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Method for Providing a Numerical Model of a Sample of Rock

20180321127 · 2018-11-08

    Inventors

    Cpc classification

    International classification

    Abstract

    The present invention relates to a method for providing a numerical model of a sample of rock that, when used for flow simulations, it reproduces the porosity and the permeability according to the measurements taken in said sample of the rock. The method is characterized in that the structure and the properties of the numerical model are populated randomly ensuring that the global behavior reproduces the measurements.

    Claims

    1. A method for providing a numerical model of a sample of rock, in particular a plug sample taken from a sample of rock, said sample of rock taken from a vertical portion of the rock core, according to its longitudinal direction, extracted from a well of an oil or gas reservoir, the method comprising: performing a CT scan of the plug sample of rock; measuring a porosity .sub.ms of the plug sample of rock; measuring a permeability k.sub.ms of the plug sample of rock; generating a 3D numerical model, at least comprising a cell discretization representing a volume of the plug sample of rock, according to the following steps: a) recovering data from the CT scan providing at least information on a statistical density function () of the value , wherein is an attenuation X-ray radiation in the volume of the plug sample of rock; b) generating a segmentation of the discretization of the first 3D model into sub-volumes responsive to the data retrieved in step a) from the plug sample of rock, said sub-volumes representing components of the plug sample of rock; c) identifying the components of each sub-volumes; d) determining a porosity statistical distribution function F() for the cells of the 3D model at least responsive to the statistical density function (); e) populating a porosity for each sub-volume of the 3D model, wherein the porosity is spatially distributed among the cells of said sub-volume by means of a Gaussian simulation algorithm responsive to the porosity statistical distribution function F() as an approximation of a Gaussian density function; f) populating a permeability among the cells of each sub-volume comprising a component contributing to the porosity, wherein the permeability is defined as a scalar function responsive to the porosity of the cell k(); g) performing a numerical simulation of the 3D model according to conditions relating to the permeability k.sub.ms in order to obtain a global permeability k.sub.num; h) iteratively repeating steps e) to g) until the global permeability measured on the plug sample of rock k.sub.ms and the permeability k.sub.num calculated from the 3D model differ less than a threshold.

    2. The method according to claim 1, wherein the porosity statistical distribution function F() at step d) is determined as follows: determining CT.sub.1=|CT.sub.meanCT.sub.min| and CT.sub.2=|CT.sub.maxCT.sub.mean| as the values ranges from CT.sub.min to CT.sub.max, and the statistical density function () has a mean value at CT.sub.mean; being .sub.1=|.sub.mean.sub.min| and .sub.2=|.sub.max.sub.mean| wherein .sub.min is a minimum value of the porosity , .sub.max is a maximum value of the porosity ; and, .sub.mean is a mean value of the porosity statistical distribution function F() fixed as the measured porosity .sub.ms; provide the porosity statistical distribution function F() as a symmetric function of the statistical density function () of the value , scaled and shifted according to: 2 1 = .Math. .Math. CT 1 .Math. .Math. CT 2

    3. The method according to claim 1, wherein for determining the porosity statistical distribution function F() on step d), the density function () is further truncated as .sub.c() by cutting-off a left tail of the density function () at a pre-specified value C[.sub.min, .sub.max] providing a truncated porosity statistical distribution function F.sub.c() corresponding to the porosity values of those components that do not contribute to the porosity of the plug sample of rock; and, when populating the porosity in step e), said step is limited to those sub-volumes representing a component contributing to the porosity of the plug sample of rock and the Gaussian simulation algorithm is responsive to the truncated porosity statistical distribution function F.sub.c().

    4. The method according to claim 1, wherein step h), when the global permeability measured on the plug sample of rock k.sub.ms and the permeability k.sub.num calculated from the 3D model differs more than the threshold, before repeating steps e) to g) step h) further comprises truncating the porosity statistical distribution function by specifying a new value of c[.sub.min, .sub.max].

    5. The method according to claim 1, wherein the Gaussian simulation algorithm of step e) is a sequential Gaussian simulation algorithm.

