SPE-121393-MS-P-IPR - SPE 121393 A New Method for Continual...

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Unformatted text preview: SPE 121393 A New Method for Continual Forecasting of Interwell Connectivity in Waterfloods Using an Extended Kalman Filter Daoyuan Zhai, Jerry M. Mendel, Feilong Liu, University of Southern California Copyright 2009, Society of Petroleum Engineers This paper was prepared for presentation at the 2009 SPE Western Regional Meeting held in San Jose, California, USA, 24–26 March 2009. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright. Abstract This paper is based on a relatively simple parametric model that characterizes the system function between a specific producer and each of its contributing injectors. The model has only two parameters for each producerinjector pair; so, if N injectors are assumed to contribute to a producer, there will be 2N unknown parameters. An adaptive strategy, using an Extended Kalman Filter (EKF), is used to estimate the 2N parameters, which are then used to generate N numeric Injector-ProducerRelationship (IPR) values for the N producer-injector pairs. The IPR values allow one to assess how well an injector influences the producer. This same model and an EKF were first used in Liu, et al [5]. The modified EKF used in this paper avoids problems that can arise when processing real data and provides additional information that is useful for future research. Our modified EKF is applied to real data from a section of an oil field. A validation strategy for the estimated IPR values is developed in terms of “prediction errors.” A strategy is also presented for choosing an optimal set of injectors that affect a producer. Finally, a simple method is presented for converting producercentric IPR values to injector-centric IPR values so that reservoir engineers can easily see which producers are being affected by a specific injector. I. Introduction One of the most important issues in water-flood management is the ability to detect preferential subsurface flow trends, i.e., where the water flows. Recently, one approach for doing this was to estimate the inter-well relationship of each injector and its contributing producers using only the production and injection rates. More specifically, the inter-well relationship, referred here as the “Injector-Producer-Relationship (IPR)”, was evaluated by estimating parameters of a model that characterizes the reservoir. After obtaining the IPR values, one can ascertain whether an injector is contributing to a producer and, if so, by how much. In addition, one can often infer other features about the reservoir from the IPR values, e.g., the directional sweep efficiency of a given pattern and presence of directional fractures. [1], [2], [4], and [7]-[11] introduce many different methods, yet all aim to achieve the above goal. The most recent works are Albertoni and Lake [1] and Yousef, et al. [11], both of which model the reservoir as a continuous impulse response that converts input signals (injection rates) to output signals (production rates). Liu, et al [5] have pointed out the following difficulties that [1] and [11] have encountered: (a) the parameters in the model and the IPRs are assumed stationary over the window of measurements for which processing occurs, and so when the IPRs change the processing needs to be redone for the new situation, and this may not be practical because the reservoir is dynamic and it may be very difficult to recognize when a change has occurred; and, (b) the CM model is somewhat complex, and although it is characterized by two parameters, to use it, the primary bottom hole pressure impact also needs to be determined. Additionally, Albertoni and Lake [1] model the production rate as a weighted sum of delayed injection rates from the injectors assumed to contribute to a producer; but, in general, how many delayed injection rates to include in the model is unknown. To overcome the above difficulties, Liu, et al [5] proposed to use a relatively simple two-parameter autoregressive model to characterize the impulse response between a single injector and a single producer. The area under the impulse response is used as a measure of the IPR. If N injectors are assumed to influence a producer, this producer-centric model contains exactly 2N parameters. This impulse response model is then used to establish an equivalent State Variable Model (SVM), which is then used by an Extended Kalman Filter (EKF) to forecast and track the IPR values. This paper continues to use the same impulse response model as in [5]; however, we modify the SVM in some important ways so that the (modified) EKF can provide some new 2 SPE 121393 information, and so that some practical difficulties can be overcome when the modified EKF is used to process real data. The rest of this paper is organized as follows: Section II provides a brief recapitulation of the SVM that is in [5]; Section III describes our modified SVM; Section IV