9 th European Conference on the Mathematics of Oil Recovery Cannes, France, 30 August - 2 September 2004

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1 1 PARALLEL WELL LOCAION OPIMIZAION USING SOCHASIC ALGORIHMS ON HE GRID COMPUAIONAL FRAMEWORK Hector Klie 1, Wolfgang Bangerth 1,2, Mary F Wheeler 1, Manish Parashar 3 and Vincent Matossian 3 1 Center for Subsurface Modeling, he University of exas at Austin, Austin, X 2 Institute for Geophysics, he University of exas at Austin, Austin, X 3 he Applied Software Systes Laboratory, Rutgers University, Piscataway, NJ Abstract he deterination of optial well locations is a challenging proble in oil production since it depends on geological and fluid properties as well as on econoic paraeters his wor addresses the efficient solution of this proble by using advanced techniques for coupling three iportant coponents of autonoic optiization: the Integrated Parallel Accurate Reservoir Siulator (IPARS) for production prediction, new optiization algoriths, in particular the Siultaneous Perturbation Stochastic Approxiation (SPSA) approach, and the Grid infrastructure to access coputational resources on the networ in a sealess way We illustrate the ethodology using nuerical results based on real data 1 Introduction Optiizing how and where wells are drilled in an oil reservoir is a proble with both high econoic ipact and high coplexity raditionally, this tas is carried out by analyzing a few scenarios with a nuerical reservoir siulator However, this ay potentially result into isleading decisions with large consequences Optiization algoriths proise to perfor a systeatic exploration of a broader set of scenarios and ai at finding the optiu under soe given conditions ogether with the experience of specialists, they also allow for a better assessent of uncertainty and reduction of ris in decision-aing he ain constraint for their use is the cost of repeatedly evaluating different exploitation scenarios via nuerical solution of a coplex set of coupled nonlinear partial differential equations on up to illions of gridblocs In this study, we present a coputational fraewor for this proble, including Grid enabled technologies [4] to autoatically discover and use available coputational resources in a distributed environent, efficient optiization algoriths, and an efficient reservoir siulator Using this fraewor, we have generated a coplete dataset for a sall but realistic test case in which only the two horizontal co-ordinates of a vertical well are subject to optiization his large dataset is at the outer reaches of what is presently feasible with today's coputational facilities he test case shows a nuber of pathologies that also need to be expected in ore realistic probles, including a large nuber of local optia In this paper, we use it to test and explore how two different optiization algoriths perfor he optiization algoriths we present in this paper are a version of the Siultaneous Perturbation Stochastic Approxiation (SPSA) algorith [] that was odified to handle the fact that the optiization doain is a bounded integer lattice, and a version of a Finite Difference Gradient algorith adapted to the sae constraints As will be shown, both algoriths need about as any function evaluations, but SPSA is significantly less prone to get caught in local axia, and is thus a ore reliable tool for optiization of probles with any local extrea he software used for these coputations builds on the DISCOVER toolit which allows agents running on geographically distributed achines to counicate with each other, to detect avail- 9 th European Conference on the Matheatics of Oil Recovery Cannes, France, 30 August - 2 Septeber 2004

2 2 able resources, and to run reservoir siulations on the [2] It is thus an ideal tool for applications that potentially run for very long ties and that need to ae use of coputational resources with availability and capabilities that ay change over tie he paper is organized as follows In the following section, we present the well placeent proble in ters of the reservoir odel and the econoic odel we use as an objective function We also introduce the two optiization algoriths we will use on this proble In Section 3, we give an overview of the Grid technology on which our software is built Nuerical results are then presented in Section 4, before concluding with Section 5 2 Description of the proble In this section, we provide a brief overview of the oil-water reservoir odel within the IPARS fraewor We then proceed to describe the objective function for the optiization proble he last part of the section is devoted to integer versions of the SPSA and FDG optiization algoriths 21 he reservoir flow odel For the purpose of this wor, we describe the reservoir odel by a set of partial differential equations for the conservation of ass of each coponent =o,w (oil and water): ( N ) φ + U = q (1) t Here, φ is the porosity of the porous ediu, N the concentration of a phase, and q the sources (production and injection rates) he fluxes U are defined using Darcy's law which, with gravity ignored, reads as U = ρ Kλ P, where ρ denotes the density of the phase, K the pereability tensor, λ the obility of a coponent, and P the pressure of a phase Additional equations specifying volue, capillary, and state constraints are added, and boundary and initial conditions copleent the syste Finally, N = S ρ with S denoting saturation of the phase (for ore inforation, see eg, [3]) he continuous proble is discretized using ixed finite eleents he resulting discrete odel is solved by the IPARS (Integrated Parallel Accurate Reservoir Siulator) software developed at the Center for Subsurface Modeling at he University of exas [6] IPARS is a parallel reservoir siulation fraewor for odeling ultiphase, ultiphysics flow in porous edia [,9] 22 he econoic revenue odel In general, the econoic value of production is a function of the tie of production and of injection and production rates in the reservoir Neglecting fixed costs, we define our objective function by suing, over the tie horizon [0,], the revenues fro produced oil over all production wells, and subtracting the costs of disposing produced water and the cost of injecting water: f = [ ] ()( o w, disp w cw, inj qw s + r) ( p) c q ( s) c q () s 0 1 dt (2) inj wells 0 prod wells he coefficients c o, c w,disp and c w,inj are the prices of oil and the costs of disposing and injecting water, in dollars per barrel each he exponential factor odels that up-front costs have to be paid off with interest Finally, we define f (p) to be the negative total revenue With the siulation odel and objective function described above, the optiization proble reads: Find the optial well location p * such that p * = argin p P f ( p) t (3)

