Fast 4D Seismic History Matching for Thin Reservoir Models

Transcription

1 Paper number: SPE-FFFFFF Fast 4D Seismic History Matching for Thin Reservoir Models Ilya Fursov, Heriot-Watt University; Colin MacBeth, Heriot-Watt University Summary We propose an approach for incorporating the time-lapse seismic data into history matching loop. The approach relies on calibrating the linear relationships between the seismic attributes and the reservoir dynamic parameters (pressures, saturations). This allows us to avoid the conventional complex petro-elastic and seismic modelling, and the associated uncertain petro-elastic parameters. To test the approach, an uncertainty estimation framework is employed, where the data error covariance matrices are assessed based on preliminary history matching, and the Randomized Maximum Likelihood method is used for generating samples from the posterior distribution. Testing of the approach on a synthetic model shows improvement of the model forecasts compared to the conventional well history matching. The improvement manifests itself as a reduction of the forecast uncertainties, which proves the ability of the method to employ 4D seismic data for constraining the simulation model. Furthermore, it is shown that proper definition of the seismic error covariance matrix which accounts for the spatial correlations results in better model forecasts. Finally, the proposed method is applied to a North Sea reservoir, demonstrating reduction of the forecast uncertainties (compared to the conventional well history matching) for the wells located close to the notable observed 4D seismic signals. The method is thus easy to use, and it has a potential of constraining the reservoir simulation models, making them produce more confident predictions. Introduction Reservoir simulation is one of the tools used in the petroleum industry to integrate the different types of the data available for the petroleum reservoir and produce the forecasts of the future reservoir behaviour. The data contributing to the reservoir simulation model are the static data (e.g. the core analyses, well logs, 3D seismic), and the dynamic data (well tests, routine field production, tracers data). Integration of the dynamic data to the initially static reservoir model is done by history matching a process of calibrating the model to make it reproduce the historic reservoir behaviour. The more static and dynamic data the simulation model is consistent with, the more reliable the model forecasts will be. 1

2 Another type of dynamic data that can be used for history matching is the time-lapse (4D) seismic data, which involves a few 3D seismic surveys shot at different moments of the field life. The difference between these seismic responses can highlight the elastic changes which took place in the reservoir rock due to the production and injection activities. For instance, pressure increase due to water injection causes the rock softening (acoustic impedance decrease), and the corresponding 4D seismic signal allows identifying the extent of the overpressured reservoir compartments. Water saturation increase during waterflood results in the rock hardening (acoustic impedance increase) in the 4D seismic response, so the swept reservoir areas can be inferred. Gas saturation increase manifests itself as the rock softening, and hence one can establish the location of the secondary gas caps from the time-lapse seismic signal. The 4D seismic data has better areal coverage of the reservoir than the well data, and responds to the pressure and saturation change (although in a non-trivial way, due to the complexities of the petro-elastic behaviour and seismic wave interference). Because of that it has the potential of further constraining the reservoir simulation model when used in history matching in addition to the other field data. Unlike the well data, where the simulation model results can be directly compared with the historical observations, the time-lapse seismic data and the simulation model initially exist in different domains, and are not comparable. There are approaches which allow converting either seismic data, or model data, or both to the same domain, so that the mismatch between the model and the observations can be evaluated quantitatively in the history matching loop. The first approach is converting the seismic response to the simulation model domain i.e. inversion of the pressure and saturation maps (or volumes) from the 4D seismic attributes. Since the pressure and saturation inversion from time-lapse seismic is quite challenging, the case studies following this approach mostly deal with the synthetic models, e.g. (Davolio et al., 11; Landa and Horne, 1997), but there are also real field applications, e.g. (Jin et al., 11). The second approach is conversion from the simulation model domain to the seismic domain, usually through the petro-elastic modelling and the subsequent modelling of the seismic wavefield. The petro-elastic modelling introduces a range of new uncertain petro-elastic parameters to the history matching. Besides, it depends on the uncertain variation of the reservoir geology (e.g. distribution of shales and porosity) in the interwell space. In (Huang et al., 1997) the history matching of a Gulf of Mexico turbidite reservoir is performed by comparing the 4D seismic with the simulation model in the domain of seismic amplitudes. The third approach employs the intermediate domain the domain of impedances, such that the seismic volumes are inverted to the impedance cubes, which are compared with the impedances obtained from the

3 simulation model through the petro-elastic modelling. This domain appears to be the most popular in the history matching literature, see e.g. (Emerick et al., 7; Gosselin et al., 3; Gosselin et al., 1; Reiso et al., 5; Roggero et al., 7). There are also approaches which avoid the direct comparison of the observed and modelled time-lapse reservoir signatures, calculating instead some correlation measures between the quantities in question (impedances, seismic attributes). Usually they are employed for the reservoirs where petro-elastic modelling is challenging, e.g. for a gas condensate reservoir (Waggoner et al., ), or a compacting chalk reservoir (Kjelstadli et al., 5). In this work a new method for integrating the time-lapse seismic data into the history matching loop is introduced. It does not involve any conversion of the simulation model or the seismic into a different domain, and does not involve any classical petro-elastic or seismic modelling or inversion. Instead, the map of time-lapse seismic attribute is treated as a linear combination of the time-lapse maps of the reservoir dynamic parameters (average pressure and saturations). The coefficients arising in the seismic-model relationship are estimated from the linear regression. As no full-physics seismic modelling is employed in this procedure, there is no need to resolve any petro-elastic uncertainties in the history matching loop, so the procedure is easy in application compared to the classical petro-elastic modelling approach. The latter approach often relies on the laboratory measurements to calibrate the petro-elastic parameters, however these measurements may not adequately describe the in-situ petro-elastic behaviour of the actual reservoir (Landro, 1). The proposed method is free of this concern, as it adjusts itself to the actual petro-elastic and seismic response of the reservoir. The other advantage of the method is that it is fast and does not require much CPU resources. Finally, formulation of the seismic part of the objective function is done in the least squares form, so it can be readily applied in the Bayesian uncertainty estimation setting (under the assumption of the Gaussian errors). Integration of 4D Seismic into the History Matching Loop In this work we only consider thin reservoirs i.e. those for which the seismic does not resolve vertically the separate reservoir units in the situation where the reservoir thickness is small compared to the seismic dominant wavelength. For such reservoirs the 4D seismic can be naturally treated in the D attribute maps form. These maps, incorporated into the history matching loop as one of the inputs, reflect the reservoir elastic response to the reservoir pressure and saturation changes. In order to quantitatively assess consistency of the observed 4D seismic attribute map with the simulation model, we start from considering a linear relationship between the seismic and the average maps of the reservoir dynamic properties: pressure, water and gas saturations. 3

4 A = a P + a Sw + a Sg. (1) P Sw Sg In this equation A, P, Sw, Sg represent the time-lapse maps of the corresponding quantities (seismic attribute, pressure, water saturation and gas saturation respectively), a, a, a are the constant numbers, and P Sw Sg the equation is treated in the point-wise sense (so in fact there is a system of linear equations). The right hand side of this equation will be also referred to as B in what follows. Generally, the coefficients a ( i i = P, Sw, or Sg ), would vary laterally depending on the reservoir thickness, porosity, and net-to-gross ratio (NTG). This variation will be accounted for below by multiplying the right hand side by a scaling map, however the coefficients themselves will still be treated as constants. When seismic data for the given reservoir is available from the multiple time-lapse seismic surveys (monitors), then all these data can be used in the equation. In this case the left hand side includes all the points of time-lapse attribute maps from all monitors, the right hand side includes the points of the reservoir dynamic property maps from the corresponding time steps. Relationships of form (1) between the 4D seismic and the reservoir dynamic parameters are considered by (Alvarez and Macbeth, 14; Falahat et al., 13; Floricich et al., 6; MacBeth et al., 6) for the purposes of pressure and saturation inversion from the 4D seismic. The coefficients a are estimated in these papers from i the data at well locations, or using a calibrated petro-elastic model. Their physical meaning in terms of porosity, rock stress sensitivity, fluid properties, and other parameters is discussed by (Alvarez and Macbeth, 14). In application to history matching, relationship (1) can be used to estimate the agreement between the seismic (the left hand side) and the simulation model (the right hand side) even without calculating the unknown constants a in advance. For that, we perform the linear regression with the points from the map i A being the dependent variables, the corresponding points from P, Sw, Sg being the independent variables, and constants a being the regression coefficients. High quality regression can then be regarded as a sign of good i agreement between the seismic and the simulation model, and the coefficient of determination R can be used as a quantitative measure of it. Having estimated the coefficients a, we can calculate the right hand side i and this essentially means performing a simplified (proxy) simulator-to-seismic modelling. Thus, relationship (1) compares the observed 4D seismic A with the modelled 4D seismic Assuming (for the moment) that the errors in seismic data are not correlated and have standard deviation σ, B. the seismic part of the objective function f can be defined as the sum of squares: B, 4

