B. Todd Hoffman and Jef Caers Stanford University, California, USA

Transcription

1 Sequential Simulation under local non-linear constraints: Application to history matching B. Todd Hoffman and Jef Caers Stanford University, California, USA Introduction Sequential simulation has emerged as a robust and fast method for generating stochastic realizations. The recent development of a sequential simulation method by drawing structures from training images allows generating a wide variety of geological styles in a controlled fashion. Various methods for conditioning realizations to secondary information from geophysical or remote sensing techniques have been developed and applied. However, the current state-of-the-art in sequential simulation lacs the ability to include data that has a strongly non-linear and multiple-point relationship with the unnown. A classical example of such data is a subsurface flow response measured in a well, over time, and may consist of pressure, fractional flow or tracer data. Incorporating such data is of crucial importance in building models with fields such as petroleum engineering and hydrology. Most co-riging or bloc-riging methods cannot incorporate such data since they require a linear or pseudo-linear, often single-point relationship, between the target variable and the data. In this paper we present a method for constraining sequential simulation method to such data. The method is an iterative approach that searches efficiently for realizations that match any type of non-linear data. The approach consists of iteratively perturbing the local conditional probability distribution used to simulate outcomes at each grid node. Theory shows that the proposed perturbation method does not destroy the intended reproduction of geological continuity as provided by a variogram model, or by a training image, nor any conditioning by primary or secondary data. The perturbations are regional, in the sense that various regions of the model can be perturbed by different amounts. Such regional approach provides great flexibility in matching a large variety of local non-linear constraints. The number of regions and their shape can be chosen arbitrarily. The magnitude of model perturbation in each region is optimized using a newly developed parallel 1D optimization method. We show the efficiency and effectiveness of this method using a realistic 3D synthetic example. As application example, we discuss the problem of history matching pressure and flow data in oil and gas fields. Sequential simulation: a recall The aim of sequential simulation, as it was originally construed, it to reproduce the properties of a given multi-variate distribution under some conditioning data. Consider a set of N random variables Z(u i ) for any set of N locations u i, not necessarily on a regular grid. Given a multi-variate law defines on the set of N variables, a decomposition is formed F( u, u, K, u,z,z, K,z ) = F( u,z (n N+ 1)) F( u,z (n N+ 2)) 1 2 N 1 2 N N N N 1 N 1 F( u,z (n + 1)) F( u,z (n)) F(u j,z j (n-j+1) is the conditional cdf (ccdf) of Z(u j ) given a set of data values (n) and the previous (j-1) realizations z ( ) (u i ). This decomposition allows to generate a sample from the multivariate law by sequentially visiting, then sampling at each node along a random path. While the framewor provided by the decomposition loos theoretically appealing, it is limited in its practical extent. The only law for which a practical decomposition is available is the multi-variate Gaussian distribution. Each ccdf is in that case estimated using a linear riging technique, yet the resulting realizations lac geological realism as they reflect the maximum entropy property (disconnected extremes) of the multi-gaussian law, constrained only by the variogram. A more ad-hoc approach to the problem is taen, by considering the fact that any product of a series of conditional distribution results in a specific multi-variate law, represented by its realizations. The multivariate law is never explicitly stated, motivated by the fact that in all practical applications the primary interest lies in the realization and the properties/statistics it exhibits. Drawing at each location along the random path from a series of conditional distributions will result in a given realization. The aim of doing so is to reproduce realizations that reproduce a certain statistic or statistics. Some important examples are: 1

