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Case Study 6 - Solutions - Part 1 The first part of this...

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1 Part 1 The first part of this case study is to complete the exercise given on page 77 of the supplementary notes to the well testing textbook. First, we assume a buildup test is occurring near a sealing fault. Step 1 The general buildup equation in SI form is: From the above equation it is apparent that the solution to a buildup is the difference between the solutions to two drawdowns. These are evaluates at times and respectively. Step 2 The terms can be expressed as the sum of two contributions: One from the real test well and one from a ghost well.
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2 The can be expanded into: Letting: This becomes: Term 1 represents the solution of the real well for an infinitely acting reservoir evaluated at , while term 2 represents the solution of the ghost well evaluated at 2R. The can be expanded into: Letting: This becomes: Term 3 represents the solution of the real well for an infinitely acting reservoir evaluated at , while term 4 represents the solution of the ghost well evaluated at 2R.
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3 Step 3 Combining the above expression into the buildup equation, the general solution to a buildup near a fault becomes:
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4 Step 4 Evaluating the equation at early times, when : As And Therefore Step 5 The above equation can be transformed into field units and base 10 logarithms as follows: Since the formula to change the base of a log is: And since:
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5 Therefore: Therefore the equation in field units and base 10 logarithms is:
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6 Step 6 As time increases, the Horner plot will again approach a straight line. At large values of the ei function can be represented by: Recall the general buildup equation: Applying the assumption for large values of : Note the ½ term has vanished, representing a doubling of the original slope on a Horner plot during late times. This came as a result of the assumption, which allowed terms to be joined and reduced. The above equation can be transformed into field units and base 10 logarithms as follows:
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7 Since the formula to change the base of a log is: And since: Therefore: Therefore the equation in field units and base 10 logarithms is:
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8 Step 7 There are several ways to determine the distance to the fault. My preferred approach would be to purely work with the shifted Horner line at early times, which has the following equation as derived in step 5: In this equation the term represents the slope of the shifted Horner line. By measuring the slope and using the typically known core and fluid properties the permeability can be determined. Providing the initial reservoir pressure is known, the only unknown in the above equation is R. Therefore by choosing any point on the shifted Horner line, selecting the corresponding wellbo re pressure and ∆t, we can reduce the above and solve for R. The actual process of solving for R is difficult, since it is included inside the exponential integral function. There are several ways to handle this, including a
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