FR2812403A1 - Process for determining, from diagraphs obtained from a tool for measuring a physical property, the uncertainties of petrophysical properties in a formation - Google Patents

Abstract

A synthetic diagraph (6) is calculated (5) to match the response of the tool in a simplified model (4) of the reservoir. Contributions of all the environmental perturbations Pi (7) are calculated and their estimated variances \=si are added to other variances to obtain a total variance \=Sm, which is projected on the axis of the cost curve at its minimum to give the uncertainty The process involves: determining (3) for a reservoir an equivalent simplified model (4); calculating (5) a synthetic diagraph (6) matching the response of the tool for the simplified model; calculating (8, 9) the contributions of all the environmental perturbations Pi (7) for which variances \=si are estimated a priori to the variance of the data, adding them together and to other variances of the data to obtain a total variance EPSILON m of the measurements; calculating (11) the minimum of the cost curve near the inversion point, where the minimum is defined as a relationship which minimizes cost between the physical property and the invasion radius ri; calculating the cost (30) as a function of the physical property in the virgin zone, assuming a relationship between the property and the invasion radius ri such that the corresponding coordinates give a minimum; and projecting on the physical property axis the value EPSILON m of the variances to obtain the uncertainty of the physical property. The physical property and the invasion radius given by the minimum are assumed to be the true parameters of the formation. Each porous bank is treated separately. If the information is insufficient to deduce all the required parameters with the desired precision, the number of parameters is reduced to simplify the model to using quadratic functions. The physical property is the conductivity.

Description

1 2812403

  METHOD FOR DETERMINING FROM DIAGRAPHS

UNCERTAINTIES ON

  PETROPHYSICAL PROPERTIES OF A TRAINING

  The present invention relates to a method for determining, from logs, uncertainties on the properties.

petrophysics of a formation.

  It is well known to carry out i0 resistivity logs in petroleum drilling using tools designed to measure the resistivity of the formations crossed by this drilling. The purpose of these logs is to estimate the quantity of hydrocarbon that could be withdrawn from the borehole, in order to determine if its exploitation

  can be economically justified.

  One can for example carry out these logs by creating in the formation of the electric currents around the tool. The tool gives to each of the depths at which the measurements are made, a set of values of apparent resistivities at this depth, functions of the radial distribution of the real resistivities of the formation. In fact, in the general case, the resistivity of a porous formation is not constant radially. This is related to the phenomenon known as invasion caused by the filtration of drilling mud in the formation. This filtration is due to the hydrostatic mud pressure which is higher than that of the fluids originally contained by the formation. By moving the fluids to be measured away from the measuring devices located in the borehole, this phenomenon will generally complicate the estimation of reserves

  of oil displaced by the invasion.

  As a first approximation, a formation can be described by a sequence of horizontal banks, invaded or not. In an impermeable area (e.g. clays), it will often suffice to describe the

  the bench with only one parameter: its resistivity, assumed to be constant.

  In a porous formation, each bank must be described by at least three parameters, also constant in the bank

2 2812403

  À The invasion radius ri: radius of a cylinder coaxial with drilling delimiting the fluids in place with respect to fluids contaminated by the mud (so-called invaded area); The resistivity of the invaded zone Rxo, radially uniform inside this cylinder; ò The resistivity of the virgin zone R. (or its inverse, the conductivity (c,) outside the invaded zone. It is this last parameter which must be determined with the best possible precision to evaluate the hydrocarbon content

training.

  In summary, in a tank carrying hydrocarbons which can contain several banks, the formation will be described by a set

  models with three unknown parameters: RI (or C (,), RxO and r.

  Theoretically, at a given depth, the formation could be estimated from three measurements, if these have a sensitivity

  sufficient for these parameters, and are not correlated with each other.

  Generally, the measurement of Rxo is done with sufficient precision with skid tools. If the bank is sufficiently thick, the parameters R. and ri can be obtained by combining two other apparent resistivity measurements. The problem to be solved can become much more difficult in the common cases of thin banks (of the order of a meter). Indeed in these cases, two effects combine to make the estimation of the petroleum parameters delicate: the reduction in the sensitivity of the measurement to the parameters to be determined R, and ri and a strong correlation of the various measurements carried out. To solve these common problems, we have to derive the training parameters from the measurements, using a process known as inversion. This process is based on the determination of an optimal synthetic training model, such that the synthetic measurement modeled using a software giving the response of the tool, comes as close as possible

  of the measurement actually recorded.

  Once the resistivity Ri of the virgin part of the formation has been calculated, the water saturation Sw of this virgin zone is deduced by an empirical formula known as the Archie formula:

3 2812403

  Rw) o - ó is the porosity of the formation, ratio of the volume of pores in the formation to the total volume of the formation, - the water saturation Sw of the virgin zone is the ratio of the volume of water in the pores to total pore volume, and

  - Rw is the resistivity of water, assumed to be known.