3 3 If no confusion is possible, we drop the subscript fro f when we copare function evaluations for the sae tie horizon but different well locations p We note that in the discrete odel, possible well locations are only cell-bloc id-points P is thus a bounded integer lattice 23 Optiization algoriths he first optiization algorith we consider is an integer version of the Siultaneous Perturbation Stochastic Approxiation (SPSA) ethod First introduced by Spall [], it uses the following idea: in each iteration, choose a rando direction in search space By using two evaluation points in this and the opposite direction, deterine if the function values increase or decrease in this direction, and get an estiate of the value of the derivative of the objective function in this direction hen tae a step in the descent direction with a step length that is the product of the approxiate value of the derivative and a factor that decreases with successive iterations In its basic for, SPSA can only operate on unbounded continuous sets, and is thus unsuited for optiization on our bounded integer lattice P A odified SPSA algorith for such probles was first proposed and analyzed in [5] While their ethod involved fixed gain step lengths and did not incorporate bounds, both points are easily integrated In order to describe our algorith, let us define to be the operator that rounds a real nuber to the next integer of larger agnitude Furtherore, let Π be the operator that aps every point outside the bounds of our optiization doain onto the closest point in P and that does not odify points inside these bounds hen the integer SPSA algorith which we will use for the coputations in this paper is stated as follows (for ore details see [1,2]): Algorith 21 (Integer SPSA) 1 Set =1, γ = 0 101, α = While < K ax or convergence has not been reached do 21 Choose a rando search direction with ( ) { 1, + 1 }, 1 l n l c a 22 Copute c =, a γ = α + 23 Evaluate f = f ( Π( + )) and f = f ( Π( )) p c p c + 24 Copute g = ( f f ) Π( p + c ) Π( p c ) 25 Set p = Π( p a g ) +1 / 26 Set = + 1 end While he second algorith is a finite difference gradient (FDG) algorith that shares ost of the properties of the SPSA algorith discussed above In particular, we use the sae ethods to deterine the step lengths for function evaluation and iterate update, and the sae convergence criterion However, instead of a rando direction, we copute the search direction by a twosided finite difference approxiation of the gradient his algorith requires 2n function evaluations per iteration, in contrast to the 2 evaluations in the SPSA ethod However, we can expect better search directions fro it 3 he Grid coputing paradig he Grid is rapidly eerging as the doinant paradig for wide area distributed coputing [4] Its goal is to provide a service-oriented infrastructure that leverages standardized protocols and services to enable pervasive access to, and coordinated sharing of geographically distributed hardware, software, and inforation resources 9 th European Conference on the Matheatics of Oil Recovery Cannes, France, 30 August - 2 Septeber 2004

