WO2010039757A1

WO2010039757A1 - Method for characterizing a geological formation

Info

Publication number: WO2010039757A1
Application number: PCT/US2009/058917
Authority: WO
Inventors: Erik Jan Banning-Geertsma
Original assignee: Shell Oil Company; Shell Internationale Research Maatschappij B.V.
Priority date: 2008-10-02
Filing date: 2009-09-30
Publication date: 2010-04-08

Abstract

This invention relates generally to methods for characterizing an environment where inversion methods are used to fit a model to data, and more specifically to apparatuses and method for characterizing a geological formation. The various embodiments of the present invention overcome the shortcomings of the prior art by providing a "turbo boosting" method that incorporates optimized power coefficients to reduce the number of necessary iterations without losing the accuracy of the interpretation. An optimum value of the power coefficient can be found by minimizing the square norm of the difference between a subset of an updated predicted response and a corresponding subset of an observed response. The updated predicted response is a function of the interval specific undetermined parameter (mi), the subset of the observed response (dobs, i), and the subset of the current predicted response (dpred, i) and accordingly so is the power coefficient.

Description

METHOD FOR CHARACTERIZING A GEOLOGICAL FORMATION

TECHNICAL FIELD

[0001] This invention relates generally to methods and apparatuses for characterizing an environment where inversion methods are used to fit a model to data, and more specifically to apparatuses and method for characterizing a geological formation.

BACKGROUND

[0002] Conventional inversion methods can be applied to response data measured in an actual system or environment (or a modeled system or environment) to determine values of undetermined model parameters. These inversion methods allow interpretation of the data in terms of the values of the undetermined parameters that characterize the environment or system in which the data were taken. [0003] In conventional inversion, the values of the undetermined parameters are determined by mathematically minimizing the difference between a calculated response of a modeled system or environment (known as a forward model calculation) and the response data. Conventional inversion typically requires many forward modeling calculations and is hence processing-intensive and slow. This is especially true where the data include many points and modeled responses include multiple undetermined parameters. More undetermined parameters moreover increase the possibility of non-unique inversion results.

[0004] Using another type of inversion, the end result is reached by executing a limited number of forward modeling steps or iterations. The model parameter values chosen at each iteration depend in a prescribed way on the mismatch between calculated and actual data responses of the previous iteration, and no explicit mathematical minimization procedures are used. Requiring only a limited number of forward modeling steps, this inversion method is fast when compared to conventional inversion. However, this inversion method uses arbitrarily chosen power coefficients that can negatively affect the speed and accuracy of the method if chosen poorly. [0005] It would be useful to have a method of quickly and accurately determining values of undetermined parameters. For example, when applying the technique of Transient ElectroMagnetics (TEM) in a geosteehng scenario, it would be useful to accurately invert TEM measurements that are obtained during drilling so as to be able to form an image of the subsurface around the drill bit. Therefore, a need exists in the industry to address the aforementioned deficiencies and inadequacies.

SUMMARY

[0006] The various embodiments of the present invention overcome the shortcomings of the prior art by providing a "turbo boosting" method that incorporates optimized power coefficients to reduce the number of necessary iterations without losing the accuracy of the interpretation. The turbo boosting method is applicable to time domain electromagnetic data collected in a layered geological formation as well as other systems and environments. Also, the turbo boosting method can be applied to observed data that have the same or different dimensionality.

[0007] According to an exemplary embodiment, a method of characterizing an environment includes operating a measuring device to measure an observed response in the environment and operating a computing unit to interpret the observed response. The computing unit generates a modeled response that is a function of undetermined parameters, determines current values of the undetermined parameters, and determines a current predicted response using the current values of the undetermined parameters. The computing unit then defines a subset of the current predicted response and a corresponding subset of the observed response that are compared to determine a value of a ratio function. One of the undetermined parameters that affects the subset of the current predicted response is selected as an interval specific undetermined parameter. A value of a power coefficient is determined and used along with the value of the ratio function to update the current value of the interval specific undetermined parameter. The interval specific undetermined parameter is typically updated for a fixed number of iterations according to this method. [0008] An optimum value of the power coefficient can be found by minimizing the square norm of the difference between a subset of an updated predicted response and a corresponding subset of an observed response. The updated predicted response is a function of the interval specific undetermined parameter (IT),), the subset of the observed response (d_Obs,ι), and the subset of the current predicted response (d_pred,i) and accordingly so is the power coefficient.

