Abstract:
Method for quantifying uncertainties related to continuous and discrete (qualitative) parameters descriptive of a medium such as an underground zone and/or for managing the selection of a scenario from a series of possible scenarios relative to this medium, by construction of experiment designs and results analysis suited to the experiment designs constructed.  
     The method essentially comprises constructing the factorial part of the experiment designs by folding a regular fraction for the quantitative factors and assignment of at least one modality of a qualitative factor to each one of the blocks formed by folding, determining the axial part of the qualitative factors according to a D-optimality criterion (preferably on a limited number of scenarios) and analysing the results by combining a sensitivity analysis and a risk analysis involving marginal models (models adjusted to each scenario) and a global model (model adjusted to all of the scenarios). The method allows for example, in an otherwise uncertain context, to compare different production scenarios (reservoir crossed by open or closed faults, enhanced recovery by water injection or WAG type alternating injection, etc.) in order to better understand the role of non-controllable discrete parameters (status of a fault, etc.) and/or to select the scenario which optimizes production in the case of controllable discrete parameters (well addition, completion levels, water injection or WAG, etc.).  
     Application: reservoir exploration or engineering for example.

Description:
FIELD OF THE INVENTION  
         [0001]    The present invention relates to a method for quantifying uncertainties related to continuous and discrete parameters descriptive of a medium such as an underground zone and/or for managing the selection of a scenario from a series of possible scenarios relative to this medium, by construction of experiment designs and results analysis suited to the experiment designs constructed.  
           [0002]    The method finds applications notably in the sphere of reservoir exploration or engineering for example, which is the sphere to which it is applied in the description hereafter.  
           [0003]    The availability of increasingly realistic numerical models of complex flows facilitates simulation and therefore opens up the way for controlled management of the field development and production schemes. However, although many data are obtained on the reservoir by different means (geology, geophysics, local measurements in wells, well testing, etc.), there still are many uncertain simulator input parameters and a large number of exploitation possibilities. Let us take the example of a reservoir consisting of six layers for which a high uncertainty remains on the imperviousness of layer  4 . To produce this reservoir, one proposes to place a producing well at the top of the reservoir, but one hesitates to drill just below layer  4  in order to drain the lower part of the reservoir at the risk of favouring an early water influx. A statistical formalism and the carrying out of experiments (simulations for different values of uncertain parameters, for example between 0.01 mD and 10 mD for the vertical permeability of layer  4 , and for various production scenarios, for example drilling or not below layer  4 ) according to an experiment design allows to compare production scenarios and to improve knowledge of the reservoir while avoiding excessive and redundant use of simulations of the numerical flow model.  
           [0004]    The goal is thus to provide reservoir engineers with a methodology allowing to compare various production scenarios (reservoir crossed by open or closed faults, enhanced recovery by water injection or WAG type alternating injection, etc.), notably upon quantification of the uncertainties in reservoir engineering, and more generally upon quantification of the uncertainties in exploration-production of oil reservoirs. Comparing scenarios, in an otherwise uncertain context, is a necessary stage:  
           [0005]    on the one hand to better understand the role of the non-controllable uncertain discrete parameters (status of a fault, etc.), and  
           [0006]    on the other hand to select the scenario that optimizes production in the case of controllable discrete parameters (well addition, completion levels, water injection or WAG, etc.).  
           [0007]    Such a methodology allows to predict the future dynamic behaviour of the field and it allows well-considered adjustment of the decisions made as regards development.  
         BACKGROUND OF THE INVENTION  
         [0008]    The documents mentioned hereafter are representative of the prior art:  
           [0009]    Benoist D., Tourbier Y. and germain-Tourbier S. (1994). Plans d&#39;Expériences: Construction et Analyse. Technique &amp; Documentation-Lavoisier, Paris;  
           [0010]    Cox D. R. (1984). Present Position and Potential Developments: Some Personal Views: Design of Experiments and Regression. J. of Royal Statistical Society, Ser. A, 147, pp.306-315;  
           [0011]    Dejean J. -P., Blanc G. (1999). Managing uncertainties on production predictions using integrated statistical methods. SPE 56696, SPE Annual Technical Conference and Exhibition, Houston, Oct. 3-6, 1999;  
           [0012]    Draper N. R. and John J. A. (1988). Response-Surface Designs for Quantitative and Qualitative Variables. Technometrics, 30(4), pp.423-428;  
           [0013]    Droesbeke J. -J., Fine J. and Saporta G. (1997). Plans d&#39;Experiences: Applications à l&#39;Entreprise. Technip, Paris;  
           [0014]    Montgomery D. C. and Peck E. A. (1992). Introduction to Linear Regression Analysis. Wiley Series in Probability and Mathematical Statistics, New York;  
           [0015]    Wu C. F. J. and Ding Y. (1998). Construction of Response Surface Designs for Qualitative and Quantitative Factors. J. of Planning and Inferences, 71, pp.331-348;  
           [0016]    Zabalza I., Dejean J. -P., Collombier D. (1998). Prediction and density estimation of a horizontal well productivity index using generalized linear model. ECMOR VI, Peebles, Sep. 8-11, 1998;  
           [0017]    Zabalza-Mezghani I. (2000). Analyse statistique et planification d&#39;expérience en ingénierie de réservoir. Thèse de doctorant de 3 ème  cycle, Université de Pau.  
