The present invention relates to a process for positioning at least a fluid production well and/or at least a fluid injection well in a field, carried out by an electronic positioning system, the process comprising:

- acquiring a geocellular model of the field, the geocellular model defining a plurality of cells, each cell being provided with fluid properties including at least a fluid density property and at least a fluid flow property for at least two fluid phases.

The method applies in particular for the positioning of injection and production wells in a field containing a hydrocarbon reservoir. It also applies in the positioning of carbon dioxide injection wells in carbon dioxide sequestration applications or in the positioning of water injection wells in hydrogeological applications. More generally, the method applies to any application in which one or more fluids are injected into or produced from a reservoir in the subsoil.

After the process is successfully carried out, wells can be bored in a field according to the positioning which has been determined with the process according to the invention.

The positioning of wells is a critical task in the production of a field containing a hydrocarbon reservoir. Indeed, the respective positions of producer wells and/or of injector wells, is a factor which may greatly affect the productivity of the field and the volume of hydrocarbon recovered, hence its profitability.

A numerical gridded model of the field is often generated to determine the properties of the reservoir contained in the field, including geology, infrastructure, and fluid properties.

Based on this model and on raw field data, a team of scientists determines the best potential locations for wells, usually based on experience, taking into account the constraints which exist in the field, such as distance to surface well head clusters or platforms. Key design parameters include spacing between wells, well drain length and well configurations. This process is time consuming and requires significant human effort and skill.

Software products have been developed to help positioning wells relative to the reservoir. These software products are usually based on calculations of geographic coordinates of the wells. Each well to be positioned is usually defined by a set of three coordinates for each end of the well drain (i.e the fraction of the well length where flow occurs between the reservoir and the wellbore). Therefore, the software must optimize at least six parameters per well. For a set of fifteen wells, the number of parameters raises to ninety, which becomes costly and lengthwise to solve, if possible.

In order to overcome this drawback, methods have been developed to improve well positioning, while decreasing the required resources in terms of computers or human force, by notably reducing the number of variables to be optimized.

For example, WO2019064037 discloses a process of the above mentioned type in which geophysical properties of each cell of the field, such as a distance to another cell or group of cells having at least an undesired property and of a fluid property of the cell are calculated for example by averaging the property among neighboring cells.

Insertion point drivers are then calculated to optimize the position of each well and successively positioning them.

The calculation of distances in the field from a source point is generally carried out using a diffusive time of flight method such as the Fast Marching Method.

In practical implementations, reservoir models used as input to reservoir flow simulators contain information pertaining to the properties of the volume of rock represented by cells and often information relative to flow properties between cells or group of cells. Information relative to the ability to flow between cell often include transmissibility multipliers for example representative of specific geological features present in the field, such as faults.

Transmissibility multipliers create challenges for applying some numerical approach on such models such as the Fast Marching Method to compute diffusive time of flight from source point. Taking into account the transmissivity multipliers is time and resource consuming and very often, the calculations do not take them into account.

This can lead to undesired consequences, such as non-optimum or inadequate positioning of the wells in the field, leading to higher investment costs or/and production inefficiencies.

One aim of the invention is therefore obtaining a process for positioning wells in a field which more accurately takes into account the geological peculiarities of the field, while still being efficient in terms of calculation time and resources.

To this aim, the subject matter of the invention is a process of the above type, characterized by:

- defining, from the fluid properties, a primary transmissivity graph of cells of the geocellular model, the cells forming nodes of the primary transmissivity graph, the primary transmissivity graph having edges for each couple of cells between which fluid flow is considered possible, each edge having a transmissivity calculated from the fluid properties without consideration of transmissivity multipliers, a face direction, and a transmissivity multiplier ; - expanding the primary transmissivity graph to split each edge between two original nodes displaying a transmissivity multiplier whose value is smaller than 1 by adding an additional node having a zero volume and a position centered at mid distance from the original nodes to obtain an expanded transmissivity graph ;

- calculating at least one geophysical property for each cell of at least a group of cells of the geocellular model, using the expanded transmissivity graph;

- positioning or predicting flow in at least a well in the field using the calculated at least one geophysical property of the cells of at least a group of cells of the geocellular model.