[0009] The foregoing has broadly outlined some of the aspects and features of the present invention, which should be construed to be merely illustrative of various potential applications of the invention. Other beneficial results can be obtained by applying the disclosed information in a different manner or by combining various aspects of the disclosed embodiments. Accordingly, other aspects and a more comprehensive understanding of the invention may be obtained by referring to the detailed description of the exemplary embodiments taken in conjunction with the accompanying drawings, in addition to the scope of the invention defined by the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

[0010] FIG. 1 illustrates a turbo boosting method, according a first exemplary embodiment of the present disclosure.

[0011] FIGs. 2 and 3 schematically illustrate a measurement device and a formation, according to a second exemplary embodiment of the present disclosure.

[0012] FIG. 4 is a graph illustrating a time domain measurement representing an observed response of the formation of FIGs. 2 and 3.

[0013] FIG. 5 is a graph illustrating apparent resistivity and rectangularized resistivity. [0014] FIG. 6 is a graph illustrating formation models. [0015] FIG. 7 is a graph illustrating a reference function.

[0016] FIG. 8 is a graph illustrating a resistivity log representing an observed response of the formation of FIGs. 2 and 3.

[0017] FIG. 9 is a graph illustrating formation models. DETAILED DESCRIPTION OF THE INVENTION

[0018] As required, detailed embodiments of the present invention are disclosed herein. It must be understood that the disclosed embodiments are merely exemplary of the invention that may be embodied in various and alternative forms, and combinations thereof. As used herein, the word "exemplary" is used expansively to refer to embodiments that serve as illustrations, specimens, models, or patterns. The figures are not necessarily to scale and some features may be exaggerated or minimized to show details of particular components. In other instances, well-known components, systems, materials, or methods have not been described in detail in order to avoid obscuring the present invention.

Therefore, specific structural and functional details disclosed herein are not to be interpreted as limiting, but merely as a basis for the claims and as a representative basis for teaching one skilled in the art to variously employ the present invention.

General Description of Turbo Boosting [0019] Where data of an observed response are measured in an environment and modeled with a modeled response, a "turbo boosting" method can be used to determine values of undetermined parameters of the modeled response such that the mismatch between a resulting predicted response and the observed response is reduced. According to exemplary turbo boosting methods, the end result of each of the values is reached by executing a limited number of forward modeling steps or iterations. The values chosen at each iteration depend in a prescribed way on the mismatch between the predicted response of the previous iteration and the observed response.

[0020] The turbo boosting method of the present disclosure develops and incorporates optimized power coefficients that improve the tradeoff between the number of iterations and the reduction in mismatch between the predicted response and the observed response. In other words, the disclosed turbo boosting method has been found to achieve a result that is comparable to conventional turbo boosting methods while using fewer iterations than conventional turbo boosting methods. Accordingly, the disclosed method is faster than conventional boosting methods. In addition, the disclosed turbo boosting methods have been found to achieve a better result than conventional boosting methods where each of the methods uses the same number of iterations.

Definitions

[0021] As used herein, the term "modeled response" refers to a function of one or more undetermined parameters that can be used to generate data that represent the response of an environment. Certain of the undetermined parameters characterize the environment. The term "predicted response" refers to data that are generated as a result of inputting values into the undetermined parameters of the modeled response. The term "observed response" refers to data measured in the actual environment or data representing an actual response. It should be understood that where the misfit between a predicted response and the observed response is relatively small, the parameter values used to generate the predicted response may substantially accurately characterize the environment in which the data are measured. [0022] The modeled response can be linear or nonlinear. For example, a modeled response can be considered linear with respect to one of the undetermined parameters where changing the one of the undetermined parameters causes a substantially proportional change to the result of the modeled response. [0023] Parameters that significantly contribute to the observed response are included in a well chosen modeled response. Certain of these parameters are undetermined parameters. The number of undetermined parameters that contribute to a data point of the observed response is referred to herein as the dimensionality of the data point. [0024] Although the observed response is described herein as the response of a physical environment, it is envisaged that the turbo boosting methods described herein can be applied to any of various environments where an observed response is measured and a modeled response is used to generate a predicted response. For example, such an environment can be an electrical system, a mechanical system, a financial system, and the like. [0025] An exemplary turbo boosting method is now described as it applies to exemplary environments. It is envisaged that the equations that govern other environments or systems can be substantially similar to those of the exemplary environments described herein or may be variations thereof. Generalized Development of a Turbo Boosting Method for an Environment where Data Points of an Observed Response have a Dimensionality of One

[0026] A first exemplary environment is generally described as a distribution of one or more undetermined parameters m. Referring to FIG. 1 , step one 1 of a first exemplary turbo boosting method is measuring an observed response d_Obs in the environment. Observed response d_Obs is measured in the environment in relation to an incremented parameter z. For example, the incremented parameter z can be time, frequency, or position. In any case, in this first example, each point of observed response d_Obs, measured in relation to one of a series of values of incremented parameter z, is characterized by one of the distribution of undetermined parameters m.