    6. The method according to claim 1, wherein the Gaussian simulation algorithm of step e) is a Gaussian random function simulation algorithm.

    7. The method according to claim 1, wherein a first porosity statistical distribution function F() responsive to the statistical density function () in step d), when represented within a normalized range [0,1], is calculated as: measuring a mean porosity .sub.mean from the plug sample of rock according to: .Math. .Math. CT 2 .Math. .Math. CT 1 = 1 2

    8. The method according to claim 1, further including evaluating a connectivity of the components contributing to the porosity of the plug sample of rock by: measuring a permeability k.sub.ms(s) of the plug sample of rock using two fluids, based on a saturation s of one fluid with respect to the other fluid for a specified range of the E variable; performing a numerical simulation according to step g) that further provides the numerical values of the permeability k.sub.num(s) for at least the specified range of the s variable; and iteratively repeating steps e) to g) according to step h) until the global permeability measured on the plug sample of rock k.sub.ms (S) and the permeability k.sub.num (s) calculated from the 3D model differ less than the specified threshold for the range of the s variable measured under a specified norm.

    9. The method for providing a 3D numerical model of the sample of rock taken from a vertical portion of the core extracted from a well of an oil or gas reservoir, comprising: performing a CT scan of the sample of rock; identifying facies and a number of components of the facies of the sample of rock; for each facies, extracting a plug sample of rock from said facies of the sample of rock; generating a first 3D numerical model according to claim 1, at least comprising a cell discretization representing a volume of the plug sample of rock; generating a second 3D numerical model, at least comprising a cell discretization representing a volume of the sample of rock, taken as a longitudinal portion of the rock core, wherein the volume of the sample of rock comprises the volumes of the plugs and the cell discretization of the second 3D numerical model is clustered according to the facies of the 3D sample; for each cluster representing a facies, the components are stochastically populated to the cell discretization of said second 3D numerical model by means of a MPS algorithm (Multipoint Statistics) responsive to patterns provided by the cell discretization of the plug sample of rock of the first 3D model of its facies and restricted to a condition that the number of components n of the sample of rock and the partial rate for each component .sub.i, i=1 . . . n is maintained; for each cluster representing a facies, the porosity are stochastically populated to the cell discretization of said second 3D numerical model by means of a Gaussian algorithm; for each cluster representing a facies, the permeability is populated among the cells as a scalar function responsive to the porosity of second 3D numerical model k().

    10. The method according to claim 9, wherein before stochastically populating to the cell discretization of the second 3D numerical model from the cell discretization of the first 3D numerical model of the plug sample of rock, said cell discretization of the plug sample of rock is coarsened.

    11. The method according to claim 10, wherein the coarsening of the cell discretization of the plug sample of rock comprises: generating a coarser cell discretization than the cell discretization of the plug wherein each cell of the coarser discretization comprises one or more cells of the cell discretization of the plug sample of rock; each cell of the coarser discretization comprises the component with higher frequency among the components of the cell discretization of the plug sample of rock; a density of each cell of the coarser discretization is calculated as a mean of the density among the cells of the cell discretization of the plug sample of rock; a porosity of each cell of the coarser discretization is calculated as a mean of the porosity among the cells of the cell discretization of the plug sample of rock; a permeability of each cell of the coarser discretization is calculated as k() from the porosity value at said cell of the coarser discretization; provide the coarser cell discretization as a new discretization of the plug sample of rock to be used in the MPS algorithm.

    12. The method according to claim 9, wherein the porosity, the permeability, or both, are calculated in a plurality of sections along the longitudinal direction of the second 3D numerical model providing one or more discrete functions.

    13. A computer program product adapted to carry out a method according to claim 1.

    Description

    DESCRIPTION OF THE DRAWINGS

    [0063] The foregoing and other features and advantages of the invention will be more clearly understood based on the following detailed description of a preferred embodiment provided by way of illustrative and non-limiting example in reference to the attached drawings.