2 Sequential indicator simulation (SISIM, Deutsch and Journel, 1998) relies on indicator riging to determine ccdfs and aims at reproducing the variogram of a series of ranges of values, rather than the overall variogram. Single normal equation simulation (SNESIM, Strebelle, 2002) relies on obtaining the ccdfs from a training image by scanning the image of configurations of the data (n-j+1). In this way, the geological heterogeneity of the training image is reproduced. Any of the above sequential simulation method can be conditioned to so-called hard data or exact sample observations or soft data or indirect observations. The latter require to perform some form of coriging during sequential simulation. The method also allows conditioning to volume-average data as long as the averaging is linear or a power average. Next, we develop a method for conditioning any type of sequential simulation method to non-linear data using a method termed probability perturbation (Caers, 2003) in a regionalized manner. Regional Probability Perturbation Conditioning geostatistical realizations to highly non-linear data requires an iterative technique. An initial realization is created and subsequently changed (perturbed) until it honors the new non-linear data. However, if the realization properties are perturbed directly, other data such as geologic continuity may not be honored. As discussed earlier, a convenient property of simulation is that the realizations are generated from probability models, which are now shortly denoted as P(A B), with A the unnown value to be simulated, and B any hard, soft or linear volume average data and previously simulated values. As D we denote any set of non-linear data. For example A could be the hydraulic conductivity at any location, B some well data and ground penetrating radar data and D some tracer test data. Given a random seed and a series of probability models, P(A B), sequential simulation will generate a unique realization. Consider using one such realization as the initial realization. One way to perturb this realization is the probability model, P(A B) and eep the same random seed. The magnitude of the perturbation of P(A B) will necessarily depend on the difference between the data D and the forward evaluation of the physical law the reproduces that data. For demonstration purposes, we will consider only the case of a binary spatial variable described by an indicator random function model 1 if a given event occurs at u I( u ) = 0 else A={i(u) = 1} could mean that sand occurs at location u, while i(u) = 0 indicates non-sand occurrence. The initial realization containing all locations u will be termed i (0) (u). In the case where the initial realization already honors the non-linear data, there should be no perturbation. Conversely, in the case where the realization does not honor the non-linear data, the realization should be perturbed considerably. To achieve a perturbation that is between no change and considerable change, another probability model, P(A D), that depends on the non-linear data, D, is introduced. A perturbation of P(A B is achieved by combining the conditional probabilities P(A B) and P(A D) into a joint probability model, P(A B,D). The combination of two probability models into a new one is achieved with Journel s conditional independence method (2002). This new probability model is used to populate the next realization. P(A D) is defined as follows: P(A D) = (1-r D ) i (0) (u) + r D P(A) [0,1] where P(A) is the marginal distribution. A set of free parameters, {r D1,,r DK } control how much the model is perturbed in a particular region,. Note that K is the total number of regions and indicates a particular region. Region definition is left to the user, and while they may have any arbitrary shape, they may not overlap. The regions of the realization will be denoted {R 1, R 2,, R K }, and the entire realization is R = (R 1 R 2 R K ). P(A D) is defined for the entire realization, R, but its local value depends on the region definition: when u is located in region R, the perturbation parameter taes on a value of r D. Therefore P(A D) can have different values for different regions of the reservoir. To better understand the relationship between r D and P(A D) and how a perturbation of some initial realization is created consider the two limiting cases all r D =1 and (2) all r D =0. When they all equal zero, P(A D) = i (0) (u) and via Journel s method, P(A B,D) = i (0) (u); hence, the initial realization, i (0) (u), is 2

