Flowing partially penetrating well: solution to a mixed-type boundary value problem

A new semi-analytic solution to the mixed-type boundary value problem for a flowing partially penetrating well with infinitesimal skin situated in an anisotropic aquifer is developed. The solution is suited to aquifers having a semi-infinite vertical extent or to packer tests with aquifer horizontal boundaries far enough from the tested area. The problem reduces to a system of dual integral equations (DE) and further to a deconvolution problem. Unlike the analogous Dagan's steady-state solution [Water Resour. Res. 1978; 14:929–34], our DE solution does not suffer from numerical oscillations. The new solution is validated by matching the corresponding finite-difference solution and is computationally much more efficient. An automated (Newton–Raphson) parameter identification algorithm is proposed for field test inversion, utilizing the DE solution for the forward model. The procedure is computationally efficient and converges to correct parameter values. A solution for the partially penetrating flowing well with no skin and a drawdown–drawdown discontinuous boundary condition, analogous to that by Novakowski [Can. Geotech. J. 1993; 30:600–6], is compared to the DE solution. The D–D solution leads to physically inconsistent infinite total flow rate to the well, when no skin effect is considered. The DE solution, on the other hand, produces accurate results.

[1]  R. C. Earlougher.  Advances in Well Test Analysis , 1977 .

[2]  J. Gillis,et al.  Mixed boundary value problems in potential theory , 1966 .

[3]  Roger Thunvik,et al.  Determination of formation permeability by double-packer tests , 1984 .

[4]  Gedeon Dagan,et al.  A note on packer, slug, and recovery tests in unconfined aquifers , 1978 .

[5]  Numerical Evaluation of Flowmeter Test Interpretation Methodologies , 1996 .

[6]  K. Novakowski.  Interpretation of the transient flow rate obtained from constant-head tests conducted in situ in clays , 1993 .

[7]  Hantush.  Drawdown Around a Partially Penetrating Well , 1961 .

[8]  M Muskat,et al.  THE FLOW OF HOMOGENEOUS FLUIDS THROUGH POROUS MEDIA: ANALOGIES WITH OTHER PHYSICAL PROBLEMS , 1937 .

[9]  D. K. Babu,et al.  Flow to a Partially Penetrating Well in a Double‐Porosity Reservoir , 1984 .

[10]  V. T. Chow,et al.  Advances in hydroscience , 1964 .

[11]  Serge Leroueil,et al.  Analysis of the in situ constant-head permeability test in clays , 1990 .

[12]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[13]  M. S. Hantush.  Nonsteady flow to flowing wells in leaky aquifers , 1959 .

[14]  Michael J. Economides,et al.  Pressure transient analysis for single well/reservoir configurations (considering partial penetration, mixed boundary, wellbore storage, and skin effects) , 1985 .

[15]  V. D. Kagawan,et al.  Pressure-transient analysis for fractured wells producing at constant pressure , 1988 .

[16]  Adiabatic-isothermal mixed boundary conditions in heat transfer , 1987 .

[17]  B. Noble,et al.  Methods Based on the Wiener-Hopf Technique. , 1960 .

[18]  P. Hsieh,et al.  Comment on “Evaluation of Slug Tests in Wells Containing a Finite‐Thickness Skin” by C. R. Faust and J. W. Mercer , 1985 .

[19]  Y. P. Chang,et al.  Anisotropic Heat Conduction With Mixed Boundary Conditions , 1984 .

[20]  S. P. Neuman,et al.  Effect of partial penetration on flow in unconfined aquifers considering delayed gravity response , 1974 .

[21]  V. I. Fabrikant,et al.  Mixed Boundary Value Problems of Potential Theory and Their Applications in Engineering , 1991 .

[22]  Louis A. Schmittroth,et al.  Numerical inversion of Laplace transforms , 1960, Commun. ACM.

[23]  Giorgio Cassiani,et al.  Hydraulics of a partially penetrating well : solution to a mixed-type boundary value problem via dual integral equations , 1998 .

[24]  Z. Kabala,et al.  Well Hydraulics with the Weber–Goldstein Transforms , 1997 .

[25]  M. S. Hantush.  Aquifer Tests on Partially Penetrating Wells , 1961 .

[26]  A. S. Odeh.  Steady-State Flow Capacity of Wells With Limited Entry to Flow , 1968 .

[27]  M. S. Hantush.  Hydraulics of Wells , 1964 .

[28]  G. E. Schneider,et al.  An Investigation Into the Heat Loss Characteristics of Buried Pipes , 1985 .

[29]  E. O. Brigham,et al.  The Fast Fourier Transform , 1967, IEEE Transactions on Systems, Man, and Cybernetics.

[30]  John R. Rice,et al.  Plastic deformation in brittle and ductile fracture , 1970 .

[31]  C. L. Tien,et al.  Analysis of conduction-controlled rewetting of a vertical surface , 1975 .

[32]  James J. Butler,et al.  Slug tests in partially penetrating wells , 1994 .

[33]  Wei Chun Chu,et al.  Analysis of Interference Test Data Influenced by Wellbore Storage and Skin at the Flowing Well , 1980 .

[34]  S. W. Lohman,et al.  Nonsteady flow to a well of constant drawdown in an extensive aquifer , 1952 .

[35]  W. B. Wilkinson,et al.  Constant Head in Situ Permeability Tests in Clay Strata , 1968 .

[36]  C. Tranter,et al.  Integral Transforms in Mathematical Physics , 1971 .

[37]  Z. Kabala,et al.  Response of a partially penetrating well in a heterogeneous aquifer: integrated well-face flux vs. uniform well-face flux boundary conditions , 1997 .

[38]  Fred J. Molz,et al.  An Analysis Technique for Multilevel and Partially Penetrating Slug Test Data , 1990 .

[39]  Paul S. Hammond,et al.  A perturbation method for mixed boundary-value problems in pressure transient testing , 1990 .