The process according to the invention may comprise one or more of the following feature(s), taken solely or according to any technical possible combination:

- expanding the primary transmissivity graph comprises, for each additional node having zero volume and a position centered at mid distance from the original nodes, calculating a permeability of the additional node using the equation:

, wherein K* and KB are respectively the permeabilities of the original nodes, M is the transmissibility multiplier across the edge between the original nodes, and a is a length factor comprised between 0 and 1 ;

- expanding the primary transmissivity graph comprises, for each additional node having zero volume and a position centered at mid distance from the original nodes, calculating the edge transmissibilities between the additional node and the original nodes using the respective equations wherein KA and KB are respectively the permeabilities of the original nodes, S is the surface of contact between the original nodes, L is the length between centers of the original nodes;

- the additional node has edges in the same direction as the edge between the original nodes;

- at least a geophysical property for each cell of at least a group of cells of the geocellular model is an optimized distance from a cell or a group of cells using a time of flight computed using the expanded transmissivity graph;

- the time of flight is calculated with a Fast Marching Method; - calculating at least a geophysical property for each cell of at least a group of cells comprises determining, for each node of the graph, conditions including (i) whether the slowness tensor in the node is isotropic and (ii) whether the node connects only to direct three dimensional neighbor nodes, the calculating step comprising if the conditions (i) and (ii) are met, applying a Sethian time of flight computation to calculate the time of flight through the node or if the conditions (i) and (ii) are not met, applying a Tsitsiklis time of flight computation to calculate the time of flight through the node;

- calculating at least a geophysical property for each cell of at least a group of cells comprises determining a weighed fluid property of each cell, using the fluid property of the cell and of neighboring cells around the cell in the expanded transmissivity graph;

- calculating at least a geophysical property for each cell of at least a group of cells comprises determining a transmissibility probability associated to each edge of the expanded transmissivity graph by scanning all edges of the expanded transmissivity graph, adding the edge transmissivities to a register, building a cumulated density function from the register and obtaining the transmissivity probability from the cumulated density function;

- calculating at least a geophysical property for each cell of at least a group of cells comprises determining at least a propagation path from each cell of the at least one group of cells, computing at each said edge of the or each propagation path, an associated share value as the product of an associated share value of an adjacent edge of the or each propagation path multiplied by the transmissibility probability of the said edge, the weighed fluid property being calculated by summing the products of the associated share value with the fluid property of the cell at each cell along the or each propagation path;

- calculating at least a geophysical property for each cell of at least a group of cells comprises determining an end of the at least one propagation path by calculating a maximal number of cells or/and a maximum cell volume in the at least one propagation path;

- determining a limit of the at least one propagation path comprises calculating a cell number along the at least one propagation path and/or calculating a cumulative value of a property representative of cell volume along the at least one propagation path , the end of the at least one propagation path being reached when the cell number or/and when the cumulative value of the property representative of cell volume reaches a predefined threshold;

- the positioning of the at least one well is carried out by selecting a cell having a maximized weighed fluid property;

- the weighed fluid property value of each cell is representative of a weighed fluid volume producible from the cell; - the process comprises displaying on a display geographical positions of the at least one well or/and fluid flow values in the at least one well obtained from the positioning or predicting flow in the at least a well in the field;

- the process comprises generating a computer file comprising geographical positions of the at least one well or/and fluid flow values in the at least one well obtained from the positioning or predicting flow in the at least a well in the field, the computer file being configured to be uploaded in a computer of a well drilling system or/and in a well production system.

The invention also relates to a system for positioning or predicting flow in at least a fluid production well and/or at least a fluid injection well in a field, comprising:

- an acquiring module configured to acquire a geocellular model of the field, the geocellular model defining a plurality of cells, each cell being provided with fluid properties including at least a fluid density property and at least a fluid flow property for at least two fluid phases,

- a defining module, configured to define, from the fluid properties, a primary transmissivity graph of cells of the geocellular model, the cells forming nodes of the primary transmissivity graph, the primary transmissivity graph having edges for each couple of cells between which fluid flow is considered possible, each edge having a transmissivity calculated from the fluid properties without consideration of transmissivity multipliers, a face direction, and a transmissivity multiplier ; the defining module being configured to expand the primary transmissivity graph to split each edge between two original nodes displaying a transmissivity multiplier whose value is smaller than 1 by adding an additional node having a zero volume and a position centered at mid distance from the original nodes to obtain an expanded transmissivity graph;

- at least a calculating module configured to calculate at least one geophysical property for each cell of at least a group of cells of the geocellular model, using the expanded transmissivity graph;

- a positioning or flow predicting module configured to position or to predict flow in at least a well in the field using the at least one geophysical property of the cells of at least a group of cells of the geocellular model.

The invention also relates to a computer program product comprising instructions, which when the program is executed by a computer, cause the computer to carry out the process as defined above.

