EP2037080B1 - Method for determining the permeability of a network of fractures based on a connectivity analysis - Google Patents

Description

The present invention relates to the field of optimization of the exploitation of underground deposits, such as hydrocarbon deposits, especially when they include a network of fractures.

The method according to the invention is particularly suitable for the study of the hydraulic properties of fractured terrains, and in particular to study hydrocarbon displacements in underground deposits.

In particular, the invention relates to a method for determining the permeability of a fracture network, so as to predict the flows of fluids that may occur through the deposit. It is then possible to simulate hydrocarbon production according to various production scenarios.

The petroleum industry, and more specifically the exploration and exploitation of oil deposits, require the acquisition of the best possible knowledge of underground geology to effectively provide a reserve assessment, a production model, or farm management. In fact, the determination of the location of a production well or an injection well, the constitution of the drilling mud, the completion characteristics, the parameters necessary for the optimal recovery of the hydrocarbons (such as injection pressure, production flow, ...) require a good knowledge of the deposit. To know the deposit means to know the petrophysical properties of the subsoil in every point of space.

To do this, the oil industry has a long history of combining technical measurements with modelizations, carried out in the laboratory and / or with software. Modeling of Oilfields are therefore an essential technical step for any exploration or exploitation of the deposit. These models are intended to provide a description of the deposit.

State of the art

The engineers in charge of the exploitation of fractured tanks, need to know perfectly the role of the fractures. A " fracture " is a plane discontinuity, very thin in relation to its extension, which represents a plane of rupture of a rock in the deposit.

On the one hand, the knowledge of the distribution and the behavior of these fractures makes it possible to optimize the location and the spacing between the wells that one intends to drill through the oil field.

On the other hand, the geometry of the fracture network conditions the displacement of the fluids both at the reservoir scale and at the local scale where it determines elementary matrix blocks in which the oil is trapped. Knowing the distribution of fractures, is therefore very useful, also, at a later stage, for the tank engineer who tries to calibrate the models he builds to simulate the deposits in order to reproduce or predict past production curves. or future.

The engineers in charge of the exploitation of fractured reservoirs, therefore, need to estimate the large-scale permeability (that of the drainage radius of a well or inter-well space for example) of fracture networks, and to predict the hydrodynamic behavior (flow, pressure, ..) of these networks, in response to external demands imposed via wells.

For these purposes, geoscientists proceed in the first place to characterize the fracture network, in the form of a set of families of fractures characterized by geometric attributes.

Then, in order to simulate flows within the fractured reservoir, a numerical model is most often used. This model is applied to a discretized representation of the deposit, that is to say that the deposit is cut into a set of meshes. The application of the numerical model requires knowledge of the flow properties of the fracture network at meshes scale, usually of hectometres. In particular, the permeabilities of the fracture network must be determined.

This can be done reliably from a flow calculation, performed on a geometric model representative of the fracture network. Such a method is described in the following patent: FR 2,757,947 ( US 6,023,656 ).

However, this numerical calculation method is expensive in computation time, for complex and / or large tanks. Often the discretization of a deposit leads to the construction of a grid with millions of meshes.

The specialist then has alternative methods. It has in fact analytical methods of calculation. An " analytical method " is one or more equations for accurately determining, without approximation or recourse to numerical techniques (iterative, etc.) the unknowns of a problem according to the data. An example of an analytical method is described for example in the following document:
M.Chen, M. Bai, and JC Roegiers, Permeability Tensors of Anisotropic Fracture Networks, Mathematical Geology, Vol.31, No. 4, 1999

However, the analytical methods are most often based on simplifying assumptions of the physical problem and do not allow to obtain the precision achieved by the numerical methods which, they allow to take into account the real complexity of the physics in its entirety. However, it is sometimes crucial to maintain high accuracy in estimating the permeabilities of fracture networks, so that the best production scenarios can be selected to optimize hydrocarbon production.

The document " SAIT I. OZKAYA, JOERG MATTNER: "Fracture connectivity from fracture intersections in borehole image logs", COMPUTERS & GEOSCIENCES, no. 29, 2003, pages 143-153, XP002472550 Describes a method of assessing connectivity by the average number of fracture-fracture intersections.

The document " LAETITIA MACÉ, LAURENT SOUCHE AND JEAN-LAURENT MALLET: "3D fracture modeling integrating geomechanics and geology data.", AAPG INTERNATIONAL CONFERENCE, October 24, 2004, - October 27, 2004, pages 1-6, XP002472551, Canc a "describes a method in which the connectivity of fractures is determined directly from a stochastic realization of the network, by means of a processing method applied to the geometry of the network.

The method according to the invention

• on détermine, au sein de chaque maille, un indice de connectivité, fonction au moins du nombre d'intersections entre fractures, au moyen de ladite description géométrique ;
• on estime la perméabilité du réseau de fractures de mailles dont l'indice de connectivité est supérieur à un seuil ;
• on affecte une valeur fixée de perméabilité, au sein des autres mailles dont l'indice de connectivité est inférieur audit seuil, de façon à limiter le nombre d'estimations de perméabilité ; et
• on optimise l'exploitation du gisement, en simulant des écoulements de fluides dans ledit gisement, en fonction des perméabilités du réseau de fractures en chaque maille.