5 f Ak Bk 1 = = (1 R ) S tot k σ σ, () where A, k Bk are the different points from the left hand side (observed 4D seismic) and the right hand side (modelled 4D seismic) respectively; these points are indexed by k. The second equality in () follows from the R definition: the total sum of squares S 1 R = S res / S, i.e. the ratio of the sum of squares of residuals S = ( A B ) and tot tot. For the real cases the errors in the seismic data are likely to be correlated, so the definition of f would require application of the full data error covariance matrix converting f into f res k k C. This can be done by s T 1 = ( A B) C ( A ), (3) s B where the "modelled" time-lapse seismic B is again given by the right-hand side of equation (1), and is calculated by finding the unknown coefficients a which minimise the value of i f in (3). The basic formula (1) is quite a rough tool, and it may require some modifications in order to provide a better link between the time-lapse seismic and the simulation model. Two modifications are suggested: firstly, point-wise multiplication of the right hand side by some scaling map A : A = a P + a Sw + a Sg) A, (4) ( P Sw Sg and secondly, introduction of the quadratic terms of the reservoir dynamic properties: A = a P + a Sw + a Sg + a P + a P Sw +...) A. (5) ( P Sw Sg PP PSw Multiplication by the scaling map allows accounting for the lateral variations of the static reservoir properties such as thickness, porosity and NTG, which affect the time-lapse seismic response of the reservoir. In (Falahat et al., 13) it is essentially proposed to use the pore volume map as A. In (MacBeth et al., 6) the left hand side of equation (1) is divided by the map A equal to the baseline seismic attribute map. However it should be noted that division operation is ill-posed compared to multiplication, and numerical problems may emerge at the points where A is close to zero. Instead, multiplication of the right hand side by A is deemed to be a more appropriate treatment of the reservoir lateral heterogeneities, as proposed in equation (4). Introduction of the quadratic term for pressure was considered by (Floricich et al., 6; Landro, 1), and water saturation quadratic term was proposed by (Meadows, 1). We also introduce the quadratic term for gas k saturation, and the mixed terms like P Sw. All the three proposed formulae (1), (4), (5) use the linear (or at most quadratic) gas term, whereas in the literature it is often believed that gas impact on the 4D seismic is 5

6 highly nonlinear, e.g. (Domenico, 1974) reports strong non-linearity from the laboratory studies, (Floricich et al., 6) use the exponential dependence on Sg, (Lumley et al., 8) claim that CO saturation cannot be resolved from 4D seismic because of the non-linear velocity dependence on the gas saturation. In our work we essentially adhere the linear treatment of gas, as established and proposed by (Falahat et al., 13). Effect of both modifications (4) and (5) on the linear regression quality was checked on the threedimensional three-phase synthetic model (Fig. 1) which is considered in detail in the subsequent section. For the synthetic seismic and the average dynamic parameter maps provided by the model (Fig. ), the basic equation (1) resulted in R =. 65, as shown in Table 1. Using the baseline seismic attribute map for scaling in (4) improved it to R =. 76. A smaller enhancement took place when two quadratic terms Sw, Sg were introduced, leading to R =.74. These two terms had the maximum contribution, and once these were added, introduction of any other quadratic term only gave a marginal improvement. The considered example together with a few other synthetic tests showed that both scaling operation in equation (4) and quadratic terms in equation (5) improve the quality of the proposed relationship between the time-lapse seismic and the simulation model, so only these two equations will be used for history matching in what follows. Among the two choices for the scaling map A discussed above the pore volume map, and the baseline attribute map synthetic tests revealed slightly better performance of the latter in terms of the regression quality. Besides, in the case of a real reservoir, one is not likely to have a reliable pore volume map, whereas the seismic attribute maps are readily available. For this reason we will only consider scaling with the baseline attributes in our history matching studies. Practice has also revealed that simply solving either equation (4), (5) does not always result in the physically meaningful signs of values of the coefficients a. Consider, for example, a simple situation when the reservoir i pressure decreases ( P < ), leading to the reservoir hardening. Suppose that the selected time-lapse seismic attribute reacts on the reservoir hardening by increasing its value ( A > ). It is obvious that the resulting derivative A / P should be negative. If, for example, we are dealing with equation (4), then we arrive at A / P = A a <. If linear regression (usually because provided poor quality input) results in positive P A a P, the numerical link between the seismic and the simulation model becomes rather meaningless. To treat this, the constraints are introduced when solving equations (4), (5) in the least squares sense, which allow maintaining the proper signs of the derivatives of A with respect to the pressure, water and gas saturations changes. The resulting constrained least squares problem is then solved by the active set method of quadratic 6

7 programming. In all seismic history matching problems considered below the following derivative signs are maintained: A / P, A / Sw, A / Sg. (6) These signs are defined according to the petro-elastic model (lower impedance sands, higher impedance shales) and the seismic attribute used (sum of negative amplitudes between the reservoir top and base horizons). Uncertainty Estimation History matching with 4D seismic data imposes additional constraints on the reservoir simulation model, which will potentially result in more reliable model forecasts compared to the conventional well history matching. However, improvement of the forecasts quality cannot be judged based on a single history matched model, because history matching, as an inverse problem, is ill-posed and its solution may be highly non-unique. So, instead, the benefits of the seismic history matching (SHM) versus conventional well history matching (WHM) should be examined by comparing the resulting model uncertainties of both history matching cases. In this work we adopt Bayesian approach to estimate the uncertainties. It establishes the link between the posterior probability density p ( m d) of model m given the measured data d and the prior probability density q (m) of the model m by means of the likelihood function L ( d m) : L( d m) q( m) p( m d) =. p( d) (7) The data marginal probability density p (d) from the denominator is not calculated directly, and can be regarded as a normalization constant. For simplicity, we consider prior q (m) as a uniform density in the region of the parameter space defined by the parameter ranges. The likelihood function is taken to be Gaussian, with either simple diagonal covariance matrix, or full covariance matrix accounting for the correlations between the different data points. 1 T L( d m) = g exp ( d g( m)) C 1 ( d g( m)), (8) where model m is essentially a vector of parameters, g (m) is the forward modelling procedure, C is the data T 1 error covariance matrix, γ is a normalizing constant. Expression f ( m) = ( d g( m)) C ( d g( m)) is the objective function, and its minimization, which is equivalent to maximization of the likelihood, leads to the least squares problems. 7

