Pressure transient response in stress-sensitive formations is obtained by solving analytically the radial flow equation with pressure-dependent rock properties. In cracked rock systems and tight formations, hydraulic permeability is very sensitive to pore pressure change. The mathematical model presented in this paper takes into account the reduction in permeability caused by an increase in effective stress. The dependence of permeability on pore pressure makes the flow equation strongly nonlinear. A perturbation technique is applied to determine approximate analytical solutions for transient flow in an infinite radial system with constant rate inner boundary. The model includes a new parameter, the permeability modulus, which measures the permeability dependency on pressure. The solution of the model leads to the construction of type curves that can be applied to drawdown and buildup analysis of well test data from stress-sensitive reservoirs. Type curve matching provides a way to estimate initial permeability and permeability modulus. From buildup data, initial permeability may be obtained from the slope of the semi-log straight line on Horner plot, as shown in an example using field data from a low permeability gas reservoir.

Summary A technique is presented to improve the treatment of wells in reservoir simulators by the use of an orthogonal curvilinear grid (cylindrical or elliptical) in well regions and a rectangular grid elsewhere in the reservoir. Special methods are developed to handle irregular blocks connecting the two types of regions. The nonlinear equations for the two regions are solved with different levels of implicitness; the simultaneous solution (implicit) method is applied to the well regions, while an implicit pressure, explicit saturation (IMPES) technique is used for the reservoir region. An efficient approach, requiring no outer iterations, between well and reservoir regions is used to couple different regions. The scheme is validated by the simulation of the flow of a slightly compressible fluid in a square domain with one boundary open to flow. For this case, the numerical solution with the hybrid grid approach matches the analytical solution exactly. A computer model based on the approach was developed for simulating two-phase flow in multiwell horizontal reservoirs. We show that the proposed technique provides much better prediction of the phase split in the producers than provides much better prediction of the phase split in the producers than the conventional approach. Introduction The treatment of wells in reservoir simulators can have a strong influence on computed results. The common approach is to relate the wellbore pressure and the reservoir block pressure through single-phase flow models in the well region. Peaceman showed that for a square block, the pressure obtained from the steady-state radial pressure pressure obtained from the steady-state radial pressure distribution equals the wellblock pressure at an equivalent radius of r 0.2 x. With this concept of equivalent radius, a relationship between wellbore flowing pressure and wellblock pressure can be derived easily. This approach has been extended to wells in the center of rectangular blocks and to off-centered wells. Strictly speaking, these well models are invalid when high rates of saturation change occur in the neighborhood of the wellbore. Improper handling of phase mobility leads to errors in the calculation of WOR and GOR. Furthermore, the use of Cartesian coordinates everywhere in the reservoir does not allow the simulation of real geometry of the nearwell flow. While multiphase flow near the well can be properly modeled through a coning simulator, it is difficult to do this in large simulation studies that involve many wells. On the other hand, in many situations, the usual approach of using pseudofunctions provides only a crude approximation of the real problem. One method of handling wells rigorously is the coupling of well coning models with a reservoir model. Akbar et al. incorporated a one-dimensional (1D), three-phase, radial coordinate well simulator within a twodimensional (2D), three-phase, Cartesian coordinate reservoir simulation model. The radial model simulated a rectangular gridblock in the areal model. The equality of the volumetrically weighted average pressure within the radial model and the corresponding reservoir block pressure was established. In addition, the summation of pressure was established. In addition, the summation of the fluxes into the four vertical faces of the well gridblock was taken as the influx into the radial system. Alternate calculations for the two models were performed until both models predicted the same production rates. However, material balance between the two models was not exactly maintained. Mrosovsky and Ridings extended the Akbar et al. approach by coupling a'2D cylindrical model with a three-dimensional (3D), rectangular-grid reservoir simulator. To obtain good resolution in the vicinity of the wellbore, a fine grid spacing around the well is often used in reservoir simulation. Rosenberg developed a method of localized mesh refinement for finite-difference techniques. Recent developments in the use of local grid refinement and adaptive implicit methods are major steps toward the improvement of well treatment in reservoir simulators. These approaches require the use of the same coordinate system everywhere in the reservoir, however, and cannot take advantage of the almost radial nature of flow near the wells. The need for an accurate and simple way to represent wells in reservoir simulators has led to the development of a hybrid grid approach. In this technique, a Cartesian grid is used for the entire reservoir with arbitrarily fine curvilinear orthogonal grids in well regions that may span one or more blocks of the Cartesian grid (see Fig. 1). An integral approach is applied to derive the discretized flow equations. The solutions for the various regions can be decoupled in such a way that different levels of implicitness in the treatment of transmissibilities can be considered. Furthermore, the well region problem can be solved in 1, 2, or 3D, as appropriate. SPERE p. 611

The assumption of constant rock properties in pressure-transient analysis of stress-sensitive reservoirs can cause significant error in determination of reservoir transmissibility and storativity. On the other hand, inclusion of pressure-dependent rock properties makes the governing equation for the pressure in the reservoir nonlinear. These nonlinearities can be treated only approximately by numerical means. If a permeability modulus is defined, the nonlinearities associated with the governing equation become weaker and an analytical solution in terms of a regular perturbation series can be obtained for a radial infinite-acting reservoir. Three terms in the perturbation series are derived to show the convergence and accuracy of the solution. The equation obtained for each order (zero, first, and second) in the perturbation series is solved exactly, and hence, the solution is exact to the third order.The effect of wellbore storage on the pressure behavior is also investigated. First-order approximation for bounded systems is presented to show qUalitative effects. A field example is analyzed to determine the permeability modulus and reservoir properties.

