DETERMINING A PARAMETER IN A PHYSICAL SYSTEM

Abstract: A method of determining a parameter of a physical model representing the electrical behaviour of a composition is provided. The method comprises defining said model by a relationship between at least one electrical property of the composition, a plurality of physical variables of the composition and said parameter, and measuring a first electrical property of the composition. A sample which is representative for said composition is selected, and a second electrical property of the sample is measured for various magnitudes of at least one of said physical variables. An incoherence function is selected, defining a difference between the measured electrical properties and the electrical properties as calculated from said relationship, said incoherence function being such that independent measurements are weighted in dependence of their accuracy. Said parameter is determined by minimisation of the incoherence function. The first and second property can be the same property.

Description: DETERMINING A PARAMETER IN A PHYSICAL SYSTEM The present invention relates to a method of determining a parameter in a physical system representing the electrical behaviour of a composition. The method can for example be applied in well-logging whereby the objective is to determine a quantity of a fluid, such as water, brine or hydrocarbon, in an earth formation in order to assess whether such fluid can be exploited economically. In the art of well-logging, physical models are generally applied to represent the electrical behaviour of the earth formation. From the well-log results and the physical model, the content of a component of the earth formation is then determined. A known method of determining parameters of such physical model is disclosed in "Electrical conductivities in oil-bearing shaly sands", Waxman M.H., and Smits L.J.M., SPE paper 1863-A presented at the 42nd Annual Fall Meeting, Houston, October 1-4, 1967. This paper discloses a method of determining a parameter of a physical model representing the electrical behaviour of an earth formation, consisting of defining said model by a relationship between the electrical conductivity of the formation, a plurality of physical variables of the formation and said parameter, selecting a sample which is representative for said formation and measuring the electrical conductivity of the sample for various magnitudes of the physical variables, and determining said parameter by applying the selected relationship to the measured conductivities of the sample. In this known method the model, which is generally referred to as the Waxman-Smits model, is defined by the relationship: wherein; C = the conductivity of the partially brine saturated formation represented by the sample C w = the conductivity of brine present in the formation; S w = the water saturation in the pore space (0..1), which equals 1-S 0 where S 0 denotes the hydro¬ carbon saturation; B = the equivalent conductance of sodium clay- exchange cations as a function of C w and temperature; Q v = the cation exchange capacity per unit pore volume; G * = a formation factor of the formation represented by the sample; G .* is represented as wherein φ = the pore space in the formation; m* = a parameter to be determined, in the form of the cementation exponent; n * = a parameter to be determined, in the form of the saturation exponent. The parameters m * and n * characterise the response of the conductivity of the earth formation to changes in physical variables such as φ and S w . In the known method these parameters are determined in a mutually independent way, namely: by using conductivity measurements in a non-hydro- carbon bearing zone in the earth formation or with laboratory conductivity measurements on a plurality of fully brine saturated samples; m * can be determined from the relation between log(C ~ ) and log(φ) or log(C, , - C ~ ) and t w t log(φ), respectively; and by using laboratory conductivity measurements on a plurality of partially brine saturated samples, n* can be determined from the relation between The results achieved with this known method are not always sufficiently accurate, probably because the parameters are determined in a non-optimal manner. It is an objective of the invention to provide an improved method of determining a parameter of a physical model representing the electrical behaviour of a composition. The method according to the invention thereto comprises: defining said model by a relationship between at least one electrical property of the composition, a plurality of physical variables of the composition and said parameter; - measuring a first electrical property of the composition; selecting a sample which is representative for said composition and measuring a second electrical property of the sample for various magnitudes of at least one of said physical variables; selecting an incoherence function defining a difference between the measured electrical properties and the electrical properties as determined from said relationship, said incoherence function being such that the measured values are weighted in dependence of their accuracy; and determining said parameter by minimisation of said incoherence function. It is to be understood that the first and second electrical property can be the same property. By minimising the incoherence function each parameter is determined in a manner that all experimental data, both from in-situ measurements of the composition and from laboratory measurements on the sample representative for the composition, are taken into account in a weighted manner, i.e. in dependence of their respective accuracies. For example, in case the composition forms an earth formation, less accurate well-log measurements are taken into account in the incoherence function with less weight than accurate sample measurements. It is thereby achieved that the parameters are determined with increased accuracy resulting in a physical model, and applications thereof, of increased accuracy. Suitably the step of determining said parameter by minimisation of the incoherence function includes an iterative process. Advantageously a plurality of said parameters are determined simultaneously in minimising said incoherence function, each parameter being represented in said relationship. The incoherence function can suitably be selected such that independent measurements of substantially equal accuracy are substantially equally represented therein. Furthermore, the incoherence function is preferably monotonically increasing with the deviation between the measured electrical properties and the calculated