Computers & Geosciences 37 (2011) 1174–1180Contents lists available at ScienceDirect Computers & Geosciences journal homepage: www.elsevier.com/locate/cageo 3D porosity prediction from seismic inversion and neural networks Emilson Pereira Leite n, Alexandre Campane Vidal ~ Pandia ´ Calo ´geras, 51 CEP, 13083-970 Campinas, SP, Brazil Department of Geology and Natural Resources, Institute of Geosciences, State University of Campinas, Joao a r t i c l e i n f o a b s t r a c t Article history: Received 10 March 2010 Received in revised form 19 August 2010 Accepted 31 August 2010 Available online 20 November 2010 In this work, we address the problem of transforming seismic reflection data into an intrinsic rock property model. Specifically, we present an application of a methodology that allows interpreters to obtain effective porosity 3D maps from post-stack 3D seismic amplitude data, using measured density and sonic well log data as constraints. In this methodology, a 3D acoustic impedance model is calculated from seismic reflection amplitudes by applying an L1-norm sparse-spike inversion algorithm in the time domain, followed by a recursive inversion performed in the frequency domain. A 3D low-frequency impedance model is estimated by kriging interpolation of impedance values calculated from well log data. This low-frequency model is added to the inversion result which otherwise provides only a relative numerical scale. To convert acoustic impedance into a single reservoir property, a feed-forward Neural Network (NN) is trained, validated and tested using gamma-ray and acoustic impedance values observed at the well log positions as input and effective porosity values as target. The trained NN is then applied for the whole reservoir volume in order to obtain a 3D effective porosity model. While the particular conclusions drawn from the results obtained in this work cannot be generalized, such results suggest that this workflow can be applied successfully as an aid in reservoir characterization, especially when there is a strong non-linear relationship between effective porosity and acoustic impedance. & 2011 Elsevier Ltd. All rights reserved. Keywords: Reservoir characterization Seismic inversion Feed-forward neural network Matlab 1. Introduction During the last decades, several methods for mapping acoustic impedance from post-stack seismic amplitude data were developed and tested with the aim of providing additional information for detailed reservoir characterization. Nowadays, most of the research efforts in this field are focused in the inversion and interpretation of variations of seismic reflection amplitude with change in distance between source and receiver (amplitude vs. offset) from pre-stack data. However, post-stack data obtained from recorded P-waves are still widely used because of their ready availability and low time-consuming processing. Because wells in a reservoir field are often spaced at hundreds or even thousands of meters, the ultimate goal of a seismic inversion procedure in the context of reservoir characterization is to provide models not only of acoustic impedance but also of other relevant physical properties, such as effective porosity and water saturation, for the interwell regions. Such quantitative interpretations may sometimes require the use of other seismic attributes additionally to the traditional seismic reflection amplitudes (Rijks and Jauffred, 1991; Lefeuvre et al., 1995; Russell, 2004; Sancevero et al., 2005; Soubotcheva, 2006). n Corresponding author. Tel.: + 55 19 35214697; fax: + 55 19 32891097. E-mail addresses:
[email protected],
[email protected] (E.P. Leite). 0098-3004/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cageo.2010.08.001 The seismic inversion method that is presented in this work is classified as a deterministic inversion method (Russell, 1988). Although many recent papers have demonstrated some advantages of geostatistical methods over deterministic methods (Francis, 2005; Robinson, 2001), the latter can still provide geologically plausible acoustic impedance models at a much lower computational cost. The first deterministic inversion methods for acoustic impedance mapping were developed in the late 70 s and became to known generally as recursive inversion (Lavergne and Willm, 1977; Lindseth, 1979). The basic premise of those and of all methods that were subsequently developed in the 1980s is the local validity of the 1-D convolutional model. During the 1980s, sparse-spike inversion methods were developed consisting of some techniques that make use of an additional premise that the reflections occur as sparsely distributed spikes within a layered Earth (Oldenburg et al., 1983; Russell, 1988). In this case the reflectivity function is mathematically represented as the product of the reflection coefficients and a Dirac delta function shifted by the two-way travel time to each layer. Two well known methods that fall in this category are the L1-norm sparse-spike inversion (Sacchi and Ulrych, 1996), which is applied in the methodology described in this work, and the maximum likelihood inversion (Hampson and Russell, 1985). Prediction of reservoir properties from acoustic impedance can also be thought as a kind of inversion and traditionally have been addressed through the application of multivariate statistics and, calculated through ordinary kriging of the kwon AI values at the well log positions. (7) low-pass filter AIres and add to the result of step (6). The first term in Eq. 2.P.. For a properly usage of the recursive inversion. Hampson et al. then it is inverted into AI according to the following sequential steps (Ferguson and Margrave. the seismic traces should be deconvolved into reflectivity series as suggested by Eq. through a recursive equation 1þ rj IAj þ 1 ¼ IAj . 