Integration of Different Types of Data for Characterization of Reservoir Heterogeneity

Transcription

1 5th Conference & Exposition on Petroleum Geophysics, Hyderabad-24, India PP Integration of ifferent Types of ata for Characterization of Reservoir Heterogeneity S. K. Mishra Mumbai High Asset, 38, Priyadarshini Bldg., ONGC, Eastern Express Highay, Mumbai ABSTRACT : This paper presents a technique for pressure transient and its derivative analysis for single porosity reservoirs in a class of geological setting that are not amenable to conventional techniques. The application of a fractal model to analyze unsteady state pressure transient data of Netonian fluids in a heterogeneous reservoir for heterogeneity characterization is considered. The fractal geometry is a very poerful method to describe most complex phenomenon, especially to scale the nonuniformity and non-sequences of heterogeneous porous media. The pressure transient response is analyzed for flo in a connected fracture netork and fracture ith matrix participation. With the help of these approaches, single-phase fluid flo in the fractal object is described by appropriate modification in diffusivity equation. The automatic type curve matching technique for both pressure and its derivative are used in evaluating reservoir parameters from pressure buildup/dradon and falloff test data of heterogeneous reservoir. The goal of the pressure transient data analysis is to establish a reasonable estimate of the reservoir parameters of interest for better understanding of the reservoir behavior. Recent experiences have brought forard more reasonable expectations of geologists, geophysicists and reservoir engineers by sharing their knoledge in integrating various sources of information in finding the reservoir heterogeneity. Thus permeability data estimated from transient ell test data are utilized to generate 3- model by combining it ith porosity derived from seismic or ell log measured porosity ith the help of geostatistical techniques. The timely acquisition of data and their continued evaluation at unknon locations ith geo-statistical techniques are foundational to sound the reservoir management to develop the field and implement the various applicable improved oil or enhanced oil recovery (IOR/EOR) schemes. INTROUCTION The development of pressure transient analysis in the oil industry spreads over more than 6 years for oil and gas ell test analysis. The availability of better measuring devices that provide accurate and reliable ell test data of the reservoir have alloed us to develop sophisticated interpretation techniques. This helps in understanding the dynamic behavior of the reservoir [1]. Computer aided ell test analysis techniques are old; so many papers have been published on this subject in the last several years. These papers are describing the soare packages for ell test analysis ith either conventional methods or automated type curve matching techniques [2, 3, 4, & 5]. The manual processes are not capable of analyzing ith good degree of confidence and incomplete data are obtained from badly designed tests, hich unfortunately are in the majority. Therefore the valuable information of ell and reservoir cannot be extracted from acquired ell test data. The use of computer in ell test analysis to date is generally reproducing the manual interpretation processes or to performing the estimation of reservoir parameters based on regression analysis of observed data ith a pre-selected reservoir model. Watson et al. [6] proposed a method for selecting the most appropriate model from a given pool of candidates and an artificial intelligence, and rule based approaches to solve the problem [7, 8]. It is ell knon that a corner stone of modern reservoir ell test interpretation involves the use of the pressure derivative or pressure integral first and second derivative techniques developed a fe years ago, is responsible for identification of reservoir model [9, 1]. The pressure transient data analysis has been given by many authors for heterogeneity characterization of fractal reservoir [11, 12]. The fractal geometry is an appropriate and poerful tool to describe complex phenomenon of porous media. If the fluid flo through porous media is studied by using fractal, the discernible and cognitive ability of porous media and its geometry size ill be raised. Chang and Yortsos [12] have given the formulation of mathematical equations for the flo of fluids in a fractal reservoir [13]. Beier [14] and Aprilian [15] have given the flo based model of fluid flo through fractal 847