           [0018]    The methods for organizing experiment designs generally aim to best plan the experiments or tests to be carried out so as to establish relations between various causes or factors (here the permeability, the porosity, interactions, etc.) and the responses studied (here the cumulative volume of oil, the water cut, etc.) and to derive, if possible, predictive models. In the description hereafter, the analytical (conventionally polynomial) model resulting from the adjustment of experimental results is referred to as &lt;&lt;response surface&gt;&gt;. These methods generally comprise the construction of experiment designs that have to be performed to establish these relations and an analysis of the results.  
           [0019]    Various studies have been carried out by Dejean J. -P. et al. (1999), Zabalza I. (1998) and Zabalza-Mezghani I. (2000) in order to quantify the uncertainties on physical parameters of underground hydrocarbon reservoirs such as the porosity, the permeability, the position of the well, the drilled well length, the structure of the heterogeneities by geostatistical modelling, etc., which use the experiment design method and statistical methods.  
           [0020]    Although the method resulting from these studies allows to deal with the continuous physical parameters (quantitative factors), it allows to take account of discrete parameters (qualitative factors) such as the status of a fault in the reservoir for example only by repeating the same experiment design as many times as there are scenarios to be compared. The simulation cost then quickly becomes prohibitive. Furthermore, the studies being carried out separately on each scenario, it is impossible to take account of the effect due to the discrete parameters and thus to quantify the uncertainty related to the scenarios. In the text hereafter, the possible states of a discrete parameter are referred to as &lt;&lt;modalities&gt;&gt;. The scenarios result from the combination of the modalities of discrete parameters. For example, a discrete parameter with two modalities and a discrete parameter with three modalities generate six scenarios. The engineer can choose a scenario if the discrete parameters are controllable (completion levels, etc.) or have no action on the scenarios if a discrete parameter is not controllable (status of a fault, etc.).  
           [0021]    The construction of designs integrating both quantitative and qualitative factors has notably been dealt with by Cox (1984). The objectives to be fulfilled by these designs are defined by Draper N. R. et al. (1988). The experiment design construction method defined by Wu C. F. J. et al. (1998) proposes for example fixing for the simulation the quantitative factor levels by means of a composite design (well-known to the man skilled in the art and described in any manual on experiment designs) to which columns representing the qualitative factors are added. These columns are determined from quantitative criteria (optimization of conventional experiment design criteria, D-optimality). In practice, the authors have constructed designs allowing to integrate a single discrete parameter with two modalities. Their construction method, based on the numerical optimization of a quality criterion, rapidly reaches its limits when the number of scenarios increases. In fact, the algorithmic cost of determination of such a design cannot be considered in practice.  
           [0022]    In the prior art, the results obtained by applying experiment designs are analysed according to a conventional scheme notably described by Benoist D. et al. (1994) or by Droesbeke J. -J. et al. (1997). This method is perfectly suited to the quantitative factors but it does not allow to fully deal with experiment designs involving quantitative and qualitative factors. There is in fact a loss of information which we think should be exploited, notably within the context of uncertainties quantification in reservoir engineering.  
           [0023]    Conventional analysis of experiment design results cannot be applied in this context, for two main reasons. First, the designs proposed allow to adjust not only one but several distinct models (global model including the various scenarios and marginal models for each scenario). The model(s) that will provide the most information during the result analysis stages therefore have to be determined. On the other hand, it is necessary to fully understand the role of the discrete parameters in the global model (simple effect and interactions of the qualitative factors, repercussion on the response).  
           [0024]    In the context defined above, the main quality required for economical experiment designs integrating both quantitative and qualitative factors is to have the necessary properties for good adjustment of the response surface to the quantitative factors. The structure of the conventional composite design is suited to this type of problem. However, it cannot be applied as it is when there are qualitative factors. In fact, it requires fixing five levels (arranged on a ratio scale) for each factor, which is not possible with the modalities of the qualitative factors (for example two modalities for the status of a fault, open or closed).  
         SUMMARY OF THE INVENTION  
         [0025]    The method according to the invention allows to quantify uncertainties related to continuous and discrete parameters descriptive of an underground zone and/or to manage the selection of a scenario from a series of scenarios, by construction of experiment designs comprising a factorial part, a central part and an axial part, which take account of quantitative and qualitative factors, and a results analysis suited to the experiment designs constructed, characterized in that  
           [0026]    the factorial part of the experiment designs is constructed by folding a factorial design fraction for the quantitative factors and assignment of at least one modality of a qualitative factor to each block formed by folding, and  
           [0027]    the results are analysed by combining a sensitivity analysis and a risk analysis involving marginal models and a global model.  
