METHOD OF DETERMINING A MAP OF HEIGHT OF LIQUID HYDROCARBON IN A RESERVOIR

Abstract

This invention relates to a method for determining a map of expectation and/or of variance of height of liquid hydrocarbon in a geological model. The method allows analytical resolution of these maps while taking account of the uncertainties in the variables allowing this calculation such as the porosity or oil saturation of the rock and also the uncertainty in the presence of certain types of facies given by the apportionment cubes of architectural elements p.sub.AE(c) and proportion cubes for each facies, these proportions then having a triangular distribution defined by the following three values p.sub.A,AE,min(c), p.sub.A,AE,max(c) and p.sub.A,AE,mode(c). In particular, the method comprises the calculation of the sum of the plurality of architectural elements of p.sub.AE(c).Math.(p.sub.A,AE,min(c)+p.sub.A,AE,max(c)+p.sub.A,AE,mode(c)) and the determining of a value of expectation of height of liquid hydrocarbon for said column as a function of the sum determined.

Claims

1. A method for determining a map of expectation of height of liquid hydrocarbon in a geological model, the geological model comprising meshes that can be associated with a facies among a plurality of facies and with an architectural element among a plurality of architectural elements, the method comprises, for at least one column of the geological model: /a/ receiving an apportionment cube for each architectural element AE among the plurality of architectural elements, with the apportionment cube describing for each mesh c of the model the probability p.sub.AE(c) that said mesh belongs to said architectural element, /b/ receiving a proportion cube for each facies among the plurality of facies and for each architectural element among the plurality of architectural elements, with the proportion cube associated with a facies A and with an architectural element AE describing for each mesh c of the model the proportion of said facies in said mesh if said mesh belongs to said architectural element AE, with said proportion being a random variable of triangular distribution defined by a minimum value p.sub.A,AE,min(c), a maximum value p.sub.A,AE,max(c), and a mode p.sub.A,AE,mode(c), /c/ determining, for each mesh c of said column and for each facies A of the plurality of facies, the sum over the plurality of architectural elements of p.sub.AE(c).Math.(p.sub.A,AE,min(c)+pA,AE,maxc+pA,AE,modec, and /d/ determining a value of expectation of height of liquid hydrocarbon for said column analytically according to the sum determined in the step /c/ for each mesh c of said column and for each facies A.

2. The method according to claim 1, wherein, with each mesh c of the model being associated with a proportion of porosity for each one of the facies of said plurality of facies, with said proportion of porosity being a random variable of triangular distribution defined by a minimum value .sub.A,min(c), a maximum value .sub.A,max(c), and a mode .sub.A,mode(c), the method further comprises: /c-1/ determining, for each mesh c of said column and for each facies A among the plurality of facies, the value (.sub.A,min(c)+.sub.A,max(c)+.sub.A,mode(c)), wherein, the determination of the step /d/ is a function of the value determined in the step /c-1/for each mesh c of said column and for each facies A among the plurality of facies.

3. The method according to claim 1, wherein, with each mesh c of the model being associated with a proportion of rock volume favourable to the presence of hydrocarbons for each one of the facies of said plurality of facies, with said proportion of rock volume favourable to the presence of hydrocarbons being a random variable of triangular distribution defined by a minimum value NTG.sub.A,min(c), a maximum value NTG.sub.A,max(c), and a mode NTG.sub.A,mode(c), the method further comprises: /c-2/ determining, for each mesh c of said column and for each facies A among the plurality of facies, the value (NTG.sub.A,min(c)+NTG.sub.A,max(c)+NTG.sub.A,mode(c)), wherein, the determination of the step /d/ is a function of the value determined in the step /c-2/ for each mesh c of said column and for each facies A among the plurality of facies.

4. The method according to claim 1, wherein, with each mesh c of the model being associated with a liquid hydrocarbon saturation, said liquid hydrocarbon saturation being characterised by a liquid hydrocarbon-water interface dimension z.sub.w having a distribution probability p(z.sub.w) between a maximum value z.sub.w,min, and a minimum value z.sub.w,max, and by a triangular distribution of the hydrocarbon saturation for each dimension h above this interface dimension, said distribution of hydrocarbon saturation being defined by a minimum value CS.sub.HC,A,min(h), a maximum value CS.sub.HC,A,max(h), and a mode CS.sub.HC,A,mode(h), the method further comprises: /c-3/ determining, for each mesh c of said column having a dimension z and for each facies A among the plurality of facies, the value ${}_{z_{w,\max }}^{z_{w,\min }}.Math.\frac{{\mathrm{cs}}_{\mathrm{HC},\min }\left(z-z_w\right)+{\mathrm{cs}}_{\mathrm{HC},\mathrm{mode}}\left(z-z_w\right)+{\mathrm{cs}}_{\mathrm{HC},\max }\left(z-z_w\right)}{3}.Math.p\left(z_w\right).Math.{\mathrm{dz}}_w$ wherein, the determination of the step /d/ is a function of the value determined in the step /c-3/ for each mesh c of said column and for each facies A among the plurality of facies.

5. The method according to claim 4, wherein, with said liquid hydrocarbon saturation being characterised by a liquid hydrocarbon-gas interface dimension z.sub.g having a probability p(z.sub.g), of distribution between a maximum value z.sub.g,min, and a minimum value z.sub.g,max, the method further comprises: /c-4/ determining, for each mesh c of said column having a dimension z, the value $\{\begin{array}{cc}0& \mathrm{if}.Math..Math.z{z}_{g,\min }\\ 1-{}_{z_{g,\max }}^z.Math.p\left(z_g\right).Math.{\mathrm{dz}}_g& \mathrm{if}.Math..Math.z_{g,\max }z{z}_{g,\min }\\ 1& \mathrm{if}.Math..Math.z{z}_{g,\max }\end{array}$ wherein, the determination of the step /d/ is a function of the value determined in the step /c-3/for each mesh c of dimension z of said column and for each facies A among the plurality of facies, multiplied by the value determined in the step /c-4/ for each mesh c of dimension z of said column.

6. The method for determining a map of variance of height of liquid hydrocarbon in a geological model, the method comprises, for at least one column of the geological model: /e/ determining a value of expectation of height of liquid hydrocarbon for said column, according to claim 1; /f/ determining a value of variance of height of liquid hydrocarbon for said column analytically according to the sum determined in the step /e/.

7. The method according to claim 6, wherein, the values of the proportion cubes for two different cells of the model and for different architectural elements are considered independently between them.

8. The method according to claim 6, wherein, the method further comprises: /g/ receiving a correlogram (A,z) according to a direction of said column for said facies A in a given architectural element; /h/ determining the expectation of the product of the presence of the facies A in said architectural element in a mesh c1 by the presence of the facies A in said architectural element in a mesh c2, with the distance between c1 and c2 according to said direction of the correlogram being z, with the probability of the presence of the facies A of said architectural element, for the mesh c1 being p.sub.A,AE,c1, with the probability of the presence of the facies A of said architectural element for the mesh c2 being p.sub.A,AE,c2, according to:
(1(A,z)).Math.(Esp(p.sub.A,AE,c1).Math.Esp(p.sub.A,AE,c2)+{square root over (Var(p.sub.A,AE,c1).Math.Var(p.sub.A,AE,c2))})+(A,z).Math.(Esp(p.sub.A,AE,c1)+.Math.Esp(p.sub.A,AE,c2)) wherein, the determination of the step /f/ is according to the determination of the step /h/.

9. A device for determining a map of expectation of height of liquid hydrocarbon in a geological model, with the geological model comprising meshes able to be associated with a facies among a plurality of facies and with an architectural element among a plurality of architectural elements, the device comprises, for at least one column of the geological model: /a/ an interface for receiving an appointment cube of architectural element for each architectural element AE among the plurality of architectural elements, with the apportionment cube describing for each mesh c of the model the probability p.sub.AE(c) that said mesh belongs to said architectural element, /b/ an interface for the receiving a proportion cube for each facies among the plurality of facies and for each architectural element among the plurality of architectural elements, with the proportion cube associated with a facies A and with an architectural element AE describing for each mesh c of the model the proportion of said facies in said mesh if said mesh belongs to said architectural element AE, with said proportion being a random variable of triangular distribution defined by a minimum value p.sub.A,AE,min(c), a maximum value p.sub.A,AE,max(c), and a mode p.sub.A,AE,mode(c), /c/ a circuit for the determining, for each mesh c of said column and for each facies A of the plurality of facies, the sum over the plurality of architectural elements of p.sub.AE(c).Math.(p.sub.A,AE,min(c)+p.sub.A,AE,max(c)+p.sub.A,AE,mode(c)). /d/ a circuit for the determining of a value of expectation of height of liquid hydrocarbon for said column analytically according to the sum determined by the circuit/c/ for each mesh c of said column and for each facies A.

