.5

3 4 5 6 7

0.0

0.1

0.2

0.3

0.4

0.5

3 4 5 6 7

0.0

0.1

0.2

0.3

0.4

0.5

3 4 5 6 7

0.0

0.1

0.2

0.3

0.4

0.5

historyrun 5run 15run 25Figure 4.3: Graphs showing bottom hole pressure for well 1 (BHP W1) and well 2 (BHPW2); and water cut for well 1 (WCUT W1) and well 2 (WCUT W2). Simulation results forrun 5 (one of the runs in the initial design), run 15 (match based on 11 runs) and run 25(match obtained after several iterations, see Table 4.5) are shown. For BHP W1 and BHPW2, run 5 lies above the other runs, then comes run 15, and run 25 and the history arealmost identical, except BHP W2 at 1 year.

10

Simulated response variables Input variables ObjectiveBHP BHP1 BHP2 KO1 KO2 KWAT KBAR function(Bar) (Coded to -1,0,1 scale) values1 204.68 117.11 92.14 1 0 -1 0 7662 220.44 47.38 161.57 -1 1 1 0 66183 218.27 84.46 19.26 0 -1 -1 -1 56264 218.52 119.75 117.24 1 1 -1 -1 16105 222.96 125.03 131.72 0 0 1 -1 29796 199.18 5.58 87.27 -1 0 -1 1 71607 197.79 83.55 118.70 0 1 -1 1 7738 218.25 157.59 75.58 1 -1 1 1 50579 204.01 93.46 97.21 0.423 0.445 -1.219 -0.118 4110 214.10 107.30 110.21 -0.141 -0.330 0.340 1.25 64711 203.89 94.45 98.41 0.409 0.423 -1.176 -0.084 6212 208.73 96.08 106.09 -0.097 -0.108 -0.375 0.476 20813 209.23 97.66 101.15 0.275 0.271 -0.958 -0.209 131H 207.03 89.59 93.28 -0.148 -0.309 -0.494 0.504 |Table 4.4: Runs that were made for the example in this article. Runs 1{8 come from aninitial D-optimal design, runs 9{11 were runs in the optima of the (kriging) response surfacesof BHP, BHP1 and BHP2, based on the 8 �rst runs, runs 12{13 were runs in the optimabased on the 11 �rst runs. The run that was used to generate the \history" is shown below(indicated with \H").5 Discussion and ConclusionsWe have presented a general method that can be used for history matchingand other optimization problems where computer intensive simulations areinvolved.We use response surfaces to predict the simulations for other combinationsof the input than those that have been run on the simulator. The responsesurface is thus a critical factor of success in our method. The quality of theresponse surface depends on the experimental design used, and on the modelused for the response surface. One question is whether one should use krig-ing, regression or something else. Another question is which terms should beincluded in the regression model, or in the trend of a kriging model.In this paper we have shown a synthetic example with 4 input variables(reservoir parameters) and 5 response variables. We have also tested themethod on other examples not included in this paper with up to 8 input vari-ables and 24 response variables with reasonably good results (match obtainedafter about 30 simulation runs). Typical real-world problem will often havehigher dimensions, both in the input and the response. Also each simulationbecomes more computer intensive. The authors believe that also for largerexamples than the tested ones, the procedure presented in this paper is betterthan the trial and error approach. 11

Simulated response variables (Coded to -1,0,1 scale) Obj.BHP BHP1 BHP2 BT1 BT2 KO1 KO2 KWAT KBAR f.(Bar) (Days) (Coded to -1,0,1 scale)1 204.68 117.11 92.14 1498 1789 1 0 -1 0 0.1062 220.44 47.38 161.57 1566 1771 -1 1 1 0 0.7813 218.27 84.46 19.26 1541 1777 0 -1 -1 -1 0.6514 218.52 119.75 117.24 1572 1883 1 1 -1 -1 0.1925 222.96 125.03 131.72 1412 1885 0 0 1 -1 0.3366 199.18 5.58 87.27 1512 1814 -1 0 -1 1 0.8957 197.79 83.55 118.70 1662 1928 0 1 -1 1 0.1038 218.25 157.59 75.58 1393 2244 1 -1 1 1 0.6329 204.01 93.46 97.21 1515 1865 0.423 0.445 -1.219 -0.118 0.01010 214.10 107.30 110.21 1422 2025 -0.141 -0.330 0.340 1.25 0.07411 203.89 94.45 98.41 1481 2017 0.409 0.423 -1.176 -0.084 0.00714 211.89 92.83 105.37 1305 1870 -0.301 -0.313 0.051 1.001 0.03315 207.41 96.66 106.25 1294 2026 -0.054 -0.075 -0.436 0.641 0.03816 200.33 91.58 97.91 1446 1901 0.272 0.270 -1.094 0.199 0.00617 214.49 68.74 98.34 1430 1815 -0.552 -0.252 -0.294 -0.200 0.06718 219.02 26.46 119.2 1597 1765 -1 0.249 -0.294 -0.701 0.60019 216.25 12.06 48.84 1381 1798 -1 -0.754 -0.795 -0.701 0.99020 213.47 34.38 77.07 1338 1891 -1 -0.754 0.208 0.302 0.42021 216.12 91.35 95.30 1274 2074 -0.462 -0.643 0.727 0.756 0.01922 214.76 110.21 62.91 1399 1970 -0.076 -1 0.341 0.370 0.16223 216.18 60.07 68.17 1333 1808 -0.847 -1 1 1 0.19824 206.30 89.20 95.68 1334 1915 -0.171 -0.291 -0.454 1.25 0.00925 199.61 88.99 95.41 1507 1919 0.129 0.062 -0.980 0.519 0.005H 207.03 89.59 93.28 1453 1995 -0.148 -0.309 -0.494 0.504 |Table 4.5: Runs that were made for the example in this article when �ve response variablesare used. Runs 1{8 come from an initial D-optimal design, runs 9{11 were runs in theoptima of the (kriging) response surfaces of BHP, BHP1 and BHP2, based on the 8 �rstruns, runs 14{16 were runs in the optima based on the 11 �rst runs and using all 5 responsevariables. To improve the matches in runs 14{16, we made several iterations with new runsin the optimum of the response surface and a small D-optimal design for new runs near theoptimum: Run 17 is a run in the optimum of the response surface based on the �rst 16runs, and runs 18{20 were designed around run 17. The next optimum was run 21, andruns 22{23 are the new local design. Finally, we made the runs 24 and 25 in the optima ofthe response surface based on 23 runs. The run that was used to generate the \history" isshown below (indicated with \H"). 12