           [0028]    A D-optimality criterion is for example used to determine the axial part of the qualitative factors.  
           [0029]    The method according to the invention is advantageous in that it saves repeating as many experiment designs as there are scenarios to be compared and it therefore allows to notably reduce the number of simulations to be carried out. Since the studies are carried out jointly on all the scenarios, the method really takes account of the effect due to the discrete parameters.  
           [0030]    More explicitly, the technique of folding a regular fraction for the quantitative factors is advantageous because it allows to eliminate certain aliases of the original regular fraction and it thus allows to introduce additional interactions between quantitative factors in the global model. Assignment thereafter of a modality of a qualitative factor to each block formed by folding allows on the one hand to manage the aliases between the qualitative factors and the quantitative factors, and on the other hand to know the aliases (of the original fraction) on each modality of the qualitative factors.  
           [0031]    This technique has for example allowed to construct so far several ten experiment designs integrating 2 to 8 continuous parameters while taking into account either a discrete parameter with two modalities (2 scenarios), or a discrete parameter with three modalities (3 scenarios), or two discrete parameters with two modalities (4 scenarios), or a discrete parameter with two modalities and a discrete parameter with three modalities (6 scenarios).  
           [0032]    The construction method according to the invention can be extended to designs taking account of a larger number of quantitative and qualitative factors with more than three modalities. In relation to the prior methods, it allows to better manage the aliases on the factorial part by means of the folding technique and to integrate several qualitative factors with two or three modalities while reducing the numerical optimization cost of the D-optimality since, unlike Wu and Ding&#39;s technique (1998), it is carried out only on the axial part of the qualitative factors and for a limited number of scenarios.  
           [0033]    According to an embodiment of the method, a sensitivity analysis is carried out by means of marginal models to detect the terms or actions (simple effects and interactions of the factors) which influence each scenario and a sensitivity analysis is carried out by means of the global model to detect the terms that globally influence all of the scenarios. This use of marginal models allows to obtain substantially the same results in terms of detected actions as the prior methods using a composite design on each scenario. Furthermore, using the global model provides additional information (effects of the discrete parameters and a richer model on the continuous parameters), and at a lower cost (observed reduction of the order of 20% in relation to the prior methods).  
           [0034]    According to an embodiment of the method, the risk is analysed by localized prediction of a response in form of a prediction interval for a set of fixed values of said parameters, by means of the global model, which leads to a lower cost in relation to the methods wherein a composite design is constructed for each scenario.  
           [0035]    According to an embodiment of the method, the risk is analysed by predicting responses from a large number of sets of values of said parameters, randomly selected in their variation range.  
           [0036]    Preferably, the global model is used when the or each discrete parameter is not controllable (status of a fault for example, which can be open or closed).  
           [0037]    When the or each discrete parameter is controllable, the global model is preferably used if it detects a very influential quantitative-quantitative interaction (in terms of concrete result in the application selected) which cannot be detected by the marginal models; in the opposite case, the marginal models are used.  
           [0038]    A consequence of this embodiment of the method is that it fines down the sensitivity analysis while removing the doubts about the respective influence of the quantitative-quantitative interactions and the aliased quantitative-qualitative interactions.  
           [0039]    The method finds applications for example in reservoir exploration and engineering. 
       
    
    
     BRIEF DESCRIPTION OF THE FIGURES  
       [0040]    Other features and advantages of the method according to the invention will be clear from reading the description hereafter of embodiments given by way of non limitative example, with reference to the accompanying drawings wherein:  
         [0041]    [0041]FIG. 1 illustrates a case of validation of a scenario concerning a reservoir model,  
         [0042]    [0042]FIG. 2 is a chart of the aliases of the experiment designs tested in the validation case where the shaded boxes show that the terms of the model cannot be estimated and the white boxes show that the terms of the model are estimated independently of the other terms (no alias),  
         [0043]    [0043]FIGS. 3A, 3B show, in a Pareto diagram, the results of a sensitivity analysis of a cumulative volume of oil showing the marginal respective roles of the marginal models (FIG. 3A) and of the global model (FIG. 3B), and a comparison with the results obtained by means of a prior method,  
         [0044]    [0044]FIG. 4 shows the results of a risk analysis (localized prediction of a GOR) and a comparison of the results obtained from the global model, the marginal models and the prior method,  
         [0045]    [0045]FIGS. 5A, 5B respectively show the results of a risk analysis (Monte Carlo type prediction) on a cumulative volume of produced oil (FIG. 5A) and on a GORn (FIG. 5B), with quantification of the uncertainty on the discrete and continuous parameters, and comparison of the results obtained from the global model, the marginal models and the prior method,  
         [0046]    [0046]FIG. 6 shows the results of a risk analysis (Monte Carlo type prediction) obtained on two different scenarios, with analysis of the global model, and the influence of the simple effects and of the quantitative-qualitative interactions on the density,  
         [0047]    [0047]FIG. 7 shows the results of a risk analysis (Monte Carlo type prediction) with an illustration of the methodology concerning the modification of the global model,  
         [0048]    [0048]FIG. 8 shows the result of a Monte Carlo type 3-year prediction of the cumulative volume of oil according to various scenarios (density vs. GOR),  
         [0049]    [0049]FIGS. 9A to  9 D show the result of a Monte Carlo type 3-year prediction of the GORn according to various scenarios (density vs. GOR), and  
         [0050]    [0050]FIG. 10 shows a conventional structure of a composite design modified to take account of qualitative factors.  
