METHOD OF ANALYZING EXPONENTIAL DATA

Abstract: A method for analyzing data modeled by a sum of exponential terms by determining weighted moments of a characteristic independent variable and amplitude for such data. In a specific embodiment, weighted moments of time and intensity are determined from the data. Corresponding weighted moments of time and intensity are also determined for a model consisting of a sum of exponential terms. Corresponding weighted data moments and modeled moments of time are equated and corresponding weighted data moments and modeled moments of intensity are equated. The resulting set of equations is solved simultaneously for the values of the time and intensity parameters which appear in the model.

Description: METHOD OF ANALYZING EXPONENTIAL DATA This invention relates to a method for analyzing exponential data and more particularly to analyzing data varying as the sum of a plurality of exponential terms. The data may vary with time, distance, or any other independent variable. A specific application of such analysis is in the characterizat on of subterranean formations with regard to fluid or mineral content, lithologic characteristics, porosity, or to provide stratigraphic correlation. A borehole logging tool is conventionally used to measure neutron or gamma radiation die-away in the formations surrounding a borehole. In neutron porosity logging, for example, a neutron source is utilized for bombard ng the formations with fast neutrons as the logging tool is advanced through the borehole. In the course of moderation, the fast neutrons reach the epithermal range and thence are further moderated until they reach the thermal neutron range. The populations of neutrons at the various energy levels die-away, or decay, with time following primary radiation and thus offer means of characterizing the formations. The rate of decay of the epithermal neutron population gives a quantitative indication of the amount of hydrogenous material present in the formations which in turn is indicative of the porosities of the formations. Examples of .both methods and apparatus for carrying out epithermal die-away porosity logging are set forth in U.S. Patent No. 4,097,737 to . R. Mills, Jr., and U.S. Patent Nos. 4,556,793 and 4,590,370 to L. S. Allen and W. R. Mills, Jr. An article entitled "Improved Methods of Signal Processing For Pulsed Neutron Capture Logging", SPWLA Twenty- Second Annual Logging Symposium, June 23-26, 1981 by R. Randall and E.C. Hopkinson discusses a method of pulsed neutron capture logging to differentiate oil , gas and saline formation water environments through casing. The method applies statistical averaging to a single exponential die-away term in the logged data to determine neutron decay rate. In a still further reference, U.S. Patent No. 4,600,838 to D. K. Steinmaπ and L. A. Jacobson, there is described a method of thermal neutron decay logging for overcoming excessive statistical fluctuations in the logged data, particularly in short neutron decay time formations. This method involves the determination of zero and first order moments of time during first and second discrete sequences of time gates respectively and the taking of the ratio of such moments to obtain a thermal neutron decay time constant for the formations surrounding the logged borehole. In yet further references, U.S. Patent No. 4,625,110 to Smith, Jr. and U.S. Patent No. 4,638,161 to Smith, Jr. and Verbout, there is described a porosity determination utilizing a two-exponential model approach to epithermal neutron die-away. While the foregoing described methods and systems for neutron die-away logging have-been utilized in characterizing subsurface formations, there is still a need for a method of improving on such characterizations when the logged data is the sum of a plurality of exponential terms, and more specifically, when there are at least three exponential terms representing differing die-away rates for borehole effects, formation effects and thermal neutron effects. The present invention provides for such a method of analysis as it relates to exponentially varying data, and particularly, to radioactive borehole data having a plurality of differing exponential terms. The present invention is directed to a method for analyzing data modeled ty a sum of exponential terms by determining weighted moments of a characteristic independent variable and amplitude from such data, identifying modeled moments of the independent variable and amplitude, and analyzing the relationships between such weighted data and modeled moments. In a particular embodiment, time and intensity parameters are determined for data assumed to vary as the sum of a plurality of exponential terms. Weighted moments of time and intensity are determined from the data, the number of such moments being twice the number of exponential terms used to represent the data. Corresponding weighted moments of time and intensity are also determined from a model consisting of a sum of exponential terms. Firstly, corresponding weighted data moments and modeled moments of time are equated, and corresponding weighted data moments and modeled moments of intensity are equated. Secondly, the resulting set of equations is solved simul aneously for the values of the parameters which appear in the model . O Figure 1 illustrates a borehole logging system for making neutron or gamma-ray die-away measurements to be analyzed ty the method of the present invention. Figure 2 illustrates a specific example of epithermal neutron die-away measurements as might be made by the borehole 5 logging system of Figure 1. Figures 3A-3H are flow charts representing the steps carried out in accordance with the method of the present invention for analyzing the sum of the plurality of exponential terms illustrated in the die-away measurements of Figure 2. 