3 retained in its entirety. When they all are equal to one, P(A D) = P(A); therefore, P(A B,D) = P(A B), and a new realization, i (u), is generated that is equally probable to i (0) (u). Therefore the set of parameters, {r D1,,r DK }, define a perturbation of an initial realization to another equiprobable realization. When parameters are not zero or one but values in between (as they usually are), the resulting realizations will be some combination of the two realizations. Figure 1A shows an initial realization of two facies (or categories), and Figure 1B is the map of P(A D). It is generated using Equation with r D1 =0.80 and r D2 =5. By combining this P(A D) with the same P(A B) used to generate the initial realization, a new realizations, i rd ( u), can be generated (Fig. 1C). Since rd1 is close to one, this region is significantly different from the initial. Conversely, since r D2 is close to zero, this region is very similar to the initial realization. Notice that the facies are in the same location and only the shape has slightly changed. Although the two regions perturbations are quite dissimilar, discontinuities along the border of Regions 1 and 2 are not observed. The reason for not creating artifact discontinuities can be explained by the nature of the sequential simulation algorithm and by the application of the perturbation method. In sequential simulation, each location is simulated based on conditioning data and any previously simulated nodes. The method searches for any such previously simulated nodes in an elliptical search neighborhood. This search neighborhood may (and should) cross the region-boundaries. When simulating a nodal location in one region, the nodal values in any other regions are used to determine P(A B,D), hence creating continuity across the boundaries. Secondly, geological continuity is assured in the perturbation method through the probability P(A B), which is not calculated per region but for all regions together. A similar approach was taen in Hu et al. (2001) in conjunction with the gradual deformation methods that have some degrees of similarity with the probability perturbation method. Yet it should be pointed out that the probability perturbation method does not aim at changes the model gradually as this often leads to slow convergence and the method can be applied to any type of variable. Optimize r D. There may exist a set of r D values, such that i rd ( u), will honor the non-linear data better than the initial realization. Finding the optimum realization, i rdopt ( u), is a problem parameterized by the free parameters, {rd1,,r DK }. Finding the optimum realization is equivalent to finding the optimum set of r D values. {rd1,..., rdk} opt = min{o(rd1,..., rdk ) = Σ KO(rD ) } (2) r D where O(r D1,, r DK ) is the overall objective function, and O(r D ) is the regional objective. O(r D ) is calculated for each region and is defined as some measure of difference between the actual data, D and S the data currently included in the realization, D (r ). D O S = D (r ) D :1, 2Κ K (3) D In general the values of O might depend on all {r D1 r DK }. However, the values in a region are mostly dependent on the properties in that region; thus, we will assume, at least explicitly, that O only depends on r D. Based on this assumption, the best r D for each region can be determined using a one-dimensional optimization routine, but since there are multiple regions, the one-dimensional problems must be solved in parallel (at the same time but without explicit reference to each other). Note that the model is not actually broen into K independent problems where each region would be solved separately. Rather, the realizations are always generated over the entire reservoir, and regions are only used for objective function calculations and perturbation parameter, r D, updating. Although the r D values are updated without explicit reference to each other, dependence is implicitly maintained by creating the models over the entire model. Using the parallel 1D search for a best realization between two equiprobable realizations, one does not expect to converge to a global minimum. Thus, a two-loop optimization routine is necessary where the previous optimum realization, i rdopt ( u), is used as the initial realization in the next step, replacing i (0) (u) in Eq.. This constitutes the outer loop of iteration that will be stopped when the error is below a sufficient tolerance. Also during each outer iteration, the random seed is changed. By doing so, a new equiprobable realization is generated when all r D =1. This allows the method to again search between two equiprobable realizations in the next inner iteration. 3

4 Example In order to obtain accurate predictions of flow in oil and gas reservoirs, a model of the subsurface is created. This model needs to be constrained to a wide variety of data such as hard data from well logs, soft data from seismic, geological conceptual models and many types of dynamic (pressure/flow) data observed from past production or well performance tests. This dynamic data from subsurface porous media is often highly non-linear. An example of how to include and honor all of the various types of data is illustrated. The example uses a synthetic 3D fluvial channel reservoir model as the reference. The reference model was created by Mao (1999) and is based on a North Sea reservoir. It is made up of three distinct facies: channel sand, crevasse splays, and mudstone. The locations of the facies were determined by using an object-based simulation technique. The petrophysical properties were generated independently for each facies using Sequential Gaussian Simulation. The model has aerial dimensions of 10,000 ft by 13,000 ft and is 450 ft thic, and it is divided into 100 gridblocs in the x-direction, 130 in the y-direction, and 30 in the z-direction. The z-direction has three major horizons each with 10 layers. The three horizons have channels of different size, different directions, and different sinuosity (Figure 2). In this example nine vertical wells (six production and three injection) were added to the model for flow simulation. A real-life situation of incomplete information is established in order to generate a reservoir simulation model. The SNESIM algorithm is used to create the geostatistical realizations for the model (Strebelle, 2002). The geological parameters required for the realizations differ from the reference truth as follows: The facies proportions are calculated from the well-logs directly. The proportion of channel sand encountered by the wells is 27% and the proportion of crevasse encountered is 6%. Note that this is different than the reference, which has channel and crevasse proportions equal to 41% and 5%, respectively. The difference in the amount of facies also effects the pore volume. While the reference pore volume is 1.76x10 9 barrels, the generated models are from 1.32 x10 9 to 1.44 x10 9 barrels. Although the reference has different channel shape parameters for different layers, the channel shape parameters for the geostatistical realizations are the same for all layers. Two layers of a five-layer training image are shown in Figure 3. The training image was generated with an object-based technique, and each layer is 200 by 200 gridblocs. The proportions for each facies in the training image come directly from the well-log information discussed in the previous bullet. The porosity and permeability are assumed constant for each facies and obtained by averaging porosity and permeability per facies from wells. The channel porosity is 29%, and its permeability equals 456 md. For the crevasse splay, the porosity is 23%, and the permeability is 253 md, and for the mudstone, the porosity equals 7%, and the permeability is 3 md. The matter of how to define regions within the reservoir remains to be discussed. For this example, streamlines are well suited for the job because they show the direct paths by which fluid will enter a production well. Every production well will have a set of streamlines entering that well. All cells hit by this set of streamlines define the drainage zone for that well, and the various drainage zones will define the region geometry. As the model is perturbed, the facies geometry will change; consequently, the drainage area of the production wells also will change. The region definitions for one stage are displayed in Figure 4 Fractional flow information at the six production wells is the non-linear data that is being incorporated into the model. Although the generated models have different channel shapes than the reference, a match with the reference data is consistently achieved. Figure 5 displays a realization that has included the fractional flow data. The reference data along with the initial and matched data are displayed for the six wells (Figure 6). Although, exact matches are not achieved, significant improvement is observed in all wells, and some wells such as Well 3 and Well 6 show dramatic improvement. The matches obtained would be adequate for most true field applications. Conclusions 1. The proposed regional probability perturbation method can efficiently include highly non-linear data while continuing to honor other pieces of data such as geologic continuity. 2. The method is quite general in the sense that any type of sequential simulation method can be used as well as any type of non-linear data as long as the forward model (e.g. a finite element or difference) is nown. 4

