Automatic History Matching by use of Response Surfaces and ...

Automatic History Matching by use of Response Surfaces and Experimental Design Alfhild L. Eide1 Lars Holden1 Edel Reiso2 Sigurd I. Aanonsen2 1 Norwegian Computing Center, Oslo, Norway 2 Norsk Hydro a.s., Bergen, Norway Presented at 4th European Conference on the Mathematics of Oil Recovery, Rros, Norway, 7-10 June, 1994 Abstract

History matching is the process of using production history to improve the estimates of geological and petrophysical parameters in the oil eld. We estimate response surfaces based on a set of reservoir simulations with dierent combinations of the reservoir parameters. A response surface is a simpli ed relation y^(x) between reservoir simulator input x and output (response) y , and gives a rough overview of the behavior of the response in the whole region of interest. These response surfaces are searched to nd parts that are close to the historical data. Since a history matching problem may have several solutions, surfaces that give overview over the whole region are of interest. History matching is done by minimizing the distance between the observed values of the response variables, and the response predicted from the response surfaces y^(x). The process is iterative: make experimental design, run the selected experiments, generate response surface, optimize, make new re ned design, run : : : and so on. One or several iterations may be performed automatically. The method is demonstrated on a synthetic reservoir simulation example.

1 Introduction History matching is the process of using production history to improve the estimates of geological and petrophysical parameters in the oil eld. These parameters should be modi ed to induce the reservoir simulator to reproduce production history. An indirect, or secondary aim is to predict future reservoir performance. 1

x ?

-

-y - y, observed history

Black box

Figure 1.1: History matching can be seen as an inverse problem. For a given x, it is

possible to calculate y = f (x). What we are looking for, is an x, or a set of x's that will reproduce the observed y-value y .

Traditionally, history matching has been performed by trial and error. The experienced reservoir engineer has changed one or a few variables at a time hoping for an improved match. With the increased computer resources available and increased understanding of the problem, it is possible to improve this process. The most straightforward improvement is to reduce the manual work in the process. There is much manual work in generating input les, starting of reservoir simulations, reading of output les and visualization of the results. The more dicult improvement task is automatic generation of a set of input data to test out. This will both reduce the required manpower and make the history match more objective. To estimate the uncertainty in the predictions based on the history match, objectivity is important. Traditional gradient techniques using numerical derivatives usually require too many simulations to be applicable to automatic history matching. This computation time may be signi cant reduced by the use of optimal control theory methods (see Chavent, Dupuy & Lemonier (1975), Palatnik & Zakirov (1992), e.g.), or by calculating sensitivity coecients parallel to the simulations (Anterion, Eymard & Karcher 1989), (Tan & Kalogerakis 1991). Our main idea is to use estimated response surfaces based on a set of reservoir simulations with dierent combinations of the reservoir parameters. The response surface is a simpli ed relation y^(x) between reservoir simulator input x and output (response) y (Figure 1.1), and gives a rough overview of the behavior of the response in the whole region of interest. These response surfaces are searched to nd parts that are close to the historical data. Since a history matching problem may have several solutions, surfaces that give overview over the whole region are of interest. If necessary, some parts of the response surfaces can be re ned by running additional reservoir simulations. History matching is done by minimizing the distance between the observed values of the response variables yobs, and the response predicted from the response surface y^(x). The process is iterative: make experimental design, run the selected experiments, generate response surface, optimize, make new re ned design, run : : : and so on. Experimental design and response surface methods are discussed in Section 2. Section 3 lists the steps in the method, and Section 4 gives an example. 2