[0027] Step two 2 of the first exemplary turbo boosting method is generating a modeled response and initial estimates of undetermined parameters. The modeled response is a function of undetermined parameters m and initial estimates of values of undetermined parameters m are input into the modeled response to generate an initial predicted response d_pred.

[0028] Step two 2 also includes establishing intervals yi, associated response subsets dobsj, d_pred,i, and associated undetermined parameters m. For purposes of teaching, in this first example, an exemplary turbo boosting method is directed to only one of intervals y,, the associated observed response subset d_Obs,ι, the associated predicted response subset d_pred,i, and the associated undetermined parameter m. Specifically, each of the data points in observed response subset dobsj have a dimensionality of one and is affected by one of the undetermined parameters m that is referred to herein as interval specific undetermined parameter rrij. [0029] In other examples described in further detail below, data points of observed response d_Obs have dimensionalities greater than one and the dimensionality of data points of observed response d_Obs can vary. [0030] Subscript "i" is used herein to indicate that subsets of data sets, parameters, intervals, etc. correspond to one another. In later described examples that include multiple subsets, parameters, intervals, etc., subscript "i" is substituted for numerals 1 , 2, 3, etc. to distinguish groups of subsets, parameters, intervals, etc. from one another.

[0031] Interval y is defined by boundary points Z_j, which are values of incremented parameter z. Boundary points Z_j correspond to points where observed response d_Obs is separated into observed response subsets d_Obs,i such that each observed response subset d_Obsj is affected by a different subset of undetermined parameters m. Subscript "j" is used to indicate specific ones of incremented parameter z values. In later described examples, subscript "j" is substituted for numerals 1 , 2, 3, etc. to distinguish the parameter values Z_j from one another.

[0032] According to a third step 3 of the first exemplary turbo boosting method, initial value of interval specific undetermined parameter rrij is updated according to _m ^(κ+1)

. Rj is a ratio function, qi is a power coefficient, and n is an iteration count that distinguishes between a current value or data set and an updated value or data set. According to the exemplary method, the value of interval specific undetermined parameter rrij is updated for a fixed number of iterations.

[0033] The values of other interval specific undetermined parameters rrij, each corresponding to another observed response subset d_Obsj, can be updated in the same manner and the resulting values can be used to generate an updated predicted response d_pred. Where the misfit between the updated predicted response d_pred and the observed response d_Obs is reduced, the values of the undetermined parameters m may be used to characterize the environment. Elements of the equation used to determine updated values of interval specific parameter rrii are now described in further detail.

Ratio Function and Field Ratio [0034] To determine ratio function Rj, values of a field ratio Fj are calculated by dividing values of predicted response subset d_pred,i by the corresponding values observed response subset dobsj. Field ratio Fi is given by F^ⁿ⁾ = d⁽"> / d_{obs i} . According to an exemplary method, ratio function Rj is determined by taking the average of field ratio Fj over interval y,. This is represented by R_t ⁽ⁿ⁾ = (F_i ⁽ⁿ⁾) .

Alternatively, ratio function Rj can be a value of field ratio Fj in interval y,. For example, ratio function Rj can be a value of field ratio Fj that corresponds to incremented parameter z value at the center of interval y,.

Optimum Value of Power Coefficient

[0035] The optimum value of power coefficient qi is that which minimizes the square norm of the difference between updated predicted response subset d_pred,i and observed response subset dobsj. This can be represented by

<P(<l,)

-d_obs , \\ = ^min - ^An exemplary formulation of power coefficient qi is now described.

[0036] Interval specific undetermined parameter rrij is represented in a general form as the sum of a known or selected background parameter rrib and an interval specific anomalous parameter Δrrii, given by m = m_b + Am . Background parameter rrib can be selected in different ways, for example as a constant or iterating value, as illustrated in the following examples. In any event, background parameter rrib provides a reference from which interval specific anomalous parameter Δrrii determines interval specific model parameter rrij. An updated interval specific undetermined parameter rrii is given by

.