0 In accordance with the present invention, weighted moments (or weighted averages) of a characteristic independent variable and amplitude are utilized over selected data intervals to determine values of the parameters that characterize such data when they are assumed to vary as the sum of exponential terms. The data may be 5 either continuous or discrete. This method is especially applicable to the analysis of data as might be obtained from the radioactivity borehole logging tool 10 of Figure 1 as it traverses along the borehole 11 through the surrounding formations 12. Such data, representing epithermal Q neutron die-away, thermal neutron capture die-away, or gamma-ray die-away, typically are assumed to be represented ty a plurality of exponential terms, such as represented by the decay curve illustrated in Figure 2. For purposes of illustration, the curve in Figure 2 is assumed to consist of the sum of two epithermal neutron decay curves and one thermal neutron decay curve. A first exponential decay curve a,e~ u 1 from 0 to t, microseconds represents predominantly the rate of decay of epithermal neutrons attributable to borehole effects. A second exponential decay curve a 2 e"μ ^ rom t l t0 *2 m i croseconds represents predominantly the rate of decay of epithermal neutrons attributable to formation effects. A third decay curve a-,e ""μ 3 from t ? to t.e microseconds represents predominantly the rate of decay attributable to thermal neutron effects. Each of these curves becomes a straight line when plotted on a semilogarithmic scale. For a more detailed description of such a plurality of die-away characteristics, as well as a description of a borehole logging apparatus 10, including surface equipment, for making the die-away data measurements, reference may be had to the aforementioned U.S. Patent Nos. 4,097,737; 4,556,793; and 4,590,370. The method of the present invention will now be described in accordance with two cases, firstly, when the exponentially varying data are measured as a continuous function of time and, secondly, when the data are measured in discrete time intervals. Either case is also applicable for exponentially .increasing data. Firstly, with respect to the continuous case, the data to be analyzed are represented by y(t) with 0 t <_ t , where y(t) is the measured intensity of the physical quantity being observed at time t. The data are to be modeled as follows: m -u.t f(t) = ∑- a. e k , (1) k=l κ where 0 <_ t <_ t - The model consists of a sum of m exponential terms with m intensities a l k,, and m time constants μ. to be determined from the data. The following are weighted moments of time and intensity of the data y(t): (t ' a„' t " κ b)' = and (2) t α y(t) dt The quantity • t β α(t α.,tι 0,) is a relative moment of t β , calculated over the range t, to t. , with a weighting function ^^ 3 D - t α . Similarly, y (t , | is a relative moment of y.t), calculated over the range t to t. , with a weighting function t α . The quantities t β (t_,t. ) andV (t,,t. ) are α a D α a b derived from the data; the notation indicates that the ca culated moment values depend on the interval (t_,t. ). The quantities a D t, and t. are within 0 < t < t . 3 D — — t For m components, m different values of t (t ,t. ) α a D and different values of y α (t a_,t D.) are calculated. These different values are obtained by choosing distinctive sets of values for the quantities α, 8, t , and t.. The choices are based on achieving a good contrast among the m terms in the model, and this in turn is based on the observed behavior of the data. Now, with the same sets of values of α, β, t and t. — - a — a D that were used to calculate t (t .t.) and y (t ,t.), α a D α a b the following relative expected values of moments from the model are calculated: *b t α f(t) dt ιτ -a For components there are values of E[t α (t 3 ,t. D )] and m values of E[f α (t " a,,t D. )]. l _ Next the corresponding model and data moments are equated: ELt α (t a,,t D. )] = t α (ta -tb. ) for m equations, (6) and E[f α (t3 = ,tD k )] = y * α (ta,,tD.) for m equations. 20 ( 7) These equations are solved simultaneously to get estimates of the parameters a. and μ. for k=l, m. Secondly, with respect to the discrete case, the data to be analyzed are represented ty an array of values Y for n = 1, N . 