Method of determining maximum stress in a well drilled in a reservoir

Abstract

A method of determining maximum stress in a well drilled in a reservoir, primarily a hydrocarbon reservoir, where there is at least one zone. Collapse regions are produced while drilling a well because the material of the wall of the well exceeds its maximum allowable stress, the material fractures and falls off, leaving a cavity. The caliper of the damaged zone is measured by devices that extend radially until coming into contact with the physical wall of the well. The disclosed method determines the maximum allowable stress based on the caliper measurements and other variables which are determinable.

Claims

1. A computer-implemented method of determining a maximum stress σ.sub.max at a point of a path {right arrow over (r)}(t) in a well drilled in a reservoir comprising collapsed regions, wherein said method comprises the steps of: a) generating a geomechanical computational model in a domain comprising the path {right arrow over (r)}(t) of the well that is drilled by at least incorporating rock data and the mechanical properties thereof, and wherein generating said geomechanical computational model includes: 1. measuring the diameter D of the well using a measurement tool configured to acquire measurements of the interior walls of the well, wherein the measurements are acquired while drilling the well or after the well is drilled, 2. using the measurement tool, obtaining the measurement of the value of the caliper C(t) measured in at least one collapsed zone, and 3. a pre-established function of a minimum stress σ.sub.min(t); b) generating a fluid computational model in the same domain by at least incorporating data with respect to a density γ(z) of the drilling fluid used in each level of vertical depth z(t) from the surface of the reservoir while drilling the well, wherein said fluid computational model models the rock as a porous medium and comprises a pore pressure p.sub.p in said porous medium; c) for a given point of the path of the well {right arrow over (r)}(t=t.sub.0) with a region of damage due to the collapse in the wall of said well, determining: 4. the level z(t.sub.0) of vertical depth measured from the surface of the reservoir, and 5. a section S having a circular configuration corresponding to a section of the well transverse to the path {right arrow over (r)}(t) in t=t.sub.0, the section which has the configuration corresponding to the case in which there was no collapse and in which the maximum stress σ.sub.max(t.sub.0) is to be determined; d) predetermining an expression of a function for the maximum stress σ.sub.max(t, par1) where par1 is the set of parameters of the function to be determined; e) pre-establishing initial values for the parameters par1; f) defining the error function between the measured caliper C(t.sub.0) and the calculated caliper C.sub.c(t.sub.0) as
E=∥C(t.sub.0)−C.sub.c(t.sub.0)∥ for a pre-established norm where the calculated caliper C.sub.c(t.sub.0) depends on σ.sub.max(t, par1) and therefore on the set of parameters par1; g) determining a collapse area, according to the cross-section in t=t.sub.0, under the hypothesis that the collapse area is bound by an elliptical section that determines with the ends of the ellipse at the semi-major axes thereof the value of the calculated caliper C.sub.c(t.sub.0) according to the following sub-steps: 6. determining a pressure of the drilling fluid used, if any, the pore pressure p.sub.p, the maximum stress σ.sub.max given by the expression σ.sub.max(t, par1), the minimum stress σ.sub.min, and the mechanical properties of the rock in section S from the geomechanical computational model at the point of the path {right arrow over (r)}(t.sub.0); 7. determining the state of stress σ(θ) of the rock along the periphery of section S of the borehole at least as a function of the data from the preceding step where: i. α is the scalar value of the equivalent stress, ii. θ the angle with respect to a system of axes located in section S of the borehole, centered on the center of said section S and with an orientation in the plane containing section S such that σ(θ=0)=σ.sub.min and σ(θ=π/2)=σ.sub.max; 8. determining the collapse angle θ.sub.br as the angle centered on θ=π/2 and covering the arc of the periphery of section S where the stress σ(θ) is greater than the maximum allowable stress of the rock; 9. defining a family of ellipses having eccentricity e, contained in the plane of section S, such that: iii. the ellipse corresponding to the value of eccentricity e=1, where the eccentricity is defined by a ratio of the value of the semi-minor axis to the semi-major axis is the circumference established by the circular section S of the well; and iv. an intersection between the ellipse and the circular section S of the well is established at least at points π/2+θ.sub.br/2 and π/2−θ.sub.br/2 as well as the symmetrical points −π/2+θ.sub.br/2 and −π/2−θ.sub.br/2 thereof, respectively; 10. defining a safety factor
F=Σσ.sub.ext/Σσ.sub.res where Σσ.sub.ext is the sum of external forces on the rock at a given point of the rock, which depend at least on the in-situ stresses, on the density of the drilling fluid (γ) if any, on the elastic properties of the rock, and on the pore pressure p.sub.p; and where Σσ.sub.res is s the sum of resistance forces of the rock at the same point, which depend on the stress tensor, on the resistance properties of the rock, and on the angle of internal friction of the rock; 11. determining a function F(θ, e) as the safety factor F evaluated at a point of the ellipse defined by the eccentricity e for a value of the angle θ; 12. establishing a cutoff threshold value θ.sub.0<π/2; 13. determining a value of the eccentricity e.sub.0 closest to one which verifies F(θ.sub.0, e.sub.0)=f.sub.0, where f.sub.0 is a pre-established reference value close to one; 14. establishing as an estimated region of damage, according to the section S of the well and at the vertical level z, the intersecting region between the ellipse of eccentricity e.sub.0 and the circumference of the section S of the well; 15. establishing as a value of the caliper C.sub.c(t.sub.0) the difference between the major side (b) of the ellipse of eccentricity e.sub.0 and the radius of circumference (D/2) of the section S of the well; h) establishing a threshold value ε>0 and iteratively determining, until achieving convergence, where an error E function is less than said threshold value ε, for a pre-established search space in the variables of the set of parameters par1 where the value of the calculated caliper C.sub.c(t.sub.0) is calculated in each iteration according to step g); i) based at least in part on the set of parameters par1 that made the error E minimum, determining the maximum stress σ.sub.max at the given point of the path of the well {right arrow over (r)}(t.sub.0) in the reservoir.