8 In definition of the covariance matrix we acknowledge the discussion from (Tarantola, 5), where modelling and theoretical errors are considered, and the total covariance matrix is derived as the sum of covariance matrices corresponding to these two error sources. In our work the total covariance matrix C is dealt with, without distinguishing the modelling and theoretical covariances. It is calculated as C = SRS, where S is a diagonal matrix containing the standard deviations for all data points, and R is the correlation matrix which accounts for the temporal (in the well data) or spatial (in the seismic data) correlations. Covariance matrix C (and correlation matrix R ) is naturally split into blocks C and w C corresponding to the wells part of objective s function f and the seismic part of the objective function 1 f. The wells part of objective function f is treated 1 in the usual least squares sense, with either diagonal covariance matrix, or a non-diagonal one. Furthermore, the wells block C of the covariance matrix is split into smaller blocks corresponding to the different well data w vectors obtained from the simulation model (e.g. water cut of each producer, etc). In practice C is commonly taken as just a diagonal matrix, implying that the errors in the well data are not w correlated, and only the standard deviations C should be accounted for. For example, (Aanonsen et al., wii 3) claim that the correlation length in the well data is around 1- days, so if the well data are reported at monthly rate, which is usually the case for the simulation models, these correlations can be ignored. In Schlumberger SimOpt user guide (Schlumberger, 1) roughly the same statement about the correlations in well data is made. It seems that in the above considerations only the measurement errors are addressed by the authors. We think that a more appropriate approach will be to consider both measurement and modelling errors, allowing non-diagonal covariance matrices for the well data. When the time-lapse seismic data is introduced into history matching, the error model for it usually accounts for the spatial correlations, as encountered in the literature, see e.g. (Aanonsen et al., 3). This entails the need to use a non-diagonal covariance matrix C in definition (3) of the seismic part of the objective function. We s use a simplified representation C = s R, so each map point is assigned the same value of the standard deviation s s s s defining the magnitude of the seismic errors. The correlation matrix s and orientation of the correlated features on the error map. R defines the spatial range s One of the criteria which may be useful in checking the validity of the covariance matrix is discussed by (Tarantola, 5): if the forward modelling procedure is linear and the covariance matrix correctly describes the data errors, then the minimum of the objective function f should follow the χ distribution with υ degrees of 8

9 freedom, where υ = N N (here N is the number of data, and d d p N is the number of parameters). This p statement is true if the inverse problem is not under-determined and the forward modelling matrix has full rank. For this χ distribution the mean value is υ, and the variance equals υ, so the following estimate on the minimum of f can be considered, as suggested by (Oliver et al., 8): υ 5 υ f min υ + 5 υ. (9) These considerations are only rigorous for the case of linear forward modelling, and the quasi linear situations are to be treated with caution. However, (Oliver et al., 8) report that for synthetic history matching problems the expected minimum value of the objective function is in reasonable agreement with the above theory. If the final value of the objective function after optimization is not consistent with the estimate (9), this may mean failure of the optimization algorithm to find the reasonable minimum, or incorrectly defined standard deviations (std s) of the data errors. Too low values of f will mean overestimated std s, too high values will min correspond to underestimated std s. To calculate the covariance matrix, we employ the following procedure: 1. Make a preliminary history match with a diagonal covariance matrix, where the standard deviations are defined in some reasonable manner, e.g. based on the magnitude of the observed data and engineering judgement.. For each historic well data vector (and each seismic attribute map) estimate the error r = d g(m) between the observation data and the history matched (preliminary) model. Among the historic data vectors we do not consider the cumulative quantities like the total water production due to the complicated covariance matrices these involve. 3. Calculate the autocovariance function c ( h) = 1/ N r( x) r( x + h), and visually find the resulting obs correlation range. Using the found range, the correlation matrix i R for each well data vector i is w calculated, e.g. employing the spherical covariance function. For the time-lapse seismic maps, the autocovariance is a D map, which typically exhibits an ellipse-like feature (cf. Fig. 3). From this map, we can visually establish the rotation angle (azimuth) of the ellipse, the major and the minor correlation ranges. The corresponding correlation matrix information and the 1D spherical covariance function. R for the seismic data is then calculated using the ellipse s 9

10 4. Define the standard deviation for each well data vector, and for the seismic data. To do this, the preliminary std s are corrected based on the χ criterion outlined above. For accomplishing this step, a few additional history matching runs are required. Once the covariance matrices (and hence the likelihood function) have been defined, the estimates based on the posterior distribution of the models can be produced. We employ the Randomized Maximum Likelihood (RML) method (Oliver et al., 8) which allows generating models from the posterior distribution. The method is only rigorous for the problems with linear forward modelling, although for a few simple non-linear examples the models sampled by the RML are reported to follow the posterior distribution reasonably well (Gao et al., 6; Liu and Oliver, 3; Oliver et al., 8). In the absence of prior data (which is essentially implied by the uniform priors), generating a model using RML involves two steps: * 1. Perturb the observed data: d = d obs + P, where perturbation P is normally distributed, with covariance matrix taken as in the likelihood definition (8): P ~ N(, C).. Find model m * * * T 1 * which minimizes f ( m) = ( d g( m)) C ( d g( m)), using some optimization algorithm. This will be the model following the posterior distribution. Although the considered prior model distributions are uniform, they are defined over some region R in the parameter space. To account for that, the minimization step of the RML procedure outlined above is performed with the parameter constraints given by region R, although this is not the most rigorous way to incorporate the prior which is uniform over R. The objective function minimization in the preliminary history matching and the RML minimization step is done by CMA-ES optimization algorithm (Covariance Matrix Adaptation Evolution Strategy) which is proposed by (Hansen and Ostermeier, 1996, 1). CMA-ES is a black box stochastic optimizer, which only requires the values of the objective function to work. It has a number of invariance properties including invariance to the rigid transforms of the search space, and invariance to the strictly monotonic (order-preserving) transformations applied to the objective function. Since the algorithm works with generations of models, it is easily parallelizable when calculating the objective function values among each generation. Synthetic Case Study Testing of the proposed seismic history matching method together with the uncertainty estimation by RML was performed for a synthetic model. The model is a three phase reservoir penetrated by two vertical wells producer and injector, which are controlled by the constant bottom-hole pressure. The reservoir has the average 1

11 thickness of 35 m, and heterogeneous properties (porosity, permeability, NTG), see Fig. 1. There is a small water saturated zone penetrated by the injector which is completed over the whole layer thickness, and a small gas cap penetrated by the producer which is completed below the gas cap. Production history covers 5 days: for the first 5 days only producer is working dropping the reservoir pressure below the bubble point, followed by 5 days of shut-in period, followed by 4 days of the producer and injector working together. The historical period is followed by days of prediction period which will be used for checking the predictive capabilities of the history matched models. For this period the controlling BHPs at wells are inherited from the preceding historical period. Two seismic surveys are considered: a baseline survey prior to the production start, and a monitor survey at the end of the historical period, i.e. at 5 days. Seismic modelling is performed using the same petro-elastic properties and the wavelet as were established for a UK Continental Shelf (UKSC) reservoir (Amini, 14), which is considered in the next section. Because of the wavelet selected, such modelling essentially results in the coloured inversion seismic (Lancaster and Whitcombe, ), where the sand units correspond to the negative amplitudes, and the shales correspond to the positive amplitudes in the seismic section. Due to the low reservoir thickness (equal to 35 m, whereas the dominant seismic wavelength is around 14 m), seismic only resolves vertically a single sand unit. For the time-lapse monitoring, we take the seismic attribute equal to the sum of negative amplitudes between the reservoir top and base horizons. The resulting time-lapse map of the attribute between the monitor at 5 days and the baseline is shown in Fig., along with the corresponding maps of the pressure change, water saturation change and gas saturation change. As can be seen from the water saturation map, water breakthrough does not occur by the end of the historical period. However, the approximate position of the water saturation front can be inferred from the time-lapse seismic map, so it is anticipated that adding the 4D seismic information to the history matching loop should reduce the uncertainty in predicting the water breakthrough and the water cut development. The grid mesh of the original cells model used to produce the well and seismic data was upscaled, resulting in a cells coarse model employed for history matching. NTG and porosity in this coarse model were taken constant, equal to.75 and.5 respectively. Horizontal permeability was taken constant for each of the four layers, equal to 1 md, md, 4 md, 15 md respectively for layers 1 4, and the vertical permeability was set equal to the horizontal one. The upscaled model is thus laterally homogeneous. It was parameterised by 4 variables capturing the main reservoir uncertainties, as follows: 11