10. A computer program product comprising instructions for the implementing of the method according to claim 1, when this program is executed by a processor.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0059] Other characteristics and advantages of the invention shall further appear when reading the following description. The latter is solely for the purposes of illustration and must be read with respect to the annexed drawings wherein:

[0060] FIG. 1a shows a possible example of a geological model;

[0061] FIG. 1b describes a possible example of a probability distribution that has a triangular distribution;

[0062] FIG. 2a describes an example of a transversal cross-section of a reservoir comprising water, gas and oil as well as the rate of water saturation of the rock of this reservoir.

[0063] FIG. 2b describes an example of a water saturation curve of the rock of a reservoir modelled as being continuous and linear in pieces;

[0064] FIG. 3 is an example of a flow chart of an embodiment of the invention;

[0065] FIG. 4 shows an example of a device allowing for the implementation of an embodiment of the invention;

[0066] FIGS. 5a and 5b are respectively examples of maps of expectation and of variance of height of liquid hydrocarbon in the model determined using multi-creation tools;

[0067] FIGS. 5c and 5d are respectively examples of maps of expectation and of variance of height of liquid hydrocarbon in the model determined using methods according to a possible embodiment of the invention.

DETAILED DESCRIPTION OF THE DRAWINGS

[0068] FIG. 1a shows a possible example of a transversal cross-section of a geological model 100.

[0069] This geological model 100 is comprised of a large number of pixels (ex. 104, 105) each one describing a facies present in the geological subsurface.

[0070] The petrol-physical data NTG, Phi and So are often determined for each facies. For each pixel c (of dimensions h.sub.pxh.sub.pxh.sub.p), it is possible to calculate a pixel value of HuPhiSo defined as follows:

[00004]$\underset{A_i}{\overset{}{.Math.}}.Math..Math.1_{{}_{A_i}}.Math.\left(x,y,z\right).Math.h_p.Math.{}_{c,A_i}.Math.{\mathrm{NTG}}_{c,A_i}.Math.{\mathrm{SO}}_{c,A_i}$

[0071] where 1.sub.A.sub.i(x, y, z) is the signalling function of the facies A.sub.i (equal to 1 if the facies for the pixel of coordinates (x,y,z) is A.sub.i, equal to 0 otherwise).

[0072] Of course, each pixel is associated with a single facies and:

[00005]$\underset{A_i}{\overset{}{.Math.}}.Math..Math.1_{{}_{A_i}}.Math.\left(x,y,z\right)=1$

[0073] Moreover, according to the geological history of the subsurface, the sedimentary filling and the geodynamic context that gave rise to the rocks of the reservoir, the sedimentation can be done heterogeneously but in a structured manner. The sedimentary environments created as such are more or less rich with certain facies.

[0074] For example, in a channel, the facies that are most present will be the coarse sands. The farther the rock is from the channel the more clayey it is.

[0075] These structures are generally called the architectural elements (AE).

[0076] For a given pixel of the model, it is considered that there is only one architectural element. The geological model 100 comprises three architectural elements 101, 102, and 103.

[0077] The interfaces between two architectural elements are often marked on maps by geophysicists. For example, the interface 102.sub.min can represent a possible interface between the architectural element 102 and the architectural element 103.

[0078] It is as such possible to introduce, as for the facies, a signalling of architectural elements 1.sub.AE.sub.j(x, y, z) indicating the presence or not of the architectural element AE.sub.j for the pixel of coordinates (x,y,z).

[0079] The proportions of facies in a given pixel c are, most often, a function of the architectural element associated with this pixel c. As such the signalling of facies A.sub.i, for a given architectural element AE.sub.j, is expressed in the form 1.sub.A.sub.i.sub.|AE.sub.j(x, y, z) (also noted as 1.sub.A.sub.i.sub.,AE.sub.j(x, y, z) or 1.sub.A.sub.i.sub.,AE.sub.j(c)).

[0080] As such, the pixel HuPhiSo value is defined as follows:

[00006]$\underset{A_i}{\overset{}{.Math.}}.Math.\underset{{\mathrm{AE}}_j}{\overset{}{.Math.}}.Math.\stackrel{}{1_{{}_{{\mathrm{AE}}_j}}.Math.\left(x,y,z\right).Math.1_{\mathrm{Ai}{\mathrm{AE}}_j}}.Math.\left(x,y,z\right).Math.h_c.Math.{}_{c,A_i}.Math.{\mathrm{NTG}}_{c,A_i}.Math.{\mathrm{SO}}_{c,A_i}$

[0081] with h.sub.c the height of the pixel c under consideration.

[0082] It is also possible to determine the column HuPhiSo values by adding the pixel HuPhiSo values for all of the columns of the model (for example, column 106). This column HuPhiSo value is given by the following formula:

[00007]$\underset{c}{\overset{}{.Math.}}.Math.\underset{A_i}{\overset{}{.Math.}}.Math.\left(\underset{{\mathrm{AE}}_j}{\overset{}{.Math.}}.Math.1_{{\mathrm{AE}}_j}.Math.\left(c\right).Math.1_{\mathrm{Ai}{\mathrm{AE}}_j}.Math.\left(c\right)\right).Math.h_c.Math.{}_{c,A_i}.Math.{\mathrm{NTG}}_{c,A_i}.Math.{\mathrm{SO}}_{c,A_i}$

[0083] with c the pixels of the column under consideration and with h.sub.c the height of the current pixel.

[0084] In the general case, the signalling 1.sub.A.sub.i.sub.|AE.sub.j(x, y, z) is not known in a precise manner, pixel by pixel: only the proportion cubes of the various facies A.sub.i are known. In order to overcome this problem, prior solutions proposed: [0085] to determine the limits of the architectural elements stochastically; [0086] to determine a map of facies by stochastically simulating a possible value of facies for each pixel thanks to the given proportion cubes for each facies of each architectural element, [0087] to determine pixel HuPhiSo value of each pixel; [0088] to determine pixel HuPhiSo values for each column of the model (i.e. HuPhiSo map); [0089] to reiterate the first four steps a large number of times so as to obtain an average and a converging variance of the HuPhiSo maps.

[0090] This invention does not choose this stochastic path.

[0091] Indeed, it is possible to determine for each column of the model, the average of the column HuPhiSo values and the variance of the column HuPhiSo values analytically, after certain hypotheses and geological simplifications presented hereinafter.

[0092] Determination of the Expectation

[0093] The expectation of the column HuPhiSo values can be expressed in the form:

[00008]$\mathrm{Esp}\left(\mathrm{HuPhiSO}\right)=\underset{c=1}{\overset{n}{.Math.}}.Math.\underset{A_i}{\overset{}{.Math.}}.Math.h_c.Math.\left(\underset{{\mathrm{AE}}_j}{\overset{}{.Math.}}.Math.\left(\mathrm{Esp}\left[1_{{\mathrm{AE}}_j}.Math.\left(c\right)\right].Math.\mathrm{Esp}\left[1_{\mathrm{Ai}{\mathrm{AE}}_j}.Math.\left(c\right)\right]\right)\right).Math.\left(\mathrm{Esp}\left({}_{c,A_i}\right).Math.\mathrm{Esp}\left({\mathrm{NTG}}_{c,A_i}\right).Math.\mathrm{Esp}\left({\mathrm{SO}}_{c,A_i}\right)\right)$

[0094] by assuming that the pixels c of the column are indexed from 1 to n (with n being a positive integer greater than 2).

[0095] The expression of this average creates certain geological hypotheses in order to simplify the expression of this average: e.g. the independence of the parameters .sub.c,A.sub.i, NTG.sub.c,A.sub.i and SO.sub.c,A.sub.i (as such, the expectation of the product of these variables is also the products of their expectation). As such, this expression is not the simple mathematical resolution of the problem posited.