2 Integration of ifferent Types of ata reservoir and explained the complex reservoir ell test result hich cannot be matched and explained by using conventional model in oil field. Chakraborty [16] reported the flo characteristics of non-netonian poer-la fluid flo in fractal reservoir. Heett s [17] basic ork uses a Euclidean reservoir ith permeability/porosity spatial distribution that obeys fractal statistics. When a porous medium is fractal in nature, its geometry and transport properties differ in non-trival ays from those of the Euclidean flo media. The ideal response of perfect objects is described as follos: Many of their basic properties, defined as averages over a region of scale r, are scale dependent and are proportional to poer of r (r ). The mass density of fractal reservoir netork of fractures around an arbitrary point decreases in a poer-la fashion ith respect to the distance r. The exponent of this is (d f -d s ), here d f is mass fractal dimension of fractal netork of fractures and d s is the Euclidean dimension of the medium in hich the fractal object is enclosed. Several types of heterogeneity may be found on different scales and they influence various rock properties. The reservoir rock properties, such as porosity and permeability are affected by geologic processes and must possess certain spatial continuity in order for the hydrocarbon accumulation to be productive reservoirs. La [18] as among the first to statistically analyze reservoir permeability and concluded that permeability has a lognormal probability density function. The early studies in hich probability theory as applied to the existence of randomness in the permeability distribution of reservoir devoted a great deal of attention to to problems. The first problem concerned the form of the probability density function found in nature for permeability. The second problem as expressed in the aim to find a technique for computing a single proper value of the average or characteristic permeability for characterization of a particular field in its entirety. Both these things ere considered by Warren & Price [19]. These investigators constructed a spatially heterogeneous permeability field by randomly choosing a ne value of permeability for each mesh point on a regular grid. The first, of the above question as approached in collecting of a large number of measurements of permeability from ildcat ells. From these, an approximated lognormal distribution as found. The mean and variance of this distribution ere then considered to characterize the natural reservoir. GEO-STATISTICAL APPROACH This is a more realistic ay to take into consideration the variability of natural processes, the details of hich are too numerous, inaccessible, and complex for direct analysis. For this reason, and because the mathematical procedures are becoming more idely knon, geo-statistical techniques are becoming popular in the petroleum industry. Geo-statistics as created at the end of the fiies by Matheron [2]. It is a part of probability theory applied to Earth Sciences. The main originality consists in the basic principle that take into account to facts: First, the data dealt ithin Earth Science are located in space and the spatial correlation beteen these data is of prime importance and is constitutive part of the theory. Secondly, data are measured on different volumes but e are usually interested in quantities measured on a support, hich is larger than the data themselves [21, 22] Fortunately, recent experience has brought forard more reasonable expectation of geologists, geophysicists and reservoir engineers by sharing their knoledge in integrating various sources of information in finding heterogeneity ith the use of geo-statistical techniques [23, 24, & 25]. These investigators have employed Kriging and conditional simulation techniques to represent the spatial distribution of permeability and porosity for simulation studies. The present paper describes the integration of pressure transient test data and ell log derived porosity data to estimate the permeability and porosity distribution ith the help of geo-statistical techniques at unknon locations of the reservoir. MATHEMATICAL FORMULATION OF THE PHYSICAL PROBLEM The mathematical formulation of the problem consists of to parts. The first is the pressure transient equation formulation for fluid flo through fractal reservoir. The second is the geo-statistical equation formulation to estimate the reservoir properties at unknon locations ith the data knon at the various drilled locations. The to methods are combined together to generate 3- heterogeneity of reservoir at reservoir scale. FORMULATION OF PRESSURE TRANSIENT EQUATION The mathematical formulation of the pressure transient equation for fluid flo through fractal reservoir has 848