3 5 i¼2 from which we can discard the high-order terms leading to the expression AIM ¼ AI1 expð2 M X rj Þ: 1175 valid for most of the practical cases where rj rj0:3j (Oldenburg et al. obtaining a residual AIres vector. Methodology 2.. the AI values at the positions of each seismic sample can be extracted from a 3D model covering the entire seismic volume. and (9) add the low-frequency trend from step (1) to the result of step (8). 2001. IAj þ 1 þIAj ð2Þ where rj is the reflection coefficient at the jth interface of a set of N superposed layers. A low cut-off for coupling the low frequency trend and a high cut-off were defined by finding where the energy content of the original seismic traces approaches to zero in the amplitude spectrum. (ii) the seismic trace s(t) can be represented by the convolution of the reflectivity coefficient series r(t) with a bandlimited wavelet w(t) and the addition of a random noise n(t): sðtÞ ¼ rðtÞwðtÞ þnðtÞ: ð1Þ For zero incident angles. AI1 is the known acoustic impedance in the top layer and AIM is that of the Mth layer.. After estimating r from the seismic amplitudes. (5) determine a scalar a to match the mean power of AIrel and AIres. ð6Þ j¼2 Eq. therefore.. and (iii) there is no need to known the underlying statistical distribution of the input data. Porosity prediction using Neural Networks The procedure outlined here can be applied to reservoirs that do not show a linear relationship between AI and the reservoir property that needs to be mapped. Other optimization algorithms can also be used to minimize Eq. Berteussen and Ursin. A. Vidal / Computers & Geosciences 37 (2011) 1174–1180 more recently. where a controls the sparsity of the solution.. (3) apply Eq. With the second term. To accomplish this. (6) is a practical formula used in recursive inversion for transformation of reflectivity into impedance. 1996): (1) compute the linear trend of a spatial correspondent AI vector and subtract it.1. 1994). ð3Þ 1rj which in turn can be generalized to provide the AI value of an arbitrary M layer by M Y 1þ rj IAM ¼ IA1 : ð4Þ 1rj j¼2 The natural logarithm is applied to both sides of Eq. respectively. we apply a constrained sparse-spike optimization procedure that minimizes the objective function JðrÞ ¼ a M X j¼1 jrj jþ 1 1 2 : ðsWrÞ: 2 s ð7Þ using the conjugate-gradient algorithm (Shewchuk. It is of course possible to include an extra constraint on impedances directly in Eq. such as Iterative ¨ Reweighted Least Squares (Bjorck. (ii) less sensitivity to the presence of noise in the data. by the approach described in this paper it is possible to keep control of the frequency contents involved and the frequency cut-offs to properly add the trend in acoustic impedance. This approximation is 2. In general. Due to the sparse distribution of wells. and IA ¼ rv where r e v are the density and P-wave velocity. (7). This characterizes the band-limited nature of the seismic data. 2002. The main advantages of NN methods over most traditional statistical methods can be summarized as follows: (i) the ability to extract nonlinear relationships between the input data and the target values. demonstrating that the former can provide higher correlation coefficient between actual and predicted reservoir property values and minimize the problem of sparse well coverage. 2001. (7) is provided in order to allow minimization of the L1norm of the reflectivities. Seismic inversion The basic premises behind all seismic inversion methods in the context of this work are as follows: (i) the Earth can be represented locally by a stack of plane and parallel layers with constant physical properties. In practice. For the particular example shown in this . this means that small wavelength features in the log impedance curve will not be recovered by the inversion and. 1983. (4) compute the Fourier spectra of AIrel. Leite. 1996) or soft-tresholding algorithms (Loubes and De Geer. (6) multiply the spectra of AIrel by a. rj is the reflection coefficient of the jth layer. (4) in order to obtain a linear approximation: " # M X r3 r5 ð5Þ lnðIAM Þ ¼ lnðIA1 Þ þ 2 ri þ i þ i þ . these papers compare performances of NN models with traditional regression methods. the interpreter has to be cautious while analyzing the inversion results. W is a wavelet coefficient matrix and s is the standard deviation of the seismic data noise. (7). 2004. 