The invention relates to a method implemented by computer to optimize the exploitation of a deposit comprising a fracture network, in which the deposit is discretized in a set of meshes. A geometrical description of the fracture network in each of the meshes is also developed. The method has the following steps:

• determining, within each mesh, a connectivity index, at least a function of the number of intersections between fractures, by means of said geometric description;
• the permeability of the network of mesh fractures whose connectivity index is greater than a threshold is estimated;
• assigning a fixed value of permeability, within the other meshes whose connectivity index is below said threshold, so as to limit the number of permeability estimates; and
• the exploitation of the deposit is optimized by simulating flows of fluids in said deposit, as a function of the permeabilities of the fracture network in each cell.

To further optimize the estimation of the permeability of the fracture network in each cell, we select, for each cell, a method for estimating the permeability of the fracture network as a function of the value of the connectivity index.

The selection of the method is then performed by defining two connectivity thresholds corresponding to two connectivity index values defining three connectivity index intervals. A different method is then selected for each of the intervals, so as to optimize the estimation of the permeability in each cell. We will choose the simplest method preserving the precision of the results.

• on dispose d'un ensemble de mailles comportant chacune un réseau de fractures pour lequel on dispose d'une description géométrique ;
• on détermine un indice de connectivité pour chacune des mailles ;
• on détermine une perméabilité du réseau en chaque maille, à l'aide d'un simulateur d'écoulement ;
• on construit une courbe de la perméabilité en fonction de l'indice de connectivité ;
• on définit les seuils en fonction de la forme de ladite courbe, de façon à ce que la perméabilité obéisse à la même loi de comportement en fonction de l'indice de connectivité au sein des trois intervalles définis par les seuils.

In this embodiment, the thresholds can be defined empirically, or by performing the following steps:

• we have a set of meshes each comprising a fracture network for which we have a geometric description;
• a connectivity index is determined for each of the cells;
• a network permeability is determined in each cell, using a flow simulator;
• a curve of permeability is constructed as a function of the connectivity index;
• the thresholds are defined according to the shape of said curve, so that the permeability obeys the same constitutive law as a function of the connectivity index within the three intervals defined by the thresholds.

In this embodiment, it is possible to choose the set of meshes for which a geometric description is available, by selecting a set of meshes resulting from the discretization of the deposit, the indices of which are distributed over the interval of the connectivity indices. calculated for all the meshes resulting from the discretization of the deposit.

• on estime la perméabilité du réseau de fractures au sein des mailles dont l'indice de connectivité est supérieur au second seuil, au moyen d'une formule analytique ;
• on estime la perméabilité du réseau au sein de mailles dont l'indice de connectivité est compris entre les deux seuils, au moyen d'un simulateur d'écoulement.

The permeability estimation methods can be chosen as follows:

• the permeability of the fracture network within the meshes whose connectivity index is greater than the second threshold is estimated by means of an analytical formula;
• it estimates the permeability of the network within meshes whose connectivity index is between the two thresholds, by means of a flow simulator.

• estimer la perméabilité du réseau de fractures au sein des mailles dont l'indice de connectivité est supérieur au second seuil, au moyen d'une formule analytique dans laquelle on considère que la perméabilité du réseau croît linéairement en fonction de l'indice de connectivité ;
• et estimer la perméabilité du réseau au sein de mailles dont l'indice de connectivité est compris entre les deux seuils, au moyen d'une méthode dans laquelle on considère que la perméabilité du réseau n'obéit plus à la même relation qu'au-delà du second seuil.

In this case, the permeability can be estimated as a function of the value of the connectivity index. For example:

• estimating the permeability of the fracture network within the mesh whose connectivity index is greater than the second threshold, by means of an analytical formula in which it is considered that the permeability of the network increases linearly as a function of the connectivity index;
• and to estimate the permeability of the network within meshes whose connectivity index lies between the two thresholds, by means of a method in which it is considered that the permeability of the network no longer obeys the same relationship as beyond the second threshold.

Other characteristics and advantages of the method according to the invention will appear on reading the following description of nonlimiting examples of embodiments, with reference to the appended figures and described below.

Brief presentation of the figures

• The figure 1 represents a network permeability curve K as a function of connectivity index I _C , from which a percolation threshold is determined $I_{\mathrm{VS}}^p$
  
  and a threshold of linearity $I_{\mathrm{VS}}^l$
  
  ( $I_{\mathrm{VS}}^p\approx 1$
  
  and $I_{\mathrm{VS}}^l\approx 3$
  
  ).
• The figure 2 illustrates the two-dimensional discretization of a deposit in a set of meshes, and indicates the meshes for which we do not calculate the permeability (zone 1, in white), the meshes for which we calculate the permeability using a flow simulator (zone 2, in gray), and the meshes for which the permeability is calculated using a linear formula (zone 3, in black).

Detailed description of the method

1. 1- Discrétisation du gisement en un ensemble de mailles
2. 2- Description géométrique du réseau de fractures
3. 3- Analyse de la connectivité du réseau de fractures
4. 4- Détermination de la perméabilité équivalente d'un réseau de fractures
5. 5- Simulation des écoulements de fluides
6. 6- Optimisation des conditions de production du gisement

The method according to the invention makes it possible to optimize the exploitation of a hydrocarbon reservoir, especially when the latter comprises a network of fractures. In particular, the method makes it possible to minimize the time required to determine the permeabilities of the fracture network, while preserving a good accuracy of the results. The method has six steps:

1. 1- Discretization of the deposit into a set of meshes
2. 2- Geometric description of the fracture network
3. 3- Analysis of the connectivity of the fracture network
4. 4- Determination of the equivalent permeability of a fracture network
5. 5- Simulation of fluid flows
6. 6- Optimization of the production conditions of the deposit

1- Discretization of the deposit into a set of meshes

For a long time, the petroleum industry has been combining technical measurements with modelizations, carried out in the laboratory and / or with software.

