Gabriela B. Savioli

Comparison of optimization techniques for automatic history matching

Abstract Reservoir parameters are estimated by adjusting simulation models to match field or laboratory data. Multivariate optimization techniques with physically realistic constraints on the parameters are used in order to obtain these estimates. Two examples are presented. The first example is the analysis of a drawndown test. Permeability and porosity are determined by minimizing an objective function which is the sum of the squares of the differences between theoretical and measured pressure-time distributions at the well. The minimization is performed by applying four different optimization techniques: Davidon-Fletcher-Powell (DFP), Fletcher-Reeves (FR), Quasi-Newton Approximation for the Least-Squares Problem (QNA) and Levenberg-Marquardt (LM). The second example is the simultaneous determination of capillary pressure and relative permeability curves of oil/water systems. It is based on the analysis of transient output data measured from a linear coreflood experiment. QNA and LM are used to match results from a numerical simulator to laboratory coreflood data. The special methods for the least-squares problem (LM, QNA) behave better than the two others (DFP, FR). LM and QNA arrive to the optimal point more frequently than DFP and FR. LM takes less computing time than QNA but is more affected by rounding errors. Therefore, QNA shows the best behavior when finding the optimum. The automatic algorithms are of particular use whenever the equations which govern the flow are too complex to be solved by the traditional analytical-graphical methods.

Statistical analysis of heterogeneities and their effect on build-up and drawdown tests

The objective of this work is to determine the effect of permeability and porosity spatial variations on well test pressure response. Data from three wells and synthetic data are used. Field data consist of permeability and porosity as functions of depth and pressure transient test measurements from the same wells. To achieve the objective, two different tools are applied: statistical characterization of heterogeneities and a well test interpretation method. For the data: (1) permeabilities are represented by exponential distribution functions; (2) constant porosity estimates that fit measured transient pressures are almost equal to the statistical arithmetic mean, while constant permeability estimates lie between the median and the arithmetic mean; and (3) minor permeability variations cause important changes in pressure response.

A parametric analysis of waves propagating in a porous solid saturated by a three-phase fluid

This paper presents an analysis of a model for the propagation of waves in a poroelastic solid saturated by a three-phase viscous, compressible fluid. The constitutive relations and the equations of motion are stated first. Then a plane wave analysis determines the phase velocities and attenuation coefficients of the four compressional waves and one shear wave that propagate in this type of medium. A procedure to compute the elastic constants in the constitutive relations is defined next. Assuming the knowledge of the shear modulus of the dry matrix, the other elastic constants in the stress-strain relations are determined by employing ideal gedanken experiments generalizing those of Biots theory for single-phase fluids. These experiments yield expressions for the elastic constants in terms of the properties of the individual solid and fluids phases. Finally the phase velocities and attenuation coefficients of all waves are computed for a sample of Berea sandstone saturated by oil, gas, and water.

On some numerical methods for solving 2D radial flow towards an oil well

In this paper we study a family of finite difference schemes in two dimensions to model the single phase flow of oil through heterogeneous porous media. That family depends on one parameter θ, 0 ≤ θ ≤ 1. Using a suitable order of equations and unknowns, a linear system of equations, with a particular structure, is obtained. The corresponding matrix, excluding the first row and column, has up to five elements in each row, arranged in five diagonals. The system of linear equations is solved by a method based on Taylor series of matrix functions (TSMF). The convergence conditions for this technique are established and the most convenient θ is selected to increase the time step Δt. Besides, TSMF is compared with two iterative methods, ADI and block-SOR, usually applied to solve multidimensional equations. Both methods, ADI and block-SOR, are adapted to this particular problem. We conclude that TSMF is the fastest technique using adequate values of θ and Δt, but the time increment Δt must remain small because of the convergence condition. On the other hand, block-SOR converges using large values of Δt, but it uses a large amount of CPU time. ADI is discarded for not presenting advantages over the other two techniques. Therefore, TSMF is recommended when a short period of time must be simulated, and block-SOR is suitable for long simulations and applying a variable time increment.

