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In this chapter, we present analytical solutions for different probe types: 1) Circular disk, 2) Circular ring, 3) Guard, 4) Rectangular, 5) Elliptical. There are also other types of probes. We will present solutions to only these types. Probe flow geometries create mixed-boundary value problems (MBVPs) where the open surface S1 is for fluid withdrawal, as shown in Fig. 7.1. Probes are set against the mudcake over the surface of the wellbore to establish hydraulic communication between the tool and formation. Probes are set on the impermeable cylindrical wellbore and make an open surface S1 by cutting the impermeable wellbore wall (normally mudcake), as shown in Fig. 7.1. The pressure over the open surface (pw) is assumed to be uniform pressure and is a priori unknown, and q is the constant-rate fluid withdrawal from the probe. It is assumed that permeabilities of the media in the principal directions are denoted by kx, ky, and kz, and the formation is transversely isotropic (kh = kx = ky and kv = kz), where kh and kv are horizontal and vertical permeabilities, respectively. Mobilities are defined as λh = kh/μ and λv = kv/μ, where μ is the fluid viscosity. Storativity is defined as φ = ϕct. The total flow rate q(t) ≡ q is assumed to be independent of time, the constant-rate case. If the rate is variable (a function of time), the solution for the time-dependent flow rate can easily be derived using Duhamel’s principle (Duhamel 1833) or the well-known convolution property of the Laplace transform. The variable-rate case, with skin and wellbore storage formulations, is given in Chapter 6.

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