[0037] Predicted response subset d_pred,i is represented as the sum of a background predicted response subset dbj and a reference response subset pi that is scaled by interval specific anomalous parameter Δrrij, given by d_pred,, = d_b>l +P₁ Am₁ . Background predicted response d_b and reference response p are known or calculated data sets. For example, background predicted response db can be data generated from the modeled response where undetermined parameters m are evaluated with initial estimates of values that may include the value of background parameter rrib. Reference response p can be data generated from the modeled response where a value of a reference parameter Δm_ref is used to evaluate each of undetermined model parameters m and the data are further divided by the value of reference parameter Δm_ref. An updated predicted response subset d_pred,i is given by d^ = d_{b l} +p_tAm^^+l) .

[0038] Updated predicted response subset d_pred,i can be given in terms of current data sets and current values of parameters by d_p ⁽ _red,ι

-m_b) . The updated predicted response subset d_pred,i can then be substituted into the equation for function Φ(qi) and the optimum value of interval specific power coefficient qi can be determined by setting the derivative of function Φ(qi) with respect to power coefficient qi to zero and solving for power coefficient qi, which gives ^_{) =} _ Ufjn_b - Al B)InV*

In R^

^A = (_.P_>,d_b, - d_{obs ι}) B = (M₁)

[0039] Here, expressions A and B are represented with inner product notation. As responses p, db, dobs are each vectors (linearly ordered sets) of values that correspond to values of an interval y,, expressions A and B can be evaluated by taking the dot product of the two vectors of the inner product that are separated by a comma. The dot product gives the sum of the product of the two vectors at each point in the interval y,, represented by

^ = ∑ PXd_btl -d_{obs ι}) y

B = ∑ Pr P, y

[0040] The interval specific power coefficient qi is optimized or determined so as to improve the performance of the turbo boosting method. The interval specific power coefficient qi is influenced by observed response subset d_Obsj, background predicted response subset d_b,i, reference response subset P₁, current predicted response subset d_pred,i, the current value of interval specific undetermined parameter rrij, and the value of background parameter rrib. [0041] Interval specific power coefficient qi is updated with each iteration n. For example, with each iteration n, the current predicted response subset d_pred,i and the current value of interval specific undetermined parameter rrii each change. According to certain formulations, background predicted response subset dbj and the value of background parameter rrib also each change.

[0042] The general formulation provided above is well suited for observed response data that have a dimensionality of substantially one and where the modeled response is substantially linear with respect to each of undetermined parameters m. The following examples demonstrate the application of an exemplary turbo boosting method to observed response data that have a dimensionality that is greater than one. In these examples, different subsets of observed response data are influenced by different subsets of undetermined parameters m.

[0043] There are a wide range of situations in which the turbo boosting method can be applied. For purposes of teaching, the following examples demonstrate the application of exemplary turbo boosting methods to observed responses measured in geophysical environments.

Time Domain Electromagnetic (TDEM) Measurement (Increasing Dimensionality Example) [0044] A second embodiment of a turbo boosting method is now described in the context of time domain electromagnetic (TDEM) data collected in a layered geological formation 4. In the second exemplary embodiment, incremented parameter z that observed response d_Obs is measured with respect to is time. Observed data d_Obs increase in dimensionality with respect to incremented parameter z, as described in further detail below. Measurement of observed data dobs is now described.

[0045] Referring to FIGs. 2 and 3, an exemplary formation 4 includes three layers Li, L₂, L3. Each layer has an associated resistivity p-i, p₂, P3 and adjacent layers Li, L₂, L3 are separated at a boundary B-i, B₂. Referring to FIG. 2, a measurement system 10 is associated with a drill bit 16 that forms a borehole 12 in formation 4 and takes measurements while drilling (MWD). In alternative embodiments, borehole 12 is drilled, the drill is removed, and a measurement device is then lowered into borehole 12 by a cable or other suitable suspension means (not shown).

[0046] To drill borehole 12, a drill bit 16 is positioned at the end of a series of tubular elements, referred to as a drill string 18. Drill bit 16 can be directed by a steering system (not shown), such as a rotatable steering system or a sliding steering system. In certain applications, previous or concurrently taken measurements facilitate directing drill bit 16, for example, toward a hydrocarbon fluid reservoir.