The invention will be better understood, based on the following description, given solely as an example, and made in reference to the appending drawings, in which: - [Fig.1 ] Figure 1 is a schematic view of a representation of a geocellular model of a field in which at least a well has to be positioned, the geocellular model comprising a plurality of cells;

- [Fig .2] Figure 2 is a schematic view of several cells of the geocellular model of figure 1 , in which the transmissibility multiplier between cells is not equal to one between at least two cells;

- [Fig.3] Figure 3 is a view similar to figure 2, in which the transmissibility multiplier between two cells has been set equal to 0.2 ;

- [Fig.4] Figure 4 is a schematic view of an expanded graph of the geocellular model in which a sub graph comprising an additional node having zero volume and a position centered at mid distance between two original nodes is defined according to the invention;

- [Fig.5] Figure 5 is a schematic view illustrating how the node with nil volume is positioned;

- [Fig.6] Figure 6 is a table illustrating a hybrid calculation of diffusive time of flight taking into account transmissivity multipliers;

- [Fig.7] Figure 7 is a schematic view illustrating how probably accessible 3D calculations can be carried out to determine a weighed fluid property at a specific cell taking into account the fluid properties at neighboring cells;

- [Fig.8] Figure 8 is a flow chart illustrating the main steps of a process according to the invention;

- [Fig.9] Figure 9 is a flow chart illustrating the hybrid fast marching calculation according to the invention;

- [Fig.10] Figure 10 is a flow chart illustrating the weighed fluid property calculation according to the invention;

- [Fig.11] Figure 11 is a schematic view of a system configured for carrying out the process according to the invention.

A first process according to the invention is carried out for defining the locations of a plurality of wells 10, 12 in a field 14 containing a fluid reservoir (see figure 1). The fluid reservoir is located in a subsurface, onshore or offshore.

The reservoir generally contains at least a first fluid to be produced, and potentially a second auxiliary fluid to be produced along with the first fluid. A third fluid and/or a fourth fluid are advantageously injected in the reservoir to drive the production of the first and/or of the second fluid.

For example, the first fluid is oil and/or gas, the second fluid being gas and/or oil. The third fluid and/or fourth fluid are generally water, gas, and/or oil. The first fluid and the second fluid are preferentially hydrocarbons. The reservoir may comprise several regions, for example at least an aquifer, an oil leg, and a gas cap. An aquifer is generally delimited upwards by a water oil contact or “WOC”. An oil leg is delimited between a water oil contact and a gas oil contact or “GOC”. The gas cap is located above the gas oil contact.

In some instances, the field 14 comprises geological features 17 capable of affecting the transfer of fluids between regions of the field 14, such as faults.

The wells 10, 12 to be positioned in the field 14 are producer wells 10 and injector wells 12.

Producer wells 10 aim at the extraction of a desired fluid, i.e. the first fluid and/or the second fluid. Injector wells 12 are also positioned for injecting the third fluid and/or the fourth fluid to enhance the production of the desired fluid at the producer wells 10.

The wells 10, 12 can be positioned using different patterns. In a dispersed pattern, injector wells 12 are located without preference for area of the reservoir where or close to where injected fluid is originally present. On the contrary, in a peripheral patter injector wells 12 are located with a preference for areas of the reservoir where or close to where injected fluid is originally present. The well positioning patter can be mixed, i.e. be peripheral relative to injectors injecting a particular fluid and dispersed relative to other types of injectors injecting a second type of fluid.

The field 14 is numerically simulated using a geocellular model 16 which is schematically illustrated on figure 1 as a two-dimensional graph forming a grid.

The geocellular model 16 comprises at least one, or sometimes several sets of model realizations, each set containing typically a unique 2D, or 3D grid geometry made of a geocellular grid. The grid geometry is advantageously structured, i.e. follows a geometrical pattern. In a variant, the grid is unstructured.

The grid comprises a plurality of cells 18 defining nodes of the graph. Each cell 18 has a specific geographical position in the model, defined by geographical coordinates. Each cell 18 moreover has a shape and a volume.

The model for example comprises more than 1000 cells 18 and generally between 100 0000 cells 18 and 5000 000 cells 18.

Each cell 18 is associated with cell infilling properties, which characterize the content of the cell 18, as well as the properties of the fluid contained in the cell 18 when applicable.

The cell properties are usually chosen among the net to growth (NTG), the porosity Phi, the total compressibility Ct, the initial saturation in the considered fluid phase Si, the minimum saturation Sm in the considered phase during reservoir flow, the permeability K defined as a XYZ tensor property, K in each direction i = X, Y , Z being noted Ki, a relative permeability, Kr at or behind front for a given injection phase which is also defined as a XYZ tensor property, Krg designating a relative permeability to gas, Krw designating a relative permeability to water and Krwg relative permeability to co-injection of water gas.

Each cell 18 is also characterized by a diffusive pressure propagation slowness Slow, which is a tensor property, by a movable accumulation Accu, which can be defined for the fluid targeted for production and noted AccuP and which can be defined for the fluid targeted for injection as AccuL.

Each cell 18 has general dimensions DX, DY, DZ which can be averaged. Each cell 18 is connected to another cell 18 by edges. Inter-cell properties can be defined by a transmissibility between cells 18.