12 - 8 parameters for the water-oil phase relative permeabilities, including 6 parameters for the two curves using Lomeland-Ebeltoft-Thomas (LET) parameterization (Lomeland et al., 5), and parameters for the end points; - 8 parameters for the gas-oil relative permeabilities (similar to the oil-water curves); - 1 pore volume multiplier, 1 k v / k ratio, well skin factors, 4 horizontal permeability multipliers (one h per each layer). We use the vectors of well oil production rate (WOPR), well water production rate (WWPR), well water injection rate (WWIR), and well gas oil ratio (WGOR) as the observed well data. For the purposes of uncertainty estimation the original well data were perturbed by adding the measurement errors, taken as uncorrelated Gaussian noise with the standard deviations defined in Table. Apart from these errors, history matching would involve the modelling errors. To estimate them, seven history matching runs were made with the 5 days of the original unperturbed history. The resulting models turned out to reproduce the well data very closely, despite not quite flexible parameterization and the fact that the models are laterally homogeneous and coarser than the original heterogeneous model. The corresponding modelling errors have standard deviations 4 8 times smaller than those of the measurement errors shown in Table. For this reason the modelling errors, which generally have non-diagonal covariance matrix, can be neglected, and the objective function will only account for the introduced measurement errors which are not correlated. The synthetic seismic to be used in history matching ( A in Fig. ) was not artificially perturbed, so there will be no measurement error. However the modelling error is likely to exist due to roughness of the upscaled simulation model, roughness of the parameterization, and roughness of the seismic modelling within the history matching loop (regression between the maps). To estimate these errors the steps outlined in the section above were undertaken. Twenty four SHM cases with diagonal C were run instead of just a single case to gather s more statistics. For each of the cases a D covariance was estimated and then we took the final covariance (Fig. 3) as the arithmetic average of the 4 covariances, which is statistically appropriate. The major and minor axes were picked visually and are shown by the black lines on the figure. To pick them only the inner most reliable contours of the ellipse were considered. The outer contours seem to be somewhat distorted compared to the inner ones, and the direction picked from the outer contours is displayed by the blue line. This direction was not o used however. The estimated ellipse parameters are as follows: rotation angle χ = 4, major correlation range R e = 14 m = 7 cells, minor correlation range r e = 8 m = 4 cells. 1

13 To examine the effect of 4D seismic on the history matching uncertainty, three different history matching (HM) setups were considered: 1. Well data (conventional) HM. The perturbed well data and the diagonal covariance matrix are used.. Seismic + well data HM, seismic without correlations. The well data are treated as in setup #1. The seismic part of the objective function f uses the diagonal covariance matrix. The regression equation between the seismic and simulation model maps involves six (all possible) quadratic terms P, Sw, Sg, P Sw, P Sg, Sw Sg, and scaling by the baseline attribute map A. Taking into account the number of seismic data (points) equal to 183, we estimated s = 6. from the χ criterion. The root mean square (RMS) of the observed 4D attribute map itself equals Seismic + well data HM, seismic with correlations. The difference with setup # is in the seismic covariance matrix C, which was calculated from the covariance ellipse of the seismic modelling s errors, having the semiaxes 7 x 4 cells, rotated at o 4. From χ criterion the estimated s = 46.. The predictive capabilities and the associated uncertainties for each history matching setup were assessed by firstly running the RML procedure for the 5 days historical period. This produces a number of models which can be regarded as the samples from the posterior distribution. The CMA-ES optimisation settings for a single RML run involve the population size λ = 15, and 4 generations altogether, resulting in the total 6 forward modelling runs. The starting point for each CMA-ES search is taken as a uniformly distributed random point within the ranges of all parameters to maximise the diversity of the found local minima. Next, each RML-matched model is run by the simulator for another days of the prediction period, where the BHP controls at wells are inherited from the preceding historical period (thus, production and injection rates are the quantities to be calculated). The resulting well data forecasts can then be compared with the data from the original fine scale model, so that the predictive power of each HM setup can be assessed. The scatter of the well data forecasts among the bundle of models will in turn indicate the degree of uncertainty present therein. To estimate how many RML-matched models are necessary for a history matching setup to give a realistic uncertainty picture, we considered the possible options of 5, 1,, 5 models per bundle, and it was found that models is a reasonable choice. Before moving further it is worth looking at the example of the 4D seismic attribute maps generated during SHM by the regression procedure introduced in the section above. Fig. 4 displays the input attribute map and two maps produced for the models from setups # and #3. As can be seen, the maps show high similarity and s s 13

14 the main 4D features corresponding to the water saturation change and gas saturation change are reproduced reasonably well. Thus, regression between the time-lapse seismic maps and the reservoir dynamic property maps can be regarded as an adequate simulator-to-seismic tool in the context of history matching. The results of uncertainty estimation with RML for each history matching setup are shown in Fig. 5. The plots display the different modelled well data (oil and water production rates, water injection rate) both for the historical and the prediction periods. The plots of GOR are not displayed here because its major fluctuations due to the reservoir depressurisation and gas cap depletion occur during the historical period and are grasped by history matching, whereas in the prediction period all the HM setups show rather small GOR uncertainty and good forecasts. To display the uncertainty estimate, we calculated the mean and std of each well data vector, and the corridor of mean ± std is shown on the plots. For each HM setup this calculation involved the bundle of models generated by RML. The observations (blue circles on the plots) are the well data from the original fine-scale model with the added perturbations. For HM setups 1 3 all the history matching runs show a good well data match for the historical period, but we are interested in the behaviour at the prediction period which is located on the right of the vertical green line on the plots. The conventional well HM (setup #1) shows moderate uncertainty, and the stripe mean ± std generally covers the observations. When the seismic data without correlations are introduced (setup #), the std s notably decrease, but at the same time the means shift upwards, especially for the water injection rate for which the prediction mostly fails. Adding the correlations of the seismic data errors (setup #3) leads to some increase in std s compared to the non-correlated case, however there is still an improvement of the std compared to the conventional well HM. This improvement is more pronounced at the early stages of prediction, with std ratio values for the different well data vectors ( std (setup#1) / std(setup#3)) starting from 3 at the early prediction period and then decreasing to at the late prediction period. Another enhancement from introduction of the correlations in seismic data concerns the mean trajectories which became less biased for setup #3 than for the uncorrelated setup #. The bias has also decreased compared to the ordinary well history matching (setup #1). This is seen most markedly for the water injection and production rates. For the oil production rate setup #3 also gives fine forecasts in the early prediction period, however some bias builds up in the late period. To make these statements mode quantitative, the root mean square (RMS) of the difference ( bias ) between the mean well data vectors estimated by setups #1 #3 and the 14

15 observed data were considered. History matching setup #3 has the lowest RMS values of the bias, which are up to 3 times lower than the values for setups #1 and #. From the history matching exercise performed on this synthetic model we conclude that the proposed algorithm of incorporating the observed time-lapse seismic attributes is capable of reducing the simulation model uncertainties. Furthermore, it was highlighted that one should properly define the correlation ranges that may be present in the data errors in order to obtain adequate future forecasts and uncertainty estimation. With these observations we move to the application of our workflow to a real field case study, which is described in the following section. North Sea Field Case Study Field description. The proposed seismic history matching procedure was applied for history matching a segment of a field located at the Atlantic margin of the UK Continental Shelf (UKSC). The field background can be found in the works of (Dobbyn and Marsh, 1; Govan et al., 6; Leach et al., 1999; Richardson et al., 1997), and we provide the brief description below. The reservoir consists of a range of layers from the Tsequence : T5, T8, T31a, T31b, T34, T35, with T31 (a and b) being the main reservoir. These are made up of the Tertiary age turbidites, with productive sands ranging from channels to sheet-like sands with different patterns of overlap and connectivity. The complex turbidite facies changes together with the normal faulting present in the field result in the complex compartmentalization and reservoir connectivity pattern. Four segments are distinguished in the field, and our work is focused on the southernmost segment 4 which is separated from the northern segment 1 by a sealing fault, see Fig. 6. The reservoir porosity and permeability are good, with average values equal to.7 and 6 md respectively. The initial reservoir pressure P init = 97 psi (at depth 194 m TVDSS) is close to the bubble point pressure P bub = 8 psi. There is a limited aquifer at the western part of the field which provides little natural support, plus small local gas caps. These characteristics result in the necessity of water injection for efficient oil recovery. The complex connectivity pattern between the sand geobodies imposes the challenge of appropriate placement of the producing and injecting wells for good pressure support. The initial estimates of the reservoir connectivity were overly optimistic, resulting in placement of producers and injectors in different sand channels to maximize the sweep. However, the producers were found to have inappropriate pressure support meaning that the actual field connectivity is considerably smaller than the estimate. Poor pressure support led to high production GOR, so in the first 3 4 years of the field life the focus of the reservoir management was on managing gas. Subsequently, the infill drilling programme was 15