3 Integration of ifferent Types of ata C been considered. The assumptions made in the present study are same as given in [26]. The hydraulic diffusivity, η, for fractal permeable netork scales as η( r ) = k( r) / µ cφ( r) r (1) ( ) θ here, θ is a parameter related to the topology of the netork (θ ); r is the distance from the ellbore; also µ is the viscosity of the fluid, c is total compressibility, k and φ are permeability and porosity of the medium respectively. The hydraulic diffusivity contains both permeability and porosity. The porosity and permeability of an area netork scales as (2 d ) ( r / ) d (2 / 1) ( / ) f d r f φ ( r) = φ, and, r s k ( r) = k (2) r here, d f is the fractal dimension and d s is spectral dimension. Alexander & Orbach [27] defined the fraction or spectral dimension as d s =2.d f / (2+θ). Later, e ill see that asymptotic slope of the pressure transient curve is related to the fractal dimension. For this reason, d s is used to describe the fractal netork instead of θ. The value of d s lies beteen < d s 2. A diffusion equation for pressure describes the flo of a single-phase, slightly compressible fluid through reservoir. 2 = ( d For f / d an s ) C areally s /( φ chr heterogeneous ) reservoir, a general form of the equation applies and includes spatial variations of permeability and porosity. For radially symmetric flo in a plane, e rite (3) An equivalent ellbore radius, r, and takes into account positive or negative skin. Substitution of equation (2) into equation (3), e get Equation (4) is a generalization of the usual diffusivity equation, since it applies to any permeable netork of fractal dimension d f embedded into to-dimensional Euclidean space. By setting d f = d s = 2, one obtains the usual diffusivity equation. form as (4) The equation (4) can be ritten in dimensionless (5) Where, P = k h( d f / d s )( Pi P) /( qb µ ),, r = r /, and r The boundary condition for a circular ellbore uses arcy s equation to calculate the production rate of reservoir fluids entering the ellbore. This don-hole production rate must equal the surface production rate plus the rate of ellbore storage. Aer riting this mass balance in dimensionless form, e have (6) For an infinite reservoir, additional boundary conditions are lim P ( r, t ) = (7) r The initial condition is as P (,) = (8) r SOLUTION OF THE PRESSURE TRANSIENT PROBLEM The above equations are solved ith the help of a Laplace transform. Applying the transform to equations (5) to (8), e obtains (1) lim P ( r, z) = (11) r P (,) = (12) r By a suitable change of variables, it is possible to transform equation (9) into a Bessel equation. To get this, introduce the folloing variables P = r γ F(ρ) (13) Where, ρ α Pr β (9) = (14) Substitution of equations (13) and (14) into equation (9), e get (15) if, α = P z, β = d f / d s, and γ = (2d f /d s -d f )/2. Equation (15) is Bessel s modified differential equation of order, 1-d s /2. The general solution is given as F ρ ) = c I ( ρ ) + c K ( ) (16) ( 1 ν 2 ν ρ 849

4 Integration of ifferent Types of ata Where I ν (ρ) & K ν (ρ) are modified Bessel functions of first and second kind, respectively, and of order ν. In this case ν = 1-d s /2. The constants c 1 and c 2 are determined from the boundary conditions (1) & (11). We apply the change of variables in equations (13) & (14) to these boundary conditions. The pressure response at ellbore ithout ellbore storage and skin effect is given as K ν ( z ) P = 3 / 2 (17) z K ν 1 ( z ) If the Laplace transform solution P for the constant rate and ithout ellbore storage and skin effect is available then the dimensionless pressure response at ellbore for the constant ellbore storage and skin effect can be obtained as [28] [ z P + S ] P = (18) z 1 + C z z P + S [ ( )] The dimensionless pressure at ellbore depends on d s through ν