  Knowledge of Sw makes it possible to determine the portion of the pores occupied by hydrocarbons, oil or gas So = 1-SW Finally, the product (p * S, gives the estimated volume of hydrocarbon

per cubic meter of training.

  On the basis of the estimated average content of the hydrocarbon formation, a choice will then be made on the usefulness or the mode of exploitation of the borehole. It is therefore essential to know the uncertainties on R. as well as possible in order to be able to derive, in a trivial manner, the uncertainty on the volume of hydrocarbons, in order to manage

  the risks of the choices made.

  Definitions: In what follows, we will call À Reconstruction errors The difference, for each measurement sample, between the actual measurement and the synthetic measurement calculated at

  based on an optimized training model.

  ò Cost: the sum of the squares of these differences (sum weighted by the number of elementary measures selected in the vicinity of the

  bench considered and used by the inversion process).

  Cost Map: represents the Cost calculated for a set of values of the optimized parameters, around the values given by the inversion process (we can for example easily represent this map in two dimensions if there are only two parameters to reverse: the coordinate axes represent

4 2812403

  respectively R, (or (C,) and ri, and each curve corresponding to a constant value of the Cost (figure 1); Cost Function: the variation of the Cost with one of the parameters inverted. This requires additional equations to link the other parameters We have already tried to estimate the uncertainties a posteriori on (C ',, but all the attempts made so far have failed due to: - unacceptable time required by a realistic treatment based on statistical techniques like so-called Monte-Carlo methods; - treatment that is too simplified and unsuitable for the problem to be solved, which ignores the true origin of the uncertainties in the measurements and / or the form of the Cost Map Software for example is commonly found inversion which provide an estimate of the posterior uncertainties based on - the curvature of the Cost Function to its minimum - the number of parameters inverted

- reconstruction errors.

  It was found that the posterior uncertainties thus calculated were considerably too optimistic. We can find at least

two reasons for this.

  First, the reconstruction errors do not represent the real uncertainties on the data due to the information available, to solve the problem, which is too poor compared to the

  number of degrees of freedom (number of parameters to be determined).

  This means that, even if the reconstruction of the measurement was very good, it could be obtained in a unambiguous way. In addition, it should be noted that the reverse logging is only one embodiment among a vast set of possibilities not available for making statistics. Indeed, ideally, it would be necessary to have measurements made with a large number of tools of the same type, but each having its own measurement error. Likewise, the training model which is assumed is also highly flawed, we note for example the uncertainties

2812403

  on the positioning of the benches, the resistivity distribution of the formation around the area of interest, the radius of the borehole, the

resistivity of mud ...

  Furthermore, the local curvature of the Cost Function, at its minimum, may not be representative of an "average" curvature that it would be necessary to know when the uncertainties on the data or the assumed training model are not

extremely weak.

  The present invention, which aims to overcome these drawbacks, is therefore based on the following considerations, which are the basis of the invention. First, the uncertainties on the logging measurements to be inverted and on the presupposed training model are difficult to quantify. Consequently, the estimation of the uncertainties on the petrophysical parameters can only be imprecise. This remark legitimizes the following hypotheses: The inversion of the logging data leads to a model which is only the most probable among a large number of possible models, depending on measurement errors and a priori on the training. However, it is not currently possible, for reasons of speed of calculation, to adopt

  a statistical approach to the problem.

  We will therefore make a first hypothesis according to which the model resulting from the inversion does indeed correspond to reality. In other words, we will first assume that the

  measures and the training model are known without error.

  In addition, statistical approaches being excluded, we will opt for an error propagation process, that is to say that we will examine what the consequences are on the parameters.

  reversed measurement and training model errors.

  But, in order for the errors to add up and propagate normally, such a method of error propagation in turn supposes on the one hand that the response of the tool is linear and, on the other hand, that the functions of Costs are quadratic and the distribution of logging or model errors

training be Gaussian.

6 2812403

  Discussion It is important to discuss these constraints which allow the addition of errors and therefore the entry into a framework which allows the credibility of the results obtained: a Linear response of the tools: this can only be an approximation which is not acceptable only if the primary uncertainties Pi are low (the primary uncertainties are the environmental uncertainties which lead to measurement errors for example an uncertainty on the radius of the borehole, on the position of the interfaces separating the different banks, on the resistivity of the wall walls ...). Gaussian distribution of errors: the distribution of primary uncertainties is generally not known, but it is generally accepted, in the absence of other information, to assume these centered and Gaussian uncertainties. It should be noted here that the uncertainties on the measurement, which are those which intervene in the determination of the a posteriori uncertainties of the parameters, will be Gaussian only if the assumption of linearity of the tool is

assumed (see paragraph above).