2 Experimental design and response surface methods Reservoir simulations can be time consuming and expensive. It is important to get maximum information from a relatively small number of reservoir simulations. An experimental design is a plan describing the dierent choices of each of a number of input variables in a series of simulation runs. Experimental design provides alternatives to the traditional \vary one at a time" strategy. Among other things, it is possible to estimate the joint eect of changing two parameters simultaneously (Box, Hunter & Hunter 1978). The theory of experimental design was developed and applied in agriculture in the 1920's. Since the mid 1980's, experimental design has also been studied and used for `computer experiments', see for instance Welch, Buck, Sacks, Wynn, Mitchell & Morris (1992), Sacks, Welch, Mitchell & Wynn (1989), Morris, Mitchell & Ylvisaker (1993). Experimental design of reservoir simulations is demonstrated by Damsleth, Hage & Volden (1992) and has been followed up by Egeland, Hatlebakk, Holden & Larsen (1992). Like Damsleth et al. (1992) and Egeland et al. (1992) we use D-optimal designs. D-optimality is a mathematical procedure to select the optimal runs from a (large) set of possible runs (the candidate set). Data from previous runs 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 input and response variables, the output is:

 the optimal design w.r.t. obtaining optimally precise estimates of the coecients in the equation given.

The regression equation may be written: (2.1)

y = F (x)b + ;

where y is a vector of response values, b is a vector of coecients to be estimated, and  is a vector of independent random variables, each with expectation 0 and variance 2. The vector function F may be a general function of x, but in this paper we limit F to be second order polynomials in x. Let the number of coecients bi be n. For a given set of m experiments, F (x) becomes an m  n matrix; the design matrix. 3

The least squares estimate for b is given by ^b = (F T F )?1F T y (2.2) while the covariance matrix is (2.3)

Cov ^b = 2(F T F )?1

It can be shown that the optimal precise estimates of b is obtained when the determinant of F T F is maximized over the region of interest, and this is why the method is called D-optimality (Determinant-optimality) (See St. John (1971) or Fedorov (1972) for a complete discussion.)

2.1 Response surfaces

A response surface is a simpli ed relation between the simulator input and output. We consider the reservoir simulator as a \black box" (Figure 1.1). Based on the results from the selected simulations, a response surface is generated which can be used to predict simulator output for other input values. The most standard way of estimating a response surface is regression. For deterministic simulations, interpolating surfaces, for instance kriging surfaces, is an interesting alternative. Our practical experience with these response surfaces suggests that in an early stage, when there are few data points, one might as well use regression surfaces, but as more data points are included, kriging gives better results. Addition of an extra data point does not change the regression surface much, but when kriging is used, the response surface will always be changed to interpolate the new data point. This is an advantage when new data points are added to improve the quality of the response surface near a possible optimum. If gradient information is available, this can also be used in the generation of response surfaces, see Morris et al. (1993).

3 Method

1. Experimental design: For each input variable to be adjusted through history matching, 3 levels are speci ed: low, base case and high level. The candidate set contains all combinations of these levels (possibly removing unfeasible combinations of input variables). Then a D-optimal design is generated (selected from the candidate set), based on the a priori equation and the number of simulations required. 2. Reservoir simulations: Run the reservoir simulations in the D-optimal design. 4

3. Analysis: Fit a regression or kriging model to the data. Automatic model selection (see Miller (1990) for a review) may be used for selection of terms in the regression model. 4. Optimization using response surface: Minimize X (3.1) F (^y(x)) = wi(^yi(x) ? yi)2 i

where y^(x) is the response surface, y is the observed history, x is a vector of input variables (reservoir parameters) and wi are weights. We use a standard multidimensional optimization routine (Powell's method from Press, Flannery, Teukolsky & Vetterling (1988)), but since there are often several optima, we select a set of dierent starting points by rst calculating the value of the object function on a grid and then selecting the best points in the grid as starting points. Optimization results are presented as a list of possible optima. 5. Reservoir simulations: Run reservoir simulations close to the optimal values. The user may select one or several optima and explore the response surface near these by performing additional simulations. 6. Iteration: If necessary, go back to step 1 to set up a new design and do additional experiments. To improve the estimate of the response surface near a predicted optimum, a new design can be generated close to this optimum. The levels in the new design can be generated as follows: The optimum input values are base case. Calculate the distance to the nearest data point, and let the dierence between base case and high (low) levels in the new design depend on this distance. Then a new Doptimal design can be generated around the optimum, using runs that have already been made. With a program system where experimental design and analysis are integrated with a possibility to start reservoir simulations and read the results back into the system, this iteration process is easy to run. The Norwegian Computing Center has developed the computer program \DECISION" for design and analysis of reservoir simulations. This program has been extended with a history matching module and it is possible to start the reservoir simulator from within \DECISION". One or several iterations may be performed automatically with default options in design and analysis.

