diff --git a/src/ert/analysis/_es_update.py b/src/ert/analysis/_es_update.py
index 8915fc02f0e..59569ebb67d 100644
--- a/src/ert/analysis/_es_update.py
+++ b/src/ert/analysis/_es_update.py
@@ -610,6 +610,24 @@ def correlation_callback(
         param_ensemble_array = _load_param_ensemble_array(
             source_ensemble, param_group, iens_active_index
         )
+
+        # Calculate variance for each parameter
+        param_variance = np.var(param_ensemble_array, axis=1)
+        # Create mask for non-zero variance parameters
+        non_zero_variance_mask = ~np.isclose(param_variance, 0.0)
+
+        log_msg = f"Updating {np.sum(non_zero_variance_mask)} parameters {'with' if module.localization else 'without'} adaptive localization."
+        logger.info(log_msg)
+        progress_callback(AnalysisStatusEvent(msg=log_msg))
+
+        log_msg = f"There are {num_obs} responses and {ensemble_size} realizations."
+        logger.info(log_msg)
+        progress_callback(AnalysisStatusEvent(msg=log_msg))
+
+        log_msg = f"There are {(~non_zero_variance_mask).sum()} parameters with 0 variance that will not be updated."
+        logger.info(log_msg)
+        progress_callback(AnalysisStatusEvent(msg=log_msg))
+
         if module.localization:
             config_node = source_ensemble.experiment.parameter_configuration[
                 param_group
@@ -618,14 +636,15 @@ def correlation_callback(
             batch_size = _calculate_adaptive_batch_size(num_params, num_obs)
             batches = _split_by_batchsize(np.arange(0, num_params), batch_size)
 
-            log_msg = f"Running localization on {num_params} parameters, {num_obs} responses, {ensemble_size} realizations and {len(batches)} batches"
+            log_msg = f"Adaptive localization has split parameters into {len(batches)} batch{'es' if len(batches) != 1 else ''}."
             logger.info(log_msg)
             progress_callback(AnalysisStatusEvent(msg=log_msg))
 
             start = time.time()
             cross_correlations: List[npt.NDArray[np.float64]] = []
             for param_batch_idx in batches:
-                X_local = param_ensemble_array[param_batch_idx, :]
+                update_idx = param_batch_idx[non_zero_variance_mask[param_batch_idx]]
+                X_local = param_ensemble_array[update_idx, :]
                 if isinstance(config_node, GenKwConfig):
                     correlation_batch_callback = functools.partial(
                         correlation_callback,
@@ -633,17 +652,15 @@ def correlation_callback(
                     )
                 else:
                     correlation_batch_callback = None
-                param_ensemble_array[param_batch_idx, :] = (
-                    smoother_adaptive_es.assimilate(
-                        X=X_local,
-                        Y=S,
-                        D=D,
-                        alpha=1.0,  # The user is responsible for scaling observation covariance (esmda usage)
-                        correlation_threshold=module.correlation_threshold,
-                        cov_YY=cov_YY,
-                        progress_callback=adaptive_localization_progress_callback,
-                        correlation_callback=correlation_batch_callback,
-                    )
+                param_ensemble_array[update_idx, :] = smoother_adaptive_es.assimilate(
+                    X=X_local,
+                    Y=S,
+                    D=D,
+                    alpha=1.0,  # The user is responsible for scaling observation covariance (esmda usage)
+                    correlation_threshold=module.correlation_threshold,
+                    cov_YY=cov_YY,
+                    progress_callback=adaptive_localization_progress_callback,
+                    correlation_callback=correlation_batch_callback,
                 )
 
             if cross_correlations:
@@ -665,9 +682,9 @@ def correlation_callback(
 
         else:
             # In-place multiplication is not yet supported, therefore avoiding @=
-            param_ensemble_array = param_ensemble_array @ T.astype(  # noqa: PLR6104
-                param_ensemble_array.dtype
-            )
+            param_ensemble_array[non_zero_variance_mask] = param_ensemble_array[  # noqa: PLR6104
+                non_zero_variance_mask
+            ] @ T.astype(param_ensemble_array.dtype)
 
         log_msg = f"Storing data for {param_group}.."
         logger.info(log_msg)
diff --git a/test-data/ert/heat_equation/heat_equation.py b/test-data/ert/heat_equation/heat_equation.py
index e36862ea1aa..63b086a492b 100755
--- a/test-data/ert/heat_equation/heat_equation.py
+++ b/test-data/ert/heat_equation/heat_equation.py
@@ -63,6 +63,8 @@ def sample_prior_conductivity(ensemble_size, nx, rng):
     rng = np.random.default_rng(iens)
     cond = sample_prior_conductivity(ensemble_size=1, nx=nx, rng=rng).reshape(nx, nx)
 
+    cond[nx // 2 :, :] = 1.0
+
     if iteration == 0:
         resfo.write(
             "cond.bgrdecl", [("COND    ", cond.flatten(order="F").astype(np.float32))]
diff --git a/tests/ert/ui_tests/cli/test_field_parameter.py b/tests/ert/ui_tests/cli/test_field_parameter.py
index b26c6495b5e..6ebc139a89e 100644
--- a/tests/ert/ui_tests/cli/test_field_parameter.py
+++ b/tests/ert/ui_tests/cli/test_field_parameter.py
@@ -296,3 +296,7 @@ def test_parameter_update_with_inactive_cells_xtgeo_grdecl(tmpdir):
             assert "nan" not in Path(
                 "simulations/realization-0/iter-1/my_param.grdecl"
             ).read_text(encoding="utf-8")
+
+
+# def test_field_parameter_update_using_heat_equation(heat_equation_storage):
+#     print("Hei")