  At Quadratic Cost Functions: This point is one of the determining points of the process, and will therefore be developed: If, in the simplified case of two parameters C, and laughed at inverting, we examine an "ideal" Cost Map (figure 1), the "equi-Cost" curves are more or less elongated ellipses. Under these conditions, the Cost Functions are effectively quadratic. If now, we compare this ideal card to a

  real map (figure 2) we can observe that the curves "of equi-

  Cost "are more like elongated and deformed rectangles, with possibly secondary minima, than ellipses. In fact, the real Cost Map corresponds in 3 dimensions to a flat valley, deep and bumpy, delimited by very steep slopes or the Cost represents the altitude. If this valley was perfectly flat, this would lead to an absolute indeterminacy of the two parameters (since the Cost would remain constant for an infinity of couples of parameters C ,, ri and that therefore only a relation between these two parameters could be obtained.) If we now respect this relationship,

7 2812403

  we can determine a Cost at the bottom of the valley, function of a single parameter C (. It has been demonstrated and verified that this Cost function is very close to a quadratic function under the following conditions: 5. The bank to be inverted is only described by two parameters (C,

and laughed.

  À The RI / Rxo ratio is greater than 2 Note: In the case of an actual formation, the hydrocarbon-carrying reservoir is only very rarely homogeneous for several reasons 1. The porosity or water saturation of the formation can vary within each elementary bank 2. The formation can also be laminated (sand and clay sandwiches) 3. The argillosity of the various banks is not constant 4. The depth of invasion is not constant

  vertically or can have a complex profile (micro-

  annulus) 5. The tank can be made up of several benches To respect the condition of having only two parameters to be inverted in the tank, we will have to reduce ourselves to a simplified bench which can be defined, in the sense of the tanker, as a homogeneous bench having the same hydrocarbon content as the actual tank and a

  depth of invasion equal to the average invasion.

  Under these conditions, the invention relates to a method of determining, from logs made using a tool for measuring a physical quantity, the uncertainties on the petrophysical properties of a formation, characterized by that it comprises the stages consisting in: - determining for a training reservoir an equivalent simplified training model;

8 2812403

  - calculate a corresponding synthetic log of the response of the tool to the simplified model; - calculate the contributions of all the environmental disturbances Pi whose variances o'i are estimated a priori, to the variance of the data, add them together and to the other variances of the data so as to obtain a total variance Em of the measurements; - calculate the valley bottom of the Cost Map in the vicinity of the reversal point, this valley bottom being defined as a relation which minimizes the Cost, between said physical quantity and the radius of invasion ri; - calculate the Cost function of said physical quantity in the virgin zone, by supposing a relation between said physical quantity and the radius of invasion ri such that the point of corresponding coordinates is in the valley bottom; - project on the axis of said physical quantity the value Zmn of the variance of the measurements to obtain the uncertainty on said physical quantity. In a particular implementation mode, it is assumed that, in the Cost Map, said physical quantity and the radius of invasion ri as given by the inversion are the true parameters

training.

  In particular, each porous bench can be treated separately.

  When the information available is insufficient to deduce all the desired parameters of the formation with the desired precision, one can reduce the number of parameters to simplify the model so as to be placed in the case of quadratic Cost functions Said physical quantity can be in particular the conductivity ('We will now describe, by way of nonlimiting example, particular embodiments of the invention, with reference to the appended schematic drawings in which - Figure 1 illustrates an ideal cost card - the figure 2 illustrates an actual cost map

9 2812403

  - Figure 3 is a flow diagram of the method according to the invention - Figure 4 illustrates a typical tank model, in its environment; FIG. 5 shows the same simplified reservoir composed of a single bank and FIG. 6 represents a Cost function as a function of C,

at the bottom of the valley.

  Figures 1 and 2 have already been described.

  Referring now to Figure 3, the first step 1 of the process is to establish a training model. This step is carried out in a conventional manner by inverting the logs recorded in this formation. This model will be thereafter

considered the true model.

  The benches, representative of the reservoir, are then chosen in 2, for example manually, then the model is simplified in 3 We thus obtain in 4 the simplified model of training in

tank level selected.

  Having a software giving the answer of the tool used, and applying in 5 this software to the simplified training model, we

  obtains in 6 a synthetic reference log.

  The primary uncertainties are determined at 7. We establish for this a standard table of primary uncertainties Pi. We will assume

  that their distribution is centered Gaussian.

  The software giving the response of the tool is again used at 8 by applying the primary uncertainties 7 to the simplified training model, then the deviations from the log obtained are calculated at 9

  compared to the synthetic reference log.

  We then make sure in 10 that Ri 2 * Rxo to be in the

quadratic case.