2 5 Y is the measured intensity in time interval n. The data are model ed by m -λ. n F_ = Σ A. e K , (8) π k=l where 30 n = 1,N. Similarly to the continuous case, the model consists of a sum of m exponential terms with m intensities A. and m time constants λ. to be estimated from the data. The following are weighted moments of interval number and intensity of the data: "b "I ( Vn II a_.' π ) = and (9) n, 10 i _ n?, " α Y n n α i 20 The quantity n (n,,n κ ) is a relative moment of n , a. a D calculated over the interval range n to n. , with a weighting function n α . Similarly, Y (n,,n k ) is a relative moment of Y_, calculated over n_ to n. , with a weighting function n . π — — a D ________ The Quantities n (n„,n ) and Y (n,,n. ) are derived z> α a D α a D from the data; the notation indicates that the calculated moment values depend on the interval (n a ,n b ). The quantities n a and n. are within 1 _< n_< N. For m components, ___ m, different values of n α (n α,,n D κ ) ,„ and m different values of Y (n,,n.) are calculated. These 30 α a D different values are obtained by choosing distinctive sets of values for the quantities α, s, n and n b . The choices are based on achieving a good contrast among the m terms in the model, and this in turn is based on the observed behavior of the data. With the same sets of values of α, β, n and n. that s a b were used to calculate n α (n a.n. D ) and Y α (n a„,n κ D), the following relative expected values of moments are calculated from the model : ∑ b n a F π a n ECF α u (n "a ua'> _ "„b)"_ (12) For m components, m values of E[n (n ,n l and m values of E[F a (n a,,n. D )] are calculated. Next, the corresponding model and data moments are equated: E[n α (na .nb. )] = n α β (na.n.b ) for m eq ^uations, (13) and E[F α (n a ,n b )] = Y Q (n a. n b ), for m equations. ( 14) These equations are solved simultaneously to get estimates of the parameters A. and λ. for k = 1, m. For the continuous and discrete cases, the following properties ex st: A. has units of counts per unit interval, λ. has units of inverse unit interval, a^ has units of counts per unit time, and μ. has units of inverse unit time. —g— Special situations may arise in which values of one or more of the model parameters can be fixed because of some prior knowledge. This is most likely to be the case for the μ^ or the λ^. Such prior knowledge is used in the model, with a corresponding reduction in the number of unknown parameters to be determined. For convenience in using Equations (2)-(5) or (9)-(12), integer values could be selected for α and β. The analysis described above includes the special case of a constant term in the model. In such case the associated time constant is zero. The analysis can also be applied to non-cont guous data, i.e. data that contain one or more gaps. This situation can be accommodated by utilizing the gapped nature of the data in the selection of the αuantities t, and t. , or n„ and a D' a For the case of discrete time intervals, the interval lengths may vary over the data set. Finally, a specific example of the method of the present invention will now be described for the case wherein the data are available as a continuous function of time and are modeled ty three exponential die-away components as shown in FIG 2 for a semi! ogarithmic plot of y(t) as a function of t. Times t=0 and t=t.p are indicated, along with the " corresponding data values y(0) andy(t^). The intensities and decay rates of the three components shown in FIG 2 are arbitrary. The data model is f(t) =Z a. e κ , (15) k=l κ where °£ 1 1 t f For convenience α and β are chosen to be unity. Next, sets of values of t and t b are selected to be used in Eαuations (2)-(5). The 1 ogarithmic difference between y(0) and y(t f ) is lπy(0) - lny(t f ) = In [y(0)/y(t f )] s (16) where In is the natural logarithm. Dividing the above log difference into 3 equal intervals, each of length Δ, yiel s the following: Δ = 1/3 In Cy(0)/y(t f )]. (17) This defines two values y(t,) and y(t-) such that: In [y Oj/yft j )] = Δ, and (18) In [y(t 1 )/y(t 2 )3 = Δ. (19) These are equivalent to y(t χ ) =y(0) e " \ and (20) y(t-) =y(t 1 ) e "Δ = y(0) e " Δ . ( 21) For a given set of data, the values y(t,) andy(t 2 ) determine times t, and t-., as shown graphically in FIG 2. The times t=0, t«, t , t f are used in calculating data moments and model moments. Thus, in Equations (2)-(5) with α=β=l, the following 3 sets of values are used for ( a _t : (o.t j ), (t ls t 2 ), (t 2 ,t f ). The equations for the weighted moments of the data are as follows: 0 t y(t) dt f t 2 ) t t y(t) dt 77 i. (t e ? ., t τ.) = -— . (23c) t t 2 t dt The expressions given in Equations (22a-c) and (23a-c) cannot be explicitly evaluated further, except for the trivial denominators in Equations (23a-c). Otherwise, these can be evaluated only numerically with a specific data furction y(t) available. With the model assumed in Equation (15), the equations for the expected values of the weighted moments of the model are as f ol 1 ows : 3 / *2 , -»k l ∑- a. t e dt ! k=l K V t χ E[t l (t P 3 = , (24b) ECf 1 11 t 2 )] , and (25b) Although tedious algebraically, all indicated operations i Equations (24a-c) and (25a-c) can be carried out explicitly. Note that each equation involves all six model parameters a,, μ., μ-, a-,, μ 3 * Equating corresponding model moments and data moments, we obtain: ECtJ.O.t j )] = tJ.O.t j ) , (26a) E[tj(t ls t 2 )] = tj(t ls t 2 ) , (26b) E[t[ct 2 ,t f )] = tj(t 2 ,t f ) , (26c) E[f 1 (t 1 ,t 2 )] = ^( l5 t 2 ) , and (26e) The above six ranscendental equations are solved simultaneously to yield values of the six model parameters. Referring now to FIGS 3A-H, there are shown detailed flow charts of an algorithm used to analyze epithermal neutron die-away data for formation porosity and having three differing decay rates attributable to borehole, formation and thermal effects. The algorithm of this flow chart may be programmed in FORTRAN or any other suitable programming language for use on any modern digital computer. While the foregoing preferred embodiment of the method of the present invention has been described and illustrated, numerous modifications or alterations may be made without departing from the spirit and scope of the invention, as set forth in the appended claims.