     
    
     DETAILED DESCRIPTION  
       [0051]    The factors dealt with by experiment designs are usually quantitative, i.e. they have an entirely defined measurement scale (porosity (%), permeability (mD), etc.). However, one may occasionally wish to introduce qualitative factors in the modelling so as to study the influence of various scenarios on the simulator response (status of a fault, well addition in a new zone to be drained, etc.). These factors have the specific feature of taking a fixed number of states called modalities (open or closed fault, addition of 2, 5 or 7 wells) that cannot be quantitatively compared with one another. They have no measurement scale. It is therefore necessary to code these states using variables indicative of the &lt;&lt;presence-absence&gt;&gt; of the modalities of the qualitative factors. They are numerical and can thus be introduced in the model.  
         [0052]    Concerning the introduction of discrete parameters in the model and the necessity of coding these discrete parameters, one may refer to the books by Montgomery and Pecks (1992) or Benoist et al. (1994).  
         [0053]    I) Construction of the Designs  
         [0054]    The goal of this first part is to construct economical experiment designs integrating both quantitative and qualitative factors. In the present context, the main quality required for these designs is to have the necessary properties allowing good response surface adjustment to the quantitative factors. Faced with such a problem, the user can consider two approaches. The first one consists in not taking account of the two types of parameters and in applying a composite design to all the factors. This solution requires 5 (or 3) levels (arranged on a ratio scale) for each factor, which is not always possible for qualitative factors. The second approach consists in using a composite design on each scenario defined by the modalities of the qualitative factors. It is in fact this second solution which has been priorly used (i.e. repetition of designs) because it allows to obtain a quality response surface thanks to the properties of the composite designs. It can however become very costly.  
         [0055]    The problem of taking account of the qualitative factors in designs for response surface was raised by Cox (1984). The first to take a real interest in the question are Draper and John (1988). They notably discuss the relations between design and model in order to define desirable and reasonable goals as regards the designs. The construction method developed in this patent is aimed to reach these goals described hereunder.  
         [0056]    Goals and Model  
         [0057]    The quantitative factors are denoted by x 1 , . . . ,x d  and the qualitative factors by z 1 , . . . ,z n .  
         [0058]    (H1) The design has to efficiently adjust an order two global model of the type  
                                                   mean   quantitative   qualitative   quant.-quant.   quant.-qual.   quadratic       eff.   simple eff.   simple eff.   interactions   interactions   eff.                   (1,   x i ,   z p ,   x i x j ,   x i z p ,   x i   2 )                  
 
         [0059]    i=1, . . . ,d and p=1, . . . ,m. The maximum determinant criterion is used to compare the designs, Δ= s {square root}{square root over (det( t XX),)} where s is the number of actions in the model.  
         [0060]    (H2) The design is divided in two parts. The first one must allow to adjust an order one global model of the type  
         (1, x i , z p , x i x j , x i z p )  
         [0061]    allowing a sensitivity study. It is therefore necessary to know the alias table of this first part of the design. The second one can be seen as an additional part so as to extend the model to order two.  
         [0062]    (H3) For each combination of the qualitative factors or for each level of a qualitative factor z k , the design must allow to efficiently adjust an order two marginal model of the type  
         (1, x i , x i ,x j , x i   2 )  
         [0063]    Description of the Design and of its Construction  
         [0064]    The construction method developed here takes up the conventional composite design structure which seems well-suited to goals (H1) and (H2). It is however modified in order to integrate the particularities due to the qualitative factors. The design thus consists of the three parts (factorial, central and axial) of the conventional composite design on the quantitative factors, with which are juxtaposed the qualitative factor columns that are naturally also divided in three parts, as shown in FIG. 10.  
         [0065]    Notations z k   f  and z k   a  designate the components of the qualitative factor z k  associated with the factorial and axial parts.  
         [0066]    ⋄ The factorial part on the quantitative factors has 2 levels −1 and 1, and corresponds to a fraction of the complete design of size N f =2 d−r . It has been decided here to use regular fractions to construct the factorial part because they have the property of being defined by r alias and thus facilitate the sensitivity analysis. The factorial part is constructed with the known folding principle which consists in doubling a regular fraction either by repeating the columns or by converting them to their opposites. Thus, after folding, a column C of the initial fraction becomes  
                         
 
         [0067]    This technique allows to simplify the alias structure of the factorial part, or even to go from a fraction of solution R to a solution R+1. The regular fraction is folded as many times as there are scenarios, and a scenario defined by the qualitative factors is associated to each block thus formed. This technique allows, on the one hand, to eliminate aliases on the quantitative-quantitative interactions of the original regular fraction (richer order one global model (H2)), and on the other hand to know the aliases on each scenario (those of the original fraction) (H3).  
