Summary new analytic solution, based on an approximate trilinear flow model, is developed to study the transient behavior of a well intercepted by a finite-conductivity vertical fracture. The solution accounts for the effects of skin, wellbore storage, and fracture storage. Both constant-pressure and constant-rate cases are considered. The solution is simple and reliable for short-time analysis. Combining this solution with a semilog asymptotic solution provides a reliable tool for analysis and formation evaluation of fractured wells. We also demonstrate that the optimization technique is a convenient means of formation parameter estimation. A set of early-time asymptotic solutions is also presented. These solutions provide qualitative and quantitative relations of the simultaneous influences of wellbore storage, fracture storage, and skin damage on early-time wellbore pressure behavior. Introduction The increased activities in exploiting tight reservoirs by means of hydraulic fracturing techniques have generated considerable interest in the development of pressure-testing procedures for evaluating fracture performance. procedures for evaluating fracture performance. Pressure or production analyses of fractured wells have been investigated with numerical as well as analytic methods. Cinco-Ley surveyed published numerical and analytic reservoir flow models for fractured wells. A numerical approach with a reservoir simulator can rigorously treat nonlinear fluid/rock properties, as well as formation heterogeneity and geometry. From the standpoint of ease of analysis, however, we prefer analytic models, if applicable, over numerical models. For pressure testing of wells, analytic models can be grouped into two types according to their solution methods: semianalytic and asymptotic analytic models. The semianalytic model was first developed by Gringarten et al. for infinite-conductivity fractured wells and was later extended to finite-conductivity fractured wells by Cinco-Ley et al. In these approaches, the governing linear partial differential equations were transformed first to a set of integral equations. Then, these integral equations were discretized in time and space to find the unknown variables of pressure and flow in the fracture. Adapting this algorithm for routine well testing purposes would require considerable computer coding and storage comparable to the requirement of a purely numerical approach. In asymptotic analytic solutions, a square-root-of-time solution and recently developed asymptotic bilinear solutions are used for formation evaluation of fractured wells. The square-root-of-time solution is applicable only to short and high-conductivity fractures. The bilinear model is applied when the influence of the flow from fracture tip is not felt within the fracture. Also, while the slope of pressure vs. the fourth root of time will provide an estimate of fracture conductivity, the fracture length cannot be obtained directly. Therefore, we conclude that no simple, suitable analytic model for formation evaluation of fractured wells, capable of providing both fracture length and conductivity, has been developed. We present a new analytic mathematical model for flow to a fractured well. On the basis of physical and mathematical reasoning, we approximate the flow between the formation and the fracture as having a trilinear behavior. This model, called the trilinear model, considers the effects of skin, wellbore storage, fracture storage, and constant-pressure and constant-rate cases. The solutions are simple and reliable for short-time analysis (the time before semilog straight-line behavior is reached) of a well intercepted by a vertical fracture. Combining our short-time solution with semilog asymptotic solutions provides a reliable tool for pressure testing of fractured provides a reliable tool for pressure testing of fractured wells. We also present a method of formation parameter estimation by means of an optimization technique. This procedure requires an optimization (or error minimization) procedure requires an optimization (or error minimization) subroutine. We demonstrate that the fracture parameters can be determined conveniently with the optimization technique and the trilinear model. Finally, we present early-time asymptotic solutions for both constant-pressure and constant-rate cases to illustrate the simultaneous influences of skin, wellbore storage, and fracture storage at early testing times. SPEFE P. 75
In this paper a number of factors contributing to mass transfer by cross flow in gas displacement are examined. The mechanisms involved are diffusion, dispersive mixing, capillary pumping, interfacial tension variation, and relative permeability modification. The use of a modified Wilke-Chang correlation for bulk diffusion coefficients in multi-component systems is suggested. The performance of this correlation is tested against the limited experimental data available on diffusion coefficients. A general formulation of the terms needed to be included in compositional simulators for cross flow effects is reviewed, and thence some of the key aspects are analyzed using measurements from the IFP-diffusion experiment. The role of the gas phase tortuosity factor is investigated, but spatially varying capillary force is also found to be a dominating mechanism. An example reservoir application with a two-layer heterogeneous character is then studied to determine the relative magnitudes of cross flow, and how these compare between a nearly miscible displacement and a dry gas vaporizing process. Introduction In the study of gas displacement processes we are necessarily much concerned with the effects of a severely adverse viscosity ratio causing viscous fingering. The viscous fingering tendencies are now