  In this case, we determine in 11 the relationship between C, and ri at the valley bottom and we successively calculate the Cost function at the valley bottom and the value of (C, corresponding to the variance of the measures em. We obtain by difference with the original C, and ri values, assumed to be true, AzI, and Ari

2812403

  If the relation Ré 2 * Rxo is not verified, we can nevertheless determine by another algorithm 11 bis, more complex than 11, the

  valley bottom and get, as before, AC, and Ari.

  If AC, in 12 is sufficiently weak, we obtain AC, and Ari in 13, representing the posterior uncertainties on the parameters. Otherwise, the linearity assumptions, originally assumed, are no longer applicable, and it will not be possible to determine a posteriori uncertainties. We will only give the information that the inversion process is not stable with the tools used. We can nevertheless deduce that it would be appropriate to add one or more complementary measures, each time this type of training is encountered. We will now describe in more detail the stages of establishing the simplified training model, calculating errors on the

  logging, and valley bottom calculation.

  FIG. 4 shows a half-plane of a formation in vertical section passing through the axis 20 of a borehole. A log 22 has also been shown schematically. The log 22 is preferably obtained with a measuring tool microresistivity or

  imaging that gives good depth resolution.

  The first step is to determine the banks 231, 232 ...

  23i of training from the log 22. This can for example be done manually based on the discontinuities of the log, or using a segmentation program. Clay walls 24, 25 will often be recognized by their low

  relative resistivity. The full description of the training will be

  determined by the reversal process, giving as detailed an analysis of the training as possible. It is determined, per bank or group of banks (for example the group of banks 23 *), corresponding to the reservoir zone to be evaluated, the total hydrocarbon content of the formation of this zone and the mean radius of invasion. We then derive a single bank (Figure 5), the same thickness as the reservoir area, but described with only three parameters RA, Rxo, ri corresponding respectively to the same estimated quantity of hydrocarbon, a conductivity of the invaded area C , 0 equal to the average of the conductivities of this zone and a

  invasion equal to the average invasion.

1 1 2812403

  The calculation of the errors is done in a simple way by taking from each measurement sample the relative difference between the synthetic reference log and the calculated synthetic logs (also from the simplified model) by successively applying the different primary disturbances Pi. These deviations are squared and summed, then normalized over the number

  of samples. This gives the total variance Em of the measurements.

  The determination of the valley bottom, in the case where R, 2 * Rxo is also simple, since this valley bottom is a straight line in the plane C ,, ri. So just invert the single parameter ri for two values of (C, (for example (, t = 1, 3 * (Co and C, 2 = O, 7 * C (, 0), C, 0 being the value of (C ,, at the level of the reservoir, in the simplified model not disturbed by Pi. To then determine the Cost as a function of (C at the bottom of the valley, it will suffice to choose pairs of values of

  parameters C, and ri which verify the linear relationship (s)

  above, and calculate this Cost at different values of C ,. In the case of RI 2 * Ro, the relation C ,, ri at the bottom of the valley is no longer linear, but the determination of the relation poses no problem. We see such a function 30 in Figure 6. We also see in this figure the line 31 of equi-cost corresponding to the value E; m of the variance of the measurements. This line intersects the curve 30 at a point 32 whose abscissa C, is representative of the uncertainty a

  posteriori on the measurement of this quantity.

Claims (4)

  1 - Method for determining, from logs made using a tool for measuring a physical quantity ((;,), the uncertainties on the petrophysical properties of a formation, characterized in that it comprises the steps consisting in - determining (3) for a training reservoir an equivalent simplified training model (4); calculating (5) a synthetic log (6) corresponding to the response of the tool to the simplified model - calculating (8 , 9) the contributions of all the environmental disturbances Pi (7) whose variances oi are estimated a priori, to the variance of the data, adding them together and to the other variances of the data so as to obtain a total variance Em of the measurements; - calculate (11; 11bis) the valley bottom of the Cost Map in the vicinity of the reversal point, this valley bottom being defined as a relation which minimizes the Cost, between said physical quantity and the radius of invasion r i; - calculate the Cost (30) as a function of said physical quantity in the virgin zone, by supposing a relation between said physical quantity and the radius of invasion ri such that the point of corresponding coordinates is located in the valley bottom; - project on the axis of said physical quantity the value Em. of the variance of the measurements to obtain the uncertainty on said physical quantity. 2 - Method according to claim 1, in which it is assumed that, in the Cost Map, said physical quantity and the radius of invasion ri as given by the inversion are the true parameters training.   3 - Method according to any one of claims 1 and 2,   in which each porous bench is treated separately.   4 - Method according to any one of claims 1 to 3,   in which the information available is insufficient to deduce all the desired parameters of the training with precision 13 2812403   desired, and in which one reduces the number of parameters to simplify the model so as to be placed in the case of quadratic Cost functions   - Next method according to any one of claims   1 to 4, in which said physical quantity is the conductivity (C.