WATER RESOURCES RESEARCH, VOL. 17, NO. 4, PAGES 921-927, AUGUST 1981 Identification of the Wairak½i GeothermalSystem LARISSA JU. FRADKIN Physics and Engineering Laboratory,Department of Scientific and IndustrialResearch Lower Hutt, New Zealand A systematic approach is used to collect information importantfor identification of the Wairakei geo- thermal system by conducting numerical experiments with its different moct•ls. The identified model is shown to possess some forecasting power. Identification in the presence of linear feedback is discussed. INTRODUCTION Identificationof a real world system consists in finding its mathematical model:firstthe plausible equations and then the estimates of the equations' parameters. Assumptions about the nature of the system made prior to any identification play an importantrole in the choice of modelstructure and parameter estimation method. The aim of this paper is to perform identi- fication of a real systemwith an emphasis on verification of such assumptions. Verification is made by comparing differ- ent models and is completed by validation of the best can- didates. These models are obtained by choosing different equationsand employing different parameter estimation al- gorithms. It is demonstrated that in this way enough informa- tion may be collectedto distinguish betweenthe failures of a model that are due to the choice of model structure and those that are due to the choice of estimation method. The system under study is the Wairakei geothermal reser- voir, New Zealand. It supplies one of the oldest geothermal power stationsof the world. The currently acceptedglobal model ascribes the Wairakei phenomenon to the presence of a column of hot water risingunderground over a hot plate in a cold water environment. This column is mainly liquid but is topped with a two-phase zone approximately500 m thick. Its overall depth is of the order of 10 km, and near the surface its area is about 15 kin'-.In this paperthe Wairakei reservoir is defined as the upper portion of the column, about 1.5 km thick. The fundamentalproblem of reservoirengineering is the assessment of the exploitation-provoked pressuredrop for various modes of production. The peculiar feature of Wai- rakei is that its pressuredrop is uniform across the field [Bolton,1970].This pressure drop induces a recharge, i.e., an inflow of water from the hot plate beneathand from the sides. Various physical models aim at explainingthe pressure drop in terms of a balance between the discharge from the wells and this recharge[Bolton, 1970; Grant, 1977;McNabb, 1975; Faustand Mercer, 1979; Pritchett et al., 1976; Robinson, 1977]. The amount of recharge is different in different models. Wairakei data (Figure 1) consists of monthly mass dis- charges from all the wells and representative monthly pres- suresat a fixed level within the reservoir (at 274 m below mean sea level). Marked fluctuations in discharge rate have been introducedby operators opening and shutting wells as part of the field operations. During the initial period the dis- chargerate increased due to the openingof new wells. Then there was the 6-monthspartial shutdownin 1968. Since that time, the total numberof wellsin production hasnot changed. This paper is not subject to U.S. copyright. Published in 1981 by the American GeophysicalUnion. Pressures have been steadilydeclining, but there was a small recovery in 1968 due to the shutdown. For the purposes of this study a PL1 computerprogram, SYSID, was developed that incorporates a few of the estima- tion algorithms described by Young [1974].This program can be used for forecasting purposes. BASIC CONCEPTS In this section, all the notionsimportant for an understand- ing of the paper are discussed for the caseof two measured zero-mean variables • and f. Let a deterministic model of a real systembe the following mathematical relationship be- tween a 'noise-free' input u and a 'noise-free' output y: where Yi= Yi T' ai i • I (1) ai r= (al •, '" , an •,bo i, '" , bm') yf --(Yi-io, '", Yi--nio, Ui, U•_i o,"' , Ui_'m•o) Here i is a time index, iois a time stepbetween different mea- surements, I = (il, il + io,'", i•_ - io,i•} is an hdex set with htegers io,i•, i• > 0; htegers n, m • 0 characterise the order of the model. The parameter vector a is real-valued and may va• with t•e. The equivalentformulation is as follows: Alt-'o] y•- B[t_•o ] u• i • I where t-•o• a back-shiR operatorand A[t%] = a•t-•o + ... + a,t-"'• B[t%] • bo + '" b•t -•ø Let the obsc•ation equations be • = u• + e• i • I (2) yi = y• + • i • I where e•and • are measurement c•ors. Equivalent to (1) - (2) is the foHowhg stochastic model: f• = y•r. a•+ •' i • I (3) where •1' ---- --11i T' ai + •,r= (r•i_,o, ..., r•i-,,,o, •,, '", •-,.•o) A family of equations of type (1) is referred to below asmodel structure (1). A model of a certainstructure is identifiedby applicationof a parameter estimationmethod to data, assuming that there 921