Applications of Simulated Annealing on Actual but Atypical Permeability Data

The usc of geostatistics is becoming recognized as a standard means of representing reservoir heterogeneity. Geostatistics has enjoyed an extensive use and a fairly well developed theoretical base. This is a little less true of simulated annealing (SA), the form of geostatistics tested here, but it is also a mature technology. Yet there remains a need to exercise these procedures under actual conditions of nonuniformly sampled data, non-Gaussian distributions and truncated data sets, Providing insights into how to deal with these nonidealitics is the objective of this work. We find that SA estimates are improved when the original data sets are power-transformed. However, SA estimates tend to deviate from the input cumulative distribution function (CDF) bccausc of cxccssive rejections, This deviation can be corrected by including the CDF into the SA objective function. Introduction Stochastic reservoir modeling refers to the generation of synthetic reservoir properties that are conditioned to observations. Ideally, the generated ‘image” of reservoir properties should honor all available data; seismic traces, geological description, core measurements, well logs, pressure test analysis, ctc Various methods have been applied to the stochastic modeling of resctvoir heterogeneities. One of them, which has been rcccntly introduced is simulated annealing (SA) 1‘2. This is a combinatorial optimization technique that involves a two-step procedure: first, an objective fimction (OF) is built, and second, the OF is minimized by an appropriate algorithm, The main advantage of SA is that it can combine data from different sources by simply adding extra information into the OF. In this work, the objective tirnction is minimized by the Metropolis algorithm, as described by Kirkpatrick et al, 1 and Sen ct al. 3, After analyzing the performance of the three algorithms, Sen et aL3 concluded that the Metropolis algorithm is the fastest when solving small problems. And this is our case. Our purpose is to test the ability of SA to generate synthetic permeability fields that represent actual heterogeneity. With that aim, the generated image of permeabilitics is compared with core measurements, Our data consist of three sets of permeability measurements as functions of depth. They correspond to three wells from different reservoirs: well A, B and C. It has been generally accepted4 that in a rock type unit, permeabilitics follow a log-normal distribution, However, there are no theoretical background for that thoughts. Jensen et al 6 proposed a power transformation of permeability data, which depends on one parameter p, Any random series-parallel arrangement of permeability elements yields a p-normal distribution such that –1 < ps 1. The normal distribution has p = I and the log-normal distribution has p = O. Jensen6 showed several data sets which fall within these limits. None of the three sets of permeability data follow log-normal distributions. Therefore, we applied the Jensen p-transformation to them. Permeabilities from well B do follow a normal distribution after p-transformation. But those from well A show an exponential distribution On the other hand, permcabilities from WCI1C could not be arranged in a known distribution either before or after the p-transformation. The goal of this work is to analyze the performance of the SA method in conjunction with the Metropolis algorithm on these three atypical sets of data.

A model for CO2 storage and seismic monitoring combining multiphase fluid flow and wave propagation simulators. The Sleipner-field case

The main objective of this paper is to use a flow simulator to represent the CO2 storage and combine it with a wave propagation simulator in order to obtain synthetic seismograms qualitatively matching time-lapse real field data. The procedure is applied to the Utsira formation at Sleipner field. The field data at the site available to us is a collection of seismic sections (time-lapse seismics) used to monitor the CO2 storage. An estimate of the CO2 injection rate and the location of the injection point are known. Using these data, we build a geological model, including intramudstone layers with openings, whose coordinates are defined by performing a qualitative match of the field seismic data. The flow simulator parameters and the petrophysical properties are updated to obtain CO2 saturation maps, including CO2 plumes, so that the synthetic seismic images resemble the real data. The geological model is based on a porous-media constitutive equation. It considers a poroelastic description of the Utsira formation (a shaly sandstone), based on porosity and clay content, and takes into account the variation of the properties with pore pressure and fluid saturation. Moreover, the model considers the geometrical features of the formations, including the presence of shale seals and fractures. We also assume fractal variations of the petrophysical properties. The numerical simulation of the CO2-brine flow is based on the Black-Oil formulation, which uses the pressure-volume-temperature (PVT) behavior as a simplified thermodynamic model. The corresponding equations are solved using a finite difference IMPES formulation. Using the resulting saturation and pore-pressure maps, we determine an equivalent viscoelastic medium at the macroscale, formulated in the space-frequency domain. Wave attenuation and velocity dispersion, caused by heterogeneities formed of gas patches, are described with White’s mesoscopic model. The viscoelastic wave equation is solved in the space-frequency domain for a collection of frequencies of interest using a finite-element iterative domain decomposition algorithm. The space-time solution is recovered by a discrete inverse Fourier transform, allowing us to obtain our synthetic seismograms. In the numerical examples, we determine a set of flow and petrophysical parameters allowing us to obtain synthetic seismograms resembling actual field data. In particular, this approach yields CO2 accumulations below the mudstone layers and synthetic seismograms which successfully reproduce the typical pushdown effect.