[0096] If the signalling of facies 1.sub.A.sub.i(c) is a random variable following a Bernouilli distribution of which the parameter is the local proportion of the facies, then the expectation of this variable knowing p.sub.A.sub.i(c) can be expressed in the following form:

Esp(1.sub.A.sub.i(c)|p.sub.A.sub.i(c))=Esp(1.sub.A.sub.i(c)=1|p.sub.A.sub.i(c))=p.sub.A.sub.i(c)

In conclusion:

Esp(1.sub.A.sub.i(c))=Esp(p.sub.A.sub.i(c))

[0097] The proportions p.sub.A.sub.i(c) are determined with certainty only on wells. There is therefore an uncertainty on the proportion of facies in the rest of the reservoir. Because of this uncertainty, the proportion of facies can be, it too, a random variable, of which the probability distribution can be modelled by a triangular distribution 150 described in FIG. 1b (for a given facies A.sub.i and for a pixel c) having p.sub.min,Ai as a minimum value, p.sub.mode,Ai as a mode value and p.sub.max,Ai as a maximum value. As such:

[00009]$\mathrm{Esp}\left(1_{\mathrm{Ai}{\mathrm{AE}}_j}.Math.\left(c\right)\right)=\frac{p_{\min ,A_i{\mathrm{AE}}_j}\left(c\right)+p_{\mathrm{mode},A_i{\mathrm{AE}}_j}\left(c\right)+p_{\max ,A_i{\mathrm{AE}}_j}\left(c\right)}{3}$

[0098] Of course, the reservoir is not necessarily stationary, i.e. its proportions p.sub.A.sub.i(c) (and P.sub.min,A.sub.i(c),p.sub.mode,A.sub.i(c),p.sub.max,A.sub.i(c)) can vary according to the cell c under consideration: for example, by approaching a limit or a particular point, a facies can tend to disappear or tend to predominate.

[0099] The same reasoning is valid for calculating the expectation of the signalling of architectural elements. However, the proportion of architectural elements does not have any uncertainty even if several positions of interfaces between architectural elements are possible (the operator is often requested to point a minimum interface 102.sub.min, and a maximum interface 102.sub.max (possibly a probable interface 102.sub.mode, if this latter interface is not determined, it is considered that this probable interface is located at an equal distance from the minimum and maximum interfaces). As such:

Esp(1.sub.AE.sub.j(c))=p.sub.AE.sub.j(c)

[0100] In conclusion, and through analogy, the expression .sub.AE.sub.jEsp[1.sub.AE.sub.j(c)]. Esp1Ai|AEjc can be expressed in the form

[00010]${}_{{\mathrm{AE}}_j}.Math.p_{{\mathrm{AE}}_j}\left(c\right).Math.\frac{p_{\min ,A_i{\mathrm{AE}}_j}\left(c\right)+p_{\mathrm{mode},A_i{\mathrm{AE}}_j}\left(c\right)+p_{\max ,A_i{\mathrm{AE}}_j}\left(c\right)}{3}$

which is in fact the proportion of facies A.sub.i for the pixel c (which takes into account the possible presence of several AE and therefore several proportion cubes for the same facies).

[0101] A rock is comprised of grains and of voids between the grains. The porosity of the rock is the ratio between the volume of the voids and the volume of the rock:

[00011]$=\frac{V_{\mathrm{void}}}{V_{\mathrm{total}}}.Math.$

[0102] This porosity is most often apportioned according to a natural distribution such as a normal distribution. Its average is, it too, uncertain and follows a triangular probability distribution.

[0103] Phi is therefore geologically the sum of two random variables:

=.sub.avg+.sub.stoch

[0104] where .sub.avg is a random variable that can follow a triangular distribution (similar to FIG. 1b), of minimum value .sub.min, of mode value .sub.mode and of maximum value .sub.max, and where .sub.stoch is a random variable which follows a centred normal distribution (with zero average).

[0105] As such:

[00012]$\mathrm{Esp}\left(\right)=\mathrm{Esp}\left({}_{\mathrm{avg}}\right)+\mathrm{Esp}\left({}_{\mathrm{stoch}}\right)=\mathrm{Esp}\left({}_{\mathrm{avg}}\right)=\frac{{}_{\min }+{}_{\mathrm{mode}}+{}_{\max }}{3}$

[0106] The variable NTG represents the fraction of a rock volume favourable to the presence of hydrocarbons. If V is a rock volume, then NTG*V is the volume favourable to the presence of hydrocarbons, it is the useful volume of the rock. The value NTG can be apportioned according to a natural distribution that generally follows (but not necessarily) a normal distribution of average NTG.sub.avg.

[0107] The average NTG.sub.avg can be, itself, uncertain and can follow a triangular probability distribution (similar to FIG. 1b), of minimum value NTG.sub.min, of mode value NTG mode and of maximum value NTG.sub.max.

[0108] The distribution NTG is therefore geologically the sum of two random variables:

NTG=NTG.sub.avg+NTG.sub.stoch

[0109] where NTG.sub.avg is a random variable which follows the triangular distribution described hereinabove and where NTG.sub.stoch is a random variable which follows a centred normal distribution (with zero average).

[00013]$\mathrm{Esp}\left(\mathrm{NTG}\right)=\mathrm{Esp}\left({\mathrm{NTG}}_{\mathrm{avg}}\right)=\frac{{\mathrm{NTG}}_{\min }+{\mathrm{NTG}}_{\mathrm{mode}}+{\mathrm{NTG}}_{\max }}{3}$

[0110] The variable SO represents the oil saturation of the rock.

[0111] The pores of the rocks can be filled with gas, water or oil (petrol). The oil saturation So, the water saturation Sw and the gas saturation Sg are therefore complementary:

So+Sw+Sg=1

[0112] In addition, the hydrocarbon saturation (i.e. gas and oil) is noted as S.sub.HC:

S.sub.HC=So+S.sub.g

There is therefore:

S.sub.HC=1Sw

[0113] In the reservoirs (porous and permeable rocks), the fluids are subjected to gravity and apportioned according to their density. As such, under a permeable rock 201, it is possible to find (see FIG. 2a) gas at the top (zone 202), oil/petrol in the middle (zone 203) and finally water (zone 204).

[0114] Underneath the oil/water contact 205 (dimension z.sub.205), the pores are filled only with water and, above the contact 205 (zone 203), they can be filled with oil but residual water can also be absorbed in the grains of the rock.

[0115] As such, below the dimension z.sub.205, the water saturation Sw is 1 (curve 211). Above the dimension z.sub.205, the water saturation Sw decreases progressively (curve 212) to a dimension (z.sub.R) at which the water saturation Sw remains substantially constant (i.e. the residual value of water saturation) and equal to V.sub.R (curve 213).

[0116] The saturation curves (211, 212, 213) can be, themselves, uncertain (for a fixed dimension z.sub.205 and known for certain). It is as such possible to determine three water saturation curves: [0117] a curve having a minimum water saturation (214), [0118] a curve having a maximum water saturation (215), [0119] a mode curve having the most likely case (212 and 213).

[0120] As such, for each height above the contact 205, it is possible to define a triangular probability distribution for the water saturation using three corresponding points of the curves defined as such.

[0121] Moreover, the dimension z.sub.w of the interface between the oil and the water (z.sub.205 in the example of FIG. 2a) may not be known for certain. This dimension z.sub.w can also be represented by a random variable that follows a law of triangular probability of maximum value z.sub.w,min, of minimum value z.sub.w,max and of mode value Z.sub.w,mode.

[0122] S.sub.HC (Z)=CS.sub.HC (zz.sub.w) is noted for revealing the fact that the curve of hydrocarbon saturation can be translated vertically due to the uncertainty on z.sub.w: [0123] the curve having a minimum of hydrocarbon saturation is noted as CS.sub.HC,min(zzw, [0124] the curve having a maximum of hydrocarbon saturation is noted as CS.sub.HC,max(zzw, [0125] the mode curve having the most likely case is noted as CS.sub.HC,mode(zz.sub.w).

[0126] Likewise, the dimension z.sub.g representing the dimension of the gas/oil interface (z.sub.206 in the example of FIG. 2a for the interface 206) cannot be known for certain. This dimension z.sub.g can be represented by a random variable that follows a law of triangular probability of maximum value z.sub.g,min, of minimum value z.sub.g,max and of mode value z.sub.g,mode

[0127] As such, the expectation of the hydrocarbon saturation S.sub.HC can be expressed in the form:

[00014]$\begin{array}{c}\mathrm{Esp}\left(S_{\mathrm{HC}}\left(z\right)\right)=.Math.\mathrm{Esp}\left(1-\mathrm{Sw}\left(z\right)\right)\\ =.Math.\mathrm{Esp}\left({\mathrm{CS}}_{\mathrm{HC}}\left(z-z_w\right)\right)\\ =.Math.\mathrm{Esp}\left(\mathrm{Esp}\left({\mathrm{CS}}_{\mathrm{HC}}\left(z-z_w\right)z_w\right),z_w\right)\end{array}$

[0128] As the variables CS.sub.HC and Zw can be considered, geologically, as two independent random variables, it is possible to write:

[00015]$\mathrm{Esp}\left(S_{\mathrm{HC}}\left(z\right)\right)={}_{z_{w,\max }}^{z_{w,\min }}.Math.{}_{{\mathrm{cs}}_{\mathrm{HC},\min }}^{{\mathrm{cs}}_{\mathrm{HC},\max }}.Math.{\mathrm{cs}}_{\mathrm{HC}}\left(z-z_w\right).Math.p\left({\mathrm{CS}}_{\mathrm{HC}}\left(z-z_w\right)z_w\right).Math.p\left(z_w\right).Math.{\mathrm{dcs}}_{\mathrm{HC}}.Math.{\mathrm{dz}}_w$$.Math.\mathrm{Esp}\left(S_{\mathrm{HC}}\left(z\right)\right)={}_{z_{w,\max }}^{z_{w,\min }}.Math.p\left(z_w\right).Math.\mathrm{Esp}\left({\mathrm{CS}}_{\mathrm{HC}}\left(z-z_w\right).Math..Math.\mathrm{knowing}.Math..Math.z_w\right).Math.{\mathrm{dz}}_w$$\mathrm{Esp}\left(S_{\mathrm{HC}}\left(z\right)\right)={}_{z_{w,\max }}^{z_{w,\min }}.Math.\frac{{\mathrm{cs}}_{\mathrm{HC},\min }\left(z-z_w\right)+{\mathrm{cs}}_{\mathrm{HC},\mathrm{mode}}\left(z-z_w\right)+{\mathrm{cs}}_{\mathrm{HC},\max }\left(z-z_w\right)}{3}.Math.p\left(z_w\right).Math.{\mathrm{dz}}_w$

[0129] As such, there are three terms to be calculated of the type

[00016]${}_{z_{w,\max }}^{z_{w,\min }}.Math.\frac{{\mathrm{cs}}_{\mathrm{HC},\mathrm{XXX}}\left(z-z_w\right)}{3}.Math.p\left(z_w\right).Math.{\mathrm{dz}}_w$

(with XXX among min, max and mode).

[0130] If the probability distribution of z.sub.w is assumed to be triangular, it is possible to establish two cases:

if z.sub.w[z.sub.w,max,z.sub.w,mode]

[00017]$p\left(z_w\right)=p_{1.Math.p}.Math.z_w+p_{1.Math.c}=\frac{2.Math.z_w}{\left(z_{w,\min }-z_{w,\max }\right).Math.\left(z_{w,\mathrm{mode}}-z_{w,\max }\right)}.Math.+\frac{-2.Math.z_{w,\max }}{\left(z_{w,\min }-z_{w,\max }\right).Math.\left(z_{w,\mathrm{mode}}-z_{w,\max }\right)}.Math.$
if z.sub.w[z.sub.w,mode,z.sub.w,min]

[00018]$p\left(z_w\right)=p_{2.Math.p}.Math.z_w+p_{2.Math.c}=\frac{-2.Math.z_w}{\left(z_{w,\min }-z_{w,\max }\right).Math.\left(z_{w,\min }-z_{w,\mathrm{mode}}\right)}+.Math.\frac{2.Math.z_{w,\min }}{\left(z_{w,\min }-z_{w,\max }\right).Math.\left(z_{w,\min }-z_{w,\mathrm{mode}}\right)}.Math..Math.$

[0131] As such:

[00019]${}_{z_{w,\max }}^{z_{w,\min }}.Math.\frac{{\mathrm{cs}}_{\mathrm{HC},\mathrm{XXX}}\left(z-z_w\right)}{3}.Math.p\left(z_w\right).Math.{\mathrm{dz}}_w={}_{z_{w,\max }}^{z_{w,\mathrm{mode}}}.Math.\left[\frac{{\mathrm{cs}}_{\mathrm{HC},\mathrm{XXX}}\left(z-z_w\right)}{3}.Math.\left(p_{1.Math.p}.Math.z_w+p_{1.Math.c}\right)\right].Math.{\mathrm{dz}}_w.Math.+{}_{z_{w,\mathrm{mode}}}^{z_{w,\min }}.Math.\left[\frac{{\mathrm{cs}}_{\mathrm{HC},\mathrm{XXX}}\left(z-z_w\right)}{3}.Math.\left(p_{2.Math.p}.Math.z_w+p_{2.Math.c}\right)\right].Math.{\mathrm{dz}}_w.Math.$

[0132] Then by carrying out a change in the variable h=zz.sub.w, it is possible to write:

[00020]${}_{z_{w,\max }}^{z_{w,\min }}.Math.\frac{{\mathrm{cs}}_{\mathrm{HC},\mathrm{XXX}}\left(z-z_w\right)}{3}.Math.p\left(z_w\right).Math.{\mathrm{dz}}_w={}_{z-z_{w,\mathrm{mode}}}^{z-z_{w,\max }}.Math.[\frac{{{\mathrm{cs}}_{\mathrm{HC},\mathrm{XXX}}}^{\left(h\right)}}{3}.Math.\left(p_{1.Math.p}\left(z-h\right)+p_{1.Math.c}\right).Math.\mathrm{dh}+{}_{z-z_{w,\min }}^{z-z_{w,\mathrm{mode}}}.Math.\left[\frac{{\mathrm{cs}}_{\mathrm{HC},\mathrm{XXX}}\left(h\right)}{3}.Math.\left(p_{2.Math.p}\left(z-h\right)+p_{2.Math.c}\right)\right].Math.\mathrm{dh}$

[0133] In practice, the curve CS.sub.HC,XXX (and therefore the curve Sw.sub.YYY with YYY=min if XXX=max, YYY=mode if XXX=mode and YYY=max if XXX=min, as Sw.sub.min=1CS.sub.HC,max) is often linear by pieces, for example, between two successive dimensions of the following lists: [0134] h1.sub.1, h1.sub.2, . . . h1.sub.n-1, h1.sub.n1 (ordinates in increasing order in the interval [zz.sub.w,max,zz.sub.w,mode], with the end dimensions corresponding to the boundaries of the interval, with these end dimensions able to not correspond to a breakpoint of the linearity of the curve, see hereinbelow) and [0135] h2.sub.1, h2.sub.2, . . . h2.sub.n2-1, h2.sub.n2 (ordinates in increasing order in the interval [zz.sub.w,mode,zzw,min, with the end dimensions corresponding to the boundaries of the interval, as such h2.sub.1=h1.sub.n1, with these end dimensions able to not correspond to a breakpoint of the linearity of the curve, see hereinbelow).

[0136] For example, FIG. 2b shows an example of a water saturation curve 1CS.sub.HC,XXX(zz.sub.w) (250), with the dimension z.sub.w represented by the line h=0.

[0137] This curve 250 is linear by piece between the successive points 251, 252, 253, 254, 255, 256 and 257.

[0138] If the point of dimension (z does not correspond to one of the preceding points, it is possible to define a new point 271 on the linear portion of the water saturation curve between the points 252 and 253. The same applies for the dimension (zz.sub.w,mode) (corresponding then to the new point 272) and for the dimension (zz.sub.w,min) (corresponding then to the new point 273).

[0139] As such:

[00021]${}_{z-z_{w,\min }}^{z-z_{w,\max }}.Math.\frac{{\mathrm{cs}}_{\mathrm{HC},\mathrm{XXX}}\left(z-z_w\right)}{3}.Math.p\left(z_w\right).Math.{\mathrm{dz}}_w=\underset{i=1}{\overset{n.Math.1-1}{.Math.}}.Math..Math.\left[{}_{{h.Math.1}_{i+1}}^{{h.Math.1}_i}.Math.\left[\frac{{\mathrm{cs}}_{\mathrm{HC},\mathrm{XXX}}\left(h\right)}{3}.Math.\left(p_{1.Math.p}\left(z-h\right)+p_{1.Math.c}\right)\right].Math..Math.\mathrm{dh}\right]+\underset{i=1}{\overset{n.Math.2-1}{.Math.}}.Math.\left[{}_{{h.Math.2}_{i+1}}^{{h.Math.2}_i}.Math.\left[\frac{{\mathrm{cs}}_{\mathrm{HC},\mathrm{XXX}}\left(h\right)}{3}.Math.\left(p_{2.Math.p}\left(z-h\right)+p_{2.Math.c}\right)\right].Math.\mathrm{dh}\right]$

[0140] In the example of FIG. 2b, each one of the two preceding sums comprises two terms:

[00022]$\begin{array}{cc}-\underset{i=1}{\overset{n.Math.1-1}{.Math.}}.Math.\left[{}_{{h.Math.1}_{i+1}}^{{h.Math.1}_i}.Math.\left[\frac{{\mathrm{cs}}_{\mathrm{HC},\mathrm{XXX}}\left(h\right)}{3}.Math.\left(p_{1.Math.p}\left(z-h\right)+p_{1.Math.c}\right)\right].Math.\mathrm{dh}\right]& \end{array}$

(interval 281) comprises a terms corresponding to the integral between the dimensions of the point 271 and of the point 253 and an integral term between the dimensions of the point 253 and of the point 272;

[00023]$\begin{array}{cc}-\underset{i=1}{\overset{n.Math.2-1}{.Math.}}.Math.\left[{}_{{h.Math.2}_{i+1}}^{{h.Math.2}_i}.Math.\left[\frac{{\mathrm{cs}}_{\mathrm{HC},\mathrm{XXX}}\left(h\right)}{3}.Math.\left(p_{2.Math.p}\left(z-h\right)+p_{2.Math.c}\right)\right].Math.\mathrm{dh}\right]& \end{array}$

(interval 282) comprises a term corresponding to the integral between the dimensions of the point 272 and of the point 254 and an integral term between the dimensions of the point 254 and of the point 273.

