Proceedings of the International Field Exploration and Development Conference 2017 pp 1850-1869 | Cite as

# Build-up Analysis of Multi-Well System in Naturally Fractured HTHP Gas Reservoirs

## Abstract

In order to clarify reservoir characteristics and identify pressure build-up characteristics resulting from the offset effect, well test models of bottomhole pressure in the multi-well system of infinite homogeneous and double porosity reservoirs were derived in consideration of effective radius. For a testing well in the model, skin and wellbore storage effect were considered, while for offset wells, skin and wellbore storage were ignored. The exact solution to Bessel function in the Laplace space was derived with Laplace transform method. Two-type curves were plotted for offset wells being online or offline simultaneously. In addition, characteristics of the type curve were analysed and the method for pressure build-up analysis was established. Under both scenarios mentioned above, long-term asymptotic solutions show that the curves of the pressure derivative rise stepwise, and there appears a multi-radial flow stabilization line. The ratio of height of each stabilization line to that of the first stabilization line is the algebraic sum of the dimensionless production combining the testing well and the effective offset wells. In the meanwhile, the curves of pressure derivatives descend under the scenario of the testing well build-up when the offset wells are producing simultaneously.

## Keywords

Multi-well system Pressure build-up Homogeneous and double porosity reservoirs Type curve Pressure derivative## Nomenclature

- \( B \)
Formation-volume factor

*C*WBS coefficient, m

^{3}/Pa*C*_{t}Total compressibility, Pa

^{−1}*h*Net-pay thickness, m

*K*Permeability, m

^{2}*K*_{0}Bessel function

*K*_{1}Bessel function

*m*\( m = 2\int {\frac{p}{\mu Z}} {\text{d}}p \), MPa

^{2}/mPa s*p*_{i}Initial pressure, Pa

*q*_{j}Rate, m

^{3}/s- \( r_{\text{w}} \)
Wellbore radius, m

*S*Skin factor

- z
Laplace variable

- \( C_{\text{D}} \)
\( C_{\text{D}} = \frac{C}{{2\uppi\varphi hC_{\text{t}} r_{\text{w}}^{2} }} \)

- \( p_{\text{D}} \)
\( p_{{j{\text{D}}}} = \frac{{2\,\uppi\,Kh\left( {p_{\text{i}} - p_{\text{wf}} } \right)}}{{q_{1} \mu B}} \)

- \( q_{{j{\text{D}}}} \)
\( q_{{j{\text{D}}}} = \frac{{q_{j} }}{{q_{1} }} \)

- \( r_{\text{D}} \)
\( r_{\text{D}} = \frac{r}{{r_{\text{w}} {\text{e}}^{ - S} }} \)

- \( t_{\text{D}} \)
Dimensionless time, \( t_{\text{D}} = \frac{Kt}{{\varphi \mu C_{\text{t}} r_{\text{w}}^{2} }} \)

- \( t_{\text{p}} \)
Production time, hour

*t*_{pD}Dimensionless production time

- Δ
*t*_{D} Dimensionless shut in time

- a
\( \frac{{\lambda C_{\text{D}} }}{{\omega \left( {1 - \omega } \right)}} \)

- \( b \)
\( \frac{{\lambda C_{\text{D}} }}{{\left( {1 - \omega } \right)}} \)

- \( D \)
Diffusivity ratio for composite reservoir, \( D = {{\left( {{k \mathord{\left/ {\vphantom {k {\phi \mu c_{t} }}} \right. \kern-0pt} {\phi \mu c_{t} }}} \right)_{2} } \mathord{\left/ {\vphantom {{\left( {{k \mathord{\left/ {\vphantom {k {\phi \mu c_{t} }}} \right. \kern-0pt} {\phi \mu c_{t} }}} \right)_{2} } {\left( {{k \mathord{\left/ {\vphantom {k {\phi \mu c_{t} }}} \right. \kern-0pt} {\phi \mu c_{t} }}} \right)_{1} }}} \right. \kern-0pt} {\left( {{k \mathord{\left/ {\vphantom {k {\phi \mu c_{t} }}} \right. \kern-0pt} {\phi \mu c_{t} }}} \right)_{1} }} \)

- \( D \)
Non-Darcy flow coefficient

- \( M \)
Mobility ratio for composite reservoir, \( M = {{\left( {{k \mathord{\left/ {\vphantom {k \mu }} \right. \kern-0pt} \mu }} \right)_{2} } \mathord{\left/ {\vphantom {{\left( {{k \mathord{\left/ {\vphantom {k \mu }} \right. \kern-0pt} \mu }} \right)_{2} } {\left( {{k \mathord{\left/ {\vphantom {k \mu }} \right. \kern-0pt} \mu }} \right)_{1} }}} \right. \kern-0pt} {\left( {{k \mathord{\left/ {\vphantom {k \mu }} \right. \kern-0pt} \mu }} \right)_{1} }} \)

- \( N \)
Number of wells

- \( r \)
Distance from the well test wellbore, m

- \( R_{i} \)
Outer radius of inner zone annulus in radial-composite model, m

- \( t \)
Time, h

- \( \Delta t \)
Shut in time, h

## Greek symbols

- \( \phi \)
Porosity

- \( \lambda \)
Interporosity flow coefficient

- \( \mu \)
Viscosity, cp

- \( \omega \)
Storativity ratio

## Subscripts

- \( {\text{D}} \)
Dimensionless variable

- \( {\text{f}} \)
Fracture

- \( {\text{m}} \)
Matrix

- \( {\text{j}} \)
Index

- \( {\text{w}} \)
Wellbore conditions

- \( {\text{wf}} \)
Flowing wellbore conditions

- \( {\text{ws}} \)
Shut in wellbore conditions

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