In the model 16, the fluid properties of each cell 18 are advantageously defined by at least a cell infilling property representative of a fluid density and by at least a cell infilling property representative of an ability of a fluid to flow.

A first cell infilling property is advantageously a diffusive slowness Slow, which can be considered on an anisotropic (XYZ tensor) or on a isotropic basis. In a typical form, the slowness Slow in each cell 18 is equal to:

Slow= Phi x [NTG, for slowness in x and y directions] x Ct/K(x,y or z)/Kr(x,y, or z) (1)

Variants include degenerated or inflated form of the typical form.

A second cell infilling property is a movable accumulation indicator Accu. In a typical form, the accumulation indicator is equal to:

Accu=Phi.Ntg.(Si-Sm) (2)

Variants include degenerated or inflated forms of the typical form.

Another cell infilling property is a dimensionless indicator of the ability of a particular fluid to flow in or out of the wells or on/into/towards neighboring wells. In the typical form, the volume weight mean transmissibility Trans in the three-direction can be written as:

(DX.DY.KZ/DZ+NTG.DY.DZ.KX/DX+NTG.DX.DZ.KY/DY)/(NTG.(DX.DY .DZ)) (3)

Alternatively, a property equal, in each cell 18, to the sum of the transmissibility of all connections to the considered cell 18 divided by the cell volume, or any other indicator of the ability to flow into wells or towards neighboring cells could be used. Using the geocellular model 16, a primary transmissivity graph can be defined including as nodes all cells 18 of the model 16 and as edges all couples of cells 18 for which flow is considered possible (non-nil transmissivity).

Each node is characterized by the cell infilling properties of the corresponding cell. Edges are characterized by a transmissivity computed from connected node properties without consideration of transmissivity multipliers, a face direction, and a transmissivity multiplier (potentially combining the effect of multiple sources of transmissivity multipliers.

In the primary transmissivity graph, when fluids are able to flow between cells 18, the transmissibility multiplier is generally set equal to 1 . The transmissibility multiplier between at least a pair of cells 18 is, in some instances, different than 1 , and in particular smaller than 1 , for example comprised between 0.1 and 0.5, in particular between 0.1 and 0.3, for example equal to 0.2.

The geological features creating multipliers smaller than 1 running along the boundaries of several cells are for example caused by faults planes containing material (e.g., calcite) deposited over geological times or resulting of clay and sand mixing by shearing during fault formation.. Another example would be the deposition at the base of a geological channel of a thin layer of sedimentary material characterized by poor petrophysical characteristics (low permeability and porosity); this so called sedimentary “basal lag” is typically represented by a transmissivity multiplier. In same channel context, a terminal channel plug (a channel filled with clay during deposition) can be represented as a transmissibility multiplier. Due to constraints related to the geomodelling process or flow simulation process, it is very frequent to rely upon models in which geological information support include both cells and cell face or fraction thereof with transmissibility multipliers associated to faces or fraction thereof.

The model 16 is for example an assembly of data obtained from a simulation done in a commercial reservoir modeling simulator such as ECLIPSE or IX (INTERSECT) from SCHLUMBERGER, STARS and IMEX from CMG, or any similar product.

In the process according to the invention, each well 10, 12 is defined within the model by at least one well location cell which is referred to as a well insertion point 20.

Advantageously, each well 10, 12 is defined by at minimum one well insertion point 20, and potentially, a well drain, which is the part of a well which is producing from or injecting into the reservoir. Well drains can be positioned serially along a common trajectory or in parallel manners. The well drain is defined as a series of consecutive intervals joining cells 18 in which flow between the reservoir and the well occurs. At minimum, one well insertion point 20 and one predefined drain length, such as a maximum drain length or a half drain length are enough to define a well drain in the process according to the invention. The predefined drain length is an input of the process.

For example, the well insertion point 20 is chosen as the center of the well drain. The predefined drain length is then a maximum half drain length between the drain center and drain ends.

In a variant, the well insertion point 20 is at one end of the well, the predefined drain length being a maximum full length of the well drain.

In another embodiment, the exact path of the well drain is defined from a first well insertion point 20, by determining other well insertion points 20 of the same well 10, 12.

The process according to the invention is carried out in a system 30 schematically represented in figure 11.

The system 30 generally comprises at least a calculator 32 provided with at least one processor 34, and at least one memory 36 containing software modules configured to be executed by the processor 34.

The system 30 further comprises a display 38 and a man-machine interface 40 generally embodied as a keyboard, a mouse and/or a touch screen.

According to the invention, the memory 36 contains at least a software module 50 for acquiring to the calculator 32 the geocellular model 16 of the field.

The memory 36 further contains a software module 52 for defining the primary transmissivity graph mentioned above from the geocellular model 16 and for expanding the the primary transmissivity graph to obtain an expanded transmissivity graph by splitting each edge displaying a transmissivity multiplier whose value is not equal to 1 and creating an additional node.

