1. Well Testing

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17. 17 1.3 Solution to Diffusivity Equation There are four solutions to Eq.(1.1) that are particularly useful in well testing: (1) The solution for a bounded cylindrical reservoir (2) The solution for an infinite reservoir with a well considered to be a line source with zero wellbore radius, (3) The pseudo steady-state solution (4) The solution that includes wellbore storage for a well in an infinite reservoir

18. 18 The assumptions that were necessary to develop Eq.(1.1) (1) Homogeneous and isotropic porous medium of uniform thickness, (2) Pressure-independent rock and fluid properties, (3) Small pressure gradient, (4) Radial flow (5) Applicability of Darcy’s law ( sometimes called laminar flow ) (6) Negligible gravity force.

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28. 28 Question: Why does p w > p i for certain t ?

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49. 49 Boundary effect time analyzed from type curves The visually deviated point from type curve analysis t D * =1.96*10 6 Closed circular reservoir with r eD = 3000 case

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60. 60 Flow Equation for Generalized Reservoir Geometry

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64. 64 Boundary effect time analyzed from type curves The visually deviated point from type curve analysis t D * =1.96*10 6 Closed circular reservoir with r eD = 3000 case

65. 65 Boundary effect time estimated from radius of investigation equation The visually deviated point from type curve analysis ( I ) ( II ) ( III ) closed circular reservoir with r eD = 3000 case

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78. 78 Development of a mathematical relationship between sandface (formation) and surface flow rates

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91. 91 1.4 Radius of investigation

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101. 101 1.5 The Principle of Superposition

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103. 103 InterferenceTest Consider three wells, well A, B, and C that start to produce at the same time from infinite reservoir (Fig. 1.8). Application of the principle of superposition shows that

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105. 105 In Eq.(1.49), there is a skin factor for well A, but does not include skin factors for wells B and C. Because most wells have a nonzero skin factor and because we are modeling pressure inside the zone of altered permeability near well A, we must include its skin factor. However, the pressure of nonzero skin factors for wells B and C affects pressure only inside their zones of altered permeability and has no influence on pressure at Well A if Well A is not within the altered zone of either Well B or Well C.

106. 106 Bounded reservoir Consider the well (in fig. 1.9) a distance, L, from a single no-flow boundary. Mathematically, this problem is identical to the problem of a two-well system; actual well and image well.

107. 107 Extensions of the imaging technique also can be used, for example, to model (1) pressure distribution for a well between two boundaries intersecting at 90°; (2) the pressure behavior of a well between two parallel boundaries; and (3) pressure behavior for wells in various locations completely surrounded by no-flow boundaries in rectangular-shape reservoirs. [ Matthews, C. S., Brons, F., and Hazebroek, P.: “A method for determination of average pressure in a bounded reservoir,” Trans, AIME (1954) 201, 182-191

108. 108 Variable flow-rate

109. 109 Proceeding in a similar way, we can model an actual well with dozens of rate changes in its history we also can model the rate history for a well with a continuously changing rate (with a sequence of constant-rate periods at the average rate during the period).

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112. 112 1.6 Horner’s Approximation In 1951, Horner reported an approximation that can be used in many cases to avoid the use of superposition in modeling the production history of a variable-rate well. With this approximation, we can replace the sequence of Ei functions, reflecting rate changes, with a single Ei function that contains a single producing time and a single producing rate. The single rate is the most recent nonzero rate at which the well was produced; we call this rate q last for now. This single producing time is found by dividing cumulative production from the well by the most recent rate; we call this producing time t p, or pseudoproducing time

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114. 114 (1) The basis for the approximation is not rigorous, but intuitive, and is founded on two criteria: (a) Use the most recent rate, such a rate, maintained for any significant period (b) Choose an effective production time such that the product of the rate and the production time results in the correct cumulative production. In this way, material balance will be maintained accurately.

115. 115 (2) If the most recent rate is maintained sufficiently long for the radius of investigation achieved at this rate to reach the drainage radius of the tested well, then Horner’s approximation is always sufficiently accurate. We find that, for a new well that undergoes a series of rather rapid rate changes, it is usually sufficient to establish the last constant rate for at least twice as long as the previous rate. When there is any doubt about whether these guidelines are satisfied, the safe approach is to use superposition to model the production history of the well.

116. 116 Example 1.6 – Application of Horner’s Approximation Given: the Production history was as follows:

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119. 119 Reference Books (A) Lee, J.W., Well Testing, Society of petroleum Engineers of AIME, Dallas, Texas,, 1982. (B) Earlougher, R.C., Jr., Advances in Well Test Analysis, Society of Petroleum Engineers, Richardson, Texas, 1977, Monograph Series, Vol. 5. (1) Carlson, M.R., Practical Reservoir Simulation: Using, Assessing, and Developing Results, PennWell Publishing Co., Houston,TX, 2003. (2) FANCHI, J.R., Principles of Applied Reservoir Simulation, Second Edition, PennWell Publishing Co., Houston,TX, 2001. (3) Ertekin, T., Basic Applied Reservoir Simulation, PennWell Publishing Co., Houston,TX, 2003. (4) Koederitz, L.F., Lecture Notes on Applied Reservoir Simulation, World Scientific Publishing Company, MD, 2005

120. 120 Introduction This course intended to explain how to use well pressures and flow rates to evaluate the formation surrounding a tested well, by analytical and numerical methods. Basis to this discussion is an understanding of (1) the theory of fluid flow in porous media, and (2) pressure-volume-temperature (PVT) relations for fluid systems of practical interest.

121. 121 Introduction (cont.) One major purpose of well testing is to determine the ability of a formation to produce fluids. Further, it is important to determine the underlying reason for a well’s productivity. A properly designed, executed, and analyzed well test usually can provide information about FORMATION PERMEABILITY, extent of WELLBORE DAMAGE (or STIMULATION), RESERVOIR PRESSURE, and (perhaps) RESERVOIR BOUNDARIES and HETEROGENEITIES.

122. 122 Introduction (cont.) The basic test method is to create a pressure drawdown in the wellbore, this causes formation fluids to enter the wellbore. If we measure the flow rate and the pressure in the wellbore during production or the pressure during a shut-in period following production, we usually will have sufficient information to characterize the tested well.

123. 123 Introduction (cont.) This course discusses (1) basic equations that describe the unsteady-state flow of fluids in porous media, (2) pressure buildup tests, (3) pressure drawdown tests, (4) other flow tests, (5) type-curve analysis, (6) gas well tests, (7) interference and pulse tests, and (8) drillstem and wireline formation tests Basic equations and examples use engineering units (field units)

124. 124 Chapter 1 Fluid Flow in Porous Media

125. 125 1.1 Introduction (a)Discussion of the differential equations that are used most often to model unsteady-state flow. (b) Discussion of some of the most useful solutions to these equations, with emphases on the exponential-integral solution describing radial, unsteady-state flow. (c) Discussion of the radius-of-investigation concept (d) Discussion of the principle of superposition Superposition, illustrated in multiwell infinite reservoirs, is used to simulate simple reservoir boundaries and to simulate variable rate production histories. (e) Discussion of “pseudo production time”.

126. 126 1.2 The ideal reservoir model Assumptions used (1) Slightly compressible liquid (small and constant compressibility) (2) Radial flow (3) Isothermal flow (4) Single phase flow Physical laws used (1) Continuity equations (mass balances) (2) Flow laws (Darcy’s law)

127. 127 Derivation of continuity equation

128. 128 (A) In Cartesian coordinate system

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133. 133 (B) In Cylindrical Polar Coordinates

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140. 140 Darcy and practical units

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