16 implemented together with the improved water injection management, which allowed decreasing the production GOR to normal values. The infill drilling decisions were driven by the effective use of 4D seismic and pressure data. As the field matured, the water production increased and the reservoir management shifted to managing sweep and water cut. Dataset description. In our history matching study only segment 4 is examined, which is considered to be isolated from the adjacent segment 1. We use the following data provided by the operator: 1. Reservoir simulation model, which has dimensions cells for segment 4. The model contains the main reservoir T31 plus the overlying (T34, T35) and underlying (T5, T8) reservoirs connected to it. The sands of layers T5, T8, T34, T35 are developed predominantly in the western part of segment 4. The model cell dimensions are 5 5 m horizontally, and the vertical dimension is on average 3. m,.4 m, 6. m for the overlying layers, T31 layer, and underlying layers respectively.. Seismic cubes (coloured inversion (Lancaster and Whitcombe, ), full offset stack) acquired at 1996,, 4, 6, 8. The 1996 seismic is the preproduction one and will be used as a baseline for the time-lapse studies. To assess the 4D seismic non-repeatability, normalized RMS (Kragh and Christie, ) was calculated in the overburden for the difference between surveys 1996 and 4. The resulting mean NRMS is.36, which is quite close to the estimate.31 reported by (Falahat et al., 14). Both figures indicate moderate time-lapse non-repeatability: from the literature the good NRMS values typically range from.1 to.3, and another extreme of NRMS = 1.41 corresponds to the seismic data consisting of the random noise (Johnston, 13). Apart from the 4D application, it is worth noting that certain information relevant to the reservoir modelling can be inferred from the regular 3D seismic cubes. Firstly, 3D seismic may be used for fairly accurate mapping of the net pay, (Dobbyn and Marsh, 1; Leach et al., 1999). Then, the reservoir simulation model used in our work was originally built based on the several hundred sand geobodies derived from the 3D seismic, let alone the conventional structural information extracted from the seismic data. 3. Seismic horizons (top and base) for reservoir T31. Since only these horizons are available, for seismic history matching we only use the attributes calculated for T31, which are compared to the reservoir dynamic property maps averaged over T31 cell layers of the simulation model. The thickness of T31 reservoir is on average 7 m, and reaches the maximum of 11-1 m. The dominant seismic wavelength is approximately 14 m, so vertically seismic mostly resolves a single sand unit (sand corresponds to the negative amplitudes), and occasionally it resolves two units. By taking the attribute 16

17 equal to the sum of negative amplitudes between the top and base horizons we can account for all the resolved sand units, and thus treat seismic data in the form of maps. 4. Historical well data for the period from the start of production at 16/7/1998 to //8, including production and injection rates, water cuts, gas/oil ratios, well bottom whole pressures. Eight producers (D1 D8) and eight injectors (N1 N8) worked during this period in segment 4. Some of the wells were completed in the reservoirs above or below T31, as reported in Table 3. Examination of the original fine-scale simulation model revealed that the model significantly overestimates the reservoir connectivity and permeability, so that the bottom-hole pressure of the injectors is underestimated, and bottom-hole pressure of the producers is overestimated by the model. Too high BHP at producers also results in underestimation of the gas production. Water production is underestimated by the model, implying too high sweep efficiency (delayed water breakthrough). Since we use the stochastic CMA-ES optimisation algorithm for the assisted history matching which relies on a large amount of forward modelling, a fast-running simulation model is essential. The original model that contains 36, active cells takes 1,8 seconds to run the full history from 1998 to 8. The model was upscaled to a coarse 7 3 cells model with 8,7 active cells and a full history run time 41 seconds. Parameterization. As mentioned earlier, one of the major uncertainties for the UKCS field is the reservoir connectivity. Inspection of the original fine-scale simulation model also suggested that the reservoir connectivity should be reduced in order to match the well data. We selected two main controls over the connectivity to be used as parameters: transmissibility multipliers between certain regions of the model, and permeability multipliers within the regions. The natural choice of these regions is provided by the seismicallymapped geobodies present in the original simulation model, see Fig. 7. The majority of these geobodies have localised pancake or channel-like shape. Each single geobody can be regarded as corresponding to an elementary episode in deposition and thus having relatively constant porosity-permeability properties. This makes it geologically-reasonable using a single permeability multiplier per geobody. On the other hand, separate geobodies are likely to have shale draping around them which acts as a thin connectivity barrier between the adjacent geobodies. Such thin barriers are naturally represented in the simulation model by transmissibility multipliers. The original model has 147 geobodies in segment 4, which makes the total number of permeability and transmissibility multipliers too large to be handled by the history matching procedure. To reduce this number the geobodies were amalgamated resulting in the total of 35 geobodies for the whole model. The amalgamation was performed manually, based on the closest neighbour 17

18 principle, and the maps of 3D and 4D seismic attributes. For the specified 35 geobodies we took 33 permeability multipliers and 4 transmissibility multipliers (very often a single multiplier was used for the connections of a particular geobody with all its neighbours). Apart from that, the following parameters were used: - v kh k / ratio for the whole model; - 6 pore volume multipliers for some geobodies, including the aquifer geobody ; - 4 skin factors for two horizontal wells (D1, D5), each well penetrating three amalgamated geobodies; - Rs / t, i.e. the maximum rate of increase of the solution gas/oil ratio in the grid cells; - 7 parameters for the oil-water and gas-oil relative permeabilities, including 4 Corey exponents (Corey, 1954; Ghedan, 7), and 3 parameters controlling the end points; - 1 oil-water contact for the whole segment, and gas-oil contacts for the small gas caps (the two gas caps were not penetrated by the wells, and were mapped based on the 3D seismic data); There are 95 parameters altogether. Their ranges were selected based on the engineering judgement, so that the perturbed model remains geologically and physically meaningful and consistent with the a priori understanding of the field. Objective function. Two history matching setups will be considered: conventional well history matching (WHM) which only uses the well data, and seismic history matching (SHM) which uses the same well data plus 4D seismic data. The total period for which the well measurements are available is 9½ years, however only a limited initial part of it from the start of production at 1998 till August 4 will be taken as a historical period to perform history matching. The remaining part will be used as a prediction period to assess the predictive capabilities of the history matched models. For the well part of objective function we use a non-diagonal covariance matrix C to account for the w correlations in the modelling errors. The historic data vectors are well water cuts (WWCT), gas-oil ratios (WGOR) and bottom-hole pressures (WBHP). The wells in the simulation model are controlled by the liquid flow rate. To estimate C, the steps outlined in the section above are taken: preliminary history matching, and w analysis of the error between the preliminary model and observations. The found correlation ranges and standard deviations for all data vectors are listed in Table 4. Higher correlation ranges, e.g. 4 days, indicate that the modelling error dominates the total error, while the lower ranges, e.g. 1 days, are more typical for the vectors with higher measurement noise. The former situation is the case for the water cut vectors, the latter situation for the GOR vectors. Higher measurement noise for the 18