The modeling of the oil fields, therefore constitute a technical step essential to any exploration or exploitation of deposits. These models aim to to provide a description of the deposit, via its sedimentary architecture and / or its petrophysical properties.

These modelizations are based on a representation of the deposit, in a set of meshes. Each of these meshes represents a given volume of the deposit. The set of meshes constitutes a discrete representation of the deposit.

2- Geometric description of the natural fracture network

The geosciences specialist carries out a characterization of the geometry of the natural fracture network: he elaborates a geometrical description of the fracture network, in each meshes, by means of relevant geometric attributes.

This geometric description requires a set of measurements, made in the field by the geologist. These measurements make it possible to characterize the fracture network, so as to arrive at a description of the network in the form of a set of N families of fractures, characterized by geometric attributes.

In what follows, for the sake of clarity, we consider the two-dimensional situation of a network of fractures of N families contained in a layer, without however calling into question the possibility of extending the domain application of the invention to three-dimensional situations of multilayer tanks and / or of great thickness relative to the vertical extension of fractures.

• un angle moyen, θ_f , d'orientation par rapport à une direction de référence et un angle moyen de dispersion, α_f , de cette orientation autour de l'angle moyen θ_f . Ces paramètres d'orientation sont généralement ajustés à une loi statistique (telle que par exemple la loi de Fisher) ;
• une longueur moyenne, L_f , et une dispersion autour de cette moyenne ;
• une densité, d_f , définie comme la longueur cumulée de fracture par unité de surface (m/m²).

In two dimensions the geometric attributes relating to a family f can be:

• a mean angle, θ _f , of orientation with respect to a reference direction and a mean dispersion angle, α _f , of this orientation around the average angle θ _f . These orientation parameters are generally adjusted to a statistical law (such as for example Fisher's law);
• an average length, L _f , and a dispersion around this mean;
• a density, d _f , defined as the cumulative length of fracture per unit area (m / m ² ).

This geometrical description of the fracture network can also be determined probabilistically. A geometric description of the fracture network is then established by assigning to each family of fractures f a probability law ρ _{θ , f} orientations in the plane of the layers with respect to a reference direction, as well as a probability law of lengths ρ _{l , f} and a density d _f . Each probability law for parameter p satisfies the relation: $${\displaystyle \int {\rho }_{p,f}\mathit{dp}=1}$$

Depending on the measurements made, or the defined probability laws, we assign to each cell of the discrete representation of the deposit, a value to each of the geometric attributes describing the fracture network at the scale of this mesh.

The following documents describe an example of techniques that can be used to accomplish this task: FR 2.725.794 ( US5,661,698 ) FR 2.725.814 ( US 5,798,768 ) FR 2.733.073 ( US 5,659,135 )
At the end of this step, we have constructed a representation of the deposit in the form of a set of meshes, and we have assigned to each of these meshes a set of geometric attributes characterizing the fracture network within each mesh. .

• un angle moyen, ψ_f , d'orientation par rapport à une direction de référence dans le plan vertical et un angle moyen de dispersion, β_f , de cette orientation autour de l'angle moyen ψ_f . Ces paramètres d'orientation sont généralement ajustés à une loi statistique ;
• une hauteur moyenne, H_f , et une dispersion autour de cette moyenne ;
• une densité, d_f , définie comme la surface cumulée de fracture par unité de volume (m²/m³).

For an application to the 3D case, the geometric attributes making it possible to establish a geometrical description of the network of natural fractures, are the same parameters as for the 2D case as well as:

• a mean angle, ψ _f , of orientation with respect to a reference direction in the vertical plane and a mean dispersion angle, β _f , of this orientation around the mean angle ψ _f . These orientation parameters are generally adjusted to a statistical law;
• an average height, H _f , and a dispersion around this mean;
• a density, d _f , defined as the cumulative fracture area per unit volume (m ² / m ³ ).

A geometric description of the fracture network is established by assigning to each family of fractures f a probability law ρ _{θ , f} orientations in the plane of the layers with respect to a reference direction, a law of probability of orientations in the plane. vertical ρ _{Ψ, f} , a law of probability of lengths ρ _{l , f} as well as a law of probability of heights ρ _{H , f} , and a density d _f .

3- Analysis of the connectivity of the fracture network

In order to optimize the exploitation of a reservoir, we take into account, not only the geometry of the fracture network, but also the role of fractures on the hydrodynamic behavior of the reservoir. To determine this role, we define whether the fracture network is connected, so that it contributes directly to the flows and transport at the reservoir scale. Knowledge of this degree of connectivity is essential for the tank engineer responsible for estimating / predicting reservoir operation.

According to the invention, before calculating the permeability of the network in each mesh, which is either expensive, when using a precise method such as a numerical method, is fast but imprecise, when using a method such as an analytical method, we evaluate the connectivity of the fracture network of the mesh considered.