describes the modified EKF that is associated with the modified SVM; Section V provides a strategy for using our EKF to process data for an entire section; Section VI sets forth a validation strategy for our modified EKF; Section VII describes a procedure for choosing the initial set of candidate injectors that may (or may not) influence a specific injector; Section VIII explains how to convert producer-centric IPR values into injector-centric IPR values; and, Section IX includes our conclusions as well as some suggested topics for further research. A. Reservoir Model Our reservoir is modeled as a collection of continuoustime impulse responses that convert injection rates into a production rate. A producer-centric reservoir model with N independent injectors and one producer is depicted in Fig.1, where i1 (t ), i2 (t ),..., iN (t ) , n1 (t ), n2 (t ),..., nN (t ) and im,1 (t ), im,2 (t ), ..., im, N (t ) are the actual injection rates that flow into the reservoir, the corresponding injection rate measurement noise, and the measured injection rates, respectively; p(t ) , n p (t ) and pm (t ) are the actual production rate, the corresponding production rate measurement noise, and the measured production rate, respectively; p cj (t ) represents the amount of production p(t ) (2) where α = e − aT and γ = bα T are two parameters that determine the model and T is the sampling period. The total production rate at a specific producer is the sum of the individual production rates contributed by injectors that influence this producer. Assume N injectors influence a producer; then, for this producer: N N j =1 j =1 P ( z ) = ∑ Pjc ( z ) = ∑ H j ( z ) f (rj , k j ) I j ( z ) (3) γ j f (rj , k j ) z −1 =∑ I j ( z) −1 2 j =1 (1 − α j z ) N where P ( z ) , Pjc ( z ) and I j ( z ) are the Z transforms of the II. Liu and Mendel’s Original SVM rate in γ z −1 (1 − α z −1 )2 H ( z) = caused by the jth injector ; and, f (rj , k j ) ( j = 1,..., N ) are scale functions which show how much of each injection rate flows in the direction toward the producer, where f (rj , k j ) is viewed as a linear or non-linear scalar function of the distance, rj , and the permeability, k j , between the jth injector and the producer. Because noise-free data, i1 (t ), i2 (t ),..., iN (t ) and p(t ) , are not directly available, we use their measured total production rate, the production rate contributed by the jth injector, and the injection rate of the jth injector, respectively. From (2), the model of the subsystem between the producer and the jth injector can be expressed as: (1 − α j z −1 ) 2 Pjc ( z ) = γ j f (rj , k j ) z −1 I j ( z ) (4) which, when transformed back into time domain, is: p c (k + 1) − 2α j p c (k ) + α j 2 p c (k − 1) = γ j f (rj , k j )i j (k ) (5) j j j Liu, et al. [5] defined the IPR as the area under the sampled impulse response for each injector, and showed that it can be calculated as: IPR j = γ j f ( rj , k j ) (1 − α j ) 2 (6) Observe that three parameters— α j , γ j and f (rj , k j ) — need to be estimated to compute this IPR j ; however, because γ j and f (rj , k j ) are multiplicative in (5) and (6), the number of parameters can be reduced from three to two by letting γ ′ ≡ γ j f (rj , k j ) . Note, also, that only j values, im,1 (t ), im,2 (t ),..., im, N (t ) and pm (t ) , for our data the measured injection rates, im, j (k ) , are available for processing. B. Injector-Producer Model for Each Subsystem processing; hence, (5) and (6) can be re-expressed, as: pc (k + 1) − 2α j pc (k ) + α j 2 pc (k − 1) j j j After comparing different models and getting advice from petroleum engineers, the following two parameter auto-regressive (AR) model was chosen in [5] to represent the impulse response between a single producer and a single injector: h(t ) = bte − at = γ ′ (im, j (k ) − n j (k )) = γ ′im, j (k ) + n pc (k ) j j IPR j = (1) Because only sampled injection and production rates are available for processing, a discrete-time version of (1) is used; its Z-transform is: (7) j γ ′j (1 − α j ) 2 (8) Note that in (7), we set n pc (k ) = γ ′j n j (k ) so that we will j be able to use im, j (k ) instead of i j (k ) as our input and n pc ( k ) is the standard additive state equation noise. j SPE 121393 3 ⎧ x11 (k ) ⎤ ⎡ ⎪ ⎥ ⎢ x12 (k ) ⎪ ⎥ ⎢ ⎪ ⎥ ⎢ x14 (k ) ⎪ ⎥ ⎢ 2 2 x11 (k ) x14 (k ) − x11 (k ) x13 (k ) + x12 (k )im ,1 (k ) ⎥ ⎪ ⎢ ⎪ ⎥ ⎢ x21 (k ) ⎪ ⎥ ⎢ x22 (k ) ⎪ ⎥ ⎢ ⎪ x(k + 1) = ⎢ ⎥ + n (k ) x24 (k ) ⎪ ⎥ x ⎢ 2 ⎪ ⎢ 2 x21 (k ) x24 (k ) − x21 (k ) x23 (k ) + x22 (k )im ,2 ( k ) ⎥ ⎪ ⎥ ⎢ ⎨ M ⎥ ⎢ ⎪ ⎢ ⎥ xN 1 ( k ) ⎪ ⎢ ⎥ ⎪ xN 2 ( k ) ⎢ ⎥ ⎪ ⎢ ⎥ ⎪ xN 4 ( k ) ⎢ ⎥ ⎪ 2 ⎢ 2 xN 1 (k ) xN 4 (k ) − xN 1 (k ) xN 3 (k ) + xN 2 ( k )im , N ( k ) ⎦ ⎥ ⎣ ⎪ ⎪ ⎪ c c ⎪ p(k + 1) = p1c (k + 1) + p2 (k + 1) + L + pN (k + 1) + n p (k + 1) ⎪ = [ 0 0 0 1 0 0 0 1 L 0 0 0 1] x(k + 1) + n p (k + 1) ⎪ ⎩ C. State-Variable Model To use an EKF, one must first construct an SVM [6]. To establish the SVM, one starts with the second-order finite-difference equation (7) that is described