1983).C. (2) compute the Fourier spectra of AIres. In other words. obtaining a relative AIrel vector.2. (8) inverse Fourier transform the result of step (7). (6) to the reflectivity series. 1983). Pramanik et al. It is important to point out that this constrained sparse-spike inversion will provide an impedance model that does not display the actual reflection series but displays only the largest reflectors (Oldenburg et al. Walls et al. NN methods have been successfully applied in a wide variety of applications in reservoir characterization such as porosity and permeability prediction from seismic and well-log data or seismic facies/attributes classification (Leiphart and Hart.E. (6). Calderon. 2007). Neural Network (NN) methods. r(t) is directly related to the contrast in the acoustic impedance (AI) of superposed layers through the expression rj ¼ IAj þ 1 IAj . the low-frequency trend of step (1) was extracted from spatial correspondent AI traces estimated by kriging. Under these conditions and assuming that multiple reflections were eliminated from the seismic data. However. the algorithm also minimizes the difference between the synthetic seismic traces (Wr) and the observed traces (s). 2002). the AI value of each layer can be calculated from the knowledge of the AI value of the layer above. This requires knowledge of the wavelet representing the seismic pulse. (1). In-lines and cross-lines are spaced at about 13 and 27 m. The overall workflow of the methodology is shown in Fig. Time interval is equal to 4 ms. so as to provide the best fits. Leite and Souza Fig. For most practical situations there is no deterministic way to choose the best number of neurons to be used in the second layer and a trial and error approach has to be applied. from the first to the third layer. These neurons are represented by weights that are iteratively updated during the training stage using a gradient descent algorithm. This conversion was done using the sonic log and the initial two-way travel time (TWT) for the first log sample that provided the highest correlation coefficient (R) between the synthetic and observed trace. Flowchart of proposed methodology. Vidal / Computers & Geosciences 37 (2011) 1174–1180 work. The employed NN is a three-layer feed-forward system where the information propagates only in one direction.P. Seismic-well ties were conducted by adjusting five traces around each well and retaining the local mean wavelet. Size of 3D matrix is 301 61 375. Fig.. . we carried out a NN analysis in order to search for a relationship between effective porosity (Phie) and other well logs such as density (RHOB). A common wavelet length was then determined to be 60 ms for the particular case of this work. 2008) is employed as a transfer function in both the second and the third layer. 3. The training process is carried out until at least one of the following conditions is met: (i) a minimization of a MSE goal is achieved. gamma-ray (GR). a global mean wavelet was calculated and used for inversion of the traces away from the wells. respectively. The first layer contains the input values extracted from the well logs. 2. The third layer outputs the results that are compared with the actual target values at the end of each training iteration (or epoch) so as to check the meansquared error (MSE) between them.1176 E. validation and test subsets. Results and discussion We carried out a depth-to-time conversion to make the vertical scale of the well log AI data match the vertical scale of the seismic data so as to allow an adequate correlation. (1) as a linear system and solving for w(t) (Broadhead. we verified that the size of the wavelet should be less than or equal to 1/5 times the length of the reflection coefficient series. A. Thus we performed a deterministic wavelet extraction by writing Eq. A hyperbolic tangent sigmoid function (Demuth et al. There are two crucial factors in this procedure that may lead to poor wavelet estimation—incorrect depth-to-time conversion and incorrect size of the wavelet. This is commonly known as a seismic-well tie. Leite. 2. by comparing synthetic with observed traces. 2008). a sample set obtained from the well logs is split into training. Filho (2009) presented a more detailed description of the NN method that was also applied in this work.C. The test subset is used only to estimate the prediction power of the NN by performing a blind test and it is not used for building the NN model. (ii) occurrence of three consecutive non-improvements in the MSE for the validation subset (early-stopping). Then. The synthetic traces were calculated using the convolutional model given by Eq. 3D seismic data and spatial location of wells. As an input for designing a NN model. 1. In this work. and the sonic log (DT). In our tests. The second (hidden) layer consists of an activation function associated with a set of neurons. we control the errors in the former by changing the initial TWT and checking the value of R. or (iii) a maximum number of iterations are completed. the estimated reflectivity. Leite. Fig. (c) synthetic traces computed through convolution of reflectivity with wavelet and (d) observed traces.C. Fig. A. 3. where the acoustic impedance log. 4. (b) wavelet. 