2. The method according to claim 1, wherein the measured caliper is measured at a plurality of points N of the path of the well which can be expressed as r(t.sub.i),i=0,1,2, . . . N−1, such that the error E is the norm of the vector, the components of which are the difference between the measured caliper and the estimated caliper according to step g) at each of the N points, where each of the components can be expressed as C.sup.i (t.sub.0)−C.sub.c.sup.i (t.sub.0) with i=0,1,2, . . . N−1 determining the valid function σ.sub.max(t, par1) along at least one segment of the path established by the parameter t.

3. The method according to claim 1, wherein the pre-established function of the minimum stress σ.sub.min(t) is pre-established as σ.sub.min(t, par2), with par2 being a second set of parameters for which there is established a first estimated value, and wherein the iterative process according to step g) establishes as the search space both parameters par1 and parameters par2.

4. The method according to claim 3, wherein the minimum stress σ.sub.min(t, par2)=
σ.sub.min(t,A′,B′,C′,D′,F′)=A′+B′t+C′t.sup.2+D′e.sup.t+F′e.sup.−t with constants A′, B′, C′, D′, F′ taking experimentally adjusted values.

5. The method according to claim 1, wherein the parameters par1 of the maximum stress σ.sub.max are expressed in the form A+Bx+Cx.sup.2+De.sup.x+Ee.sup.−x with constants, A, B, C D and E to be adjusted, and where x is the variable with respect to which the correlation is established, being one of the following: parameter t, depth z(t), minimum stress σ.sub.min(t), shear stress, or a combination of two or more of the preceding variables.

6. The method according to claim 1, wherein the expression of the function σ.sub.max(t, par1) as a function of parameters par1 corresponds to an expression of the elastic solution of the maximum stress as a function of the tectonic shifts according to the direction of the maximum stress and the direction of the minimum stress.

7. The method according to claim 6, wherein the expression for σ.sub.max(t, par1) is as follows: ${\sigma }_{\max }\left(t,\mathrm{par}1\right)=\frac{\upsilon }{1-\upsilon }{\sigma }_{*}+E_1{D}_1+E_2{D}_2$ where par1 is the following set of parameters: ν is Poisson's ratio, σ is vertical stress, constants E.sub.1 and E.sub.2 are Young's moduli in directions 1 and 2, respectively, and, D.sub.1 and D.sub.2 are tectonic horizontal deformations according to directions 1 and 2, with E.sub.1, E.sub.2, D.sub.1, and D.sub.2 being the two parameters of par1.

