Summary This paper presents a new procedure to determine interwell connectivity in a reservoir on the basis of fluctuations of bottomhole pressure of both injectors and producers in a waterflood. The method uses a constrained multivariate linear-regression (MLR) analysis to obtain information about permeability trends, channels, and barriers. Previous authors applied the same analysis to injection and production rates to infer connectivity between wells. In order to obtain good results, however, they applied various diffusivity filters to the flow-rate data to account for the time lags and the attenuation. This was a tedious process that requires subjective judgment. Shut-in periods in the data, usually unavoidable when a large number of data points were used, created significant errors in the results and were often eliminated from the analysis. This new method yielded better results compared with the results obtained when production data were used. Its advantages include:no diffusivity filters needed for the analysis,minimal number of data points required to obtain good results,and flexible plan to collect data because all constraints can be controlled at the surface. The new procedure was tested by use of a numerical reservoir simulator. Thus, different cases were run on two fields, one with five injectors and four producers and the other with 25 injectors and 16 producers. For a large waterflood system, multiple wells are present and most of them are active at the same time. In this case, pulse tests or interference tests between two wells are difficult to conduct because the signal can be distorted by other active wells in the reservoir. In the proposed method, interwell connectivity can be obtained quantitatively from multiwell pressure fluctuations without running interference tests. Introduction Well testing is a common and important tool of reservoir characterization. Many well-testing methods have been developed in order to obtain various reservoir properties. Interference tests and pulse tests are used to quantify communication between wells. These methods are often applied to two wells such that one well sending the signals (by changing flow rates) and the other is receiving them (Lee et al. 2003). For a large field such as a waterflood system, however, multiple wells are present, and most of them are active at the same time. In that case, pulse tests or interference tests between two wells are difficult to conduct because the signal can be distorted by other active wells in the reservoir. In this method, data can be obtained from multiwell pressure tests that resemble interference tests. Thus, we can have several wells sending signals and the others receiving the signals at the same time. The wells that are receiving the signal, however, can either be shut in or kept at constant producing rates. The pressures at all wells are recorded simultaneously within a constant time interval. The length of the test will depend on the length of the time interval and the number of data points. Results of this method can be used to optimize operations and economics and enhance oil recovery of existing waterfloods by changing well patterns, changing injection rates, recompletion of wells, and infill drilling. This work is based on previous work conducted by Albertoni and Lake (2003) by use of injection and production rates. In their work, Albertoni and Lake developed and tested different approaches by use of constrained MLR analysis with a numerical simulator and then applied it to a waterflooded field in Argentina. They used diffusivity filters to account for the time lag and attenuation of the data. In his thesis, Dinh (2003) verified the method by use of a different reservoir simulator and applied it to a waterflooded field in Nowata, Oklahoma. He also investigated the effect of shut-in periods and vertical distances on the results. The main objectives of this work are to verify the results obtained from pressure data with results from flow-rate data to propose a new method to determine interwell connectivity and to suggest further research and study on the method. Similar to the method that uses production rates, we will concentrate on a waterflood system only. The reservoir is considered as a system that processes a stimulus (i.e., a well that is sending signals) and returns a response (i.e., a well that is receiving the signals). The effect of the reservoir on the input signal will depend on the location and the orientation of each stimulus/response pair. Because the total pressure changes at active and observation wells are not equal, only the MLR (Albertoni and Lake 2003; Dinh 2003; Albertoni 2002) was used. The effect of diffusion was not significant, thus the diffusivity filters were not used. The method was applied to two synthetic fields, one with five injectors and four producers and the other with 25 injectors and 16 producers.

