Nonlinear Parameter Estimation Based on History Matching of Temperature Measurements for Single-Phase Liquid-Water Geothermal Reservoirs

Transcription

1 Proceedings World Geothermal Congress 015 Melbourne, Australia, 19-5 Aril 015 Nonlinear Parameter Estimation Based on History Matching of Temerature Measurements for Single-Phase Liquid-Water Geothermal Reservoirs Mustafa Onur and Yildiray Palabiyik Deartment of Petroleum and Natural Gas Engineering, Istanbul Technical University, Maslak, Istanbul, Turkey Keyords: History Matching, Temerature Data, Analytical Solution, Single-Phase Liquid-Water Geothermal Reservoirs ABSTRACT In this study, the main objective is to investigate the use of temerature data through history matching for estimating the reservoir arameters related to fluid and heat flo. The investigation is conducted by using an analytical solution based on the modification of the analytical solution of Chekalyuk (1965) as a forard model to comute both transient ressure and temerature behaviors of a single-hase, liquid-ater infinite-acting geothermal system. Our modification involves the incororation of the skin factor into the original solution of Chekalyuk (1965) ho did not consider the skin effects in temerature solution. Only, the constant-rate roduction tests ere considered. Both the gradient based Levenberg-Marquardt and non-gradient Ensemble Kalman Filters (EnKF) have been considered as the non-linear arameter estimation methods. To simulate real cases, temerature data generated from the analytical solution ere corruted ith normally distributed noise. The results sho hich of the reservoir arameters (e.g., skin, orosity, ermeability, fluid and rock densities and heat caacities) can be reliably estimated from the sandface temerature transient data in the resence of noise, recorded during constant-rate dradon tests. A comarison of the Levenberg- Marquardt and the EnKF in terms of estimation rocedure and comutational erformance for the roblem of interest is also rovided. 1. INTRODUCTION The reservoir characterization through integration of dynamic data such as ressure, rate, etc., through history matching has become commonlace throughout the etroleum and geothermal industries. Although temerature data are routinely recorded in ell test alications, the use of temerature data in addition to ressure for estimating the arameters controlling the fluid and heat flo for the urose of reservoir characterization has been ignored in the ast. The temerature data for history matching has recently attracted the attention of various researchers. In the etroleum and geothermal literature, it has been shon that temerature in addition to ressure can be a good source of data for reservoir characterization by the use of simle both lumed-arameter and distributed-arameter (1D, D and 3D) flo models (Onur et al. 008a, b; Sui et al. 008a, b; Ramazanov et al. 010; Duru and Horne 010a, b). To the best of the authors knoledge, Onur et al. (008a) ere to first to investigate the use of temerature data together ith ressure data in history matching for estimating reservoir arameters of fluid and heat flo or geothermal related flo roblems. Their investigation as based on a non-isothermal lumed-arameter model caable of estimating both ressure and temerature resonses of a single-hase liquid-dominated geothermal reservoir idealized as a single-closed or recharged tank. Their model is solely based on the convection and neglected the conduction. Then, using this lumed-arameter model as a forard model in a gradient based history-matching algorithm [the Levenberg-Marquardt method ith the restricted ste rocedure as described by Fletcher (197)], they shoed that one could estimate reservoir arameters such as the bulk volume and orosity of reservoir confidently if the average reservoir temerature data along ith average ressure data is used for the urose of history-matching. In another ork based on the same non-isothermal lumed-arameter model, Onur et al. (008b) also shoed that if the secific heat caacity and density of the reservoir rock, recharge coefficient, recharge source temerature are added to the unknon arameter set, then the arameter estimation roblem based on history matching roblem becomes ill-conditioned, but one could estimate the orosity and the bulk volume of the reservoir in confidence by history matching temerature and ressure data. Later, Tureyen et al. (009) extended the non-isothermal lumed-arameter model of Onur et al. (008b) to a more general nonisothermal model of multile tanks and investigated the use of temerature data in history matching for estimation of reservoir arameters from such a non-isothermal forard model. The main conclusion of Tureyen et al. (009) is that hile ressure allos estimation for only the initial ressure, ore volume, and the recharge index, the additional temerature information allos estimation for other arameters, articularly for the bulk volumes and orosities of the reservoir and aquifers. Although both the Onur et al. (008a) and Tureyen et al. (009) models rovide useful insights on understanding of information content of