Indeed, if the fractures of a network, within a mesh, are not connected between them, the permeability of the network is null. On the other hand, if the fractures of a network, within a mesh, are all connected, the permeability is important. Indeed, a fluid has no difficulty crossing the mesh in the latter case.

In order to determine the degree of connectivity of the fractures of a network within a mesh, an index representative of the number of intersections between the fractures of the network is calculated according to the invention. Indeed, more the fractures of a network comprise intersections, more they are connected.

This index is called "connectivity index" and is noted as I _C. The index of connectivity I _C is then a parameter function of the number of intersections between the fractures of the network. It is determined in each cell, from the information from the geometric description.

avec :

• g ₁ : une fonction linéaire dépendant de la densité de fracturation du réseau ;
• g ₂ : une fonction dépendant de la dispersion des orientations (α) dans le plan horizontal
• g ₃ : une fonction linéaire dépendant de la longueur moyenne (L) des fractures ;
• g ₄ : une fonction dépendant de la dispersion des orientations (ψ) dans le plan vertical.

In general, the following formulation can be used to define the connectivity index I _C. : $$\mathit{Ic}={\mathrm{boy\; Wut}}_1\left(\mathrm{<figure-callout\; id="2"\; label="d\; boy\; Wut"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">d\; </figure-callout>},\right){\mathrm{boy\; Wut}}_{}{}_2\left(\theta ,\mathrm{<figure-callout\; id="3"\; label="\alpha \; boy\; Wut"\; filenames="imgf0001.png"\; state="\{\{state\}\}">\alpha \; </figure-callout>},\right){\mathrm{boy\; Wut}}_{}{}_3\left(\mathrm{The}\right)$$

with:

• g ₁ : a linear function depending on the fracturing density of the network;
• g ₂ : a function dependent on the dispersion of orientations ( α ) in the horizontal plane
• g ₃ : a linear function depending on the average length ( L ) of the fractures;
• g ₄ : a function depending on the dispersion of the orientations ( ψ ) in the vertical plane.

• Cas de 2 familles (i,j) d'orientations θ_i ,θ_j constantes: $${\mathit{Ic}}_{\mathit{ij}}=\frac{d_i.d_j.L_j.L_j.\sin \left(\left|{\theta }_i-{\theta }_j\right|\right)}{d_i{L}_j+d_j{L}_i}$$
  
• Cas d'une famille f dont la dispersion d'orientations est non négligeable et définie par une loi géostatistique ρ _θ,f : $${\mathit{Ic}}_f=d_f\times L_f\times {\displaystyle \underset{{\mathrm{<figure-callout\; id="1"\; label="\theta "\; filenames="imgf0002.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_1=0}{\overset{2\mathrm{\Pi }}{\int }}{\displaystyle \underset{{\mathrm{<figure-callout\; id="2"\; label="\theta "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_2>{\mathrm{<figure-callout\; id="1"\; label="\theta "\; filenames="imgf0002.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_1}{\overset{2\mathrm{\Pi }}{\int }}{\rho }_{\theta ,\mathrm{<figure-callout\; id="1"\; label="f\; \theta "\; filenames="imgf0002.png"\; state="\{\{state\}\}">f</figure-callout>}}\left({\theta }_{}\right)\left({}_1\right){\rho }_{\theta ,f}\left(\theta \right)\left|\sin \left({\theta }_1-{\mathrm{<figure-callout\; id="2"\; label="\theta "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_2\right)\right|{\mathit{<figure-callout\; id="1"\; label="d\theta "\; filenames="imgf0002.png"\; state="\{\{state\}\}">d\theta </figure-callout>}}_1{\mathit{<figure-callout\; id="2"\; label="d\theta "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">d\theta </figure-callout>}}_2}}$$
  

These functions can be evaluated empirically. We can also define the connectivity index from the average values of the geometric attributes defined above, as follows:

• Case of 2 families ( i , j ) of orientations θ _i , θ _j constants: $${\mathit{Ic}}_{\mathit{ij}}=\frac{d_i.d_j.{\mathrm{The}}_j.{\mathrm{The}}_j.\sin \left(\left|{\theta }_i-{\theta }_j\right|\right)}{d_i{\mathrm{The}}_j+d_j{\mathrm{The}}_i}$$
  
• Case of a family f whose dispersion of orientations is not negligible and defined by a geostatistical law ρ _{θ , f} : $${\mathit{Ic}}_f=d_f\times {\mathrm{The}}_f\times {\displaystyle \underset{{\mathrm{<figure-callout\; id="1"\; label="\theta "\; filenames="imgf0002.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_1=0}{\overset{2\mathrm{\Pi }}{\int }}{\displaystyle \underset{{\mathrm{<figure-callout\; id="2"\; label="\theta "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_2>{\mathrm{<figure-callout\; id="1"\; label="\theta "\; filenames="imgf0002.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_1}{\overset{2\mathrm{\Pi }}{\int }}{\rho }_{\theta ,\mathrm{<figure-callout\; id="1"\; label="f\; \theta "\; filenames="imgf0002.png"\; state="\{\{state\}\}">f\; </figure-callout>}}\left({\theta }_{}\right)\left({}_1\right){\rho }_{\theta ,f}\left(\theta \right)\left|\sin \left({\theta }_1-{\mathrm{<figure-callout\; id="2"\; label="\theta "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_2\right)\right|{\mathit{<figure-callout\; id="1"\; label="d\theta "\; filenames="imgf0002.png"\; state="\{\{state\}\}">d\theta </figure-callout>}}_1{\mathit{<figure-callout\; id="2"\; label="d\theta "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">d\theta </figure-callout>}}_2}}$$
  

