Rate Superposition for Generating Pressure Falloff Solutions for Vertical and 
Horizontal Wells

Peres, Alvaro; Boughrara, Amina; Reynolds, A.C.

doi:10.2523/90907-ms

Cited by 3 publications

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“…Even though accurate approximate analytical solutions for the injection and falloff pressure response for radial flow problems have been presented previously in the petroleum engineering literature 5,6,7,8,9 , all data considered in this paper are analyzed by nonlinear regression using a commercial reservoir simulator as the forward model 10 . Of the analyical solutions for the injection/falloff pressure, the one provided by Levitan 7 is most general and most useful as it applies for a multirate injectivity test where the rate during one or more of the periods can be zero to simulate shutin periods for falloff testing.…”

Section: Introductionmentioning

confidence: 99%

A Well Test for In-Situ Determination of Relative Permeability Curves

Chen¹,

Li²,

Peres³

et al. 2005

Proceedings of SPE Annual Technical Conference and Exhibition

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fax 01-972-952-9435. AbstractWe propose a novel well-test for in situ estimation of relative permeabilities under two-phase (oil-water) flow conditions. The test consists of three periods, (i) injection of water into an oil reservoir operating above bubble point pressure, (ii) a falloff test and (iii) a producing period. The producing period is critical as it yields production data that reflects changes in sandface mobility and thus is highly sensitive to the parameters used to model relative permeability curves, whereas, our results indicate that injection/falloff pressure data by themselves are not as reliable for defining relative permeability curves. We have developed optimization code based on the LevenbergMarquardt algorithm and coupled it with a commercial reservoir simulator to obtain a procedure for data analysis where the reservoir simulator is used as the forward model. By matching data by minimization of a weighted least squares objective function, we generate estimates of absolute permeability, relative permeabilities and the well skin factor. We show the method can be applied with either power law models or B-splines. We introduce a variable transformation that can be used to ensure that the estimated relative permeabilities are monotonic and concave up when B-splines are used.

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Section: Introductionmentioning

confidence: 99%

A Well Test for In-Situ Determination of Relative Permeability Curves

Chen¹,

Li²,

Peres³

et al. 2005

Proceedings of SPE Annual Technical Conference and Exhibition

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Approximate Analytical Solutions for the Pressure Response at a Water Injection Well

Peres

Boughrara

Chen

et al. 2004

SPE Annual Technical Conference and Exhibition

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fax 01-972-952-9435. AbstractThe first and fifth authors have previously used the Thompson-Reynolds steady-state theory to derive solutions for the pressure response at a water injection well. In the vertical well case, their solutions assumed a completepenetration well; in the horizontal well case, they assumed that the well is equidistant from the top and bottom of the formation. Here, we construct approximate analytical pressure solutions for the restricted-entry vertical well case and for a horizontal well for the case where the well's axis is not equidistant from the top and bottom boundaries. The solutions are based on adding to the single-phase solution, a multiphase term which represents the existence of the two-phase zone and the movement of the water front. We present models for the movement of water based on a combination of Buckley-Leverett equations that allow us to accurately compute the multiphase flow component of the analytical solution. By comparison with a finite-difference simulator using grid refinement and a hybrid grid, it is shown that our multiphase flow solutions are accurate.The analytical solutions provide insight into the behavior of injectivity tests at horizontal an vertical wells. For example, for a restricted-entry case, it is shown that the pressure derivative may be negative throughout an injection test even when the duration of the test exceeds ten or more days. We also show that for a well near a fault, the ratio of slopes reflected by derivative data will not in general be equal to two but is given by a formula that involves the mobility ratio. In the restricted-entry vertical well case, we provide the equations for three flow regimes.

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A Well Test for In-Situ Determination of Relative Permeability Curves

Chen

Peres

et al. 2005

SPE Annual Technical Conference and Exhibition

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We propose a novel well-test for in situ estimation of relative permeabilities under two-phase (oil-water) flow conditions. The test consists of three periods, (i) injection of water into an oil reservoir operating above bubble point pressure, (ii) a falloff test and (iii) a producing period. The producing period is critical as it yields production data that reflects changes in sandface mobility and thus is highly sensitive to the parameters used to model relative permeability curves, whereas, our results indicate that injection/falloff pressure data by themselves are not as reliable for defining relative permeability curves. We have developed optimization code based on the Levenberg-Marquardt algorithm and coupled it with a commercial reservoir simulator to obtain a procedure for data analysis where the reservoir simulator is used as the forward model. By matching data by minimization of a weighted least squares objective function, we generate estimates of absolute permeability, relative permeabilities and the well skin factor. We show the method can be applied with either power law models or B-splines. We introduce a variable transformation that can be used to ensure that the estimated relative permeabilities are monotonic and concave up when B-splines are used. Introduction Numerous papers from the 1970's and 1980's discuss techniques for the estimation of relative permeability curves by matching pressure and rate (or displaced volume) data from laboratory core floods using an optimization algorithm to minimize a least squares type objective function[1,2,3,4]. Assuming power law relative permeability curves yields a small number of parameters to be estimated and a well-conditioned optimization problem. However, relative permeability curves may not be well represented by power law models. Because of this, various authors used splines, especially B-spines, to parameterize relative permeability curves, but then special techniques were required to regularize the optimization process and ensure that monotonic relative permeability curves are obtained[4]. Although many papers have also proposed using similar techniques to adjust relative permeability curves by matching long-time production data, our objective is to generate a procedure to estimate relative permeability curves from a well test which has a duration of a few hours to a very few days. Intuitively, to do this successfully, we should generate data that is sensitive to a wide range of saturations, i.e., generate data similar to that which can be obtained in a laboratory core flood. To do so, we propose an injection/falloff test followed by a production period. The idea is that, during the flow back period, the sandface will be exposed to a wide range of water saturations and the associated pressure and phase rate data or water cut data should be sufficient to obtain good estimates of relative permeability curves. We compare results obtained from the proposed test with those obtained from a more standard injection falloff test. Even though accurate approximate analytical solutions for the injection and falloff pressure response for radial flow problems have been presented previously in the petroleum engineering literature[5,6,7,8,9], all data considered in this paper are analyzed by nonlinear regression using a commercial reservoir simulator as the forward model[10]. Of the analyical solutions for the injection/falloff pressure, the one provided by Levitan[7] is most general and most useful as it applies for a multirate injectivity test where the rate during one or more of the periods can be zero to simulate shutin periods for falloff testing. His solution, however, does not apply during a subsequent production period and thus can not be used to analyze the test proposed here.

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Rate Superposition for Generating Pressure Falloff Solutions for Vertical and Horizontal Wells

Cited by 3 publications

References 0 publications

A Well Test for In-Situ Determination of Relative Permeability Curves

A Well Test for In-Situ Determination of Relative Permeability Curves

Approximate Analytical Solutions for the Pressure Response at a Water Injection Well

A Well Test for In-Situ Determination of Relative Permeability Curves

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