average reservoir temerature in terms of model arameter estimation; nonetheless, satial changes in ressure and temerature cannot be modeled using such lumed-arameter models, and hence cannot be used to investigate the information content of the bottom-hole ressure and temerature at a ell for arameter estimation uroses. In etroleum engineering literature, a D (r-z) radial simulator modeling temerature resonse for the case of single-hase slightly comressible fluid flo in D stratified systems has been resented by Sui et al. (008a). Sui et al. (008a) have indicated that the ellbore temerature is sensitive to the radius and ermeability of damage zone around the ellbore in stratified systems. In their ork, although the develoed energy balance equation contains the effects of Joule-Thomson and thermal exansion, ressure used in energy balance equation has been obtained from the mass balance equation that makes the assumtions of isothermal flo and slightly comressible fluid. This is the same assumtion as the one used in Ramazanov et al. (010) and Duru and Horne (010a) s studies. An algorithm for an inverse solution formulated as a non-linear least-squares regression roblem to estimate ermeability (region outside damage zone), orosity, radius and ermeability of damage zone by history matching observed temerature and ressure data has been given by Sui et al. (008b) as another study. The claim that the mentioned arameters could reliably be 1

2 estimated by history matching temerature data if noise in temerature measurements is not very high is the imortant result of this study. Besides, a transient, 1D radial model couling the mass and energy balance equations to estimate the ressure and temerature resonses of a system ith the comonents oil, connate ater and rock has been given by A (010) ho has shon the ressure and temerature resonses in the reservoir because of Joule-Thomson exansion of fluids under non-isothermal conditions. Duru and Horne (010b) have realized another study about the information content of transient temerature data. An inverse solution of ermeability and orosity distributions in reservoir has been resented by history matching to temerature data ith the method of Ensemble Kalman Filter (EnKF) using a forard model based on a couled numerical solution of mass and energy balance equations for a 3D (x-y-z) system. Duru and Horne (010b) have indicated ith this study that temerature data contain more information about orosity distribution than that of ermeability distribution. In this ork, e consider a relatively simle roblem; a fully-enetrating vertical ell roducing at a constant-rate under 1D radial non-isothermal flo in an infinite-acting single-hase liquid ater geothermal reservoir for investigating the use of the temerature transients in history matching. To different arameter estimation methods; gradient based Levenberg-Marquardt and non-gradient Ensemble Kalman Filters (EnKF), are used. For our investigation, the simle analytical solution of Chekalyuk (1965) as the forard model for comuting ressure and temerature transients is used. The aer is organized as follos: a brief descrition of the modified version of the Chekalyuk (1965) analytical solution, incororating the skin effects near the ellbore, hich is used as the forard model, is rovided next, and then the details of the history-matching methods used for erforming match of the observed ressure and temerature data are discussed. Finally, the alications for arameter estimation ith the synthetic transient ressure and temerature data in the resence of noise are demonstrated.. ANALYTICAL SOLUTION The forard model considered in this ork is based on the modified version of the analytical solution of Chekalyuk (1965). Chekalyuk s original solution did not include the effect of skin factor (S) near ellbore due to stimulation and damage (see Earlougher s classical SPE monograh for the concet of skin factor; Earlougher 1977). In this ork, e have modified the original Chekalyuk temerature solution to incororate the skin factor. Later in this section, e check the validity of the ressure and temerature solutions comuted from the analytical solution ith and ithout skin effects ith the corresonding solutions comuted from a D (r-z) simulator develoed by Palabiyik et al. (013). The analytical solution assumes 1D radial fluid and heat flo toards a fully-enetrating line-source vertical ell of uniform thickness roducing at constant-rate in an infinite acting reservoir ith no-fluid and heat flo across its to and bottom boundaries. Only convective heat transfer for hich the effects of Joule-Thomson and adiabatic exansions are included on temerature transients. The effect of conduction is neglected as it has usually not have a significant effect on temerature transients during roduction eriods. The solution also assumes a single-hase flo of a slightly comressible fluid of constant viscosity and constant isothermal comressibility in an infinite-acting homogeneous reservoir. Chekalyuk (1965) decoules the diffusivity equation for ressure and convective heat flo equation for temerature by assuming that the