The memory 36 further contains at least a software module 54, 56 for calculating at least a geophysical property of each cell 18, using the expanded transmissivity graph, the geophysical property being for example a distance to a predefined cell or groups of cells, and/or a weighed fluid property of the cell 18, taking into account neighboring cells 18, using the expanded transmissivity graph.

The memory 36 further comprises a software module 58 for positioning at least a well 10, 12, preferentially several wells 10, 12 one after another, in a group of potential cells 18 using the geophysical properties calculated by the or each calculating module 54, 56 for each cell 18.

The acquiring module 42 is for example configured to acquire data relative to at least one realization of the model 16 obtained by a reservoir modeling simulator. Each model realization includes the definition of the cells 18 of the grid, and at least the cell infilling properties associated with each cell 18, as defined above.

The acquiring module 42 is advantageously configured to select a group of potential cells among the cells 18 of each model realization, in which a well insertion point 20 can be defined, and to exclude cells in which a well insertion point 20 cannot be defined. Cells in which a well position cannot be defined include for example cells already containing a well, or inaccessible cells given predefined constraints, such as geometrical constraints.

Advantageously, the acquiring module 42 is configured to provide a Boolean indicator to each cell 18, the Boolean indicator being 1 when the cell 18 belongs to the group of potential cells, the Boolean indicator being 0, when the cell 18 is excluded.

In reference to figure 3, the graph defining module 52 is configured, using the acquired data relative to each cell 18 acquired by the acquiring module 42, to define the primary transmissivity graph including as nodes 60A, 60B, all defined cells 18 of the geo-cellular model 16 and as edges 61 , all couples of cells 18 for which flow is considered possible with a non-nil transmissivity.

Each node GOA, GOB is associated with cell infilling properties corresponding to the cell 18 in which the node is located.

In reference to figure 3, the graph defining module 52 is configured to calculate for each node GOA, GOB, a transmissivity computed from connected node properties, without consideration of transmissivity multipliers, a face direction, and a transmissivity multiplier M, potentially combining the effect of multiple sources of transmissivity multipliers, thus defining a primary transmissivity graph as described above.

The graph defining module 52 is then configured to expand, according to the invention, the primary transmissivity graph, to create an expanded transmissivity graph shown in figure 4, in which the transmissivity multipliers are taken into account in the graph.

For all edges 61 displaying a transmissivity multiplier M whose value is greater than 0 and smaller than 1 , the graph defining module 52 is configured to replace the edge with an additional node 62 (see figures 4 and 5), the additional node 62 having a volume equal to zero and a position centered at mid distance from the original nodes GOA, GOB.

The graph defining module 52 is configured to define the additional node 62 with edges 61 A, 61 B in the same direction as the original edge of the original nodes GOA, GOB (see figure 4).

The graph defining module 52 is configured to calculate the permeability of the new additional node 62 using the following equation (1):

The graph defining module 52 is configured to calculate the respective transmissivities TA,TB to the former nodes 60A, 60B using the following equations (2) and (3) :

In equations (1 ) to (3), L is the length between center of cells 18A, 18B (defining nodes 60A, 60B of the expanded transmissivity graph), M is the transmissibility multiplier across edge 61 between nodes 60A, 60B in the primary transmissivity graph, TA, TB are edge transmissivities of edges 61 A, 61 B, K ₁ is the permeability of the additional node 62, S is the surface of contact between cells 18A, 18B defining nodes 60A, 606, and a is a length fraction factor of the additional node 62. The value of a is comprised between 0 and 1 , preferentially smaller than 0.5, in particular between 0.05 and 0.15. An example of value for a is 0.1.

The graph defining module 52 is therefore configured to define the expanded transmissivity graph having nodes 60A, 60B of the primary transmissivity graph, along with additional nodes 62, located at the edges of the primary transmissivity graph having a transmissivity multiplier M strictly greater than 0 and strictly smaller than 1.

In the example of figure 11 , the memory 36 contains at least a first software calculating module 54 to calculate a distance between a given cell 18 and predefined features in the field 14, and at least a second software calculating module 56 to calculate a weighed fluid property P of the given cell 18, taking into account the values of the fluid property in the given cell 18 and in neighboring cells 18 of the given cell 18.

The first calculating module 54 is configured to calculate a distance of each given cell 18 from predefined features in the field 14, preferentially having undesired properties, such as aquifers, barriers, faults and/or facilities such as wells.

The calculation is advantageously done by computing a diffusive time of flight using preferentially a Fast Marching Method computing.

The diffusive time of flight is for example defined as the time of arrival at the predetermined cell or group of cells of a pressured wave propagating in a porous medium from a source point being located at the given cell 18. The calculating module 54 is configured to calculate the diffusive time of flight using the slowness and geometry of each intermediate cell between the cell 18 and the predetermined cell or group of cells, using either a Sethian calculation or a Tsitsiklis calculation.