8. The method according to claim 1, wherein an estimate of the width angle of the damage in the wall is calculated as the angles covering the intersecting points between the ellipse of eccentricity e.sub.0 and the circumference of the section of the well.

9. The method according to claim 1, wherein the path of the well {right arrow over (r)}(t), rock data, and caliper measurements for the generation of a numerical model are obtained while drilling.

10. A non-transitory computer program product stored on a computer-readable medium and comprising computer-implementable instructions that, when executed by a computer, cause the computer to carry out the method according to claim 1.

11. The method according to claim 1, wherein the measurement tool is a sensor configured to measure by sensing the walls of the well.

12. The method according to claim 1, wherein the measurement tool includes at least one touch probe that extends radially within the interior of the well until the probe contacts the physical walls of the well.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) These and other features and advantages of the invention will be more clearly understood based on the following detailed description of a preferred embodiment, given solely by way of non-limiting illustrative example, in reference to the attached drawings.

(2) FIG. 1 shows a scheme of an embodiment of a well in an oil reservoir, defined by a drill path, where at a given point of the path a section S is established, on which section there is a region of damage and in which the maximum stress σ.sub.max is calculated.

(3) FIG. 2 shows a scheme of an embodiment of a well seen in a sectional view, as well as a pair of ellipses with a different eccentricity used in the steps of calculating the calculated caliper according to the invention, which is subsequently used for comparison with the measured caliper in the section of the well.

(4) FIG. 3 shows a graph depicting the stresses along the periphery of a family of ellipses. The family of ellipses is represented by means of a plurality of curves identified with an arrow in which the direction in which eccentricity increases is shown. The abscissas show the angle in the section taking the point of minimum stress as a reference.

(5) FIG. 4 shows an image of a vertical segment of the well in which the width of damage has been measured as a function of the vertical level and the depth of the damage. The regions with damage are darker in the drawing.

(6) FIG. 5 shows two graphs related to one another, the graph on the left showing a figure with the safety factor F as a function of the eccentricity e with a value of one for considering the equilibrium factor between the external forces and the resistance forces. Once the eccentricity is determined, the ellipse is shown on the right with said eccentricity which determines the transverse area of the damage.

(7) FIG. 6 partially reproduces FIG. 1 as an embodiment in which the determination of the maximum stress σ.sub.max is carried out in a discrete set of points of the drill path.

DETAILED DESCRIPTION OF THE ILLUSTRATED EMBODIMENTS

(8) According to the first inventive aspect, the present invention relates to a method of determining the maximum stress σ.sub.max, at a point of the path which describes a drilled well, for example in a well for the injection or production of a gas or oil reservoir.

(9) FIG. 1 schematically shows the section of a reservoir with oil reserves, where the top line represents the surface of the reservoir and the volume of the reserve identified by the bottom line (Rs) inside of which there is a well (P).

(10) The well (P) is a borehole of circular section S which extends along a path presented by a curve. The curve is shown in FIG. 1, starting from the surface, descending in an almost vertical path, and after increasing its slope, ending in an almost horizontal segment.

(11) According to the invention, step a) of the method establishes a geomechanical computational model which incorporates, among others, data about the drilled well through the curve {right arrow over (r)}(t) which defines the path. Other data such as the configuration of the domain, the upper surface, properties of the rock, are also properties that are part of the geomechanical computational model.

(12) Along this path, the maximum stress σ.sub.max in a section S located at a level z in which there is a region of damage on which measurements of the depth of this region of damage have been taken, is to be calculated. Region of damage is understood as a region where the rock has exceeded the value of maximum allowable stress, causing it to break and subsequently fall off, leaving an empty volume. This volume results in a larger wall with respect to the cylindrical reference which the drilling tool would leave behind with the diameter of the circular section S.

(13) The measurement tools can be based, for example, on touch probes which are supported on the wall in a set of points or along the entire perimeter for each level. FIG. 4 shows a customized grayscale graph, a measurement along a segment of the borehole (according to the vertical coordinate of the graph) and is developed 360°, displaying the damage along the entire perimeter. In the upper part of the graph, the indications of North, East, South, West, and North, (the letters N, E, S, W, and N, respectively) are used to indicate the orientation of the measured data.