coefficients in the diffusivity and heat flo equations are indeendent of ressure and temerature. The solution ignores the ellbore storage and non-isothermal thermal effects in the ellbore. Here, e modified the Chekalyuk (1965) equation to include the skin factor by simly treating it as a steady-state ressure dro and adding this ressure dro into the ellbore ressure solution ith zero skin (i.e., no damage or no stimulation at the sandface). Under the aforementioned assumtions, the Chekalyuk solution for the sandface temerature is given by: T q 0 ( t ) T t (1 ) Ei ( ) (1) JT, 4 kh here Ei(-x) is the exonential integral function given by r 4 t u e Ei ( x ) du () x u and is the bottom-hole ressure dro at the sandface and is given by: t ( t ) q r - Ei 4 kh 4 t 0 S (3) Here, S is the skin factor, and is the diffusivity constant: k (4) c t The variables and in Eq. 1 are given by, resectively:

3 JT, 1 C JT,, C, C, 1 C, s s (5) and q 4 h C, C, 1 C, s s (6) In Eqs. 1-6, q is ell s volumetric roduction flo rate (m 3 /s), h is the formation thickness (m), S is the skin factor (dimensionless), JT, is the Joule-Thomson coefficient (K/Pa), is the fluid viscosity (Pa s), is orosity (fraction), k is the ermeability (m ), c t is the total isothermal comressibility of the liquid-ater and rock system (c t = c r + c, here c r is the effective isothermal rock comressibility and c is the isothermal ater comressibility), C, is the secific heat caacity of ater (J kg/k), C,s is the secific heat caacity of the solid rock (J kg/k), is the density of the liquid-ater (kg/m 3 ) and s is the density of the solid rock (kg/m 3 ). T 0 and 0 are the initial reservoir temerature (K) and ressure (Pa). No, e check the agreement of the temerature and ressure solutions given by Eqs. 1 and 3 ith the corresonding solutions comuted from a D (r-z) numerical simulator develoed by Palabiyik et al. (013) for to different data sets. Both data sets are based on the same basic inut data given in Table 1. The only difference beteen the to cases is that the value of the skin factor; S = 0 and S = 5). In the simulator, the effect of skin is incororated through the use of the effective ellbore radius concet; i.e., r r e S, here r is the effective ellbore radius. It is imortant to note that the numerical simulator is quite general in that it rigorously solves the mass and energy balances, treat rock and fluid roerties as a function of both ressure and temerature, include the effects of Joule-Thomson exansion for the fluid and adiabatic exansions for fluid and rock and the conductive heat transfer on ressure and temerature transients. In Table 1, β s is the isobaric thermal exansion coefficient of rock and set it to zero in the simulator because the analytical solutions (Eq. 1 and 3) do not consider the effects of isobaric thermal exansion on ressure and temerature transients. t is the total system thermal conductivity and again, set it to zero in the simulator because Eqs. 1 and 3 do not consider the effects of conductive heat transfer both in the rock and fluid. Comarisons of ressure and temerature numerical and analytical solutions are shon in Figs. 1 and. The number of grid blocks used in the radial direction to simulate the ressure and temerature data from the simulator is 00. As can be seen from Fig. 1, e have erfect matches of the analytical ressure solutions ith the numerical ressure solutions for both cases of skin factor. Although the matches of analytical temerature solutions for the corresonding numerical temerature solutions seem to be not good for both cases of the skin factor, hoever, the differences beteen the solutions are quite small and less than 0.01%. Note also that the late-time behaviors of both the numerical and analytical solutions are quite similar and exhibit the same linearly increasing trends ith the logarithm of time as redicted by the analytical solution of Eq. 1. Hence, e can state that the agreement of the analytical ressure and temerature solutions ith those of the numerical simulator are accetable considering the fact that the numerical simulator rigorously solve the mass and thermal balance equations by accounting for the variations of the coefficients of the balance equations (e.g., density, secific heat of ater, isobaric thermal exansion of the ater ith ressure and temerature, hereas the analytical solution treats those coefficients as constant. Table 1: Inut arameter used for comarison of the Chekalyuk (1965) solution. Parameter Value q, m 3 /s 0, kpa T 0, K ( 0 C) (140) r, m 0.1 r e, m h, m k, m (md) (100) c r, 1/kPa c, 1/kPa s, 1/K 0.0 C,s, J/kg-K 1000 C,, J/kg-K 455 ρ s, kg/m 3 650, kg/m , Pa.s t, J/m-s-K 0.0 JT,, K/Pa Finally, it is orth noting that, as can be seen from Fig. 1, a semi-log lot of floing bottom-hole ressure versus time for both cases of skin factor yields a ell-defined straight line, as in the case of isothermal flo (Earlougher 1977). This indicates that one 3

