Phase behavior is important in the calculation of hydrocarbons in place and in the flow of phases through the rocks. Pore sizes can be on the order of nanometers for shale and tight-rock formations. Such small pores can affect the phase behavior of in-situ oil and gas because of increased capillary pressure. Not accounting for increased capillary pressure in small pores can lead to inaccurate estimates of ultimate recovery, and of saturation pressures. In this paper, capillary pressure is coupled with phase equilibrium equations, and the resulting system of nonlinear fugacity equations is solved to present a comprehensive examination of the effect of small pores on saturation pressures and fluid densities. Binary mixtures of methane with heavier hydrocarbons and a real reservoir fluid from the Bakken shale are considered.The results show that accounting for the impact of small pore throats on pressure/volume/temperature (PVT) properties explains the inconsistent gas/oil-ratio (GOR) behavior, high flowing bottomhole pressures, and low gas-flow rate observed in the tight Bakken formation. The small pores decrease bubble-point pressures and either decrease or increase dew-point pressures, depending on which part of the two-phase envelope is examined. Large capillary pressure also decreases the oil density in situ, which affects the oil formation volume factor and ultimate reserves calculations. A good history match for wells in the middle Bakken formation is obtained only after considering a suppressed bubblepoint pressure. The results show that the change in saturation pressures, fluid densities, and viscosities is highly dependent on the values of interfacial tension (IFT) (capillary pressure) used in the calculations.

Recently, we have shown that reservoir descriptions conditioned to multiwell pressure data and univariate and bivariate statistics for permeability and porosity can be obtained by techniques developed from inverse problem theory. The techniques yield estimates of well skin factors and porosity and permeability fields which honor both the spatial statistics and the pressure data. Imbedded in the methodology is the application of the Gauss-Newton method to construct the maximum a posteriori estimate of the reservoir parameters. If one wishes to determine permeability and porosity values at thousands of grid-blocks for use in a reservoir simulator, then inversion of the Hessian matrix-at each iteration of the Gauss-Newton procedure becomes computationally expensive. In this work, we present two methods to reparameterize the reservoir model to improve the computational efficiency. The first method uses spectral (eigenvalue/eigenvector) decomposition of the prior model. The second method uses a subspace method to reduce the size of the matrix problem that must be solved at each iteration of the Gauss-Newton method. It is shown that proper implementation of the reparameterization techniques significantly decreases the computational time required to generate realizations of the reservoir model, i.e., the porosity and permeability fields and well skin factors, conditioned to prior information on porosity and permeability and multiwell pressure data. Introduction Proper integration of static data (core, log, seismic, and geologic information) with dynamic data (production and well tests) is critical for reservoir characterization. It is known that ignoring prior information obtained from static data when history matching production data yields nonunique solutions, i.e., widely different estimates of the set of reservoir parameters may all yield an acceptable match of the production history. As early as 1976, Gavalas et al. recognized that incorporating prior data when history matching production data would reduce the variation in the estimates of gridblock values of porosity and permeability. Inverse problem theory provides a methodology to incorporate prior information when history matching production data. The standard application of inverse problem theory depends on the assumption that prior information on the model (set of reservoir parameters to be estimated) satisfies a multinormal distribution and that measurement errors in production data can be considered as Gaussian random variables with zero mean and known variance. Under these assumptions, the most probable model (the maximum a posteriori estimate) conditioned to both prior information and production data can be obtained by minimizing an objective function derived directly from the a posteriori probability density function. Since the a posteriori probability density function is derived from Bayes's theorem, this approach is often referred to as Bayesian estimation. It is convenient to minimize the objective function by a gradient method to obtain an approximation to the most probable model which is referred to as the maximum a posteriori estimate. Gavalas et al. used Gaussian type expressions for the covariance functions of porosity and permeability, the cross covariance between them, and the prior estimates of the means of porosity and permeability to incorporate prior information in the objective function when history matching multiwell pressure data obtained in a synthetic one-dimensional reservoir under single-phase flow conditions. They showed that incorporating the prior information reduced the errors in the estimates of permeability and porosity and also improved the convergence properties of the minimization algorithms considered.

Phase behavior is important in the calculation of hydrocarbons-in-place and in the flow of phases through the rocks. Pore sizes can be on the order of nanometers for shale and tight rock formations. Such small pores can affect the phase behavior of in situ oil and gas owing to increased capillary pressure. Not accounting for increased capillary pressure in small pores can lead to inaccurate estimates of ultimate recovery, and saturation pressures. In this paper, capillary pressure is coupled with phase equilibrium equations and the resulting system of nonlinear fugacity equations is solved to present a comprehensive examination of the effect of small pores on saturation pressures and fluid densities. Binary mixtures of methane with heavier hydrocarbons, and a real reservoir fluid from the Bakken shale are considered. The results show that understanding the impact of small pore throats on PVT properties explains the inconsistent GOR behavior, high flowing bottomhole pressures, and low gas flow rate observed in the tight Bakken formation. The small pores decrease bubble-point pressures and either decrease or increase dew-point pressures depending on which part of the two-phase envelope is examined. For the pore radius of 10 nanometers in the Bakken shale, the calculations show that there is more than a 900 psi reduction in the bubble-point pressure as the reservoir is depleted. Further, reduction of oil density due to small pores can impact the formation oil factor and ultimate reserve calculations. The results also show that the change in saturation pressures and fluid densities are very dependent on the values of the interfacial tension used in the calculations.