4 Example 4.1 Model

A 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 permeability. There is a barrier between the two producers, perpendicular to the oil/water contact, modeled as a row of thin blocks. The two oil zones communicate through the aquifer, and in addition there is a small opening in the barrier at the top of the reservoir covering two simulator blocks. The degree of communication through this opening was also considered as unknown in the history matching process (Table 4.1). The following production strategy is chosen: Produce well 1 for 1 year, then open well 2 and produce both wells for 1/2 year. Then start water injection and run production and injection until water breakthrough. As matching variables 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 in the two producers (Table 4.2). Due to the shape of the water cut curves, it turned out to be dicult to nd a consistent way of de ning the time of water breakthrough automatically. In the nal optimization, water breakthrough was de ned as the time when water cut exceeded 0.1. However, other de nitions would have given dierent results (see discussion below).

4.2 Experimental design and response surfaces

For each input variable, low, high and base case levels were selected. These levels were transformed to a -1,1,0 scale. This makes it possible to get an orthogonal design (i.e. the matrix F in Eq.(2.1) is orthogonal), and it is easier to interpret the regression coecients because all variables are on the same scale and symmetric around zero. The levels used, both untransformed and transformed, can be seen in Table 4.3. The initial design consisted of 8 runs. Automatic model selection was used to select a regression model for the overall trend. This trend was used in the kriging model for the response surface. Equations 4.1{4.3 give an example of what the trend may look like. The regression equations given are based on the rst 8 runs. (4.1) (4.2) (4.3)

BHP = 214:12 + 6:43  KWAT ? 7:42  KBAR + (x) BHP1 = 90:64 + 54:04  KO1 + 19:36  KWAT + (x) BHP2 = 100:45 + 22:51  KWAT + 44:95  KO2 + (x)

4.3 Optima

In history matching, it often happens that more and more data become available as time passes. In this example we rst used 3 response variables, namely BHP, BHP1 and BHP2, where history data would have been available after 1 21 year of production. Later we added BT1 and BT2 to see how the addition of more data reduces the uncertainty and the number of possible optima. 6

Water/oil contact

P2

KO2

I1

KBAR KO1

KWAT

P1

Figure 4.1: Example reservoir. Only part of the aquifer is shown. The position of the barrier is indicated with a horizontal line.

KO1 KO2 KWAT KBAR

permeability near oil producer 1 permeability near oil producer 2 permeability near injector permeability in the two uppermost barrier blocks Table 4.1: Input variables.

BHP BHP1 BHP2 BT1 BT2

Bottom hole pressure in producer 2 after 1 year Bottom hole pressure in producer 1 after 1.5 years Bottom hole pressure in producer 2 after 1.5 years Breakthrough time well 1 Breakthrough time well 2 Table 4.2: Response variables. 7

Variable name

scale low level base case high level coded -1 0 1 KO1, KO2, KWAT uncoded log 4.38 5.19 6.00 uncoded real 79.84 179.5 403.4 coded -1 0 1 KBAR uncoded log -2.3 1.15 4.6 uncoded real 0.10 3.16 99.5 Table 4.3: Levels in the initial design for the 4 input variables. Variables are transformed

from the real scale by rst taking logarithms, then coding so that zero becomes the base case level.