4 Pressure (kpa) Temerature (K) Onur and Palabiyik can estimate ermeability and skin accurately from semi-log straight line analysis based on the assumtion that the fluid roerties such as viscosity, isothermal comressibility of ater, as constant ith temerature. In another ords, this indicates that nonisothermal effects at a constant-rate roduction do not cause significant changes in floing bottom-hole temerature/ressure data. As can be clearly seen from the temerature data shon in Fig., temerature changes are quite small for this case. Interestingly, a semi-log lot of floing bottom-hole temerature versus time also yields a straight-line roortional to Joule-Thomson coefficient. The increase in temerature though small for the synthetic roblem considered here is due to Joule-Thomson heating of ater. The magnitude of the increase in the floing bottom-hole (or sandface) temerature change deends on significantly on flo rate, ermeability and orosity. For instance, decreasing ermeability increases the magnitude of the temerature change over a time. Details can be found in a study conducted by Palabiyik et al. (015). Although not resented here, the aroximate equations for the floing bottom-hole temerature and ressure can be derived from Eqs. 1 and 3 by relacing the exonential integral functions in Eqs. 1 and 3 by their logarithmic aroximations; i.e., -Ei(-x) = -ln(xe ), here = and reresents the Euler constant (Abramoitz and Stegun 197). These equations indicate that a semi-log lot of sandface temerature data versus time should exhibit to semi-log straight lines; one at early times and the other at late times; as can be noticed on the temerature data comuted from the analytical solution of Eq. 1 in Fig.. The sloe of the late-time semi-log straight line may be used to comute the Joule-Thomson coefficient µ JT, if the ermeability of the system is knon or comuted from semi-log analysis of ressure versus time data. The sloe of the early-time semi-log straight line may be used to comute the constant given by Eq. 5 rovided the Joule-Thomson coefficient and ermeability could be estimated from late-time semi-log analyses of ressure and temerature data. Of course, if the rock and fluid secific heat caacities and densities are knon a riori, one could estimate orosity from the value of estimated from the sloe of the early-time semi-log line of the temerature data. If the rock and fluid secific heat caacities and densities are not knon, it is aarent from Eqs. 1 and 3 that it is very difficult if not imossible to uniquely estimate these arameters from temerature transient data, as our results to be given in the section entitled Synthetic Alications also confirm this observation Analytical Solution Numerical Simulator Analytical Solution Numerical Simulator Analytical Solution Numerical Simulator Analytical Solution Numerical Simulator S = S = S = S = Time (days) Time (days) Figure 1: Comarison of modified Chekalyuk Figure : Comarison of modified Chekalyuk ressure solutions ith those of a numerical temerature solutions ith those of a numerical simulator (Palabiyik et al. 013). simulator (Palabiyik et al. 013). 3. HISTORY MATCHING METHODS Here, a brief descrition of the history-matching methods considered in this study is given. As stated before, e consider both gradient and non-gradient methods for history matching temerature and/or ressure data. The gradient method used in this study is the Levenberg-Marquardt (L-M) method based on a restricted rocedure as described by Fletcher (1987). It is a quite efficient method for over-determined roblems (the number of model arameters to be estimated is far less than the number of observed/measured data to be history matched) as the one considered here. The other method is a non-gradient Ensemble-Kalman filter (EnKF). Although the use of EnKF is advantageous for underdetermined roblems of arameter estimation; i.e., the number of model arameters are far more than the number of observed data and EnKF rovides rigorous samling of osterior arameter distribution for linear roblems, here e investigate ho it orks for an over-determined nonlinear roblem based on the analytical temerature solution considered here. In the folloing to subsections, some details of each arameter estimation method are given. 3.1 Gradient Based L-M Method The gradient based L-M method is used to minimize a eighted least-squares objective function hich is designed in a general ay as to be able to match ressure or temerature or both data sets simultaneously: T N 1 ( m, t ) N T T T t obs, i cal i 1 ( m, ) obs, j cal j O ( m ) I I T i T j (7),, i 1 j 1 T 4