parameter but is independent of d f. The inverse Laplace transforms of equations (17) & (18) are calculated numerically to find P, by using ell knon Stehfest algorithm [29]. REGRESSION ANALYSIS TECHNIQUE There are a large number of methods available in literature for performing the estimation of parameters in model fitting. The most popular is that of least squares method. This method calls for the minimization of the sum of the squares of the residuals defined as N 2 S(ϕ ) = ( Y i Y i ) (19) i= 1 Where, Y i = experimental values of the dependent variable for ith observation, Y i = to be predicted value of dependent variable for i th observation, N=number of observation points, & ϕ = ( ϕ1, ϕ 2, ϕ 3,..., ϕ n ) is a vector of the order n p. The best values of the model parameters are obtained hen the objective function is minimized. In the present problem, Y i is equivalent to P = (P f -P ), hich is function of ellbore loading, mechanical skin, reservoir permeability and inner and outer boundary parameters of reservoir. The minimization of S (ϕ) is done ith respect to the parameters (C, S, k, d f, d s, & r e ). The minimization of objective function, S(ϕ ) results into solving a non-linear optimization problem. The iterative technique fits a data set that depends on ho appropriate a function is selected for reservoir of different characteristics. If the initial estimates of parameters are close to the true value of the parameters, the convergence is very fast and the number of iterations ill be less. The parameters estimated from observed pressure transient test data of the ells of some field are given in Fig The permeability data obtained from the ell test data interpretation is integrated ith the measured ell log porosity to generate the 3- permeability distribution (modified Kozeny-Carman correlation or log (k)-ö crossplots) in log scale for characterization of the reservoir heterogeneity ith the use of ell knon geo-statistical techniques. Pressure P & P' in psi --> Pressure P & P' in psi --> C =15.5 S=1.146 k=11.15 md X =81.7 P obs P'obs P cal P'cal Time t in hrs --> Figure 1 : Interpreted match data of ell no. A C = S=14.19 k=46.99 md X =61.5 Y =61.5 P obs P'obs P cal P'cal Time t in hrs --> Figure 2 : Interpreted match data of ell no. B HETEROGENEITY MOELING WITH GEO-STATISTICAL METHOS The understanding of heterogeneity implies a better knoledge of connectivity beteen permeable and nonpermeable zones, and a better forecast of seep efficiency and oil saturation in the partially sept zones. Therefore, in many cases, improper modeling of geological heterogeneity 85

5 Integration of ifferent Types of ata Figure 3 : Interpreted match data of ello no. C Figure 5b : Interpreted match data of ell no. E Pressure P P & & P' P' in P' in psi in psi --> psi --> --> C =347.4 =387.7 P P obs S=-.83 P'obs k=3.7 k=3.75 P P cal md md X 1 2 =9.2 =99.5 P'cal X e =192.5 e = Y = C = S=-2.99 C =47.7 C k=325.9 md S= = S=13.17 k=3.75 md P obs X =178.5 C 1 = ω=.35 P k=138.5 obs P'obs md S=.95 λ=7.7e-6 P'obs ω=.2 P P obs cal k=69.3 md X P'obs =85.1 P λ=1.3e-6 cal P'cal Y X =545 =85.1 P'cal X =315. P cal 1 1 Y =315. P'cal Time -1 Time t t in in 1 hrs --> Time Time Time t t in in hrs t in hrs --> hrs --> --> Figure 4 : Interpreted match data of ell no. Figure 5c : Interpreted match data of ell no. E Figure 5a : Interpreted match data of ell no. E Figure 6a : Interpreted match data of ell no. F 851

6 Integration of ifferent Types of ata Figure 6b : Interpreted match data of ell no. F Figure 9 : Interpreted match data of ell no. I can have a direct impact on capital expenditure and threaten, in unfavorable oil price scenarios, the profitability of an oil field. Variogram analysis is the main component of geostatistical techniques. Kriging and conditional simulation are used on generating spatial distribution of data at unknon locations. KRIGING METHOS Knoing a number of data points and the variogram model, Kriging is the best linear estimation taking the spatial correlation into account. We estimate Z (x) at point x by (2) Figure 7 : Interpreted match data of ell no. G Z (x i ) is the knon data points and λ i are the eighting factor. These are determined by the location of the data points in relation to each other, the location of the point to be estimated in relation to the knon points, and by variogram model, simply by solving the Kriging equations. Kriging is a tool suited to a large number of problems, e.g. the estimation of the volume of the hydrocarbons in place, the estimation of porosities in large cells, plugs measurements and log measurement to obtain point estimation of permeabilities using porosities. CONITIONAL SIMULATION TECHNIQUE Figure 8: Interpreted match data of ell no. H The objective of a simulation is not to provide the best estimation, but to produce various possible versions of a partly knon reality. On the other hand, hen using simulations to produce the variability of reality, e overlook the notion of the precision of the estimation. If e allo the 852