19 GOR does make sense because gas production from the wells producing below the bubble point is a relatively irregular and unstable process involving accumulation of the gas volumes near the well and their spontaneous discharge to the well. The correlation ranges of the errors for the BHP at producers are larger than those at injectors, which is consistent with the visually higher scatter of the BHP data at the injectors compared to the producers. This is presumably related to the more stable flow conditions at the producers where gas lift is implemented (Leach et al., 1999), hence, a softer flow control, as opposed to the water injectors where the flow conditions are affected directly by the water pumps (more rigid flow control). The standard deviations of the errors for the injectors BHP are higher compared to the producers, firstly, due to the larger scatter of the measured injectors BHP mentioned above. Secondly, the absolute BHPs at injectors (4 psi 5 psi) are higher than those at producers (15 psi 3 psi). In practice, matching the higher values is more difficult and results in larger modelling errors. Well D3 is occasionally used to inject gas for local gas storage (Govan et al., 6) with subsequent production of that gas. This results in high and unstable GOR behaviour which is difficult to match in form of WGOR vector the high peaks in its misfit dominate the other well data vectors which unfavourably biases the whole optimization workflow. The gas production of well D3 was incorporated by using the quantity which behaves more smoothly the vector of the total gas production WGPT, taking only one measurement at the end of historic period (1/8/4). Producer D8 and injector N8 do not work during the historical period, so they are not included into the error model. However, these wells will appear in the prediction period, and will be used in the forecasts calculation. For well D4 no BHP measurements were used because of their absence. The seismic part of objective function f is defined by formula (3), with B given by the right hand side of (5). For the considered historic period two seismic monitors are taken: at July and August 4. The seismic attribute we use is the sum of negative amplitudes between the reservoir top and base horizons. The time-lapse attribute maps for the two monitors are shown in Fig. 8. The regression procedure applied to the seismic data uses six (all possible) quadratic terms: P, Sw, Sg, P Sw, P Sg, Sw Sg. For scaling in equation (5), the baseline attribute map is employed (Fig. 6). As mentioned above, 4D seismic is to be compared with the reservoir dynamic parameters averaged over layer T31 only, ignoring the overlying and underlying layers. Furthermore, the water-saturated zone of reservoir T31 is also excluded from consideration because it exhibits a 4D seismic response quite different from the oil-saturated zone due to the different saturation character. As can be seen in historical maps in Fig. 8, the water saturated zone (outside the black contour) shows 19

20 a very weak time-lapse seismic response. Next, the response near injector N4 is different from that near injector N1 where the reservoir softening due to pressure increase is seen. The net pay thickness which affects the timelapse seismic response is similar at the two locations, as can be inferred from the baseline attribute map (Fig. 6). The bottom-hole pressure changes at and 4 for the two wells are also very close, so the only likely explanation for the different 4D signal is the different initial saturations. The seismic covariance matrix C was assessed from analysis of the error between the preliminary history s matched model (with diagonal covariance matrix) and observations. The estimated correlation ellipse parameters are as follows: rotation angle is degrees, both major and minor correlation ranges are equal to 4 m, or.7 grid cells. From the χ criterion outlined in the section above, the standard deviation was estimated as s = 167, while the RMS for the time-lapse attribute data equals 194. s For each of the history matching setups (conventional and seismic) reservoir models from the posterior distribution were generated by the RML approach. Each RML run involved a CMA-ES optimization run with population size λ = 5, and 1 generations (i.e. 5, model evaluations). The starting point for CMA-ES was selected as a uniform random point within the parameter ranges in order to diversify the local minima found. Results. The obtained models were then run for the prediction period staring at 1/8/4 with wells controlled by the actual observed liquid rates. The calculated well data at prediction period: water cuts, GOR s, cumulative production of oil, water, gas, and the bottom-hole pressures were then compared with the corresponding actual observations. The following questions are of interest here: 1. How well is the historical data matched for both HM setups?. How well is the well data at the prediction period forecasted? 3. Does the introduction of the 4D seismic reduce the uncertainties? 4. How well are the observed 4D seismic attributes reproduced by the history matched models (for the seismic HM)? To simplify the analysis, for each HM setup the mean and std of each well data vector were calculated from the models. All the subsequent discussion is done based on these mean and std. In terms of matching the historical data and reproducing the predictions both conventional and seismic HM gave close results, so the following applies equally to both HM cases. For the historical period there were problems in history matching the water cut for well D6, which is likely because of not very flexible parameterisation. For some wells it was somewhat challenging to match the

21 historical GOR s: wells D1, D, D7 (see Fig. 9 for the well D7 GOR). Here is should be noted that matching the wells gas production for a reservoir producing below the bubble point may be a challenging problem, especially if a rather coarse simulation model is used. For wells D5, D6 located at the South-East part of segment 4 the reported historical GOR (.3. mscf/stb) is below the initial reservoir GOR (.35 mscf/stb) used in the original simulation model provided by the operator. For that reason the generated models overestimated the WGOR for these two wells (see Fig. 9 for the well D5 GOR). For the prediction period, both history matching setups could not properly forecast the BHP of well D1. The generated models predicted a notable decrease of the BHP at years 5 8 which is consistent with the increased production rate at that time. Yet, the observed BHP during this period increases, probably because of more complex interaction of the reservoir compartments or activation of a sealing fault. Prediction of water cut for well D1 also failed, because the well does not produce water in the historical period, and the 4D seismic attribute maps do not show any clear water front approaching the well. The challenges with history matching the GOR for a number of wells continued in predicting the future GOR, so the GOR forecast failed for wells D, D4 D7. BHP of injectors N1, N5 was not properly predicted, although the prediction errors were smaller than for the BHP of well D1. The bulk of the other vectors (among WBHP, WWCT, WGOR, WOPT, WWPT, WGOR) for all the wells were history matched reasonably well, and besides showed the behaviour in the prediction period consistent with the observations, which means that the future observations are contained within the estimated mean ± 3 std. An example of such vectors is given in Fig. 9, where the BHP and the water cut of well D are displayed. For the seismic history matching case we can also look at the quality of history match of the 4D attributes. For that, the observed time-lapse attributes (Fig. 8) are to be compared with the modelled attributes, i.e. those calculated from the maps of the dynamic reservoir properties by linear regression (5). The modelled time-lapse attribute maps for the monitors and 4 for one of the models generated by RML in the seismic HM setup are shown in the same figure. These maps reproduce the main features on the input 4D attribute maps: the gas softening signal near producers and pressure-up softening near injectors at and 4. Only very weak waterflood hardening signal can be seen near wells D6 and N. The maps look smoother and have lower amplitude of the values than the observed 4D attribute maps. This is natural since the linear regression is used for the attribute modelling, and it looks at the whole picture and smooths out any inconsistencies between the dependent and independent variables. 1

22 The modelled time-lapse attribute at 6, i.e. the future forecast of the 4D map, does not reproduce the actual observed attribute at 6. The modelled attribute shows considerably more extensive softening signal due to gas exsolution as compared to the observed 4D seismic. This behaviour is explained by misprediction of the BHP at producer D1 and too low reservoir pressure forecasts in the nearby compartments, causing gas to come out of solution in the model predictions. To examine the impact of time-lapse seismic data on the forecast uncertainty, we consider the std ratio for all well data vectors between the two setups, i.e. std / std. This quantity, if it is greater than 1, shows WHM SHM how much the prediction uncertainties reduce after introduction of the 4D seismic data into history matching. The std ratios for all vectors are displayed in Fig. 1. Looking at the std ratio of the water cut vectors at the prediction period, we can notice the uncertainties decrease for the seismic HM case for wells D and D4 which are completed in T31 and for which the 4D seismic maps show the water saturation signal nearby (reservoir hardening, blue colours on the map). The WWCT uncertainties increased for wells D3, D5, D7, D8. All these wells are completed either above reservoir T31, or below it (D5 is completed above, below and in T31). The 4D seismic attributes, on the other hand, were calculated only for T31 interval, and allow constraining the fluid flow essentially for this reservoir. So, the seismic HM setup provided tighter control over the wells completed only in T31, while the control over the other wells was somewhat loosened. This situation will also be observed for some other vectors related to the wells completed in T31, or outside T31. The WGOR vectors generally show uncertainty decrease for most wells. This is consistent with the fact that the historic 4D seismic maps show a lot of reservoir softening signals due to the gas coming out of solution. Thus, the 4D seismic data allows better constraining the model behaviour with respect to the gas production. Looking at the std ratio of the producers WBHP, we can see that it became worse for wells D and D3. The 4D signature near well D is quite complex as one half of the well shows reservoir hardening signal from the encroached water, the other half shows the reservoir softening due to gas. The pressure signal most likely was totally masked by these two, and pressure prediction became worse. Well D3 is completed above T31 layer, so it is not covered by the considered 4D seismic attributes. Reduction of uncertainties in the WBHP forecast can be seen for wells D4, D8, and more notably for wells D5, D6. The latter two wells are located in the South-West part of segment 4, and the 4D seismic indicates reservoir hardening due to water saturation increase in the vicinity of these wells. So, absence of the softening signal in that area was probably beneficial for constraining the pressure behaviour of the producers.