[0047] Measurement system 10 includes a measurement device 24 that is generally described as an array of transmitters and receivers and a corresponding support structure. The illustrated measurement device 24 includes a transmitter 26 and a receiver 28. Referring to FIG. 3, measurement device 24 is positioned in borehole 12 in first layer Li of formation 4, at a first distance Hi from first boundary B-i, and at a second distance H₂ from second boundary B₂. [0048] In the second exemplary embodiment, each of transmitter 26 and receiver 28 includes a coil antenna. Transmitter 26 and receiver 28 are arranged to be substantially coaxial. This arrangement is used for purposes of teaching. In alternative embodiments, transmitters and/or receivers can be multi-axial so as to send and receive signals along multiple axes. [0049] Measurement system 10 further includes a data acquisition unit 40 and a computing unit 50. Data acquisition unit 40 controls the output of transmitter 26 and collects the response at receiver 28. The response and/or data representative thereof are provided to computing unit 50 to be processed according to the methods described herein. Computing unit 50 includes computer components including a data acquisition unit interface 52, an operator interface 54, a processor unit 56, a memory 58 for storing information, and a bus 60 that couples various system components including memory 58 to processor unit 56. Memory 58 is a computer readable medium that can store instructions for performing a turbo boosting method as described herein. Processor 56 reads and executes the instructions. Operator interface 54 allows a user or other device, for example, to manipulate data; to select, change, or delete values of model parameters; or to program computing unit 50 with new or different instructions. [0050] Computing unit 50 can be positioned at the surface or at a remote location such that information collected by measurement device 24 while in borehole 12 is readily available. For example, a telemetry system can connect measurement device 24, data acquisition unit 40, and computing unit 50. In alternative embodiments, data acquisition unit 40 and/or computing unit 50 is combined with or integral to measurement device 24 and processes data while in borehole 12.

[0051] Step one 1 of the second exemplary turbo boosting method is measuring observed response of the formation 4. In this exemplary embodiment, undetermined parameters m that significantly affect observed response d_Obs are resistivities pi, p₂, P3 and boundary distances H-i, H₂.

[0052] Referring to FIG. 4, a graph of observed response d_Obs is illustrated. In the exemplary embodiment, step two 2 is at least partially accomplished as observed response d_Obs is used to determine boundary points z and initial values of undetermined parameters m. As the incremented parameter z is time, boundary points z-i, z₂, z₃, z₄ are boundary times and intervals y-i, y₂, y3 are time intervals.

[0053] The selection of boundary points z-i, z₂, z₃, z₄ is now described in further detail. Boundary points z<\, z₂, z₃, z₄ can be determined according to a boundary detection algorithm. For example, the boundary detection algorithm can select points where observed response d_Obs deviates from a substantially constant decaying slope (log-log graph) or can select points where an apparent resistivity curve a, described in further detail below, deviates from a substantially constant value. Boundary points z-\, z₂, Z₃, z₄ separate observed response dobs into observed response subsets d_Obs,-ι, d_Obs,2, d_Obs,3 that correspond to intervals y-i, y₂, ys-

[0054] In the second exemplary embodiment, the dimensionality of observed response d_Obs increases with respect to incremented parameter z. Specifically, the number of undetermined parameters m that affect each of data points of observed response subset d_Obs,₂ is greater than the number of undetermined parameters m that affect each of data points of observed response subset d_Obs,i and the number of undetermined parameters m that affect each of data points of observed response subset d_Obs,3 is greater than the number of undetermined parameters m that affect each of data points of observed response subset d_Obs,₂-

[0055] Undetermined parameters m that influence data points of each observed response subset d_Obs,i, d_Obs,2, d_Obs,3 are now described in further detail. Each observed response subset d_Obs,i, d_Obs,2, d_Obs,3 is affected by a subset of undetermined parameters m. One of undetermined parameters m in the subset is selected as interval specific undetermined parameter m-i, rτi₂, 1TI3 that is updated using the corresponding observed response subset d_Obs,-ι, d_Obs,2, d_Obs,3- Undetermined parameters m in a subset of undetermined parameters m other than interval specific undetermined parameter m-i, rτi₂, 1TI3 that affect observed response subset d_Obs,-ι, d_Obs,2, d_Obs,3 are referred to herein as auxiliary undetermined parameters m.