5 The model arameter vector, m, in Eq. 7 can, in general, contain 13 model arameters as unknon as given by k h S c C C T T 0 0 m,,,,,,,,,,, (8) t,, s, s JT, Here, T is the transose of the vector. The I and I T terms in Eq. 7 are indicators hich can only be either a 1 or a 0. These indicators are used for matching either ressure, temerature or both. T obs and obs reresent the observed (measured) temerature and ressure times t T and t, resectively. T cal and cal are the corresonding comuted temeratures and ressures from the Chekalyuk ressure and temerature solutions given by Eqs. 1 and 3. N and N T are the total number of data to match for ressure and temerature, resectively. and T reresent the standard deviations of the errors associated ith the ressure data and the temerature data, resectively.,i and T,j are the eights assigned to the ressure and temerature data, resectively. These eights are assigned by the user as 0 or a ositive number. The gradient based Levenberg-Marquardt method based on a restricted rocedure described by Fletcher (1987) is quite efficient to minimize Eq. 7. We have also constrained the arameters based on the rocedure of Carvalho et al. (199). It should be noted that e have included almost all of the individual arameters aearing in Eqs Hoever, e realize that it ill be very difficult if not imossible to estimate uniquely each of these individual arameters. Instead, one may ork ith the folloing 10 groued arameters to make the arameter estimation roblem better ell-osed, although e have not ursued this model arameter set in this study: C, C 0 0 m k /, S,, c,,, h, T, (9) t JT,, r Here, C r is the ratio of thermal caacity of fluid to thermal caacity of the total system (note that the roduct of density and secific heat is referred to as the thermal caacity) defined by: C r C, C, 1 C, s s. (10) 3. Non-Gradient Based EnKF Method EnKF is a Monte Carlo method here uncertainty is reresented by an ensemble of realizations. The rediction of the uncertainty is erformed by ensemble integration using the forard model (Evensen 003). An overvie of the alications of EnKF in the etroleum industry is given in Aanonsen et al. (009). Our imlementation of the EnKF in this ork follos those of Zafari and Reynolds (007) and Li et al. (010). It should be noted that the EnKF is based on assumtions of the linear model, Gaussian distribution of the model arameters, and Gaussian errors in observed data. Although our model considered here is nonlinear, e ould like to see the alicability of the EnKF for the nonlinear model considered here. Previously, Tureyen et al. (007) alied the EnKF method for history-matching ressure data by the use of isothermal lumed-arameter models for single-hase liquidater geothermal reservoirs. They shoed that the EnKF can ork for such non-linear models. In the EnKF method, e define a state vector. Then, the method roceeds ith a forecast ste (steing forard in time) and an assimilation ste in hich the state vector including the model arameters of the system are udated or corrected to honor the measured data. In our alication, the initial ensemble is an ensemble of realizations of the rior robability distribution function of a articular uncertain arameter (e.g. ermeability, orosity, etc.) generated by random fields. For the roblem considered here, e use the folloing EnKF udate equation for the jth ensemble member at time t r, here e have observed data (Li et al. 010 and Morton et al. 011): ( ) ( ) (11) for j = 1,,,N e. Here, N e is the number of ensemble members. In Eq. 11, the suerscrits u and refer to udated and redicted quantities, resectively. y is the N y -dimensional state vector at time t r defined by [ ] (1) here m reresents the N m -dimensional vector of model arameters (see Eq. 8); here N m reresents the total number of unknon model arameters, and T reresent N T -dimensional vector of ellbore ressures and/or temeratures comuted from the analytical ressure and temerature solutions of Chekalyuk (see Eqs. 1-6); here N T is the total number of comuted ressures and/or temeratures. Note that N y = N T + N m. is the N r -dimension vector of erturbed observed data, generated by adding Gaussian noise of mean zero and covariance to the observed data at time t r, and is the N r -dimensional vector of observed data at the data assimilation time t r. N r is the number of observed data to be used in assimilation time t r. In our formulation, e exressed the observed data to be (t r ) and/or T (t r ) measured at time t r ith a knon, uncorrelated Gaussian, error variance of σ m(t r ). So, the matrix is a diagonal matrix. In Eq. 11, the covariance matrices and are comuted from the ensemble redictions by: ( ) ( ) ( ) (13) and 5

6 F lo in g B o tto m h o le P re s s u re (k P a ) F lo in g B o tto m h o le T e m e ra tu re (K ) Onur and Palabiyik ( ) ( ) ( ) (14) here and reresent the ensemble averages of the redicted data and state vectors, comuted from: (15) and (16) Note that is an N y N r symmetric matrix, and is an N y N r matrix. The nice feature of this EnKF imlementation is that these covariance matrices do not need to exlicitly comuted or stored. Further details of the EnKF method can be found in Evensen (003), Aanonsen et al. (009), and Li et al. (010). 