If we consider the statistic distribution distributions of the geometrical parameters of each of the families f, we can use the following expression of I _c for a case with N Families: $$\mathit{Ic}=\frac{1}{{\displaystyle \underset{k=1}{\overset{\mathrm{NOT}}{\Sigma }}\frac{d_k}{{\mathrm{The}}_k}}}\left({\displaystyle \underset{i=1}{\overset{\mathrm{NOT}}{\Sigma }}\left(d_i^2\cdot {\displaystyle \underset{{\mathrm{<figure-callout\; id="1"\; label="\theta "\; filenames="imgf0002.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_1=0}{\overset{2\mathrm{\Pi }}{\int }}{\displaystyle \underset{{\mathrm{<figure-callout\; id="2"\; label="\theta "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_2>{\mathrm{<figure-callout\; id="1"\; label="\theta "\; filenames="imgf0002.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_1}{\overset{2\mathrm{\Pi }}{\int }}{\rho }_{\theta ,i}\left({\theta }_1\right){\rho }_{\theta ,i}\left({\theta }_2\right)\left|\sin \left({\theta }_1-{\mathrm{<figure-callout\; id="2"\; label="\theta "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_2\right)\right|}}{\mathit{<figure-callout\; id="1"\; label="d\theta "\; filenames="imgf0002.png"\; state="\{\{state\}\}">d\theta </figure-callout>}}_1{\mathit{<figure-callout\; id="2"\; label="d\theta "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">d\theta </figure-callout>}}_2\right)}+{\displaystyle \underset{i=1}{\overset{\mathrm{NOT}}{\Sigma }}{\displaystyle \underset{j=i+1}{\overset{\mathrm{NOT}}{\Sigma }}{d}_i{d}_j{\displaystyle \underset{{\mathrm{<figure-callout\; id="1"\; label="\theta "\; filenames="imgf0002.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_1=0}{\overset{2\mathrm{\Pi }}{\int }}{\displaystyle \underset{{\mathrm{<figure-callout\; id="2"\; label="\theta "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_2=0}{\overset{2\mathrm{\Pi }}{\int }}{\rho }_{\theta ,i}\left({\theta }_1\right){\rho }_{\theta ,\mathrm{<figure-callout\; id="2"\; label="j\; \theta "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">j\; </figure-callout>}}\left({\theta }_{}\right)\left({}_2\right)\left|\sin \left({\theta }_1-{\mathrm{<figure-callout\; id="2"\; label="\theta "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">\theta </figure-callout>}}_2\right)\right|{\mathit{<figure-callout\; id="1"\; label="d\theta "\; filenames="imgf0002.png"\; state="\{\{state\}\}">d\theta </figure-callout>}}_1{\mathit{<figure-callout\; id="2"\; label="d\theta "\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">d\theta </figure-callout>}}_2}}}}\right)$$

4- Determination of the equivalent permeability of the fracture network

To determine the hydrodynamic behavior of a reservoir, it is necessary to evaluate a permeability of the fracture network on a large scale. In each cell, a so-called "equivalent" permeability of the fracture network contained in this mesh is then calculated.

Two methods exist: one digital, expensive in computing resources for large tanks (many meshes), the other, analytical, fast but approximate because based on simplifying assumptions, relating to the geometry of the network for example.

Thanks to the invention, the reservoir engineer can optimize in cost (time) and quality (accuracy) the calculation of the fracture permeabilities.

The calculation of the permeabilities according to the invention is carried out by first analyzing the value of the connectivity index I _C.

Selection of meshes for which it is necessary to determine the permeability

• Si l'indice de connectivité I_C de la maille considérée indique que les fractures sont déconnectées entre elles, on considère la perméabilité du réseau nulle à grande échelle. Hormis le cas de fractures/failles de grande extension, le rôle des fractures sur le comportement hydrodynamique à grande échelle du réservoir est négligeable (perméabilité du réseau nulle). Dans ce cas, il n'est donc pas nécessaire de calculer la perméabilité du réseau. Ceci peut concerner des millions de mailles dans un modèle de réservoir fracturé, et l'on évite ainsi un très grand nombre de calculs inutiles.
• Si l'indice de connectivité I_C indique que les fractures sont connectées entre elles, on considère que le réseau acquiert une perméabilité à grande échelle. Le rôle hydraulique des fractures risque d'être sensible et ces dernières doivent être intégrées dans l'étude de la dynamique du réservoir.

The principle is as follows:

• If the connectivity index I _C of the mesh in question indicates that the fractures are disconnected from one another, the permeability of the null network on a large scale is considered. Apart from the case of large-scale fractures / faults, the role of fractures on the large-scale hydrodynamic behavior of the reservoir is negligible (zero network permeability). In this case, it is not necessary to calculate the permeability of the network. This can involve millions of meshes in a fractured reservoir model, thus avoiding a very large number of unnecessary calculations.
• If the connectivity index I _C indicates that the fractures are connected to each other, it is considered that the network acquires a permeability on a large scale. The hydraulic role of fractures is likely to be sensitive and these should be integrated into the study of reservoir dynamics.