23 The injectors WBHP shows uncertainty increase for wells N1 and N. For well N1 the conventional WHM case resulted in a smaller std, but the forecast did not reproduce properly the actual BHP observations. The seismic HM also did not reproduce the future observations, but gave larger std, which means that it made a more reasonable estimate of the future BHP behaviour (see also Fig. 11 and the relevant discussion below). Well N is completed both in T31 and above it. The improvement of the WBHP uncertainty of injectors can be seen for wells N4, N8, and especially for wells N5, N6, N7. Uncertainty decrease for N5 and N7 is natural since the 4D maps show notable pressure-up signal for these wells (see Fig. 8). When examining the uncertainties of the forecasted well data not only should we look at the decrease or increase of the standard deviations between two history matching cases, but also at the precision of the forecasts. * Consider some predicted well data vector d, and denote by d its actual observations (true values), d the mean estimated mean, σ the estimated standard deviation. The std assessed by a HM setup can be regarded as a promise of the HM setup that the forecasted error e = d * d will not exceed 3 σ (or other reasonable mean multiple of σ ). If the HM case promised smaller std but did not deliver on that promise by producing a large prediction error, then the uncertainties estimated cannot be trusted, and it makes little point to report any uncertainty reduction. Thus, for the two HM setups considered in this work, apart from checking how did the std s change after introducing 4D seismic into history matching, we also examined how both HM setups deliver on their promises. For that, we calculated standard deviation σ. The quantity smaller it is, the more the estimate The crossplots of e / σ for each predicted well data vector and plotted it versus the e / σ shows the un-reliability (or normalized error) of the forecasts, the * d is consistent with the true data mean d for the given σ. e / σ versus σ showed the following (see Fig. 11 for the examples): 1. e / σ is generally less than 3, except for the few well data vectors where the prediction obviously failed, as discussed above. This is the quantitative confirmation of the statement made before that the majority of the vectors reasonably predicted the actual future observations.. For the majority of non-cumulative vectors (WWCT, WGOR, WBHP) the un-reliability of the forecasts e / σ for the two HM cases is approximately the same, whereas the uncertainty σ may differ. This simplifies comparison of the HM cases as essentially σ s should be compared. To give an example, Fig. 11 shows the crossplot for vector D4 WWCT, for which the seismic HM resulted in lower uncertainties, whereas the forecasts un-reliability e / σ basically did not change. A 3

24 counterexample is provided in the same figure by the crossplot of N1 WBHP discussed above, for which seismic HM produced higher std, but at the same time lower forecasts un-reliability. To conclude on the history matching conducted for segment 4 of the UKSC field, we have obtained the history matched models which both reasonably match the historic data and provide a sensible forecast for the future production. Introduction of the 4D seismic data has resulted in reduction of the forecast uncertainties where it might be naturally expected i.e. for the wells completed in T31 reservoir for which there exist notable 4D seismic signals nearby associated with water saturation increase or pressure increase. The gas signal widely present in the time-lapse seismic maps helped in reducing the overall gas production uncertainty, however accurate prediction of the gas production itself remained the unresolved challenge for both conventional and seismic history matching cases. Conclusions We developed a novel seismic history matching procedure which allows avoiding the classical petro-elastic and seismic trace modelling. The procedure relies on the linear data-driven relationships between the 4D seismic data and the reservoir dynamic parameters that were previously suggested in the literature for the problems of pressure and saturation inversion from the time-lapse seismic attributes. The relationships were adapted to the history matching setting, and two enhancing modifications were proposed: multiplication by a scaling map to account for the lateral reservoir heterogeneities, and introduction of the quadratic terms to the relationships to increase flexibility of the modelled seismic responses. The seismic history matching procedure was tested on a simple synthetic model where it showed decrease of the forecast uncertainties for the seismic history matching case compared to the conventional well history matching. This uncertainty decrease demonstrates the value of 4D seismic in constraining the reservoir simulation model, and proves validity of the proposed seismic history matching algorithm. Furthermore, it was established that proper definition of the seismic covariance matrix in the objective function is essential for obtaining accurate predictions. The procedure was further tested on a segment of the UKSC field, considering two major cases: conventional well history matching and seismic history matching. Both cases had a reasonable performance in reproducing the observed data at the historical period and at the subsequent prediction period. Comparing the seismic case with the well case at the prediction period, some uncertainty decrease was observed for the wells which have notable 4D seismic signals nearby (the signals corresponding to the reservoir softening from gas breakout or pressure increase, and reservoir hardening due to water encroachment). This behaviour is 4

25 qualitatively in line with the expectations, and confirms the working potential of the proposed seismic history matching method. Nomenclature A = baseline seismic attribute (map) a P, asw, asg = constant coefficients c = autocovariance function C = covariance matrix d = data vector e = error f 1, f = wells and seismic parts of objective function g (m) = forward modelling (simulator) h = thickness, m k, = vertical and horizontal permeabilities, md v k h L = likelihood function P = pressure, psi R = correlation matrix R s = solution gas oil ratio, mscf/stb r = residue; error t = time, days A = time-lapse seismic attribute (map) P = time-lapse average pressure (map), psi Sw = time-lapse average water saturation (map) Sg = time-lapse average gas saturation (map) σ = standard deviation ϕ = porosity Acknowledgements We thank sponsors of the Edinburgh Time Lapse Project Phase V (BG, BP, CGG, Chevron, ConocoPhillips, ENI, ExxonMobil, Hess, Ikon Science, Landmark, Maersk, Nexen, Norsar, Petoro, Petrobras, RSI, Shell, Statoil, Suncor, Taqa, TGS and Total) for supporting this research. References Aanonsen, S. I., Aavatsmark, I., Barkve, T., Cominelli, A., Gonard, R., Gosselin, O., Kolasinski, M., and Reme, H. 3. Effect of Scale Dependent Data Correlations in an Integrated History Matching Loop Combining Production Data and 4D Seismic Data. Presented at the SPE Conference. SPE MS. Alvarez, E., and Macbeth, C. 14. An insightful parametrization for the flatlander's interpretation of timelapsed seismic data. Geophysical Prospecting 6(1):

26 Amini, H. 14. A pragmatic approach to simulator to seismic modelling for 4D seismic interpretation. PhD thesis, Heriot-Watt University. Corey, A The Interrelation between Gas and Oil Relative Permeability. Producers Monthly 19. Davolio, A., Maschio, C., and Schiozer, D. J. 11. Incorporating 4D Seismic Attributes Into History Matching Process Through An Inversion Scheme. Presented at the SPE Conference. SPE MS. Dobbyn, A., and Marsh, M. 1. Material Balance: A Powerful Tool for Understanding The Early Performance of The Schiehallion Field. Presented at the SPE Conference. SPE MS. Domenico, S. N Effect of water saturation on seismic reflectivity of sand reservoirs encased in shale. Geophysics 39(6): Emerick, A. A., Moraes, R., and Rodrigues, J. 7. History Matching 4D Seismic Data with Efficient Gradient Based Methods. Presented at the SPE Conference. SPE MS. Falahat, R., Obidegwu, D., Shams, A., and MacBeth, C. 14. The interpretation of amplitude changes in 4D seismic data arising from gas exsolution and dissolution. Petroleum Geoscience (3): Falahat, R., Shams, A., and Macbeth, C. 13. Adaptive scaling for an enhanced dynamic interpretation of 4D seismic data. Geophysical Prospecting 61(1): Floricich, M., MacBeth, C., Stammeijer, J., Staples, R., Evans, A., and Dijksman, C. 6. A new technique for pressure - saturation separation from time-lapse seismic - Schiehallion case study. Presented at the 68th EAGE Conference and Exhibition, incorporating SPE EUROPEC, Vienna, Austria, June 1-June 15. Gao, G., Zafari, M., and Reynolds, A. C. 6. Quantifying Uncertainty for the PUNQ-S3 Problem in a Bayesian Setting With RML and EnKF. SPE Journal 11(4): SPE-9334-PA. Ghedan, S. G. 7. Dynamic Rock Types for Generating Reliable and Consistent Saturation Functions for Simulation Models. Presented at the SPE Conference. SPE MS. Gosselin, O., Aanonsen, S. I., Aavatsmark, I., Cominelli, A., Gonard, R., Kolasinski, M., Ferdinandi, F., Kovacic, L., and Neylon, K. 3. History Matching Using Time-lapse Seismic (HUTS). Presented at the SPE Conference. SPE MS. Gosselin, O., van den Berg, S., and Cominelli, A. 1. Integrated History-Matching of Production and 4D Seismic Data. Presented at the SPE Conference. SPE MS. Govan, A. H., Primmer, T., Douglas, C. C., Moodie, N., Davies, M., and Nieuwland, F. 6. Reservoir Management in a Deepwater Subsea Field - The Schiehallion Experience. SPE Reservoir Evaluation and Engineering 9(4): SPE-9661-PA. 6