When the number of input variables increases, the number of simulationsincreases drastically if the same precision in the response surface is wanted.Before making an experimental design, it is clearly important to limit the num-ber of input variables under study. The experienced reservoir engineer mayselect the variables that are thought to be of importance based on previous ex-perience. Another possibility is to split the problem into several sub-problems.When the dimension of the response increases, the number of responsesurfaces to be estimated also increases. Often, the response variable of interestis a curve y(t), consisting of a large number of observations/predictions of acertain quantity at di�erent points in time. Instead of looking separately ateach point on the curve, it may be wise to use some kind of parametrizationthat reduces the amount of data and focuses on the important parts of thecurve. The curve shape can be represented as a function of a parameter vector� plus some random error. We may then estimate a relation �(x) and use thisresponse surface in prediction and history matching.The procedure described can be run in a fully automatic mode, but it isalso possible to go in and look at the program output at each step, and alterfrom the default options if wanted. The experienced reservoir engineer maybe able to reduce the number of simulations needed by using knowledge aboutwhich variables should be varied, in which direction, or which variables shouldbe included in the response surface model.So far, the method is promising, but it needs more case studies, bothsmall ones where it is easy to make many simulations to evaluate the qualityof the results, and large, realistic case studies. We see also several possibleextensions of the methods: A bayesian approach where prior information aboutthe distributions is used together with the response surface, can be used toconcentrate the search on the most probable areas of the input variables, andit can be used to say something about the uncertainty in the input variablesafter matching, and to give uncertainty in the prediction estimates.We believe that the increase in computer power will make \computer ex-periments" more and more important, simulation models will become morecomplex, and the design and analysis of such computer experiments is worthstudying.AcknowledgmentsThese methods have been developed in several projects �nanced by NorskHydro and Statoil. We thank Thore Egeland at the Norwegian ComputingCenter for valuable suggestions and comments.13

ReferencesAnterion, F., Eymard, R. & Karcher, B. (1989), Use of parameter gradients forreservoir history matching, in `SPE Symposium on Reservoir Simulation',Society of Petroleum Engineers, Houston, Texas. SPE 18433.Box, G. E. P., Hunter, W. G. & Hunter, J. S. (1978), Statistics for experi-menters., John Wiley & Sons, New York.Chavent, G., Dupuy, D. & Lemonier, P. (1975), `History matching by use ofoptimal control theory', Soc. Pet. Eng. J. pp. 74{86.Damsleth, E., Hage, A. & Volden, R. (1992), `Maximum information at min-imum cost: A north sea �eld development study with an experimentaldesign', J. Pet. Techn. pp. 1350{1356.Egeland, T., Hatlebakk, E., Holden, L. & Larsen, E. A. (1992), Designingbetter decisions, in `European Petroleum Computer Conference', Societyof Petroleum Engineers, Stavanger, Norway. SPE 24275.Fedorov, V. V. (1972), Theory of optimal experiments, Academic Press Inc.,New York.Miller, A. J. (1990), Subset Selection in Regression, Chapman and Hall, NewYork.Morris, M. D., Mitchell, T. J. & Ylvisaker, D. (1993), `Bayesian design andanalysis of computer experiments: Use of derivatives in surface predic-tion', Technometrics 35(3), 243{255.Palatnik, B. & Zakirov, I. (1992), Multiphase history matching by �nite ele-ment approximation in porous and naturally fractured reservoirs, inM. A.Christie & et al., eds, `3rd European Conference on the Mathematics ofOil Recovery', Delft University Press, Delft, the Netherlands.Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. (1988),Numerical Recipes in C, second edn, Cambridge University Press, Cam-bridge.Sacks, J., Welch, W. J., Mitchell, T. J. & Wynn, H. P. (1989), `Computerexperiments', Statistical Science 4(4), 409{423.St. John, R. (1971), `D-optimality for regressions designs: A review', Techno-metrics 17(21), 15{23.Tan, T. & Kalogerakis, N. (1991), A fully implicit three-dimensional three-phase simulator with automatic history matching capability, in `SPESymposium on Reservoir Simulation', Society of Petroleum Engineers,Anaheim. SPE 21205.Welch, W. J., Buck, R. J., Sacks, J., Wynn, H. P., Mitchell, T. & Morris, M.(1992), `Screening, predicting and computer experiments', Technometrics34(1), 15{25. 14