WATER RESOURCES RESEARCH, VOL. 21, NO. 3, PAGES 305-310, MARCH 1985 Detection of Linear Boundaries by Drawdown Tests' A Semilog Type Curve Matching Approach ABRAHAM SAGEEV, ROLANDN. HOR.NE, AND HENRY J. RAMEY, JR.. Department of Petroleum Engineering, School of Earth Sciences, Stanford University, California A new method for pressure drawdown analysis in reservoirs affected by linear boundaries is developed. This methodis based on a semilog curvematching approach and may be appliedto both constant pressure and impermeable linear boundary cases. A semilogtype curve is presented by which the distancebetween a well and the linear boundary may be estimated. The new analysis method is an improvement over the current doublestraight line analysis method. The pressure response of the well is matched without the needto find two straightlines, and several advantages result.First, the analysis may be performed for interference tests for whichthe first semilog straight line neverdevelops. Second, the duration of the test requiredfor the new method is shorter by an order of magnitudethan that required for existing semilog methods, since it is not necessary to wait for the second straight line to develop. INTRODUCTION The detectionof linear boundaries of reservoirs is of great importancein predictingthe behavior of these reservoirs underexploitation. Reservoirs may be bounded by imperme- ableor leakyfaults, which may be approximated asno-flow or constantpressure linear boundaries, respectively. Constant pressure linear boundaries may occur in shallow aquifers fed by neighboring rivers or linear lake shores. I [ICSC 1111Ci::tl IJtYlAllt. l•tllU• 111dy OU UUtU•tUU allu lv•at•u u•v tis• analysis of their effect on nearby wells during pump tests, Severalconfigurations of linear boundaries have been dis- cussed in the literature. Stallman [1952] and Davis and Haw- kins [1963] considered a singlelinear boundary.Ferris et. al. [1962], Kruseman and De Ridder[1970], Ramey et. al. [1973], and Tiab and Crichlow [1979] discussed two, three, and four linear boundaries around a well. Tiab and Kumar [1980] con- sidered a well between two parallel sealing faults.Common to all these studies is the use of the method of imaging line sources and sinks to generate the effects of linear boundaries. This study concentrates on determiningthe distancebe- tween a pumping well or an observation well to a single linear boundary using data from a pressure drawdown test. The effect of the linear boundary may be generated by locating an image well symmetrically on opposite sides of the boundary from the real well (seeFigure 1). A source and sink produce the same response as a constant pressure linear boundary, and the dimensionless pressure at a givenpoint is PD -•' -- «[Ei(--X1) -- Ei(--X2)] (1) Two sources produce the same response as an impermeable boundary where the dimensionless pressure at a given point is p,, = - + ,(-x33 (2) The dimensionless variables are defined in the conventional manner' 2•T(p i -- p) Pt) = (3) Q tn = Tt/Srw 2 (4) rDi--- ri/rw (5) Copyright1985by the American Geophysical Union. Paper number 4W 1389. 0043-1397/85/004W-1389505.00 Ei is the exponential integral,and X i = rni2/4tn (6) The definitions of T and S are presented in the nomenclature. These terms are defined in this manner, since this linear boundary problemis solved in termsof pressure. Two methods for determining the distance between a well and a linear boundary have beendescribed in the literature: a log-logmethodand a semfiog method. Stallman [1952] presented log-log type curves for a constant rate line source well near a linear boundary(see Figure 2). The outlined curve in the center is the Theis [1935] curve, also referred to as the line source or exponential integral curve. The curves below the line source curve approach a steady state pressure and represent the response affectedby constant pressure linear boundaries. The curvesthat deviate above the line sourcecurve are for impermeablelinear boundaries.The parameter of the various curves is the ratio of the distance between the pressure point and the image well to the distance between the pressure point and the productionwell, r2/r • (see Figure 1). The pressure time response can be matched to Stallman's [1952] log-log type curves,and the ratio r2/r • may be esti- mated. For an observationwell, r• is known, hence the dis- tance between the interference well and the image well may be determined. For a productionwell the ratio r2/r• is approxi- mately twice the dimensionless distance between the well and the linear boundary,dn. The dimensionless distance dn is de- fined as the ratio of the distance between the well and the linear boundaryto the radiusof the well, d/rw. The estimationof the ratio r2/r• using a log-log match is difficult. The curves are closely spaced and interpolating be- tween them may lead to a great deal of uncertainty. Address- ing this difficulty, Davis and Hawkins [1963], Witherspo øn et. al. [1967], and Earlougher [1977] extended Stallman's[1952] log-loganalysis by developing the doublesemilog straightline method. When the pressure-time data are graphedin a semilog fash- ion, two straight linesmay be seen. Figure 3 presents an exam- p!e of thesetwo straight lines for a well near a sealing fault, after Witherspoon et. al. [1970]. The first straight line repre- sentsthe infinite acting flow period of the production well. 305