by two state variables p c (k − 1) and p cj ( k ) . In addition, the j unknown parameters α j and γ ′ are also treated as state j variables so that they can be estimated by an EKF; hence, (7) is actually described by the following 4 × 1 state vector: x j (k ) = ⎡ x j1 (k ), x j 2 ( k ), x j 3 ( k ), x j 4 (k ) ⎤ ' ⎣ ⎦ = ⎡α j ( k ), γ ′j (k ), p c (k − 1), p c (k ) ⎤ ' j j ⎣ ⎦ (9) Aggregating the state vectors of all N injectors, one obtains a complete 4N × 1 state vector, as: x(k ) = [ x1 (k ) ', x2 (k ) ',L , xN (k ) '] ' III. New Modified SVM The SVM in (11) has been modified by us in two important ways: (1) IPR is treated as a state variable, and is estimated directly. Its error (pseudo-) variance can be used to provide upper and lower bounds for the IPR estimates. (2) Because α and IPR must be positive numbers, and sometimes their estimates become negative due to strong noises and other uncertainty factors, α and IPR are used as state variables instead of α and γ ′ . Regardless of the sign of the estimated values of α and IPR , their squared values always give positive estimates for α and IPR. Consequently, in this paper c c ⎡ IPR j (k ) α j ( k ) p j (k − 1) p j (k ) ⎤ ' is the state ⎣ ⎦ vector, (10) Using standard techniques [6], the SVM for the entire producer-centric system is given in (11) where nx (k ) = ⎡ nx1 (k ) ' nx2 (k ) ' K nxN (k ) '⎤ ' and n p (k + 1) ⎣ ⎦ are additive zero-mean white noises with covariance matrix Qx and variance rk +1 , respectively. Each component nx j (k ) is ⎡ nα j (k ) nγ ′j (k ) 0 n pc (k + 1) ⎤ ' . ⎢ ⎥ j ⎣ ⎦ The explicit form of Qx is: x j (k ) , for the jth injector. As in the previous section, the complete state x(k ) = [ x1 (k ) ', x2 (k ) ',L , xN ( k ) '] ' . L , rnα N , rnγ ′ , 0, rnpc ⎤ N N ⎦ vector is Using (8), γ ′ (k ) can be expressed in terms of IPR j (k ) j and α j (k ) as: γ ′j (k ) = IPR j (k )(1 − α j (k )) 2 (13) which allows us to re-express (7) in terms of the new state variables as: 2 Qx = diag ⎡ rnα 1 , rnγ ′ , 0, rnpc , rnα 2 , rnγ ′ , 0, rnpc ,L 1 2 1 2 ⎣ (11) 4 p c (k + 1) − 2 α j p c (k ) + α j p c (k − 1) j j j 2 (12) 2 = IPR j (1 − α j ) 2 im, j (k ) + n pc (k ) j (14) 4 SPE 121393 x11 (k ) ⎧ ⎡ ⎤ ⎪ ⎢ ⎥ x12 (k ) ⎪ ⎢ ⎥ ⎪ ⎢ ⎥ x14 (k ) ⎪ ⎢ ⎥ 2 4 2 2 2 x12 (k ) x14 (k ) − x12 (k ) x13 (k ) + x11 ( k )(1 + x12 ( k )) 2 im ,1 ( k ) ⎥ ⎪ ⎢ ⎪ ⎢ ⎥ x21 (k ) ⎪ ⎢ ⎥ x22 (k ) ⎪ ⎢ ⎥ ⎪ ⎢ ⎥ + n (k ) x24 (k ) ⎪ x(k + 1) = ⎢ ⎥ x 2 4 2 2 2 ⎪ ⎢ 2 x22 (k ) x24 (k ) − x22 (k ) x23 (k ) + x21 (k )(1 + x22 (k )) im ,2 (k ) ⎥ ⎪ ⎢ ⎥ ⎨ M ⎢ ⎥ ⎪ ⎢ ⎥ xN 1 ( k ) ⎪ ⎢ ⎥ ⎪ xN 2 ( k ) ⎢ ⎥ ⎪ ⎢ ⎥ ⎪ xN 4 ( k ) ⎢ 2 ⎥ ⎪ 2 2 2 4 ⎢ 2 xN 2 (k ) xN 4 (k ) − xN 2 (k ) xN 3 (k ) + xN 1 (k )(1 + xN 2 (k )) im , N (k ) ⎥ ⎣ ⎦ ⎪ ⎪ ⎪ c c ⎪ p(k + 1) = p1c (k + 1) + p2 (k + 1) + L + pN (k + 1) + n p (k + 1) ⎪ = [ 0 0 0 1 0 0 0 1 L 0 0 0 1] x (k + 1) + n p (k + 1) ⎪ ⎩ 1 0 0 0 ⎤ ⎡ ⎢ 0 1 0 0 ⎥ ⎢ ⎥ 0 0 0 1 ⎥ Aj = ⎢ ⎢ ⎥ 4 x j 2 ( k ) x j 4 ( k ) − 4 x 32 ( k ) x j 3 ( k ) + ⎢ ⎥ j 2 2 4 2 − x j 2 (k ) 2 x j 2 (k ) ⎥ ⎢ 2 x j1 (k )(1 + x j 2 (k )) im , j (k ) 2 2 4 x j1 (k )(1 + x j 2 (k )) x j 2 (k )im , j (k ) ⎢ ⎥ ⎣ ⎦ Consequently, our modified SVM is given (15) where, nx ( k ) and n p (k + 1) are as described in Section II.C. matrix, the explicit forms of which are given below for our NL state equation in (15): ⎛ A1 ⎜ 0 Fx = ⎜ ⎜ M ⎜ ⎝0 (16) (18) What distinguishes our SVM in (15) from the more general SVM in (16) is that in (15) h [•] is linear, and can be expressed as: y(k + 1) = Hx(k + 1) + ny (k + 1) of the state vector, and nx ( k ) and ny (k ) correspond to the additive zero-mean white noises for the state and measurement equations, respectively. In the EKF, f [•] (20) where H = [ 0 0 0 1 0 0 0 1 L 0 0 0 1] (21) ˆ( is linearized about x k | k ) , i.e., ˆ [ x ( k ) − x ( k | k ) ] + nx ( k ) 0 ⎞ ⎟ M ⎟ O O 0 ⎟ ⎟ L 0 AN ⎠ 0 L A2 O where the explicit form of Aj is given in (19). where, in general, f [•] and h [•] are nonlinear functions ˆ ˆ x(k + 1) ≈ f [ x(k | k ), k ] + Fx [ x(k | k ), k ] × (19) where Fx = ∂f [ x(k ), k ] ∂x(k ) is a 4N × 4N Jacobian IV. EKF Processing Details about the EKF can be found in [3] and [6]. Because the EKF for our modified SVM is quite similar to that of the original SVM (see [5]) we briefly state its structure. The nonlinear (NL) SVM for an EKF has the following general form: ⎧ x(k + 1) = f [ x(k ), k ] + nx (k ) ⎪ ⎨ ⎪ y (k + 1) = h [ x(k + 1), k + 1] + n y (k ) ⎩ (15) (17) is a 1 × 4N vector. The EKF has two stages, Predictor and Corrector, and is summarized as follows: ˆ(0 1. Initialize the EKF with x | 0) , P (0 | 0) , Qk and rk . SPE 121393 5 Predictor ( k = 0,1,L ): 2. ˆ ˆ x (k + 1 | k ) = f [ x ( k | k ), k ] (22) ˆ ˆ P (k + 1| k ) = Fx [ x(k | k ), k ] P (k | k ) Fx′ [ x( k | k ), k ] + Qk (23) Corrector ( k = 0,1,L ): 3. ˆ ˆ x(k + 1| k + 1) = x(k + 1| k ) + K (k + 1) × ˆ { y (k + 1) − Hx(k + 1| k )} K (k + 1) = P(k + 1| k ) H ′ HP(k + 1| k ) H ′ + rk +1 P (k + 1 | k + 1) = [ I − K (k + 1) H ] P( k + 1| k ) (24) (25) (26) V. Real Data Processing The data used in this paper