A model for wave propagation in a porous solid saturated by a three-phase fluid

This paper presents a model to describe the propagation of waves in a poroelastic medium saturated by a three-phase viscous, compressible fluid. Two capillary relations between the three fluid phases are included in the model by introducing Lagrange multipliers in the principle of virtual complementary work. This approach generalizes that of Biot for single-phase fluids and allows to determine the strain energy density, identify the generalized strains and stresses, and derive the constitutive relations of the system. The kinetic and dissipative energy density functions are obtained assuming that the relative flow within the pore space is of laminar type and obeys Darcys law for three-phase flow in porous media. After deriving the equations of motion, a plane wave analysis predicts the existence of four compressional waves, denoted as type I, II, III, and IV waves, and one shear wave. Numerical examples showing the behavior of all waves as function of saturation and frequency are presented.

Stochastic Modeling of Rock Heterogeneities Applying New Autocorrelation Estimators and Simulated Annealing

Two new integral estimators of spatial autocorrelation are put forward and studied. The first is LV (Local Variance), based on the variance of the distribution. It is also an estimator of the average value of the semivariogram within a region. The second, SLV (Semivariogram from the Local Variance), is related to the derivatives of the former. It is an estimator of the semivariogram itself. The behavior of these two estimators is compared to that of the classical semivariogram estimator (CSV) using different data sets. SLV behavior is similar to that of CSV. Both are noisy and present fluctuations that increase with lag distance. Instead, LV is smoother and more resistant to outliers, making it easier to be represented by a theoretical model. Also, LV is less affected by individual values of samples, honoring the general statistics of data. Afterwards, the ability of the three estimators to model actual spatial variability is tested using the Simulated Annealing optimization technique. This technique has a good performance when any of the three estimators is included in the objective function. It is able to reproduce actual porosity or permeability fields, even when there is a specific spatial structure (such as cyclic, staircase-like, etc). However, LV has a main advantage: it still preserves the main features of the spatial structure even though it is very smooth. Introduction Geostatistics has become a useful tool to model the main features of oil and gas reservoirs. Many stochastic simulation techniques have been applied to describe the spatial structure of rock properties, such as porosity or permeability. A stochastic simulation tool that has proved its adequacy to obtain reservoir descriptions is Simulated Annealing (SA). SA was introduced in our field for its ability to incorporate different reservoir properties in the objective function. In particular, to characterize permeability distributions, SA uses geostatistical constraints such as a given histogram (or cumulative distribution function, CDF), the autocorrelation and conditioning data values at their locations. The autocorrelation is usually measured by the semivariogram (also called variogram), which is an estimator of the semivariance. SA is named after the modeling of the process of annealing, a procedure for reducing the temperature of a system to its minimum energy. That minimum is the global optimum for a given objective function. An additional advantage of SA, is its capacity to match any shape of a semivariogram. Deutsch and Deutsch and Journel matched theoretical models of the semivariogram. On the other hand, Ouenes, Sen et al. and Savioli et al. applied experimental semivariograms because oscillations, holes and bumps which are difficult to model may be realities of the reservoir that should be taken into account. In either case, SA achieves a good semivariogram matching. There are several estimators of the semivariance. The classical estimator (CSV) was first defined by Matheron in 1962. Its main drawback is inaccuracy at large separation distances, because the number of observation pairs decreases as the separation distance increases. The lack of precision also occurs for data sets with a significant proportion of missing measurements. Advantageously, the CSV is unbiased. The estimators robustness refers to its ability to be unaffected by errors in a small proportion of the data, and its resistance to data sparcity and outliers. Two autocorrelation estimators have been introduced by Li and Lake. Both are integral estimators. Because of this, they are more robust than the previous ones. Their disadvantage is that they take more CPU time than CSV. Elsewhere, we have analyzed the behavior of Li and Lakes integral estimators by comparison with several previous point to point estimators. Also, we have studied the performance of those integral estimators when they are included in the objective function of SA. The first purpose of this paper is to present and study two new autocorrelation estimators. The first is LV (Local Variance), based on the variance of the distribution. It is also SPE 69654 Stochastic Modeling of Rock Heterogeneities Applying New Autocorrelation Estimators and Simulated Annealing A.F. Saccomano, SPE, G.B. Savioli, SPE, and M.S. Bidner, SPE, University of Buenos Aires 2 A.F. SACCOMANO, G.B. SAVIOLI, M.S. BIDNER SPE 69654 an estimator of the average value of the semivariogram within a region. LV is very robust and resistant to extreme values. It is also very smooth and easy to be represented by a theoretical model. The second, SLV (Semivariogram from the Local Variance), is related to the derivatives of the former. It is an estimator of the semivariogram itself. LV and SLV perform similarly to Li and Lakes integral estimators, and their main advantage is that they are three times faster to compute. The behavior of LV, SLV and CSV estimators is compared here using normally distributed and specific spatial structured data. The second purpose of this paper is to test the ability of the three estimators to model actual heterogeneity when they are included in the objective function of the Simulated Annealing technique. For each particular data set, that objective function was built applying either an experimental semivariogram or a theoretical model. Theory Spatial autocorrelation. Autocorrelation is the degree of similarity between values of the same property Z at different locations i x . A common measure of autocorrelation is the semivariance, defined as, ( ) ( ) ( ) ( ) { } 2 2 1 j i j i x Z x Z E x , x − = γ . ............................ (1) Under the hypothesis of second-order stationarity, the semivariance becomes a function of a single argument h, the separation or lag distance, ( ) ( ) ( ) ( ) ( ) { } 2 2 1 x Z h x Z E h x , h x − + = = + γ γ ............. (2) A plot of γ versus h is called the semivariogram. The objective of this paper is to introduce two new estimators of spatial autocorrelation, LV (Local Variance) and SLV (Semivariogram from the Local Variance). To do this, a new autocorrelation measure is defined. New autocorrelation measure. The dispersion variance of the variable Z in a volume V is defined as, ( ) { } ∫ − = V V V dx Z ) x ( Z V E 2 2 1 σ ................................... (3) where ZV is the mean value of Z in the volume V, ∫ = V V dy ) y ( Z V Z 1 ........................................................ (4) This dispersion variance, computed in a ball of diameter h, is proposed as the new autocorrelation measure, ) h ( V V 2 2 σ σ = . It describes the degree of similarity between values of Z separated, at the most, by a distance h. There is a relationship between ) h ( V 2 σ and the semivariance, Eq. 2. The mean value of the semivariance within the volume V is defined as, ∫ ∫ − = V V VV dy dx ) y x ( V γ γ 2 1 ................................... (5) Using Eqs. 3 and 5, it can be proved that, VV V γ σ = 2 ..................................................................... (6) In the one-dimensional case, where the volume V becomes a segment of length h, the semivariance can be related to the derivatives of the new autocorrelation measure,