(14) The dark zones correspond to a greater depth with respect to the reference which establishes the position of the wall when there is no damage, i.e., when it corresponds to the cylindrical surface of circular section which the tool would theoretically generate. The grayscale represents values starting from 0 according to a predefined scale, with 0 being white. In a given section, the value of greatest depth is the caliper. The caliper thereby obtained is what will be referred to as measured caliper.

(15) In this level z shown in FIG. 1, the tangent n to the path coincides with the normal to the cross-section plane where the region of damage, among others, will be determined as part of the calculation steps.

(16) The plane transverse to the drill path of the well at a pre-established point is represented by means of a discontinuous line.

(17) FIG. 2 schematically shows a circumference with a thick line which represents the theoretical wall having a circular borehole configuration in the plane of section S.

(18) As indicated above, there is a step among the steps of the method in which it is necessary to determine the region of damage, and for that purpose a geomechanical computational model of the reservoir is generated in a computational system by at least incorporating the rock data and the mechanical properties thereof, and the in situ stress field is also established. The geomechanical computational model establishes a relationship between the force field with forces acting at a given point of the domain and the properties of the material.

(19) The method requires an initial step of acquiring data, primarily the drill path, properties of rock and fluids, data about the drilling fluid if it was used, and data about at least one region of damage. This type of data can be acquired by means of measuring apparatus. The remaining steps, including the generation of numerical models, can be performed in a computational system which finally determines the maximum stress at points of the domain.

(20) For each case it is necessary to determine the forces acting at one point, and they include, among others, those caused by the fluids stored under pressure in porous rocks, or also pressures due to the drilling fluid injected while drilling the well, if there is any, are found.

(21) Additionally, a fluid computational model of the same reservoir is generated in the computational system in this example, which at least models the porous medium suitable for holding liquid. If drilling fluid is used in drilling, then the fluid model must likewise incorporate this fluid and the interaction with the walls of the well. The fluid model includes pore pressure in the porous medium.

(22) Given that the deformations of the porous media give rise to changes in the fluid computational model, and the forces of the latter influence the geomechanical computational model, both models must be coupled to one another.

(23) Through the geomechanical and fluid computational models it is possible, for example, to determine the pore pressure p.sub.p and the mechanical properties of the rock in section S.

(24) In particular, it is also possible to determine the pressure of the drilling fluid, if one is used, and at the moment corresponding to the drilling operation. Although the method determines the maximum stress in an already drilled well, if the damage occurs while drilling due to the drilling conditions, the state of stress of the region of damage involved in the calculation steps must be calculated according to the conditions that caused the damage, i.e., at the time that drilling is performed.

(25) Step c) establishes the point where at least the maximum stress σ.sub.max will be determined, and step d) proposes the functional expression thereof with the parameters to be adjusted.

(26) Steps e) and f) define the values for starting the iterative process used for the adjustment as well as the measurement of the error used in order to know when convergence has been reached.

(27) The iterative process starts from a proposal for the expression for σ.sub.max as a function of a set of parameters. After having determined the parameters, the value of maximum stress is given. If either the level of the depth of the point of the well or parameter t are among the parameters, then the value of maximum stress for all the values of the range oft for which the adjustment is valid is given as a result.

(28) The core of the iterative process is step g), in which step the caliper is calculated, said caliper being referred to as “calculated caliper” and identified as C.sub.c, is compared with the caliper obtained by measuring the wall of the well, and if it gives rise to a measurement of the error above a pre-established value, then the parameters of the expression of the maximum stress are modified in order to iterate again.

(29) The modification of the parameters is carried out by an optimization algorithm which introduces increases in value for each parameter by observing the variation of the error. Practical examples of algorithms used for reaching convergence in this optimization process are those provided by the “Matlab” calculation tool through what is referred to as the “Optimization Toolbox.” The algorithms provided by this toolbox include, among others, “Solve linear,” “quadratic,” “integer,” and “nonlinear optimization problems”. There are also other tools available in numerical computation libraries.

(30) In step g), in which the calculated caliper is specifically calculated, the method establishes a specific way of determining the region of damage in the wall of the well by taking as a hypothesis that the shape that this region of damage adopts is elliptical. The method proposes a family of ellipses from among which it determines one ellipse. Once the ellipse corresponding to the state of stress under given criteria has been established, the caliper is located at the end of the semi-major axis of the ellipse.

(31) Having seen the method in a general manner, the details of the method are described below in a more specific manner.