4. SYNTHETIC APPLICATIONS Here, the alications for arameter estimation ith the synthetic transient ressure and temerature data in the resence of noise are demonstrated. First, e simulate ressure and temerature data from Eqs. 1-6 for a constant-rate dradon test of duration 10 days. The mass flo rate is 30 kg/s hich corresonds to aroximately 0.03 m 3 /s at the conditions of initial ressure and temerature. All other inut data used to simulate this dradon test data are as given in Table 1. We corruted each simulated ressure and temerature data set by using normal random errors ith different standard deviations [ressure and temerature containing errors of standard deviations of = 5000 Pa (= 0.05 bar = 0.75 si) and T = 0.00 K (= 0.00 o C), resectively] and based the arameter estimation (i.e., nonlinear regression analysis) on these noisy ressure and temerature data sets. Shon in Figs. 3 and 4 are the semi-log lots of noisy ressure and temerature data used for nonlinear arameter estimation. In alications, e assume that e kno or could estimate the variances (or std. deviation) of errors in ressure and temerature data to be used in the L-M and EnKF methods. As lane radial flo (or infinite-acting radial fo) should aly for this synthetic data set, a semi-log lot of ressure vs. time data as shon in Fig. 3 yields a straight line ith the sloe from hich the ermeability value can be comuted rovided that the fluid viscosity and the thickness are knon and ith the intercet from hich skin factor can be comuted rovided that orosity and the total isothermal comressibility is knon (Earlougher 1977). Semi-log analysis of the data yielded k = m and S = 0.03, assuming that the orosity and the total system isothermal comressibility (c t ) are knon a riori. The values of k and S estimated from the semi-log analysis are in very good agreement ith the true (inut) values (see Table 1). So, for this synthetic test case, bottom-hole ressure data rovide very good estimates of k and S. Our results (not shon here) based on the nonlinear regression analysis of only ressure data by using both the L-M method and EnKF also confirm that history matching floing bottom-hole ressure data yield reliable (confident) estimates of k and S if other inut arameters such as orosity (), viscosity ( ) of the fluid and the total system isothermal comressibility (c t ) are knon a riori. In the alications, e ket skin factor as knon at its inut value of zero, as one can obtain its value from semi-log analysis of ressure data accurately. Next e history matched only the bottom-hole floing temerature by using the gradient based L-M method to estimate five arameters; namely,, k, C,s, C, and µ JT,. The values of the estimated arameters from this alication along ith 95% confidence intervals (hich are given in ±) of the estimated arameters are given in Table. Although the temerature match obtained ith the model temerature is not shon here, the match as quite accetable ith a root-mean-square (RMS) value equal to K, hich is very close to the inut value of standard deviation of errors used to corrut the temerature data E E Figure 3: Noisy bottom-hole ressure data for synthetic dradon test. Figure 4: Noisy bottom-hole temerature data for synthetic dradon test. It is clear that only the value of ermeability is confidently estimated. Although the estimated values of the other four arameters (, C,s, C, and µ JT, ) are close to the true, unknon inut values, their confidence interval are quite high, hich indicates that the 6

7 uncertainty in these estimated arameters are large. This may be due to small sensitivity of the temerature to these arameters and/or due to correlation among these arameters. Next, e history matched both ressure and temerature data simultaneously to estimate the same five arameters,, k, C,s, C, and µ JT, by using the L-M method. The values of the estimated arameters from this alication along ith 95% confidence intervals (hich are given in ±) of the estimated arameters are given in Table 3. Although not shon here both temerature and ressure matches obtained ith the corresonding model data are quite accetable and the values of RMS for ressure and temerature are RMS= K (for temerature) and RMS= Pa (for ressure). Again these RMS values are very close to the inut standard deviations of errors used to corrut ressure and temerature data. Comaring the results given in Table and 3, e can state that using both ressure and temerature data in history matching imroves estimation and confidence esecially for ermeability and orosity. Hoever, the confidence intervals are still quite large for the three thermal arameters; namely, C,s, C, and µ JT,,. This is most likely due to high correlation among these arameters (see Eqs. 5 and 6). Table : Estimated model arameters by history matching only temerature data; the L-M method. Model Parameter (Unknon) True Value Initial Guesses Estimated by history matching, fraction k, m C,s, J/kg-K x10 8 C,, J/kg-K x10 8 JT,, K/Pa x10 8 RMS = K Table 3: Estimated model arameters by history matching both ressure and temerature data; the L-M method. Model Parameter (Unknon) True Value Initial Guesses Estimated by history matching, fraction k, m C,s, J/kg-K x10 14 C,, J/kg-K x10 14 JT,, K/Pa x10 8 RMS = K (for temerature) and RMS= Pa (for ressure) Next, e history matched both ressure and temerature data by using EnKF method. As mentioned before, in EnKF method, e have to start ith a number of model arameter realizations and a same number of unconditional realizations of the observed data. Here, e resent our results only ertinent to a case here the number of realizations used is 100. The tyes of distributions and their corresonding