is from a section of a real oil field. Most of the injectors in this section have two completions; and because the injection rates of different completions for the same injector are separate, each completion is treated independently, i.e., each completion is treated as an independent “injector” in our SVM. The total number of producers in this field is denoted by N and the total number of injector completions is denoted by M; and the well names have been removed and re-labeled. The data starts from Jan. 1st, 2005, which is labeled as day 1 in our figures, and ends on Jul. 31st, 2008, which is labeled as day 1308. Both injection and production rates are well-test data 1 . The sampling rate of injection rate is once per day and the sampling rate of production rate is, on average, 15 days. For days when production rate measurements were not available, we used the value from the last available data until the next available data. Our strategy for processing the entire section has been to apply our producer-centric EKF to each producer separately. We choose a producer labeled as P-130 to give an example of this process. A very important first step is to choose an initial set of injectors that are possibly influencing a producer. A relatively simple and commonly used way for doing this is to use expert knowledge provided by petroleum engineers that are familiar with this field. For this section it is known that there are parallel fractures along each well that align 45o to NE. It is believed that injectors located along this fracture alignment are more likely to contribute to a producer. Knowing this, one strategy for choosing the injectors that contribute to a producer is to draw an ellipse centered about the producer whose major axis is along the fracture alignment, and to assume the injectors inside the ellipse are influencing that producer. Additional expert knowledge was provided about the size of the ellipse, i.e., in general the major axis of the ellipse 1 Pump Off Controller (POC) data for production rates are presently not available to us. should be 700 feet long and its minor axis should 500 feet long. Because these lengths are subjective, they may vary from producer to producer; hence, a strategy for choosing an optimal ellipse size is presented in Section VII. We began by using a 700′ × 500′ ellipse for a producer. Completions inside this ellipse are included in our EKF model. To give the reader a clearer idea of this, P-130 and injectors inside the ellipse are depicted in Fig. 2(a) by circle and triangles, respectively. Note that the upper and lower triangles at the same location represent short and long completion of an injector, respectively. And there are 46 completions in this model. After applying our EKF to this single producer-46 completions model, we obtained the IPR curves that are depicted in Fig. 3(a). 46 completions is not a small number, so it is very likely that some of these completions may be irrelevant to P-130. Observe, from Fig. 3(a), that there are many IPR curves that only have very small values. We assumed, therefore, that the completions that have very low IPR values do not influence P-130 and should be eliminated from the model. More specifically, we computed the mean of the IPR values for the most recent month (i.e., days 1278-1308) for each completion and compared it with a chosen threshold. A completion was kept in the model only if its IPR value was greater than or equal to the threshold; otherwise, that completion was eliminated from the model. Our strategy for choosing a good threshold is dependent on the number of completions inside the ellipse, and was inspired by the fact that an existing common, simple and practical way for deciding the impacts of injectors in water-flood management is to assign equal weights to them. For example, if N injectors are thought to be impacting a producer, then one assumes each of them has an impact weight of 1/N. We modified this by using ρ / N as the threshold, where we found, by trial and error, that 80% of 1/N does a good job as a threshold. The IPR curves in Fig. 3(a) are in absolute values, but, to use a percentage threshold they need to be normalized. This is done by dividing each IPR curve by the sum of all the IPR curves. These IPR curves in percentages are depicted in Fig. 3(b). Also shown on that figure is the 80%/N =1.74% threshold line (dotted) when N = 46. After eliminating all of the completions whose IPRs fell below 1.74%, only 17 completions were left. The remaining completions and eliminated completions are depicted in Fig. 2(b) by regular size triangles and smaller triangles, respectively. Our EKF was then re-applied to this single producer, but for only the 17 remaining completions. The resulting IPR values are depicted in Fig. 4. Observe that, although some portion of the three lowest IPR curves are still below the threshold, we are only considering the mean IPRs of the most recent month, which occurs at the rightend of the data, and