An Advanced Coupled Rock–Fluid Computational Model for Dynamic–Radial Mud Filtration with Experimental Validation

Abstract

Drilling mud filtration occurs during an overbalanced drilling activity and concurrently with mud loss through pore throats and fractures. Mud loss and filtration are increased when the wellbore fluid condition is in a dynamic mode (pipe rotation and/or fluid circulation), rather than static. Formation damage is a critical industry challenge that results from mud loss and filtration. There is a considerable amount of experimental studies with only a few modeling approaches for characterizing dynamic mud filtration. Most of these studies have not accounted for factors that can exacerbate mud filtration which includes but not limited to: temperature, pipe rotation, pip/wellbore geometry/eccentricity, and porous media complexity. In this study, two mathematical and computational modeling approaches that can be used to predict dynamic drilling mud filtration in a radial coordinate system are presented. In the first modeling approach, a mechanistic model that is based on a material balance of filter cake evolution is presented. Critical factors that impact dynamic–radial mud filtration (temperature, rotary speed, rock permeability, and rock porosity) and other factors (wellbore/reservoir dimensions, filter cake properties, and mud/filtrate rheological properties at reservoir temperature) were accounted for. The model was solved with a numerical approach and commercial software. In the second approach, a scanning electron microscopy image of selected dry core samples, combined with image processing, was used to estimate the pore size and porosity of the internal filter cake. The pore structure of the rock samples and filter cake was modeled using the bundle of curved tubes approach. The deposition probability of mud particles was considered through filtration theories. The modeling results were validated with dynamic–radial filtration experiments. The results from both models closely matched the experimental results. On average, the models revealed no more than 4% relative error in predicting dynamic mud filtration. The novelty in both approaches is the incorporation of critical parameters in the models over a wide range and their responses to cumulative filtrate invasion in different rock types.

Appendix

Filter cake material balance is given by (Civan 1994; 1996) as:

$$- \in_{\text{s}} {\raise0.7ex\hbox{${{\text{d}}r_{\text{c}} }$} \!\mathord{\left/ {\vphantom {{{\text{d}}r_{\text{c}} } {{\text{d}}t}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${{\text{d}}t}$}} = {\raise0.7ex\hbox{${R_{\text{ps}} }$} \!\mathord{\left/ {\vphantom {{R_{\text{ps}} } {\rho_{\text{p}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\rho_{\text{p}} }$}}.$$

(15)

The volume mass fraction of solids as a function of porosity is given as:

$$\in_{\text{s}} = 1 - \emptyset_{\text{c}} .$$

(16)

The net deposition rate of particles to form an external filter cake is given as:

$$R_{\text{ps}} = k_{\text{d}} u_{\text{c}} c_{\text{p}} - k_{\text{e}} \left( {\tau_{\text{s}} - \tau_{\text{cr}} } \right),$$

(17)

where $\tau_{s}$ is the mud (slurry) shear stress and it is given as:

$$\tau_{\text{s}} = k\left( {\frac{\pi RS}{{15n\left( {1 - G^{{{\raise0.7ex\hbox{${ - 2}$} \!\mathord{\left/ {\vphantom {{ - 2} n}}\right.\kern-0pt} \!\lower0.7ex\hbox{$n$}}}} } \right)}}} \right)^{n} .$$

(18)

In Eq. (18),

$$G = {\raise0.7ex\hbox{${r_{\text{dp}} }$} \!\mathord{\left/ {\vphantom {{r_{\text{dp}} } {r_{\text{c}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${r_{\text{c}} }$}}, \quad {\text{for}}\quad 0.5 \le {\raise0.7ex\hbox{${r_{\text{dp}} }$} \!\mathord{\left/ {\vphantom {{r_{\text{dp}} } {r_{\text{c}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${r_{\text{c}} }$}} \le 0.9.$$

(19)

Substituting Eqs. (16) and (17) into (15) yields

$${\raise0.7ex\hbox{${ - {\text{d}}r_{\text{c}} }$} \!\mathord{\left/ {\vphantom {{ - {\text{d}}r_{\text{c}} } {{\text{d}}t}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${{\text{d}}t}$}} = \frac{{k_{\text{d}} u_{\text{c}} c_{\text{p}} - k_{\text{e}} \left( {\tau_{\text{s}} - \tau_{\text{cr}} } \right)}}{{\left( {1 - \emptyset_{\text{c}} } \right)\rho_{\text{p}} }}.$$

(20)

In terms of the filtration rate, the radial volumetric flux at the external cake surface is:

$$u_{\text{c}} = \frac{q}{{2\pi hr_{\text{c}} }}.$$

(21)

Substituting Eq. (21) into (20) gives:

$$- \frac{{{\text{d}}\left( {{\raise0.7ex\hbox{${r_{\text{c}} }$} \!\mathord{\left/ {\vphantom {{r_{\text{c}} } {r_{\text{w}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${r_{\text{w}} }$}}} \right)}}{{{\text{d}}t}} = A\frac{q}{{{\raise0.7ex\hbox{${r_{\text{c}} }$} \!\mathord{\left/ {\vphantom {{r_{\text{c}} } {r_{\text{w}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${r_{\text{w}} }$}}}} -B {\quad0} \le {\raise0.7ex\hbox{${r_{\text{c}} }$} \!\mathord{\left/ {\vphantom {{r_{\text{c}} } {r_{\text{w}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${r_{\text{w}} }$}} \le 1,$$