7 Integration of ifferent Types of ata fluctuations to occur only beteen data points, and e build every possible version of the reality so as to match the data points, the simulation is said to be conditional. This provides possible versions of the partly knon reality ith the properties; (1) They have the same spatial variability as reality, i.e., if e compare the experimental variogram model of simulation, e must be very close to the experimental variogram model of the data. (2) The simulation and data have the same histogram (pdf). The estimated permeability data from observed pressure transient data of the several oil and gas ells of the reservoir are used here for further analysis. The permeability data obtained from dynamic pressure transient test data of the ells is integrated ith measured ell log porosity data of the same ells to serve as a support to study and illustrate Kriging and conditional simulation techniques on a real anisotropic reservoir for its characterization. The major draback of geo-statistical simulation is that there is no objective ay to choose one realization a priori or to rank them ith respect to some attributes of interests. It is important to note that the differences beteen these stochastic images provide a directly usable visualization of the uncertainty about reservoir heterogeneity. Where all the different realizations agree, there is little or no uncertainty, here they differ most there is maximum uncertainty. If the uncertain streaks happen to be in the areas that are highly consequential to reservoir performance, the need for additional data has been established together ith the locations at hich these data are needed. CONCLUSION AN RECOMMENATION We shoed procedures; (1) to estimate the distribution of permeability and porosity in the reservoir by using dynamic data; (2) to assess the uncertainty associated ith the calculated permeability and porosity and; (3) to determine the value of each data type from the point of vie of the inverse problem. The procedures allo us to integrate information from several sources to compute distributions of permeability and porosity. The information included are pressure transient test data (pressure buildup/dradon or pressure falloff), production history (rate and ater cut), ell log data (neutron porosity, density, sonic, resistivity and gamma ray logs, etc.), permeability-porosity correlations, interpreted 3- seismic map (depth structure map including faults), variogram models and some geological information. The more interesting conclusions obtained from the present ork are about the roles of pressure transient testing, ell logs and 3- seismic data and their integration. Pressure transient testing has been considered here an important tool to solve complex reservoir models in the neighborhood of the ells and to improve the productivity. These are very important considerations from the engineering and economical point of vie. The reservoir parameters like permeability, faults (sealing, partial sealing and non-sealing), non-permeable boundaries, mechanical skin, and closed and constant pressure boundaries are estimated from the observed pressure transient data of the ells (Fig. 1-9). The uncertainty or non-uniqueness is observed in the ell test data interpretation of the ell nos. E and F (Fig. 5a-5c & 6a and 6b). The same has been resolved by considering vertical heterogeneity model. There is a very good match beteen the observed and predicted pressure data of the ells. The faults and the other types of the boundaries (closed and constant pressure) estimated from the ell test data are of the same pattern as predicted by 3- seismic data of the reservoir. Heterogeneity, the spatial variation in the properties, is a ubiquitous feature in all reservoirs and one of the most