This study is an extension of a novel technique to determine interwell connectivity in a reservoir based on fluctuations of bottom hole pressure of both injectors and producers in a waterflood system. The technique uses a constrained multivariate linear regression analysis to obtain information about permeability trends, channel and barriers. Some of the advantages of this new technique are simplified one-step calculation of interwell connectivity coefficients, small number of data points and flexible testing plan. However, the previous study did not provide either in-depth understanding or any relationship between the interwell connectivity coefficients and other reservoir parameters. This paper presents a mathematical model for bottom hole pressure responses of injectors and producers in a waterflood system. The model is based on available solution for fully penetrating vertical wells in a closed rectangular reservoir. It is then used to calculate interwell relative permeability, average reservoir pressure change and total reservoir pore volume using data from interwell connectivity test described in previous study. Reservoir compartmentalization can be inferred from the results. Cases of producers as signal wells, injectors as response wells and shut-in wells as response wells are also presented. Summary of results for these cases are provided. Reservoir behaviors and effect of skin factors are also discussed in this study. Some of the conclusions drawn from this study are:The mathematical model works well with interwell connectivity coefficients to quantify reservoir parameters;The procedure provides in-depth understanding of the multi-well system with water injection in the presence of heterogeneity;Injectors and producers have the same effect in terms of calculating interwell connectivity and thus, their roles can be interchanged. This study provides flexibility and understanding to the method of inferring interwell connectivity from bottom-hole pressure fluctuations. Interwell connectivity tests allow us to quantify accurately various reservoir properties in order to optimize reservoir performance. Different synthetic reservoir models were analyzed including: homogeneous, anisotropic reservoirs, reservoirs with high permeability channel, partially sealing fault and sealing fault. The results are presented in details in the paper. A step-bystep procedure, charts, tables, and derivations are included in the paper. Introduction Previous study carried out by Dinh and Tiab (2007) has introduced a new technique to infer interwell connectivity from bottom-hole pressure fluctuations in a waterflood system. The technique was proven to yield good results based on numerical simulation models of various cases of heterogeneity. In this study, an analytical model for multi-well system with water injection was derived for the technique. The model is based on available solution for a fully penetrating vertical well in a closed rectangular multi-well system and uses the principle of superposition in space. Based on analytical analysis, a new technique to analyze data of interwell connectivity test was developed. This technique utilizes the least squares regression method to calculate the average pressure change. Thus, reservoir pore volume, average reservoir pressure and total average porosity can be estimated from available input data. The results were verified using a commercial black oil numerical simulator.

The purpose of this study is to develop a technique, based on the pressure derivative concept, for interpreting pressure transient tests in wells with an inclined hydraulic fracture. Detailed analysis of unsteady-state pressure behavior of fully penetrating inclined fracture in an infinite slab reservoir was provided. Both uniform flux and infinite conductivity models were considered. The study has shown that inclined fracture pressure data exhibit similar flow regimes as for vertical fracture counterpart. Those flow regimes are linear and pseudo-radial flow for both uniform flux and infinite conductivity models. However, for infinite conductivity model, a bi-radial flow regime is also observed. In the case of high formation thickness to fracture half length ratio and high angle of inclination, both uniform flux and infinite conductivity inclined fracture model exhibit an additional flow regime called early radial flow. Both bi-radial flow and early radial flow regimes for inclined hydraulic fracture have not been mentioned in the literature before.A step by step procedure based on Tiab's Direct Synthesis (TDS) was developed in this study. Fracture properties such as half fracture length, inclination angle, formation permeability and pseudo-skin factor can be obtained from the direct interpretation of the log-log plot of pressure and pressure derivative without the need of any type curve matching. Several unique features of the pressure and pressure derivative plots of both uniform flux and infinite conductivity inclined fracture models were identified including the points of intersection of straight lines for different flow regimes. These points can be used to verify the results or to calculate unknown parameters. Equations associated with these features were derived and their usefulness was demonstrated. Numerical examples with both pressure build-up and drawdown data were also demonstrated for this procedure.

