Slide 1 Well Testing Slide 2 2 Slide 3 3 Slide 4 4 Slide 5 5 Slide 6 6 Slide 7 7 Slide 8 8 Slide 9 9 Slide 10 10 Slide 11 11 Slide 12 12 Slide 13 13 Slide 14 14 Slide 15 15 Slide 16 16 Slide 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 Slide 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. Slide 19 19 Slide 20 20 Slide 21 21 Slide 22 22 Slide 23 23 Slide 24 24 Slide 25 25 Slide 26 26 Slide 27 27 Slide 28 28 Question: Why does p w > p i for certain t ? Slide 29 29 Slide 30 30 Slide 31 31 Slide 32 32 Slide 33 33 Slide 34 34 Slide 35 35 Slide 36 36 Slide 37 37 Slide 38 38 Slide 39 39 Slide 40 40 Slide 41 41 Slide 42 42 Slide 43 43 Slide 44 44 Slide 45 45 Slide 46 46 Slide 47 47 Slide 48 48 Slide 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 Slide 50 50 Slide 51 51 Slide 52 52 Slide 53 53 Slide 54 54 Slide 55 55 Slide 56 56 Slide 57 57 Slide 58 58 Slide 59 59 Slide 60 60 Flow Equation for Generalized Reservoir Geometry Slide 61 61 Slide 62 62 Slide 63 63 Slide 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 Slide 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 Slide 66 66 Slide 67 67 Slide 68 68 Slide 69 69 rere Slide 70 70 xexe xexe Slide 71 71 Slide 72 72 Slide 73 73 Slide 74 74 Slide 75 75 Slide 76 76 Slide 77 77 Slide 78 78 Development of a mathematical relationship between sandface (formation) and surface flow rates Slide 79 79 Slide 80 80 Slide 81 81 Slide 82 82 Slide 83 83 Slide 84 84 Slide 85 85 Slide 86 86 Slide 87 87 Slide 88 88 Slide 89 89 Slide 90 90 Slide 91 91 1.4 Radius of investigation Slide 92 92 Slide 93 93 Slide 94 94 Slide 95 95 Slide 96 96 Slide 97 97 Slide 98 98 Slide 99 99 Slide 100 100 Slide 101 101 1.5 The Principle of Superposition Slide 102 102 Slide 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 Slide 104 104 Slide 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. Slide 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. Slide 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 Slide 108 108 Variable flow-rate Slide 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). Slide 110 110 Slide 111 111 Slide 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 Slide 113 113 Slide 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. Slide 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. Slide 116 116 Example 1.6 – Application of Horner’s Approximation Given: the Production history was as follows: Slide 117 117 Slide 118 118 Slide 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 Slide 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. Slide 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. Slide 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. Slide 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) Slide 124 124 Chapter 1 Fluid Flow in Porous Media Slide 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”. Slide 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) Slide 127 127 Derivation of continuity equation Slide 128 128 (A) In Cartesian coordinate system Slide 129 129 Slide 130 130 Slide 131 131 Slide 132 132 Slide 133 133 (B) In Cylindrical Polar Coordinates Slide 134 134 Slide 135 135 Slide 136 136 Slide 137 137 Slide 138 138 Slide 139 139 Slide 140 140 Darcy and practical units Slide 141 141 Slide 142 142 Slide 143 143