important factors governing the fluid flo in porous media. The most pronounced form of heterogeneity involves permeability and porosity. The permeability obtained from the interpretation of the pressure transient test data of the ell is integrated ith the measured ell log porosity (neutron porosity) of that ell to generate the vertical permeability ith the help of modified Kozney-Carman correlation. This calculated permeability and the porosity derived from neutron log and other logs are used to generate the 3- permeability and porosity distribution by the use of ell knon geo-statistical techniques (Kriging and conditional simulation methods) to characterize the reservoir heterogeneity (Fig. 1-15). Proper quantification of the spatial distribution of these properties and associated uncertainties is important for reservoir simulation problem. The histogram of the permeability and porosity data (pdf) and their simulated grid values (pdf) histogram of both permeability and porosity are almost similar in the nature (Fig. 14 and 15). 853

8 Integration of ifferent Types of ata Base map of the reservoir ith ells AT36 Northing irection, meters -----> AT236 AT157 AT367 AT258 AT153 AT168 AT282 AT271 AT268 AT275 AT277 AT283 AT221 AT281 AT134 AT274 AT15 AT375 AT25 AT336 AT138 AT361 AT269 AT x1 6 2.x1 6 2.x1 6 2.x1 6 2.x1 6 2.x1 6 2.x1 6 2.x1 6 Easting irection, meters -----> Figure 1 : Layout of the ells for characterization Z X Y Easting d Grid Vie Northing Figure 12 : Permeability distribution by Kriging Variogram Separation istance (M) Vertical Variogram top13-hazbotm Figure 11 : Variogram of Permeability data values. Figure 13 : Porosity distribution by Conditional Simulation. 854

9 Integration of ifferent Types of ata 2.7 AT36 AT236 PF AT134 AT367 AT157 AT258 AT153 AT15 AT168 AT274 AT268 AT275 AT271 AT282 AT283 AT277 AT221 AT281 AT375 AT25 AT138 AT336 AT361 AT AT18 AT E-5 1E-4 1E Easting Northing Grid PF 3d Grid Vie PF NPHI ata PF Figure 14: Permeability data value PF, Conditional simulation grid value PF & Permeability distribution by Conditional Simulation. Figure15 : Porosity data value PF, Conditional simulation grid value PF & Porosity distribution by Conditional Simulation. PF PF ACKNOWLEGEMENT -35 The author is thankful to Shri J. L. Narasimham, GGM SSM-MH asset for valuable suggestions in carrying 19.8 out the present ork. The author is also thankful to r. B. L. Lohar,.621E GM (Maths.) for encouraging to carry out the present study. REFERENCES 1E-5 1E-4 1E PERM ata PF 1E-4 1E Z Y X Easting d Grid Vie Fig Northing Ramey, H. J. Jr.: J. of Petroleum Technology, 1982, p.147. Earlougher, R. C. Jr.: advances in ell test analysis, Monograph Series, SPE, allas, 1975, 5. Rosa, A. J. & Horne R. N.: SPE paper no , Barua, J. et al.: J. SPE of Formation Evaluation, Gringarten, A. C.: SPE paper no. 1499, Watson, A. T. & Lee, W. J.: SPE paper no , Erdle, J. C. et al.: SPE paper no. 1539, Allian, O. F. & Horne, R. N.: J. of Petroleum Technology, 199. Bourdet,. et al.: J. SPE of Formation Evaluation, ouglas, A. A.: Paper presented at SPE Aberdeen section meeting, Grid PF Aberdeen, Hardy, H. H. & Beier, R.: Fractals in Reservoir Engineering. World Scientific, Ne Jersey, Chang, J. & Yortsos, Y. C.:J. SPE Formation Evaluation 5, 31-38, 199. Beier, R. A.: SPE paper 21553, presented at CIM/SPE International Tech. Meeting, Calgary, 199. Beier, R. A.: J. SPE Formational Evaluation 9, , Aprilian, S. et al.: Proc. Annual Conf. of SPE paper no , Chakraborty, C. et al.: Proc. Annual Conf. of CIM/SPE paper no. 2475, Heett, T. A.: Proc. Annual Tech. Conf. of SPE at Ne Orleans, SPE paper 15386, La, J.: Trans., AIME , Warren, J. E. & Price, H.: Trans., AIME , Matheron, G.: traite de geostatistique Tome, Ed. Technip, Paris, Issaks, E. H. & Srivastva, R. M.: An introduction to Applied Geostatistics. Oxford Press, Journal, A. G. & Huijbregts, G.: Mining Geostatistics. Academic Press, Laherrere, J.: Comptes Rendus de l Acade mie des Science, Publie t. 322, Se rie II, Lake, L. W., and et al.: Reservoir characterization, Vol. 1 & 2. Academic Press, 1986, Sagar, R. K., et al.: Proc. Annual Conf. of SPE paper 26462, Acuna, J. A. et al.: SPE paper no presented at SPE Ann. Tech, Conf., Washington, Alexander, S. & Orbach, R.: Journal de Physique-Letters, 43, , Van Everdingen, A. F. & Hurst, W.: Petroleum Trans., AIME, ecember Stehfest, H.: Algorithm 368. Communication of ACM 13, 1, 47-49,