The threshold value of the connectivity index, from which it is considered that it is necessary to calculate the permeability, can be obtained empirically, or by simulations. Those skilled in the art may in particular use a flow simulator, a software well known to specialists, to define this threshold. This threshold is called the percolation threshold. It's noted $I_{\mathrm{VS}}^p.$

Thus, the evaluation of the connectivity of the fracture network in each mesh, makes it possible to select the cells of the discretization of the deposit, for which it is necessary to determine the network permeability by an appropriate calculation method. The other meshes have a null value of network permeability.

Selection of a method for determining the permeability of the fracture network

According to one embodiment, the connectivity index thus calculated can be used more. In fact, by establishing a permeability curve as a function of the connectivity index, it is possible to define permeability behaviors, making it possible to define the most suitable determination technique.

According to a general embodiment , the permeability calculation method is selected by defining connectivity thresholds corresponding to connectivity index values defining connectivity index intervals. A method is selected for each of said intervals.

1. i- on dispose d'un ensemble de mailles comportant chacune un réseau de fractures pour lequel on dispose d'une description géométrique ;
2. ii- on calcule l'indice de connectivité de chacune de ces mailles ;
3. iii- on détermine une perméabilité du réseau en chaque maille, à l'aide d'un simulateur d'écoulement par exemple ;
4. iv- on construit une courbe de la perméabilité en fonction de l'indice de connectivité ;
5. v- on définit les seuils en fonction de la forme de cette courbe, de façon à ce que la perméabilité varie en fonction de l'indice de connectivité selon un comportement homogène au sein des trois intervalles définis par les seuils.

These thresholds can be defined empirically, or for example, by performing the following steps:

1. i- we have a set of meshes each comprising a fracture network for which we have a geometric description;
2. we calculate the connectivity index of each of these meshes;
3. a network permeability is determined in each mesh, using a flow simulator for example;
4. iv- a permeability curve is constructed as a function of the connectivity index;
5. The thresholds are defined according to the shape of this curve, so that the permeability varies according to the connectivity index according to a homogeneous behavior within the three intervals defined by the thresholds.

Homogeneous behavior means that, over an interval, the permeability curve obeys the same constitutive law as a function of the connectivity index. The permeability curve as a function of the connectivity index can then be modeled by a single analytical formula (linear law, polynomial, etc.). In other words, over an interval, the network has the same law of flow behavior, ie the same law of permeability (hydraulic behavior) as a function of the connectivity index.

In practice, the set of meshes of step i can be defined in the following way: having calculated the index of connectivity for all the meshes of the discretization of the deposit, one selects a set of meshes, whose indices are distributed over the range of connectivity indices calculated for all the meshes of the deposit.

According to a particular embodiment , two connectivity thresholds are defined, defining three connectivity index intervals.

Il est défini sur la figure 1 par $I_C^p\approx 1.$

Il existe un second seuil, noté $I_C^l,$

appelé seuil de linéarité. Il est défini sur la figure 1 par $I_C^l\approx 3.$

Au-delà de ce seuil, la courbe est une droite.The figure 1 illustrates such an approach. This figure represents a network permeability curve, K , as a function of the connectivity index I _C. We see that there is a first threshold. This threshold corresponds to the percolation threshold $I_{\mathrm{VS}}^p.$

It is defined on the figure 1 by $I_{\mathrm{VS}}^p\approx 1.$

There is a second threshold, noted $I_{\mathrm{VS}}^l,$

called the linearity threshold. It is defined on the figure 1 by $I_{\mathrm{VS}}^l\approx 3.$

Beyond this threshold, the curve is a straight line.

la perméabilité est constante (nulle), au dessus de $I_C^l,$

la perméabilité croît linéairement. Entre les deux seuils, la perméabilité évolue en fonction de l'indice de connectivité selon une relation unique et non linéaire.These two thresholds define three intervals over which the permeability varies according to a homogeneous behavior as a function of the connectivity index: below $I_{\mathrm{VS}}^p,$

the permeability is constant (zero), above $I_{\mathrm{VS}}^l,$

the permeability increases linearly. Between the two thresholds, the permeability changes according to the connectivity index in a unique and non-linear relationship.

• 
  $I_C\le I_C^p:$
  
  K(I_C ) = 0
• 
  $I_C\ge I_C^l:$
  
  sur cet intervalle, la perméabilité du réseau de fractures croît linéairement en fonction de l'indice de connectivité (I_C ) ou de la densité de fractures (d). On peut alors utiliser une méthode de calcul, permettant de déterminer la perméabilité à partir d'une formule analytique. On peut par exemple définir la formule suivante : $$K\left(I_c\right)=a.I_c+b$$
  
  Les coefficients a et b peuvent être déterminés par une simple régression linéaire.
• 
  $I_C^p\le I_C\le I_C^l:$
  
  sur cet intervalle de transition, la perméabilité K du réseau évolue suivant une certaine fonction g dépendant de l'indice de connectivité (I_C ) ou de la densité de fractures (d) : $$K\left(I_c\right)=g\left(I_c\right)$$
  
  La fonction g est une fonction distincte de la fonction linéaire définie sur l'intervalle $I_C\ge I_C^l.$
  
  Elle est fixée pour un type de réseau donné, c'est-à-dire pour les réseaux dont seule la densité varie (à nombre N de familles, orientations et longueurs de fractures de chaque famille fixés).