         [0068]    ⋄ The central part consists of a point at the centre of the variation range of the quantitative factors for each scenario defined by the qualitative factors. Its size is thus N c =lev 1 x . . . xlev m , where lev k  represents the number of modalities of the kth qualitative factor. It allows to estimate the variability of the response on the quantitative factors and to test the appropriateness of the order one model.  
         [0069]    ⋄ The axial part has 3 levels −α, 0 and α on the quantitative factors. It is of constant size N a =2d, whatever the number of qualitative factors. It allows to estimate the quadratic effects of the quantitative factors. Once the factorial part fixed, the axial part on the qualitative factors, z k   a , is selected in such a way that the design meets hypothesis (H1), i.e. the maximization of the determinant criterion for the model of hypothesis (H1). In general, optimization of this criterion is not performed numerically, unlike Wu and Ding (1998), but it is deduced from the particular form of matrix  t XX, while favouring a limited number of scenarios.  
         [0070]    The total size of the design is N=N f +N c +N a =2 d−r +lev 1 x . . . xlev m +2d.  
         [0071]    The prior method as used by Wu and Ding (1998) mentioned above uses an optimization on the whole of the components of the qualitative factors, which rapidly leads to a high algorithmic cost when the number of scenarios increases. The construction method defined here involves on the contrary an optimization criterion (determinant) only on the axial part of the qualitative factors and on a limited number of scenarios. The optimization range considered in the present method being more reduced, it is possible in return to construct, at an acceptable cost, for example designs for 2 to 8 quantitative factors and for a discrete parameter with two modalities (2 scenarios), a discrete parameter with three modalities (3 scenarios), two discrete parameters with two modalities (4 scenarios) or a discrete parameter with two modalities and a discrete parameter with three modalities (6 scenarios). This construction technique can be extended to designs that take account of a larger number of quantitative and qualitative factors with more than three modalities.  
         [0072]    II) Analysis of the Results  
         [0073]    The conventional analysis of experiment design results cannot be applied in this context for two main reasons. First, the designs allow adjustment of not only one but of several distinct models (global model to the various scenarios and models for each scenario). The model(s) that will provide the most information therefore has to be determined to analyse the results. Besides, it is necessary to fully understand the role fulfilled by the discrete parameters in the global model (simple effect and interactions of the qualitative factors, repercussion on the response). We provide here a methodology which answers these questions. It is established according to two main lines: sensitivity analysis and risk analysis.  
         [0074]    Sensitivity Analysis  
         [0075]    The goal of sensitivity analysis is to detect the terms of the model (simple effects, quantitative-quantitative interactions, quantitative-qualitative interactions) that influence the response. For this first stage, the factorial part of the design is used to adjust models without quadratic terms.  
         [0076]    Model Selection  
         [0077]    A complete sensitivity analysis requires simultaneous use of the global and marginal models.  
         [0078]    The marginal models allow to detect the terms or actions that influence each scenario. These models are often quite poor but they allow at the minimum an analysis of the simple effects of the quantitative factors (FIG. 2).  
         [0079]    The global model detects the influential terms, all scenarios being taken into account. It may be noted that an action can be considered to be negligible by the global model whereas it is very influential but on only one scenario (FIG. 3). In fact, the global model proceeds in a way by averaging the influence on all the scenarios, hence the importance of the marginal models. The global model has two main qualities:  
         [0080]    It allows to detect if a discrete parameter is influential via its simple effect. In other words, if there is a discrepancy in the average response from one scenario to the other (FIG. 6). Furthermore, if permitted by the design, it allows to detect influential quantitative-qualitative interactions, i.e. to take account of the difference in the behaviour of the simple effects of the quantitative factors between the scenarios (FIG. 6).  
         [0081]    The factorial part of the design has been constructed to enrich the global model in relation to the marginal models. It therefore allows to detect many more globally influential actions (FIG. 3).  
         [0082]    Comparison with the prior method (repetition of a composite design on each scenario) (FIG. 3)  
         [0083]    1) The marginal models give results that are in accordance with those of the prior method (same terms detected).  
         [0084]    2) The global model provides additional information (effects of the discrete parameters and richer model on the continuous parameters).  
         [0085]    3) Cost reduction observed: 20% in relation to the prior method.  
         [0086]    Risk Analysis  
         [0087]    Localized Prediction  
         [0088]    Localized prediction consists in predicting the response in form of a prediction interval for a fixed set of parameters. The goal in the end can be to determine a set of parameters that will optimize the response surface. Prediction is performed from a model integrating the quadratic effects adjusted by means of the design including the factorial, central and axial parts.  