arameters (mean and std. dev.) used for generating the unconditional realizations of the model arameters are given in Table 4. Table 4: Assumed distributions of the model arameters for the EnKF method. Model Parameter Mean Standard Deviation Distribution Tye, fraction log-normal k, m log-normal C,s, J/kg-K log-normal C,, J/kg-K log-normal JT,, K/Pa normal Figures 5 and 6 sho the comuted model resonses from initial ensembles for ressure and temerature data, resectively. As can be seen from Figures 5 and 6, bottom-hole ressure and temerature data redicted by using 100 initial ensembles of the model arameters are quite different from the observed data (shon as data oints). Figures 7 and 8 sho the history matched ressure and temerature resonses, resectively, after udating each initial ensemble of the model arameters by the EnKF rocedure. The results are accetable because the ensemble means of 100 realizations of model arameters conditioned (or otimized) by history matching to observed data are close to the true unknon values (see Table 5). Furthermore, the comuted ensemble means of 7

8 F lo in g B o tto m h o le P re s s u re (M P a ) F lo in g B o tto m h o le T e m e ra tu re (K ) F lo in g B o tto m h o le P re s s u re (M P a ) F lo in g B o tto m h o le T e m e ra tu re (K ) Onur and Palabiyik bottom-hole ressure and temerature data ith each of 100 conditioned realizations of model arameters from t = 0 to the last time ste of the observed data exhibit an excellent agreement ith bottom-hole ressure and temerature curves hich do not contain error. Moreover, from t = 0 to the last time ste of the observed data, standard deviations beteen bottom-hole ressure and temerature data containing error and the ensemble means of bottom-hole ressure and temerature data comuted is Pa and K for ressure and temerature, resectively. These values are in an excellent agreement ith the standard deviations (5000 Pa for ressure and 0.00 K for temerature) of random errors used to corrut the simulated bottom-hole ressure and temerature data ithout error (or noise) E n s e m b le s (c o m u te d ith u n c o n d itio n e d re a liz a tio n s o f m o d e l a ra m e te rs ) "O b s e rv e d " E n s e m b le s (c o m u te d fro m u n c o n d itio n a l re a liz a tio n s o f m o d e l a ra m e te rs ) "O b s e rv e d " E Figure 5: Bottom-hole ressure data comuted from 100 unconditioned realizations of model arameters. Figure 6: Bottom-hole temerature data comuted from 100 unconditioned realizations of model arameters re ru n s ith c o n d itio n e d re a liz a tio n s o f m o d e l a ra m e te rs "o b s e rv e d " e n s e m b le m e a n o f re ru n s 9.6 Exact data (no noise ) re ru n s ith c o n d itio n e d re a liz a tio n s o f m o d e l a ra m e te rs "O b s e rv e d " 9.0 e n s e m b le m e a n o f re ru n s exact data (no noise) E Figure 7: Bottom-hole ressure curves comuted Figure 8: Bottom-hole temerature curves by the model starting from t = 0 for comuted by the model starting from t = 0 for 100 conditioned realizations of model arameters. 100 conditioned realizations of model arameters The values of the estimated model arameters ith their estimated mean and std. dev. before and after the EnKF rocedure method in comarison ith the true (unknon) values of the model arameters are given in Table 5. The values given in the nd and 3 rd columns in Table 5 reresent the standard deviations (uncertainties) of the ensemble of each model arameter before and after history matching by EnKF. Before history-matching to measured data, as exected, the standard deviations of model arameters are high (see the nd column in Table 5). Uncertainties of model arameters are exected to decrease by history matching to measured ressure and temerature data. If values given for model arameters in the 3 rd column in Table 5 are insected, e see that the standard deviations of ensembles of model arameters conditioned by history matching to ressure and temerature data decreases in comarison ith the standard deviations of the unconditioned model arameter ensembles. This is an exected and desired situation. Hoever, reduction in the standard deviation of errors is arameter deendent; in other ords, reduction may be significant for some arameters, hile reduction may not be significant for other arameters. It is clear from Table 5 that the largest reduction in standard deviation is observed for ermeability (in comarison ith the standard deviation of the ensemble unconditioned). As can be seen from Table 5, history matching bottom-hole ressure and temerature data by the EnKF method reduces the standard deviation of ensemble of ermeability conditioned by factor of 80. Indeed, this result is in agreement ith the results obtained from the L-M method. Moreover, ermeability is the arameter that has the smallest 95% confidence interval among the model arameters otimized by history matching ith the L-M method (see Table 3). 8