4 shows the spectral content of the reflectivity. (c) synthetic traces and (d) observed traces near Well 2. .E. Example of seismic-well tie for Well 2: (a) impedance log converted to two-way time and resampled to interval of 4 ms. the wavelet. (b) reflectivity obtained after deterministic wavelet extraction.P. 3. Normalized amplitude spectrum of (a) reflectivity. The spectral content is similar for the other four wells in the area and Fig. the synthetic traces and the observed traces are 1177 shown. Vidal / Computers & Geosciences 37 (2011) 1174–1180 An example of this procedure for Well 2 can be visualized in Fig. the synthetic traces and the observed traces. 1. (7)).02 to 0. AI log profile (red curves) vs. as 5 and 60 Hz.P. 6.1178 E. Fig.035. Fig. Leite. Fig. In real applications it is commonly difficult to estimate the standard error of the noise in the seismic data (s in Eq. respectively. A. 5 shows the AI lowfrequency model obtained by kriging the AI well logs at the wells depicted in Fig. Correlation coefficient between observed seismic traces and synthetic traces obtained from inverted models around Well 2. We used a value of 5% in this work.) . AI low-frequency model obtained by kriging of AI well logs. Fig. The lateral boundaries were defined so as to embrace wells that were previously found to have some oil or gas content in the field. The reservoir top and base were estimated from well log markers allowing the definition of minimum and maximum time values. Vidal / Computers & Geosciences 37 (2011) 1174–1180 low and high cut-offs were defined. the reader is referred to the web version of this article. inverted AI profiles (blue curves) for Well 2 as a varies from 0. 7.C. 5. A reasonable assumption is that it corresponds to some percentage of the peak amplitude of the traces. after some tests. thus establishing the vertical boundaries of the seismic 3D grid shown in the subsequent figures. (For interpretation of the references to color in this figure legend. are presented in Figs. Color bar is in m/s g/cm3. the inverted model is enriched in details and can be used for posterior prediction of the reservoir properties. A synthetic seismic model calculated from this inverted model is highly correlated with the observed seismic data (average R is equal to 0.025 was considered to be adequate because the inverted curve follows the main trends that appear in the log profile while R is sufficiently large. The training must be performed in the time scale instead of the depth scale to allow posterior prediction in the entire seismic volume using the inverted AI model as input.035.) 1179 smooth. it lacks too much vertical detail and it gives a low R. validation and test. Four results for Well 2 considering a particular range of a. For a ¼0.P. where the transition of the behavior of the inversion solution is quite noticeable. In this work we have found that a feed-forward NN can successfully predict Phie from the joint use of GR and AI logs.1]. The graphs in Fig.035 the solution is very Fig. . Vidal / Computers & Geosciences 37 (2011) 1174–1180 The parameter a has to be estimated empirically.02 r a r0.C.0 in steps of 0. When a is smaller than 0. All values were normalized into range [ 1. 6 and 7.84). This was checked by an iterative cross-validation scheme where the samples that compose the three subsets were randomly interchanged and a new NN model was obtained at each iteration. 8.92). AI obtained through proposed inversion methodology. The behavior of the inverted curve does not change significantly outside of the range 0.005 and comparing the inversion results with the log profile. which in turn can only be obtained in the time scale. 9. In spite of this. A. the NN models were able to map the test samples into Phie within an acceptable level of accuracy (R¼0. validation (20%) and test (20%) of the NN.025. as well as checking the value of R between synthetic and observed traces around the wells. While the range of AI values is about the same. NN training. (For interpretation of the references to color in this figure legend. the reader is referred to the web version of this article. Leite. The 3D AI inverted model is shown in Fig.01 to 1. R is correlation coefficient between outputs and actual target Phie values. The small amount of log values is due to the depth-to-time conversion.E. A sample set consisting of 32 log values was extracted from the well logs for training (60%). 8. 9 show the results for the NN model that provided the highest overall Fig. We performed tests by varying a from 0. overfitting seems to occur. A value of a ¼0. which unavoidably reduces the vertical resolution of the original well log data. N. 11). Comparing deterministic and stochastic seismic inversion for thin-bed reservoir characterization in a turbidite synthetic reference model of Campos Basin. M. H. Ulrych. Ph. A. 8.. M.. The Netherlands. 1995.. Alberta. Lakings. Beale.S.G. Reservoir Property Prediction from Well-logs. 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