[0141] It is considered that the equation describing the curve cs.sub.HC,XXX between two points h1.sub.i and h1.sub.i+1 is A1.sub.i,XXXh+B1.sub.i,XXX and that the equation describing the curve cs.sub.HC,XXX between two points h2.sub.i and h2.sub.i+1 is A2.sub.i,XXXh+B2.sub.i,XXXX.

[0142] In other words, the curve cs.sub.HC,XXX is linear by piece between two following successive points: (C1.sub.1,XXX, h1.sub.1), (C1.sub.2,XXX, h1.sub.2), . . . (C1.sub.n1-1,XXX, h1.sub.n1-1), (C1.sub.n1,XXX, h1.sub.n1)=(C2.sub.1,XXX, h2.sub.1), (C2.sub.2,XXX, h2.sub.2), . . . (C2.sub.n2-1,XXX, h2.sub.n2-1), (C2.sub.n2,XXX, h2.sub.n2).

[0143] Then:

[00024]$\frac{{C.Math.1}_{i+1,\mathrm{XXX}}-{C.Math.1}_{i,\mathrm{XXX}}}{{h.Math.1}_{i+1}-{h.Math.1}_i}={A.Math.1}_{i,\mathrm{XXX}.Math.}.Math.\mathrm{and}$$.Math.{C.Math.1}_{i,\mathrm{XXX}}-\frac{{C.Math.1}_{i+1,\mathrm{XXX}}-{C.Math.1}_{i,\mathrm{XXX}}}{{h.Math.1}_{i+1}-{h.Math.1}_i}.Math.{h.Math.1}_i={B.Math.1}_{i,\mathrm{XXX}}.Math..Math..Math.\mathrm{and}$$.Math.\frac{{C.Math.2}_{i+1,\mathrm{XXX}}-{C.Math.2}_{i,\mathrm{XXX}}}{{h.Math.2}_{i+1}-{h.Math.2}_i}={A.Math.2}_{i,\mathrm{XXX}.Math.}.Math.\mathrm{and}.Math.$$.Math.{C.Math.2}_{i,\mathrm{XXX}}-\frac{{C.Math.2}_{i+1,\mathrm{XXX}}-{C.Math.2}_{i,\mathrm{XXX}}}{{h.Math.2}_{i+1}-{h.Math.2}_i}.Math.{h.Math.2}_i={B.Math.2}_{i,\mathrm{XXX}}.Math..Math.$

and

[0144] As such:

[00025]$\begin{array}{cc}{}_{z_{w,\max }}^{z_{w,\min }}.Math.\frac{{\mathrm{cs}}_{\mathrm{HC},\mathrm{XXX}}\left(z-z_w\right)}{3}.Math.p\left(z_w\right).Math.{\mathrm{dz}}_w=\underset{i=1}{\overset{n.Math.1-1}{.Math.}}.Math..Math.\left[{}_{{h.Math.1}_i}^{{h.Math.1}_{i+1}}.Math.\left[\frac{{A.Math.1}_{i,\mathrm{XXX}}.Math.h+{B.Math.1}_{i,\mathrm{XXX}}}{3}.Math.\left(p_{1.Math.p}\left(z-h\right)+p_{1.Math.c}\right)\right].Math..Math.\mathrm{dh}\right]+\underset{i=1}{\overset{n.Math.2-1}{.Math.}}.Math.\left[{}_{{h.Math.2}_i}^{{h.Math.2}_{i+1}}.Math.\left[\frac{{{A.Math.2}_i}_{,\mathrm{XXX}}.Math.h+{B.Math.2}_{i,\mathrm{XXX}}}{3}.Math.\left(p_{2.Math.p}\left(z-h\right)+p_{2.Math.c}\right)\right].Math.\mathrm{dh}\right].Math.& \\ .Math.\mathrm{or}.Math.\text{:}.Math..Math..Math.{}_{z_{w,\max }}^{z_{w,\min }}.Math.\frac{{\mathrm{cs}}_{\mathrm{HC},\mathrm{XXX}}\left(z-z_w\right)}{3}.Math.p\left(z_w\right).Math.{\mathrm{dz}}_w=\frac{1}{3}.Math.\underset{i=1}{\overset{n.Math.1-1}{.Math.}}.Math..Math.\left[\left(-p_{1.Math.p}.Math.{A.Math.1}_{i,\mathrm{XXX}}.Math.\frac{{h.Math.1}_{i+1}^3}{3}+\left({A.Math.1}_{i,\mathrm{XXX}}.Math.\left(p_{1.Math.c}+p_{1.Math.p}\right).Math..Math.z\right)-p_{1.Math.p}.Math.{B.Math.1}_{i,\mathrm{XXX}}\right).Math.\frac{{h.Math.1}_{i+1}^2}{2}+{B.Math.1}_{i,\mathrm{XXX}}.Math.\left(p_{1.Math.c}+p_{1.Math.p}.Math.z\right).Math.{h.Math.1}_{i+1}\right)-\left(-p_{1.Math.p}.Math.{A.Math.1}_{i,\mathrm{XXX}}.Math.\frac{{h.Math.1}_i^3}{3}+\left({A.Math.1}_{i,\mathrm{XXX}}.Math.\left(p_{1.Math.c}+p_{1.Math.p}.Math.z\right)-p_{1.Math.p}.Math.z\right)-p_{1.Math.p}.Math.{B.Math.1}_{i,\mathrm{XXX}}\right).Math.\frac{{h.Math.1}_i^2}{2}+{B.Math.1}_{i,\mathrm{XXX}}.Math.\left(p_{1.Math.c}+p_{1.Math.p}.Math.z\right).Math.{h.Math.1}_i)]+\frac{1}{3}.Math.\underset{i=1}{\overset{n.Math.2-1}{.Math.}}.Math..Math.\left[\left(-p_{2.Math.p}.Math.{A.Math.2}_{i,\mathrm{XXX}}.Math.\frac{{h.Math.2}_{i+1}^3}{3}+{A.Math.2}_{i,\mathrm{XXX}}.Math.\left(p_{2.Math.c}+p_{2.Math.p}\right).Math..Math.z\right)-p_{2.Math.p}.Math.{B.Math.2}_{i,\mathrm{XXX}}\right).Math.\frac{{h.Math.2}_{i+1}^2}{2}+{B.Math.2}_{i,\mathrm{XXX}}.Math.\left(p_{2.Math.c}+p_{2.Math.p}.Math.z\right).Math.{h.Math.2}_{i+1})-\left(-p_{2.Math.p}.Math.{A.Math.2}_{i,\mathrm{XXX}}.Math.\frac{{h.Math.2}_i^3}{3}+{A.Math.2}_{i,\mathrm{XXX}}.Math.\left(p_{2.Math.c}+p_{2.Math.p}.Math.z\right)-p_{2.Math.p}.Math.{B.Math.2}_{i,\mathrm{XXX}}\right).Math.\frac{{h.Math.2}_i^2}{2}+{B.Math.2}_{i,\mathrm{XXX}}.Math.\left(p_{2.Math.c}+p_{2.Math.p}.Math.z\right).Math..Math.{h.Math.2}_i)]& \end{array}$

[0145] As such, each one of the terms of Esp(S.sub.HC(z)) can be calculated easily

[00026]$(i.e.{}_{z_{w,\max }}^{z_{w,\min }}.Math.\frac{{\mathrm{cs}}_{\mathrm{HC},\mathrm{XXX}}\left(z-z_w\right)}{3}.Math.p\left(z_w\right).Math.{\mathrm{dz}}_w$

with XXX among min, max and mode).