none of the mean IPR values for that month fall below this threshold. Consequently, none of the 17 injectors were eliminated, and therefore, our 6 SPE 121393 processing for P-130 is completed. If, perchance, some of the 17 injectors had been eliminated, we would have then repeated this procedure until a situation was reached where no more of the injectors are eliminated 2 . The EKF procedure that we have described is performed for each of the producers in the section. If parallel processing is available then this processing can be done in parallel because of its producer-centric nature. VI. Validation Using History Matching How does one validate our EKF method? Unless a field test is performed, there are no truth data available as there would be when synthetic data or reservoir simulation data are used (as in [5]). Although field tests can be performed, they are disruptive and expensive. Another approach, the one we have used is history matching, i.e., we go back in history and locate a time point at which a significant step change in an injection rate has occurred (as suggested to us by petroleum engineers), and use the estimated IPR at that point to “forecast” the production rate at some future time. This forecasted production rate is then compared with the real production rate, since the latter is available to us as a part of the historical data record we began with. Forecasting of the production rate is accomplished by using the predictor equation (21) of our EKF. We illustrate this process next by an example. By carefully inspecting the historical injection rates for all the remaining 17 completions for P-130, we noticed that most of them had a significant rise in their injection rates around day 1045; therefore, day M=1045 was chosen as the starting day for our forecasts. Starting from day M, we used the estimated parameter values obtained at that day, i.e.: ˆ ˆ ˆ x j ( M | M ) = ⎡ IPR j ( M | M ), α j ( M | M ), ⎢ ⎣ (27) ˆ c ( M − 1| M ), p cj ( M | M ) ⎤ ', j = 1,...N ˆ pj ⎦ and the planned injection rates, i.e., i j ( M ),L, i j ( M + 30) ( j = 1,..., N ) , to forecast the next month’s production rates, day-by-day, i.e.: ˆ ⎧ x( M + 1| M ) = f [ x( M | M ) ] ⎪ˆ ⎨ ˆ ⎪ p ( M + 1| M ) = Hx( M + 1| M ) ⎩ˆ ˆ ⎧ x( M + 2 | M ) = f [ x( M + 1 | M ) ] ⎪ˆ ⎨ ˆ ⎪ p ( M + 2 | M ) = Hx( M + 2 | M ) ⎩ˆ M (28) ˆ ⎧ x ( M + 30 | M ) = f [ x( M + 29 | M ) ] ⎪ˆ ⎨ ˆ ⎪ p ( M + 30 | M ) = Hx( M + 30 | M ) ⎩ˆ where H is the same as in (21). 2 During such an iteration process, the threshold would be kept at 80%/N, where N is the initial number of injectors, in this case, 46. N is not chosen to be the surviving number of injectors because the resulting threshold would become so large that it is possible that too many injectors would be eliminated. After one month, new injection and production rate measurements [ im, j (k ) and pm (k ) ( j = 1,..., N ; k = M + 1,..., M + 30) ] are available, so we update our Section IV EKF using those new measurements, to obtain ˆ( the updated state vector, i.e., x M + 30 | M + 30) , that are then used to forecast another month’s production rates by using (28) again. This process is repeated until we have run out of data history. We observed that IPR curves are not flat during an entire month (e.g., see Fig. 4); hence, we computed their gradients at the day the forecast starts, and assumed the IPRs would follow the gradients during the following one month’s forecasts, e.g., the first month’s forecasts start at ˆ M=1045, and the gradient of IPR ( M | M ) and j ˆ α j (M | M ) , g IPR j ( M ) and g αj ( M ) ( j = 1,..., N ) , can be approximated as: ⎧ ˆ ˆ IPR j ( M + 1 | M ) − IPR j ( M | M ) ⎪g (M ) = ⎪ IPR j ( M + 1) − M (29) ⎨ ˆ ˆ α j (M + 1 | M ) − α j (M | M ) ⎪ ⎪ g α j (M ) = ( M + 1) − M ⎩ It follows, therefore, that: ⎧ IPR ( M + 1 | M ) = g ˆ ˆ ( M ) + IPR j ( M | M ) j IPR j ⎪ (30) ⎨ ˆ ˆ ⎪ α j ( M + 1| M ) = g α j ( M ) + α j ( M | M ) ⎩ Because we assume IPR j and α j follow the gradient for the entire month, (30) provides the following daily interpolation formulas that were used by us ( d = 1,...,30 ): ⎧ IPR ( M + d | M ) = g ˆ (M ) + j IPR j ⎪ ⎪ ˆ IPR j ( M + d − 1| M ) ⎪ ⎨ ˆ ⎪ α j (M + d | M ) = g α (M ) + j ⎪ ⎪ ˆ α j (M + d − 1 | M ) ⎩ (31) Results have shown that forecasts incorporating these interpolations outperformed the forecasts that did not use them, so they have been used in all of our results that are described below. The overall historical production rate for P-130 is plotted as the solid line in Fig. 5, whereas the dotted line shows the monthly forecasts, starting at day 1045. Observe that the production starts to increase after day 1045 and that the forecasted production has detected this increase, although there are some forecasting errors. For viewing convenience, we zoom in to the prediction interval and this is plotted in Fig. 6(a), and, the prediction error is plotted in Fig. 6(b). SPE 121393 Some summaries of the monthly errors are given in Table 1. It contains monthly error summaries