Summary A physically and mathematically rigorous transient-state equilibrium diffusion model is applied for simultaneous determination of the gas-diffusion and interface-mass-transfer coefficients from pressure de-cline by dissolution of gas in quiescent liquids involving petroleum reservoirs. The short- and long-time analytical solutions of this model are reformulated to enable direct determination of the best-estimate values of these parameters by regression of experimental data. Typical experimental data are then analyzed by means of the present improved methods, and the values obtained are compared with the re-ported values. The present methodology is proven practical and yields unique and accurate parameter values. Introduction Gas-diffusivity and interface-mass-transfer coefficients are important parameters determining the rate of dissolution of the injection gases in oil during secondary recovery, and the rate of dissolution and separation of light gases in reservoir oil and brine, water tables associated with depleted-reservoir gas storage, drilling mud, and completion fluids (Hill and Lacey 1934; O'Bryan et al. 1988; O'Bryan and Bourgoyne 1990; Bodwadkar and Chenevert 1997; Bradley et al. 2002; Liu and Civan 2005). In order to develop proper gas-injection strategies, accurate values of these parameters are required for reservoir simulation and prediction of oil recovery by miscible flooding and the optimization of miscibility for best recovery. Laboratory measurement of gas diffusivity in quiescent liquids is usually accomplished through the measurement of the pressure of gas in contact with certain liquids, such as oil, brine, drilling mud, and completion fluids in a closed PVT cell (see Fig. 1) during gas dissolution in the liquid phase. The accuracies of the available models, including those by Riazi (1996), Sachs (1997, 1998), and Zhang et al. (2000), are limited by the inherent simplifying assumptions involved in the analytic treatment and the subsequent interpretation of such experimental data. As judged by the reported studies, there appears to be no consensus among the available analytical approaches used for diffusivity measurement. In addition, the previous studies focused mostly on the determination of gas diffusivity and did not account for interface-mass-transfer effects. The methodology offered by Civan and Rasmussen (2001, 2002, 2003), and further elaborated in the present paper, allows for both interface mass-transfer effects and for bulk diffusivity. It is a novel and practical approach that determines parameters describing both effects from a given set of pressure-decline data. The best estimate of the coefficient of diffusion of gas species (solute) in a given liquid medium (solvent) is usually inferred indirectly by matching the prediction of a suitable mathematical model involving the species transfer by diffusion to experimental data under prescribed conditions. For this purpose, Sachs (1998) resorts to the numerical solution of the nonlinear model equations incorporating the dependency of the diffusion coefficient on concentration without clearly describing the boundary conditions used in the solution.

in Wiley InterScience (www.interscience.wiley.com).A rapid and effective data analysis and interpretation approach is developed and validated for simultaneous determination of the film-mass-transfer and diffusion coefficients from time-limited experimental data obtained by dissolving gas in liquids by the pressure-decay method under isothermal conditions. Whereas previous approaches require experimental data until equilibrium and only determine the diffusion coefficient, accurate and rapid estimation of both parameters are achieved using a shorter set of time-limited data, thereby reducing the errors owing to swelling by significant gas dissolution at later times. The equilibrium conditions can be predicted theoretically stemming from an analysis of the time-limited data. This provides the estimates of the equilibrium pressure and gas solubility. This methodology not only yields accurate parameter values, but also alleviates the sufficiently large-time collection of pressuredecay data needed to essentially achieve equilibrium.