9 J o u le -T h o m s o n C o e ffic ie n t (K /P a ) P e rm e a b ility, k (m ) P o ro s ity (fra c tio n ) Onur and Palabiyik Figures 9-11 sho the comarisons of changes of osterior ensembles (100 realizations) ertinent to ermeability, orosity and Joule-Thomson coefficient ith their true values before and after history matching (conditioning), resectively. For instance, as can be seen from Fig. 9, each of 100 realizations is very close to true value of ermeability after history matching. This is because bottom-hole ressure and temerature data sho large sensitivity to ermeability in comarison ith the other model arameters. As to the other arameters, it is observed that the standard deviations of the arameters (, C,s, C,, and JT,) by the EnKF rocedure decrease in comarison ith those of their rior standard deviations. Hoever, this decrease is not as much as in the standard deviation of the ermeability (see Figures 9-11 and Table 5). Table 5: Values and statistics of the estimated arameters from the EnKF method. Model Parameter (Unknon) True Value Mean (and Std. Deviation) of Initial Ensembles Mean (and Std. Deviation) of Ensembles Assimilated by EnKF, fraction (= 0.05) k, m (= ) C,s, J/kg-K (= 455.7) C,, J/kg-K (=1968.6) JT,, K/Pa (= ) 0.1 (= 0.04) (= ) (= 319.) (=1409.) (= ) 6 E E Prior (unconditioned) realization C o n d itio n e d re a liz a tio n b y E n K F T ru e u n k n o n Prior (unconditioned) realization C o n d itio n e d re a liz a tio n b y E n K F T ru e u n k n o n 4 E E E E E E R e a liz a tio n n o R e a liz a tio n n o. Figure 9: Comarison of unconditioned (rior) and conditioned ensemble realizations of ermeability by EnkF method. Figure 10: Comarison of unconditioned (rior) and conditioned ensemble realizations of orosity by EnKF method. -1. E Prior (unconditioned) realization C o n d ito n e d re a liz a tio n b y E n K F T ru e u n k n o n -1.4 E E E E R e a liz a tio n n o. Figure 11: Comarisons of unconditioned (rior) and conditioned ensemble realizations of Joule-Thomson coefficient by EnKF method. 9

10 As to the comutational efficiency of the gradient based L-M method and the non-gradient EnKF method, our results sho that both methods are quite fast for this simle over-determined (model ith five unknon model arameters) case studied here, though the EnKF requires a slightly more comutation time. For the set of 100 realizations of the five unknon model arameters, the EnKF method only requires a fe seconds for comletion. Although e have not yet ursued, e think that the real advantage of the EnKF over the gradient based methods ill be for the underdetermined arameter estimation roblems based on the numerical models here each grid-block rock/fluid roerty is treated as unknon as ursued by Duru and Horne (010b). 5. CONCLUSIONS The main conclusions of this ork can be stated as follos: i. Based on the modified analytical solution of the Chekalyuk (1965), it is shon that the sandface temerature data during roduction from a vertical ell at a constant-rate from an infinite acting reservoir exhibit a semi-log straight line at late times. This sloe is roortional to the roduct of the ermeability and the Joule-Thomson coefficient. The intercet of this straight line contain information about the skin factor and the secific heat caacities of the rock and fluid. Hoever, a unique estimation of the individual values of ermeability, orosity, skin, Joule-Thomson coefficient, secific heat caacities from temerature data alone in the resence of noise seems to be very difficult if not imossible due to strong correlation among the model arameters. ii. As the ressure data for the ell/reservoir model considered in this study should exhibit a semi-log straight line from hich the ermeability and skin factor can be reliably determined, hence, using ressure data ith the temerature data in history matching imroves the arameter estimation roblem and allos one to estimate orosity and the Joule-Thomson coefficient confidently rovided that the secific heat caacities and densities are assumed to be knon a riori. iii. It is observed that for the simle over-determined case studied here, both gradient-based L-M and EnKF methods are comutationally efficient to use to history match ressure and temerature data and both methods give very similar results although the L-M method yields closer estimations to the true values of the model arameters. iv. For the synthetic case study considered here, both otimization methods sho that ermeability, orosity, and the Joule- Thomson coefficient can be determined more reliably from ressure and temerature data in contrast to the other model arameters such as the secific heat caacities of rock and fluid roerties if observed data contain error. ACKNOWLEDGEMENTS We ould like to thank the Scientific and Technological Research Council of Turkey (TUBITAK, Project No: 110M48) and the Unit of ITU (Istanbul Technical University) Scientific Research Projects for the suorts that they suly to finance this ork. REFERENCES A, J.F.: Nonisothermal and Productivity Behavior of High Pressure Reservoirs, SPE Journal, 15, (010), Aanonsen, S. I., Noevdal, G., Oliver, D. S., Reynolds, A. C., and Valles, B.: The Ensemble Kalman filter in reservoir engineering - a revie. 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