    [0064] FIG. 1 This figure shows a schematic representation of a sectional view of a reservoir located under the surface to be explored.

    [0065] FIG. 2 The figure shows the sample of rock extracted from the well with several facies corresponding to the facies of the reservoir; and, it also shows plug samples taken from each facies of said sample of rock.

    [0066] FIG. 3a, 3b, 3c FIGS. 3a, 3b and 3c show three functions, the statistical density function () obtained from the CT scan, the porosity density function F() obtained from () and the truncated statistical density function .sub.c().

    DETAILED DESCRIPTION OF THE INVENTION

    [0067] A detailed description of the invention is disclosed wherein the invention is used over a particular reservoir as it is shown in the schematic sectional view in FIG. 1. The reservoir is limited by a surface (S) that is being explored using methods for retrieving limited data from the subsoil.

    [0068] Wells (1) are a common practice for retrieving data from the subsoil. The well (1) provides a large sample of rock (2) along the vertical direction Z-Z, shown as a dashed line, at certain location on the surface (S) to be explored.

    [0069] Numerical models of the reservoir are built by discretizing certain volume (V) under the surface (S) to be explored and, soil properties must be defined in the numerical model. Such discretization and the equations involved in modeling the volume (V), being V the numerical domain, are chosen depending on the facies located within the volume (V) and its properties.

    [0070] During this process, the large sample of rock (2) obtained from the well (1) is used for retrieving valuable information for defining facies (F.sub.1, F.sub.2, F.sub.3, F.sub.4, F.sub.5) and the physical properties of the components.

    [0071] According to an embodiment of the present invention, a 3D numerical model of the large sample of rock (2) obtained by drilling along the vertical direction Z-Z will be generated in a two steps method, a first step provides numerical models of plugs extracted from each facies (F.sub.1, F.sub.2, F.sub.3, F.sub.4, F.sub.5) and a second step populates said numerical models to the entire domain of the sample of rock (2).

    [0072] In this example, the large sample of rock (2), the core, after the extraction is qualitatively analyzed by a skilled person, for instance by using a CT scan. The 3D image of the CT scan allows the skilled person to identify facies (F.sub.1, F.sub.2, F.sub.3) and regions of different components. Additional laboratory experiments, such as measurement of natural radiation, provide a more accurate qualitative description of the large sample or rock.

    [0073] FIG. 2 shows the sample of rock (2) extracted from the well (1) with three facies (F.sub.1, F.sub.2, F.sub.3) corresponding to the structure of the underground facies (F.sub.1, F.sub.2, F.sub.3, F.sub.4, F.sub.5), each facies (F.sub.1, F.sub.2, F.sub.3) separated by an interface identified as I.sub.1 and I.sub.2.

    [0074] This qualitative description is used for obtaining plugs (P.sub.1, P.sub.2, P.sub.3), small samples of rock from each facies (F.sub.1, F.sub.2, F.sub.3) previously identified.

    [0075] According to the first aspect of the invention, for each plug (P.sub.i, i=1 . . . N, being N the total number of facies) a 3D numerical model is built wherein said 3D numerical models, according to an embodiment, will have a high resolution discretization of the volume of the plug; that is, the numerical model comprises a large number of cells in order to provide an accurate model of the sample of rock.

    [0076] A laboratory measurement is carried out over the plug, wherein a low viscosity fluid, usually helium or nitrogen, is injected into the plug wherein the injected volume determines the porosity value .sub.ms of the plug.

    [0077] A second one is carried out over the same plug in order to obtain the permeability k.sub.ms along a certain direction. In this embodiment the plug has a cylindrical shape and the direction is the longitudinal. The plug is closed in its lateral wall. Then, a fluid is injected into one of the circular surfaces at certain pressure in respect to the pressure at the opposite circular surface. The differential pressure imposes a flow which is driven in the longitudinal direction of the plug with an internal configuration of the velocity field depending on the porosity and the permeability of said plug. The total flow provides a measurement of the permeability of the plug k.sub.ms.