A Sethian calculation is defined for example in the publication: J.A. Sethian. A Fast Marching Level Set Method for Monotonically Advancing Fronts, Proc. Natl. Acad. Sci., 93, 4, pp.1591-1595, 1996. It is applicable for a given intermediate cell under the following conditions (i) the slowness tensor in the intermediate cell towards which the pressure wave propagates is isotropic and (ii) the intermediate cell edges connect to direct three dimensional neighboring cells.

A Tsitsiklis calculation is defined for example in the publication: Tsitsiklis, J. 1995. Efficient Algorithms for Globally Optimal Trajectories. IEEE Trans, on Automatic Control. It is applicable if the slowness tensor in the intermediate cell towards which the pressure wave propagates is anisotropic or if the intermediate cell edges contains non-neighboring connections

Advantageously, the calculating module 54 is configured to test, for each intermediate cell between the cell 18 and the predetermined cell or group of cells whether conditions (i) and (ii) are met, in which case it is compatible with a Sethian calculation or whether conditions (i) or (ii) are not met, in which case a Tsitsiklis calculation using the expanded transmissivity graph properties.

The results are for example illustrated in column 70 of the table disclosed in figure 6.

Therefore, the calculating module 54 is configured to calculate the time of flight through each intermediate cell (see column 72) using a hybrid approach in which locally, each intermediate cell is tested for its compatibility with Sethian calculation or Tsitsiklis calculation using the expanded transmissivity graph and then, depending on the compatibility conditions, the calculation of time of flight from the intermediate cell to another cell is carried out using the Sethian calculation if compatible or using the Tsitsiklis calculation is the cell is incompatible with the Sethian calculation.

The second calculating module 56 is configured to calculate for each given cell 18 a weighed fluid property P, using the expanded transmissivity graph and the value of the fluid property in the cell 18, and in neighboring cells 18 over a maximum number of cells or a maximum volume of cells around the given cell 18.

To this aim, the second calculating module 56 is configured to initially build an edge transmissibility inverse cumulated density function associated to the expanded transmissivity graph. This is done by scanning all edges of the expanded transmissivity graph, adding the edge transmissivities to a register, classifying the edges by order of transmissivity value in the register to associate a ranking number to each edge and building a cumulated density function using the ranking number.

Then, the calculating module 56 is configured to calculate an inversed cumulated density function representing a transmissivity probability comprised between 0 and 1 to propagate through the edge. The edge having the highest transmissivity and thus the highest ranking number Nmax is associated to a probability of 1 , the edge having the lowest transmissivity and thus the lowest ranking number being associated to a probability of 0.

The edge having the ranking number N is associated with a probability PC given for example by equation (4) in which CDF designates the inverse cumulated distribution function of the transmissibility:

In the example of Figure 7, the calculations can be summed up in the following table:

The calculating module 56 is then configured to determine at least a propagation path 80, 82 in the geocellular model 16 from each given cell 18, and to compute at each edge along the or each propagation path 80, 82, an associated share value AS as the product of an associated share value AS of an adjacent edge of the or each propagation path 80, 82 closer to the given cell 18 multiplied by the transmissibility probability PC of the edge.

In parallel, the calculating module 56 is configured to calculate the weighed fluid property P of the given cell 18 by calculating at each cell along the or each propagation path 80, 82 from the given cell 18, the product PR of the associated share value AS with the fluid property value of the cell 18, and by summing the products PR along the propagation path 80, 82.

The calculating module 56 is configured to determine an end of the at least one propagation path by calculating a maximal number of cells 18 in the or each propagation path 80, 82.

This comprises calculating a cell number along the or each propagation path 80, 82 and/or calculating a cumulative value of a property V representative of cell size along the or each propagation path 80, 82. The end of the at least one propagation path 80, 82 is reached when the cell number or/and when the cumulative value V of the property representative of cell size reaches a predefined threshold.

The weighed fluid property P is thus calculated by summing the products of each cell fluid property with the associated share value at each edge along the at least one propagation path 80, 82 until a predefined number of cells or/and a predefined cumulative value V has been reached.

Such a calculation is carried out for all cells 18 of the selected group of cells. Then, the value of P is normalized over the whole group of cells 18.

In the example of Figure 7, the calculation is summed up in the following table:

The positioning module 58 is configured to calculate at least an optimal position for a well 10, 12 using at least one geophysical property calculated by the calculating modules 54, 56.

For example, the geophysical properties calculated by the calculation module 54 are used to position the well insertion point 20 of the well 10, 12. In particular, the well insertion point 20 is positioned to maximize the distance from wells of a given type, in particular wells of the same type as the type of well 10, 12 which is to be positioned, and also to maximize the distance to a undesired fluid phase present in the field 14, or to minimize the distance or optimize the distance to wells of another type.