4 4 he software ipleenting the various steps in the coputations outlined in the previous section consists of a nuber of odules that are coupled together using this Grid technology hese odules are (see Fig 1): (1) he optiization service; (2) the IPARS factory, that upon request starts IPARS for a given well location or retrieves previous solutions fro a database syste; (3) IPARS itself, perforing the parallel siulation; (4) the econoic revenue odel that taes the IPARS output and coputes the econoic value of the oil produced in the scenario for which the siulation was run; and, (5) the clients requesting the optiization service hese odules can run on different achines at physically different locations, and counicate with each other as Grid enabled services using the DISCOVER Coputational Collaboratory that provides the echaniss for distributed applications [2] In particular, DISCOVER provides the tools to find, interact with, and enquire the capabilities of services running on distributed achines coupled via the Internet It builds on the Globus toolit, CORBA Coodity Grid, and a nuber of other standardized services Figure 1 Optiization process on the Grid he advantages of this approach are apparent: while the optiization server can run on a user s destop achine, it counicates with a central IPARS factory instance that can start IPARS runs on other achines, including on cluster coputers and worstations, thus distributing the actual worload to available resources in the ost efficient way [2] 4 Nuerical results We consider a relatively siple reservoir = [ 0,40] [ 0,5120] Ω of roughly 25 illion ft 2, which is discretized by spatial grid blocs of 0 ft length along each horizontal direction, and a depth of 30 ft Hence, the odel consists of 3904 grid blocs he reservoir under study is located at a depth of 1 and corresponds to a 2D section extracted fro a real field in the Gulf of Mexico he porosity has been fixed at φ = 0 2 but the reservoir has a heterogeneous pereability field It is assued to be surrounded by ipereable roc he fluids are initially in equilibriu with water pressures set to 2600 psi and oil saturation to 0 he reservoir odel consists of 5 fixed wells: 2 water injectors at the botto left and 3 oil producers at the top right corner of the doain Injection and production rates are coputed by specifying a fixed botto hole pressure he location of a third injection well is subject to optiization he set of possible well locations in the discrete odel is the integer lattice of grid bloc idpoints, ie { 40,120,200,,440} { 40,120,200,,500} P = In all cases, we ran our siulations for =2000 days We undertoo to generate a large realistic data set for the evaluation of optiization algoriths by coputing econoic revenues for all 3904 possible well locations, and for 200 different tie horizons between 200 and 2000 days

5 5 he data set therefore consists of f ( p) for 200 values of and 3904 values for p A plot of f 2000 ( p) is shown in Fig 3 Each siulation for a particular p too inutes on a Linux PC with a 2GHz AMD Athlon processor, for a total of alost 2000 CPU hours Figure 2 Pressure (left) and saturation (right) contours after 2000 days of siulation Note that while we would in general lie to copute the global optiu, we will usually be content if the algorith finds a solution that is alost as good his is iportant in the present context where the revenue surface plotted in Fig 3 has 2 local optia, with the global optiu t being f ( p = ( 2920,920) ) = However, there are 5 ore local extrea within only half a percent of this optial value, which aes finding the global optiu rather coplicated he white ars in Fig 3 (right) indicate the best well positions found by the SPSA algorith when started fro different points As can be seen, SPSA is able to find good locations fro arbitrary starting points, even tough it is unable to find the global optiu every tie A typical optiization run showing both the path SPSA taes and convergence of the objective function is depicted in Fig 4 In order to evaluate the Integer SPSA algorith, we start it fro every possible location p i in the set P, and for each of these 3904 SPSA runs record the point pˆ where it terinated, the function value ( ) p i f ˆ at this point, and the nuber K i of function evaluations Since ultiple function evaluations at the sae point can be cached and are therefore inexpensive, we also record the nuber L i of unique function evaluations i Figure 3 Surface view of the search space (left) and contour view with solutions obtained by SPSA with different initial guesses (right, with solutions indicated by white diaonds) 9 th European Conference on the Matheatics of Oil Recovery Cannes, France, 30 August - 2 Septeber 2004

6 6 We consider the following statistical properties of SPSA: (1) What is the average value 3904 * f = = f ( pˆ )/ 3904, i 1 i and how close is it to ( p ) 50 ( ˆ ) ϕ f? (2) what is the value ϕ 50 such that 95 f p i in 50% of runs (and siilar for the 95th percentile, ϕ )? And, (3) what are the average nuber K = =1 K / 3940 i i and L = = L / 3904 i 1 i of function evaluations and unique function evaluations to obtain convergence? Figure 4 Coputed well positions and corresponding revenue value during optiization he answer to the first two questions is closely related to the probability with which the algorith terinates at any given point p P For the two functions f 500 ( p) and f 2000 ( p), this probability of stopping at p is shown in Fig 5 It is obvious that the locations where the algorith stops are close to the (local) optia of the solution surface Figure 5 Probability surface for the SPSA algorith at 500 days (top-left),2000 days (topright) and for the FDG algorith at 500 days (botto-left) and 2000 days (botto-right) of siulation