5 3. The method has been shown to wor for the practical example of including production data in a petroleum reservoir model. A good history match was attained on a channel reservoir that has 9 wells, 10 years of production and 50,000 gridblocs. References Caers, J.: Geostatistical history matching under training-image based geological model constraints, paper SPE presented 77 th Annual Fall Meeting of SPE in San Antonio, TX, Sept. 29 Oct. 2, see also SPE 74716, to be printed in SPE Journal, Deutsch, C.V and Journel, A.G. (1998), GSLIB: the geostatistical software library. Oxford University Press. Hu, L-Y., Blanc, G., Noetinger, B., Gradual deformation and iterative calibration of sequential simulations. Mathematical Geology, 33, Journel, A. G., Combining Knowledge from Diverse Sources: an Alternative to Traditional Data Independence Hypothesis, Mathematical Geology, v. 34, No. 1, January Mao, S., Generation of a reference petrophysical/seismic data set: the Stanford V reservoir, Stanford Center for Reservoir Forcasting (SCRF) Report No. 12, May Strebelle, S.: Conditioning Simulation of Complex Structures Multiple-Point Statistics Mathematical Geology, v. 34, No. 1, January Initial Realization (A) P(A D) (B) Perturbed Realization (C) 0.5 P(A) r D1 = 0.80 r D2 = 5 Figure 1: Perturbation of initial realization, (A) to get new realization, (C). Perturbation completed by combining initial probability model, P(A B), with new probability model P(A D), (B). Porosity Layer 6 Layer Layer Figure 2: Reference model for 3D example. Porosity is shown and channels can be distinguished. 5

6 I1 I1 200 Gridblocs 200 Gridblocs Figure 3: Two layers from a five-layer training image. Region Geometry Region 1 Layer 1 Layer 15 Region 2 Region 3 Region 4 Region 5 I1 P2 P1 P3 I2 P4 P6 I3 P5 I1 P2 P1 P3 I2 P4 P6 I3 P5 Region 6 Figure 4: Dynamic region geometry for one stage defined using streamlines. Layer 3 Layer 8 65 Layer Figure 5: History matched realization for 3D example. Water Cut Time (days) 1 ref 1 init 1 match 3 ref 3 init 3 match Water Cut Time (days) 2 ref 2 init 2 match 5 ref 5 init 5 match Water Cut Time (days) 4 ref 4 init 4 match 6 ref 6 init 6 match Figure 6: Water cut reference, initial match and final match of six production wells for one realization. 6