    [0078] A CT scan of the plug provides a gray scale image and the voxels of the image provides a discretization of the plug. In this embodiment, the discretization of the image is used as the discretization of the numerical model of the plug.

    [0079] The image, according to this embodiment, is used for clustering the discretization according to the components of the plug by means of a segmentation algorithm and, those clusters identified with components not having porosity will not be used for propagating the porosity and the permeability reducing the computational cost of the simulations that will be carried out at a later stage.

    [0080] The CT scan provides information related to the porosity of the plug as the attenuation of the X-rays radiation is inversely proportional to the density of the component. FIG. 3a shows the statistical density function of the CT scan measurement, (). The CT scan statistical density function ranges between CT.sub.min and CT.sub.max, and a mean value CT.sub.mean is calculated from () therefore it is a known value.

    [0081] As it has been indicated, the CT scan statistical density function is related with the porosity function. Therefore the CT.sub.mean value is equal to the experimental porosity measurement .sub.ms.

    [0082] As disclosed, the porosity statistical distribution function F() at step e) is determined as follows: [0083] determining CT.sub.1=|CT.sub.meanCT.sub.min| and CT.sub.2=|CT.sub.maxCT.sub.mean| as the values ranges from CT.sub.min to CT.sub.max, and the statistical density function () has a mean value at CT.sub.mean; [0084] being .sub.1=|.sub.mean.sub.min| and .sub.2=|.sub.max.sub.mean| wherein .sub.min is the minimum value of the porosity , .sub.max is the maximum value of the porosity ; and, .sub.mean is the mean value of the porosity statistical distribution function F() fixed as the measured porosity .sub.ms; [0085] provide the porosity statistical distribution function F() as the symmetric function of the statistical density function () of the value , scaled and shifted such as:

    [00002] 2 1 = .Math. .Math. CT 1 .Math. .Math. CT 2

    [0086] Under this condition, and as it is shown in FIGS. 3a and 3b, the porosity statistical distribution function F() is the statistical density function () after being flipped according to a vertical axis and properly scaled according to the former relationship .sub.2/.sub.1=CT.sub.1/CT.sub.2. Once all the values are known the curve is normalized between .sub.min and .sub.max, these values can range [0, 1].

    [0087] After determining if there are components which do not contribute to the porosity distribution, the density function (), in an embodiment, a further truncated density function .sub.c() is determined by truncation cutting-off the left tail of the function () at a pre-specified value c[CT.sub.minCT.sub.max] for the determination of the porosity statistical distribution function F(). The procedure explained in step e) is performed to get a distribution as in the FIG. 3c.

    [0088] The pre-specified value c corresponds to the attenuation cut-off value identifying those components that do not contribute to the porosity of the plug sample of rock; that is, those components represented by the values located in the truncated left tail.

    [0089] When populating the porosity in step e) of the method, according to the first aspect of the invention, said step is limited to those sub-volumes representing a component contributing to the porosity of the plug sample of rock. In the same step, the Gaussian simulation algorithm is applied as being responsive to the truncated density statistical distribution function .sub.c(). The modified porosity statistical distribution function F() will be identified as the truncated porosity statistical distribution function F.sub.c(). In a particular embodiment, a Gaussian simulation algorithm is implemented by using an interface comprising a pointer to a statistical distribution function for passing the function by reference. The instantiated porosity statistical distribution function F.sub.c() is allocated to said pointer in such a way the Gaussian simulation algorithm directly uses said porosity statistical distribution function F.sub.c().

    [0090] Once the porosity statistical distribution function F() or F.sub.c() is determined it is populated for each sub-volume of the 3D model by spatially distributing it by means of a Gaussian simulation algorithm responsive to said porosity statistical distribution function (F() or F.sub.c()).

    [0091] The permeability k is deemed to depend on the porosity according to a continuous function k(), so the permeability is populated for each cell of the sub-volume responsive to the porosity already calculated, that is, the permeability is obtained as responsive to the porosity according to a predetermined function k().