[0146] Moreover, it is possible to define the oil+water signalling (or the liquid-gas signalling, with the liquid here being the oil and the water) 1.sub.OW by:

[00027]$1_{\mathrm{OW}}.Math.\left(z\right)=\{\begin{array}{c}1.Math..Math.\mathrm{if}.Math..Math.z<z_g\\ 0.Math..Math.\mathrm{otherwise}\end{array}$

[0147] As such, the oil saturation can be expressed in the form:

So(z)=S.sub.HC(z).Math.1.sub.OW(z)

In conclusion:

Esp(So(z))=Esp(S.sub.HC(z)).Math.Esp(1.sub.OW(z))

[0148] The liquid-gas signalling can be determined as follows:

[00028]$\mathrm{Esp}\left(1_{\mathrm{OW}}.Math.\left(z\right)\right)=\{\begin{array}{c}0.Math..Math.\mathrm{if}.Math..Math.z{z}_{g,\min }\\ 1-{}_{Z_{G,\min }}^z.Math.p\left(z_g\right).Math.{\mathrm{dz}}_g.Math..Math.\mathrm{if}.Math..Math.z_{g,\max }z{z}_{g,\min }\\ 1.Math..Math.\mathrm{if}.Math..Math.z{z}_{g,\max }\end{array}$

[0149] with p(z.sub.g) the probability that the dimension of the gas-oil interface z.sub.g is located at this dimension, and with this probability having a distribution between the boundaries z.sub.g,max and z.sub.g,min.

[0150] FIG. 3 shows a flow chart that allows for the determining of an expectation of the HuPhiSo map, in an embodiment of the invention.

[0151] During the receiving of the receiving of the n.sub.xm proportion cubes 301.sub.1,1, 301.sub.2,1, . . . , 301.sub.n,m (respectively describing the proportions of the facies A.sub.1|AE.sub.1, A.sub.2|AE.sub.1, . . . A.sub.n|AE.sub.m) and of the apportionment cubes of the m architectural elements 302.sub.1, 302.sub.2, . . . , 302.sub.m (respectively describing the architectural element AE.sub.1, AE.sub.2, . . . AE.sub.m in the model), it is possible to create (step 303) an empty HuPhiSo model. This model is a model that comprises cells/pixels. This model is of dimensions similar to the dimensions of the proportion cubes and/or of the apportionment cubes describing the apportionment of the architectural elements.

[0152] The proportion cube 301.sub.i,j comprises a plurality of cells c each associated with three proportion values as such describing a triangular distribution of proportion: a minimum value p.sub.min,A.sub.i.sub.|AE.sub.j(c), a maximum value p.sub.max,A.sub.i.sub.|AE.sub.j(c) and a mode value p.sub.mode,A.sub.i.sub.|AE.sub.j(c).

[0153] The apportionment cube 302.sub.j comprises a plurality of cells c with each one associated with a proportion value p.sub.AE.sub.j(c).

[0154] If a cell of the model does not have any associated value of expectation of HuPhiSo (test 304, output OK), this cell is selected.

[0155] It is as such possible to determine (step 305) the expectation of the HuPhiSo value for the selected cell:

[00029]$\underset{A_i}{\overset{}{.Math.}}.Math..Math.h_c.Math.\left(\underset{{\mathrm{AE}}_j}{\overset{}{.Math.}}.Math..Math.\left(p_{{\mathrm{AE}}_j}\left(c\right).Math.\frac{p_{\min ,A_i{\mathrm{AE}}_j}.Math.\left(c\right)+p_{\mathrm{mode},A_i{\mathrm{AE}}_j}\left(c\right)+p_{\max ,A_i{\mathrm{AE}}_j}\left(c\right)}{3}\right)\right).Math.\left(\mathrm{Esp}\left({}_{c,A_i}\right).Math.\mathrm{Esp}\left({\mathrm{NTG}}_{c,A_i}\right).Math.\mathrm{Esp}\left({\mathrm{SO}}_{c,A_i}\right)\right)$

[0156] This formula creates a certain number of simplifications based on geological considerations mentioned hereinabove.

[0157] If it is considered that the variables .sub.c,A.sub.i, NTG.sub.c,A.sub.i, or SO.sub.c,A.sub.i are certain (i.e. fixed values without uncertainty) for the cells c and for the facies Ai, then the expectation of these values is equal to these certain values and the value determined in the step 305 can be directly associated with the cell selected in the model.

[0158] If ever the variable .sub.c,A.sub.i is considered to be uncertain, three values that represent a triangular probability distribution are provided for this cell selected and for each facies A.sub.i: a minimum value .sub.c,A.sub.i.sub.,min, a maximum value .sub.c,A.sub.i.sub.,max, and a mode value .sub.c,A.sub.i.sub.,mode. Then, the expectation of the porosity for the selected cell and for the facies A.sub.i is (step 306a):

[00030]$\mathrm{Esp}\left({}_{c,A_i}\right)=\frac{{}_{c,A_i,\min }+{}_{c,A_i,\mathrm{mode}}+{}_{c,A_i,\max }}{3}$

[0159] Likewise, if the variable NTG.sub.c,A.sub.i is considered to be uncertain, three values representing a triangular distribution of probability are provided for this selected cell and for each facies A: a minimum value NTG.sub.c,A.sub.i.sub.,min, a maximum value NTG.sub.c,A.sub.i.sub.,max, and a mode value NTG.sub.c,A.sub.i.sub.,mode. Then, the expectation of the value NTG for the selected cell and for the facies A.sub.i is (step 306b):

[00031]$\mathrm{Esp}\left({\mathrm{NTG}}_{c,A_i}\right)=\frac{{\mathrm{NTG}}_{c,A_i,\min }+{\mathrm{NTG}}_{c,A_i,\mathrm{mode}}+{\mathrm{NTG}}_{c,A_i,\max }}{3}$

[0160] Finally, if the variable SO.sub.c,A.sub.i is considered to be uncertain, three water saturation curves are provided as a function of the depth z of the mesh c, a minimum, maximum and mode curve, linear by piece and respectively defined by a plurality of points and by three values representing a triangular distribution of probability for the value of the dimension of a water-oil interface: a minimum value z.sub.w,max, a maximum value z.sub.w,min, and a mode value Z.sub.w,mode. Then, the expectation of the value SO for the selected cell is (step 306c) calculated as indicated hereinabove.

[0161] A probability distribution p(z.sub.g) of the oil-gas dimension z.sub.g can also be provided.

[0162] The value determined in the step 305 and in the step 306a and/or 306b and/or 306c can be used to calculate a value of the expectation of the variable HuPhiSo. This expectation can then be associated with this cell.

[0163] If all of the cells of the model have an associated value of expectation of HuPhiSo (test 304, output KO), it is possible to add (step 307) the expectations of HuPhiSo of the cells that belong to the same column in order to obtain an expectation of HuPhiSo column for said column under consideration.

[0164] This map 308 is then provided to an operator for viewing and/or later processing.

[0165] Determination of the Variance

[0166] Moreover, the variance of the HuPhiSo column values can be expressed in the form:

Var(HuPhiSo.sub.column)=Esp(HuPhiSo.sub.column.sup.2)Esp(HuPhiSo.sub.column).sup.2

[0167] The calculation of the term Esp(HuPhiSo.sub.column) and therefore of Esp(HuPhiSo.sub.column).sub.2 is detailed hereinabove and the latter can be calculated for the calculation of the variance.

[0168] The formula hereinbelow makes the hypothesis that, geologically, the variables .sub.c,A.sub.i, NTG.sub.c,A.sub.i and SO.sub.c,A.sub.i are independent variables:

[00032]$\mathrm{Esp}\left({\mathrm{HuPhiSo}}^2\right)=\underset{A}{\overset{}{.Math.}}.Math.\underset{B=A}{\overset{}{.Math.}}.Math.\underset{c.Math..Math.1}{\overset{}{.Math.}}.Math.\underset{c.Math..Math.2=c.Math..Math.1}{\overset{}{.Math.}}.Math.h_{c.Math..Math.1}.Math.h_{c.Math..Math.2}.Math.\mathrm{Esp}\left(1_A.Math.\left(c.Math..Math.1\right).Math.1_B.Math.\left(c.Math..Math.2\right)\right).Math.\mathrm{Esp}\left({}_{c.Math..Math.1,A}.Math.{}_{c.Math..Math.2,B}\right).Math.\mathrm{Esp}\left({\mathrm{NTG}}_{c.Math..Math.1,A}.Math.{\mathrm{NTG}}_{c.Math..Math.2,B}\right).Math.\mathrm{Esp}\left({\mathrm{So}}_{c.Math..Math.1,A}.Math.{\mathrm{So}}_{c.Math..Math.2,B}\right)+\underset{A}{\overset{}{.Math.}}.Math.\underset{BA}{\overset{}{.Math.}}.Math.\underset{c.Math..Math.1}{\overset{}{.Math.}}.Math.\underset{c.Math..Math.2=c.Math..Math.1}{\overset{}{.Math.}}.Math.h_{c.Math..Math.1}.Math.h_{c.Math..Math.2}.Math.\mathrm{Esp}\left(1_A.Math.\left(c.Math..Math.1\right).Math.1_B.Math.\left(c.Math..Math.2\right)\right).Math.\mathrm{Esp}\left({}_{c.Math..Math.1,A}.Math.{}_{c.Math..Math.2,B}\right).Math.\mathrm{Esp}.Math.\left({\mathrm{NTG}}_{c.Math..Math.1,A}.Math.{\mathrm{NTG}}_{c.Math..Math.2,B}\right).Math.\mathrm{Esp}\left({\mathrm{So}}_{c.Math..Math.1,A}.Math.{\mathrm{So}}_{c.Math..Math.2,B}\right)+\underset{A}{\overset{}{.Math.}}.Math.\underset{B=A}{\overset{}{.Math.}}.Math.\underset{c.Math..Math.1}{\overset{}{.Math.}}.Math.\underset{c.Math..Math.2c.Math..Math.1}{\overset{}{.Math.}}.Math.h_{c.Math..Math.1}.Math.h_{c.Math..Math.2}.Math.\mathrm{Esp}\left(1_A.Math.\left(c.Math..Math.1\right).Math.1_B.Math.\left(c.Math..Math.2\right)\right).Math.\mathrm{Esp}\left({}_{c.Math..Math.1,A}.Math.{}_{c.Math..Math.2,B}\right).Math.\mathrm{Esp}\left({\mathrm{NTG}}_{c.Math..Math.1,A}.Math.{\mathrm{NTG}}_{c.Math..Math.2,B}\right).Math.\mathrm{Esp}\left({\mathrm{So}}_{c.Math..Math.1,A}.Math.{\mathrm{So}}_{c.Math..Math.2,B}\right)+\underset{A}{\overset{}{.Math.}}.Math.\underset{BA}{\overset{}{.Math.}}.Math.\underset{c.Math..Math.1}{\overset{}{.Math.}}.Math.\underset{c.Math..Math.2c.Math..Math.1}{\overset{}{.Math.}}.Math.h_{c.Math..Math.1}.Math.h_{c.Math..Math.2}.Math.\mathrm{Esp}\left(1_A.Math.\left(c.Math..Math.1\right).Math.1_B.Math.\left(c.Math..Math.2\right)\right).Math.\mathrm{Esp}\left({}_{c.Math..Math.1,A}.Math.{}_{c.Math..Math.2,B}\right).Math.\mathrm{Esp}\left({\mathrm{NTG}}_{c.Math..Math.1,A}.Math.{\mathrm{NTG}}_{c.Math..Math.2,B}\right).Math.\mathrm{Esp}\left({\mathrm{So}}_{c.Math..Math.1,A}.Math.{\mathrm{So}}_{c.Math..Math.2,B}\right)$

[0169] with A, B a facies (possibly identical) among the various facies possible, and c1, c2 a cell (possible identical) of the column under consideration for the calculation of the column HuPhiSO.

[0170] The first quadruple sum corresponds to the terms where the facies A and B are identical and the cells c1 and c2 are identical, the second sum corresponds to the terms where the facies A and B are different but the cells c1 and c2 identical, the third sum corresponds to the cases where the facies A and B are the same but the cells c1 and c2 different and finally the last sum corresponds to the cases where the facies A and B are different and the cells c1 and c2 also.

[0171] The geological hypothesis is made that there cannot be two different facies in the same cell, i.e.:

Esp(1.sub.A(c1).Math.1.sub.B(c2))=0 if c1=c2 and AB

[0172] As such:

[00033]$\mathrm{Esp}.Math.\left({\mathrm{HuPhiSo}}^2\right)=\underset{c}{\overset{}{.Math.}}.Math.\underset{A}{\overset{}{.Math.}}.Math.h_c^2.Math.\mathrm{Esp}\left(1_A.Math.{\left(c\right)}^2\right).Math.\mathrm{Esp}\left({}_{c,A}^2\right).Math.\mathrm{Esp}\left({\mathrm{NTG}}_{c,A}^2\right).Math.\mathrm{Esp}\left({\mathrm{So}}_{c,A}^2\right)+\underset{A}{\overset{}{.Math.}}.Math.\underset{c.Math..Math.1}{\overset{}{.Math.}}.Math.\underset{c.Math..Math.2c.Math..Math.1}{\overset{}{.Math.}}.Math.h_{c.Math..Math.1}.Math.h_{c.Math..Math.2}.Math.\mathrm{Esp}\left(1_A.Math.\left(c.Math..Math.1\right).Math.1_A.Math.\left(c.Math..Math.2\right)\right).Math.\mathrm{Esp}\left({}_{c.Math..Math.1,A}.Math.{}_{c.Math..Math.2,A}\right).Math.\mathrm{Esp}\left({\mathrm{NTG}}_{c.Math..Math.1,A}.Math.{\mathrm{NTG}}_{c.Math..Math.2,A}\right).Math.\mathrm{Esp}\left({\mathrm{So}}_{c.Math..Math.1,A}.Math.{\mathrm{So}}_{c.Math..Math.2,A}\right)+\underset{A}{\overset{}{.Math.}}.Math.\underset{BA}{\overset{}{.Math.}}.Math.\underset{c.Math..Math.1}{\overset{}{.Math.}}.Math.\underset{c.Math..Math.2c.Math..Math.1}{\overset{}{.Math.}}.Math..Math.h_{c.Math..Math.1}.Math.h_{c.Math..Math.2}.Math.\mathrm{Esp}\left(1_A.Math.\left(c.Math..Math.1\right).Math.1_B.Math.\left(c.Math..Math.2\right)\right).Math.\mathrm{Esp}\left({}_{c.Math..Math.1,A}.Math.{}_{c.Math..Math.2,B}\right).Math.\mathrm{Esp}\left({\mathrm{NTG}}_{c.Math..Math.1..Math.A}.Math.{\mathrm{NTG}}_{c.Math..Math.2,B}\right).Math.\mathrm{Esp}\left({\mathrm{So}}_{c.Math..Math.1,A}.Math.{\mathrm{So}}_{c.Math..Math.2,B}\right).Math.$

[0173] This sum reveals a certain number of identical terms that it is possible to group together in order to decrease the required calculation time.

[0174] If the column under consideration comprises n cells and if there are N facies possible (the latter then being indexed), it is possible to write:

[00034]$\mathrm{Esp}\left({\mathrm{HuPhiSo}}^2\right)=\underset{c}{\overset{}{.Math.}}.Math.\underset{A}{\overset{}{.Math.}}.Math.\left[h_c^2.Math.\mathrm{Esp}\left({}_{c,A}^2\right).Math.\mathrm{Esp}\left({\mathrm{NTG}}_{c,A}^2\right).Math.\mathrm{Esp}\left({\mathrm{So}}_{c,A}^2\right).Math.\underset{\mathrm{AE}}{\overset{}{.Math.}}.Math..Math.\mathrm{Esp}\left(p_{\mathrm{AE}}\left(c\right)\right).Math.\mathrm{Esp}\left(p_{A,\mathrm{AE}}\left(c\right)\right)\right]+2.Math.\underset{A=1}{\overset{N}{.Math.}}.Math.\underset{c.Math..Math.1=1}{\overset{n-1}{.Math.}}.Math.\underset{c.Math..Math.2=c.Math..Math.1+1}{\overset{n}{.Math.}}.Math..Math.\left[h_{c.Math..Math.1}.Math.h_{c.Math..Math.2}.Math.\mathrm{Esp}\left(1_A.Math.\left(c.Math..Math.1\right).Math.1_A.Math.\left(c.Math..Math.2\right)\right).Math.\mathrm{Esp}.Math..Math.\left({}_{c.Math..Math.1,A}.Math.{}_{c.Math..Math.2,A}\right).Math.\mathrm{Esp}\left({\mathrm{NTG}}_{c.Math..Math.1,A}.Math.{\mathrm{NTG}}_{c.Math..Math.2,A}\right).Math..Math.\mathrm{Esp}\left({\mathrm{So}}_{c.Math..Math.1,A}.Math.{\mathrm{So}}_{c.Math..Math.2,A}\right).Math.\underset{\mathrm{AE}.Math..Math.1}{\overset{}{.Math.}}.Math.\underset{\mathrm{AE}.Math..Math.2}{\overset{}{.Math.}}.Math..Math.\mathrm{Esp}\left(1_{\mathrm{AE}.Math..Math.1}.Math.\left(c.Math..Math.1\right).Math.1_{\mathrm{AE}.Math..Math.2}.Math.\left(c.Math..Math.2\right)\right).Math.\mathrm{Esp}\left(1_{A,\mathrm{AE}.Math..Math.1}.Math.\left(c.Math..Math.1\right).Math.1_{A,\mathrm{AE}.Math..Math.2}.Math.\left(\mathrm{c2}\right)\right)\right]+2.Math.\underset{c.Math..Math.1=1}{\overset{n-1}{.Math.}}.Math.\underset{c.Math..Math.2=c.Math..Math.1+1}{\overset{n}{.Math.}}.Math.\underset{A=1}{\overset{N}{.Math.}}.Math.\underset{BA}{\overset{N}{.Math.}}.Math.[h_{c.Math..Math.1}.Math.h_{c.Math..Math.2}.Math.\mathrm{Esp}\left(1_A.Math.\left(c.Math..Math.1\right).Math.1_B.Math.\left(c.Math..Math.2\right)\right).Math.\mathrm{Esp}\left({}_{c.Math..Math.1,A}.Math.{}_{c.Math..Math.2,B}\right).Math..Math.\mathrm{Esp}\left({\mathrm{NTG}}_{c.Math..Math.1,A}.Math.{\mathrm{NTG}}_{c.Math..Math.2,B}\right).Math.\mathrm{Esp}(.Math.{\mathrm{So}}_{c.Math..Math.1,A}.Math.{\mathrm{So}}_{c.Math..Math.2,B}).Math..Math.\underset{\mathrm{AE}.Math..Math.1}{\overset{}{.Math.}}.Math..Math.\underset{\mathrm{AE}.Math..Math.2}{\overset{}{.Math.}}.Math.\mathrm{Esp}(.Math.1_{\mathrm{AE}.Math..Math.1}.Math.\left(c.Math..Math.1\right).Math..Math.1_{\mathrm{AE}.Math..Math.2}.Math..Math.\left(c.Math..Math.2\right)).Math..Math.\stackrel{}{.Math.}.Math..Math..Math.\mathrm{Esp}(.Math.1_{A,\mathrm{AE}.Math..Math.1}.Math.\left(c.Math..Math.1\right).Math.1_{B,\mathrm{AE}.Math..Math.2}.Math.\left(\mathrm{c2}\right))].Math.$

[0175] The term .sub.AEEsp(p.sub.AE(c)).Math.Esp(p.sub.A,AE(c)) was calculated previously and, following the geological simplifications made, is:

[00035]$\underset{\mathrm{AE}}{\overset{}{.Math.}}.Math..Math.\frac{p_{\mathrm{AE}}\left(c\right).Math.(p_{A,\mathrm{AE}.Math.\min }\left(c\right)+p_{A,\mathrm{AE}.Math.\mathrm{mode}}\left(c\right)+p_{A,\mathrm{AE}.Math.\max }\left(c\right)}{3}$

[0176] The term Esp(1.sub.AE1(c1)1.sub.AE2(c2)) can be expressed as follows if the geological hypothesis is made according to which the uncertainties concerning the architectural elements are only horizontal (there are no vertical uncertainties):

[00036]$\mathrm{Esp}\left(1_{\mathrm{AE}.Math..Math.1}.Math.\left(c.Math..Math.1\right).Math.1_{\mathrm{AE}.Math..Math.2}.Math.\left(c.Math..Math.2\right)\right)=\{\begin{array}{c}{p}_{\mathrm{AE}.Math..Math.1}\left(c.Math..Math.1\right).Math.p_{\mathrm{AE}.Math..Math.2}\left(c.Math..Math.2\right).Math..Math.\mathrm{if}.Math..Math.\mathrm{at}.Math..Math.\mathrm{least}.Math..Math.\mathrm{one}.Math..Math..Math.\mathrm{of}.Math..Math.\mathrm{the}.Math..Math.\mathrm{AE}.Math..Math..Math.\mathrm{is}.Math..Math.\mathrm{certain}\\ p_{\mathrm{AE}}\left(c.Math..Math.1\right).Math..Math.\mathrm{if}.Math..Math.\mathrm{the}.Math..Math..Math.\mathrm{AE}.Math..Math..Math.\mathrm{are}.Math..Math.\mathrm{all}.Math..Math..Math.\mathrm{uncertain}.Math..Math..Math.\mathrm{and}.Math..Math.\mathrm{AE}.Math..Math.1=\mathrm{AE}.Math..Math.2\\ 0.Math..Math..Math.\mathrm{if}.Math..Math.\mathrm{the}.Math..Math.\mathrm{AE}.Math..Math.\mathrm{are}.Math..Math.\mathrm{all}.Math..Math.\mathrm{uncertain}.Math..Math.\mathrm{and}.Math..Math.\mathrm{AE}.Math..Math.1\mathrm{AE}.Math..Math.2\end{array}$

[0177] That is to say:

[00037]$\mathrm{Esp}\left(1_{\mathrm{AE}.Math..Math.1}.Math.\left(c.Math..Math.1\right).Math.1_{\mathrm{AE}.Math..Math.2}.Math.\left(c.Math..Math.2\right)\right)=\{\begin{array}{c}{p}_{\mathrm{AE}.Math..Math.2}\left(c.Math..Math.2\right).Math..Math.\mathrm{if}.Math..Math.\left(p_{\mathrm{AE}.Math..Math.1}\left(c.Math..Math.1\right)=1\right)\\ p_{\mathrm{AE}.Math..Math.1}\left(c.Math..Math.1\right).Math..Math..Math.\mathrm{if}.Math..Math.\left(p_{\mathrm{AE}.Math..Math.2}\left(c.Math..Math.1\right)=1\right)\\ 0.Math..Math.\mathrm{if}.Math..Math.\left(p_{\mathrm{AE}.Math..Math.1}\left(c.Math..Math.1\right)=0\right).Math..Math.\mathrm{or}.Math..Math..Math.\mathrm{if}.Math..Math.\left(p_{\mathrm{AE}.Math..Math.2}\left(c.Math..Math.2\right)=0\right)\\ \mathrm{otherwise}.Math..Math.\{\begin{array}{c}{p}_{\mathrm{AE}}\left(c.Math..Math.1\right).Math..Math.\mathrm{if}.Math..Math.\mathrm{AE}.Math..Math.1=\mathrm{AE}.Math..Math.2\\ 0.Math..Math..Math.\mathrm{if}.Math..Math.\mathrm{AE}.Math..Math.1\mathrm{AE}.Math..Math.2\end{array}\end{array}$

[0178] Then, we have:

[00038]$\underset{\mathrm{AE}.Math..Math.1}{\overset{}{.Math.}}.Math.\underset{\mathrm{AE}.Math..Math.2}{\overset{}{.Math.}}.Math..Math.\mathrm{Esp}\left(1_{\mathrm{AE}.Math..Math.1}.Math.\left(c.Math..Math.1\right).Math.1_{\mathrm{AE}.Math..Math.2}.Math.\left(c.Math..Math.2\right)\right).Math.\mathrm{Esp}\left(1_{A,\mathrm{AE}.Math..Math.1}.Math.\left(c.Math..Math.1\right).Math.1_{B,\mathrm{AE}.Math..Math.2}.Math.\left(c.Math..Math.2\right)\right)=\underset{\mathrm{AE}.Math..Math.1}{\overset{}{.Math.}}.Math..Math.\mathrm{Esp}\left(1_{\mathrm{AE}.Math..Math.1}.Math.\left(c.Math..Math.1\right).Math.1_{\mathrm{AE}.Math..Math.1}.Math.\left(c.Math..Math.2\right)\right).Math.\mathrm{Esp}\left(1_{A,\mathrm{AE}.Math..Math.1}.Math.\left(c.Math..Math.1\right).Math.1_{B,\mathrm{AE}.Math..Math.1}.Math.\left(c.Math..Math.2\right)\right)+\underset{\mathrm{AE}.Math..Math.1}{\overset{}{.Math.}}.Math..Math.\underset{\mathrm{AE}.Math..Math.2\mathrm{AE}.Math..Math.1}{\overset{}{.Math.}}.Math..Math.\mathrm{Esp}\left(1_{\mathrm{AE}.Math..Math.1}.Math.\left(c.Math..Math.1\right).Math.1_{\mathrm{AE}.Math..Math.2}.Math.\left(c.Math..Math.2\right)\right).Math.\mathrm{Esp}\left(1_{A,\mathrm{AE}.Math..Math.1}.Math.\left(c.Math..Math.1\right).Math.1_{B,\mathrm{AE}.Math..Math.2}.Math.\left(c.Math..Math.2\right)\right)$