27 Hansen, N., and Ostermeier, A Adapting arbitrary normal mutation distributions in evolution strategies: the covariance matrix adaptation. Presented at the IEEE International Conference on Evolutionary Computation, Nagoya, Japan, May - May. Hansen, N., and Ostermeier, A. 1. Completely derandomized self-adaptation in evolution strategies. Evolutionary computation 9(): Huang, X., Meister, L., and Workman, R Reservoir Characterization by Integration of Time-lapse Seismic and Production Data. Presented at the SPE Conference. SPE MS. Jin, L., Alpak, F. O., van den Hoek, P., Pirmez, C., Fehintola, T., Tendo, F., and Olaniyan, E. E. 11. A Comparison of Stochastic Data-Integration Algorithms for the Joint History Matching of Production and Time-Lapse Seismic Data. Presented at the SPE Conference. SPE MS. Johnston, D. H. 13. Practical applications of time-lapse seismic data. Kjelstadli, R. M., Lane, H. S., Johnson, D. T., Barkved, O. I., Buer, K., and Kristiansen, T. G. 5. Quantitative History Match of 4D Seismic Response and Production Data in the Valhall Field. Presented at the SPE Conference. SPE MS. Kragh, E., and Christie, P.. Seismic repeatability, normalized rms, and predictability. The Leading Edge 1(7): Lancaster, S., and Whitcombe, D.. Fast-track 'coloured' inversion. Presented at the SEG Conference. Landa, J. L., and Horne, R. N A Procedure to Integrate Well Test Data, Reservoir Performance History and 4-D Seismic Information into a Reservoir Description. Presented at the SPE Conference. SPE MS. Landro, M. 1. Discrimination between pressure and fluid saturation changes from time-lapse seismic data. Geophysics 66(3): Leach, H., Herbert, N., Los, A., and Smith, R The Schiehallion development. Presented at the Petroleum Geology Conference. Liu, N., and Oliver, D. S. 3. Evaluation of Monte Carlo Methods for Assessing Uncertainty. SPE Journal 8(): SPE PA. Lomeland, F., Ebeltoft, E., and Thomas, W. H. 5. A new versatile relative permeability correlation. Presented at the International Symposium of the Society of Core Analysts, Toronto, Canada. Lumley, D., Adams, D., Wright, R., Markus, D., and Cole, S. 8. Seismic monitoring of CO geosequestration: realistic capabilities and limitations. Presented at the SEG, Las Vegas. 7

28 MacBeth, C., Floricich, M., and Soldo, J. 6. Going quantitative with 4D seismic analysis. Geophysical Prospecting 54(3): Meadows, M. A. 1. Enhancements to Landro s method for separating time-lapse pressure and saturation changes. Presented at the SEG International Exposition and Annual Meeting, San Antonio. Oliver, D. S., Reynolds, A. C., and Liu, N. 8. Inverse theory for petroleum reservoir characterization and history matching. Cambridge: Cambridge University Press. Reiso, E., Haver, M. C., and Aga, M. 5. Integrated workflow for quantitative use of time-lapse seismic data in history matching: A North Sea field case. SPE MS. Richardson, S. M., Herbert, N., and Leach, H. M How Well Connected Is the Schiehallion Reservoir? SPE-3856-MS. Roggero, F., Ding, D. Y., Berthet, P., Lerat, O., Cap, J., and Schreiber, P.-E. 7. Matching of Production History and 4D Seismic Data - Application to the Girassol Field, Offshore Angola. SPE-1999-MS. Schlumberger. 1. SimOpt user guide 1.. Tarantola, A. 5. Inverse problem theory and methods for model parameter estimation: SIAM. Waggoner, J. R., Cominelli, A., and Seymour, R. H.. Improved Reservoir Modeling With Time-Lapse Seismic in a Gulf of Mexico Gas Condensate Reservoir. SPE MS. Author Biographies Ilya Fursov is a PhD student at Edinburgh Time-Lapse Project, Heriot-Watt University. His current research is on the different methods of integration of time-lapse seismic data and reservoir fluid flow modelling, including seismic history matching approaches. Fursov holds MSc in applied mathematics from Novosibirsk State University and MSc in petroleum engineering from Heriot-Watt University. Colin MacBeth 8

29 Figures, Tables Fig. 1 3D properties of the synthetic model: permeability, initial water saturation (Sw), porosity, NTG, initial gas saturation (Sgas). The injector is on the left, the producer is on the right. 9

30 Fig. Map of ϕ h NTG, baseline attribute map (A ), time-lapse (monitor at 5 days minus baseline) attribute map (ΔA), and the corresponding time-lapse average maps of pressure (ΔP), water saturation (ΔSw), gas saturation (ΔSg). 3

31 Fig. 3 Estimated D covariace map for the synthetic model seismic error. The major and minor axes of the ellipse shown by the black lines. The outer thick contour line corresponds to zero value on the map. 31

32 Fig. 4 Time-lapse seismic attribute maps for the synthetic model: the noiseless input map (top), output map for an RML-matched model of setup # (middle), output map for an RML-matched model of setup #3 (bottom). 3

33 sm3/day Setup #1, WOPR, sm 3 /day time, days sm3/day Setup #1, WWPR, sm 3 /day time, days sm3/day Setup #1, WWIR, sm 3 /day time, days sm3/day Setup #, WOPR, sm 3 /day time, days sm3/day Setup #, WWPR, sm 3 /day time, days sm3/day Setup #, WWIR, sm 3 /day time, days sm3/day Setup #3, WOPR, sm 3 /day sm3/day Setup #3, WWPR, sm 3 /day sm3/day Setup #3, WWIR, sm 3 /day 1 5 time, days time, days time, days Fig. 5 Plots of oil production rate (left column), water production rate (middle column), water injection rate (right column) for HM setups #1, #, and #3 (rows). Blue circles are the observed data, black line is the mean, red lines show mean ± std, the vertical green line shows the end of the 5 days history. 33

34 Fig. 6 Map of the sum of negative amplitudes attribute for T31 reservoir of the UKSC field, with sands corresponding to red colours, shales blue colours (1996 seismic survey). The field s major segments 1 and 4 are displayed, together with the two major East-West faults (solid black lines). The wells completed in T31 reservoir of segment 4 are also shown. 34

35 Fig. 7 The fine 3D grid of the UKSC field simulation model with seismically-mapped geobodies, showing the top of T31 reservoir. 35

36 Fig. 8 Segment 4 historical and modelled time-lapse attributes (sum of negative amplitudes), top to bottom: 1996 (historical), (historical), 1996 (modelled), (modelled). The modelled maps were produced from a history matched model (seismic HM setup). The oil-saturated zone is outlined with the black solid line. Red colours correspond to the reservoir softening, blue colours correspond to the reservoir hardening. 36