         [0089]    Model Selection  
         [0090]    For this type of study, it is recommended to use the global model because it gives more stable results from one scenario to the other. Predictions using marginal models can be more unstable in some cases (FIG. 4) because only a small number of terms can be introduced in these models.  
         [0091]    Comparison With the Prior Method  
         [0092]    1) The global model gives results whose quality is equivalent to the results of the prior method (FIG. 4).  
         [0093]    2) For a globally equivalent model quality, the designs that integrate the qualitative factors allow to reduce the number of simulations of the prior method (the example of FIG. 4 shows a 20% reduction from 2 composite designs of size 29, i.e. 58 simulations, to a design with qualitative factors with 46 simulations).  
         [0094]    Monte Carlo Prediction  
         [0095]    The Monte Carlo prediction consists in predicting responses from a large number of sets of parameters randomly selected in their variation range. The density and the quantiles of the responses obtained are then calculated in order to quantify the uncertainty on the continuous and discrete parameters.  
         [0096]    Role of the Discrete-Parameter Terms in the Global Model  
         [0097]    The density obtained from the Monte Carlo predictions can vary from one scenario to the other. The point is then to know which terms of the global model allow to retranscribe this variation. Paragraph 1 of the first part describes the role of the simple effects of the qualitative factors and of the quantitative-qualitative interactions in the model. This role affects the density as follows:  
         [0098]    The simple effect of the discrete parameter(s) of the global model induces a difference in the average response between the scenarios (FIG. 6)  
         [0099]    The quantitative-qualitative interactions of the global model induce the uncertainty difference (forms of the marginal densities) (FIG. 6).  
         [0100]    Modification of the Global Model  
         [0101]    However, all the designs do not allow to introduce the quantitative-qualitative interactions in the model. These interactions are then aliased with quantitative-quantitative interactions (for example design P30 of the validation case). It is therefore important to ensure that the really influential interactions appear in the model among the aliases. For example, the quantitative-qualitative interaction replaces the quantitative-quantitative interaction with which it is aliased if knowledge of the physical phenomenon suggests that the quantitative-quantitative interaction is negligible. In cases where knowledge of the physical phenomenon does not allow to take a decision, the following procedure will be carried out (FIG. 7):  
         [0102]    For each scenario, the densities constructed from the global model and the marginal models are drawn,  
         [0103]    if an uncertainty difference is observed (amplitude and form of the densities) between the global model and the marginal models, one deduces therefrom that there is a lack of quantitative-qualitative interactions in the global model,  
         [0104]    an iterative procedure is then carried out by replacing the quantitative-quantitative interactions of the global model by the quantitative-qualitative interactions with which they are aliased until the behaviour of the uncertainty obtained by the global model is stabilized.  
         [0105]    This technique also allows to fine down the sensitivity analysis by removing the doubt about the respective influence of the aliased quantitative-quantitative and quantitative-qualitative interactions. In fact, if replacing in the global model an influential quantitative-quantitative interaction by the aliased quantitative-qualitative interaction significantly modifies the density, it can be deduced that it is the quantitative-qualitative interaction which influences the response and not the quantitative-quantitative interaction.  
         [0106]    Model Selection  
         [0107]    The model is selected after modifying the global model if necessary.  
         [0108]    In cases where the discrete parameter is not controllable (status of a fault, etc.), the global model allows to correctly quantify the uncertainty on the continuous and discrete parameters. In some cases, the marginal models tend to overestimate the uncertainty interval defined by the quantiles (FIG. 5), considering the small number of terms that can be introduced in the model.  
         [0109]    In cases where the discrete parameter is controllable (well addition, etc.), two situations can arise. Either the global model allows to detect a very influential quantitative-quantitative interaction that cannot be estimated with the marginal models and the global model is used (FIG. 8), or the additional quantitative-quantitative interactions of the global model are detected as negligible and the marginal models are used (FIG. 9). In both situations, quantification of the uncertainty on the continuous parameters for each scenario allows to select the scenario that optimizes production.  
         [0110]    Comparison with the Prior Method  
         [0111]    1) The designs of the present invention give results whose quality is equivalent to those of the prior method (FIGS. 5, 6,  8  and  9 ), but at a lower cost.  
         [0112]    2) The global model allows quantification of the uncertainty on the scenarios and the continuous parameters, which is impossible when carrying out separate studies.  
         [0113]    3) The marginal models allow to remove the doubt on the respective influence of the aliased quantitative-quantitative and quantitative-qualitative interactions.  
         [0114]    III) Validation Case and Figures  
         [0115]    Description of the Validation Case  
         [0116]    We present here a validation case for a design with 6 continuous parameters and a 2-modality discrete parameter.  
         [0117]    For this validation stage, a synthetic reservoir model has been constructed. It consists of 6 layers, as illustrated in FIG. 1. The lower layer is permeable and water-saturated. Layers  2 ,  3 ,  5  and  6  have good reservoir properties whereas layer  4  has less favourable petrophysical properties. Layers  2  to  6  are impregnated with oil. Besides, there is a high uncertainty on the imperviousness of layer  4  due to clay banks, which might prevent any vertical transmissivity between layers  1 - 2 - 3  and  5 - 6 .  