Summary This work considers the analysis of pressure data, both drawdown and buildup, obtained at a well producing a reservoir in which the absolute permeability varies with position. A new inverse-solution algorithm is presented that can be applied to estimate the reservoir permeability distribution as a function of distance from the well. Introduction The emergence of reservoir characterization has stimulated efforts to obtain improved information on reservoir heterogeneities. This work considers single-phase flow to a well in a reservoir where permeability varies with distance from the well. We consider methods for estimating the permeability distribution from well-test pressure data. The methods considered were obtained by modifying and extending elegant seminal works of Oliver and Yeh and Agarwal. Oliver used a perturbation theory technique to obtain the wellbore pressure drawdown solution at a single well in an infinite-acting reservoir where absolute permeability varies with position. His solution assumes 2D flow in an (r,) coordinate system and that permeability is a function of r and i.e., k = k (r,). As presented, his solution assumes that permeability varies slightly about a reference, base, or "average" value, kref. In Ref. 3, Oliver used the Backus-Gilbert method to approximate the permeability distribution under the assumption that a reference permeability value can be determined from a semilog plot of pressure vs. time. He applied the method to a three-zone, composite, infinite-acting reservoir where the permeability in the inner and outer zones is k = 2,000 md and in the middle zone is k = 1,500 md. In Refs. 1 and 3, Oliver considers only the analysis of pressure-drawdown data. In this work, we remove Oliver's restrictions and consider the analysis of both drawdown and buildup data obtained at a well with an arbitrary variation in absolute permeability in the radial direction. Most importantly, we derive an inverse-solution algorithm to estimate this permeability distribution directly from well-test pressure data. Unlike Oliver's application of the Backus-Gilbert procedure, our inverse-solution algorithm does not assume that we can compute a reference or base permeability value from a semilog plot of pressure vs. time. In fact, we show that the base permeability value controls only the shifting of the time scale used to evaluate the kernel weighting function in Oliver's solution. Our inverse-solution algorithm, which is recursive but stable, can be applied for large variations in permeabilities and in cases where pressure data exhibit no semilog straight lines. Rosa and Horne examined the same problem as Oliver. While they noted that the pressure response for a multirate test was more sensitive to reservoir heterogeneities, like Oliver, they concluded that the inverse problem (i.e., the determination of permeability distributions) does not have a unique solution. Ref. 5 indicated that, for a multicomposite reservoir, the permeability distribution could be determined (by nonlinear regression analysis) only if the inner and outer radii of each zone were known. Kamal et al. also used a multicomposite model consisting of a few zones to analyze data from a damaged well. They matched pressure-buildup data with the model using nonlinear regression analysis to determine estimates of permeability in each zone.

This paper presents an innovative integrated methodology and working procedure for characterizing and simulating the strong non-linearity and non-stationariness caused by changes in confined pressure-volume temperature (PVT) properties over time related to pore-throat size, the pressure-dependent permeability, and the intervened multiple porous media created by multi-stage fracture stimulation. The complicated physics behind the observed phenomena are explored. More specifically, this paper demonstrates and discusses the following: 1) a new rate-transient analysis (RTA) procedure to infer the stimulated reservoir volume (SRV) and fracture parameters; 2) the impact of the non-stationary feature, compaction effect, and pore-throat related PVT properties on the flow regime and well performance; 3) how to incorporate the nonstationary and non-linear features into the reservoir model; 4) the integrated procedure for history matching, performance forecast, and recovery assessment; 5) several field examples in the Bakken to illustrate the procedure.The proposed procedure has been successfully applied for the following: 1) constructing the non-stationary, nonequilibrium, and highly non-linear simulation models; 2) facilitating the history matching by addressing permeability reduction and PVT property variations caused by compaction and capillary pressure; 3) and ensuring more reliable performance forecasts and recovery assessments.The study shows that the reduction of the bubblepoint pressure could be several hundred psi in the typical Bakken rock; moreover, such reduction continues following the depletion via the compaction effect. The compaction effect could impair the matrix permeability by up to one order of magnitude.The study reveals the following: 1) the confined PVT properties could widen the favored operation window, whereas the compaction effect could significantly impair the ultimate recovery of the wells; 2) the RTA-inferred SRV-related parameters are the key input for capturing the non-stationary features; 3) the impact on recovery could be over 50% without addressing the aforementioned non-stationary and non-linear issues.This paper explores several unique phenomena in unconventional oil reservoirs which have not previously been published. The proposed analysis and assessment procedure greatly enhances the understanding of the unconventional assets and we feel will improve the accuracy of long-term rate and recovery forecasts.