[0056] In the second exemplary embodiment, selection of interval specific undetermined parameter m-i, rτi₂, 1TI3 is facilitated by the physics of the measurement of observed response d_Obs- During interval y-i, a formation response signal Si received by receiver 28 has traveled only through first layer Li in which measurement device 24 is located. Observed response subset d_Obs,i is influenced by first layer resistivity pi but not by any other undetermined parameters m. The interval specific undetermined parameter mi is first layer resistivity pi and there are no auxiliary undetermined parameters m that affect observed response subset dobs,1 -

[0057] During interval y₂, a formation response signal S₂ will have traveled through first and second layers Li, L₂ and observed response subset d_Obs,₂ will be influenced by first and second layer resistivities p-i, p₂ as well as first boundary distance H-i. Since an updated value of first layer resistivity pi is determined as first layer resistivity pi has been selected as interval specific undetermined parameter m-i, and since an updated value of boundary distance Hi can be calculated using the updated value of first layer resistivity pi and the value of boundary point z₂, interval specific undetermined parameter m₂ for observed response subset d_Obs,₂ is selected as second layer resistivity p₂. First layer resistivity pi and first boundary distance Hi are considered auxiliary undetermined parameters m for observed response subset d_Obs,2- [0058] During time interval y3, a formation response signal S3 will have traveled through all three layers Li, L₂, L₃ and observed response subset d_Obs,3 will be influenced by first, second, and third layer resistivities p-i, p₂, p₃ as well as first and second boundary distances H-i, H₂. Updated values of first layer resistivity pi, second layer resisitivity p₂, first boundary distance H-i, and second boundary distance H₂ have been determined through selection of first layer resistivity pi as undetermined parameter mi and second layer resistivity p₂ as undetermined parameter m₂. Accordingly, interval specific undetermined parameter 1TI₃ for observed response subset d_Obs,3 is third layer resistivity P3. First and second layer resistivities pi, p₂ and first and second boundary distances H-i, H₂ are considered auxiliary undetermined parameters m for observed response subset d_Obs,3- [0059] Referring to FIG. 5, step two 2 continues as initial values of undetermined parameters m are determined by transforming observed response dobs into apparent resistivity curve a and then transforming apparent resistivity curve a into rectangulahzed resistivity curve r. Alternatively, the initial values of the resistivities pi, p₂, P3 can be selected arbitrarily.

[0060] Apparent resistivity subsets a-i, a₂, a₃ and rectangulahzed resistivity subsets r-i, r₂, r3 correspond to intervals y-i, y₂, y3. The value of each rectangulahzed resistivity subset r-i, r₂, r₃ is determined by taking the average of the corresponding apparent resistivity subset a-i, a₂, 83. The values of rectangulahzed resistivity subsets r-i, r₂, r3 provide initial values of resistivities pi,

[0061] Initial values of resistivities pi, p₂ are combined with values of boundary points z₂, Z3 to determine initial values of boundary distances H-i, H₂. The values of the resistivities pi, p₂, P3 and boundary distances H-i, H₂ can represent a formation model, as shown in FIG. 6.

[0062] Referring again to FIG. 4, a predicted response d_pred is generated with values of the initial formation model. Predicted response subsets d_pred,i, d_pred,2, d_Pred,3 correspond to intervals y-i, y₂, y₃. [0063] According to step three 3, to update the formation model and predicted response d_pred, a bi-modal turbo boosting method is used. One mode of the method is focused on updating the resistivities pi, p₂, P3 and another mode of the method is focused on correction of the boundary distances H-i, H₂.

[0064] A value of each interval specific unknown parameter m-i, rτi₂, 1TI3, selected as a corresponding resistivity pi, p₂, P3, is updated as above using a field ratio F-i, F₂, F3 and a ratio function R-i, R₂, R3 that incorporate current predicted response subsets d_pred,i, d_pred,2, d_pred,3 and observed response subsets d_Obs,i, d_Obs,2, d_Obs,3- Specifically, an updated value of resistivity pi, p₂, P3 can be determined according to

. Once updated values of the resistivities p-i, p₂, P3 have been determined, updated values of boundary distances H-i, H₂ can be calculated. A formation model with updated values of resistivity p-i, p₂, P3 and boundary distances H-i, H₂ is shown in FIG. 6. The updated formation model can be an output representing the characteristics of the formation 4. Alternatively, the output can include any of a visual representation, a graph, a table, a chart, audio information, machine readable information, a list, combinations thereof, and the like.