for different values of a scalar parameter, s, which is used in Section VII to determine the size of the ellipse for establishing the initial set of injectors. Our present case corresponds to the row in Table 1 for which s=1.0. The monthly forecasting began on day 1045 and continued for 8 months. In Table 1, the average daily prediction errors for each month, and the average daily error over the entire eight-month period, have been computed. Additionally, Error-Production-Ratio (EPR), which is the ratio of the average daily prediction error to the average daily production rate over 8 months, has been computed. EPR is useful because petroleum engineers have told us that the measurement-noise level for this section is around 10-20%, e.g. if the measured production is 300 barrels, then about 30-60 barrels is noise. The EPR for s = 1 is 10.23% which is a very good value. VII. Optimization of Initial Set of Injectors Using History Matching An issue that plays a very critical role in our EKF processing is how to choose a set of initial injectors for the producer-centric model. Although it has been previously mentioned that an ellipse with 700 feet major axis and 500 feet minor axis is used to select the initial set of injectors, in general, an optimal size for the ellipse may be very different for different producers. If the ellipse is too small, one would miss some influential injectors from the very beginning. On the other hand, if the ellipse is too large it could include many irrelevant injectors. Such injectors may significantly bias the first round of EKF processing, because they can affect the threshold elimination process that has been described in Section V. Inspired by techniques used in standard optimization problems, and the history matching results described Section V, our strategy for finding an optimal ellipse size (for each producer) was to use the average daily prediction error as an objective function and to minimize it with respect to ellipse size. To do this, we used an ellipse with 700 feet major axis and 500 feet minor axis as a standard ellipse and then scaled its major and minor axes by the same scalar, s; hence, as different values of s are used one obtains ellipses of different sizes but with the same ratio of major- to minor-axes. In theory, s should be discretized very finely and also range from a small enough value to a large enough value so that the computed minimal point of the objective function is its global minimum. Unfortunately, using too many values of s is computationally very costly; hence, in this study we chose s = {0.5, 0.6, 0.7, 0.8, 0.9, 1.0}. We have observed that when s = 1.0 (which is the case discussed in Section V) the ellipse is already large enough and usually includes more than 20 injectors, and (as we will see below), when s = 0.5 the ellipse only includes a few neighboring injectors. Additionally, we observed, after several tests, that s does not have to be discretized very finely, e.g., when s changes from 0.95 to 0.9 the ellipse only shrinks a little and no injectors from the s = 0.95 ellipse are left out of the s = 0.9 ellipse. 7 EKF processing for different values of s proceeds in exactly the same manner as described in Section IV for s = 1. Results for s = 0.9, 0.6 and 0.5 are given in Figs. 715. Summaries of the prediction errors for s = {0.5, 0.6, 0.7, 0.8, 0.9, 1.0} are also given in Table 1. Observe that the objective function is minimized when s = 0.6. Thus, for P-130, the optimal ellipse used to choose the initial injector set is one with 0.6 × 700 = 420 feet major axis and 0.6 × 500 = 300 feet minor axis. VIII. IPR Table: Producer Centric to Injector Centric Conversion The previous sections have presented a complete procedure to process only one producer-centric model. Of course, this procedure has to be applied to every producer in the entire section (or in the entire field, if such data are available) The IPR results of doing this can then be summarized in a table, which in our case has N rows (producers) and M columns (injectors). A small portion of this table is depicted in Table 2. The non-zero values in this table are for those injectors that have survived our EKF processing-procedure. Zero values are for the remaining injectors. This table allows one to quickly look up the IPR value between any producer-injector pair. Reservoir engineers are more interested in an injectorcentric viewpoint than in a producer-centric viewpoint, because they have control over water allocation at each injector; hence, for them it would be better if the results were injector-centric. Observe that Table 2 can be viewed in two different ways. Data are entered into it one row at a time (producer-centric), but it can then be viewed column-wise (injector-centric). Unfortunately, the numerical IPR values are still difficult to interpret. One solution to this is to normalize the IPR values into percentage values by dividing the values in each column by the sum of values in the column. This