Hydraulic fracturing is an important well-stimulation technique that has been widely used in the oil and gas industry. Most of the pressure-transient-analysis techniques to analyze pressure responses of fractured wells are based on the assumption that the fracture is either vertical or horizontal. However, a hydraulic fracture could be inclined with a nonzero angle with respect to the vertical direction. Field studies have shown that most hydraulic fractures are not perfectly vertical. Thus, for an inclined hydraulic fracture, the vertical-orientation assumption may lead to erroneous results in welltest analysis, especially when the inclination angle is significant. However, there are very few studies concerning pressure-transient analysis of inclined hydraulic fractures, and there is no applicable well-test-analysis procedure available for inclined fractures.The purpose of this study is to develop a technique, on the basis of the pressure-derivative concept, for interpreting pressuretransient tests in wells with an inclined hydraulic fracture. Detailed analysis of unsteady-state pressure behavior of a fully penetrating inclined fracture in an infinite-slab reservoir was provided. Both uniform-flux and infinite-conductivity models were considered. The study has shown that inclined-fracture pressure data exhibit flow regimes similar to those for vertical fractures. Those flow regimes are linear and pseudoradial flow for both uniform-flux and infinite-conductivity models. However, for the infinite-conductivity model, a biradial-(or elliptical) flow regime is also observed. In the case of a high formation-thickness/fracture-half-length ratio and high angle of inclination, both uniform-flux and infiniteconductivity inclined-fracture models exhibit an additional flow regime, called early radial (ER) flow in this paper. This ER-flow regime for an inclined hydraulic fracture has not been mentioned in the literature before.A type-curve-matching technique was developed in this study using both pressure and pressure-derivative curves. This typecurve-matching procedure can be used to obtain the following parameters: fracture half-length, inclination angle, formation permeability, and the pseudoskin factor. The results should be verified with other pressure plots such as the semilog plot of ⌬p vs. t and the ⌬p-vs.-t 1/2 plot. A set of type curves with associated data was also provided for uniform-flux and infinite-conductivity inclinedfracture models. Detailed explanations, tables, figures, and a numerical example are included in this paper.

Hydraulic fracturing is an important well stimulation technique that has been widely used in the oil and gas industry. Most of the pressure transient analysis techniques to analyze pressure responses of fractured wells are based on the assumption that the fracture is either vertical or horizontal. However, a hydraulic fracture could be inclined with a non-zero angle with respect to the vertical direction. Field studies have shown that most hydraulic fractures are never perfectly vertical. Thus, for an inclined hydraulic fracture, the vertical orientation assumption may lead to erroneous results in well test analysis especially when the inclination angle is significant. However, there are very few studies concerning pressure transient analysis of inclined hydraulic fracture and there is no applicable well test analysis procedure available for inclined fractures. For this reason, it is important to develop well test analysis procedures for this type of fracture. The purpose of this study is to develop a technique, based on the pressure derivative concept, for interpreting pressure transient tests in wells with an inclined hydraulic fracture. Detailed analysis of unsteady-state pressure behavior of fully penetrating inclined fracture in an infinite slab reservoir was provided. Both uniform flux and infinite conductivity models were considered. The study has shown that inclined fracture pressure data exhibit similar flow regimes as for vertical fracture counterpart. Those flow regimes are linear and pseudo-radial flow for both uniform flux and infinite conductivity models. However, for infinite conductivity model, a bi-radial flow regime is also observed. In the case of high formation thickness to fracture half length ratio and high angle of inclination, both uniform flux and infinite conductivity inclined fracture model exhibit an additional flow regime called early radial flow. Both bi-radial flow and early radial flow regimes for inclined hydraulic fracture have not been mentioned in the literature before. A type curve matching technique was developed in this study using both pressure and pressure derivative curves. This type curve matching procedures can be used to obtain the following parameters: half fracture length, inclination angle, formation permeability and the pseudo-skin factor. The results should be verified with other pressure plots such as semi-log plot of ?P vs. t and ?P vs. t 1/2 plot. A set of type curves with associated data was also provided for uniform flux and infinite conductivity inclined fracture models. Detailed explanations, tables, figures and numerical examples are included in this paper. Introduction Hydraulic fracturing technique involves creation of fracture or fracture system in porous medium to overcome wellbore damage, to improve oil and gas productivity in low permeability reservoirs or to increase production in secondary recovery operations. In this study, a new type curve matching procedure, based on the pressure derivative concept, was developed for interpreting pressure transient tests in a well with an inclined hydraulic fracture. A general literature review on inclined hydraulic fracture will be provided. Both uniform flux and infinite conductivity models will be discussed in this paper. Only the case where the fracture is symmetric in both lateral and horizontal directions was considered in this study. Analytical solution for each flow regime is explained in details. The calculation procedure and analytical model verification are illustrated using numerical examples.