         [0118]    It is assumed that major quantitative type uncertainties remain on the field, notably on the absolute and relative permeability values. For this study, the following 6 uncertain continuous parameters were selected:  
         [0119]    x 1  represents the horizontal permeability of layers  1 ,  2 ,  3 ,  5  and  6 , denoted by Khi Uncertainty range: 700 mD-1300 mD  
         [0120]    x 2  represents the vertical permeability of layer  4 , denoted by Kz4 Uncertainty range: 0.01 mD-10 mD  
         [0121]    x 3  represents the residual oil saturation after sweeping with water, denoted by Sorw Uncertainty range: 0.1-0.3  
         [0122]    x 4  represents the residual oil saturation after sweeping with gas, denoted by Sorg Uncertainty range: 0.1-0.3  
         [0123]    x 5  represents the maximum point of relative water permeability, denoted by Krwm Uncertainty range: 0.2-0.7  
         [0124]    x 6  represents the maximum point of relative water permeability, denoted by Krgm Uncertainty range: 0.1-0.3.  
         [0125]    To produce this reservoir, we propose placing a producing well at the top of the reservoir and an injection well at the aquifer level. The producing well drilling levels remain uncertain. One hesitates to drill just below layer  4  so as to drain the lower part of the reservoir in case of imperviousness of layer  4 , at the risk of favouring an early water influx that would penalize the total productivity of the well. It has therefore been decided to introduce a 2-modality qualitative factor for this completion:  
         [0126]    z represents the completion at layers  5  and  6  (z=+1) or at layers  3 ,  5  and  6  (z=−1).  
         [0127]    The simulator responses studied at different times after production start are  
         [0128]    the cumulative volume of oil (COS),  
         [0129]    the normalized  
         the                 normalized                 GOR                   (       0   ≤   GORn     =         2        Q   G         (       Q   G     +     RS   ×     Q   O         )       &lt;   2       )       ,                         
 
         [0130]    the water cut (FW).  
         [0131]    The technical goals of this study are to:  
         [0132]    identify the influential parameters (continuous and discrete) or the influential interactions via a sensitivity analysis,  
         [0133]    quantify, via a risk analysis, the uncertainty induced by the influential parameters on the production estimates, and  
         [0134]    recommend a choice for the completion type according to the probability densities obtained.  
         [0135]    The Experiment Designs Tested With Their Aliases  
         [0136]    The chart of FIG. 2 shows the alias structures of the designs tested in the validation case with the models that can be adjusted therewith. Designs P30 and P46 integrate a discrete parameter. They have been constructed according to the method of this invention. Designs C42 and C58 correspond to the prior method, i.e. to the repetition of a composite design of size 21 and 29 on each one of the two scenarios.  
         [0137]    A Posteriori Systematic Validation  
         [0138]    In order to validate the results and to compare the prior method with the method developed in this patent, a series of a posteriori simulations has been carried out. We thus have 100 simulations performed in the entire range of uncertainty of the continuous parameters (50 for each scenario), distinct from the simulations carried out at the points of the experiment designs. We shall thus compare the results of these simulations (considered to be the absolute reference) with the predictions provided, on the one hand by means of the designs which are the object of the present invention, and on the other hand by means of the repeated composite designs (current state of the art).  
         [0139]    Sensitivity Analysis  
         [0140]    It can be seen in FIG. 3 that the results obtained with the marginal models of design P46 are of the same nature as those of design C58. The same terms are globally detected on each scenario but with 12 simulations less.  
         [0141]    Furthermore, the global model of design P46 allows to analyse additional quantitative-quantitative interactions (Kz4*sorw and Kz4*sorg are detected influential), as well as the effect of the discrete parameter (z) and of the quantitative-qualitative interactions (sorg*z and Kz4*z are detected influential, an effect expected for Kz4*z).  
         [0142]    Risk Analysis: Localized Prediction  
         [0143]    In FIG. 4, the bars represent the average on the 100 simulations of the relative error in percent:  
         ∑     i   =   1     100            (       y   i     -       y   ^     i       )       y   i                             
 
         [0144]    where y i  is the ith validation simulation and ŷ i  its prediction. The grey areas represent the relative error, all the scenarios being taken into account, and the hatched areas represent the relative error per scenario. The curve represents the number (%) of validation simulations that belong to their prediction interval.  
         [0145]    The relative error allows to appreciate the quality of the prediction, and the percentage of validation simulations in its interval allows to appreciate the quality of the prediction interval. It can be noted that the global model gives more stable results that the marginal models. In fact, the marginal model for z=−1 seems to be of very bad quality (5% relative error and only 20% of the validation simulations in their prediction interval). The global model of design P46 gives results of the same quality as design C58 with 12 simulations less.  