Mathematical modelling of flow towards an oil well

We describe the numerical approximations and applications of a mathematical model that governs the flow of oil towards a well. The flow of a single-phase fluid in a porous medium is governed by a parabolic equation obtained by combining the Darcys and the continuity equations. In order to account for the spatial variations of porosity and permeability, and for permeability anisotropy, a two-dimensional model is put forward. A mixed initial–boundary value problem is numerically solved by a finite-difference family of numerical schemes, which depends on a parameter θ. The stability—conditional or unconditional, depending on θ—and the convergence of the schemes have been proved. The linear system originated at each time step is solved by the iterative ADI and block-SOR methods, and by a Taylor series of matrix functions (TSMF). These methods are compared and their relative efficiencies are carefully assessed. TSMF is the fastest technique given that adequate values of θ and time step Δt are used—but Δt must remain small. A combination of TSMF and block-SOR with variable Δt seems to be the best policy. Our numerical simulator is tested by reproducing the existing analytical solutions for limiting cases, and then applied in well test analysis. The contributions of this work are: (1) we introduce the TSMF technique to reservoir simulation and (2) vertical permeability and permeability spatial variations are included in a well test simulator for further developments. Copyright

Modeling of CO2 storage in aquifers

Storage of CO2 in geological formations is a means of mitigating the greenhouse effect. Saline aquifers are a good alternative as storage sites due to their large volume and their common occurrence in nature. The first commercial CO2 injection project is that of the Sleipner field in the Utsira Sand aquifer (North Sea). Nevertheless, very little was known about the effectiveness of CO2 sequestration over very long periods of time. In this way, numerical modeling of CO2 injection and seismic monitoring is an important tool to understand the behavior of CO2 after injection and to make long term predictions in order to prevent CO2 leaks from the storage into the atmosphere. The description of CO2 injection into subsurface formations requires an accurate fluid-flow model. To simulate the simultaneous flow of brine and CO2 we apply the Black-Oil formulation for two phase flow in porous media, which uses the PVT data as a simplified thermodynamic model. Seismic monitoring is modeled using Biots equations of motion describing wave propagation in fluid-saturated poroviscoelastic solids. Numerical examples of CO2 injection and time-lapse seismics using data of the Utsira formation show the capability of this methodology to monitor the migration and dispersal of CO2 after injection.