[0065] Values of resistivities pi, p₂, p₃ can be updated in a selected order. For example, where a first undetermined parameter m is an interval specific undetermined parameter rrij with respect to one of observed response subsets d_Pred,i, d_pred,2, d_pred,3 and an auxiliary undetermined parameter m with respect to another of observed response subsets d_pred,i, d_pred,₂, d_pred,3, an updated value of first undetermined parameter m can be used in the determination of an updated value of a second undetermined parameter m, which is the interval specific undetermined parameter rrij with respect to the one of observed response subsets d_Pred,i, d_pred,₂, d_pred,3 for which first undetermined parameter rrii is an auxiliary undetermined parameter m. An updated value of first undetermined parameter m can be used to provide an updated predicted response d_prΘd before determining an updated value of second undetermined parameter m.

[0066] Power coefficients q-i, q₂, q3 are determined according to

where A = (P₁, d^^ -d_obs>1) , B = (P₁₉P₁) . In this formulation, background parameter

(rrib in the general formulation) is selected as the current value of the resistivity p-i, p₂, P3 such that updated interval specific anomalous resistivity Δpi, Δp₂, Δp₃ (interval specific anomalous parameter Am; in the general formulation) can be given as

■ Also, background predicted response (d_b in the general formulation) is selected as the current predicted response d_pred such that the updated predicted response subset d_predj can be given by d⁽^" ^] = 41_,, +MA⁽"^+1> ■ Reference response subset pi is the data generated where a value of a reference resistivitiy Δp_ref is used to evaluate each of the resistivities pi, P₂, P3 of modeled response and the result is divided by the value of reference resistivity Δp_ref. Exemplary reference response p, shown in FIG. 7, is the normalized response of a homogeneous formation having a resistivity equal to the value of reference resistivity Δp_ref.

Geophysical Log - Variable Dimensionality Example [0067] Referring to FIG. 8, according to step one 1 of a third exemplary turbo boosting method, an observed response d_Obs is measured as a resistivity log. In this case, incremented parameter z is position and intervals y-i, y₂, y3 are depth intervals. Observed response dobs is affected by undetermined parameters m that include resistivities p-i, p₂, P3 of layers L-i, L₂, L3 and boundary positions z-\, z₂, Z3, Z₄. It should be understood that the turbo boosting method described herein can be applied to other geophysical logs.

[0068] According to step two 2 of the third exemplary turbo boosting method, boundary positions z-i, z₂, Z3, z₄ are determined from a measurement of a secondary observed response x_Obs such as a shallow resistivity log. Specifically, boundary positions z-i, z₂, z₃, z₄ can be selected to correspond to inflection points of secondary observed response x_Obs-

[0069] Continuing with step two 2, an initial formation model is determined by rectangulahzing observed response d_Obs- Referring to FIG. 9, the value of each rectangulahzed resistivity subset r-i, r₂, r3 is equated to the maximum or minimum of the corresponding observed response subset d_Obs,-ι, d_Obs,2, d_Obs,3- Alternatively, the value of each rectangulahzed resistivity subset r-i, r₂, rs can be set equal to the average of the corresponding observed response subset d_Obs,-ι, d_Obs,2, d_Obs,3- The rectangulahzed resistivity subset η, r₂, r3 values and the estimated boundary positions z-i, z₂, Z3, z₄ provide an initial formation model and are used to determine an initial predicted response d_prΘd- [0070] Interval specific unknown parameters m-i, rri₂, 1TI3 are selected as resistivities p-i, p₂, P3 since intervals y-i, y₂, y3 correspond to positions in layers Li, L₂, L₃. Such a selection assumes that observed response subset d_Obs,i, d_Obs,2, d_Obs,3 is biased towards the resistivity p-i, p₂, P3 θf layer the Li, L₂, L3 in which a measurement is made. [0071] According to step three 3, referring to FIGs. 8 and 9, to update the formation model, values of resistivities pi, p₂, p₃ are updated while values of boundary positions z-i, z₂, Z3, z₄ are held constant. Field ratios F-i, F₂, Fs and ratio functions Ri, R₂, Rs are determined as above with observed response subsets d_Obs,i, d_Obs,2, d_Obs,3 and predicted response subsets d_pred,i, d_pred,2, d_pred,3- Updated values of the resistivities pi, p₂, P3 are determined as above according to