7 * ( p ) f f 9 th European Conference on the Matheatics of Oil Recovery Cannes, France, 30 August - 2 Septeber ϕ 95 ϕ K able 1 Statistical properties of terination points of the integer SPSA algorith * ( p ) f f 50 ϕ 95 ϕ K able 2 Statistical properties of terination points of the integer FDG algorith he statistical qualities of terination points of the integer SPSA algorith are suarized in able 1 for the four data sets at =500, 1000, 1500, 2000 days he table shows that on average the stopping position is only a few percent worse than the global optiu heϕ 50 andϕ 95 values are iportant in the decision how often to restart the optiization algorith fro different starting positions Such restarts ay be deeed necessary since the algorith does not always stop at the global optiu, and we ay want to iprove on the result of a single run by starting 95 fro an iproved initial guess While ϕ reveals what value of f ( pˆ i ) we can expect fro a 50 single run in 95% of cases ( alost certainly ), the ϕ value indicates what value we can expect fro the better of two runs started independently he conclusion fro these values is that to be within a few percent of the optial value one run is not enough, while two are Finally, the last two coluns indicate that the algorith, on average, only needs 3-52 function evaluations to converge; whereas, if we cache previously coputed values, only function evaluations are required able 2 show the results obtained for the Finite Difference Gradient algorith Fro the table, it is obvious that for earlier ties, the algorith needs less function evaluations However, it also produces worse results, being significantly farther away fro the global optiu on average he reason for this is seen in the botto row of Fig 5 where we show the terination points of the algorith: it is apparent that the algorith quite frequently gets stuc in local optia, at least uch ore frequently than the SPSA algorith his leads to an early terination of the algorith and suboptial overall results 5 Conclusions We have presented the application of integer versions of the SPSA and the FDG optiization algoriths to find the optial well location on a realistic dataset fro the Gulf of Mexico In order to validate their solutions, an exhaustive evaluation of the search space was perfored We have copared the quality of solutions as well as their efficiency In particular, we have shown L L

8 that the SPSA algorith yields very good solutions with relatively few function evaluations, and is capable of handling situations with any local extrea significantly better than a siple gradient-based algorith his nowledge is useful for tacling larger probles for which the exact optiu is unnown and ipossible to copute Given the size of the proble and the expensiveness of functions evaluations (20-30 inutes of CPU tie per function evaluation), the coputations were perfored using a Grid enabled software fraewor, in which optiizers, siulators, and function evaluators are separate agents on different, geographically distributed achines hese agents are able to autoatically detect the availability of coputational resources, and start instances of the IPARS reservoir siulation software on appropriate achines while still providing the researcher with the ability to control the various distributed processes fro the destop We presently prepare a ore in depth coparative analysis of SPSA with other optiization algoriths for the well placeent proble [1] Acnowledgeents he research was supported by the National Science Foundation (NSF) via grants ACI (CAREERS), EIA (NGS), NSF EIA /EIA (IR), ANI (NR), CNS (NGS) and by DOE ASCI/ASAP (Caltech) via grant References [1] W Bangerth, H Klie, and M F Wheeler On the Reservoir Oil Placeent Proble using the Siultaneous Perturbation Stochastic Approxiation Method, in preparation, 2004 [2] W Bangerth, H Klie, V Matossian, M Parashar, and M F Wheeler An Autoatic Reservoir Fraewor for the Stochastic Optiization of Well Placeent Accepted for publication in Cluster Coputing: he Journal of Networs, Software ools and Applications, 2004 [3] J R Fanchi, Principles of Applied Reservoir Siulation, Gulf Professional Publishing, 2nd edition, 2001 [4] I Foster and C Kesselan he Grid: Blueprint for a New Coputing Infrastructure Morgan Kaufann Publishers, 1999 [5] L Gerencsér, S D Hill, and Z Vágó Discrete optiization via SPSA In Proceedings of the Aerican Control Conference, Arlington, 2001, pp , 2001 [6] IPARS: Integrated Parallel Reservoir Siulator [] Q Lu, M Peszynsa, and M F Wheeler A parallel ulti-bloc blac-oil odel in ultiodel ipleentation SPE Journal, (3):2--2, Septeber 2002 SPE 9535 [] J C Spall Introduction to stochastic search and optiization: Estiation, siulation and control John Wiley & Sons, Inc, Publication, New Jersey, 2003 [9] M F Wheeler, J A Wheeler, and M Peszynsa A distributed coputing portal for coupling ulti-physics and ultiple doains in porous edia In L R Bentley, J F Syes, C A Brebbia, W G Gray, and G F Pinder, editors, Coputational Methods in Water Resources, pp A A Balea, 2000