electrical properties. For example, the incoherence function can be of the form: G(Xi) - G(Yi) N ∑wiF i=l H (Xi) wherein i=l..N, N being the number of physical measurements, X_ represents the measured electrical property, Yj_ represents the calculated electrical property, G and H represent functions of the electrical property (H(X )>0) F represents a function with /X ≠0 [F(X) > F(0)] f wi represents the weight attributed to the physical measurement (w-j_>0) . The functions F, G and H can be selected as follows: F(X) = |X| α ,α e R, α > 0 G(X) = χβ,β e R, β ≠ 0 H(X) = 1 or H(X)= G (X) . Another suitable form of the incoherence function is: ∑ N wi [Xi _ Yi ] 2 i=l wherein i=l..N, N being the number of physical measurements, Xi represents the measured electrical property, Yi represents the calculated electrical property, W- L represents the weight attributed to the physical measurement (w^>0) . Furthermore, the incoherence function can be of the form: wherein i=l..N, N being the number of physical measurements, Xi represents the measured electrical property, Y represents the calculated electrical property, Wi represents the weight attributed to the physical measurement (wi≥O) . In a practical embodiment of the invention said at least one electrical property of the composition as defined in said relationship and the measured second electrical property of the sample are selected from the conductivity, the resistivity and the membrane potential. Furthermore, it is suitable to select the measured first electrical property of the composition from the conductivity and the resistivity. The composition can include, for example, a fluid and the model then forms a saturation model for the fluid in the composition. When applying the invention in the technique of wellbore logging, said composition comprises an earth formation, and the model forms a saturation model for a selected one of brine and hydrocarbon fluid contained in the formation. In selecting said relationship it is preferred to include therein the electrical conductivity of the composition and a plurality of composition parameters including, for each component in the composition, physical parameters representing the electrical conductivity and the volume fraction of the component, said relationship being such that the components are substantially equally represented in said relationship by means of said physical parameters. The model used in applying the invention can be, for example, the Waxman-Smits model, the Archie model, the Poupon-Leveaux model, the Simandoux model, the Dual-Water model of Clavier-Coates-Dumanoir, or the effective medium model of Spalburg. Suitably said relationship is selected to include the electrical conductivity of the composition and a plurality of composition parameters including, for each component in the composition, physical parameters representing the electrical conductivity and the volume fraction of the component, said relationship being such that the components are substantially equally represented in said relationship by means of said physical parameters. It is to be understood that each one of the components is represented in the relationship in substantially the same manner as any other of the components. For example, the relationship can be selected as: (σ e ff - an) . (Lσ e ff + (I-DOQ) "1 = Σ φ k (o"k - OQ ) . (Lσ k + (l-Dσn) "1 wherein OQ represents an auxiliary parameter in the form of a conductivity tensor k = 1 ... N, N being the number components in the composition σ e ff represents the conductivity tensor of the sample σ k represents the conductivity tensor of component k φ k represents the volume fraction of component k L represents a depolarisation tensor. Advantageously said auxiliary parameter is selected to be: σ 0 = Σ h k σ k wherein h k represents a mixing coefficient tensor pertaining to component k. Preferably each mixing coefficient is selected as h k - λ k φk v ( ∑λ n Φn v)_1 wherein k,n = 1 ... N, N being the number components in said plurality of components λ k represents a percolation rate tensor pertaining to component k φ k represents the volume fraction of component k v k n represents a percolation exponent pertaining to component k, n. The invention will be described hereinafter by way of example in more detail. Consider a set of measurements made on an earth formation which essentially consists of rock, brine, clay, and hydrocarbons. The measurements include: A) logging data on a water-bearing interval in the formation, the data consisting essentially of conductivity and porosity measurements under known C w , S w (=l), and T conditions; B) logging data on an interval in said formation susceptible of being hydrocarbon-bearing, which data essentially consist of conductivity and porosity measurements under known C w , and T condition; and C) laboratory data on core plugs representative for said formation, the data consisting essentially of conductivity, porosity, and Q v measurements under controlled C w , S w , and T conditions. Then, from the laboratory data, a standard correlation between the porosity and Q v can be made, providing Q v information along with the logging data. As saturation model the Waxman-Smits model referred to in the description of the prior art method, will be used, i.e.: For the data mentioned under points A) and C) an incoherence function P(m * ,n * ) is defined as follows wherein i=l..N, N = the total number of independent measurements referred to under A) ; W = the weight attributed to the i-th measured conductivity of the rock; C^ i = the i-th calculated conductivity of the rock whose value depends on m* and n*; κ t , i = the i-th measured conductivity of the rock; j =l . .M, M = the total number of measurements referred to under C) ; W J = the weight attributed to the j-th measured conductivity of the rock; C t, j = the j-th calculated conductivity of the rock whose value depends on m* and n*; κ K tt,.ij = the j-th measured conductivity of the rock; * = a parameter in the form of the cementation exponent, to be determined; n * = a parameter in the form of the saturation exponent, to be determined. The weights are chosen proportional to the inverse of the estimated accuracy variation of the measurement. The values of m* and n* are determined by minimising P(m * ,n * ) using a multi-dimensional down-hill-simplex method, which is an iterative mathematical method well-known to those skilled in the art. The resulting values for m* and n* can then be used in the Waxman-Smits model to solve for S w given the data referred to under B) , thus providing the water-saturation (hence also the hydrocarbon saturation) in the earth formation.