    [0092] The proposal of the 3D numerical model obtained is numerically simulated according to the same conditions used in the laboratory for measuring the porosity and the permeability. Said simulations provide the calculated porosity and the calculated permeability.

    [0093] If the difference between the calculated and the measured properties are less than a pre-specified threshold then the proposed 3D model is deemed to be valid. If not, the porosity and the permeability is recalculated as disclosed wherein the process of generating a new scalar field for the porosity and permeability is iteratively repeated until the calculated porosity and the calculated permeability differ less than the pre-specified threshold.

    [0094] This iterative method may converge very slowly or even not converge. If this is the case, before generating a new porosity and permeability property for the discretized domain the domain is reclustered trying a different cluster.

    [0095] From the computed statistical density function F(), a truncated function F.sub.c() is proposed as the probability density function of the porosity. Another variable used for avoiding the slow convergence is the cut-off c value of the attenuation. These criteria, the cut-off value c or the clustering, may be recalculated if the iterations carried out generating the porosity and the permeability are more than a pre-specified value.

    [0096] Once the method has converged, the 3D numerical model reproduces the porosity and the permeability of the plug for the .sub.ms and the k.sub.ms measurements.

    [0097] The method further comprises additional checks in the connectivity of the components that contribute to the porosity of the plug sample of rock by further comprising: [0098] measuring the permeability k.sub.ms(s) of the plug sample of rock by an experiment with two fluids, depending on the saturation s of one fluid in respect to the other fluid for a pre-specified range of the s variable, [0099] the numerical simulation according to step g) is extended to provide the numerical values of the permeability k.sub.num(s) for at least the same range of the s variable, [0100] the condition of the iterative process in step h) is further limited to the condition that the global permeability measured on the plug sample of rock k.sub.ms (s) and the permeability k.sub.num(s) calculated from the 3D model differ less than a pre-specified threshold for the range of the s variable measured under a pre-specified norm.

    [0101] The use of two fluids ranging with different values of the saturation s of one fluid in respect to the other fluid provides a function representing the permeability k.sub.ms (s) in the range of s being measured.

    [0102] The iterative process applied has a stricter criterion as the permeability of the numerical model must fulfill the measurements in a range of the saturation variable, not only in just one value.

    [0103] The 3D model obtained for the plug will be used as the basis for generating the corresponding facies of the 3D model of the large sample of rock, the core. If the same resolution of the plug is used in the 3D model of the large sample, the discretization for the core would be too large.

    [0104] The size of the sample of rock taken from this well is usually quite large and the generation of a 3D numerical model for it may require a computational cost that may be non-affordable.

    [0105] In this embodiment, according to the 3D numerical model of the plug sample taken from the core or sample of rock (2) before stochastically populating to the cell discretization of the second 3D numerical model from the cell discretization of the first 3D numerical model of the plug sample of rock, said cell discretization of the plug sample of rock is coarsened in order to reduce its discretization size.

    [0106] The method, when including the coarsening of the cell discretization of the plug sample of rock, comprises: [0107] generating a coarser cell discretization than the cell discretization of the plug wherein each cell of the coarser discretization comprises one or more cells of the cell discretization of the plug sample of rock, [0108] each cell of the coarser discretization comprises the component with higher frequency among the components of the cell discretization of the plug sample of rock, [0109] the density of each cell of the coarser discretization is calculated as the mean of the density among the cells of the cell discretization of the plug sample of rock, [0110] the porosity of each cell of the coarser discretization is calculated as the mean of the porosity among the cells of the cell discretization of the plug sample of rock, [0111] the permeability of each cell of the coarser discretization is calculated as k() from the porosity value at said cell of the coarser discretization, [0112] make available the coarser cell discretization as the new discretization of the plug sample of rock to be used in the Multipoint Statistics (MPS) algorithm.

    [0113] The coarsened 3D numerical model is used as training images to propagate the spatial component structure at core scale. Using Gaussian simulation methods the porosity is populated as a function of the components and respecting the experimental results. The permeability is estimated according to a continuous function k(). The 3D numerical model provides a pattern of the internal structure of the components and the components itself that is propagated along a bigger structure such as the core.

