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Summary This paper presents a new analytical solution to obtain the shut-in wellbore pressure at a vertical well that had been producing at a constant wellbore hole pressure from a closed-boundary reservoir with multiple wells producing at constant, different wellbore pressures. The analytical solution is a set of coupled simultaneous equations in the Laplace space (one for each well). These are obtained by using separation of variables, Green's functions, and the unit-step function with the methods of Rodriguez and Cinco-Ley1 and Camacho-V. et al.2 The equation for the shut-in well is nonlinear, preventing us from obtaining the formal solution of the set of equations either in real time or in Laplace space. A number of approaches were used to make the solution tractable. These included expansion of Green's function for large times, a Taylor series expansion of the rates around the time of shut-in, and application of the concepts of Najurieta.3 None of the approaches gave satisfactory results.

The purpose of this paper is to document transient behavior at a well that produces a fractured reservoir by solution-gas drive. The authors consider the dual-porosity model Barenblatt et al. and Warren and Root introduced and the dual-permeability model Bourdet presented. These idealizations are considered to demonstrate the consequences of production from the matrix system. They document pressure and saturation behaviors in the fracture and matrix systems. The interaction between the matrix and the fracture systems under multiphase flow conditions also is discussed. For convenience, they refer to the Warren and Root model as the dual-porosity model and to the Bourdet idealization as the dual-permeability model.

Summary This paper presents results pertinent to the analysis of well performance in solution-gas-drive systems. The characteristics of wells producing solution-gas-drive systems are documented. Procedures to correlate multiphase flow solutions with single-phase flow systems for both transient and boundary-dominated flows are presented.

Summary This paper presents procedures to obtain reservoir parameters from constant-pressure drawdown data in solution-gas-drive reservoirs. A novel procedure to determine the mechanical skin factor is introduced. Examples, including a field case, illustrate the use of this procedure. An estimate of the drainage area can be obtained with the derivative of rate data. A theoretical basis for analyzing data by the pressure-squared, p2, approach is presented; this procedure permits the approximate determination of sandface effective permeabilities in the transient flow period. For damaged wells, it is possible to obtain rough estimates of the size of the skin zone and the ratio of reservoir/skin-zone permeability when early transient data are available. The expression of the appropriate dimensionless rate in terms of physical properties for solution-gas-drive systems is presented. Finally, this paper presents a procedure to obtain an estimate of the change in sandface saturation during the transient flow period.