The connectivity index, calculated for each cell of the fractured field model, is then used as follows:

• 
  $I_{\mathrm{VS}}\le I_{\mathrm{VS}}^p:$
  
  K ( I _C ) = 0
• 
  $I_{\mathrm{VS}}\ge I_{\mathrm{VS}}^l:$
  
  over this interval, the permeability of the fracture network increases linearly as a function of the connectivity index ( I _C ) or the fracture density ( d ). We can then use a calculation method to determine the permeability from an analytical formula. For example, we can define the following formula: $$K\left(I_{\mathrm{vs}}\right)=\mathrm{at}.I_{\mathrm{vs}}+b$$
  
  The coefficients a and b can be determined by a simple linear regression.
• 
  $I_{\mathrm{VS}}^p\le I_{\mathrm{VS}}\le I_{\mathrm{VS}}^l:$
  
  on this transition interval, the permeability K of the network evolves according to a certain function g depending on the connectivity index ( I _C ) or the fracture density (d): $$K\left(I_{\mathrm{vs}}\right)=\mathrm{boy\; Wut}\left(I_{\mathrm{vs}}\right)$$
  
  The function g is a function distinct from the linear function defined on the interval $I_{\mathrm{VS}}\ge I_{\mathrm{VS}}^l.$
  
  It is fixed for a given type of network, that is to say for the networks whose only density varies (with number N of families, orientations and lengths of fractures of each fixed family).

In this case, a numerical method for calculating the fracture permeabilities makes it possible to obtain a precise value of permeability. Such a method is described in the following document: FR 2,757,947 ( US 6,023,656 ).

However, an alternative to the numerical method can be adopted in order to increase the speed of permeability calculations. It consists in using an approximation, such as an analytical formula giving the evolution of the permeability as a function of the connectivity index.

In conclusion, the evaluation of the connectivity of the fracture network in each mesh, allows to select a method of determination of the permeability adapted to the need required for each mesh (ie a reliable method on the one hand, fast and inexpensive in computing time on the other hand). At the same time, it makes it possible to define three regions of the field (or set of meshes) each having a homogeneous behavior in fracture permeability.

5- Simulation of fluid flows

At this stage, the reservoir engineer has a discretized representation (set of meshes) of the hydrocarbon deposit, from which he wishes to extract the hydrocarbons. This representation is indicated fracture network permeability, that is to say that each cell is associated with a permeability value.

The reservoir engineer chooses a production process, for example the water injection recovery process, the optimal implementation scenario for the field in question. The definition of an optimal water injection scenario will consist, for example, in determining the number and location (position and spacing) of injection and production wells in order to best take into account the impact of fractures on the progression of fluids within the tank.

Depending on the chosen scenario, and fracture network permeabilities, it is then possible to simulate the expected hydrocarbon production, using a tool well known to specialists: a flow simulator. Such software makes it possible to simulate the flows of fluids within the reservoir.

6- Optimization of the production conditions of the deposit

By selecting various scenarios characterized for example by various respective locations of the injectors and producers, and by simulating the production of hydrocarbons for each of them according to step 5, one can select the scenario to obtain optimal production of the deposit. . We then optimize the exploitation of the deposit, by implementing, on the spot, the production scenario thus selected.

Application example.

A hydrocarbon deposit is discretized with a network of fractures. The figure 2 illustrates the result of this mesh in two dimensions.

A geometric description of the fracture network is developed in each of the meshes, using information from geological measurements and analyzes.

Within each mesh, a connectivity index is defined, defined for example by the following formulation (formula based on the mean attributes of each of the families of fractures): $$\mathit{Ic}=\frac{{\displaystyle \underset{i=1}{\overset{\mathrm{NOT}}{\Sigma }}{\displaystyle \underset{j=i+1}{\overset{\mathrm{NOT}}{\Sigma }}{d}_i.d_j.\left|\sin \left({\theta }_i-{\theta }_j\right)\right|}}}{{\displaystyle \underset{i=1}{\overset{\mathrm{NOT}}{\Sigma }}\frac{d_i}{{\mathrm{The}}_i}}}$$

This index defines the average intersection number between fractures, within each mesh.

et de linéarité, $I_C^l\approx 3,$

pour classer les mailles en fonction de leur indice de connectivité et définir ainsi trois zones (ou régions) du champ caractérisées respectivement par des valeurs de cet indice inférieures à $I_C^p,$

supérieures à $I_C^l,$

et comprises entre $I_C^p$

et $I_C^l.$

Cette définition de zones détermine le choix de la méthode de calcul des perméabilités de fracture.We then take into account the existence of the two percolation thresholds, $I_{\mathrm{vs}}^p\approx 1,$

and linearity, $I_{\mathrm{VS}}^l\approx 3,$

to classify the meshes according to their connectivity index and thus define three zones (or regions) of the field characterized respectively by values of this index lower than $I_{\mathrm{VS}}^p,$

greater than $I_{\mathrm{VS}}^l,$

and between $I_{\mathrm{VS}}^p$

and $I_{\mathrm{VS}}^l.$

This definition of zones determines the choice of the method for calculating the fracture permeabilities.