(22)

where

$$A = \frac{{c_{\text{p}} k_{\text{d}} }}{{\left( {1 - \emptyset_{\text{c}} } \right)\rho_{\text{p}} 2\pi hr_{\text{w}}^{2} }},$$

(23)

and

$$B = \frac{{k_{\text{e}} \left( {\tau_{\text{s}} - \tau_{\text{cr}} } \right)}}{{\left( {1 - \emptyset_{\text{c}} } \right)\rho_{\text{p}} r_{\text{w}} }}.$$

(24)

The Forchheimer’s (1901) equation for radial flow of the mud is given as:

$$- \frac{\partial p}{\partial r} = \frac{\mu }{K}u + \beta \rho u^{2} .$$

(25)

Combining Eqs. (25) and (21) yields:

$$- \frac{\partial p}{\partial r} = \frac{\mu }{2\pi hK}\frac{q}{r} + \frac{\beta \rho }{{2\pi h^{2} }}\left( {\frac{q}{r}} \right)^{2} ,$$

(26)

where $\beta_{i}$ is the inertial flow coefficient for the rock or filter cake and per Liu et al. (1995), it is given as:

$$\beta_{i} = \frac{{2.92 \times 10^{4} T_{\text{f/ c}} }}{{\emptyset_{\text{f/c}} K_{\text{f/ c}} }}.$$

(27)

In Eq. (27), T_f/c means tortuosity of formation or filter cake. The same goes for the porosity and permeability symbols (please check the Greek symbols). Integrating Eq. (26) for conditions before and during evolution of external filter cake results in Eqs. (28) and (29), respectively:

$$P_{\text{c}} - P_{\text{e}} = \frac{{q_{\text{o}} \mu }}{{2\pi hK_{\text{f}} }}ln\left( {\frac{{r_{\text{e}} }}{{r_{\text{w}} }}} \right) + \frac{{\beta_{\text{f}} \rho q_{\text{o}}^{2} }}{{\left( {2\pi h} \right)^{2} }}\left( {\frac{1}{{r_{\text{w}} }} - \frac{1}{{r_{\text{e}} }}} \right)^{2} ,$$

(28)

$$- P_{\text{e}} = \frac{q\mu }{{2\pi hK_{\text{f}} }}\left[ {ln\left( {\frac{{r_{\text{e}} }}{{r_{\text{w}} }}} \right) + \frac{{K_{\text{f}} }}{{K_{\text{c}} }}ln\left( {\frac{{r_{\text{w}} }}{{r_{\text{c}} }}} \right)} \right] + \frac{{\beta_{\text{f}} \rho q}}{{\left( {2\pi h} \right)^{2} }}\left[ {\frac{1}{{r_{\text{w}} }} - \frac{1}{{r_{\text{e}} }} + \frac{{\beta_{\text{c}} }}{{\beta_{\text{f}} }}\left( {\frac{1}{{r_{\text{w}} }} - \frac{1}{{r_{\text{e}} }}} \right)} \right].$$

(29)

Combining Eqs. (28) and (29) eliminates (P_c–P_e) and yields the quadratic Eq. (30) for non-Darcy mud filtrate flow rate:

$$\alpha q^{2} + \beta q + \gamma = 0,$$

(30)

whose solution is in Eq. (31) (Civan 2007)

$$q = \frac{{ - \beta + \sqrt {\beta^{2} - 4\alpha \gamma } }}{2\alpha },$$

(31)

where

$$\alpha = \frac{\rho }{{\left( {2\pi h} \right)^{2} }}\left[ {\beta_{\text{f}} \left( {\frac{1}{{r_{\text{w}} }} - \frac{1}{{r_{\text{e}} }}} \right) + \beta_{\text{c}} \left( {\frac{1}{{r_{{{\text{w}} - \delta }} }} - \frac{1}{{r_{\text{w}} }}} \right)} \right], .$$

(32)

$$\beta = \frac{\mu }{{2\pi hK_{\text{f}} }}\left[ {\ln \left( {\frac{{r_{\text{e}} }}{{r_{\text{w}} }}} \right) + \frac{{K_{\text{f}} }}{{K_{\text{c}} }}ln\left( {\frac{{r_{\text{w}} }}{{r_{\text{w}} - \delta }}} \right)} \right],$$

(33)

$$\gamma = - \left[ {\frac{{q_{\text{o}} \mu }}{{2\pi hK_{\text{f}} }}ln\left( {\frac{{r_{\text{e}} }}{{r_{\text{w}} }}} \right) + \frac{{\beta_{\text{f}} \rho q_{\text{o}}^{2} }}{{\left( {2\pi h} \right)^{2} }}\left( {\frac{1}{{r_{\text{w}} }} - \frac{1}{{r_{\text{e}} }}} \right)} \right].$$

(34)

The filter cake thickness (δ) is a function of wellbore radius and filter cake radius (δ = r_w − r_c).