Claims: CLAIMS: 1. A method for determining characteristic decay parameters and amplitudes of exponentially varying data comprising the steps of: a) determining weighted moments of a characteristic independent variable and amplitude from said data, b) determining corresponding weighted moments of the character stic independent variable and amplitude for a model consisting of a sum of exponential terms, c) equating said weighted data moments of the characteristic independent variable and said modeled moments of the characteristic independent variable, d) equating said weighted data moments of ampl tude and said modeled moments of amplitude, and e) simultaneously solving the equations resulting from steps c) and d) for values of the characteristic decay parameters and amplitudes appearing in said model. 2. The method of claim 1 wherein said characteristic independent variable is time. 3. The method of claim 2 wherein said data are obtained during a pulsed neutron logging survey of the formations surrounding a bocehole. 4. The method of claim 3 wherein said exponentially varying data includes terms representing a borehole effect on the decay of neutrons, a formation effect on the decay of neutrons, and a time-dependent background effect on the measurement. 5. The method of claim 4 wherein said exponential y vary ng terms identify the decay of epithermal neutrons resulting from the pulsed neutron logging of said formations. 6. The method of claim 4 wherein said exponentially varying terms identify the decay of thermal neutrons resulting from the pulsed neutron logging of said formation. 7. The method of claim 4 wherein said exponentially varying terms identify the decay of thermal neutron capture gamma rays resulting from the pulsed neutron logging of said formation. 8. The method of claim 2 wherein said data are continuous, said characteristic decay parameters having units of inverse unit time, and said amplitudes having units of counts per unit time. ' 9. The method of claim 2 wherein said data are in discrete interval , said characteristic decay parameters having units of inverse unit interval, and said amplitudes having units of counts per unit interval . 10. The method of claim 2 wherein said data are obtained from a gamma radiation survey. 11. The method of claim 1 wherein said characteristic independent variable is distance. 12. A method for determining the intensities and time constants of data modeled to vary as the sum of a plurality of exponential terms, comprising the steps of: a) determining a weighted moment of time t α 8 (ta ,t. ) from said data in accordance with the following: where y(t) = exponentially varying data, t α - weighting function, t β = time moment, t α, = initial time, and t b = final time, b) determining a weighted moment of intensity " y~(t ,t b ) from said data in accordance with the following: c) determining a weighted moment of time E[t α (t α.t κ D)3 for a model consisting of a sum of exponential terms in accordance with the following expression: where f(t) is a model represented by the summation of exponential terms as follows: m -μ-± f(t) = Σ a, e ; for 0 < t < t f k=l κ where a^ = intensity, ut < = time constant, and k = number of exponential terms from 1 to , d) determining a weighted moment of intensity E[f Ct (t α.t D.)3 for a model consisting of a sum of exponential terms in accordance with the following expression: e) equating the weighted moments of time t (t ,t. ) from step (a) with the corresponding modeled moments of time E[t β (t a ,t b )3 from step (c): E[t α (t a_,t D.)3 = t α β (t a ,t D. ) for m equations, f) equating the weighted moments of intensity y Ct (t α_,t D κ ) from step (b) with the corresponding weighted data moments of intensity E[f (t ,t b )3 from step (d): E[f α (t a.t. D)3 =y α (t a .t D.) fo m equations, and g) simultaneously solving the equations resulting from steps (e) and (f) for values of the time and intensity parameters appearing in said model.

Inventors: ALLEN LINUS SCOTT, DION ERIC PAUL, MILLS WILLIAM RAYMOND JR, ALLEN, LINUS, SCOTT, DION, ERIC, PAUL, MILLS, WILLIAM, RAYMOND, JR.
Applicants: MOBIL OIL CORP, MOBIL OIL CORPORATION
Application year: 1989
Main classification: G01V5
IPC codes: G01V5