Optimization with 8 runs and 3 response variables gave the optima shown in Table 4.4. The simulation runs in these 3 optima were then used to improve the response surface, which now exhibited 2 distinct optima (runs 12 and 13 of Table 4.4). However, the match itself is not signi cantly improved. Figure 4.2 shows 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, water breakthrough occurs and BT1 and BT2 are now available. We expect that this will reduce the number of possible optima. We re-run optimization based on the 11 rst runs, but using 5 response variables instead of 3. To compensate for the dierence in absolute values between the dierent variables, we used relative dierences in the object function, that is X y^i (x) ? yi !2 (4.4) : F (^y(x)) = yi i From the list of possible optima, we selected 3 dierent points for new simulations. Results are shown in Table 4.5. The results are now improved, but the match is still not satisfactory, and 3 additional iterations with runs in 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 25 compared with the history run. For the pressures, the match was signi cantly improved, while the results are less satisfactory for water cut. As mentioned, breakthrough was de ned as the time when water cut exceeded 0.1. From Figure 4.3 we see that even if the match is reasonably good at one point in time, the trend is not well reproduced in any of the runs. This shows that to obtain a better match of the water cut over time|and a better prediction, more than one measurement is needed. 8

BHP(0, 0, KWAT, KBAR)

BHP A

T

K

W

B

A

R

K 223. 219. 214. 210. 205. 200. 00 - 1. 600 - 0.

00 .2 -0

0 20 0.

0 60 0.

- 1.

00 1.

1. 0 0 0. 6 00 0. 2 00 - 0. 200 - 0. 60 0 00

BHP(0, 0, KWAT, KBAR)

BHP A

T

K

W

B

A

R

K 225. 219. 212. 206. 200. 193.

1. 0 0 0. 6 00

00 - 1. 600 - 0.

0 . 20 -0

- 0. 0 20 0.

- 0. 0 60 0.

- 1.

00 1.

0. 2 00 200

60 0

00

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

Water Cut W1 0.0 0.1 0.2 0.3 0.4 0.5

BHP W1 (Bar) 100 150 200 250

history run 5 run 15 run 25

2

3 4 Years

5

6

100

history run 5 run 15 run 25

0

1

2

3 4 Years

5

2

Water Cut W2 0.0 0.1 0.2 0.3 0.4 0.5

1

BHP W2 (Bar) 150 200 250

0

history run 5 run 15 run 25

6

3

4 Years

5

6

6

7

history run 5 run 15 run 25

3

4

5 Years

Figure 4.3: Graphs showing bottom hole pressure for well 1 (BHP W1) and well 2 (BHP W2); and water cut for well 1 (WCUT W1) and well 2 (WCUT W2). Simulation results for run 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 BHP W2, run 5 lies above the other runs, then comes run 15, and run 25 and the history are almost identical, except BHP W2 at 1 year.

10

1 2 3 4 5 6 7 8 9 10 11 12 13 H

Simulated response variables Input variables Objective BHP BHP1 BHP2 KO1 KO2 KWAT KBAR function (Bar) (Coded to -1,0,1 scale) values 204.68 117.11 92.14 1 0 -1 0 766 220.44 47.38 161.57 -1 1 1 0 6618 218.27 84.46 19.26 0 -1 -1 -1 5626 218.52 119.75 117.24 1 1 -1 -1 1610 222.96 125.03 131.72 0 0 1 -1 2979 199.18 5.58 87.27 -1 0 -1 1 7160 197.79 83.55 118.70 0 1 -1 1 773 218.25 157.59 75.58 1 -1 1 1 5057 204.01 93.46 97.21 0.423 0.445 -1.219 -0.118 41 214.10 107.30 110.21 -0.141 -0.330 0.340 1.25 647 203.89 94.45 98.41 0.409 0.423 -1.176 -0.084 62 208.73 96.08 106.09 -0.097 -0.108 -0.375 0.476 208 209.23 97.66 101.15 0.275 0.271 -0.958 -0.209 131 207.03 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 an initial D-optimal design, runs 9{11 were runs in the optima of the (kriging) response surfaces of BHP, BHP1 and BHP2, based on the 8 rst runs, runs 12{13 were runs in the optima based on the 11 rst runs. The run that was used to generate the \history" is shown below (indicated with \H").