(32) Given the direction normal to the plane of section S, by means of a rotation about said normal, a direction in which the stress is minimum σ.sub.min and a direction, which is perpendicular to the previous direction, in which the stress is maximum σ.sub.max, are established. These directions are used as axes of reference for establishing the site where the damage occurs and its extent.

(33) During the iterative process, the value of minimum stress σ.sub.min is known. The value of maximum stress σ.sub.max to be calculated is the value of the previous iteration or the value proposed for starting the iterative process. This data determines the state of stress for calculating the region of damage.

(34) In the method according to the invention, the parameters determining the maximum stress are updated in each iteration until reaching convergence. In this same instance, the minimum stress σ.sub.min is known and it does not have to change. In other words, if the minimum stress σ.sub.min is known, the data is used in all the iterative process steps.

(35) Nevertheless, according to one embodiment the minimum stress σ.sub.min is also a value which is updated in the iterative process such that upon reaching convergence, the method also provides a more precise value of minimum stress σ.sub.min. In this instance, it is also a known value in a given iteration because the same occurs as with the maximum stress σ.sub.max. In other words, either it is the value of the previous iteration or it is the starting value taken for starting the iterative process.

(36) Once the axes are established, the state of stress in the rock along the curve defined by the circumference corresponding to the wall of the borehole is determined. Based on the state of stress, the value of equivalent stress is calculated by determining the arc of curve in which said equivalent stress is greater than the allowable stress of the rock.

(37) This arc is centered on π/2 due to the way of constructing the axes of reference, and the width thereof is the collapse angle θ.sub.br.

(38) FIG. 2 shows both axes, which are the axes that will correspond to the major and minor sides of a family of ellipses. This family of ellipses is parameterized by means of the eccentricity e defined as the ratio of the semi-minor axis of the ellipse a and the semi-major axis of the ellipse b. For a value of eccentricity equal to 1, the ellipse is the circumference of radius R which corresponds to the circumference representing the wall of the well according to section S. For decreasing values of eccentricity e, ellipses having one end of the major side penetrating the rock, whereas the minor side is smaller than the radius of the well R, are obtained. In relation to the ellipse thereby obtained, the part of the ellipse penetrating the rock and which will be the curve that defines the region of damage will be of particular interest.

(39) The points where the collapse angle starts and finishes are the points where the intersection between the circumference and any of the ellipses of the family parameterized in e is established.

(40) The values of 2b and 2a in FIG. 2 identify the length and width, respectively, of given ellipse. Two ellipses of eccentricity e.sub.1 and e.sub.0 are likewise shown.

(41) For determining the ellipse which defines the region of damage, the safety factor
F=Σσ.sub.ext/Σσ.sub.res
is used, where Σσ.sub.ext is the sum of external forces on the rock at a given point of the rock, which depend at least on the in situ stresses, on the density of the drilling fluid (γ) should there be any, on the elastic properties of the rock, and on the pore pressure p.sub.p; and
where Σσ.sub.res is the sum of resistance forces of the rock at the same point, which depend on the stress tensor, on the resistance properties of the rock, and on the angle of internal friction of the rock.

(42) This safety factor depends the angle and on the factor of eccentricity, where the value of one identifies the balance between the forces and the resistance capacity. When this balance is broken, damage is considered to exist. Nevertheless, it is possible for one skilled in the art to chose values f.sub.0 other than one, though close to it, for example, as a safety factor. Valid values of f.sub.0 are comprised in the [0.7, 1.3] range, and more preferably in the [0.8, 1.2] range, and more preferably in the [0.9, 1.1] range, and more preferably in the [0.95, 1.05] range.

(43) FIG. 3 shows a graph of the stress as a function of the angle θ, where for values close to π/2, identified in the drawing as close to 90 given that it is expressed in degrees instead of radians, the stress acquires asymptotically high values as the degree of eccentricity increases.

(44) This fact renders the approach according to the state of the art for the estimation of damage useless since in no case would it be considered that a safe situation exists.

(45) With this hypothesis, the zone of the end of the ellipse reaches values that are not allowable in virtually any instance, which would invalidate this method of determining the region of damage. Nevertheless, it has been found that if this drawback is overcome by eliminating values above the previously specified value θ.sub.0<π/2, then the method predicts the region of damage with great precision.