The positioning module 58 is also configured to use the result of the calculating module 56 to localize the insertion point of well 10, 12 at a position where the fluid production can be maximized and/or the pore volume is suitable to accommodate injected fluid. Preferentially, the positioning of the wells 10, 12 is sequential, each well 10, 12 being positioned one after the other. More preferably, the positioning comprises calculating at least a distance maximization insertion point driver from the distances calculated by calculation module 54, at least a fluid property insertion point driver calculated from the weighed fluid properties calculated by calculation module 56 and potentially a combined insertion point driver using the or each distance maximization insertion point driver or/and the or each fluid property insertion point driver. Such a method is disclosed for example in WO 2019/064037.

Once the position of the well 10, 12 has been calculated in the field 14, the system 30 may be configured to display on the display 38 a graph of the field illustrating geographical positions of the at least one well 10, 12 or/and fluid flow values in the at least one well 10, 12 obtained from the positioning in the at least a well 10, 12 in the field from the positioning module 58.

The system 30 may also be configured to generate a computer file comprising geographical positions of the at least one well 10, 12 or/and fluid flow values in the at least one well 10, 12 obtained from the positioning or predicting flow in the at least a well 10, 12 in the field. The computer file is configured to be uploaded in a computer of a well drilling system or/and in a well production system.

Finally, an actual well can then be bored in the field 14 at the determined position.

An example of process for positioning at least a fluid production well 10 and/or fluid injection well 12 in a field 14, carried out by the electronic positioning system 30, will now be described in reference to figures 8 to 10.

Initially, at step 100, the acquiring module 42 acquires data relative to at least one realization of the model 16 obtained by a reservoir modeling simulator, including the definition of the cells 18 of the grid, and at least the cell infilling properties associated with each cell 18, as defined above.

The acquiring module 42 advantageously selects a group of potential cells among the cells 18 of each model realization in which a well insertion point 20 can be defined.

At step 102, the graph defining module 52 uses the acquired data relative to each cell 18 acquired by the acquiring module 42, and defines the primary transmissivity graph including as nodes 60A, 60B, all defined cells 18 of the geo-cellular model 16 and as edges 61 , all couples of cells 18 for which flow is considered possible with a non-nil transmissivity, as shown schematically on figure 3.

At step 104, the graph defining module 52 then expands, according to the invention, the primary transmissivity graph, to create an expanded transmissivity graph shown in figure 4. For all edges 61 of the expanded transmissivity graph displaying a transmissivity multiplier M whose value is greater than 0 and smaller than 1 , the edge is replaced with an additional node 62 (see figures 4 and 5). The additional node 62 has a volume equal to zero and a position centered at mid distance from the original nodes 60A, 60B.

The graph defining module 52 calculates the permeability of the new additional node 62 and the transmissivities to the original nodes 60A, 60B using equations (1) to (3).

Then, at step 106, the software module 54, 56 calculates at least a geophysical property of each cell 18, using the expanded transmissivity graph.

At step 106A shown in figure 9, in sub step 110, the first calculating module 54 selects a given cell 18 and calculates a distance of the given cell 18 from predefined features in the field 14, preferentially having undesired properties, such as aquifers, barriers, faults and/or facilities such as wells.

The calculation is advantageously done by computing a diffusive time of flight using preferentially a Fast Marching Method computing.

At step 112, the calculating module 54 carries out a test, for each intermediate cell between the given cell 18 and the predetermined cell or group of cells whether the above conditions (i) and (ii) are met, in which case it is compatible with a Sethian calculation or whether conditions (i) or (ii) are not met, in which case a Tsitsiklis calculation using the expanded transmissivity graph properties.

The results are for example illustrated in column 70 of the table disclosed in figure 6.

Therefore, depending on the compatibility conditions, it carries out the calculation of time of flight through the intermediate cell to another cell using the Sethian calculation (sub step 114) if compatible with the Sethian calculation or using the Tsitsiklis calculation (sub step 116) is the intermediate cell is incompatible with the Sethian calculation.

At step 118, the calculating module 54 adds to the previously calculated time of flight, the time of flight calculated through the intermediate cell and loops until the predetermined cell or group of cells is reached (sub step 120).

At step 106B, shown in figure 10, the second calculating module 56 calculates for each given cell 18 a weighed fluid property P, using the expanded transmissivity graph and the value of the fluid property in the cell 18, and in neighboring cells 18 over a maximum number of cells or volume of cells around the given cell 18.

At sub step 128, the second calculating module 56 builds an edge transmissibility inverse cumulated density function associated to the expanded transmissivity graphas explained above.