5 Discussion and Conclusions We have presented a general method that can be used for history matching and other optimization problems where computer intensive simulations are involved. We use response surfaces to predict the simulations for other combinations of the input than those that have been run on the simulator. The response surface is thus a critical factor of success in our method. The quality of the response surface depends on the experimental design used, and on the model used for the response surface. One question is whether one should use kriging, regression or something else. Another question is which terms should be included 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 the method on other examples not included in this paper with up to 8 input variables and 24 response variables with reasonably good results (match obtained after about 30 simulation runs). Typical real-world problem will often have higher dimensions, both in the input and the response. Also each simulation becomes more computer intensive. The authors believe that also for larger examples than the tested ones, the procedure presented in this paper is better than the trial and error approach. 11

1 2 3 4 5 6 7 8 9 10 11 14 15 16 17 18 19 20 21 22 23 24 25 H

Simulated response variables BHP BHP1 BHP2 BT1 BT2 (Bar) (Days) 204.68 117.11 92.14 1498 1789 220.44 47.38 161.57 1566 1771 218.27 84.46 19.26 1541 1777 218.52 119.75 117.24 1572 1883 222.96 125.03 131.72 1412 1885 199.18 5.58 87.27 1512 1814 197.79 83.55 118.70 1662 1928 218.25 157.59 75.58 1393 2244 204.01 93.46 97.21 1515 1865 214.10 107.30 110.21 1422 2025 203.89 94.45 98.41 1481 2017 211.89 92.83 105.37 1305 1870 207.41 96.66 106.25 1294 2026 200.33 91.58 97.91 1446 1901 214.49 68.74 98.34 1430 1815 219.02 26.46 119.2 1597 1765 216.25 12.06 48.84 1381 1798 213.47 34.38 77.07 1338 1891 216.12 91.35 95.30 1274 2074 214.76 110.21 62.91 1399 1970 216.18 60.07 68.17 1333 1808 206.30 89.20 95.68 1334 1915 199.61 88.99 95.41 1507 1919 207.03 89.59 93.28 1453 1995

(Coded to -1,0,1 scale) KO1 KO2 KWAT KBAR (Coded to -1,0,1 scale) 1 0 -1 0 -1 1 1 0 0 -1 -1 -1 1 1 -1 -1 0 0 1 -1 -1 0 -1 1 0 1 -1 1 1 -1 1 1 0.423 0.445 -1.219 -0.118 -0.141 -0.330 0.340 1.25 0.409 0.423 -1.176 -0.084 -0.301 -0.313 0.051 1.001 -0.054 -0.075 -0.436 0.641 0.272 0.270 -1.094 0.199 -0.552 -0.252 -0.294 -0.200 -1 0.249 -0.294 -0.701 -1 -0.754 -0.795 -0.701 -1 -0.754 0.208 0.302 -0.462 -0.643 0.727 0.756 -0.076 -1 0.341 0.370 -0.847 -1 1 1 -0.171 -0.291 -0.454 1.25 0.129 0.062 -0.980 0.519 -0.148 -0.309 -0.494 0.504

Obj. f. 0.106 0.781 0.651 0.192 0.336 0.895 0.103 0.632 0.010 0.074 0.007 0.033 0.038 0.006 0.067 0.600 0.990 0.420 0.019 0.162 0.198 0.009 0.005 |