(46) Once θ.sub.0<π/2 has been established, the value of the eccentricity e.sub.0 closest to one which verifies F(θ.sub.0, e.sub.0)=f.sub.00 is determined, where f.sub.0 is the pre-established reference value close to one.

(47) As described above, FIG. 4 shows an image of the drilled wall in a well showing the zones where damage has occurred. The letters N, E, S, and W identify North, East, South, and West, respectively, and correspond to a perimetral development of 360 degrees (2π radians).

(48) The image is taken a posteriori, once the well has been drilled or obtained by sensing while drilling. The values shown allow obtaining the value of the measured caliper which will be compared with the value of the calculated caliper.

(49) FIG. 5 shows a graph of the function F(θ.sub.0, e)=f.sub.0=1 with the eccentricity e as a free parameter. It is where the function takes this value f.sub.0=1, which determines the eccentricity e which in turn defines a single ellipse of the previously defined family of ellipses.

(50) In this embodiment, the ellipse has an eccentricity of 0.4. The right side of the drawing shows a quarter circumference, the circumference representing the section of the wall of the well, and also a quarter of the ellipse having an eccentricity of 0.4. The inner area of the ellipse having an eccentricity of 0.4 is established as the region of damage.

(51) Once the ellipse has been determined, the caliper is also determined as the calculated caliper.

(52) The use of more than one point along the well allows calculating at the same time the maximum stress in a segment of the well. In this instance, one of the parameters of the function to be adjusted is the parameter t or the level z of depth. In this instance, the expression obtained for the maximum stress σ.sub.max(t, par1) is valid for the segment of the well in which the expression has its domain of definition.

(53) By means of step g), in the final iteration the region or regions of damage is obtained for the points at which the measurement of the measured caliper is provided. With these regions and under the hypothesis of the elliptical-shaped region of damage, it is also possible to determine the angle of the region of damage as the angle between the points of the circumference of the section where the intersection with the ellipse takes place.

(54) The use of specific expressions for maximum stress where σ.sub.max(t, par1) uses values having a physical interpretation is of particular interest. As a specific instance, the expression of the function σ.sub.max(t, par1) as a function of the parameters corresponds to an expression of the elastic solution of the maximum stress as a function of the tectonic shifts according to the direction of the maximum stress and the direction of the minimum stress.

(55) In one embodiment, the expression for σ.sub.max(t, par1) is as follows:

(56) ${\sigma }_{\max }\left(t,\mathrm{par}1\right)=\frac{\upsilon }{1-\upsilon }{\sigma }_{*}+E_1{D}_1+E_2{D}_2$
where par1 is the following set of parameters:
ν is Poisson's ratio,
σ.sub.* is vertical stress,
constants E.sub.1 and E.sub.2 are Young's moduli in directions 1 and 2, respectively, and,
D.sub.1 and D.sub.2 are tectonic horizontal deformations according to directions 1 and 2, with E.sub.1, E.sub.2, D.sub.1, and D.sub.2 being the two parameters of par1.

(57) Both the minimum stress and the maximum stress can adopt other expressions for correlation with other variables. A sufficiently generic expression can be expressed as A+Bx+Cx.sup.2+De.sup.x+Ee.sup.−x with the constants A, B, C, D, and E to be adjusted, where x is the variable with respect to which the correlation is established.

(58) In the case of maximum stress, the variable x may be one of the following: parameter t, the depth z(t), minimum stress σ.sub.min(t), shear stress, or a combination of two or more of the preceding variables.

(59) In the case of minimum stress, the variable x is preferably the parameter t, such that the minimum stress σ.sub.min can be written as
σ.sub.min(t,A′,B′,C′,D′,F′)=A′+B′t+C′t.sup.2+D′e.sup.t+F′e.sup.−t
with the constants A′, B′, C′, D′, F′ taking experimentally adjusted values.

(60) To start the iterative process, it is necessary for the expressions of the correlations to have an initial value assigned thereto. One way of giving an initial value is to start from a value which is increasing according to depth. In instances where estimated data about stress is available, said data can be used to determine the values of the constants before iteration starts. The iterative process, which has been proven to be convergent in all cases, will modify these values until giving rise to the correlation which best explains the variation in stress once convergence has been reached.

(61) Any of the expressions used in the correlations can be defined by fragments.