         [0146]    Risk Analysis: Uncertainty Quantification by Monte Carlo Prediction  
         [0147]    The cones of FIG. 5 are determined from the quantiles of the density of the Monte Carlo predictions. They represent the uncertainty on the production (Cos and GORn) due to the continuous and discrete parameters. The points are the a posteriori simulations. It can be observed that all of the designs are of very good quality (80% to 100% of the a posteriori simulations are in the uncertainty interval), even for design P30 which requires 28 simulations less than C58.  
         [0148]    By definition, the normalized GOR cannot exceed value 2, and the upper boundary of the uncertainty interval is often above 2. There is an interval overestimation whatever the design. It can however be noted that the marginal models exaggerate this overestimation, which is why the global model is preferably used.  
         [0149]    Role of the Discrete Parameters in the Model and Modification of the Global Model  
         [0150]    The curves of FIG. 6 represent the densities of the Monte Carlo predictions obtained on each scenario. In the global model, the average response difference between two scenarios is taken into account by an important simple effect of z (26.1% of the total influence of the terms of the model according to the sensitivity analysis). The difference in the behaviour of the quantitative factors between the scenarios is taken into account by interactions x i z (8.4% of the total influence of the terms of the model) which are integrated in the global model for design P46.  
         [0151]    The global model of design P30 does not allow to estimate the quantitative-qualitative interactions (see alias table). In its initial form (1), it therefore does not allow to take account of the uncertainty difference that can be observed by means of the marginal models: an amplitude difference can be seen in FIG. 7 in the graph on the right (marginal models), which does not appear in the graph on the left (global model). It is then necessary to return to the Pareto diagram of the sensitivity analysis to find a quantitative-quantitative interaction(s) which logically seems to be negligible and which has however been detected as influential, and to replace it by the quantitative-qualitative interaction to which it is related. For example, for the validation case, it may be assumed that interaction Khi*krgm has physically no reason to be influential, it is therefore replaced by interaction z*Kz4 to which it is related. In fact, it seems obvious that the interaction between the completion level and the permeability of layer  4  will play an important role in the production. After the global model has been modified, an uncertainty behaviour identical to the behaviour obtained by the marginal models can be found.  
         [0152]    This technique can be used in order to fine down the sensitivity analysis. When two terms are aliased, only one appears in the model and therefore in the Pareto diagram. It is then impossible to know (quantitatively) which one of the two terms is actually influential. The densities on each scenario obtained with the global model and the marginal models can then be drawn. If an uncertainty difference is observed between the two of them, a quantitative-quantitative interaction is replaced in the global model by the quantitative-qualitative interaction with which it is aliased. If the uncertainty difference decreases, one may conclude that the quantitative-qualitative interaction is more influential in the model than the quantitative-quantitative interaction and conversely.  
         [0153]    Choosing Between Global Model or Marginal Models for the Monte Carlo Prediction  
         [0154]    The densities obtained by design C58 are considered as references here.  
         [0155]    In the Pareto diagram of FIG. 8, one can see that the global model of design P46 detects very influential quantitative-quantitative interactions that cannot be estimated by the marginal models (Kz4*sorw, Kz4*sorg and sorg*krwm). Therefore, the densities on each scenario obtained by means of the global model of design P46 are more in accordance with those of design C58 than the densities of the marginal models of P46.  
         [0156]    In the Pareto diagram of FIG. 9, it can be seen that the global model of design P46 detects quantitative-quantitative interactions that cannot be estimated by the marginal models (Kz4*sorw, Kz4*sorg), but they are very weakly influential. The densities on each scenario obtained by means of the marginal models of P46 are then more in accordance with those of design C58 than the densities of the global model of P46.  
         [0157]    Conclusion  
         [0158]    Concerning the sensitivity analysis, designs P46 and P30 have allowed to detect substantially the same influential terms as designs C42 and C58 by means of the marginal models. Furthermore, for a lower number of simulations (20% reduction), using the global model of designs P30 and P46 provides additional information by detecting quantitative-quantitative interactions that neither design C42 nor design C58 can estimate, and by analysing the effect due to the completion via the simple effect or the interactions of the qualitative factor.  
         [0159]    Concerning the risk analysis by localized prediction, designs P30 and P46, by means of the global model, give results whose quality is equivalent to the prior designs C42 and C58, at a lower cost.  
         [0160]    Concerning the Monte Carlo type prediction, designs P30 and P46, by means of the global model or of the marginal models, allow both to quantify the uncertainty as well as designs C42 and C58 with less simulations and to remove the doubt about the respective influence of the aliased quantitative-quantitative and quantitative-qualitative interactions (here Khixkrgm and zxKz4).  
         [0161]    The contribution of the designs of this invention and of the methodology developed is therefore significant and allows to improve the results of the prior method while reducing the number of simulations.  
         [0162]    The method has been described in applications to reservoir exploration or engineering. It is clear that it could also be used in other contexts such as, for example, medicine or agronomy.