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Summary A generalized methodology is presented that obtains analytical solutions for imbibition waterfloods in naturally fractured oil reservoirs undergoing multistep matrix-to-fracture transfer processes. The phenomenological representation of the oil transfer from matrix to fracture is based on a three-exponential matrix-to-fracture transfer function, the necessity for which is seen by examination of experimental data. The resulting integro differential equation is converted to a fourth-order partial differential equation, linearized by invoking the unit endpoint mobility-ratio approximation, and then solved analytically by asymptotic means. It is shown that the asymptotic-approximation approach significantly reduces the complexity of the solution process and yields adequate solutions for long-time evaluation of waterfloods in naturally fractured reservoirs. The solution is not only computationally advantageous, but it also provides a physically meaningful interpretation of the propagation speed and diffusive spreading of the progressing wave front, which could not be readily obtained from the usual type of solution methods, such as the direct application and inversion of the Laplace transformation. Introduction The permeability of the fracture system in most naturally fractured reservoirs is much greater than that of the porous matrix, and therefore the fractures form the preferential flow paths while the matrix acts as the source of oil for fractures.1 Therefore, modeling oil recovery by waterflooding of naturally fractured reservoirs by the fracture-porosity and matrix-source approach provides a computationally convenient method.1–4 For example, the application of Civan's4 method of weighted sums, a generalization of the quadrature and cubature methods, enables the numerical solution of such models more accurately and rapidly than the finite-difference method while using 10-fold greater gridpoint spacing and a 3,000-fold larger time increment without the inherent numerical dispersion problems of the finite-difference method. The accuracy of the fracture-porosity and matrix-source approach, however, depends on the implementation of a properly defined matrix-to-fracture transfer function in the Buckley-Leverett formulation. For this purpose, Aronofsky et al.2 used a one-parameter empirical function obtained by correlation of the cumulative matrix-to-fracture oil transfer given byEquation 1 where ? I=an empirical constant and V8 I=the volume of the movable oil initially available in the porous matrix, measured per unit bulk volume, although they recommended adding more exponential terms for better correlation of the experimental data. Hence, Kazemi et al.5 proposed an empirical function composed of an infinite series of exponential terms, given byEquation 2 but they did not provide theoretical justification or apply it to their work. Civan1,6 and then Gupta and Civan7 perceived naturally fractured porous media as having primary-fracture and secondary-matrix porosity. The porous structure of the matrix consists of interconnected and dead-end pores (see Fig. 1). Because the fractures have relatively larger permeability than the matrix, the flow is considered to occur essentially through the fracture network, and the matrix feeds oil into the fractures owing to the imbibition of water and thus acts as a source. The imbibition of water into the matrix causes oil to discharge into the fractures. The oil existing in the dead-end pores passes into the interconnected pores and then to the matrix-fracture interface to accumulate over the fracture face, where it is entrained and removed by the fluid system flowing through the fracture. Civan1,6 and then Gupta and Civan7 derived two- and then three-exponential matrix-to-fracture transfer functions, respectively, based on the principle that dynamic processes occur at rates proportional to the governing driving forces. The proportionality factors are called the rate constants. Thus, for the present case, the oil transfers at various points of naturally fractured formations were assumed to occur at rates proportional to the oil available at those sites. Thus, Gupta and Civan7 theoretically derived and verified by a variety of reported experimental data that a maximum of three exponential terms is sufficient for accurate phenomenological representation of the oil transfer from matrix to fracture. As indicated by Table 1, however, the contribution of the dead-end pores varies for different types of porous media. Analytic solutions of the 1D Buckley-Leverett flow problem involving a one-parameter matrix-to-fracture transfer function have been presented by deSwaan,3 Davis and Hill,8 Kazemi et al.,5 and Luan and Kleppe.9 These analytical solutions have been possible after linearizing the governing integro differential equation by invoking the unit endpoint mobility-ratio assumption, which allows for approximating the fractional flowing water, fw, of the fracture medium by the normalized water saturation, Sw, n, according toEquation 3 where Swc and Sor=the fracture-medium connate water and residual oil saturations, and Sw=the fracture-medium water saturation. The analytical solutions have resulted in complicated mathematical forms involving modified Bessel functions and quadratures and thus require tedious and frequently inaccurate procedures to generate numerical values. Rasmussen and Civan10 significantly reduced the complexity of numerical calculations by deriving an analytical solution based on an asymptotic approximation approach. Their solution is valid for long-time evaluation of water-floods in naturally fractured reservoirs. This solution is not only computationally advantageous because of its simplified mathematical form, and of interest for field applications, but it also provides a physically meaningful interpretation of the propagation speed and diffusive spreading of the progressing wave front, which could not be readily obtained from the previous analytical solutions, such as by the direct inversion of the Laplace transformation. Civan1,4 obtained computationally efficient numerical-quadrature solutions for the waterflooding problem with and without invoking the unit endpoint mobility-ratio assumption, and solutions by the time-space method of weighted sums.