en blanc), les mailles pour lesquelles on calcule la perméabilité à l'aide d'un simulateur d'écoulement (zone 2 où $I_C^p\le I_C\le I_C^l,$

en gris), et les mailles pour lesquelles on calcule la perméabilité à l'aide d'une formule linéaire (zone 2 où $I_C\ge I_C^l,$

en gris).The figure 2 illustrates, in two dimensions, the meshs of the representation of the deposit for which one does not calculate by the permeability (zone 1 where $I_{\mathrm{VS}}\le I_{\mathrm{VS}}^p,$

in white), the meshes for which the permeability is calculated using a flow simulator (zone 2 where $I_{\mathrm{VS}}^p\le I_{\mathrm{VS}}\le I_{\mathrm{VS}}^l,$

in gray), and the meshes for which the permeability is calculated using a linear formula (zone 2 where $I_{\mathrm{VS}}\ge I_{\mathrm{VS}}^l,$

in gray).

Zone 1 does not calculate permeability. We therefore gain valuable computing time. In zone 3, we perform a linear calculation that gives us the same precision as a numerical simulation. On zone 2, to obtain an important precision, one uses a flow simulator.

A production process is then selected, for example water injection. The mode of implementation of this method for the field considered however remains to be specified, and more particularly if this field proves to be fractured. Different implementation scenarios, differing from each other by the position of the wells, for example, are then defined and compared on the basis of quantitative criteria of production / recovery of the fluids in place. The evaluation (forecasting) of these production criteria requires the use of a field simulator able to reproduce (simulate) each of the scenarios. In the case of fractured reservoirs, the permeabilities of the fracture network at the simulator resolution scale (the reservoir mesh) constitute basic information essential for performing these simulations, and decisive for guaranteeing the reliability of the production forecasts.

Advantages

The invention makes it possible to estimate the large-scale permeability (scale of the drainage radius of a well or the inter-well space for example) of these fractures, in a fast and precise manner.

It is then possible to predict the hydrodynamic behavior (flow, pressure, ..) in response to external stresses imposed via wells during the production of hydrocarbons.

The engineers in charge of the exploitation of the deposit then have a tool allowing them to quickly evaluate the performance of different production scenarios, and thus, to select the one that optimizes the exploitation with regard to the criteria selected by the operator, such as ensure optimum hydrocarbon production.

Thus, the invention finds an industrial application in the exploitation of underground deposits, comprising a network of fractures. It may be a hydrocarbon deposit for which it is desired to optimize production, or a gas storage reservoir for example, for which it is desired to optimize the injection or the storage conditions.

Claims (7)

1. Computer-implemented method for optimizing the exploitation of a deposit comprising a fracture network, in which method the deposit is discretized into a set of unit cells and a geometric description of the fracture network in each of the unit cells is generated, the method comprising the following steps:

   - within each cell, a connectivity index that is dependent at least on the number of intersections between fractures is determined by means of said geometric description;

   - the permeability of the fracture network of unit cells the connectivity index of which is higher than a threshold is estimated by selecting, for each unit cell, a method for estimating the permeability of the fracture network depending on the value of the connectivity index, by defining two connectivity thresholds corresponding to two connectivity-index values defining three connectivity-index intervals, and a different method is selected for each of said intervals, so as to optimize the estimation of the permeability of each unit cell;

   - a set permeability value is assigned to the other unit cells the connectivity index of which is lower than said threshold, so as to limit the number of permeability estimations; and

   - the exploitation of the deposit is optimized by simulating fluid flows in said deposit, depending on the permeabilities of the fracture network in each unit cell.
2. Method according to Claim 1, wherein the thresholds are defined empirically.
3. Method according to Claim 1, wherein the thresholds are defined by carrying out the following steps:

   - a set of unit cells each comprising a fracture network for which a geometric description is available is found;

   - a connectivity index is determined for each of the unit cells;

   - a permeability of the network in each unit cell is determined using a flow simulator;

   - a curve of the permeability as a function of the connectivity index is constructed;

   - the thresholds are defined depending on the shape of said curve, so that the permeability obeys the same behavioural law as a function of connectivity index within the three intervals defined by the thresholds.
4. Method according to Claim 3, wherein the set of unit cells each comprising a fracture network for which a geometric description is available is determined by selecting a set of unit cells issued from the discretization of the deposit, the indices of which are distributed over the interval of the connectivity indices computed for all of the unit cells issued from the discretization of the deposit.
5. Method according to one of Claims 1 to 4, wherein:

   - the permeability of the fracture network within the unit cells the connectivity index of which is higher than the second threshold is estimated by means of an analytical formula;

   - the permeability of the network within unit cells the connectivity index of which is comprised between the two thresholds is estimated by means of a flow simulator.
6. Method according to Claim 5, wherein the permeability is estimated depending on the value of the connectivity index.
7. Method according to Claim 6, wherein:

   - the permeability of the fracture network within the unit cells the connectivity index of which is higher than the second threshold is estimated by means of an analytical formula in which the permeability of the network is considered to increase linearly as a function of connectivity index;

   - the permeability of the network within unit cells the connectivity index of which is comprised between the two thresholds is estimated by means of a method in which the permeability of the network is considered to no longer obey the same relationship as above the second threshold.

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