allows us to observe what percent of the total water that is allocated to an injector goes to a specific producer. We show such results in Table 3 for two injectors. Observe that very large amounts of water from injectors I-322 and I-344 go to producers P-124 and P-146, respectively. Fig. 16 depicts a novel graphical way to represent these injector-centric results. Observe that we have connected an injector to all of the producers that have nonzero IPRs using lines of different widths. The higher the %IPR value is, the thicker the line is, and the lower the %IPR value is the thinner the line is. IX. Conclusion and Future Research Topics This paper has continued the work of [5]. After extensive testing of the EKF on real data, important modifications have been made by us to the SVM, allowing the square root of IPR values to be estimated directly, which avoids estimating negative results for IPR values. We have also: described a way to integrate important expert knowledge about the oil field during the initialization of the EKF processing; presented a method to eliminate unimportant injectors from the model; and shown how to validate the accuracy of our EKF results by using production-rate history-matching. Our validation process has also led us 8 SPE 121393 [2] A. N. Araqye-Martinez, “Estimation of autocorrelation and its use in sweep efficiency calculation,” MS thesis, U. of Texas at Austin, Austin, Texas, 1993. [3] S. Haykin, Kalman Filtering and Neural Networks, John Wiley & Sons, New Jersey, 2001. [4] K. Heffer, K. J., Fox, R. J., and McGill, C. A., “Novel Techniques Show Links Between Reservoir Flow Directionality, Earth Stress, Fault Structure and Geomechanical Changes in Mature Waterfloods,” Paper SPE30711 Presented at SPE Annual Technical Conference and Exhibition, Dallas, Oct., 1995. [5] F. Liu, J. M. Mendel and A. M. Nejad “Forecasting InjectorProducer Relationships from Production and Injection Rates Using an Extended Kalman Filter”, SPE 110520, Presented at SPE Annual Technical Conference and Exhibition, Anaheim, CA, Nov. 2007. [6] J. M. Mendel, Lessons in Estimation Theory for Signal Processing, Communications, and Control. Prentice Hall PTR, Upper Saddle River, NJ, 1995. [7] M. N. Panda and A. K. Chopra, “An integrated approach to estimate well interaction, ” Paper SPE 39563, presented at 1998 SPE India Oil and Gas Conference and Exhibition, New Delhi, Feb., 17-19, 1998. [8] De Sant’Anna Pizarro, J. O., “Estimating injectivity and lateral autocorrelation in heterogeneous media,” Ph.D dissertation, U. of Texas Austin, Texas, 1998. [9] B. T. Refunjol, “Reservoir characterization of North Buck Draw field based on tracer response and production/injection analysis,” MS Thesis, Univ. of Texas Austin, Texas, 1996. [10] T. Soeriawinata and M. Kelkar, “Reservoir management using production data,” Paper SPE 52224, presented at 1999 SPE Mid-Continent Operation Symposium, Oklahoma City, Oklahoma, March 1999. [11] A. A. Yousef, P. Gentil, J. L. Jensen and L. W. Lake, “A Capacity Model to Infer Interwell Connectivity From Production and Injection Rate Fluctuatuions,” SPE 95322, Presented at SPE ATCE 2005, Dallas, Oct., 2005. to develop a method for selecting an optimum ellipse size (for each producer) for choosing the initial set of injectors by minimizing history-matching errors. Finally, we showed how to convert a table of producer-centric IPR values to a table of injector centric IPR values. Although the EKF gives good results, there still is room for improvements. Using more frequently sampled POC data will very likely improve the estimation. By using POC data, it may be possible to group several producers together, in which case our reservoir model changes from a single-producer–multiple-injector model to a multiple producer–multiple injector model. The EKF is still applicable to such a model. Finally, it may be possible to treat each actual injection rate as a state variable so that noisy injection rates do not have to be used in the predictor equations. The downside to doing this is an increase in the dimension of the EKF. Acknowledgments This study was funded by the Center of Excellence for Research and Academic Training on Interactive Smart Oilfield Technologies (CiSoft). CiSoft is a joint University of Southern California-Chevron initiative. The authors would like to thank Prof. Iraj Ershaghi, Dave Tuk, Jim Brink and John Houghton for their valuable discussions. The authors would also like to thank Hyokeyong Lee for generously providing computer programs that pre-process the raw data, and other members from Prof. Cyrus Shahabi’s group for their useful discussions on data issues. References [1] A. Albertoni and L. W. Lake, “Inferring Interwell Connectivity only from Well-Rate Fluctuations in Waterfloods,” SPE Reservoir Evaluation & Engineering, Vol. 6, Number 1, Feb., 2003. Table 1. Measures of average daily prediction error of production rates for eight months, including: overall average daily prediction error (barrels/day), and average daily error to average daily production rate ratio (EPR), all for different size ellipses (s). st S=1.0 S=0.9 S=0.8 S=0.7 S=0.6 S=0.5 1 month 7.3067 3.7455 7.2293 4.4729 5.8975 9.7345 nd rd th th th th th 2 month 3 month 4 month 5 month 6 month 7 month 8 month 16.259 66.977 34.464 34.588 23.108 75.625 19.180 15.506 38.252 26.247 38.229 22.287 43.321 13.253 15.273 33.060 16.235 43.111 13.786 47.965 13.034 14.829 26.305 17.878 43.224 14.721 35.270 13.643 14.161 26.667 18.717 43.360 14.663 51.121 20.796 16.310 26.735 18.209 48.188 10.872 56.535 11.865 average 34.688 28.779 26.583 13.268 13.234 13.833 EPR 10.23% 8.49% 7.84% 3.91% 3.90% 4.08% SPE 121393 9 Table2. Upper left portion of the IPR table for the entire section of 191 producers (rows) and 374 injectors (columns) 3 . (Well names have been re-labled.) P-1 P-2 P-3 P-4 P-5 P-6 P-7 P-8 P-9 P-10 P-11 P-12 P-13 P-14 P-15 P-16 P-17 P-18 P-19 P-20 P-21 P-22 P-23 P-24 P-25 P-26 P-27 I-1 0.0211 0 0 0 0 0.0683 0 0 0 0.0946 0.1062 0.0678 0.0512 0 0 0 0 0 0.0360 0 0 0 0 0 0 0 0 I-2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0654 I-3 0.1571 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I-4 0 0 0 0 0 0 0 0 0 0.0852 0.0782 0.0541 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0536 0 I-5 0 0 0 0 0 0 0 0 0.1429 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I-6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I-7 0.0692 0 0 0 0 0 0.0339 0 0 0.1477 0 0.1014 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0504 0 I-8 0 0 0 0 0 0 0 0 0.1334 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I-9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I-10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0402 0.0329 0.0543 0 0 0 0 0 0.0911 0 0 0 0 Table3. Normalized IPRs for injectors I-322 and I-344. Entries are in %. (Well names have re-labled.) P-124 I-322 P-170 P-5 P-116 P-167 P-191 41.77 21.42 20.51 16.16 0.54 0.20 P-146 I-344 P-139 P-180 P-39 P-135 41.77 17.70 14.32 14.14 12.07 Figure 1: The Reservoir Model for reservoir with N injectors and single producer [5]. 3 All the entries in the IPR table have not been normalized yet. 10 SPE 121393 Figure 2: When s = 1, (a) P-130’s local area and 46 initial completions included in our model; and (b) completions after our elimination process. (Well names have been re-labeled.) Figure 3: When s = 1, (a) IPR curves for the 46 initial completions; and (b) normalized IPR curves (solid lines) and the elimination threshold (dotted line). Figure 4: When s = 1, (a) IPR curves for the 17 remaining completions after the elimination process; and, (b) normalized IPR curves (solid lines) and the elimination threshold (dotted line). SPE 121393 11 Figure 5: Historical production rate for P-130 (solid line) and predicted production rate (dotted line) when s = 1. Predictions begin at day 1045. Figure 6: When s = 1, (a) Historical production rate forP-130 (solid line), and predicted production rate (dotted line); and, (b) error between the real production rate and the predicted production rate. Figure 7: When s = 0.9, (a) P-130’s local area and initial completions included in our model; and (b) completions after our elimination process. (Well names have been re-labeled.) 12 SPE 121393 Figure 8: When s = 0.9, (a) IPR curves for the 10 remaining completions after the elimination process; and, (b) normalized IPR curves (solid lines) and the elimination threshold (dotted line). Figure 9: When s = 0.9, (a) Historical production rate for P-130 (solid line), and predicted production rate (dotted line); and (b) error between the real and the predicted production rate. Figure 10: When s = 0.6, (a) P-130’s local area and initial completions included in our model; and (b) completions after our elimination process. (Well names have been re-labeled.) SPE 121393 13 Figure 11: When s = 0.6, (a) IPR curves for the 9 remaining completions after the elimination process; and, (b) normalized IPR curves (solid lines) and the elimination threshold (dotted line). Figure 12: When s = 0.6, (a) Historical production rate for P-130 (solid line), and predicted production rate (dotted line); and (b) error between the real and the predicted production rate. Figure 13: When s = 0.5, (a) P-130’s local area and initial completions included in our model; and (b) completions after our elimination process. (Well names have been re-labeled.) 14 SPE 121393 Figure 14: When s = 0.5, (a) IPR curves for the 8 remaining completions after the elimination process; and, (b) normalized IPR curves (solid lines) and the elimination threshold (dotted line). Figure 15: When s = 0.5, (a) Historical production rate for P-130 (solid line), and predicted production rate (dotted line); and (b) error between the real and the predicted production rate. Figure 16: Graphical representations of normalized IPRs for (a) injector I-322 and (b) injector I-344 (see Table 3). (Well names have been re-labeled.) ...
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