In order to get desired accuracy in reservoir simulation, one must refine the grid in certain regions, for example, the regions around wells. When this is done in conventional simulators, unnecessary refinement occurs in some regions. This problem is avoided by the use of "Local Grid Refinement" (LGR) techniques, but the resulting matrix is large and of irregular sparsity. In this paper we address several practical aspects of some methods of LGR, including practical aspects of some methods of LGR, including the use of hybrid grid, and then present three highly efficient methods of solving resulting problems with some Domain Decomposition (DD) techniques. In the first method, a relaxation in the unknowns is performed. In the second one, we use overlapping performed. In the second one, we use overlapping boundaries between the subdomains. In the last one, a coarse grid solution is obtained first and it provides the boundary conditions for the fine grid solution. The LGR and DD methods developed in this work were implemented and tested in a three-dimensional, three-phase, black-oil model. The results are presented in this paper for three examples. It is also presented in this paper for three examples. It is also shown that the DD technique with overlapping boundaries is extremely efficient for solving problems containing one or more LGR regions. The method problems containing one or more LGR regions. The method proposed in this paper can easily take advantage of proposed in this paper can easily take advantage of parallel processors. Also, it is easy to implement the parallel processors. Also, it is easy to implement the proposed techniques in existing simulators. proposed techniques in existing simulators. The LGR and DD techniques discussed in this paper improve the accuracy of reservoir simulation paper improve the accuracy of reservoir simulation around wells and in other regions of high activity. Using these techniques, one not only obtains the correct well pressure, but also the correct saturation near the well. A consequence of this is that WOR and GOR are predicted accurately without well pseudofunctions. pseudofunctions Introduction Reservoir performance forecasting is accomplished by reservoir simulators that solve coupled non-linear partial differential equations describing multicomponent, multiphase flow in the reservoir. In most cases the flow equations are reduced to a system of non-linear algebraic equations by finite-difference techniques and then linearized by the Newton-Raphson method. The resulting linear system has a sparse matrix that can be solved by direct methods for small problems and by iterative methods for larger problems. In the practical application of this tool the number of equations can be more than 100,000 and can even approach 1,000,000. Special gridding and solution techniques are required to take advantage of the physical characteristics of the problem. Recently, there has been a lot of interest in the application of LGR2-10 in reservoir simulation. This technique gives high resolution in regions where the flow rates are high, such as near the wells, without introducing small blocks in regions of low activity. The structure of the Jacobian resulting from LGR can be exploited by the use of DD techniques. P. 245

SPE Members Abstract Phase injectivities for multiphase injection processes are studied using an isothermal black-oil numerical model that properly treats wellbore/reservoir interaction. A fully implicit technique for the coupling of wellbore and reservoir flow equations is described. The effect of gravity segregation in the wellbore is taken into account to simulate cocurrent water and gas injection. It is shown that the current techniques used in reservoir simulation for assigning phase injectivities yield inconsistent results when wellbore phase redistribution takes place. The effect of multiphase flow in the well and in the reservoir during well testing is also investigated. The fully coupled wellbore/reservoir model is employed to study the effect of wellbore phase segregation on buildup pressure response. pressure response. Introduction Multiphase injection of fluids in reservoirs may occur in a variety of oil field operations such as steam and cocurrent water and gas injection processes. In enhanced recovery processes, special techniques are usually required to control and modify injection profiles. Laboratory experiments reported by Elson have showed the effect of phase separation in the wellbore. Uneven injection profiles in the perforated intervals have been clearly observed. Field observations have also shown this behavior with the lighter and less viscous phase being preferentially injected into the top of the formation while the heavier phase goes into the lower part. phase goes into the lower part. In reservoir simulation, phase injectivity is commonly handled by means of simplified approaches which usually ignores the gravity segregation in the wellbore. Multiphase flow in the wellbore and in the surrounding formation may also occur during well tests. Anomalous pressure buildup behavior, as a result of wellbore phase pressure buildup behavior, as a result of wellbore phase segregation, has been observed in high GOR wells. A general review of well test analysis with multiphase flow was presented by Raghavan. However, the effect of wellbore phase segregation was not taken into account. An attempt to account for this effect in pressure buildup analysis was made by Fair. In his work, an empirical relationship for pressure change was introduced in the wellbore storage equation in order to describe phase redistribution in the well. Recently, Winterfeld presented a model that treats rigorously multiphase flow in the wellbore during pressure buildup. The model solves simultaneously for transient multiphase flow in the wellbore and in the reservoir. Counter-current two-phase flow in the wellbore during shut-in is allowed by the use of semi-empirical expressions to account for phase to phase viscous forces. To allow for phase segregation in the simulation of multiphase injection processes and multiphase well tests, a wellbore flow model must be coupled with the reservoir model. Several papers dealing with the simulation of transient multiphase flow in pipes have been published in the technical literature. Liles and Reed developed a semi-implicit drift-flux model for simulating unsteady state two-phase flow in pipes. Later, Millers presented a similar model to study wellbore storage effects during geothermal well testing. Sharma et al. simulated transient gas-oil flow in pipelines using a black-oil type model. A thermal pipelines using a black-oil type model. A thermal compositional model for simulating transient gas-liquid flow in natural gas pipelines was reported by Kohda et al.. P. 123