Summary In this theoretical study, a numerical model was used to examine the influence of pressure level and skin factor on the inflow performance relationships (IPR's) of wells producing under solution-gas-drive systems. Examination of the synthetic deliverability curves suggests that the exponent of the deliverability curve is a function of time and that the exponent is usually greater than unity. The implication of this observation to field data is discussed. The accuracy of procedures given in the literature to predict oilwell deliverabilities is also examined. It is shown that these methods can used to predict future performance provided that the exponent of the deliverability curve is known and that extrapolations over large time ranges are avoided. If single-point tests are used to predict future performance (such tests assume that the exponent of the deliverability curve is constant), then errors in predictions will be minimized if test conditions correspond closely to actual producing conditions. Under these circumstances, errors in rate predictions will be directly influenced by the magnitude of the error that results from assuming that the exponent of the deliverability curve is known. Although relative permeability and fluid property data are required, the Muskat material-balance equation and the assumption that GOR is independent of distance can be used to predict future production rates. This method avoids problems associated with other methods in the literature and always yields reliable results. New methods to modify the IPR curve to incorporate changes in skin factor are presented. A new flow-efficiency definition based on the structure of the deliverability equations for solution-gas-drive reservoirs is proposed. This definition avoids problems that result when the currently available methods are applied to heavily stimulated wells. Introduction This paper addresses three concerns pertinent to IPR's under solution-gas drive. First, by means of numerical simulations, we examine the IPR when a skin region exists around the wellbore. This phase of our work is intended to address the question of whether the generalized form of the IPR curve is governed by the existence of a skin region. Although the skin region has a dominant effect on the well response, the influence of a skin region has been ignored for all practical purposes. (The seminal work of Vogel, however, briefly addresses the influence of a skin region.) Second, we address the prediction of future IPR's from test data. Many methods have been proposed in the literature to predict future performance. The methods of Standing and Fetkovich are based on empirical observations of solution-gas-drive reservoir performance. All methods presented in the literature assume that the mobility function, kro(uoBo), is a linear function of pressure, p, where k, is the relative permeability to oil. uo, is the viscosity of oil, and B, is the FVF of oil. Although the methods of Standing and Fetkovich can be justified on the basis of this assumption, this requirement (mobility function is a linear function of pressure) is not a necessary condition for the methods given in Refs. 2 and 3 (or for that matter, in the other references cited above) to be valid. We examine the validity of the procedures suggested by Standing and Fetkovich by means of simulations for a wide range of producing conditions. We show that the success of these methods rests primarily on the fact that the deliverability equations suggested by Vogel and Fetkovich incorporate features that reflect the nonlinear aspects of flow in the reservoir and, more importantly, are calibrated by measurements. Their procedures attempt to predict only the endpoint-the final answer. We demonstrate this point by comparing mobility-function profiles predicted by methods given in the literature. The third part of this paper examines procedures to modify the IPR curve to incorporate the influence of the skin factor, s. Brown shows that Standing's method to modify the IPR curve can result in inconsistent predictions if s is negative. Whitson says that Standing's method leads to nonphysical results if s less than 0. Whitson's observations suggest that Standing's procedure should be incorrect if s is greater than 0. Here, we show that inconsistent or nonphysical predictions are a result of the definition of flow efficiency used by Standing. Standing defined flow efficiency on the basis of liquid flow and not on the quadratic form suggested by Vogel's equation. In this work, we redefine flow efficiency, Ef, on the basis of the quadratic forms of the deliverability equations suggested by Vogel and Fetkovich and show that the inconsistencies mentioned by Brown can be eliminated. Unfortunately, the curves developed by Standing need to be corrected for all values of s. Incidentally, the definition we present is consistent with the definition of Ef used for gas wells. A new chart to modify the IPR curve based on the appropriate deliverability equation (Vogel or Fetkovich) is presented. Numerical Model We consider a homogeneous, closed, circular reservoir with the well located at the center. The well is capable of producing either at constant oil rate or at constant wellbore pressure. Gravity effects are considered to be negligible. The reservoir initially is assumed to be at the bubblepoint. An annular region concentric with the wellbore, with a permeability different from formation permeability, is used to incorporate the influence of a skin region. Several sets of relative permeability and fluid properties were used in this study. Figs. 1 and 2 show the fluid properties for two of the data sets we considered, and Fig. 3 presents the relative permeability data used in this study. Table 1 presents other information regarding reservoir properties used in this work. The results given in this work, however, can be applied over a wider range of conditions. A finite-difference model was used to obtain the results presented here. The procedures we followed to ensure that the results are accurate are similar to those presented in Refs. 10 and 11 and will not be considered here. Non-Darcy flow effects are not considered. The influence of nonDarcy flow for wells producing by solution-gas drive represents a formidable analytical problem. Many fundamental considerations are involved, which are beyond the scope of this work. Background From numerical simulations, Vogel showed that the deliverability curve for a well producing under solution-gas drive is given by ..........................................(1) Here qo/rna, is the maximum possible flow rate (rate corresponding to pwf=0), and q, is the flow rate corresponding to average pressure. p, and wellbore flowing pressure, pwf. In 1973, on the basis of field experiments, Fetkovich suggested that the deliverability curve for solution-gas drive systems is given by the following relation. ..........................................(2) Here J o(t) is the productivity index for a well producing under solution-gas drive. Field tests presented in Ref.. 3 indicate that the exponent n should be in the range 0.5: - 1. JPT P. 541^

Summary The performance of wells in solution-gas-drive reservoirs during the boundary-dominated flow period is examined. Both constant-wellbore-pressure and constant-oil-rate production modes in closed systems are considered. For the constant-wellbore-pressure production mode, Arps' performance-prediction equations are examined, and predictions of future performance are shown to be strong functions of well spacing, well condition, and fluid properties. The parameters b (the decline exponent) and di (the initial decline rate) in the Arps equations are expressed in terms of physical properties. The conditions under which these equations can be used are specified. An empirical procedure to predict production rates is also presented. In the case of constant-oil-rate production, an expression to correlate the pressure distribution in the reservoir is presented. The correlating function permits us to extend the definition of pseudosteady-state flow to solution-gas-drive systems. Its use also allows the simultaneous computation of average properties (pressure and saturation) during boundary-dominated flow from wellbore information.

Summary The response of fractured wells producing noncommunicating layered reservoirs is the focus of this work. The conductivity of the fracture is assumed to be finite. The fracture length is assumed to vary from layer to layer. Two modes of production—constant wellbore pressure and constant rate—are considered. In the first part, the fractures are assumed to communicate only at the wellbore. The results given in this section are intended to provide engineers with analytical capabilities to examine responses in wells where the layers that have been stimulated are separated by considerable distances. Procedures to interpret the results of pressure buildup and/or production tests (drawdown responses) in terms of layer properties are presented. Criteria to ensure maximum productivity are specified. The second part examines well responses when the fractures are in communication at points other than the wellbore. All other things being identical, we show that communication between fractures increases productivity.

New analytical solutions for the response at a well intercepting a layered reservoir are derived. The well is assumed to produce at a constant rate or a constant pressure. We examine reservoir systems without in-terlayer communication and document the usefulness of these solutions, which enable us to obtain increased physical understanding of the performance of fractured wells in layered reservoirs. The influence of vertical variations in fracture conductivity is also considered. Example applications of the approximations derived here are also presented.