Table 4.5: Runs that were made for the example in this article when ve response variables

are used. Runs 1{8 come from an initial D-optimal design, runs 9{11 were runs in the optima of the (kriging) response surfaces of BHP, BHP1 and BHP2, based on the 8 rst runs, runs 14{16 were runs in the optima based on the 11 rst runs and using all 5 response variables. To improve the matches in runs 14{16, we made several iterations with new runs in the optimum of the response surface and a small D-optimal design for new runs near the optimum: Run 17 is a run in the optimum of the response surface based on the rst 16 runs, and runs 18{20 were designed around run 17. The next optimum was run 21, and runs 22{23 are the new local design. Finally, we made the runs 24 and 25 in the optima of the response surface based on 23 runs. The run that was used to generate the \history" is shown below (indicated with \H").

12

When the number of input variables increases, the number of simulations increases drastically if the same precision in the response surface is wanted. Before making an experimental design, it is clearly important to limit the number of input variables under study. The experienced reservoir engineer may select the variables that are thought to be of importance based on previous experience. Another possibility is to split the problem into several sub-problems. When the dimension of the response increases, the number of response surfaces to be estimated also increases. Often, the response variable of interest is a curve y(t), consisting of a large number of observations/predictions of a certain quantity at dierent points in time. Instead of looking separately at each point on the curve, it may be wise to use some kind of parametrization that reduces the amount of data and focuses on the important parts of the curve. 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 this response surface in prediction and history matching. The procedure described can be run in a fully automatic mode, but it is also possible to go in and look at the program output at each step, and alter from the default options if wanted. The experienced reservoir engineer may be able to reduce the number of simulations needed by using knowledge about which variables should be varied, in which direction, or which variables should be included in the response surface model. So far, the method is promising, but it needs more case studies, both small ones where it is easy to make many simulations to evaluate the quality of the results, and large, realistic case studies. We see also several possible extensions of the methods: A bayesian approach where prior information about the distributions is used together with the response surface, can be used to concentrate the search on the most probable areas of the input variables, and it can be used to say something about the uncertainty in the input variables after matching, and to give uncertainty in the prediction estimates. We believe that the increase in computer power will make \computer experiments" more and more important, simulation models will become more complex, and the design and analysis of such computer experiments is worth studying.

Acknowledgments These methods have been developed in several projects nanced by Norsk Hydro and Statoil. We thank Thore Egeland at the Norwegian Computing Center for valuable suggestions and comments.

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References

Anterion, F., Eymard, R. & Karcher, B. (1989), Use of parameter gradients for reservoir 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 experimenters., John Wiley & Sons, New York. Chavent, G., Dupuy, D. & Lemonier, P. (1975), `History matching by use of optimal control theory', Soc. Pet. Eng. J. pp. 74{86. Damsleth, E., Hage, A. & Volden, R. (1992), `Maximum information at minimum cost: A north sea eld development study with an experimental design', J. Pet. Techn. pp. 1350{1356. Egeland, T., Hatlebakk, E., Holden, L. & Larsen, E. A. (1992), Designing better decisions, in `European Petroleum Computer Conference', Society of 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, New York. Morris, M. D., Mitchell, T. J. & Ylvisaker, D. (1993), `Bayesian design and analysis of computer experiments: Use of derivatives in surface prediction', Technometrics 35(3), 243{255. Palatnik, B. & Zakirov, I. (1992), Multiphase history matching by nite element approximation in porous and naturally fractured reservoirs, in M. A. Christie & et al., eds, `3rd European Conference on the Mathematics of Oil 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, Cambridge. Sacks, J., Welch, W. J., Mitchell, T. J. & Wynn, H. P. (1989), `Computer experiments', Statistical Science 4(4), 409{423. St. John, R. (1971), `D-optimality for regressions designs: A review', Technometrics 17(21), 15{23. Tan, T. & Kalogerakis, N. (1991), A fully implicit three-dimensional threephase simulator with automatic history matching capability, in `SPE Symposium 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', Technometrics 34(1), 15{25. 14