I have 3 implicit equations in 5 variables: $f_1(S, y_h, y_d, x_h, J)=0$, $f_2(S, y_h, y_d, x_h, J)=0$, and $f_3(S, y_h, y_d, x_h, J)=0$. These equations determine a 3D surface in $S-y_h-y_d$ coordinate system. The equations are given below.

M = 1;
roo = 0.01;
xd = 3;

f1[S_, yh_, yd_, xh_, J_]:= 2 M (xh^2 - (yh + 1)^2)/(xh^2 + (yh + 1)^2)^2 + 
  2 S ((xh - xd)^2 - (yh + yd)^2)/((xh - xd)^2 + (yh + yd)^2)^2 + 
  J/(2 yh) == 0,

f2[S_, yh_, yd_, xh_, J_]:= M*xh (yh + 1)/(xh^2 + (yh + 1)^2)^2 + 
  S (xh - xd) (yh + yd)/((xh - xd)^2 + (yh + yd)^2)^2 == 0,

f3[S_, yh_, yd_, xh_, J_]:= J*Log[2 yh*J/roo] + 2 M (yh + 1)/(xh^2 + (yh + 1)^2) + 
  2 S (yh + yd)/((xh - xd)^2 + (yh + yd)^2) - (Log[2/roo] + 1) == 0,

where $S, y_h, y_d, x_h, J$ are Reals, $yh>0$, and $yd>0$.

I am trying to obtain the surface in the $S-yh-yd$ coordinate system like the following image. How to plot it?

What I have tried: For 1 equation in 3 variables: $f(S, y_h, y_d) = 0$, I can use ContourPlot3D to get the surface; For equations expressed in an explicit form: $S = S(x_h, J)$, $y_h = y_h(x_h, J)$, $y_d = y_d(x_h, J)$, I can use ParametricPlot3D. But I failed when I faced this issue introduced above. Please help me. Thanks a lot for your support!

Background Information

This problem is from the paper Prominence eruptions and coronal mass ejections triggered by newly emerging flux by J. Lin, T. G. Forbes, P. A. Isenberg.