    [0114] As a summary, a method for providing a 3D numerical model of the sample of rock taken from a vertical portion of the core extracted from a well of an oil or gas reservoir is proposed according to an embodiment of the invention, this method comprises the following steps: [0115] carrying out a CT scan of the sample of rock, [0116] identifying the facies and the number of components of the facies of the sample of rock, [0117] for each facies, [0118] extracting a plug sample of rock from said facies of the sample of rock, [0119] generating a first 3D numerical model for the plug sample of rock according to the first aspect of the invention, at least comprising a cell discretization representing the volume of said plug sample of rock, [0120] generating a second 3D numerical model, at least comprising a cell discretization representing the volume of the sample of rock, taken as the longitudinal portion of the core, wherein the volume of the sample of rock comprises the volumes of the plugs and wherein: [0121] the cell discretization of the second 3D numerical model is clustered according to the facies of the 3D sample, [0122] for each cluster representing a facies, the components are stochastically populated to the cell discretization of said second 3D numerical model by means of a MPS algorithm (Multipoint Statistics) responsive to the patterns provided by the cell discretization of the plug sample of rock of the first 3D model of its facies and restricted to the condition that the number of components n of the sample of rock and the partial rate for each component .sub.i, i=1 . . . n is kept, [0123] for each cluster representing a facies, the porosity are stochastically populated to the cell discretization of said second 3D numerical model by means of a Gaussian algorithm, [0124] for each cluster representing a facies, the permeability is populated among the cells as an scalar function responsive of the porosity of second 3D numerical model k(), [0125] making the second 3D numerical model available.

    [0126] Therefore, the proposed method generates a 3D numerical model of the sample of rock in two steps: [0127] first generating 3D numerical models of plug samples of rock taken from each facies of the sample of rock, wherein the 3D numerical model of each plug is quite smaller than the size of the sample of rock; and [0128] second generating the 3D numerical model of the entire sample of rock populating data from the 3D models of the plug corresponding to each facies while keeping two constraints: the number of components of the sample of the rock and the partial rate for each component is kept.

    [0129] The 3D numerical model of the sample of rock (2) taken from the well (1) generated as disclosed reproduces the porosity and the permeability as it has been generated propagating the patterns of the individual 3D models of each plug sample taken for each corresponding facies.

    [0130] Nonetheless, the 3D numerical model may no reproduce the dynamical behavior of the sample of rock. If this is the case, the developed models are dynamically simulated to validate laboratory results and, if not, the method is iteratively carried out until it is validated.

    [0131] In this case, when a plurality of patterns taken from the 3D numerical model of the plug are populated within the discretization of the large 3D model of the sample of rock, the combination of patterns provide flow paths that shows dynamical properties that may differ from those measured in the sample of rock (2) extracted from the well (1). These paths are randomly regenerated when the plurality of patterns are statistically propagated. The 3D numerical model of the core or sample of rock (2) is generated as many times as needed to reach the convergence of the iterative method.

    [0132] According to a preferred embodiment, a plurality of 3D models generated as disclosed and, iteratively compared with the measurements of the samples of rock for a range of the saturation variable.

    [0133] According to an embodiment, the comparison is as follows: [0134] i) in a first step the simulation of the fluid flow over the 3D numerical model at a specific saturation generates a relative permeability value; [0135] ii) this relative permeability value is compared with the value of the experiment measured under the same conditions; [0136] iii) If the 3D model shows a permeability value (flow properties) differing less than a predetermined threshold, the permeability value is kept; [0137] iv) otherwise the permeability is modified by repeating from i) to iii).

    [0138] In this case, the validated 3D model will be that showing flow properties that differ less than a predetermined threshold measured for the entire range.

    [0139] The method further comprises any calculation of the porosity, the permeability or both in a plurality of sections along the longitudinal direction of the second 3D numerical model providing one or more discrete functions.