Then, the calculating module 56 calculates an inversed cumulated density function representing a transmissivity probability comprised between 0 and 1 to propagate through the edge The edge having the highest transmissivity and thus the highest ranking number Nmax is associated to a probability of 1 , the edge having the lowest transmissivity and thus the lowest ranking number being associated to a probability of 0.

The edge having the ranking number N is associated with a probability PC defined in equation (4).

At sub step 130, the second calculating module 56 selects a given cell 18 at which the weighed fluid property P has to be calculated.

It then initializes

- a “stop counter value” V to zero, corresponding to the cell number or/and cumulative value of a property V representative of cell size;

- an empty “map of neighbors" associating references to nodes to values;

- an empty “list of visited nodes” ;

- a “start node fluid property value” at zero;

- the “target node” as the “start node” with an “associated share value” of one

At sub step 132, the second calculating module 56 determines if the “stop counter value” is below the a priori defined threshold. If yes, at sub step 134, for every neighbor of the target node not part of the list of visited nodes, the second calculating module 56 computes a new “associated share value" AS as the product of the “associated share value” AS of the target node multiplied by the calculated probability PC of the neighbor to target node transmissivity.

At sub step 136, if the neighbor is already part the “list of neighbors”, the second calculating module 56 checks whether its associated share value AS is higher than the new “associated share value” AS. If true, then it keep the existing associated share value, and if false, it updates the value to the “associated share value” AS. If the neighbor is not already part of the “list of neighbors”, it adds it with the new associated share value.

At sub step 138, the second calculating module 56 selects amongst the list of neighboring nodes, the nearest neighbor node displaying the largest associated share value to define an additional cell of the propagation path 80, 82.

The second calculating module 56 adds the product of the nearest neighbor node associated share AS by the nearest neighbor node property P to the “start node property value".

It also increments a cell number or/and adds the nearest neighbor node property V representative of cell size to the stop counter value. The second calculating module 56 adds the nearest neighbor node and its associated share value to the list of visited nodes and to a propagation path 80, 82.

It then loops back to sub step 132. At sub step 132, if the second calculating module 56 determines that the “stop counter value” is above the predefined threshold, an end of the or each propagation path 80, 82 from the given cell 18 is defined. The weighed fluid property P for the given cell 18 is then equal to the accumulated “start node property value".

At sub step 140, the second calculating module 56 then loops to sub step 130 to carry out again steps 132 to 138 with another given cell 18, until all given cells 18 of the selected group of cells are associated with a weighed fluid property P, at sub step 142.

In reference to figure 8, the positioning module 58 then calculates at least an optimal position for a well 10, 12 using at least one geophysical property calculated by the calculating modules 54, 56, as described above.

Thanks to the creation of an expanded transmissivity graph in which each edge 61 between two original nodes 60A, 60B displaying a transmissivity multiplier whose value is smaller than 1 is replaced with an additional node 62 having a zero volume and a position centered at mid distance from the original nodes 60A, 60B, it is possible to very easily take into account complex transmissivity features such as resulting from geological faults in the positioning of wells 10, 12 in a field.

Indeed, the expanded transmissivity graph can be used to characterize the quality of each cell by computing diffusive time of flight from peculiar geological features or facilities (e.g. aquifers or wells) by a Fast Marching Method. The process according to the invention factors transmissivity multiplier information in the diffusive time of flight computations, which improves the accuracy of the computation as a quality indicator and in the algorithmic steps taken to adapt the computation to some peculiar reservoir configurations to maintain computational efficiency.

The expanded transmissivity graph can also be used to characterize quality of a cell defined as a weighed fraction of any fluid property characterizing the cells in which neighboring cells and their weights are computed recursively by propagation to the nearest neighbor and distances computed by convolution of the distance to the parent cell and a probability of propagation function of the transmissivity between cells.

The methods thus establishes reservoir quality indicators for cells relative to suitability to positioning wells capturing heuristics such as the drive to locate wells in region where there is fluid to produce or displace, or where the ability to flow is high.

Producer wells 10 and/or of injector wells 12 can then be bored in the field based on positions obtained by the process according to the invention, to impove the productivity of the field and the volume of hydrocarbon recovered, hence its profitability.

In a variant or in complement, the memory 36 of the system further comprises a software module 58 for predicting flow in at least a well 10, 12 of the field, using the values of diffusive time of flight determined by the calculation module 54 or/and using the values of weighed fluid property of the cells 18.

For example, the fluid flow in the well 10, 12 is calculated using the diffusive time of flight from an injector well 12 to each cell 18 of a group of cells between the injector well 12 and the producer well 10 and the diffusive time of flight between each cell 18 of the group of cells and the producer well.

Based on the predicted flow, actual wells 10, 12 can be bored and fluid can be injected in well 10 and produced from well 12.