JNTU B.Tech IV Year RESERVOIR STIMULATION Previous Question Paper (July/August - 2022 - R15)

The radial diffusivity equation for a slightly compressible fluid in a homogeneous, isotropic reservoir is given by:

To solve this for a well producing at a constant rate q in an infinite-acting reservoir, we define the following initial and boundary conditions:

1. Initial condition: p(r, 0) = pᵢ for all r.
2. Inner boundary condition: lim(r→0) [r(∂p/∂r)] = (qμB)/(2πkh).
3. Outer boundary condition: p(r, t) = pᵢ as r → ∞.

Using the Boltzmann transformation variable η = r² / (4αt), where α = k / (φμcₜ), the partial differential equation is transformed into an ordinary differential equation:

Integrating this equation twice and applying the boundary conditions yields the exponential integral solution:

For small values of the argument (r²/4αt < 0.01), which corresponds to long production times or points near the wellbore, the exponential integral is approximated by the logarithmic solution:

This is the standard line-source solution used in pressure transient analysis to determine reservoir properties like permeability and skin factor.

The performance of reservoir stimulation, such as hydraulic fracturing or acidizing, is fundamentally governed by the physical and mechanical properties of the reservoir rock. The pertinent properties include porosity, permeability, mineralogical composition, and geomechanical properties (Young's modulus, Poisson's ratio, and fracture toughness).

1. Porosity and Permeability: These define the storage capacity and fluid flow potential. Measurement is typically conducted in the laboratory using core plugs. Porosity is determined via the gas expansion method (Boyle's Law) using a helium porosimeter, where the volume of the rock matrix is calculated by measuring the pressure change in a known reference cell. Permeability is measured using a core flood apparatus by applying Darcy's Law, where a fluid of known viscosity is flowed through the core at a constant rate while measuring the pressure drop across the sample.

2. Mineralogical Composition: The presence of reactive minerals like clays or carbonates dictates the chemical compatibility of stimulation fluids. This is measured using X-Ray Diffraction (XRD) to identify crystalline phases and Scanning Electron Microscopy (SEM) combined with Energy Dispersive X-ray Spectroscopy (EDS) to determine the spatial distribution and morphology of minerals within the pore space.

3. Geomechanical Properties: These are critical for predicting fracture initiation and propagation.

• Young's Modulus and Poisson's Ratio: Measured through triaxial compression tests on cylindrical core samples. The sample is subjected to confining pressure while axial stress is increased. Stress-strain curves are generated to calculate the static elastic moduli.
• Fracture Toughness: This measures the rock's resistance to crack propagation. It is typically determined using the Chevron-notched specimen test or the Semi-Circular Bend (SCB) test, where a notched core sample is loaded to failure to calculate the critical stress intensity factor.

These laboratory measurements are often integrated with well-log data (such as sonic logs for dynamic moduli) to calibrate models across the entire reservoir interval, ensuring accurate design of stimulation treatments.