As above, power coefficients q-i, q₂, qs can be given by

where A = (_p,d^_ι -d_obsv) , B = (_Pι,_Pι) . [0072] In alternative embodiments, predicted response d_pred and observed response d_Obs can be compared subset by subset whereby only interval specific undetermined parameters rrii that correspond to intervals y, where predicted response subset d_prΘd,i and observed response subset dobsj substantially differ are updated according to the turbo boosting method described herein. [0073] In still another embodiment, the invention can be described as the following: a method of transient deep electromagnetic reading or logging, comprising the steps of a) collecting a raw transient response signal that is indicative of a formation response as a function of time, b) finding response times at which relatively abrupt changes in the time derivative of the signal occur, c) constructing an initial formation bed model, including the sub-steps of c)1 ) assuming that these response times correspond to bed boundaries, and c)2) finding initial values for the resistivity of each bed by fitting model solutions to the part of the signal having response times between two bed boundaries, d) modifying the initial formation bed model by executing a number of iterations employing at least the following steps: d)1 ) adjusting the locations of the bed boundaries in the formation bed model based on the found values of the resistivity of each bed, d)2) calculating a computed signal representing what the measured signal would look like if the modified formation bed model corresponded to the actual formation, d)3) determining values of the computed signal and values of the measured signal at response times corresponding to bed centers in between two neighboring adjusted bed boundaries, d)4) finding new values of resistivity of each bed by dividing the values of the computed signal and values of the measured signal at the bed centers and raising the quotient to the power of a predetermined number and equating the results to the new values of resistivity for each bed, d)5) using the new resistivity as the found values for the resistivity of each bed, and reiterating step d) at least once.

[0074] The present invention has been illustrated in relation to a particular embodiment which is intended in all respects to be illustrative rather than restrictive. Those skilled in the art will recognize that the present invention is capable of many modifications and variations without departing from the scope of the invention.

[0075] The law does not require and it is economically prohibitive to illustrate and teach every possible embodiment of the present claims. Hence, the above- described embodiments are merely exemplary illustrations of implementations set forth for a clear understanding of the principles of the invention. Variations, modifications, and combinations may be made to the above-described embodiments without departing from the scope of the claims. All such variations, modifications, and combinations are included herein by the scope of this disclosure and the following claims.

Claims

C L A I M S

1. A method of characterizing a geological formation based on an observed response measured in the formation, the observed response being an electromagnetic response measured as a function of time, the method comprising: using a computing unit, fitting a modeled response to the observed response, the modeled response being a function of at least one undetermined parameter (m), a current predicted response being determined using the modeled response and a current value of the at least one undetermined parameter (m), and one of the at least one undetermined parameter (m) being an interval-specific undetermined parameter (m,), the fitting method comprising: i) determining a value of a ratio function (R,) by comparing the current predicted response and the observed response; ii) determining a value of a power coefficient (q,), the power coefficient (q,) being determined so as to minimize the result of a function of the difference between the observed response and an updated predicted response, the updated predicted response being a function of the interval- specific undetermined parameter (m,), the observed response, and the current predicted response; iii) determining an updated value of the interval-specific undetermined parameter (m,) according to:

m ( »₊₁, _{= m} (») _{( Λ} (», ) -,<•^■ ' _{; a nd}

generating an output representing the characteristics of the environment, said output being based on the updated values of interval specific parameter (m,).

2. The method of claim 1 wherein the power coefficient is determined so as to minimize the square norm of the difference between the observed response subset and an updated predicted response.

3. The method of claim 1 wherein the fitting method further comprises determining an updated predicted response using the modeled response and the updated value of the interval-specific undetermined parameter.

4. The method of claim 1 wherein the updated value of the interval-specific undetermined parameter is determined by repeating step iii) for a number of iterations.

5. The method of claim 1 wherein the fitting method further comprises defining the current predicted response and the observed response by identifying at least one point where an apparent resistivity curve deviates from a substantially constant value.

6. The method of claim 1 wherein the fitting method further comprises defining the current predicted response and the observed response by identifying at least one point where a log-log graph of the observed response deviates from a substantially constant decaying slope.

7. The method of claim 6 wherein the at least one point corresponds to a boundary between layers of a formation.

8. The method of claim 1 wherein the observed response is a geophysical log corresponds to a length interval.

9. The method of claim 1 wherein the observed response is a geophysical log and corresponds to a time interval.

10. The method of claim 9 wherein the fitting method further comprises defining the current predicted response and the observed response by identifying at least one inflection point of a shallow log that corresponds to the geophysical log.