recognized to be augmented by, or perhaps dominated by, channeling through the higher permeability pathways of a heterogeneous medium. These effects take their most severe form when the mobility ratio is large, as is the case for heavy oils. Fractured systems provide examples of highly heterogeneous porous media where the fractures create pathways for a displacing gas to bypass and leave oil behind in the matrix system. The adverse fingering or channeling tendencies will be mitigated to varying degrees by cross flow effects induced by the following physical mechanisms:mass diffusion,dispersive mixing,capillary forces,gravity forces. All these factors are rate dependent, so that the effectiveness of a gas displacement process needs to be analyzed in a manner which takes quantitative account of these interacting mechanisms. We will discuss some aspects of the first three of the above factors in this paper. Gravity effects are also extremely important, but are not included in the scope of this discussion. In immiscible displacement it is expected that viscous fingering or channeling should be less dominant, because of the effects of relative permeabilities. The mobility ratio of the Buckley-Leverett front will often lie in the range 1.0 MBL2.0, even though / greater than, greater than 1.0. The further stabilizing effect usually associated with capillary forces when the flow rate is low, may not apply to a nonwetting phase invasion, since the capillary forces lead to preferential selection of the larger pores or higher permeability pathways for the displacement. However, an important mechanism contributing to the cross flow, and therefore potentially to the overall sweep factor, is that of capillary pumping. In the latter, substantial reduction of the oil saturation in the flow pathway causes a high saturation gradient with respect to an adjacent stagnant region. This implies that (Pc/ So) (So/ dn) is large (where n is normal to the flow), and oil as the wetting phase is induced to flow into the pathway. Necessarily gas counterflows and invades the stagnant region to reduce the saturation gradient (see Figure 1). The magnitude of this effect depends on go, the interfacial tension, which varies with phase behavior. Thus in considering the merits of multiple contact miscible processes, relative to nearly miscible processes, or to vaporization by dry gas cycling, there are a number of trade-offs in terms of economic performance to be considered. Because of the complexity of the cross flow phenomena, which necessarily interact with phase behavior and density differences causing gravity segregation, the only satisfactory route available for computing gas displacement is through compositional simulators. Care must be exercised in selecting both the grid details and the compositional representation to achieve a solution which is not overcome by numerical truncation errors. We will give illustrative solutions of some of the cross flow effects using a modified version of a commercial compositional simulator. Modeling of the mass diffusion term has uncertainties because of the lack of a good theory and measurements for bulk diffusion coefficients in multi-component systems. P. 827^
Thi s paper presents a mult i component surface tension correlation based on scaling theory. In addition to particular exponents employed. the correlation contains two new features: (1) A corresponding-states equation is derived for a correlation coefficient. commonly referred to as a parachor.As a result. the hydrocarbon pseudocomponent parachors can be calculated through this equation. once their pseudocritical properties are estimated.(2) An approach is proposed to calcu-1 ate the parachors of mi xtures. In contrast to the conventional approach. which calculates the mixture parachor via molar mixing of component parachors, this approach first obtains the pseudocrtical properties of the mixture and then employs the correspondi ng-states equat i on to cal cul ate the mixture parachor.
Summary Pt. McIntyre field operates an enriched-gas-injection scheme that displaces oil in a multicontact miscible (MCM) displacement achieved through a combined condensing/vaporizing (C/V) mechanism. A range of miscible-injectant (MI) compositions, varying from the minimum miscible enrichment (MME) to substantially above the MME, is available to the project. Field-scale numerical-simulation studies for the Pt. McIntyre field show that incremental oil recovery is nearly doubled when using the richest available MI composition. At the MME, dispersion can act to substantially reduce oil recovery by reducing the concentration of enriching components in the near-miscible zone. Increasing enrichment above the MME compensates for this action. Accurate predictions of incremental oil depend on the numerical dispersion in a simulator being able to match the impact of physical dispersion. We show how compositional core data from an MI-swept interval provide confirmation of the impact of dispersion at field scale and demonstrate the appropriateness of the simulation model. The benefit of enrichment appears to be robust to the variation of reservoir description found at Pt. McIntyre and to whether the MI application targets incremental oil through liquid- or vapor-phase recovery. Previous studies into MI enrichment have reported that for a four-component system, the mechanism changes from C/V to purely condensing as the enrichment level approaches first-contact miscibility (FCM). We show that with the addition of a small amount of heavy component, the C/V behavior is retained with increasing enrichment until transition into FCM. This makes it unlikely that a transition to a purely condensing mechanism will occur in real reservoirs.
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