Claims: C L A I M S 1. A method of determining a parameter of a physical model representing the electrical behaviour of a composition, the method comprising: defining said model by a relationship between at least one electrical property of the composition, a plurality of physical variables of the composition and said parameter; measuring a first electrical property of the composition; - selecting a sample which is representative for said composition and measuring a second electrical property of the sample for various magnitudes of at least one of said physical variables; selecting an incoherence function defining a difference between the measured electrical properties and the electrical properties as determined from said relationship, said incoherence function being such that the measured values are weighted in dependence of their accuracy; and - determining said parameter by minimisation of said incoherence function. 2. The method of claim 1, wherein the step of determining said parameter by minimisation of said incoherence function includes an iterative process. 3. The method of claim 1 or 2, wherein a plurality of said parameters are determined simultaneously by minimising the incoherence function, each parameter being represented in said relationship. 4. The method of any of claims 1-3, wherein said incoherence function is selected such that independent measurements of substantially equal accuracy are substantially equally represented therein. 5. The method of any claims 1-4, wherein the at least one electrical property of the composition as defined in said relationship and the measured second electrical property of the sample are selected from the con¬ ductivity, the resistivity and the membrane potential. 6. The method of any of claims 1-5, wherein the measured first electrical property of the composition is selected from the conductivity and the resistivity. 7. The method of any of claims 1-6, wherein said composition includes a fluid and the model forms a saturation model for the fluid in the composition. 8. The method of claim 7, wherein said composition comprises an earth formation, and the model forms a saturation model for a selected one of brine and hydrocarbon fluid contained in the formation. 9. The method of any of claims 1-8, wherein said relationship is selected to include the electrical conductivity of the composition and a plurality of composition parameters including, for each component in the composition, physical parameters representing the electrical conductivity and the volume fraction of the component, said relationship being such that the components are substantially equally represented in said relationship by means of said physical parameters. 10. The method of claim 9, wherein the relationship is selected as: < σ eff " σ θ ) • < Lσ eff + (I-D O Q ) "1 = Σ Φk ( °k " σ 0> • (Lσ k + (l-L)σo) "1 wherein OQ represents an auxiliary parameter in the form of a conductivity tensor k = 1 ... N, N being the number components in the composition σ eff represents the conductivity tensor of the sample σ k represents the conductivity tensor of component k φ k represents the volume fraction of component k L represents a depolarisation tensor. 11. The method of any of claims 1-8, wherein said model forms one of the Waxman-Smits model, the Archie model, the Poupon-Leveaux model, the Simandoux model, the Dual- Water model of Clavier-Coates-Dumanoir, and the effective medium model of Spalburg. 12. The method of any of claims 1-11, wherein the step of determining said parameter by minimisation of said incoherence function is carried out in a minimisation algorithm. 13. The method of any of claims 1-12, wherein said incoherence function is monotonically increasing with the deviation between the measured electrical properties and the calculated electrical properties. 14. The method of any of claims 1-13, wherein said incoherence function is of the form: G(Xi) - G(Yi) N ∑wiF i=l H (Xi) wherein i=l..N, N being the number of physical measurements, Xi represents the measured electrical property, Yi represents the calculated electrical property, G and H represent functions of the electrical property (H(Xi)>0) F represents a function with VX ≠ O [F(X) > F(0)], Wi represents the weight attributed to the physical measurement (wi>0) . 15. The method of claim 14, wherein F(X) = |X| α ,α e R, α > 0 G(X) = χβ,β e R, β ≠ 0 H (X ) = 1 or H (X ) = G (X ) . 16. The method of any of claims 1-13 wherein said incoherence function is of the form: ∑ N wi [Xi - 2 i=l wherein i=l..N, N being the number of physical measurements, Xi represents the measured electrical property, Yi represents the calculated electrical property, Wi represents the weight attributed to the physical measurement (w ≥O) . 17. The method of any of claims 1-13, wherein said inco tion is of the form: wherein i=l..N, N being the number of physical measurements, Xi represents the measured electrical property, Yi represents the calculated electrical property, wi represents the weight attributed to the physical measurement (w^O) .

Inventors: DE KUIJPER ANDRE, KOELMAN JOHANNES MARIA VIANNEY, SANDOR ROBERT KARL JOSEF, DE KUIJPER, ANDRE, KOELMAN, JOHANNES, MARIA, VIANNEY, ANTONIUS, SANDOR, ROBERT, KARL, JOSEF
Applicants: SHELL CANADA LTD, SHELL INT RESEARCH, SHELL CANADA LIMITED, SHELL INTERNATIONALE RESEARCH MAATSCHAPPIJ B.V.
Application year: 1996
Main classification: G01V3
IPC codes: G01N33, G01V3